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author | Norbert Preining <norbert@preining.info> | 2023-04-15 03:01:57 +0000 |
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committer | Norbert Preining <norbert@preining.info> | 2023-04-15 03:01:57 +0000 |
commit | 78e6b19d4678a9d641c4afa4ca800288cfa0a9d5 (patch) | |
tree | ad774593055c02ade73402379b9df52568f6940f /info/drawing-with-metapost | |
parent | b17717e3829bc08efb71ec2537d9d894c8b82d45 (diff) |
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diff --git a/info/drawing-with-metapost/Drawing-with-Metapost.pdf b/info/drawing-with-metapost/Drawing-with-Metapost.pdf Binary files differnew file mode 100644 index 0000000000..780071b759 --- /dev/null +++ b/info/drawing-with-metapost/Drawing-with-Metapost.pdf diff --git a/info/drawing-with-metapost/README.md b/info/drawing-with-metapost/README.md new file mode 100644 index 0000000000..5b84892242 --- /dev/null +++ b/info/drawing-with-metapost/README.md @@ -0,0 +1,37 @@ +# Drawing-with-Metapost + +This project provides a document that discusses how to draw technical diagrams +with John Hobby's Metapost language. It includes over 200 illustrations +created with Metapost, complete with source code as inspiration and examples. +The intended level is for intermediate to advanced users rather than complete +beginners. For introductions, tutorials, and other articles about Metapost, +see http://www.tug.org/metapost.html + +Start with "Drawing-with-Metapost.pdf" in the top directory. + +The `src` directory contains +- the TeX source for the main document +- the style file used for marking up Metapost source code +- the Metapost source for each illustration used in the main document +- the corresponding PDF file created from the MP source + +You might like to read the main document first, but you might also like to +browse through the PDFs in the src directory, and when you find one that is +interesting, have a look at the corresponding MP source file. There is a +one-to-one match between the PDF names and the MP source names, so +"apollonius.pdf" is created from "apollonius.mp". The src directory contains +a few drawings that are not (yet) included in the main document. + +To update the main PDF document I follow these steps + +- build any new or updated Metapost source files with `lualatex` to create PDFs in the src directory +- build the main tex file with `lualatex -output-directory=.. -recorder Drawing-with-Metapost` +- run a Python script to read the .fls and `git add` all the files used +- git commit and push + +Toby Thurston -- 14 Apr 2023 + +Copyright (c) 2023 by Toby Thurston. This material may be distributed only +subject to the terms and conditions set forth in the Open Publication License, +v1.0 or later (the latest version is presently available at +http://www.opencontent.org/openpub/). diff --git a/info/drawing-with-metapost/src/Drawing-with-Metapost.tex b/info/drawing-with-metapost/src/Drawing-with-Metapost.tex new file mode 100644 index 0000000000..e7143f4333 --- /dev/null +++ b/info/drawing-with-metapost/src/Drawing-with-Metapost.tex @@ -0,0 +1,8193 @@ +% Copyright (c) 2023 by Toby Thurston. This material may be distributed only +% subject to the terms and conditions set forth in the Open Publication License, +% v1.0 or later (the latest version is presently available at +% http://www.opencontent.org/openpub/). +\documentclass[a4paper,landscape]{article} +\usepackage{luamplib} +\mplibtextextlabel{enable} +\usepackage{fontspec} +\newfontfamily{\demo}{TeX Gyre Pagella} +\DeclareTextFontCommand{\textdemo}{\demo} +\usepackage{unicode-math} +\usepackage[inner=1in,textwidth=5in,textheight=6in, marginparsep=0.5in, marginparwidth=4.2in]{geometry} +\usepackage{sectsty} +\allsectionsfont{\normalfont\sffamily\bfseries} +\usepackage{graphicx} +\usepackage{mflogo} +\def\mfbook{\textsl{The \MF\kern1pt book}} +\def\texbook{\textsl{The \TeX\kern1pt book}} +\usepackage{shortvrb}\MakeShortVerb{"} +\usepackage{dwmpcode} +\title{Drawing with Metapost} +\author{Toby Thurston} +\date{March 2017 – April 2023} +\overfullrule=2pt +\def\kw#1{\begingroup\def\_{\kern.04em + \vbox{\hrule width.3em height .6pt}\kern.08em}% +\ifmmode\mathop{\textbf{#1}}\else\hbox{\bf#1\/}\fi\endgroup} +\def\op#1{\begingroup\def\_{\kern.04em + \vbox{\hrule width.3em height .6pt}\kern.08em}% +\ifmmode\mathop{\textrm{#1}}\else\hbox{\rm#1\/}\fi\endgroup} +\def\id#1{\begingroup\def\_{\kern.04em + \vbox{\hrule width.3em height .6pt}\kern.08em}% +\ifmmode\mathop{\textit{#1}}\else\hbox{\it#1\/}\fi\endgroup} +\def\cycle{{\rm cycle}} +\def\vpic#1#2{\moveright5.5in\vbox to 0pt{\hsize4in\vskip#1\centerline{\includegraphics{#2}}\vss}} +\def\mpic#1#2{\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip#1\centerline{\includegraphics{#2}}\vss}}} +\def\mnote#1#2{\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip#1\noindent #2\par\vss}}} +\def\mwpic#1#2{\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip#1\centerline{\includegraphics[width=4in]{#2}}\vss}}} +\def\mxpic#1#2#3{\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip#1\centerline{\includegraphics[width=#2]{#3}}\vss}}} +\def\nb{$\vcenter{\begin{mplibcode}beginfig(1); +for t=0 upto 3: fill unitsquare scaled 4 shifted right shifted up rotated 45 rotated 90t; +endfor currentpicture := currentpicture scaled 1/2; endfig;\end{mplibcode}}$\ } +\def\unit#1{\ifmmode\,\else\thinspace\fi\hbox{#1}} +\def\to{\mathrel{\ldotp\ldotp}} +\def\<#1>{\ensuremath{\langle\hbox{#1}\rangle}} +\newfontface\ding{Zapf Dingbats} +\def\hey{\llap{\ding ☞\ }} +% +\begin{document} +\let\folio\thepage +\renewcommand{\thepage}{\rlap{\hbox to 5in{\hss\small\folio}}} +\pagestyle{myheadings} +\raggedbottom +% title & toc in the margin +\makeatletter +\moveright 6in\vbox to +0pt{\vskip23pt\noindent\sffamily{\Large\bfseries\@title}\par\bigskip + \noindent\@author\ --- \@date\par + \bigskip\noindent + \includegraphics{archimedes.pdf} +\vss} +\makeatother +\thispagestyle{empty} +\section{Start here} + +This document contains a collection of ideas and techniques for producing attractive +technical drawings with John Hobby’s \MP\ language. I’m assuming that you already +know the basics of the language, that you have it installed as part of your up to +date \TeX\ ecosystem, and that you have established a reasonable workflow that +let’s you write a Metapost program, compile it, and include the results in your +\TeX\ document. If not, you might like to start at the \MP\ page on CTAN, and read +some of the excellent tutorials, including "mpintro.pdf". If you have already done +this, please read on. + +These notes are partly based on the examples I have developed as answers +to questions about technical drawing on the \TeX\ Stack Exchange site. In +accordance with their terms and conditions, I’ve only included material here that +I’ve written myself --- if you want other people’s code then visit the site; +while most answers there focus on writing \LaTeX\ documents, there are a great +many questions about drawing, and some of the answers are very illuminating. + +My approach here will be to explore plain \MP, with examples grouped +into themes. One approach to using this document would be to read it end to end. +Another would be to flick through until you see something that looks like it might +be useful and then see how it’s done. + +And when I say \textit{plain} \MP\ I mean \MP\ with the default format (as defined +in the file "plain.mp") loaded and only a few simple external packages (like +"boxes.mp" occasionally). Nearly all of the examples here are supposed to be self +contained, and any macros are defined locally so you can get to grips with what’s +going on. \MP\ is a very subtle language, and it’s possible to do some very clever +and completely inscrutable things with it; in contrast I’ve tried to be as clear as +possible in my examples. + +\newpage +\section{Some features of the syntax} + +\begin{itemize} + \item Assignment or equation: the equation "a=3;" means “"a" is the same as "3" throughout the + current scope”; the assignment "a:=3;" means “update the value of "a" to the value "3" + immediately”. The difference becomes apparent when you try to update a + variable in the same scope. + \mwpic{0pt}{tiling} + + This difference also lets you write linear equations + like "a=-b;". After this, as soon as you give a value to "a", \MP\ immediately works + out the value of "b". + This is clever but has its limitations. As the following snippet reveals: +\begin{code} + % if you run this you will get this in the log + a + b = 0; show (a,b); % >> (a,-a) + a=42; show (a,b); % >> (42,-42) + a:=43; show (a,b); % >> (43,-42) +\end{code} + As soon as you assign to variable with ":=" \MP\ breaks any previously + established equations. + + \item Variable types: + + — + \<numeric> "a", + \<pair> "(a,b)" + + — + \<color> "(r,g,b)", + \<cmykcolor> "(c,m,y,k)" + \<transform> "(x,y,xx,xy,yx,yy)" + + — + \<string>, + \<path>, + \<picture> + + + If you don’t declare a variable, it’s assumed that it’s a \<numeric>. + When you do declare a variable --- \<numeric> or otherwise — any value that it + already had in the current scope is removed. + + \item Implicit multiplication: \MP\ inherits a rich set of rules about numerical + expressions from \MF, and of special interest is the scalar multiplication + operator. Any simple number, like "42", "3.1415", or ".6931", or any simple + fraction like "1/2" or "355/113" standing on it’s own (technically at the + primary level) and not followed by "+" or "-" becomes a scalar + multiplication operator that applies to the next token (which should be variable + of some appropriate type). + So you can write things like "3a", or even "1/2 a" + (the space between the number and the variable name is optional). This + lets you write very readable mathematical expressions. It’s quite + addictive after a while. + +\end{itemize} + +\vskip -22pt +\moveright5.5in\vbox to 0pt{\vss\hsize 4in\noindent +The \mpl{sqrt} operator is defined at the same (top) level of precedence, so +that \mpl{sqrt2+1} is read as \mpl{(sqrt2)+1} and not \mpl{sqrt(2+1)}, but fractions +trump even that, so \mpl{sqrt 1/2 = 0.7071} is true.} + +\newpage +\section{Workflow} + +This document is not meant for beginners, so you won't find step by step tutorials +for something so simple as running \MP. But since you might not find it all that +simple, and since the basic tutorials can go out of date, here are descriptions of +my own workflows that you might find helpful. You might also think I'm being really +inefficient; if so please drop me a line and suggest an improvement. + +The common features of each of these workflows are: mac os, the MacVim editor to +edit \MP\ source code, and +Skim.app to view PostScript and PDF files. I have the complete MacTeX distribution +installed; any commands mentioned below are supplied by MacTeX. + +\subsection{Stand alone graphics with plain \MP}\label{sec:plain-flow} + +\MP\ source files have the extension ".mp", when I open a file in MacVim that +matches "*.mp", my editor profile sets the file type to "mp" (which picks up the +highlight and indentation rules supplied with MacVim), and adds some relevant +directories to the search tree. Finally, if the file is a new file, then the +profile loads this template: +\begin{smallcode} +prologues := 3; +outputtemplate := "%j%c.%{outputformat}"; +beginfig(1); + +endfig; +end. +\end{smallcode} +The first two lines are important: "prologues := 3;" makes \MP\ put the full font +details in the output so that the files are self-contained; the "outputtemplate" +line means that the output will be written to files with an extension that matches +the chosen output format, which will be "png", "svg", or more usually "eps", which +is the default (and suggests that the output is Encapsulated PostScript). + +I then add drawing and label commands, using all the traditional facilities for +typesetting labels described in section \ref{sec:trad-labels}. I compile the source +with "mpost". I usually do this from within MacVim using the command line ":!mpost %" +where "!" means “this is an external command” and the "%" picks up the current file +name. Usually I need several attempts to get a diagram right, so I open Skim to preview the +output with ":!open -a Skim %:r1.eps". I have Skim set up so that when I recompile +the source, it automatically updates the view of the PostScript output. +\vadjust{\moveright5.4in\vbox to 0pt{\vss\hsize 4in\noindent +$$\includegraphics[width=4in]{workflow-plain.pdf}$$ +If I want to use the diagram in a \LaTeX\ document I can include the EPS file +directly with +\begin{texcode} + \includegraphics{some-diagram1.eps} +\end{texcode} +but usually I prefer to convert the EPS to PDF using "epstopdf" rather than rely on +the automatic conversion. This is mainly because the PDFs are generally more useful +files to have about (I can include them in presentations etc). Sometimes I do this +manually but usually I use a small Python script to automate this process: run +"mpost" with the "-recorder" option; scan the list of files to see what got +produced; check which ones are PostScript; call "esptopdf" to make each one into a +PDF file; remove each EPS file if successful. Your mileage may vary.\par +}} + +\subsection{Stand alone graphics with Lua\LaTeX}\label{sec:sa-lua-flow} + +For graphics with more complicated text formatting, I prefer now to use +\texttt{lualatex} with the \texttt{luamplib} package. \mpic{0cm}{workflow} +The work flow is a bit simpler because there are no intermediate EPS files to worry +about. Instead of compiling with plain "mpost" I use "lualatex" with the "luamplib" +package, which calls \MP\ from within the Lua environment. The \MP\ engine actually +used is exactly the same. Here is +the template I use: +\begin{texcode} +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +\end{texcode} +\vskip-12pt +\begin{smallcode} + beginfig(1); + + endfig; +\end{smallcode} +\vskip-12pt +\begin{texcode} +\end{mplibcode} +\end{document} +\end{texcode} +As you can see, we have \MP\ source code wrapped up in a minimal \LaTeX\ document +using the \texttt{standalone} class, which automatically adjusts the page size to +fit the contents of document, so is ideal for single diagrams. One small +disadvantage is that you can only produce a single PDF output file, so you need to have a +separate file for each picture, but the good news is that you get a much simpler and +more effective integration with \LaTeX, in particular with the font environment. +Since this only works with "lualatex" you have to use the "fontspec" package, as +explained in section \ref{sec:neo-labels}. + +\subsection{Integrated graphics with Lua\LaTeX} + +If you are ready to use "lualatex" for processing your entire document, then you can +directly embed your \MP\ drawings in a series of "mplibcode" environments. Each one +produces a horizontal-mode box. For details try "texdoc luamplib". The only +drawback of this all-in-one approach is that you have to compile all the drawings +every time you compile the document, which might slow you down --- although on a +modern machine this is not really an issue any more. + +\newpage +\section{Making and using paths} +\label{plain-shapes} + +In \MP\ there are two sorts of paths: open and closed. +\mpic{-12pt}{closed-triangles} +A closed path is called a cycle, and is created with the \mpl{cycle} primitive like this: +\begin{code} +path t; t = origin -- (55,0) -- (55,34) -- cycle; +\end{code} +You can think of \mpl{cycle} as meaning ‘connect back to the start and close the path’. +Note that you have explicitly put \mpl{cycle} to make a closed path. If you wrote +\begin{code}[xrightmargin=-32pt] +path u; u = origin {right} .. (55,0) .. (55,34) .. {-2,-1} origin; +\end{code} +then $u$ would be an open path even though the last point is the same as the +first. Any path that does not have \mpl{cycle} at the end is an open +path.\mpic{-12pt}{open-triangle} + +You can use \mpl{draw} with either sort of path, but you can only use \mpl{fill} with a +cycle. This concept is common to most drawing languages but it’s often hidden: an +open path might be automatically closed for you when you try to fill it. \MP\ takes +a more cautious approach; if you pass an open path to \mpl{fill} you will get an error +that says ‘Not a cycle’, even if the first and last points are the same like path +$u$ above. + +If you want to write a macro that deals differently with the two types +of path, then you can use \mpl{cycle} in a boolean context to test +whether a given path $p$ is closed: +\begin{code} + if cycle p: + % do something for closed path p + else: + % do something for open path p + fi +\end{code} + +\smallskip\noindent +\MP\ inherits the rich path-making syntax directly from \MF, so if you want a +general refresher, or you are not +quite sure what the five joiners do, $\longrightarrow$\marginpar{% + \vbox to 0pt{\vss\halign{\kern .5in\mpl{#}\hfil\quad&#\hfil\cr +..&free curve\cr +...&bounded curve\cr +--&straight line\cr +---&tense line\cr +\&&splice.\cr}\vss}} or you would like to bone up on exactly what \mpl{curl} and +\mpl{tension} are for, then you are recommended to review Chapter 14 of \mfbook. + +\vfill\noindent +Most of the examples in this document use only the two simple joiners \mpl{--} and +\mpl{..} with the occasional use of a direction-specifying pair before or after a +point. + +\clearpage +\subsection{Predefined closed paths} + +There are several closed paths defined for you in plain \MP. +\mpic{0pt}{closed-standards} +\begin{itemize} + \item "unitsquare" is defined as the path + \mpl{(0,0)--(1,0)--(1,1)--(0,1)--cycle}. It runs counter-clockwise from the + origin, and you can use it to draw any rectangle with appropriate + use of \mpl{xscaled} and \mpl{yscaled}, or a parallelogram with \mpl{slanted}, + or a diamond with \mpl{rotated} --- but note that the definition means + that is it centred on point \mpl{(1/2, 1/2)} so you might want to + shift it by \mpl{-(1/2,1/2)} before you transform it. + + \item "fullcircle" which you can use to draw any circle or ellipse with + appropriate use of \mpl{xscaled} and \mpl{yscaled}. Unlike the square, it is defined so that it is centred + at the origin. But beware that it has unit \textit{diameter}, so its radius + is $0.5\unit{bp}$ long. The path runs counter-clockwise and starts at 3 + o’clock; which means \mpl{point 0 of fullcircle = 1/2 right} is true. + + \item "superellipse()" which creates the shape beloved of the Danish designer Piet Hein. Unlike the + other two, this one is defined as a function rather than a \<path> constant, so you need to call it + like this: +\begin{code} +path s; +s = superellipse(1/2 right, 1/2 up, 1/2 left, 1/2 down, .8); +\end{code} + to create a ‘unit’ shape that matches \mpl{fullcircle} as shown on the right. + The fifth parameter is the ‘superness’: the value + 1 makes it look almost square, 0.8 is about right, 0.5 gives you a diamond, + and values outside the range $(0.5,1)$ give you rather weird propeller + shapes. + + Note that, unaccountably, $\id{superellipse}()$ is defined +in \texttt{plain.mp} with a \kw{def} rather than a \kw{vardef}. +This means you need to enclose it in a group before you can +transform it in any way. One way to do this is to use parentheses; or you +can assign it to a \<path> variable, as shown above. + +\end{itemize} + + +\newpage\subsection{Points on the standard closed paths} + +\noindent\mpic{-10pt}{closed-points}% +\textsc{Here are the three shapes} centred on the origin and labelled to show the points +along them. \textbf{Note} that the \textit{unitsquare} shape has been shifted so that it is +centred on the origin in all of these examples. The small red circle marks the +\textit{origin}, and the labelled red dots are the points of each path. The +\textit{unitsquare} has four points, while the other two shapes both have eight. +The small +arrows between point~0 and point~1 of each shape indicate the direction of the path +that makes up the shape. + +\vfill\noindent +If you want to highlight a segment of your shape, there’s a neat way to define it +using "subpath". Assuming "p" is the path of your shape, then this: +\begin{code} +center p -- subpath(1,2) of p -- cycle +\end{code} +creates a useful wedge shape which looks like this in our three ‘standard’ +shapes. + +\vfill\noindent +Better still, you are not limited to integer points along the path of your closed shape. +So if you wanted a wedge that was exactly $1/5$ of the area of your shape, you +could try +\begin{code} +center p -- subpath(0,1/5 length p) of p -- cycle +\end{code} +Clearly this works rather better with more circular shapes. Indeed for a circle you +can convert directly between circumference angle and points along the path. So +you have defined path $c$ to be scaled copy of \mpl{fullcircle}, then +\mpl{point 1 of c} is $45^\circ$ round and 1~radian is \mpl{point 1.27324 of c}, (because +$4/\pi \simeq 1.27324$). + +\vfill\noindent +In a closed path, the point numbering in \MP\ wraps round: so in a +circle, point $n$ is the same as point $n+8$; and in general point $n$ is the same as +point $n+\hbox{\bf length }p$. This works with negative numbers too, so we could use +\begin{code} +center p -- subpath(-1,1) of p -- cycle +\end{code} +to get wedge that extends either side of point 0. +The same idea was used to draw the arrows in the first row: +\begin{code} +drawarrow subpath(1/2, length p + 1/2) of p; +\end{code} + +\newpage +\subsection{Regular polygons of a given radius}\label{polygons} + +\noindent\mpic{-10pt}{closed-polygons}% +\textsc{Regular polygons} with a given radius can be defined or drawn directly with a simple inline loop: +\vadjust{\moveright -60pt \vbox to 0pt{\vskip-3pt\includegraphics{little-hexagon.pdf}\vss}} +\begin{code} +draw for i=0 upto 5: 20 dir 60i -- endfor cycle; +\end{code} +which works because \mpl{dir d} expands to \mpl{right rotated d}. But +you might prefer to make a macro: +\begin{code} +vardef polygon(expr n, r) = + for i=0 upto n-1: (r, 0) rotated (360/n * i) -- endfor cycle +enddef; +\end{code} +This produces a closed path to represent an $n$-sided polygon that fits in a circle +of radius $r$ centred at the origin and that starts at \mpl{(r, 0)}, like the +corresponding circular path, as shown in this polygonal version of the +previous segment chart.\rlap{\ $\rightarrow$}\\ +If you need polygon paths that start at the top, you can just swap the +coordinates: +\begin{code} +vardef polygon(expr n, r) = + for i=0 upto n-1: (0, r) rotated (360/n * i) -- endfor cycle +enddef; +\end{code} +$$\includegraphics[width=0.66\textwidth]{closed-polygon-tops.pdf}$$ +\hey Note also that some extra care is required to find the centres of these shapes. The +\mpl{center} macro defined in "plain.mp" gives you the centre of the bounding box, +but this is not the same as the centre of the polygon when the number of sides is +odd. What you need instead is the geometric median: +\begin{code} +vardef median primary p = + origin for i=1 upto length p: + point i of p / length p endfor +enddef; +\end{code} +This should work for any closed path, not just regular polygons. + +\newpage +\subsection{Regular polygons of a given side length}\label{polygons-given-side} +But you might want a polygon with a fixed side instead of a fixed radius. This needs +a little trigonometry, using the sine rule:\mpic{-42pt}{closed-fixed-polygon} +\begin{code} +vardef polygon_with_side(expr n, s) = + save a, b, r; numeric a, b, r; + a * n = 360; a + 2b = 180; r = s * sind(b) / sind(a); + for i = 0 upto n-1: (0, r) rotated (a * i) -- endfor cycle +enddef; +\end{code} +Which you can use like this to produce a nest of polygons $\rightarrow$ +\begin{smallcode} +for n = 11 downto 3: path p; p = polygon_with_side(n, 72); + fill p withcolor (n/32)[white, 3/4 if odd n: red else: blue fi]; + draw p; +endfor +\end{smallcode} +These polygon paths are centred on $(0, 0)$ but +sometimes it is more convenient to construct a polygon on a known segment +rather than working out how to rotate and shift it into place. +$$\includegraphics[width=0.9\textwidth]{closed-polygon-chain.pdf}$$ +Here is a way to do that using the \mpl{of} syntax in the macro construction +$\longrightarrow$ +\vadjust{\moveright 384pt\vbox to 0pt{\vss +\begin{smallcode} +vardef poly expr n of p = save x, y; + z0 = point 0 of p; z1 = point 1 of p; + for i=2 upto n-1: + z[i] = z[i-2] rotatedabout(z[i-1], 180(2/n-1)); + endfor + for i=0 upto n-1: z[i] -- endfor cycle +enddef; +path P[]; P3 = for i=0 upto 2: 6 up rotated 120i -- endfor cycle; +fill P3 withcolor 3/4 red; draw P3; +for n = 4 upto 23: + numeric m; m = floor(n / 2); + P[n] = poly n of subpath (m, m-1) of P[n-1]; + fill P[n] withcolor (n/32)[3/4 if odd n: red else: blue fi, white]; + draw P[n]; label(decimal n, median(P[n])); +endfor +\end{smallcode}\vskip -12pt}} + +\newpage\subsection{Curved polygons} + +\textsc{The regular polygons} above are all defined with straight edges using the +\mpl{--} connector that makes a tense path. If you changed each connector to +\mpl{..} you would get a circle, and contrariwise, if you try +\mpl{tensepath(fullcircle scaled 20)} you will get a regular octagon. But we can +also adjust the directions at the corners to make a variety of closed polygon shapes +with closed edges. + +One of the most pleasing is the Reuleaux polygon, with circular arcs for edges. +$$\includegraphics[width=0.9\textwidth]{closed-reuleaux-set.pdf}$$ +The figure on the right attempts to explain the geometry.\mwpic{-160pt}{closed-reuleaux-geometry} +\begin{code} +vardef reuleaux(expr n, r) = + save a; numeric a; a = 90/n; + for t = 0 step 4a until 359: + (0,r) rotated t {left rotated (a+t)} .. {left rotated (3a+t)} + endfor cycle +enddef; +\end{code} +If you swap the directions at each point you get shapes that are not quite like +hypocycloids; play about a bit more to get flower shapes or windmills. +$$\includegraphics[width=0.9\textwidth]{closed-antireuleaux-set.pdf}$$ + +\newpage +\subsection{A triangle of Schläfli polygons}\label{sec:gcd} + +Apart from the curious polygon patterns in the display, the main \MP\ point of interest +is the recursive "gcd" macro to find the greatest common divisor.\mpic{-108pt}{gons} + +\begin{code} +input colorbrewer-rgb + +vardef gcd(expr a, b) = + if b = 0: a else: gcd(b, a mod b) fi +enddef; + +beginfig(1); +for n=2 upto 24: + for s=1 upto floor n/2: + pair p; p = (12n - 24s, -24n); + path gon; gon = + for t=0 upto n/gcd(s,n) - 1: + 10 up rotated (360/n * s * t) -- + endfor cycle; + if (n mod s = 0): + fill gon shifted p withcolor Blues 9 2; + label("$" & decimal (n/s) & "$", p); + fi + draw gon shifted p withpen pencircle scaled 1/8; + endfor +endfor +endfig; +\end{code} + +\noindent +The macro also leads directly to an efficient way to find the least common multiple: +\begin{code} +vardef lcm(expr a, b) = a / gcd(a, b) * b enddef; +\end{code} +As always in \MP, it is safer to divide as early as possible to reduce the chance of arithmetic +overflow. + + +\newpage\subsection{Building cycles from parts of other paths} + +Plain \MP\ has a built-in function to compute the intersection points of two paths, and +there’s a handy high level function called "buildcycle" that uses this function to +create an arbitrary closed path. +\mpic{0pt}{area-under-graph} +The arguments to the function are just a list of paths, and providing the paths all +intersect sensibly, it +returns a closed path that can be filled or drawn. This is often used for colouring an +area under a function in a graph. +Here is an example. The red line has been defined +as path "f" and the two axes as paths "xx", and "yy". The blue area was defined +with +\begin{code} +buildcycle(yy shifted (1u,0), f, yy shifted (2.71828u,0), xx) +\end{code} +Note the re-use of the $y$-axis path shifted along by different amounts. + +\smallskip\noindent +There are similar examples in the \MP\ manual, but "buildcycle" can also +be useful in more creative graphics. +Here’s a second example that uses closed paths to give an illusion of depth to a simple +graphic of the planet Saturn. +\mpic{0pt}{saturn} +\marginpar{\hbox{}\vskip1.3in\raggedright\noindent\textbf{Notes}\begin{itemize} + \item The first five paths are just circles and ellipses based on "fullcircle". + \item The drawing is done inside an "image" simply so that the final result can + be drawn at an angle + \item "unfill gap" is shorthand for "fill gap withcolor background" + \item The subpaths passed to "buildcycle" are chosen carefully to make sure we + get the intersections at the right points and so that the component paths + all run in the same direction. Note that "subpath (8,4) of globe" runs + clockwise (that is backwards) from point 8 to point 4. +\end{itemize}} +\mpexternal{saturn-code.mp} + +\newpage\subsection{The implementation of \texttt{buildcycle}} +\textsc{The implementation} of "buildcycle" in plain \MP\ is interesting for a number of +reasons. Here it is copied from "plain.mp" (with minor simplifications) $\longrightarrow$ +\vadjust{\moveright5.5in\vbox to 0pt{\kern-1cm% +\begin{code} +vardef buildcycle(text input_path_list) = + save ta, tb, k, j, pp; path pp[]; + k=0; + for p=input_path_list: pp[incr k]=p; endfor + j=k; + for i=1 upto k: + (ta[i], length pp[j]-tb[j]) + = pp[i] intersectiontimes reverse pp[j]; + if ta[i]<0: + errmessage("Paths " & decimal i & + " and " & decimal j & " don't intersect"); + fi + j := i; + endfor + for i=1 upto k: + subpath (ta[i],tb[i]) of pp[i] .. + endfor cycle +enddef; +\end{code} +\vss}} + +\noindent +Notice how freely the indentation can vary; this is both a blessing +(because you can line up things clearly) and a curse (because the syntax may not +be very obvious at first glance). Notice also the different ways we can use a +$\kw{for}$-loop. The first two are used at the ‘outer’ level to repeat complete +statements (that end with semi-colons); the third one is used at the ‘inner’ level +to build up a single statement. + +The use of a "text" parameter allows us to pass a comma-separated list as an +argument; in this case the list is supposed to be a list of path expressions that +(we hope) will make up a cycle. The first "for" loop provides us with a standard +idiom to split a list; in this case the comma-separated value of "input_path_list" +is separated into into a more convenient array of paths called "pp" indexed by "k". +Note that the declaration of the array as "path" forces the argument to be a list of +paths. + +The second "for" loop steps through this array of paths looking for intersections. +The index "j" is set to be "k" when "i=1", and then set to the previous value of "i" +at the end of the loop; in this way +"pp[j]" is the path before "pp[i]" in what is supposed to be a cycle. +The macro uses the primitive operator \mpl{intersectiontimes} to find the intersection +points, if any. Note that we are looking for two path times: the time to start a +subpath of the current path and the time to end a subpath of the previous path; the +macro does this neatly +by reversing the previous path and setting the $b$-point indirectly by subtracting +the time returned from the length of the path. + +If all has gone well, then "ta" will hold all the start points of the desired +subpaths, and "tb" all the corresponding end +points. +The third and final "for" loop assumes that this is indeed the case, and tries to +connect them all together. Note that it uses ".." rather than "&" just in case the +points are not quite co-incident; finally it finishes with a +\mpl{cycle} to close the path even though point "tb" of path "k" should be identical (or +at least very close) to +point "ta" of path "0". + +This implementation of \mpl{buildcycle} works well in most cases, provided that there +are enough components to the cycle of paths. If you only have two paths, then the +two paths need to be running the same direction, and the start of each path must not +be contained within the other. This is explored in the next section. + +\newpage +\subsection{Strange behaviour of \texttt{buildcycle} with two closed paths} + +The implementation of "buildcycle" in plain \MP\ can get confused if you use it with +just two paths. Consider the following example: \mpic{0pt}{overlaps-missing-filler} +\begin{code} +beginfig(1); + path A, B; + A = fullcircle scaled 2.5cm; + B = fullcircle scaled 1.8cm shifted (1cm,0); + fill buildcycle(A,B) withcolor .8[blue,white]; + drawarrow A; drawarrow B; +endfig; +\end{code} +When we compile this example, we get no error message from "buildcycle", but there +is no fill colour visible in the output. The problem is that the points found by +"buildcycle" are the same both times that it steps through the middle loop, so +the closed path it returns consists of two identical (or very close) points and the +so the fill has zero area. + +Now observe what happens when we rotate and reverse each of the paths in +turn.\mpic{-24pt}{overlaps-default-fillers} +Number 1 corresponds to the example shown above; point~0 of~$A$ is inside the closed +path $B$. In~2 we have rotated path $A$ by 180° so that the start of path~$A$ is no +longer inside $B$, and now "buildcycle" works ‘properly’ --- but this is the only +time it does so. In~3, we've rotated $B$ by 180° as well, so that $B$ starts inside +$A$ and as expected "buildcycle" fails. In 4 we've rotated $A$ back to it's +original position, so that both paths start inside each other; and we get the +union of the two shapes. In 5--8, we've repeated the exercise with path $A$ +reversed, and "buildcycle" fails in yet more interesting ways. + +You could use this behaviour as a feature if you need to treat $A$ and $B$ as sets +and you wanted to fill the intersection, union, or set differences, but if you just +wanted the overlap, then you need to ensure that both paths are running in the same +direction and that neither of them starts inside the other. + +\newpage +\subsection{Find the overlap of two closed paths} + +As we have seen, in order to get the overlap of two closed paths from "buildcycle", +we need both paths to be running in the same direction, and neither path should +start inside the other one. It's not hard to create an "overlap" macro that does +this automatically for us. The first element we need is a macro to determine if a +given point is inside a given closed path. Following Robert Sedgwick's +\textit{Algorithms in C} we can write a generic "inside" function that works with any +simple closed path. The approach is to extend a horizontal ray from +the point towards the right margin and to count how many times it crosses the closed +path; if the number is odd, the point must be inside.\vadjust{\moveright5.5in\vbox + to 0pt{\kern-4.05cm +\begin{code} +vardef inside(expr p, ring) = + save t, count, test_line; + count := 0; + path test_line; + test_line = p -- (infinity, ypart p); + for i = 1 upto length ring: + t := xpart(subpath(i-1,i) of ring + intersectiontimes test_line); + if ((0<=t) and (t<1)): count := count + 1; fi + endfor + odd(count) +enddef; +\end{code} +\vss}}\label{function:inside} + +Equipped with this function we can create an "overlap" function that first uses the +handy "counterclockwise" function to ensure the given paths are running in the +same direction, and then uses "inside" to determine where the start points are. +\begin{smallcode} +vardef front_half primary p = subpath(0, 1/2 length p) of p enddef; +vardef back_half primary p = subpath(1/2 length p, length p) of p enddef; +% a and b should be closed paths... +vardef overlap(expr a, b) = + save A, B, p, q; + path A, B; boolean p, q; + A = counterclockwise a; + B = counterclockwise b; + p = not inside(point 0 of A, B); + q = not inside(point 0 of B, A); + if (p and q): + buildcycle(A,B) + elseif p: + buildcycle(front_half B, A, back_half B) + elseif q: + buildcycle(front_half A, B, back_half A) + else: + buildcycle(front_half A, back_half B, front_half B, back_half A) + fi +enddef; +\end{smallcode} +Using this "overlap" macro in place of "buildcycle" produces less surprising +results.\mpic{-2in}{overlaps} + +\newpage +\subsection{Fitting dashed lines to a path} + +\textsc{This one is} perhaps for perfectionists, but also shows how simple it +is to make useful adjustments in \MP. As you will know, plain \MP\ provides two +built-in dash patterns, so that you can draw a path \mpl{dashed withdots} or +\mpl{dashed evenly}. +\vadjust{% +\moveright384pt\vbox to 0pt{\vskip-46pt\hsize 4.2in\small +$$ +\begin{mplibcode} +vardef exactly(expr a) = + save m; numeric m; 2m = (a-6) / round(a/6); + dashpattern(on m off m) +enddef; +vardef gooddots(expr a) = + save m; numeric m; 2m = (a-5) / round(a/5); + dashpattern(off m on 0 off m) +enddef; + beginfig(1); + for i=2 upto 8: + path c[]; + c1 = fullcircle scaled 16i shifted 80 left; + c2 = fullcircle scaled 16i shifted 80 right; + draw c1 dashed evenly withcolor 2/3 blue; + draw c2 dashed exactly(arclength c2) withcolor 1/2 red; + %draw c1 shifted 160 down dashed withdots withcolor 2/3 blue; + %draw c2 shifted 160 down dashed gooddots(arclength c2) withcolor 1/2 red; + endfor + endfig; +\end{mplibcode} +$$ + +\noindent +The blue circles on the left were drawn with \mpl{dashed evenly}, and the uneven +gaps are noticeable at the “three o’clock” positions where the paths begin and end. +As you can see the default dash spacing looks fine at some sizes but bad on others. On the right you can see the +same circular paths coloured red, and drawn with \mpl{dashed exactly(arclength c, +6)}. + +$$ +\begin{mplibcode} +vardef exactly(expr a, u) = + save m; numeric m; 2m = a / round(a/u); + dashpattern(on m off m) +enddef; + beginfig(1); + for i=2 upto 8: + path c[]; + c1 = unitsquare shifted -(1/2, 1/2) scaled 16i shifted 80 left; + c2 = unitsquare shifted -(1/2, 1/2) scaled 16i shifted 80 right; + draw c1 dashed evenly withcolor 2/3 blue; + draw c2 dashed (exactly(arclength c2, 8) shifted 6 right) withcolor 1/2 red; + endfor + endfig; +\end{mplibcode} +$$ +Other paths may require a bit more ingenuity and thought. Because the square paths +have four equal sides, they work better with a target dash length that is a multiple +of 4. Here the blue squares on the left use the default \mpl{dashed evenly}, and the +red on the right, \mpl{exactly(arclength c2, 8)} \mpl{shifted 6 right}. The shift makes +the corners look better. + + +\vss}} + +The keyword \mpl{dashed} gives you access to the PostScript "setdash" command, its +argument is a special \<picture> defined with the \MP\ \mpl{dashpattern} function. +If you look in "plain.mp" you will find these declarations: +\begin{code} +picture evenly,withdots; +evenly = dashpattern(on 3 off 3); % dashed evenly +withdots = dashpattern(off 2.5 on 0 off 2.5); % dashed withdots +\end{code} +The detailed syntax is explained in §9.4 of the \MP\ manual, but essentially +\mpl{withdots} creates a unit 5 points long with a dot in the middle, and +\mpl{evenly} creates a unit 6 points long with the dashes 3pt long (plus the round +bit at the end of each dash, unless you have changed \mpl{linecap}) and gaps 3pt +long (minus any round bits). + +You probably also know that you can adjust these sizes by applying a suitable +scaling, so that \mpl{dashed withdots scaled 1/2} gives a rather denser dotted line, +and \mpl{dashed evenly scaled 4} will give you very long dashes. But you may also +notice that the dash patterns (particularly the longer ones) do not always fit your +paths exactly -- this is especially noticeable with closed paths, where you may end +up with one unsightly long dash or a very short gap a the point where the path +begins and ends. + +Fortunately there is a very simple solution: adjust the length of the dash pattern +so that an integer number of dash units exactly fit your path. +\begin{code} +vardef exactly(expr a, u) = + save m; numeric m; 2m = (a-u) / round(a/u); + dashpattern(on m off m) +enddef; +\end{code} +Here "a" is supposed to be the \mpl{arclength} of your path, and "u" the desired +unit size, so you can use it like this: +\begin{code} +path c; c = fullcircle scaled 200; +draw c dashed exactly(arclength c, 6); +\end{code} +to get a close approximation to \mpl{dashed evenly} that exactly fits the path. + +\newpage +\section{Numbers} + +This section discusses plain \MP's scalar numeric variables +and what you can do with them. +\MP\ inherits its unusual native system of scaled numbers from \MF; like many of +Knuth's creations it is slightly quirky, but works very well once you get the hang +of it. The original objective was to make \MF\ produce identical results on a wide +variety of computers. By default all arithmetic is carried out using 28-bit +integers in units of $1/65536$. This is done automatically for you, so you don’t +need to worry about it, but you should be aware of a couple of practical +implications: +\begin{itemize} + \item All fractions are rounded to the nearest multiple of $1\over65536$, so + negative powers of 2 ($1\over2$, + $1\over4$, $1\over8$, $\dots$) are exact, but other common fractions are not: + for example $1\over3$ is represented as + ${21845\over65536} \simeq 0.333328$, and $1\over10$ as + ${6554\over65536} \simeq 0.100006$. + You should bear this in mind particularly when you + choose fractional step-values in a "for" loop; the errors can accumulate so that + you may miss your expected terminal value.\vadjust{\moveright5.5in\vbox to + 0pt{\kern-2in\hsize4in\noindent + Compare the following two snippets: + $$\vbox{\halign{#\hfil\quad&#\hfil\cr + Code&Output\cr\noalign{\smallskip\hrule\bigskip} + \vtop{\parindent0pt\hsize2.2in + \bgroup\obeylines + $\kw{for}$ $i=0$ \kw{step} $1/10$ \kw{until} $1$: + \quad$\kw{show}$ $i$; + $\kw{endfor}$ + \egroup + +} + & + \vtop{\parindent0pt\parskip-2pt\obeylines\hsize60pt\tt +>> 0 +>> 0.1 +>> 0.20001 +>> 0.30002 +>> 0.40002 +>> 0.50003 +>> 0.60004 +>> 0.70004 +>> 0.80005 +>> 0.90005 +} +\cr\noalign{\bigskip} + \vtop{\parindent0pt\hsize2.2in + \bgroup\obeylines + $\kw{for}$ $i=0$ \kw{step} $1$ \kw{until} $10$: + \quad$\kw{show}$ $i/10$; + $\kw{endfor}$ + \egroup +} + & + \vtop{\parindent0pt\parskip-2pt\obeylines\hsize60pt\tt +>> 0 +>> 0.1 +>> 0.2 +>> 0.3 +>> 0.4 +>> 0.5 +>> 0.6 +>> 0.7 +>> 0.8 +>> 0.9 +>> 1 +}\cr\noalign{\bigskip\hrule}\cr +}}$$ +You get 11 iterations in the second but only 10 with the first. +\vss}} + + \item The system limits you to numbers that are less than 4096 in absolute value. + This can be an irritation if you are trying to plot data with large values, + but the solution is simple: scale your values to a reasonable range first. + + \item Intermediate calculations are allowed to be up to 32768 in absolute value + before an error occurs. You can sometimes avoid problems by using the + special Pythagorean addition and subtraction operators, but the general + approach should be to do your calculations before you scale a path + for filling or drawing. + + \item You can turn a number up to 32768 into a string using the "decimal" + command, and then you could append zeros to it using string concatenation. + +\end{itemize} + +If you are using a recent version of \MP\ you can avoid all these issues by choosing one of +the three new number systems: double, binary, or decimal, with the "numbersystem" +command line switch. But beware that if you write programs that depend on these new +systems, they might not be so portable as others. It's nice to have these new +approaches just in case, but you will not need to use them very often. + +\newpage\subsection{Numeric constants} + +Alongside the quirky number system, +plain \MP\ also inherits three numeric constants from \MF: \id{infinity}, \id{epsilon}, +and \id{eps}: +\begin{itemize} + \item $\id{eps}$ is defined to be a +small amount that is noticeable to \MF’s rounding algorithms, namely +${32\over65536}={1\over2048}\simeq 0.00049$. As a distance on the page or screen it's invisible at +any resolution less than 150,000 dots per square inch. If you were designing fonts + in \MF, $\id{eps}$ could help you avoid bad choices of pixels at low resolutions, but in +\MP\ it's only really useful in comparisons that might suffer from rounding errors. +\id{eps} is tiny, but it's bigger than any rounding error you may encounter, so +you can safely test for equality with: $\kw{abs}(\id{a}-\id{b})<\id{eps}$. + +\item $\id{epsilon}$ is defined to be $1\over65536$, the smallest positive scaled + number. + +\item $\id{infinity}$ is defined to be $4096-\id{epsilon}$, which is the largest + number you will normally deal with. This is useful when you just want a quantity + larger than any other in the immediate vicinity. For an example, look at the + definition of the "inside" function in + section~\ref{function:inside}. +\end{itemize} +These three quantities retain (approximately) the same value even if you choose one of the +alternative, higher precision, number systems. This is probably the most sane +approach, but the constants lose their status as the smallest and largest numbers you can +have. +\vadjust{\moveright5.5in\vbox to 0pt{\kern-221pt\hsize 4.25in +\noindent +Running the toy program: +\par\bigskip +$\kw{show} \id{numbersystem}, \id{eps}, \id{epsilon}, \id{infinity}; \kw{end}\!.$ +\par\bigskip\noindent +gives the following results with the different +number systems: + +\begin{code} +>> "scaled" +>> 0.00049 +>> 0.00002 +>> 4095.99998 + +>> "double" +>> 0.00048999999999999998 +>> 1.52587890625e-05 +>> 4095.9999800000001 + +>> "binary" +>> 0.00048999999999999999999999999999999993 +>> 0.0000152587890625 +>> 4095.9999800000000000000000000000001 + +>> "decimal" +>> 0.00049 +>> 0.0000152587890625 +>> 4095.99998 +\end{code} +\vss}} + +The messy set of results shown on the right arises because "plain.mp" defines these constants +like this (in version 1.005, which is current at the time of writing): +\begin{code} +eps := .00049; % this is a pretty small positive number +epsilon := 1/256/256; % but this is the smallest +infinity := 4095.99998; % and this is the largest +\end{code} +If you want cleaner constants, feel free to redefine the two decimals as: +\begin{code} +eps := 1/2048; +infinity := 64*64-epsilon; +\end{code} +These definitions are equivalent with "scaled" numbers, but more consistent at +higher precision. In particular they ensure that we always have +$4096 = \id{infinity} +\id{epsilon}$ whichever number system is in use. + +\newpage\subsection{Units of measure} +In addition to the very small and very large numeric variables, plain \MP\ inherits +eight more that provide a system of units of measure compatible with \TeX. +The definitions in "plain.mp" are very simple: $\longrightarrow$ +\vadjust{\moveright 384pt\vbox to 0pt{\kern-24pt +\begin{code} +mm=2.83464; pt=0.99626; dd=1.06601; bp:=1; +cm=28.34645; pc=11.95517; cc=12.79213; in:=72; +\end{code} +\vss}} + +When the output of \MP\ is set to be PostScript, then the basic unit of measure is +the PostScript point. This is what \TeX\ calls a "bp" (for `big point'), and it is +defined so that $1\unit{inch}=72\unit{bp}$. The traditional printers' point, which \TeX\ +calls a~"pt", is slightly smaller so that $1\unit{inch}=72.27\unit{pt}$. + +Normal use of these units relies on \MP's implicit multiplication feature. If you write +`$\id{w}=10\,\id{cm};$' in a program, then the variable \id{w} will be set to the value 283.4645. +The advantage is that your lengths should be more intuitively understandable, but if +you are comfortable thinking in PostScript points (72 to the inch, 28.35 to the +centimetre) then there is no real need to use any of the units.\marginpar{Bizarrely, 28.35 +is also the number of grammes to the ounce.} + +It is sometimes useful to define your own units; in particular many \MP\ programs +define something like `$\id{u}=1\,\unit{cm};$' near the start, and then define all +other lengths in terms of \id{u}. If you later wish to make a smaller or larger +version of the drawing then you can adjust the definition of \id{u} accordingly. +Two points to note: +\begin{itemize} + \item If you want different vertical units, you can define something like + `$\id{v}=8\,{mm}$' and specify horizontal lengths in terms of \id{u}, but + verticals in terms of \id{v}. + \item If you want to change the definition of \id{u} or \id{v} from one figure + to the next, you will either have to use `$\kw{numeric} \id{u},\id{v};$' at + the start of the your program in order to reset them, or + use the assignment operator instead of the + equality operator to overwrite the previous values. +\end{itemize} + +The unit definitions in "plain.mp" are designed for use with the default scaled +number system; if you want higher precision definitions, then you can update them by +including something like this at the top of your program: $\longrightarrow$ +\vadjust{\moveright 5.5in \vbox to 0pt{\kern-24pt +\begin{code} +% exact values to re-define the plain.mp units +numeric bp, in, mm, cm, pt, pc, dd, cc; +72 = 72 bp = 1 in; +800 = 803 pt = 803/12 pc; +3600 = 1270 mm = 127 cm; +1238 pt = 1157 dd = 1157/12 cc; +\end{code} +%\bgroup\obeylines\parindent0pt +%$\kw{numeric} \id{bp}, \id{in}, \id{mm}, \id{cm}, \id{pt}, \id{pc}, \id{dd}, \id{cc};$ +%$72 = 72\id{bp} = 1\id{in}$; +%$800 = 803 \id{pt} = 803/12 \id{pc};$ +%$3600 = 1270 \id{mm} = 127 \id{cm};$ +%$1238 \id{pt} = 1157 \id{dd} = 1157/12 \id{cc};$ +%\egroup +\vss}} + +The effect of the $\kw{numeric}$ keyword is to remove the previous definitions; the +four equation lines then re-establish the units with very slightly more accurate +definitions. You can safely use these definitions with "scaled", as they are +equivalent to the decimals currently given in "plain.mp", but the main point of the +example is to show how you can do implicit definitions with equations. + + +\newpage +\subsection{Integer arithmetic, clocks, and rounding} + +Native \MP\ provides nothing but a "floor" function, but "plain.mp" provides several +more useful functions based on this. +\begin{itemize} + \item `$\mathop{\kw{floor}} x$' returns $\lfloor x\rfloor$, $\hbox{the + largest integer} \le x$. You can use "x=floor x" to check that $x$ is an + integer. + \item `$\mathop{\kw{ceiling}} x$' returns $\lceil x\rceil$, $\hbox{the + smallest integer} \ge x$. + + \item `$x \mathbin{\kw{div}} y$' returns $\lfloor x/y \rfloor$, integer + division. + \item `$x \mathbin{\kw{mod}} y$' returns $x-y\times\lfloor x/y \rfloor$, + integer remainder. + +\end{itemize} +Note that $\kw{mod}$ preserves any fractional part, so $355/113 \mathrel{\kw{mod}} 3 = 0.14159$. + +\smallskip +\parshape=1 0pt 3.4in +This behaviour is usually what you want. +\vadjust{\moveright 266pt \vbox to 0pt{\noindent +\begin{mplibcode} +input clocks +beginfig(1); draw clock(hour, minute) scaled 0.8; endfig; +\end{mplibcode}\vss}} +For example we can use it to turn the time of day into an appropriate rotation for +the hands of a clock.% +\vadjust{\moveright 384pt\vbox to 0pt{\kern-196pt +\begin{code} +path hand[]; +hand1 = origin .. (.257,1/50) .. (.377,1/60) + & (.377,1/60) {up} .. (.40,3/50) + .. (.60, 1/40) .. {right} (.75,0); +hand1 := (hand1 .. reverse hand1 reflectedabout(left,right) + .. cycle) scaled 50; +hand2 = origin .. (.60, 1/64) .. {right} (.925,0); +hand2 := (hand2 .. reverse hand2 reflectedabout(left,right) + .. cycle) scaled 50; + +% hour of the day to degrees +vardef htod(expr hours) = 30*((15-hours) mod 12) enddef; +vardef mtod(expr minutes) = 6*((75-minutes) mod 60) enddef; + +vardef clock(expr hours, minutes) = image( + % face and outer ring + fill fullcircle scaled 100 withcolor 1/256(240, 240, 230); + draw fullcircle scaled 99 withcolor .8 white; + draw fullcircle scaled 100 withpen pencircle scaled 7/8; + % numerals + for h=1 upto 12: + label( decimal h infont "bchr8r", (40,0) rotated htod(h)); + endfor + % hour and minute marks + for t=0 step 6 until 359: + draw ((48,0)--(49,0)) rotated t; + endfor + drawoptions(withpen pencircle scaled 7/8); + for t=0 step 30 until 359: + draw ((47,0)--(49,0)) rotated t; + endfor + % hands rotated to the given time + filldraw hand1 rotated htod(hours+minutes/60); + filldraw hand2 rotated mtod(minutes); + % draw the center on top + fill fullcircle scaled 5; + fill fullcircle scaled 3 withcolor .4 white; +) enddef; +\end{code} +\vss}} +In the program given on the right, this idea +is used to define functions that convert from hours and minutes +to degrees of rotation on the clock. +\MP\ provides two internal variables +\id{hour} and \id{minute} that tell you the time of day when the +current job started. The clock face shown here was generated using +$$\kw{beginfig}(1);{}\mathbin{\kw{draw}}\id{clock}(\id{hour},\id{minute}); \kw{endfig};$$ +to give a sort of graphical time stamp. + +\vfill + +There is also a "round" function that rounds a number to the nearest integer. It is +essentially defined as $\mathop{\kw{floor}}(x+0.5)$ except that it is enhanced to +deal with $\kw{pair}$ variables as well. If you round a pair the $x$-part and +the $y$-part are rounded separately, so that $\mathop{\kw{round}}(3.14159, 2.71828) += (3,3)$. + +The "round" function only takes a single argument, but you can use it to round to a +given number of places by multiplying by the precision you want, rounding, and then +dividing the result. So to round to the nearest eighth you might use +`$\kw{round}(x\times8)/8$', and to round to two decimal places +`$\kw{round}(x\times100)/100$'. The only restriction is that the intermediate value +must remain less than 32767 if you are using the default number system. + +\newpage +\subsection{Integer powers} + +\textsc{Even if you use} one of the new number systems, you may occasionally get +caught out by the \mpl{**} operator. As the table on the right shows, you may get +an approximate answer from \mpl{x ** y} even when $x$ and $y$ are both integers. +\vadjust{\moveright 384pt\vbox to 0pt{\kern -2pt +$$\begin{mplibcode} +beginfig(0); +%primarydef x ** y = 1 for n=1 upto y: * x endfor enddef; +for x=1 upto 19: + for y = 1 upto 7: + if x = 1: + label("$x" if y>1: & "^{" & decimal y & "}" fi & "$", (52y, -20x)); + elseif y * mlog(x) < mlog(infinity): + numeric r; r = x ** y; + label(decimal r, (52y, -20x)) withcolor 3/4 if r = round(r): blue + else: red fi; + fi + endfor +endfor +label.lft("Results of \mpl{x**y} for small values, using", lrcorner currentpicture shifted 36 up); +label.lft("the default \mpl{scaled} number system", lrcorner currentpicture shifted 24 up); +currentpicture := currentpicture scaled 0.8; +endfig; +\end{mplibcode}$$\vss}} +Note that the squares are all integers, and the powers of two appear to be ok +(although if the page was wider you would see that \mpl{2**9} is $512.00002$), but +that with a couple of exceptions cubes and higher powers are slightly off. Changing +the number system makes it worse; even $x^1$ is not always an integer. The reason +can be found in the way that the \mpl{**} operator is defined in "plain.mp". +\begin{smallcode}[xleftmargin=0pt, xrightmargin=-36pt] +primarydef x ** y = if y = 2: x * x else: takepower y of x fi enddef; +def takepower expr y of x = + if x > 0: + mexp(y * mlog x) + elseif (x = 0) and (y > 0): + 0 + else: + if y = floor y: + if y >= 0: 1 for n=1 upto y: * x endfor + else: 1 for n=-1 downto y: / x endfor + fi + else: + hide(errmessage "Undefined power: " & decimal x & "**" & decimal y) + fi + fi +enddef; +\end{smallcode} +This is inherited direction from plain \MF, and as it says in the \mfbook, it is +optimized for $x^2$ and takes care to handle correctly negative numbers and zeros. +But for all positive values of $x$ other than 2 it is implemented using logs, and +the results are therefore only approximate. To avoid confusion where this might +matter (such as a particular offset into a recursively defined path) you could +simply use \mpl{round(7**3)} to get a whole number, or if you are sure that your $y$ +values are all non-negative integers, you could temporarily replace the definition: +\begin{smallcode} +primarydef x ** y = 1 for n=1 upto y: * x endfor enddef; +\end{smallcode} + + +\newpage +\section{Pairs, triples, and other tuples} + +\vpic{7pt}{random-selection} + +\noindent +\MP\ inherits a generalized concept of number from \MF\ that includes ordered pairs. +Pairs are primarily used as Cartesian coordinates, but can also be used as complex +numbers, as discussed below. \MP\ extends this gener\-al\-ization with 3-tuples and +4-tuples. Just like pairs, the elements in these tuples can take any numeric value, +so in theory it would be possible to use them for three- and four-dimensional +coordinates, but there are no built-in facilities for this in plain \MP, so some +external library is needed. \textit{All of the various attempts at three dimensions +in \MP\ are rather difficult to use, so none of them is discussed in this document}. + +\smallskip\noindent +Unlike simple numerics, the extended tuple variables are not automatically +declared for you, so if you want to define points $A$ and $B$ you need to explicitly +write `$\kw{pair} \id{A},\id{B};$' before you assign values to them. Once you have +declared them, you can equate them to an appropriate tuple using $=$ as normal. + +\begin{code} + pair A,B; A = B = (1,2); + color R; R = (1,2,3); + cmykcolor C; C = (1,2,3,4); +\end{code} + +The normal use of triples and quads is for colours (RGB colours and CMYK colours); +Triples are type \kw{color}, quads are type \kw{cmykcolor}. +You can't have tuples of any other length, not even as constants, except for +transforms. + +A transform is how \MP\ represents +an affine transformation such as "rotated 45 shifted (10,20)". +They are represented as 6-tuples, but if you try to write: +\begin{code} + transform T; T = (1,2,3,4,5,6); % <-- doesn't work +\end{code} +you will get a parsing error (that complains about a missing parenthesis after the 4). +You can examine and assign the individual parts using `$\kw{xpart} \id{T}$' etc. +More details below, and full details in the \MF\ book. + + +\newpage\subsection{Pairs and coordinates} +Now \textbf{pairs}: if you enclose two numerics in parentheses, you get a \<pair>. A +pair generally represents a particular position in your drawing with normal, orthogonal +Cartesian $x$- and $y$-coordinates, but you can use a pair variable for other +purposes if you wish. As far as \MP\ is concerned it's just a pair of numerics. + +\MP\ provides a simple, but slightly cumbersome, way to refer to each half of a +pair. The syntax `$\kw{xpart} \id{A}$' returns a numeric equal to the first number in +the pair, while `$\kw{ypart} \id{A}$' returns the second. The names refer to the +intended usage of pair variable to represent pairs of $x$ and $y$-coordinates. +Note that they are read-only; you can't +assign a value to an $\kw{xpart}$ or a $\kw{ypart}$. So if you want to update only one +part of a pair, you have to do something like this: $\id{A} \mathrel{:}= (42, +\kw{ypart}\id{A});$ + +In addition there is a neat macro definition in plain \MP\ that allows you do deal +with the $x$- and $y$-parts of pairs rather more succinctly. +\vadjust{\moveright 384pt\vbox to 0pt{\kern-140pt +\noindent Plain \MP\ provides this definition +\begin{code} +vardef z@#=(x@#,y@#) enddef; +\end{code} +which you can use to find orthogonal points. + +\bigskip\noindent +\includegraphics{random-function} +\vss}}% +The deceptively simple definition of $\id{z}$ as a subscripted macro allows you to +write "z1 = (10,20);" and have it automatically expanded into the equivalent of +"x1=10;" and "y1=20;". You can then use "x1" and "y1" as independent numerics or +refer to them as a pair with "z1". A common usage is to find the orthogonal points +on the axes in graphs, like so $\longrightarrow$ + +\smallskip +There is also a simple way to write coordinates using a polar notation +using\label{polar} +\mpl{dir}. This macro is defined so that \mpl{dir 30} expands to \mpl{right rotated +30} and then to \mpl{(1,0) rotated 30}, which becomes \mpl{(cosd(30), sind(30)} +or \mpl{(0.86603, 0.5)}. So to get the polar notation +point $(r,\theta)$, where $r$ is the radius and $\theta$ is the angle in degrees +counter-clockwise from the positive $x$-axis, you can write `\mpl{r * dir +theta}'. As usual, with a constant you can omit the multiplication sign, so `\mpl{2 +dir 30}' provides another way to define the point "(sqrt(3),1)". + +\smallskip +Plain \MP\ defines five useful pair variables: \id{origin}, \id{right}, \id{up}, +\id{left}, and \id{down}. As so often, Knuth-Hobby definitions in "plain.mp" are +quite illuminating +$\longrightarrow$ +\vadjust{\moveright 384pt\vbox to 0pt{\kern-36pt +\begin{code} +% pair constants +pair right,left,up,down,origin; +origin=(0,0); up=-down=(0,1); right=-left=(1,0); +\end{code} +\vss}}% +As you can see, pair variables can be used in implicit equations. + +They can also be scaled using implicit multiplication, so writing +`$144 \id{right}$' is equivalent to writing `$(144,0)$' but possibly a bit more +readable. In particular the idiom `$\kw{shifted} 200 \id{up};$' +works well when applied to a point, a path, +or an image. +Unfortunately, this convenient notation does not work well with units of +measure. This is because implicit multiplication only works between a numeric constant and a +variable. So `$2 \id{in}\, \id{right}$' does not work as you might expect; you can +write `$2\id{in} \mathrel{\ast}\id{right}$' but by that stage it's probably simpler to +write `$(2\id{in},0)$' or even just `$(144,0)$'. + + +\newpage +\subsection{Pairs as complex numbers} + +As you might expect in a language designed by mathematicians, \MP's pair variables +work rather well as complex numbers. To represent the number $3+4i$ you can write +"(3,4)". To get its modulus, you write "abs (3,4)" (which gives $5$ in this case), +and to get its argument, you write "angle (3,4)" (which gives $53.1301$). Note that +"angle" returns the argument in degrees rather than radians, and that the result is +normalized so that $-180 < \kw{angle} (x,y) \le 180$. + +The standard notation for points supports this usage. You can write "z0=(3,4);" and then +extract or set the real part with "x0" and the imaginary part with "y0". If you +want to use other letters for your variable names, you can use "xpart" and "ypart" +to do the same thing. So after `\mpl{pair w; w=(3,4);}' you can get the real part with +"xpart w" and the imaginary part with "ypart w". +You can also use the polar notation shown above to write complex numbers. For +$re^{i\theta}$ you can write `\mpl{r * dir theta}' where "r" is the modulus and +"theta" is the argument in degrees. + +The predefined constants \mpl{up}, +\mpl{down}, +\mpl{left}, and +\mpl{right} also provide points on the unit circle corresponding to $i$, $-i$, +$-1$, and $+1$ respectively. It's tempting to define `\mpl{pair i; i=(0,1);}', so that +you can write constants like "4i" directly, but this is not very helpful, because +"3+4i" will give you an error since \MP\ does not let you add a "numeric" to a "pair". + +However \MP\ does let you add (and subtract) two pairs, so complex addition and +subtraction are just done with the +normal operators. +\vadjust{\moveright 384pt\vbox to 0pt{\vskip -3.74in +$$\includegraphics{complex-operators}$$ +\begin{code} +beginfig(1); + numeric u; u = 1cm; + z1 = 2 dir 15; z2 = 1.2 dir 60; + z3 = z1+z2; z4 = z1 zscaled z2; z5 = (x1,-y1); + drawoptions(withcolor 2/3 white); + draw (1/2 left -- 3 right) scaled u ; + draw (1/2 down -- 3 up ) scaled u ; + draw subpath (0,3) of fullcircle scaled 2u rotated -22.5; + drawoptions(); + dotlabel.lrt (btex $\scriptstyle 1$ etex, (u,0)); + dotlabel.ulft(btex $\scriptstyle i$ etex, (0,u)); + interim ahangle := 30; + forsuffixes @=1,2,3,4,5: + x@ := x@ * u; y@ := y@ * u; + drawarrow origin -- z@ + cutafter fullcircle scaled 5 shifted z@ + withcolor 2/3 if @ < 3: blue else: red fi; + endfor + fill fullcircle scaled dotlabeldiam; + dotlabel.rt (btex $A$ etex, z1); + dotlabel.urt(btex $B$ etex, z2); + dotlabel.top(btex $A+B$ etex, z3); + dotlabel.top(btex $A \times B$ etex, z4); + dotlabel.rt (btex $\bar{A}$ etex, z5); +endfig; +\end{code} +\vss}}% +To get the complex conjugate you +could use "reflectedabout(left,right)", but it's probably easier just to write +"(x0,-y0)" or define a simple function: +\begin{code} + def conj(expr z) = (xpart z, -ypart z) enddef; +\end{code} +Complex multiplication is provided as part of the core language by the "zscaled" +operator. This is defined with the same precedence as "scaled" or normal scalar +multiplication (which is what you usually want). So "(3,4) zscaled (1,2)" gives +"(-5,10)" because $(3+4i)\times(1+2i) = 3+6i+4i-8 = -5+10i$. +"zscaled" is only defined to work on two "pair" variables, so you can't write +\mpl{(3,4) zscaled 4}. To get that effect with "zscaled" you would have to write +\mpl{(3,4) zscaled (4,0)}, but this is the same as +\mpl{(3,4) scaled 4}, which is usually simpler to write. If your pair is +stored as a variable you can write (for example) \mpl{4 z0} to get the same +effect. Or \mpl{1/4 z0} or \mpl{z0/4} for scalar division. + +There are no other complex operators available, but it is not hard to implement the +usual operations when they are required\dots + +\newpage +\subsubsection{Extra operators for complex arithmetic} + +Since multiplication by $z$ can be thought of as a transformation consisting of +rotation by the argument of $z$ and scaling by $|z|$, you can define the complex +inverse and complex square root simply using \mpl{angle} and \mpl{abs}. +\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip -24pt +\centerline{\includegraphics{complex-inverse-and-sqrt}} +The interesting part of the left hand figure was drawn as follows: +\begin{smallcode} +input colorbrewer-rgb +for i=1 upto 3: + z[i] = (3/2 + 1/4 normaldeviate) * dir (16 i + normaldeviate); + path a; + a = z[i] scaled 42 -- origin -- zinverse(z[i]) scaled 42; + drawarrow a cutafter fullcircle scaled 5 shifted point 2 of a + withcolor SetTwo[7][2+i]; + drawdot point 0 of a withpen pencircle scaled dotlabeldiam; + drawdot point 2 of a withpen pencircle scaled dotlabeldiam; + drawdot point 2 of a withpen pencircle scaled 2 withcolor white; +endfor +\end{smallcode} +The only difference on the right is that the drawing used "zsqrt()". +\vss}} + +\smallskip\noindent +First an inverse function. The idea here is to find a function that is the opposite +of complex multiplication, so we want something that gives +\begin{code} +z zscaled zinverse(z) = (1,0) +\end{code} +In other words you need to find a complex number with an argument that is the +negative of the argument of $z$ and a modulus that will scale $|z|$ to 1. +You can use the polar notation with "dir" to write this directly: +\begin{code} +vardef zinverse(expr z) = 1/abs z * dir - angle z enddef; +\end{code} +The complex division, $z/w$, can now be done as: \mpl{z zscaled zinverse(w)}. +The only difficulty with this function is how it deals with zero, or rather with +the point $(0,0)$. Since `\mpl{abs (0,0)}' gives $0$, the function will give you +a `divide by zero' error if it's called with $(0,0)$. But this is probably what you want +it to do, since there is no easy way to represent the point at infinity in the +extended complex plane on paper. + +\medskip\noindent +For square root, you want a function `\mpl{zsqrt(z)}' that returns +a complex number with half +the argument of $z$ and a modulus that is the square root of the modulus of $z$, so that +`\mpl{zsqrt(z) zscaled zsqrt(z) = z}'. This does the trick: +\begin{code} +def zsqrt(expr z) = sqrt(abs z) * dir 1/2 angle z enddef; +\end{code} +This function also has a difficulty with the point $(0,0)$, because "angle (0,0)" is +not well defined, and so \MP\ throws an error. If you want a function that +correctly returns $(0,0)$ as its own square root, then try something like this: +\begin{code} +def zsqrt(expr z) = + if abs z > 0: sqrt(abs z) * dir 1/2 angle fi z +enddef; +\end{code} + +\newpage +\subsubsection{Using complex numbers to draw fractals} + +As an example of what you can do with complex arithmetic, here is a version of the +diagram from §4.1 of Knuth's \textsl{Seminumerical Algorithms} showing $S$, the set +of all points that can be written as $\sum_{k\ge1}a_k(i-1)^{-k}$. +\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip -81pt +\begin{smallcode} +vardef fizz(expr X) = + pair m, n; + m = right; + n = origin; + numeric a, x; + x = X; + forever: + exitif x = 0; + m := m zscaled zinverse((-1, 1)); + a := x mod 2; + n := n + a * m; + x := x div 2; + endfor + n +enddef; +input colorbrewer-rgb +color shade[]; +shade0 = Reds 5 4; shade1 = Oranges 5 4; +shade2 = Greens 5 4; shade3 = Blues 5 4; + +beginfig(1); + numeric s, t; s = 256; t = 4; + for n=0 upto (s/t*s/t-1): + numeric h, v; + h = floor 1/8 (n mod 32); + v = n mod 4; + fill fullcircle scaled t shifted (fizz(n) scaled s) + withcolor (1/2 + 1/8 v)[white, shade[h]]; + endfor; + + path xx, yy; + xx = (left--right) scaled (s+8); + yy = xx rotated 90; + for i=-1 upto 1: + draw xx shifted (0, s*i) withpen pencircle scaled 1/8; + draw yy shifted (s*i, 0) withpen pencircle scaled 1/8; + endfor + dotlabel.lrt(btex $-1-i$ etex, (-1, -1) scaled s); + dotlabel.lrt(btex $-1$ etex, (-1, 0) scaled s); + dotlabel.lrt(btex $-1+i$ etex, (-1, 1) scaled s); + dotlabel.lrt(btex $-i$ etex, (0, -1) scaled s); + dotlabel.lrt(btex $+i$ etex, (0, 1) scaled s); + dotlabel.lrt(btex $+1-i$ etex, (1, -1) scaled s); + dotlabel.lrt(btex $+1$ etex, (1, 0) scaled s); + dotlabel.lrt(btex $+1+i$ etex, (1, 1) scaled s); +endfig; +\end{smallcode} +\vss}} +$$\includegraphics[width=\textwidth]{double-dragon.pdf}$$ +\vbox to 0pt{\noindent\small\textbf{Note}: you can adjust the “resolution” with the parameter $t$, but don't +make it smaller than 1 if you are using the default number system; the diagram looks +a bit strange unless $t$ is an integer power of 2.\vss} + +\newpage +\section{Colours} + +\MP\ implements colours as simple numerics, or tuples of three or four numeric values. +Three-tuples (which are type \mpl{color}) represent RGB colours; four-tuples +(which are type \mpl{cmykcolor}) represent CMYK colours. Simple numerics are used to +represent grey scale colours. + +The numeric values of the colours can take any \kw{numeric} value, but \MP\ only considers the +range 0 to 1 --- values less than zero are treated as zero, values greater than 1 are +treated as 1. +So British Racing Green with RGB code "(1,66,37)", +or Pillar Box Red with code "(223,52,57)", can be defined like this: +\begin{code} + color brg, pbr; + brg = (0.00390625, 0.2578125, 0.14453125); + pbr = (0.87109375, 0.203125, 0.22265625); +\end{code} +or, slightly more idiomatically: +\begin{code} + brg = 1/256 (1, 66, 37); + pbr = 1/256 (223, 52, 57); +\end{code} +As you can see, you can apply implicit multiplication to a \mpl{color}, so after +the declaration above "2 brg" would be a valid colour, although you have to think +a bit to know what that means in terms of colour in your drawings. +\vadjust{\moveright 384pt\vbox to 0pt{\kern-72pt +To use RGB hex strings, you'll need to write a function: +\begin{code} +vardef hexrgb(expr Spec) = + save r, g, b; + numeric r, g, b; + r = hex(substring (1,3) of Spec); + g = hex(substring (3,5) of Spec); + b = hex(substring (5,7) of Spec); + 1/256(r,g,b) +enddef; +brg = hexrgb("#014225"); +pbr = hexrgb("#df3439"); +\end{code} +\vss}}% + +Plain \MP\ defines five basic colour constants: \mpl{red}, \mpl{green}, +\mpl{blue}, \mpl{white}, \mpl{black}. These are quite useful with leading +fractions: \mpl{2/3 red} gives a nice dark red, that's good for drawing lines you +want to emphasize; \mpl{1/2 white} gives you a shade of grey; and so on. But since +\mpl{black} is defined as \mpl{(0,0,0)}, \mpl{1/2 black} just gives you \mpl{black}. + +You can also add up \mpl{colors}. So \mpl{red + 1/2 green} gives you a shade of +orange; this is more long-winded than writing \mpl{(1, 0.5, 0)} but maybe slightly +easier to read. Much more usefully, you can use the mediation notation to get a +colour that is part way between two others. So \mpl{1/2[red, white]} gives you a +shade of pink, and \mpl{2/3[blue, white]} a sort of sky blue. You can also use this +idea to vary colour with data, as in \mpl{(r)[red, blue]} where \mpl{r} is some +calculated value. \vadjust{\moveright 384pt\vbox to 0pt{\kern-36pt +\begin{code} +color brg; brg = 1/256 (1, 66, 37); +color pbr; pbr = 1/256 (223, 52, 57); +N = 5; n = 0; +for y=1 upto N: + for x=1 upto N: + fill fullcircle scaled 16 shifted 20(x,y) + withpen pencircle scaled 2 + withcolor (n/N/N)[pbr, brg]; + label(decimal incr n infont "phvr8r", 20(x,y)) + withcolor white; + endfor +endfor +\end{code}\vss}} +Here's a toy example: + +\vbox to 0pt{\centerline{\includegraphics[scale=0.8]{color-blend-toy}}\vss} + +\newpage +\subsection{CMYK colours} + +\MP\ also implements a CMYK colour model, using tuples of four numerics. +This is more or less a direct mapping onto the PostScript "cmykcolor" functions. +In this model the four components represent cyan, magenta, yellow, and black. +White is \mpl{(0,0,0,0)} and black is anything where the last component is 1. +Beware however that the constants, \mpl{white} and \mpl{black} are defined in +"plain.mp" as RGB colours, and you can't mix the two models, so anything like +\mpl{1/2[(1,1,0,0), white]} will not work. If you want to do lots of work with CMYK colours +you might like to redefine the color constants. +$$\includegraphics[width=0.7\textwidth]{blended-color-circles}$$ +\hey The apparent blending of colours here is done by calculating the overlaps +and filling them in order. In plain \MP, there is no support for transparency in any of the +colour models; this is because they are inherited directly from PostScript. + +\moveright 384pt\vbox to 0pt{\vss +\begin{code} +% the illusion of blended colours is helped by buildcycle + +path C[], B[]; + +% arrange each circle so that point 0 is outside the others +C1 = fullcircle scaled 120 rotated 90 shifted 40 up; +C2 = C1 rotated 120; +C3 = C2 rotated 120; + +B0 = buildcycle(C1, C2, C3); +B1 = buildcycle(C1, C2); +B2 = buildcycle(C2, C3); +B3 = buildcycle(C3, C1); + +picture P; +for x=0 upto 1: + for y=0 upto 1: + P := image( + s := 1/4 + x/2; + k := 0 + y/4; + fill C1 withcolor s*(1,0,0,k); + fill C2 withcolor s*(0,1,0,k); + fill C3 withcolor s*(0,0,1,k); + fill B3 withcolor s*(1,0,1,k); + fill B2 withcolor s*(0,1,1,k); + fill B1 withcolor s*(1,1,0,k); + fill B0 withcolor s*(1,1,1,k); + undraw C1; undraw C2; undraw C3; + ) shifted -(200x, 200y); + draw P; + label.bot(("shade: " & decimal s & ", k: " & decimal k) + infont "phvr8r", point 1/2 of bbox P); + endfor +endfor +\end{code}} + +\newpage +\subsection{HSV colours} + +HSV colours are colours defined by a triple of hue, saturation, and value. +Unlike RGB and CMYK colours there is no native support in \MP\ but it is possible to +write a routine that maps HSV triples into RGB colours: +\begin{code} +vardef hsv_color(expr h,s,v) = + save chroma, hh, x, m; + chroma = v*s; + hh = h/60; + x = chroma * (1-abs(hh mod 2 - 1)); + m = v - chroma; + if hh < 1: (chroma,x,0)+(m,m,m) + elseif hh < 2: (x,chroma,0)+(m,m,m) + elseif hh < 3: (0,chroma,x)+(m,m,m) + elseif hh < 4: (0,x,chroma)+(m,m,m) + elseif hh < 5: (x,0,chroma)+(m,m,m) + else: (chroma,0,x)+(m,m,m) + fi +enddef; +\end{code} +This is based on information from the Wikipedia article on +on “HSL and HSV”. + +\medskip\noindent +The hue values in HSV colours map nicely to the familiar spectrum +of the rainbow. In the model used here 0 is red, 120 green, and 240 blue: +$$\includegraphics[width=0.85\textwidth]{color-hsv-gamut}$$ +With less saturation the colours look faded; if you lower the value they get +darker. Once you get the hang of them, they make choosing colours rather easier. +You can produce ranges of colour by changing hue, or make gradations of a single +colour by changing the saturation or value. + +\moveright 384pt\vbox to 0pt{\vss +\begin{code} +defaultfont := "phvr8r"; + +numeric s[], v[]; +s0 = 1/2; v0 = 7/8; +s1 = 7/8; v1 = 7/8; +s2 = 7/8; v2 = 1/2; +for y=0 upto 2: + for h=0 step 15 until 360: + fill fullcircle scaled 24 shifted (h, -32y) + withcolor hsv_color(h, s[y], v[y]); + draw fullcircle scaled 24 shifted (h, -32y) + withcolor white; + if y=1: + label(decimal h infont defaultfont scaled 1/2, (h,-16)); + fi + endfor +endfor + +label.urt("Less saturation", (-20,12)); +label.lrt("Lower value", (-20,-76)); + +drawarrow (-15, -12) -- (-15,12); +drawarrow (-15, -52) -- (-15,-76); +\end{code}} + +\newpage +\subsubsection{An HSV example of a graduated scale} + +\noindent +This example requires the \mpl{hsv_color} routine from the previous page. +\mpic{12pt}{color-hsv-bathymetric} +\begin{code}[xleftmargin=0pt] +defaultfont := "phvr8r"; defaultscale := 3/4; +path h,d,b; numeric n; n = 10; +h = ((-2,0)--(0,0)--(-1,3)--(-2,3)--cycle) scaled 60; +d = h rotated 180; +b = subpath (0,1) of h -- point 1+1/n of d -- + (xpart point 0 of h, ypart point 1+1/n of d) -- cycle; +fill b withcolor hsv_color(123, 1/8, 7/8); +draw subpath (2.13,4) of b; +for i=1 upto n: + fill point 4-(i-1)/n of h -- point 1+(i-1)/n of h + -- point 1+i/n of h -- point 4-i/n of h -- cycle + withcolor hsv_color(42, 1/4 + 3/4 * i/n, 1 - i/3n); + fill point 4-(i-1)/n of d -- point 1+(i-1)/n of d + -- point 1+i/n of d -- point 4-i/n of d -- cycle + withcolor hsv_color(200, i/n - 1/n, 1 - i/3n); +endfor +string s; +for i=1 upto n-1: + draw point 4-i/n of h -- point 1+i/n of h; + draw point 4-i/n of d -- point 1+i/n of d; + s := decimal if i < 4: (i**2+1) else: (10 + (i-3)*10) fi & "00"; + label.rt(s, point 1+i/n of h); + label.lft(s, point 1+i/n of d); +endfor +label.rt("Metres", point 2 of h); +label.lft("Metres", point 2 of d); +label.lft("Hypsometric tints" infont defaultfont + scaled defaultscale rotated 90, point 7/2 of h); +label.rt("Bathymetric tints" infont defaultfont + scaled defaultscale rotated -90, point 7/2 of d); +label.lft("sea level", point 0 of h); +label("areas below sea level", center b); +draw h; draw d; +\end{code} + + + +\newpage +\subsection{Grey scale} + +\moveright5.5in\vbox to 0pt{\vskip6pt +\begin{code}[xleftmargin=0pt] +numeric s; s = 13; +path atom; +atom = origin + -- (2s,0) rotated -30 -- (2s,0) rotated -30 + (0,s) + -- ( s,0) rotated 30 -- ( s,0) rotated 30 + (0,s) + -- (0,2s) -- cycle; + +picture p[]; +for i=0 upto 2: + p[i] = image( + fill atom rotated -120i withcolor (7/8 - 1/8i) ; + draw atom rotated -120i; + ); +endfor + +n = 13; +for i=-n upto n: + for j=-n upto n: + forsuffixes $=0,1,2: + draw p$ shifted ((3i*s,0) rotated -30 + + (0,floor(1/2i)*3s + 3j*s)); + endfor + endfor +endfor + +clip currentpicture to (unitsquare shifted -(1/2,1/2) + xscaled 55.425s yscaled 30s); +\end{code} +\vss} +\noindent +The \mpl{withcolor} command will also take a single \mpl{numeric} instead of a 3-tuple or +a 4-tuple. This produces a colour in grey scale (or gray scale if you prefer the +Webster spellings). Just as for the other colour types, values below 0 count as +zero and values above 1 count as one. And since the smallest possible positive +number in plain \MP\ is: $\id{epsilon} = 1/256/256;$ then you can have at most 65,536 shades in +between. + +Grey scale is appropriate for some printed media, and can make effective textures +and patterns. +The code on the right was used to produce this: +$$\includegraphics[width=\textwidth]{escher}$$ +First a basic path (named $\id{atom}$) is defined, then in the first loop three +picture variables, $p_1$, $p_2$, and $p_3$, are defined, each one rotated +120° from the previous and filled with a slightly darker shade of grey. +The double loop then draws the three versions of the shape on an up-and-down grid. +Finally the picture is clipped to a neat rectangle. + +\newpage +\subsubsection{Drawing algorithmic shadows} +\moveright5.5in\vbox to 0pt{\vskip6pt +\begin{code}[xleftmargin=0pt] +path b, w; +b = ((-3,-4)--(3,-2)--(3,+2)--(-3,4)--cycle) scaled 5; +w = b reflectedabout(up, down); + +numeric n; +n = 128; + +picture B, W; +B = image(for i=0 step 1/n until 1: + draw point 4-i of b -- point 1+i**2 of b + withcolor 1-i**8; + endfor); + + W = image(for i=0 step 1/n until 1: + draw point 4-i of w -- point 1+i**2 of w + withcolor 3/4-i**8; + endfor); + +for i=-9 upto 9: + for j=-4 upto 4: + draw if odd (i+j): W else: B fi shifted (i*30,j*30); + endfor +endfor + +clip currentpicture to bbox currentpicture yscaled 7/8; +\end{code} +\vss} +\noindent +Here is a more complex pattern, showing one way to create an +illusion of shadows with multiple fine lines. +$$\includegraphics[width=\textwidth]{shadows}$$ +The first part defines two wedge-shaped closed paths, $\id{w}$ being +the mirror image of $\id{b}$. Like the standard \id{unitsquare} path, the +path $\id{b}$ is defined so that point 0 is the bottom left corner. + +The two $\kw{picture}$ variables are produced +by drawing lines across the shapes from bottom to top. By setting $n$ high enough, +these multiple lines blend smoothly to give an even colour. And by using higher +powers of the index variable, an effective shadow can be drawn ‘bunched up’ into the +top of each shape. + +By repeating them alternately in a grid, we get an effective texture, which is +clipped at the end to a neat rectangle again. + + +\newpage +\subsection{Colorbrewer palettes}\label{colorbrewer} + +\moveright 384pt\vbox to 0pt{\vskip144pt\raggedright\hsize4in\noindent +This map shows the "RdYlBu[9]" palette in action on a map of the Brexit vote in +London. The outlines are the 33 London boroughs, and the colours show how we voted, +faded by turnout. The data and the outlines are from publicly-available UK +government sources. They were prepared for \MP\ using various Python scripts, and +they are available in the source for this document. + +\vskip 72pt +Here is the code for the palette used as the legend: +\begin{smallcode} +input colorbrewer-rgb +numeric s; s = 10; +for i = 1 upto 9: + fill unitsquare scaled s shifted (i*s, 0) withcolor RdYlBu[9][i]; + if i > 1: draw (i*s, 0) -- (i*s, s); fi +endfor +draw unitsquare xscaled 9s yscaled s shifted (s,0); +label.top("Leave" infont "phvr8r", (s, s)); +label.top("Remain" infont "phvr8r", (10s, s)); +\end{smallcode} +\vss} + +\noindent +The well-known Colorbrewer website ("http://colorbrewer2.org") provides a useful set +of colour palettes that are suitable for a wide range of applications. They were +originally written for maps, but they are useful for many other types of drawing. +If you are using an up-to-date, and complete, \TeX\ distribution, you should find +that my implementation of them for \MP\ is already installed on your system, +otherwise you can get it from "https://ctan.org/pkg/metapost-colorbrewer". The +package provides two files that define all the colour ranges; one for CMYK and +another for RGB; an example of usage is shown below on the right. + +$$\includegraphics[width=\textwidth]{brexit-map.pdf}$$ + + +%-------------------------------------------- +\newpage +\section{Random numbers} + +\MP\ provides us with two built-in functions to generate random numbers. +\vadjust{\moveright 384pt\vbox to 0pt{\kern-24pt +\begin{code} +vardef dice(expr pip_count, pip_color) = + save d,r,p, ul, ur, lr, ll; + r=1/8; path d; picture p; + d = for i=0 upto 3: + quartercircle scaled 3 shifted (15,15) rotated 90i -- + endfor cycle; + p = image(draw fullcircle scaled 6; + fill fullcircle scaled 6 withcolor pip_color); + pair ul, ur, ll, lr; + ul = 1/5[ulcorner d, lrcorner d]; + lr = 4/5[ulcorner d, lrcorner d]; + ur = 1/5[urcorner d, llcorner d]; + ll = 4/5[urcorner d, llcorner d]; + image(fill d withcolor background; draw d; + if odd(pip_count): + draw p shifted center d; + fi; + if pip_count > 1: + draw p shifted ul; draw p shifted lr; + fi; + if pip_count > 3: + draw p shifted ur; draw p shifted ll; + fi; + if pip_count = 6: + draw p shifted 1/2[ul,ur]; + draw p shifted 1/2[ll,lr]; + fi) +enddef; + +beginfig(1); +for i=0 upto 4: + draw dice(1+floor uniformdeviate 6, red) + rotated (2 normaldeviate) + shifted (36i,0); +endfor +endfig; +\end{code} + $$\includegraphics{dice}$$ +\vss}} +\begin{itemize} + \item `$\kw{uniformdeviate}\,n$' generates a random real number between $0$ and + $n$. + + Note that the $n$ is required. It can be negative, in which case you get negative random + numbers; or it can be zero, but then you just get $0$ every time. In other words the + implementation generates a number $r$ such that $0\le r<1$ and then + multiplies $r$ by + $n$. + + If you want a random whole number, use `$\kw{floor}$' on the result. + So to simulate six-sided dice, you can use `$1+\kw{floor}\,\kw{uniformdeviate}6$'. + + If you use the new number systems, and you find that the numbers generated + are all multiples of $n/4096$, so $\kw{uniformdeviate} 8192$ (for example) + generates even integers instead of random real numbers, then you should + update your \TeX\ distribution. This `feature' was an accident of the + original way that the scaled arithmetic routines were adapted. + + \item `\kw{normaldeviate}' generates a random real number that follows the + familiar normal distribution. The algorithm used is discussed in \textsl{The + Art of Computer Programming}, section~3.4.1. + If you generate enough samples, the mean should + be approximately zero, and the variance about 1. + The chance of getting a number between $-1$ and 1 is + about 68.3\%; between $-2$ and 2, about 95.4\%. + \vadjust{\moveright 3.6in\vbox to 0pt{\hsize 1.6in \vskip21pt \noindent + \small 10000 samples suggest\\\kw{normaldeviate} works.\par\vss}} + $$\includegraphics[width=4.6in]{gaussian}$$ + To relocate the mean, just add a constant. To rescale the distribution, + multiply by the desired standard deviation (the square root of the + desired variance). + + +\end{itemize} + + +\newpage\subsection{Random numbers from other distributions} + +The \kw{normaldeviate} function is provided as a primitive \MP\ operation. The +implementation is based on the `Ratio method' presented in \textsl{The Art of +Computer Programming}, section~3.4.1. It turns out to be very straightforward to +implement the algorithm for this method as a user-level program $\longrightarrow$ +\vadjust{\moveright5.5in\vbox to 0pt{\kern -64pt +\begin{smallcode} +vardef normaldeviate = + save u, v, xa; + forever: + forever: + u := uniformdeviate 1; + exitif (u>1/64); + endfor + v := sqrt(8/mexp(256)) * ( -1/2 + uniformdeviate 1 ); + xa := v/u; + exitif ( xa**2 <= -mlog(u)/64 ); + endfor + xa +enddef; + +vardef exponentialdeviate = + save u; + forever: + u := uniformdeviate 1; + exitif (u>0); + endfor + -mlog(u)/256 +enddef; + +vardef gammadeviate(expr a,b) = + save y, x, v, s, accept; boolean accept; + s = sqrt(2a-1); + forever: + forever: + y := tand(uniformdeviate 180); + exitif y<64; + endfor + x := s * y + a - 1; + accept := false; + if x>0: + v := uniformdeviate 1; + if (v <= (1+y**2)*mexp((a-1)*mlog(x/(a-1))-(256*s*y))): + accept := true; + fi + fi + exitif accept; + endfor + x/b + enddef; +\end{smallcode} +\vss}}% + +There are a couple points here. First, the inner loop around the assignment to $u$ is +designed to avoid very small values that would cause $v/u$ to be larger than 64, and +hence make "xa**2" overflow. This is a useful general technique, and justified in +terms of the algorithm since large values of $v/u$ are rejected anyway. Secondly, +the expression "sqrt(8/mexp(256))" is a constant ($ \sqrt{8/e} \simeq 1.71553 $) and could be +replaced by it's value, but this does not make an appreciable improvement to the +speed of the routine. On a modern machine, this routine is only very slightly +slower than using the primitive function. + +It is also fairly straightforward to implement random number generators that follow other statistical +distributions. The mathematical details are in the section of \textsl{TOACP} +referenced above. Two examples, for the exponential distribution and the gamma +distribution, are shown on the right. In both cases, note the care required to avoid +arithmetic overflow (and see section~\ref{trig} for the "tand" function). + +\medskip\label{mexp}\noindent +You can also see the special nature of \MP's \kw{mexp} and \kw{mlog} +functions. They are defined so that $\kw{mexp} x = \exp(x/256)$ and $\kw{mlog} x =256\log(x)$. +This is another artefact of the scaled number system. \MP\ computes $x^y$ using the +formula "mexp(y*mlog(x))", and the adjusted log values give more accurate results. +Note that this means that you have $e=\kw{mexp}(256)$. + +\medskip\noindent +At the start of each job, \MP\ automatically sets a new seed for the random number +generator, so that the sequence of numbers is different each time. But you can set +this yourself if you need the same sequence each time. At the start of your program +you should put "randomseed:=3.14;" (or whatever value you prefer). According to +\textsl{The Metafont Book}, the default value is +$\id{day}+\id{time}\ast\id{epsilon}$, but in \MP\ the exact value used depends on +the resolution of the timers available on your system; essentially the value +should be different every time you run \MP. + + +\newpage\subsection{Random walks} + +You can use the random number generation routines to produce visualizations of +random walks, with various levels of analysis. +\vadjust{\moveright5.5in\vbox to 0pt{\hsize 4.4in\kern -2\baselineskip +\begin{code} +beginfig(1); +numeric w, h, n; w = 377; h = 80; n = 500; +draw (origin--right) scaled w; +draw (origin--right) scaled w shifted (0,+h/2) withcolor 3/4; +draw (origin--right) scaled w shifted (0,-h/2) withcolor 3/4; +pair zenith, nadir; zenith = nadir = origin; +path walk[]; +for i=1 upto 8: + numeric y; y = 0; + walk[i] = origin for x=w/n step w/n until w: + hide( + y := y if uniformdeviate 1 < 1/2: + else: - fi 1; + if y > ypart zenith: zenith := (x,y) ; fi + if y < ypart nadir: nadir := (x,y) ; fi + ) + -- (x,y) + endfor; + undraw walk[i] withpen pencircle scaled 3/4; + draw walk[i] withcolor (1/2+y/h)[red, blue] +endfor +drawarrow (12 up -- 2 up) shifted zenith withcolor blue; +drawarrow (12 down -- 2 down) shifted nadir withcolor red; +endfig; +\end{code} + +\smallskip\noindent +\hey Note the \mpl{undraw} line using a slightly thicker pen; this makes it +easier to follow the lines as they cross each other. +\vss}} +$$\includegraphics[width=\textwidth]{random-walks-red-blue}$$ +In this example the random walk lines are coloured according to the final $y$-value, +and the global maximum and minimum points are marked. + +Each walk is created with an `inline' for-loop; the loop is effectively expanded +before the assignment, so that each \id{walk} variable becomes a chain of connected $(x,y)$ +pairs. Inside the loop you can conceal yet more instructions in a `\kw{hide}' block. +These instructions contribute nothing to the assignment, but can change the values +of variables outside the block. + +Note the first line of the \kw{hide} block adds $\pm1$ to $y$ with equal probability. +You can (of course) create different kinds of random walks, by changing the way you +set this delta value, for example by using a different type of random variate, or scaling +the value, or changing the odds in favour of one direction or the other. For +example: +\begin{code} +y := y if uniformdeviate 1 < p: + 2 else: - 1 fi; +\end{code} +will set the delta to $+2$ with probability $p$ and and to $-1$ with probability $1-p$. + +\newpage\subsection{Brownian motion} + +A random walk is normally constrained to move one unit at a time, but if you relax +that constraint and use `\kw{normaldeviate}' in place of `\kw{uniformdeviate}' you +can get rather more interesting patterns.\vadjust{\moveright5.5in\vbox to 0pt{ +\hsize4in\kern -20pt +\begin{code} +beginfig(2); +for n=1 upto 4: + x:=y:=0; + draw (x,y) for i=1 upto 2000: + hide(x:=x+4normaldeviate; y:=y+4normaldeviate;) + .. (x,y) + endfor withcolor ((n+2)/9)[blue,white]; + fill fullcircle scaled 3 shifted (x,y) withcolor red; +endfor +% mark the origin +fill fullcircle scaled 3 withcolor green; +endfig; +\end{code} + +\kern 120pt +\noindent +Using these random number generators means that the output is +different each time because \MP\ produces a different sequence of numbers. You may +find yourself running the program a few times until you find one you like. At this +point you will wish that you knew what "randomseed" had been used, so that you can +re-create picture. Unfortunately \MP\ does not log the value used, unless you set it +manually. So here's a trick to use in this situation: set your own random seed +using a random number at the top of your program. +\begin{code} + randomseed := uniformdeviate infinity; +\end{code} +Now \MP\ writes the (random) value used in the log for you to copy. Note that if you are using +"luamplib" you need to add the "\mplibshowlog{enable}" option. +\vss}} +If you also allow the $x$-coordinates to +wander at random as well as the $y$-coordinates you get two-dimensional random +patterns. And if you replace the straight line segments "--" with ".." so that \MP\ +draws a smooth curve through the points, as well as vary the colour each time +you draw a new curve, then the result is almost artistic. + +\medskip\noindent +\hbox to \textwidth{\includegraphics[width=1.2\textwidth]{brown3a}\hss} + +\newpage\subsection{Drawing freehand} + +This idea is shamelessly stolen from the wonderful collection of \MP\ examples +available at "http://melusine.eu.org/syracuse/metapost/". But since the +examples there are all in French (including all the names of the custom macros), +perhaps it would be better to say `translated' rather than `stolen'; moreover my +implementations are easier to use with plain \MP.\vadjust{\moveright5.5in\vbox to 0pt{ +\hsize4in\kern -5.5\baselineskip +\begin{smallcode} +def freehand_segment(expr p) = + point 0 of p {direction 0 of p rotated (4+normaldeviate)} .. + point 1 of p {direction 1 of p rotated (4+normaldeviate)} +enddef; +def freehand_path(expr p) = + freehand_segment(subpath(0,1) of p) + for i=1 upto length(p)-1: + & freehand_segment(subpath(i,i+1) of p) + endfor + if cycle p: & cycle fi +enddef; +defaultfont := "eurm10"; +color sepia; sepia = (0.44, 0.26, 0.08); +picture marker; marker = image(for s=-1/2, 1/2: + draw (left--right) scaled 2 rotated 60 shifted (s,0); +endfor); +def moved_along expr x of p = rotated angle direction x of p + shifted point x of p enddef; + +beginfig(1); +pair A, B, C, D; +A = (0,-30); B = (180,0); C = (120,90); +D = (1/2 + 1/40 normaldeviate)[A, B]; +path triangle, circumcircle, bisector; +triangle = freehand_path(A--D--B--C--cycle); +bisector = freehand_segment(C--D); +circumcircle = freehand_path(A..B..C..cycle); +draw triangle; +draw bisector; +draw circumcircle withcolor .67 red; +draw marker moved_along 1/2 of triangle withcolor .67 red; +draw marker moved_along 3/2 of triangle withcolor .67 red; +label.lft("A", A); +label.rt ("B", B); +label.top("C", C); +label.bot("D", D); +endfig; +\end{smallcode} +\vss}} + +\subsubsection{Making curves and straight lines look hand drawn} + +$$\includegraphics{random-freehand-circumcircle}$$ +A small amount of random wiggle makes the drawing come out charmingly wonky. Notice +that the "freehand_path" macro will transform a path whether it is straight or curved, +and open or closed. Notice also that to find $D$ the mid-point of a $AB$, you need +to find the +point along the freehand path; if you simply put "1/2[A,B]" there's no guarantee +that the point would actually be on the free hand path between $A$ and $B$. In this +case a little extra randomness has been added, and the +two segments $AD$ and $DB$ have been marked with traditional +markers to show that they are equal. The "moved_along" macro combines shifted and +rotating to make the markers fit the wonky lines properly. +The Euler font complements the hand-drawn look; but +you might find that a little of this type of decoration goes a long way. + +\newpage +\subsubsection{Extending straight lines slightly}\label{euler} + +This second freehand figure uses a macro to draw a wonky line through two points +with a bit of overlap at each end. The overlap size is given using the suffix +syntax. The lines are drawn in sepia ink to enhance the hand-drawn look. The angle +labels are positioned on invisible arcs between neighbouring wonky lines. +\vadjust{\moveright5.5in\vbox to 0pt{ +\hsize4in\kern -4.5\baselineskip +\begin{code}[xleftmargin=0pt] +vardef freehand_through@#(expr a, b) = + save t; pair t; + t = @# * unitvector(b - a) rotated (4 + normaldeviate); + a - t .. a {t} .. b {t} .. b + t +enddef; + +vardef mid_arc@#(expr p, a, b) = + save c; path c; + c = fullcircle scaled (2*@#) shifted p cutbefore a cutafter b; + point arctime 1/2 arclength c of c of c +enddef; + +beginfig(1); +defaultfont := "eurm10"; color sepia; sepia = (0.44, 0.26, 0.08); +pair A, B, C; A = (0,-30); B = (180,0); C = (120,90); + +path a, b, c; +a = freehand_through 7 (A, B); +b = freehand_through 8 (B, C); +c = freehand_through 6 (C, A); + +drawoptions(withcolor sepia); +draw a; draw b; draw c; + +drawoptions(withcolor .67 blue); +label.bot ("a", point 3/2 of a); +label.rt ("b", point 3/2 of b); +label.ulft("c", point 3/2 of c); + +drawoptions(withcolor .5 red); +label(char 11, mid_arc 16 (A, a, c)); +label(char 12, mid_arc 14 (B, b, a)); +label(char 13, mid_arc 14 (C, c, b)); + +drawoptions(); +endfig; +\end{code} +\vss}} + +$$\includegraphics{random-freehand-through}$$ + +\vfill +\noindent\llap{\nb\ }The AMS Euler font available to \MP\ as "eurm10" is encoded as a subset of the \TeX\ +math italic layout --- essentially it has all the Greek letters but none of the +arrows, nor the musical notation. +$$\includegraphics{euler-sampler}$$ +If you can't get the upper case $\Gamma$ at \mpl{char 0}, then you might be running +an old out-of-date version of "luamplib". + +\newpage\subsection{Increasingly random shapes of the same size} + +If you want a random-looking shape, the general approach is to find a method to make +a path that allows you to inject some random noise at each point of the path. +$$\hbox to \textwidth{\includegraphics{random-shapes}\hss}$$ +For these shapes the objective was to make them increasingly random, but to keep +them all the same length.\vadjust{\moveright5.5in\vbox to 0pt{ +\hsize4in +\begin{code} +beginfig(1); +numeric desired_length, n, s; +desired_length = 180; n = 30; s = 80; + +for r=0 upto 8: + + path shape; + shape = for i=1 upto n: + (s + r * normaldeviate, 0) rotated (360/n*i) .. + endfor cycle; + + shape := shape scaled (desired_length/arclength shape); + + draw shape shifted (r*s, 0) withcolor (r/8)[black,red]; + label(decimal r, (r*s, 0)); + +endfor +endfig; +\end{code} +\vss}} +Each time round the outer loop the \id{shape} is redeclared to clear it, and +then redefined by an inline-loop with $n$ steps like +this: +\begin{code} +shape = for i=1 upto n: (s,0) rotated (360/n*i) .. endfor cycle; +\end{code} +except that some random noise is added to the $s$ at each step: when the noise is +zero ($\id{r}=0$) you get a circle; as the noise increases the circle is +increasingly distorted. + +The scaling is done using the \mpl{arclength} operator. This works like +\mpl{length} but instead of telling you the number of points in a path, it returns +the actual length as a dimension. Dividing the desired length by this dimension +gives the required scaling factor for the random shape just defined. Notice that +you have to do this in two steps, and update the shape using ":=". This is because +you need to have defined \id{shape} before you can refer to it. + + +\newpage\subsection{Explosions and splashes} + +Random numbers are also useful to make eye catching banners for posters, +presentations, and infographics. Here are two simple example shapes: $\to$ +\mpic{18pt}{explode} + +\begin{smallcode} +string heavy_font; +heavy_font = "PlayfairDisplay-Black-osf-t1--base"; + +randomseed:=2128.5073; + +beginfig(1); +n = 40; r = 10; s = 50; +path explosion, splash; +explosion = for i=1 upto n: + (s if odd(i): - else: + fi r + uniformdeviate r,0) rotated (i*360/n) -- +endfor cycle; + +splash = for i=1 upto n: + (s if odd(i): - else: + fi r + uniformdeviate r,0) rotated (i*360/n) .. +endfor cycle; +splash := splash shifted (3s,0); + +fill explosion withcolor 1/2 green + red; +draw explosion withpen pencircle scaled 2 withcolor 2/3 red; +label("BOOM!" infont heavy_font scaled 2, center explosion) + withcolor red; +fill splash withcolor 1/2 green + blue; +draw splash withpen pencircle scaled 2 withcolor 2/3 blue; +label("SPLAT!" infont heavy_font scaled 2, center splash) + withcolor blue; +endfig; +\end{smallcode} + +\noindent +In this figure "n" is the number of points in the shape, "r" is the amount of +randomness, and :"s" is the radius used. +In order to get a clear zig-zag outline, the loop alternately adds or subtracts "r"; +and then adds a random amount on top to make it look random. +Notice that the only difference between the "explosion" and "splash" is that how +the connecting lines are constrained to be straight or allowed to make smooth +curves.\vadjust{\moveright5.5in\vtop to -2pt{\hsize 4in\vss\noindent +The display font used here is one of the gems hidden away in "psfonts.map". +If you run \MP\ with the "-recorder" option, it will create a list of all the files +used, with the current job name and an extension of ".fls". This file will include +a line which tells you exactly which version of "psfonts.map" is being used. + +The DVIPS documentation explains the format of the file, but for \MP's purposes the +first word of each non-comment line defines a font name you can try. However beware +that just because a name is defined in your map file, does not necessarily mean that +you actually have the required PostScript font files installed as well. But if you have a +full TexLive installation you will find that very many of them are already +installed.\strut +}} + + +\newpage\subsection{Simulating jagged edges or rough surfaces} + +You can use the idea of adding a little bit of noise to simulate a rough surface. +$$\includegraphics[width=0.95\textwidth]{qed}$$ +These diagrams are supposed to represent light rays reflecting from a surface: on +the left the surface is smooth ($r=0$) and on the right it's rough ($r=0.42$). +The parameter $r$ is used in the \MP\ program as a scaling factor for the random +noise added to each point along the rough surface; the only difference in the code +to produce the two figures was the value of $r$.% +\vadjust{\moveright5.5in\vbox to + 0pt{\hsize 4in\vss +\begin{smallcode} +def perpendicular expr t of p = + direction t of p rotated 90 shifted point t of p + enddef; + +beginfig(1); +u = 5mm; r = 0.42; n = 32; s = 8u; theta = -45; + +path base; +base = origin + for i=1 upto n-1: -- (i/n*s,r*normaldeviate) endfor + -- (s,0) -- (s,-u) -- (0,-u) -- cycle; +fill base withcolor .8[blue,white]; +draw base withcolor .67 blue; + +path ray[]; +for i=2 upto 6: + ray[i] = (left--right) scaled 2/3 s rotated theta shifted (i*u,0); + b := ypart (ray[i] intersectiontimes base); + ray[i] := point 0 of ray[i] + -- point b of base + -- point 0 of ray[i] + reflectedabout(point b of base, perpendicular b of base); + drawarrow ray[i]; +endfor + +label("r=" & decimal r, center base); +endfig; +\end{smallcode} +\vss}} +First the base block is created with some noise on the upper side. Then five rays +are created. Applying \mpl{ypart} to the pair of times returned by +\mpl{intersectiontimes} gives us the point of the base where the incident ray hits +it. This point and the perpendicular at that point are then used to get the angle +for the reflected ray. The diagrams are effective because the rays are reflected at +realistic looking angles. + +The simple approach to adding noise along a path works well in most cases provided +there's not too much noise, but it is always possible that you'll get two consecutive +values at opposite extremes that will show up as an obtrusive jag in your line. To +fix this you can simply run your program again to use a different random seed value; +or you could try using ".." instead of "--" to connect each point, but beware that +sometimes this can create unexpected loops. + +\newpage\subsubsection{Walking along a torn edge} + +It's also possible to use a random +walk approach so that each random step takes account of the previous one to avoid +any big jumps. Here's one way to do that. +$$\includegraphics{torn-edge-straight}$$ +\begin{code} +path t; numeric x, y; +x = 0; y=0; +t = (x, -20) -- (x, y) for i=1 upto 288: + -- (incr x, walkr y) + endfor -- (x, -20) -- cycle; +draw t withcolor .67 blue; +\end{code} +The "walkr" routine works like the "incr" and "decr" commands; it updates the value of the +argument. The idea is that the further away from zero you are, the more likely is +that the next value will take you back towards zero. +\begin{code} +vardef walkr suffix $ = + $ := $ if uniformdeviate 1 < (2**-abs($)): + else: - fi + signr $; $ + enddef; +vardef signr suffix $ = + if $<0: - else: + fi uniformdeviate 1 + enddef; +\end{code} +You can use this to produce more realistic torn edges. You can also apply this as a +form of jitter to a curved path, by adding a suitably rotated vector to enough +points along the path. + +\moveright5.5in\vbox to 0pt{\kern-4in +\begin{code} +path c; c = fullcircle scaled 200; +draw c withcolor .8 white; + +y=0; n = 600; +path t; t = for i=0 upto n-1: + point i/n*length(c) of c + + (0, walkr y) rotated angle direction i/n*length(c) of c + -- +endfor cycle; +draw t withcolor .67 red; +\end{code} +$$\includegraphics{torn-edge-circle}$$ +\vss} + + +\newpage +\section{Plane geometry} + +\noindent\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\kern-9pt\noindent +Here is the equilateral triangle point macro in action. +$$\includegraphics{geometry-triangles-on-circle}$$ +\vskip -12pt +\begin{code} +beginfig(1); +path c; c = fullcircle scaled 144; +pair a,b,p,q; +for i=0 upto 7: + a := point i of c; + b := point i+1 of c; + p := equilateral_triangle_point(a,b); + q := equilateral_triangle_point(b,a); + draw a -- p -- b withcolor .67 green; + draw a -- q -- b withcolor .67 red; +endfor +draw c withcolor .53 blue; +endfig; +\end{code} +\vss}}% +This section deals with drawing geometrical figures that involve lines, +angles, polygons, and circles. Plain \MP\ provides very few tools that are +explicitly designed to help draw geometric figures, but it is usually possible to +find an elegant construction using these tools and the relevant primitive commands. +It is tempting to build up your own library of special purpose macros, but +experience suggests that it is often better to adapt a general technique to the task +in hand, and to create a specific solution to your current problem. One of the main +issues is catching exceptions; since it is hard to write completely general macros, +I have tried simply to present each technique so that you can understand it and adapt +it as required. + +The classical constructions from Euclid's \textsl{Elements} are often useful +sources of inspiration for macros, but they do not always point in the right +direction. For example consider the first proposition: \textit{given two points +find a third point, so that the three points make an equilateral triangle}. +Euclid's construction is to draw an arc, with radius equal to the length of the +segment between the two points, at each point and find the intersection. This might +lead us to a function like this: +\begin{code} +vardef equilateral_triangle_point(expr a, b) = + save c; path c; c = fullcircle scaled 2 abs(b-a); + (c shifted a intersectionpoint c shifted b) +enddef; +\end{code} +This works but has a couple of issues. First using \mpl{intersectionpoint} feels a +bit like cheating; secondly, and more seriously, the point returned depends on the +orientation of the points "a" and "b". In some configurations the first +intersection found will be on the left, in others on the right. We could fix this +by rotating the circle "c" by "angle (b-a)", but we can do better with a simple +rotation of the second point about the first: +\begin{code} +vardef equilateral_triangle_point(expr a, b) = + b rotatedabout(a,60) +enddef; +\end{code} +And if you want to get right back to primitives you could even write that as: +\begin{code} +vardef equilateral_triangle_point(expr a, b) = + b shifted -a rotated 60 shifted a +enddef; +\end{code} + + +\newpage +\subsection{Bisecting lines and paths} + +\moveright5.5in\vbox to 0pt{\hsize4in\noindent +$$\includegraphics{mediation-pitfall}$$ +\vskip 50pt +$$\includegraphics[width=4in]{mediation-shapes}$$ +\bigskip +$$\includegraphics[width=4in]{mediation-sallows}$$ +\centerline{\textsl{Lee Sallows' theorem of median triangles}} +\vss} +\noindent +The best way to bisect a line depends on how you have defined it. +If you have two pairs $a$ and $b$, then the simplest way to find +the pair that bisects them is to write "1/2[a,b]". This mediation +mechanism is entirely general, so you can write \mpl{1/3[a,b]}, "1/4[a,b]", and +so on to define other pairs that are part of the way from $a$ to $b$. +The expression "0[a,b]" is equal to $a$, and "1[a,b]" is equal to $b$; +but the number before the left bracket does not have to +be confined to the range $(0,1)$. If you write "3/2[a,b]" you will get a pair on +the extension of the line from $a$ to $b$ beyond $b$. To get a pair going the +other way you can either reverse $a$ and $b$, or use a negative number; but don't +get caught out by the \MP\ precedence rules: "-1/2[a,b]" is interpreted as +"-(1/2[a,b])" and not as "(-1/2)[a,b]", so either put in the parentheses or swap +the order of the pairs: "(3/2)[b,a]". See $\longrightarrow$. + +If you want to work with a \kw{path} variable, rather than separate \kw{pair} +variables, you can +use the \mpl{point t of p} +notation to do mediation along the path. For a simple straight path $p$ of length 1 +then \mpl{point 1/2 of p} will give you the midpoint. More generally, +\mpl{point 1/2 length p of p} will give you the midpoint of a path of any length. +This works fine for simple paths, along which \MP's time moves evenly, +but for more complicated, curved paths you have to use this rather cumbersome notation: +\begin{code} + point arctime 1/2 arclength p of p of p +\end{code} +If your path is closed, and makes a triangle or a regular polygon, +then you can bisect it with the line +\begin{code} + point t of p -- point t + 1/2 length p of p +\end{code} +\textbf{NB}: if the polygon has an odd number of sides, then $2t$ must be a whole +number. + +\smallskip\noindent +In a triangle these bisecting lines are called medians. The three medians intersect +at the centroid of the triangle. The centroid is a good place to put a label on a +triangle. You could find it \mpl{intersectionpoint} or with +a construction using \mpl{whatever} on any two medians, but since we know that the centroid +divides each median in the ratio $2:1$ we can find the centroid of a triangle path +$p$ most simply with: +\begin{code} + z0 = 2/3[point 0 of p, point 3/2 of p]; +\end{code} +The median is the basis for several beautiful theorems about the geometry of the +triangle. The theorem shown here was first published in 2014. + +\newpage +\subsection{Bisecting angles}\label{sec:bisect} +\moveright5.5in\vbox to 0pt{\hsize4in\noindent +$$\includegraphics{bisection-euclidean}$$ +\smallskip +$$\includegraphics{bisection-vector}$$ +\vss} +\noindent +In an equilateral triangle the medians also bisect the angles at each vertex; this +is the basis of Euclid's method of bisecting an angle set out in the Second +Proposition. You can do the same in \MP, but it might not always be the best way. +Whatever approach you take, an angle is defined by three points; one that defines +the corner and two that define the lines extending from that corner. In this +exploration I've used $a$, $b$, and $c$ to represent the points, with $b$ being the +one in the middle, and at the corner. + +Euclid's method is to draw an arc centred at the corner, and then construct an +equilateral triangle on the two points where the arc crosses the lines. This is +shown on the right, with a macro that re-uses the equilateral triangle point macro +given above. But if your aim were to find any point on the line bisecting $\angle +ABC$, then you could simplify this and make it more efficient by using +$e = \frac12[p,q]$ +instead of calling the triangle macro at all. However the macro is still making two +calls to \mpl{intersectionpoint}. If you wanted to eliminate this you could use the +useful plain \MP\ macro "unitvector" to produce a solution based on adding two equal +length vectors from the corner to the two other points. Another approach is to +exploit another geometric theorem that states that the bisector of an angle in a +triangle divides the opposite side in the ratio of the two other sides. +So if sides $AB$ and $BC$ have lengths $p$ and $q$ then the bisector will +be ${p\over p+q}={1\over1+q/p}$ from $A$ to $C$, and you can express this +simply using \MP's mediation syntax: +\vbox to 0pt{ +$$\includegraphics{bisection-interior}$$ +\vss} + +\newpage +\subsection{Trisections and general sections of angles} + +There is no classical method to trisect an arbitrary angle, so you need to resort +to measuring and arithmetic in \MP. If the angle is a given this is trivial: +\mpic{-12pt}{trisection-simple.pdf} +\begin{smallcode} +path ray; +numeric theta; +ray = origin -- 200 right; +theta = 42; +draw ray; +draw ray rotated 1/3 theta withcolor 2/3 red; +draw ray rotated 2/3 theta withcolor 2/3 red; +draw ray rotated theta; +dotlabel.llft("$0$", origin); +label("$\theta/3$", 72 right rotated 1/6 theta); +label("$\theta/3$", 72 right rotated 3/6 theta); +label("$\theta/3$", 72 right rotated 5/6 theta); +\end{smallcode} +But if you have only the coordinates of some points then you need to use the +\mpl{angle} primitive to measure the angle first; \mpl{angle} takes a \kw{pair} +argument and returns a numeric representing the angle in degrees measured clockwise +from the $x$-axis to a line through the origin and the point represented by the pair. +This definition means that if you have three points $A$, $B$, and $C$, then you can +measure $\angle ABC$ with \mpl{angle(C-B)-angle(A-B)}. Following the usual +convention this gives you the angle at $B$; if you list the points in clockwise +order you will get a positive result. If you don't care about the order, you +can make this into a more robust macro: +\begin{smallcode} +vardef measured_angle(expr P, Q, R) = + (angle (P-Q) - angle (R-Q)) * turningnumber (P--Q--R--cycle) mod 360 +enddef; +\end{smallcode} +The primitive \mpl{turningnumber} is explained on p.\thinspace 111 of \textsl{The +METAFONTbook}. It takes a closed path and returns number of times that you would +turn through 360${}^\circ$ if you traversed the path. We use this here to +negate the measured angle if necessary, so that you always get the interior angle. +The \mpl{mod 360} on the end ensures that the result is in the range $0 \le \theta < +360$. Armed with a measured angle, all you then need is arithmetic. +\mpic{-160pt}{trisection-classical.pdf} +It might be possible to use the \id{solve} macro to simulate the Neusis construction +(that allows you to measure a length) illustrated on the right, but measuring the +angles is rather easier. + +\newpage +\subsection{Intersections}\label{sec:intersect} + +\moveright5.5in\vbox to 0pt{\hsize4in\noindent +\centerline{A puzzle square featuring some intersections} +$$\includegraphics[width=3in]{magic-square-14}$$ +The points were defined like this (the order was important). +\begin{code} +z1 = (10,10); +z4 = 144 right rotated 12; +z5 = z4 shifted (2, 78); +z7 = z4 reflectedabout(origin, (1,1)); + +z2 = 1/2 [z5, z7]; +z9 = whatever [z1, z4]; +z2-z9 = whatever * (z7-z1); +z8 = whatever [z1, z5] = whatever [z2, z4]; +z3 = whatever [z2, z9] = whatever [z4, z7]; +z6 = whatever [z1, z7] = whatever [z3, z5]; +\end{code} +\vss} +\noindent +If you have line segments defined by their endpoints, then the +canonical way to find their intersection, is to use +the mediation syntax with \id{whatever} twice: +$$\includegraphics[width=\textwidth]{whatever.pdf}$$ +The mediation syntax works even if the intersection point does not actually lie on +either of the two line segments. The intersection will be the point where the two +(infinite) lines through the pairs of points meet. If the two lines are parallel, +you'll get an `inconsistent equation' error. If you want to capture the calculated +values, then use undefined numeric variables instead of \id{whatever}: +\begin{code} + z0 = alpha [z1, z2] = beta [z3, z4]; +\end{code} +In this example you would find $\alpha=0.286$ and $\beta=0.5$. +If you are trying to find where the line through your points intersects a horizontal or vertical, then you only need one +mediation and a simple equation for the relevant $x$ or $y$ coordinate: +\begin{code} + z0 = alpha [z1, z2]; x0 = 0; % for example +\end{code} +If you have defined your lines as paths, and especially if they are more complicated +than straight lines, you need to use the \mpl{intersectiontimes} primitive or the +\mpl{intersectionpoint} macro, as explained on pp.136–137 of \mfbook. + +\newpage +\subsubsection{The intersection algorithm} + +\MP\ inherits a fast algorithm for finding the intersection between two paths from +\MF. It is explained rather more gnomically than usual at the end of Chapter 14 of +\mfbook, with more detail given in the web source for \MF. The core algorithm works +on paths of length 1. If you have longer paths, \MP\ works its way along the paths +applying the core algorithm to successive pairs of unit subpaths. It does this is +lexicographic order; this means that, if you have two circles $A$ and $B$, and you do this: +\begin{code} + (t, u) = A intersectiontimes B; +\end{code} +then \MP\ will first look for an intersection between subpath $(0,1)$ of $A$ and +subpath $(0,1)$ of $B$, then subpath $(0,1)$ of $A$ and subpath $(1,2)$ of $B$, and +so on, with $B$ varying faster, until you get to subpath $(7,8)$ of $A$ and subpath +$(7,8)$ of $B$. But you may never get that far, as the process stops as soon as the +first intersection is found. The upshot of this is that the intersection point +found will always be as early as possible on $A$. Note that after the call above +point $t$ of $A$ will be very close to point $u$ of $B$, as they both refer to the +same intersection point. If you want the alternative point that is earlier on $B$, +then use `$B \mathbin{\textrm{intersectiontimes}} A$' +instead.\mpic{-222pt}{intersection-AB-or-BA} + +\vfill\noindent +When we get down to paths of length 1, the algorithm works something like this: + +$$ +\includegraphics{intersection-algorithm} +$$ +\vadjust{\moveright 384pt\vbox to 0pt{\hsize 4.2in\vss \noindent +The two paths are represented as rectangles that enclose the end points and the +control points for each path. If these rectangles don't overlap then there is +certainly no intersection. Otherwise \MP\ bisects each path and considers four +smaller rectangles, in the order $(1,2)$, $(1,4)$, $(3,2)$, $(3,4)$ (as shown). In +this case it will pick $(1,2)$, discard 4, and push 3 onto a stack. It carries on +doing this, back tracking as required, until it finds sufficiently small overlapping +rectangles. The two times returned by \mpl{intersectiontimes} are the midpoints of +the subpaths enclosed by these two tiny rectangles, which is why they do not always +refer to exactly the same point.}} + +\subsubsection{Finding all intersection points} + +As noted above, the \mpl{intersectiontimes} algorithm will stop at the first +intersection of the two paths, but it is possible that the two paths will intersect +again further along. If you want to find all the intersection points then the +simplest technique is just to unwrap the algorithm slightly, and loop through all +the unit subpaths applying \mpl{intersectiontimes} to each pair. Using an array to +hold the points and a counter, you can get them with something like this: +\begin{smallcode} +pair P[], times; numeric n; n = 0; +for i = 1 upto length(A): + for j = 1 upto length(B): + times := subpath (i-1,i) of A intersectiontimes subpath (j-1,j) of B; + if xpart times > -1: + P[incr n] = 1/2[point xpart times of subpath (i-1,i) of A, + point ypart times of subpath (j-1,j) of B]; + fi + endfor +endfor +\end{smallcode} +and then use them like this: +\begin{smallcode} +for i=1 upto n: + draw fullcircle scaled 4 shifted P[i]; % or whatever +endfor +\end{smallcode} +There are a couple of \MP\ technical points to note. The \mpl{intersectiontimes} +operation returns a pair, which we assign to a pair variable $\id{times}$ +above; we have to use \mpl{:=} to re-assign it in each loop, and we have to +use an explicit pair variable because you can't assign to a literal pair; +\MP\ will give you an error if you try \mpl{(t, u) := A intersectiontimes B;}. +This may come as a surprise, because you \textit{can} legally do \mpl{(t, u) = A +intersectiontimes B}, but in a loop this causes an inconsistent equation error on +the second iteration. If you need to avoid the repeated use of \mpl{xpart} and +\mpl{ypart}, one alternative is to do this inside the loop: +\begin{smallcode} + ... + numeric t, u; + (t, u) = A intersectiontimes B; + ... +\end{smallcode} +Now the numerics are reset each time and the equation is not inconsistent. +\vadjust{\moveright 384pt\vbox to 0pt{\hsize 4.2in\vss \noindent +T\textsc{he} technique discussed on the left, works well on paths where the points on one or +both of the paths are close together, so that the unit subpaths are short; +But it +is possible to create quite long paths of unit \mpl{length} and these may intersect +each other more than once, like so: +$$\includegraphics{intersection-only-two}$$ +Here the two paths $A$ and $B$ are Bézier splines of with \mpl{length=1}, so the +normal \MP\ algorithm is only ever going to give you one of the intersections. In +the diagram above, the +red circle marks the point given by \mpl{A intersectiontimes B}. We can try +reversing the first path, and in this case you get the point marked in blue, +but what about the one in the middle? + +The most reliable approach is to take a copy of one of the paths, and snip it off at +the intersection and try again until there is nothing left to snip. +$$\includegraphics{intersection-all-three}$$ +The three points marked here were captured like this: +\begin{code} +pair P[]; numeric n; n=0; +path R; R := A; % take a copy of A +forever: + R := R cutbefore B; % snip where we cross B + exitif length cuttings = 0; % stop if nothing was cut + P[incr n] = point 0 of R; % capture the point + R := subpath (epsilon, infinity) of R; % nudge along +endfor +\end{code} +This technique also works on paths with \mpl{length} greater than one, +so you may prefer it as your general “get all the intersections” approach. +Note that the \mpl{cutbefore} macro is defined using \mpl{intersectiontimes}. + +\kern 6.5pt +}} + + +\subsection{Parallel and orthogonal or whatever}\label{sec:parallel} + +Given five known points --- $A$, $B$, $C$, $D$, and $E$ --- \MP\ can find the point $F$ +on the line $A \to B$, so that $E \to F$ is parallel to $C \to D$ like this: +\mpic{0pt}{parallel} +\begin{code} + F = whatever[A, B]; % F is on the line A..B + E-F = whatever * (C-D) % E..F || C..D +\end{code} +In the second line the expressions \mpl{E-F} and \mpl{C-D} return \<pair> +variables and the equation with \mpl{whatever} says that they must be scalar +multiples of each other. With the first equation, this is enough for \MP\ to work +out where $F$ should go. Note that \mpl{whatever} can take any real value, +positive or negative, so it does not matter whether you put \mpl{E-F} or \mpl{F-E}. +Note also that for the same reason, while $F$ will lie on the \textit{line} through +$A$ and $B$, it might not lie on the \textit{segment} from $A$ to $B$. +But also note that you \textit{cannot} write the second equation as +\begin{code} + E-F * whatever = (C-D); % <--- gives an error +\end{code} +because you can only apply \mpl{whatever} to known quantities. + +\bigskip\noindent\llap{\nb}% +To define a line perpendicular to $C\to D$ rather than parallel, then you can write: +\begin{code} + G = whatever[A, B]; + E-G = whatever * (C-D) rotated 90; +\end{code} +and obviously the "90" can be adjusted to whatever angle you please, if you want +something between parallel and orthogonal. + +\smallskip\noindent\llap{\nb}% +To define the line through $E$ that is perpendicular to $A\to B$, you should just +use $(A-B)$ instead of $(C-D)$. The diagram shows $H$, the point on $A\to B$ +that is closest to $E$; you can (I trust) work out how to define that yourself. + +\smallskip\noindent\llap{\nb}% +If you just need to compute the perpendicular distance from the point $E$ to a line $A\to +B$, rather than defining the point $H$, then you can use Knuth's ‘slick’ formula: +\begin{code} + abs ypart ((E-A) rotated -angle (B-A)) +\end{code} +This effectively rotates $E$ about $A$ by the angle of the line, so that the problem +is reduced to measuring the height of a point above the $x$-axis, which is what +\mpl{ypart} does, of course. + +\moveright5.5in\vbox to 0pt{\hsize 4in\vss\noindent +There are some limitations to what you can do with \MP's linear equations; for one +thing you can't generally say things like \mpl{length(C-A) = 72}. If you want to +find the two points on a line that are a given distance from an external point, it's +often simpler to find the intersection points of the line with a suitably scaled and +shifted circle, even if you don't actually then draw the circle. You can usually +find the other point by reversing the circle.\par} + +\newpage +\subsection{Drawing circles}\label{sec:circles} + +The canonical way to draw a circle in plain \MP\ is to use the pre-defined path +\mpl{fullcircle} with a suitable transformation. +\mpic{-12pt}{drawing-circles.pdf} +The path is defined (in "plain.mp") using two \MP\ primitive commands: +\begin{code} +path fullcircle; fullcircle = makepath pencircle; +\end{code} +A \kw{pencircle} is the basic nib that is used to draw lines that digitize neatly; +it represents a true circle of diameter 1, passing through the points $(\pm.5, 0)$ +and $(0, \pm.5)$. When processed with \kw{makepath} it turns into a closed +polygonal path with eight points that closely approximates a circle with diameter +1\unit{bp} centred on the point $(0, 0)$. To use it, you can scale it and shift it. +To draw a circle with radius 2\unit{cm} at the point $(34, 21)$ you would do: +\begin{code} +draw fullcircle scaled 4cm shifted (34, 21); +\end{code} +Remember to scale before you shift, and that \id{fullcircle} has unit +\textit{diameter}, not unit \textit{radius}. To draw a circle centred at point $A$ +that passes through point $B$ [\red{I}] try: +\begin{code} +draw fullcircle scaled 2 abs (B-A) shifted A; +\end{code} +There are of course an infinite number of circles that you can draw through two +points, but if the line between the two points is a diameter [\blue{II}], then you can do: +\begin{code} +draw fullcircle scaled abs (B-A) shifted 1/2[A,B]; +\end{code} +Finally three points define a unique circle [\green{III}]: +\begin{code} +vardef circle_through(expr A, B, C) = + save o; pair o; + o = whatever * (A-B) rotated 90 shifted 1/2 [A,B] + = whatever * (B-C) rotated 90 shifted 1/2 [B,C]; + fullcircle scaled 2 abs (A-o) shifted o +enddef; +\end{code} +Plain \MP\ also defines \id{halfcircle} and \id{quartercircle}, as the appropriate +subpaths of \id{fullcircle}, both starting at point 0 (3 o'clock). Curiously, this +differs from \MF\ where \id{quartercircle} is defined first, and the other two +composed from it. The reference point of these two paths is the center of the +complete circle of which they would be part; so if you did “\kw{draw} +\id{quartercircle} \kw{shifted} $(34, 21)$;”, you would get an quarter-circle arc from +$(34.5,21)$ to $(34,21.5)$. + +\newpage +\subsection{Incircle and excircle of a triangle} + +The incircle of a triangle is the largest circle contained in the triangle. +The centre of the incircle lies at the intersection of the internal angle bisectors. +So we can use ideas from §\ref{sec:bisect} and §\ref{sec:intersect} to define a +macro that returns the required path given points $A$, $B$, and $C$: +\mxpic{-1in}{5in}{incircle} +\begin{code} +vardef incircle(expr A,B,C) = + save a, b, m, t; pair a, b, m, t; + a = A + unitvector (C-A) + unitvector (B-A); + b = B + unitvector (A-B) + unitvector (C-B); + m = whatever[A,a] = whatever [B,b]; t = whatever[A,B]; + t-m = whatever * (B-A) rotated 90; + fullcircle scaled 2 abs (t-m) shifted m +enddef; +\end{code} + +\bigskip\noindent +The excircles of a triangle are the three circles lying outside the triangle and +tangent to one edge and the extensions of the other two. The centres of each +excircle lie at the intersection of one internal angle bisector and the external +angle bisector of one of the other corners. + +To get the external angle bisector, +all you have to do is reverse the direction of one of the \mpl{unitvector} calls: +\mxpic{0pt}{5in}{excircle} +\begin{code} +vardef excircle(expr A,B,C) = + save a, b, m, t; pair a, b, m, t; + a = A + unitvector (C-A) - unitvector (B-A); + b = B + unitvector (A-B) + unitvector (C-B); + m = whatever[A,a] = whatever [B,b]; t = whatever[A,B]; + t-m = whatever * (B-A) rotated 90; + fullcircle scaled 2 abs (t-m) shifted m +enddef; +\end{code} + +\smallskip\noindent +To get the other excircles, call the macro with the points in a different order. + +\newpage +\subsection{Circumcircle of a triangle} + +The circumcircle of a triangle is the circle through the three corners, so if you +already have the corners of your triangle as separate \<pair> variables, you can +use the \mpl{circle_through} macro from §\ref{sec:circles}. Or you can adapt the +macro to take a single triangular path: +\begin{smallcode}[xleftmargin=0pt, xrightmargin=-60pt] +vardef circumcircle(expr T) = + save m; pair m; + m = whatever * (point 0 of T - point 1 of T) rotated 90 shifted point 1/2 of T + = whatever * (point 1 of T - point 2 of T) rotated 90 shifted point 3/2 of T; + fullcircle scaled 2 abs (point 0 of T - m) shifted m +enddef; +\end{smallcode} +Note that as the diagram on the right shows,\mxpic{-2in}{4.6in}{circumcircle} +the centre of the circumcircle is the intersection of all three of the perpendicular bisectors of +sides, but for the purposes of drawing in \MP\ you only need to find the +intersection of two of them. You could write +\begin{smallcode}[xleftmargin=0pt, xrightmargin=-60pt] + m = whatever * (point 0 of T - point 1 of T) rotated 90 shifted point 1/2 of T + = whatever * (point 1 of T - point 2 of T) rotated 90 shifted point 3/2 of T + = whatever * (point 2 of T - point 3 of T) rotated 90 shifted point 5/2 of T; +\end{smallcode} +but this does not add any more information to the equation for $m$, and +\MP\ will sometimes give you an “inconsistent equation” error if your triangle +is long and thin. + +\vfill +\noindent +\hey The marks that show line segments are equal were created by this macro. +\begin{code} +vardef mark_equal(expr a, b, n) = + save m, s; numeric s; 2s = n - 1; + picture m; m = image(for t=-s upto s: + draw (down--up) scaled 2 rotated -13 shifted (t,0) + withpen pencircle scaled 1/4; + endfor); + draw m rotated angle (b-a) shifted 1/4[a,b]; + draw m rotated angle (b-a) shifted 3/4[a,b]; +enddef; +\end{code} +Given the triangular \<path> $T$, the macro was used like this: +$\longrightarrow$\vadjust{\moveright5.5in\vbox to 0pt{\vss +\begin{code} +mark_equal(point 0 of T, point 1 of T, 1); +mark_equal(point 1 of T, point 2 of T, 2); +mark_equal(point 2 of T, point 0 of T, 3); +\end{code}}} + +\newpage +\subsection{The nine-point circle of a triangle} + +The orthocentre of a triangle is the point is the intersection of the three +altitudes, shown as point $D$ below. +The point $N$, half-way from $D$ to the +circumcentre $M$ is the centre of the remarkable nine-point circle which passes +through the bases of the three altitudes and bisects the six line segments $AB$, $AC$, $AD$, +$BC$, $BD$, and $CD$.\vadjust{\moveright5.2in\vbox to 0pt{\hsize 4in\vskip-80pt\noindent +\begin{smallcode} +pair A, B, C, D, N, M, p, q, r; +A = origin; B = 377 dir 10; C = 233 dir 70; +% pedal points (not labelled) +p = whatever[B, C]; A - p = whatever * (B-C) rotated 90; +q = whatever[C, A]; B - q = whatever * (C-A) rotated 90; +r = whatever[A, B]; C - r = whatever * (A-B) rotated 90; + +D = whatever[A, p] = whatever[B, q]; +N = 1/4(A + B + C + D); % remarkably... +M = D rotatedabout(N, 180); % M is also the circumcentre + +path circumcircle, nine_point_circle; +nine_point_circle = fullcircle scaled 2 abs(N - 1/2[A, B]) shifted N; +circumcircle = fullcircle scaled 2 abs(M - A) shifted M; + +draw nine_point_circle withcolor 3/4 red; +draw circumcircle withcolor 1/2[3/4 blue, white]; + +drawoptions(dashed evenly scaled 1/4 withcolor 1/2); +draw 1/2[A,B] -- M -- 1/2[B, C]; +draw 1/2[C,A] -- M -- D; +draw A -- p; +draw B -- q; +draw C -- r; + +% show the nine points with small circle markers +drawoptions(withpen pencircle scaled 1/4); +draw fullcircle scaled 2 shifted 1/2[A, B]; +draw fullcircle scaled 2 shifted 1/2[A, C]; +draw fullcircle scaled 2 shifted 1/2[A, D]; +draw fullcircle scaled 2 shifted 1/2[B, C]; +draw fullcircle scaled 2 shifted 1/2[B, D]; +draw fullcircle scaled 2 shifted 1/2[C, D]; +draw fullcircle scaled 2 shifted p; +draw fullcircle scaled 2 shifted q; +draw fullcircle scaled 2 shifted r; + +drawoptions(); +draw A--B--C--cycle; +dotlabel.llft("$A$", A); +dotlabel.rt("$B$", B); +dotlabel.ulft("$C$", C); +dotlabel.urt("\ $D$", D); +dotlabel.llft("$M$", M); +dotlabel.llft("$N$", N); +\end{smallcode}\vss}} +$$\includegraphics[width=\textwidth]{nine-point-circle}$$ + +\newpage +\subsection{Lines tangent to a point on a path} + +\MP\ represents paths internally as a sequence of nodes. Each node consists of three +pairs: the pre-control point, the point itself, and the post-control point. +\mpic{0pt}{tangents-on-path}% +For a given path $p$ you can extract these points at time $t$ with these operators: +\begin{code} +precontrol t of p +point t of p +postcontrol t of p +\end{code} +Unless you explicitly set them differently, \MP’s curve fitting will make these +three points co-linear, so you can draw a tangent at point $t$ with +\begin{code} +draw precontrol t of p -- postcontrol t of p; +\end{code} +The length of the tangent line drawn +like this depends on the size and shape of the curve, but it is somewhat arbitrary. +So you may prefer to extract a \<pair> representing the tangent at point $t$ with +\begin{code} +pair d; d = postcontrol t of p - precontrol t of p; +\end{code} +In fact, this is so useful that +"plain.mp" provides \mpl{direction} as a shorthand: +\begin{smallcode}[xleftmargin=0pt, xrightmargin=-20pt] +vardef direction expr t of p = postcontrol t of p - precontrol t of p enddef; +\end{smallcode} +which can save you some typing. But the clever bit is that $t$ does not have to be a whole +number. If you set $t=\frac14$ (say), \MP\ works out the corresponding fractional +control points, so that you can use \mpl{direction t of p} to get a tangent at any point. + +The vector pairs returned have the right direction, but still have rather arbitrary magnitudes, so +the usual idiom is something like this: +\begin{smallcode} +path s; s = origin -- 36 unitvector(direction t of p); +drawarrow s shifted point t of p; +\end{smallcode} +or the snippet shown on the right $\longrightarrow$ + +\newpage +\subsection{Lines tangent to a circle}\label{sec:tangent-times} + +The techniques of the preceding section can be used to add a tangent line to a +given point on a circular path, but not to find the tangent lines from a given point +outside a circle. To do this, you need to adapt the standard geometrical +construction: for a given circle $C$ and a point $p$, find the midpoint of $p$ and +the center of $C$; draw a semicircle through $p$, centred on this midpoint; the tangent +point is where the semicircle intersects $C$. % +\mpic{-110pt}{tangents-point-to-circle.pdf} +Given a suitable \mpl{path C} and \mpl{pair p} you can do this: +\begin{code} +pair o, m, t, t'; o = center C; m = 1/2[o, p]; +t = C intersectionpoint halfcircle zscaled (p-o) shifted m; +t' = C intersectionpoint halfcircle zscaled (o-p) shifted m; +\end{code} +No parentheses are needed around the second path, because \mpl{intersectionpoint} is +defined with \mpl{secondarydef}. + +\medskip\noindent +Things are a little more complicated if you want the points as times along the path +$C$ and you care about which tangent point is which. Here is a routine that returns +the tangent points from $p$ as two times $a$ and $b$ on $C$, with $b$ adjusted so +that $b > a$ in all cases regardless of the relative rotation of $C$ and $p$. This +means that \mpl{subpath (a, b) of C} is always the “long way round” C, on the +opposite side from $p$, and \mpl{subpath (a, b-8) of C} is always the shorter +segment. +\begin{code} +vardef tangent_times(expr C, p) = + % return the two times on C that correspond + % to the external tangents from p to C + save o, a, b, G, H; + pair o; numeric a, b; path G, H; + o = center C; + H = halfcircle zscaled (p-o) shifted 1/2[o, p]; + G = halfcircle zscaled (o-p) shifted 1/2[o, p]; + (a, whatever) = C intersectiontimes H; + (b, whatever) = C intersectiontimes G; + (a, b if b < a: + 8 fi) +enddef; +\end{code} +Note the elegant syntax here; +if \mpl{z} is a \<pair> then the operation \mpl{zscaled z} is equivalent +to \mpl{scaled abs z rotated angle z}. + +\vpic{-200pt\noindent Here is the macro in action. +Having obtained the two times $a$ and $b$ from the macro, the dashed line +was drawn along a path that was composed with: +\vrule depth 20pt width 0pt height 2pt +\mpl{p -- subpath (a,b) of C -- cycle} +}{tangent-times-on-circle} + +\newpage +\subsection{Lines tangent to two circles (exterior)}\label{sec:adjust-times} + +The same \mpl{tangent_times} macro can be reused to find the tangents that touch two +circles, using an approach like this: \mwpic{-12pt}{tangents-two-circles-exterior} +\begin{code} +path A, B; +A = fullcircle scaled 144; +B = fullcircle scaled 60 shifted (200, 140); + +numeric R, r; +R = abs (point 0 of A - center A); +r = abs (point 0 of B - center B); + +path C; numeric t, u; +C = fullcircle scaled (2R-2r) shifted center A; +(t, u) = tangent_times(C, center B); + +draw A withpen pencircle scaled 2 withcolor 3/4[blue, white]; +draw B withpen pencircle scaled 2 withcolor 3/4[blue, white]; + +draw subpath (t, u) of A -- subpath (u-8, t) of B -- cycle; +\end{code} +Here $A$ and $B$ are the two circles you want to connect, and $A$ is larger than +$B$. $R$ is the radius of the larger, $r$ of the smaller. $C$ is an auxiliary circle +centred at the same point as $A$ and scaled so that its radius is $R-r$. If we +then find the tangent points on $C$ from the center of $B$, the points we want are +the corresponding points on $A$ and $B$. + +\medskip\noindent +This works well, provided that none of three +circles $A$, $B$, or $C$ has been rotated (that is that point 0 is at 3 o'clock in +all of them). But +this may not always be the case. For example, you might have written +\begin{code} +B = fullcircle scaled 60 shifted 240 right rotated 36; +\end{code} +and then point $t$ of $B$ would \textit{not} correspond to point $t$ of the +auxiliary circle. So to make the code above more general you have to adjust the +tangent times to take account of the relative rotation of the circles. +\vadjust{\moveright5.5in\vbox to 0pt{\hsize 4in\vskip-24pt\noindent +\begin{code} +vardef adjust_time(expr tt, AA, BB) = + tt + 1/45 angle (point 0 of AA - center AA) + - 1/45 angle (point 0 of BB - center BB) +enddef; +\end{code} +\vss}} +The in the figure $t$ was adjusted to $t'$ for $A$ and $t''$ for $B$, using the +routine shown on the right. This routine shows the relationship between \mpl{angle} +and points around a circle: $360^\circ = 8\:\hbox{points}$. + +\newpage +\subsection{Lines tangent to two circles (interior)} + +To find the interior tangents, you just need to add the smaller radius rather than +subtract it, and add 4 to the times on the smaller circles, so that they are on the +other side: +$$\includegraphics[width=\textwidth]{tangents-two-circles-interior}$$ + +\bigskip\noindent +The complete code for this is shown on the right. It uses the same +routines given above; \mpl{tangent_times} from section \ref{sec:tangent-times}, and +\mpl{adjust_time} from section \ref{sec:adjust-times}. +\vadjust{\moveright5.5in\vbox to 0pt{\hsize 4in\vss\noindent +\begin{smallcode} +path A, B; +A = fullcircle scaled 144 rotated uniformdeviate 360; +B = fullcircle scaled 60 shifted 240 right rotated 36; + +numeric R, r; +R = abs (point 0 of A - center A); +r = abs (point 0 of B - center B); + +path C; +C = fullcircle scaled (2R+2r) shifted center A; % NB +ve + +numeric t, t', t'', u, u', u''; +(t, u) = tangent_times(C, center B); +t' = adjust_time(t, C, A); +u' = adjust_time(u, C, A); +t'' = adjust_time(t + 4, C, B); % Note the plus fours +u'' = adjust_time(u + 4, C, B); + +draw A withpen pencircle scaled 2 withcolor 3/4[blue, white]; +draw B withpen pencircle scaled 2 withcolor 3/4[blue, white]; +draw C withpen pencircle scaled 1 withcolor 3/4[blue, white]; + +draw subpath (t', u') of A -- subpath (u'', t'') of B -- cycle; +draw center B -- subpath (t, u) of C -- cycle dashed evenly; + +draw center B -- point t'' of B dashed withdots scaled 1/2; +draw center B -- point u'' of B dashed withdots scaled 1/2; +draw point t of C -- point t' of A dashed withdots scaled 1/2; +draw point u of C -- point u' of A dashed withdots scaled 1/2; + +dotlabel.ulft(btex $t$ etex, point t of C); +dotlabel.lrt (btex $t'$ etex, point t' of A); +dotlabel.lrt (btex $t''$ etex, point t'' of B); +dotlabel.lrt (btex $u$ etex, point u of C); +dotlabel.ulft(btex $u'$ etex, point u' of A); +dotlabel.ulft(btex $u''$ etex, point u'' of B); +drawdot center B withpen pencircle scaled dotlabeldiam; + +drawoptions(withcolor 1/2[blue, white]); + label.urt(btex $A$ etex, point 1/2(t'+u'- 7.6) of A); + label.rt (btex $B$ etex, point 1/2(t''+u''- 2) of B); + label.urt(btex $C$ etex, point 1/2(t+u-8) of C); +drawoptions(); +\end{smallcode}\vskip -1in +}} + +\subsection{Axis of similitude}\label{sec:axosim} + +\bigskip +\noindent\hbox to \textwidth{\includegraphics[scale=0.92]{axis-of-similitude}\hss} +\vadjust{\moveright5.5in\vbox to 0pt{\hsize 4in\vss\noindent +Given three circles taken in pairs, you can use the techniques of the preceding +sections to find the three points where the common external tangents +intersect (shown here as $E_{12}$, $E_{31}$, and $E_{23}$) and the +three points where the common internal tangents intersect ($I_{12}$, +$I_{31}$, and $I_{23}$). These points have a pleasing collinearity. +The line common to the three $E$ points is known as the \textit{Axis of +Similitude}. + +The drawing is left as an exercise for the reader, except +to note that if $r_1$ and $r_2$ are \mpl{numeric} variables representing +the radius of the circles centred at the \mpl{pair} variables $C_1$ and $C_2$, +then we have: +$$\vbox{\openup3pt\halign{\hfil$#$&${}=#$\hfil\cr +E_{12}&({r_1\over r_1 - r_2})[C_1, C_2];\cr +I_{12}&({r_1\over r_1 + r_2})[C_1, C_2];\cr +}}$$ +which is a bit quicker than working out all the tangent points.}} + +\newpage +\subsection{Inversion, pole, and polar}\label{sec:inversion} + +Inversion in a circle is a generalization of reflection in a line. +\mpic{-68pt}{pole-and-polar} +It is useful for certain constructions in geometry, and easy to +implement as a macro \MP. For given circle, and a +given point $P$ lying outside the circle, the inverted point $P'$ lies inside the +circle at the intersection of the line from $P$ to the centre of the circle, and the +line between the tangent points [§\ref{sec:tangent-times}] from $P$, shown here as +$Q$ and $R$.\rlap{\quad$\longrightarrow$} + +But $OPQ$ and $OQP'$ are similar triangles, so $r/OP=OP'/r$ and so $OP' = +r^2/OP$, and since $P'$ must lie on the line through $O$ and $P$, this is enough to +define a macro directly to find $P'$ given $P$, $O$, and $r$: +\begin{code} +P' = O + unitvector(P-O) scaled r * r / abs (P-O); +\end{code} +But examining \mpl{plain.mp} shows that \mpl{unitvector} is a macro defined like this: +\begin{code} +vardef unitvector primary z = z/abs z enddef; +\end{code} +which suggests this alternative (safer) approach: +\begin{code} +vardef invert(expr P, O, r) = + save s; numeric s; s = r / abs (P-O); + O + (P-O) * s * s +enddef; +\end{code} +This works well provided that $|P| > \frac1{180} r$, which is usually the case, but +you could also consider checking that $|t|>0$ and that $s$ was not too large. + +If it was more convenient to deal with the \<path> of the circle of inversion +instead of the centre and the radius, you get the macro to work out the centre and +the radius for you. +\begin{code} +vardef invert(expr P, C) = + save o; pair o; o = 1/2[point 0 of C, point 4 of C]; + save r; numeric r; r = abs (point 0 of C - o); + save s; numeric s; s = r / abs (P - o); + o + (P - o) * s * s +enddef; +\end{code} + +\vfill + +\moveright5.5in\vbox to 0pt{\vss\hsize 4in\noindent +\noindent\llap{\nb}Inversion is reciprocal, so $P$ is the inverse of $P'$ above. Points +on the circle of inversion invert to themselves. + +\smallskip +\noindent\llap{\nb}For any given line, the \blue{\textit{pole}} of +the line with respect to a circle, is the inverse of the point on the line closest +to the centre of the circle. + +\smallskip +\noindent\llap{\nb}For any given point, the \blue{\textit{polar}} of +the point with respect to a circle, is the line through the inverse of the point +perpendicular to the line through the point and the center of the circle of +inversion. + +\smallskip +\noindent\llap{\nb}The small dotted circle through $O$ and $P'$ above is the inversion +of the whole polar line (infinitely extended).} + +\newpage +\subsection{Radical axis and radical centre}\label{sec:radical} + +The \textit{radical axis} of two circles is the line, orthogonal to the line between +the centres of the two circles which is the locus of points which have equal power +with respect to both circles; that is the points from which the tangents to each circle +are of equal length. A circle centred at any point on the axis, and drawn with radius equal to the +length of the tangent will cut both circles at right angles. + +\medskip\noindent\centerline{\includegraphics{radical-axis}} + +\medskip\noindent +In a system of three circles as shown, the \textit{radical centre} is the +intersection of the three mutual radical axes. The tangents from this point to all three circles +have the same length, so a circle with this radius cuts all three circles at right +angles. + +\moveright5.5in\vbox to 0pt{\vss\hsize 4in\begin{smallcode} +vardef radical_axis(expr ca, cb) = + numeric t, d, ra, rb; + ra = abs(center ca - point 0 of ca); + rb = abs(center cb - point 0 of cb); + d = abs(center cb - center ca); + 2t = 1 + (ra+rb) / d * (ra-rb) / d; + (up -- down) scaled 89 + rotated angle (center cb - center ca) + shifted t[center ca, center cb] +enddef; +beginfig(1); + path c[], a[]; + z1 = origin; z2 = 233 right rotated 4; z3 = 209 right rotated -42; + c1 = fullcircle scaled 202 shifted z1; + c2 = fullcircle scaled 106 shifted z2; + c3 = fullcircle scaled 62 shifted z3; + a1 = radical_axis(c1, c2); + a2 = radical_axis(c2, c3); + a3 = radical_axis(c3, c1); + z0 = whatever [point 0 of a1, point 1 of a1] + = whatever [point 0 of a2, point 1 of a2]; + numeric t; (t, whatever) = tangent_times(c1, z0); + drawoptions(withpen pencircle scaled 1 withcolor 3/4[blue, white]); + draw c1; draw c2; draw c3; + drawoptions(withcolor 3/4[blue, white]); + draw z1 -- point t of c1 dashed evenly; + drawoptions(withpen pencircle scaled 1/4); + draw z1 -- z2 -- z3 -- cycle; + drawoptions(); + draw a1; draw a2; draw a3; + drawoptions(withcolor 2/3 red); + draw fullcircle scaled 2 abs (point t of c1 - z0) shifted z0; + draw z0 -- point t of c1 dashed evenly; + drawdot z0 withpen pencircle scaled dotlabeldiam; + drawoptions(withcolor 1/4[blue, white]); + drawdot z1 withpen pencircle scaled dotlabeldiam; + drawdot z2 withpen pencircle scaled dotlabeldiam; + drawdot z3 withpen pencircle scaled dotlabeldiam; + label.urt(btex $C_1$ etex, point 1 of c1); + label.urt(btex $C_2$ etex, point 1 of c2); + label.rt (btex $C_3$ etex, point 0 of c3); +endfig; +\end{smallcode}\vskip -1in} + +\newpage +\subsection{Circles tangent to other circles} +\bigskip +\noindent\hbox to \textwidth{\includegraphics[height=\textheight]{apollonius}\hss} +\vadjust{\moveright5.5in\vbox to 0pt{\hsize 4in\vss\noindent +The classical Problem of Apollonius is to find a circle tangent to three others. +All of the approaches are rather involved, but Gergonne's is probably the simplest +to follow in \MP. + +\smallskip\noindent For three given circles \blue{$C_1$, $C_2$, and +$C_3$}, you first find the external \blue{axis of similitude} +[§\ref{sec:axosim}]; then find the \textcolor{carrot}{poles} [§\ref{sec:inversion}] of this line with +respect to each of the three circles; and thirdly find the +\textcolor{squash}{radical centre} +[§\ref{sec:radical}]. + +\smallskip\noindent +The lines from the radical centre through each of the three +poles cut each circle in two places. These six points show the tangent points for +the two \red{tangent circles}, and you can draw the circles using the three point circle +technique [§\ref{sec:circles}]. + +\smallskip\noindent +{\small +The drawing is also left as an exercise for the reader, although you can find my +version in the source code for this document. You might like to try to make a more +robust version or to find all the other tangent circles.} +}} + + +\newpage +\subsection{Coordinate geometry examples} + +\kern-\baselineskip +\vpic{-24pt}{desargues} +\kern-.5\baselineskip +\begin{code}[xleftmargin=0pt, xrightmargin=-72pt] +beginfig(1); + z.P = 200 up rotated 21; z.A = 100 left rotated -21; + z.B = origin; z.C = 90 right rotated 42; + + z.A' = 3/8[z.P, z.A]; + z.B' = 1/2[z.P, z.B]; + z.C' = 5/8[z.P, z.C]; + + z.R = whatever [z.A, z.B] = whatever [z.A', z.B']; + z.S = whatever [z.B, z.C] = whatever [z.B', z.C']; + z.T = whatever [z.C, z.A] = whatever [z.C', z.A']; + + path t[]; + t1 = z.A--z.B--z.C--cycle; + t2 = z.A'--z.B'--z.C'--cycle; + + fill t1 withcolor 7/8[red, white]; + fill t2 withcolor 7/8[blue, white]; + draw t1 withcolor 1/2 white; + draw t2 withcolor 1/2 white; + + drawoptions(dashed withdots scaled 1/2); + draw z.P--z.A; + draw z.P--z.B; + draw z.P--z.C; + + drawoptions(dashed evenly scaled 1/2); + draw z.B--z.R--z.B'; + draw z.C--z.S--z.C'; + undraw subpath (1/4, 3/4) of (z.C'--z.T) withpen + pencircle scaled 5; + draw z.C--z.T--z.C'; +\end{code} +\moveright 384pt \vbox to 0pt{\vss +\begin{code}[xleftmargin=0pt, xrightmargin=-72pt] + + drawoptions(withcolor 2/3 red); + draw 9/8[z.S,z.R] -- 9/8[z.R,z.S]; + picture pp; pp = thelabel("perspectrix", origin); + draw pp shifted 7 down rotated angle (z.S-z.R) + shifted 1/2[z.R, z.T]; + dotlabel.urt("perspector", z.P); + + drawoptions(); + dotlabel.lft (btex $A$ etex, z.A); + dotlabel.llft(btex $B$ etex, z.B); + dotlabel.lrt (btex $C$ etex, z.C); + dotlabel.lft (btex $A'$ etex, z.A'); + dotlabel.llft(btex $B'$ etex, z.B'); + dotlabel.bot (btex $C'$ etex, z.C'); + label.rt(btex Desargues' Theorem etex, (x.C', 1/2(y.P+y.C'))); +endfig; +\end{code} +\vskip -42pt} + +\newpage +\moveright5.5in\vbox to 0pt{\vskip10pt\includegraphics[width=4.2in]{trisection-triangles}\vss} +\vskip-\baselineskip +\begin{code}[xleftmargin=0pt, xrightmargin=-80pt] +randomseed := 2485.81543; +vardef measured_angle(expr p, o, q) = + (angle (p-o) - angle (q-o)) mod 360 +enddef; +beginfig(1); +picture T; +for i=0 upto 1: + for j=0 upto 1: + clearxy; + T := image( + z1 = (120 + uniformdeviate 21, 0); + z2 = (120 + uniformdeviate 21, 0) rotated 120 rotated 21 normaldeviate; + z3 = (120 + uniformdeviate 21, 0) rotated 240 rotated 21 normaldeviate; + numeric a, b, c; + a = measured_angle(z3, z1, z2); + b = measured_angle(z1, z2, z3); + c = measured_angle(z2, z3, z1); + z4 = whatever [z1, z2 rotatedabout(z1, 1/3 a)] + = whatever [z2, z3 rotatedabout(z2, 2/3 b)]; + z5 = whatever [z2, z3 rotatedabout(z2, 1/3 b)] + = whatever [z3, z1 rotatedabout(z3, 2/3 c)]; + z6 = whatever [z3, z1 rotatedabout(z3, 1/3 c)] + = whatever [z1, z2 rotatedabout(z1, 2/3 a)]; + fill z4--z5--z6--cycle withcolor 3/4[red + 1/2 green, white]; + draw z4--z5--z6--cycle; + draw z1 -- z4 -- z2 -- z5 -- z3 -- z6 -- cycle + dashed withdots scaled 1/4; + draw z1 -- z2 -- z3 -- cycle; + ); + draw T shifted (200i, 240j); + endfor +endfor +label.rt(btex \vbox{\halign{#\hfil\cr The trisectors of each angle\cr +in any given triangle form a\cr central equilateral triangle.\cr}} etex, (24, 128)); +endfig; +\end{code} +\newpage +\kern-\baselineskip +\vpic{1in}{projections} +\mpexternal[xleftmargin=0pt]{projections-code.mp} + +\newpage +\moveright5.5in\vbox to 0pt{\hsize4in\noindent This needs an \mpl{invert} macro, such +as the one from [§\ref{sec:inversion}]. + +\smallskip\noindent +\includegraphics[width=288pt]{arbelos} + +\vskip 1.414in\noindent +\begingroup +\raggedleft\fontsize{8}{10}\selectfont\textsf{% +One must also recognize that any attempt to illustrate geometry\\ +involves a basic fallacy. For example, a straight line is unbounded\\ +and infinitely thin and smooth, while any illustration is unavoidably\\ +of finite length, of positive thickness, and rough edged.\\[2pt] +— Benoit Mandelbrot, \textsl{The Fractal Geometry of Nature}} +\par\endgroup +\vss} +\begin{smallcode}[xleftmargin=0pt, xrightmargin=-72pt] +beginfig(1); +pair A,B,C; A = origin; C = 244 right; B = 7/8[A, C]; +path c[]; +c1 = fullcircle scaled 2 abs(A-C); % large circle for the inversions +c2 = fullcircle scaled abs(A-C) shifted 1/2[A,C]; +c3 = fullcircle scaled abs(A-B) shifted 1/2[A,B]; +c4 = fullcircle scaled abs(B-C) shifted 1/2[B,C]; +c5 = invert(c4,c1); +numeric d; d = abs(point 0 of c5 - point 4 of c5); % diameter of c5 + +drawoptions(withcolor 3/4 white); +draw c4; draw c5; +draw invert(subpath(0, 3/2) of c2, c1); +draw invert(subpath(0, 3/2) of c3, c1); +drawoptions(); +draw subpath(-1/4,1) of c1 withcolor 3/4[1/2 red,white]; +label.bot("\textit{circle of inversion}", point -1/4 of c1) + withcolor 1/4[1/2 red, white]; + +for i=1 upto 72: + path c, c'; + c = c5 shifted (0, i*d); c' = invert(c, c1); + if i < 5: + drawoptions(withcolor 3/4 white); + draw c; draw origin -- center c; + fill fullcircle scaled dotlabeldiam shifted center c; + fill fullcircle scaled dotlabeldiam shifted center c'; + drawoptions(); + fi + draw c' withpen pencircle scaled 1/4 withcolor 2/3 blue; +endfor +forsuffixes $=2,3,4: draw subpath (0,4) of c$ + withpen pencircle scaled 1/4 withcolor 2/3 blue; +endfor +draw A--C; +dotlabel.lft("$A$", A); dotlabel.llft("$B$", B); dotlabel.rt("$C$", C); +endfig; +\end{smallcode} + +\newpage +\section{Trigonometry functions}\label{trig} + +\MP\ provides only two basic trigonometry functions, "sind" and "cosd". This lack +appears to be a deliberate design; in general it's much easier to use the "rotated" +and "angle" functions than to work out all the sine, cosines and arc-tangents +involved in rotating parts of your picture. But if you really want the `missing' +functions they are not hard to implement. + +First you might want versions that accept arguments in radians instead of degrees. +For this you need to know the value of $\pi$, but this is not built into plain \MP. +If you are using the default number system then it's enough to define it to five +decimal digits,% +\vadjust{\moveright 384pt\vbox to 0pt{\kern-144pt +\begin{code} +numeric pi; +% approximate value +pi := 3.14159; +% measure round a circular arc... +pi := 1/4 arclength (quartercircle scaled 16); +% up to 32 digits of precision +pi := 3.1415926535897932384626433832795; +% as many digits as are needed... +vardef getpi = + numeric lasts, t, s, n, na, d, da; + lasts=0; s=t=3; n=1; na=0; d=0; da=24; + forever: + exitif lasts=s; + lasts := s; + n := n+na; na := na+8; + d := d+da; da := da+32; + t := t*n/d; + s := s+t; + endfor + s +enddef; +pi := getpi; + +% conversions +vardef degrees(expr theta) = 180 / pi * theta enddef; +vardef radians(expr theta) = pi / 180 * theta enddef; +% trig functions that expect radians +vardef sin(expr theta) = sind(degrees(theta)) enddef; +vardef cos(expr theta) = cosd(degrees(theta)) enddef; +% inverse trig functions +vardef acosd(expr a) = angle (a,1+-+a) enddef; +vardef asind(expr a) = angle (1+-+a,a) enddef; +vardef acos(expr a) = radians(acosd(a)) enddef; +vardef asin(expr a) = radians(asind(a)) enddef; +% tangents +vardef tand(expr theta) = save x,y; (x,y)=dir theta; y/x enddef; +vardef atand(expr a) = angle (1,a) enddef; +\end{code} +\vss}} +but if you are using one of the new number systems you might want more digits of +precision. In fact there's no harm in always defining these extra digits; even when +you are using the default "scaled" number system, \MP\ will happily read as many +extra digits of $\pi$ as you supply, before it rounds the value to the nearest +multiple of $1\over65536$ (which turns out to be $3.14159$). The same applies to the +"double" number system, but the "binary" and "decimal" number systems will give you +an error if you supply more digits that the default precision. So in general +it's best to use no more than 32 digits. It's also possible, but not really worth +the trouble, to define a routine to calculate $\pi$ to the current +precision.\rlap{\raise1ex\hbox{\ $\smash{\nearrow}$}} However you define it, once you are armed with a value for $\pi$ you can +then define functions to convert between degrees and radians, and some more `normal' +versions of sine and cosine. + +There's no built-in $\arccos$ or $\arcsin$ function but each is very easy to +implement using a combination of the "angle" function and the Pythagorean difference +operator. + +\MP\ does have built-in functions for tangents; but they are called "angle" and +"dir" and they are designed for pairs. So $\kw{angle}\,(x,y)=\arctan(y/x)$ while +$\kw{dir}\,30$ gives you the point $(x,y)$ on the unit circle such that $\tan +30^{\circ} = y/x$. You can use these ideas to define tangent and arctan functions +if you really need them, but often "angle" and "dir" are more directly useful +for drawing. + +You should also be aware that the tangent function shown here does not +check whether $x=0$; if this is an issue, then add something like this +at the appropriate point: +$$\kw{if}\,x=0\!:\id{infinity}\: \kw{else}\!: y/x \kw{fi}$$ + +\newpage +\section{Traditional labels and annotations}\label{sec:trad-labels} + +\moveright384pt\vbox to 0pt{\hsize 4.2in\vss\textcolor{blue}{\itshape This section describes labels and +annotations in what can be called the traditional \MP\ environment, where your +figures are compiled with "mpost". The section after this describes labels \& +annotations in the newfangled (but better) world of "lualatex" and the +"luamplib" package.}\par\kern 16pt} + +\noindent +\MP\ does not draw text directly; but it provides two different mechanisms to +turn some text into a \<picture>, which can then be treated like any other; +saved as a variable, drawn directly, or transformed in some way with a +scaling, a reflection, or a rotation. The first mechanism is described below, +the other in §\ref{btex}. + +\subsection{Simple strings in PostScript fonts with \texttt{infont}}\label{infont} + +The first mechanism is the primitive binary operation "infont". As explained in +section~8.3 of the \MP\ manual, it takes two strings as arguments: the left hand +argument is the string of text to be printed; the right hand argument is the name of +the font to use; and the result is a "picture" primary.\vadjust{ +\moveright384pt\vbox to 0pt{\kern -126.5 bp + \hsize 4.2in\noindent +To find the name of a suitable font, you have to consult your local "psfonts.map" +file, and probably the PSNFSS documentation. +Here are a few of the many fonts available on my local \TeX\ installation; the name +to use with "infont" is in the first column. +$$\includegraphics{trad-font-samples}$$ +The text example in the first line +of this table was produced with +\begin{smallcode} +draw "Hand in glove 42" infont "pagk8r" shifted (124,144); +\end{smallcode} +Note that in PostScript terms each of these font names refers to a combination of +three files: an encoding that maps the characters you type to the glyphs in the +font; a font metrics file that defines the sizes of the virtual boxes surrounding +these glyphs; and a set of PostScript routines that actually draw them. In a \TeX\ +installation +these combinations are defined in a font map file, usually called "psfonts.map". +If you run "mpost" with the "-recorder" switch it will write an extra log file (with +a ".fls" extension) that lists the names of all the files used in a job. +The actual font map file in use will be one of these. You can then browse it to +find a definitive list of the font names you can use with your local \MP. +\vss}} + +To make a suitable string you can enclose your text +in double quotes to make a string token, or to refer to a \<string> variable, or do +one of these: +\begin{itemize} + \item Concatenate two other strings with \mpl{&}. + + \item Use \mpl{substring (a,b) of s} to get a substring of string \mpl{s}. + + \item Use \mpl{min(a,b,...)} or \mpl{max(a,b,..)} to find the lexicographically smallest + (or largest) string in the list \mpl{a,b,...}. The list must have at least two + entries, and they must all be strings. + + \item Use \mpl{char} to convert a numeric expression to the corresponding ASCII + code; + the numeric expression is rounded to the nearest integer modulo + 256. + + \item Use \mpl{decimal} to get a string representing + the value of a numeric expression. + + \item Apply \mpl{str} to any suffix (and hence to any variable). You get back a + string representation of the suffix or variable name. + + \item Use \mpl{readfrom} to read one line from a file as a string. + + \item Use \mpl{fontpart} to extract the name of the font used in a picture created + with \mpl{infont} --- the string will be empty if there's no text element in the + picture. + + \item Use \mpl{textpart} to get the text used in a picture created by \mpl{infont} --- + the string will be empty if there's no text element in the picture. + +\end{itemize} + +\newpage +\subsubsection{Character sets used by \texttt{infont} to set +text}\label{sec:charsets} + +Standard \MP\ is configured to accept as input only space and the usual 94 visible +ASCII characters (that is the characters numbered 32 to 126 in the tables at the +right), but you can use any 8-bit characters as the payload of a string. +However, plain \MP\ is set by default to use "cmr10", the familiar Computer Modern typeface +developed by Knuth for \TeX, and unfortunately, this is encoded using the \TeX\ text +font encoding (also known as `OT1', and as shown in the first table in Appendix~F of the {\sl \TeX book}). +\mpic{-108pt}{trad-font-tables} +From the point of +view of using "infont" to make simple labels, this means that the characters for +space and seven other characters ("< > \ _ { | }") are in the wrong place. +You are likely to notice this first if you try to set a label with two words; the +space will come out as a small diagonal stroke accent that is used in plain \TeX\ +to make the characters Ł and ł, used in Polish and other Slavic languages. + +To fix this you should change the default font at the start of your program: +\begin{code} +defaultfont := "texnansi-lmr10"; % for Computer Modern Roman +\end{code} +If you want "cmss10", use "texnansi-lmss10" and so on. The encoding is shown on the +right. The characters printed in black correspond to the widely used ISO Latin~1 +encoding. If you want to use one of the standard PostScript fonts listed on the +previous page, then the encoding to use is either "8y" to get the same "texnansi" +arrangement or "8r" to get the arrangement shown in the lower table. + +Choosing a font with one of these encodings means that if you use Windows code page +1252 or ISO Latin 1 as the encoding for your text editor, you can create labels with +accented characters using "infont" and without resorting to "btex" \dots\ "etex". +But if you are using UTF-8 characters (as many of us are now), then you have to do +some extra work to get them printed correctly with "infont". A solution is shown on +the next page. + +In the normal course of labelling a drawing, it is always possible to +use "btex" \dots\ "etex" to produce your accented characters as discussed in section +\ref{btex} below; but it may be that you are using \MP\ to represent data and labels +supplied from some other program or a website. In this case it can be useful to be +able to work with at least a subset of UTF-8 input. + +\newpage +\subsubsection{Mapping a subset of UTF-8 for \texttt{infont}} + +UTF-8 is a way of +representing 16-bit Unicode characters with sequences of 8-bit characters. So your +UTF-8 aware editor may show you an é but \MP, knowing nothing about UTF-8, +will see this as é. But you can write a fairly simple routine to decode a +commonly-used subset of +UTF-8.\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip -9.5pt\parindent 0pt +\def\item{\leavevmode\llap{•~}} +\item You can extend this idea to cope with other UTF-8 characters, including + those that use three bytes. The UTF-8 page on Wikipedia shows you how it + works. Essentially you look at the values of the next 2 or 3 characters and + then pick the appropriate character in your encoding with "char". But your + output is still, of course, limited to the 256 characters in your encoded font. + +\item If you get tired of writing "decode", you could define a short cut with a + shorter name. You could even write it as a primary without parentheses like + this: +\begin{code} +def U primary s = if string s: decode(s) fi enddef; +\end{code} +which would let you write: +\begin{code} +label.rt(U"café à la möde", (x,y)); +\end{code} + +\item However, there's no point in making any of this too elaborate. If you + really want proper Unicode support you should use \MP\ with Lua\TeX. (See + below in §\ref{sec:neo-labels}). +\vss}} +\begin{code}[xrightmargin=-10pt] +vardef decode(expr given) = + save a,i,s,out; string s, out; numeric a, i; + out = ""; i=0; + forever: + i := i+1; s := substring (i-1,i) of given; a := ASCII s; + if a < 128: + elseif a = 194: + i := i+1; s := substring (i-1,i) of given; + elseif a = 195: + i := i+1; s := char (64 + ASCII substring (i-1,i) of given); + else: + s := "?"; + fi + out := out & s; + exitif i >= length given; + endfor + out +enddef; +\end{code} +Use it with "infont" like this: \verb|decode("café") infont "ptmr8r"| to produce a +normal "picture" that can be passed to \mpl{draw} or saved as +usual.\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip -9.5pt +\noindent The fragment on the left produces: +$$\includegraphics{utf8}$$ +The \texttt{label} macro automatically calls \texttt{infont} with the current value +of \texttt{defaultfont}; notice how it also adds some extra space. +\vss}} +\begin{code} +draw "café noir £2.50" infont "pncr8r"; +draw decode("café noir £2.50") infont "pncr8r" shifted 12 down; +defaultfont := "pncr8r"; +label.rt("café noir £2.50", 24 down); +label.rt(decode("café noir £2.50"), 36 down); +\end{code} +Note that you can't just use \mpl{draw} with a string variable; you have to use +"infont" to turn the string into a picture. +On the other hand, "label" calls "infont" automatically, but you must explicitly set +the default font, preferably to one with an encoding that +is compatible with ISO Latin~1. + + +\newpage +\subsubsection{Typographical minus signs with \texttt{infont}} + +If you are producing labels for a numeric reference scale, like the axis of a chart, +it is convenient to be able to write a loop like this: +\begin{code} +for x=1 upto 3: label.bot(decimal x, (x*cm, 0)); endfor +\end{code} +to produce your labels, however if $x$ is negative this does not come out so well, +because the first character of the string produced by \mpl{decimal -1} is an +\mpl{ASCII 45}, +which is the hyphen character. What we need is the mathematical minus sign instead; +this is what you get with \mpl{btex $-1$ etex} of course, but that's harder to put in a +loop with traditional \MP. Instead you can do this:\vadjust{\moveright5.5in\vbox to 0pt{\vss\hsize4in +$$\includegraphics{minus}$$ +\vss}} +\begin{code} +string minus_sign; +minus_sign := char 143; % if you are using the texnansi encoding +minus_sign := char 12; % if you are using the 8r encoding +for x = -3 upto 3: + label.bot(if x<0: minus_sign & fi decimal abs(x), (x*cm, 0)); +endfor +\end{code} +Note that this does not work with the default encoding used in "cmr10" because +there is no minus sign available in that font. Plain TeX uses "char 0" from +"cmsy10". + +\subsubsection{Bounding boxes and clipping with \texttt{infont}} + +Once the encoding is fixed, the other two parts of a PostScript font are the font +metrics and the programs that draw the actual glyphs. The font metrics define the +width of each character and provide a kerning table to adjust the space between +particular pairs.\vadjust{\moveright5.5in\vbox to 0pt{\vss\hsize4in +$$\includegraphics{infont-example}$$ +}} This means that certain characters will overlap each other or stick out +beyond the bounding box of the picture produced by "infont". This is not normally a +problem unless the picture happens to be at the edge of your figure. In the first +example observe how the last letter sticks out to the right; in the second a +wider baseline has been added to prevent this. +If you want this effect, but you +don't want to see the baseline, then draw it +using the colour \mpl{background}. + +\subsubsection{But what about the \texttt{label} command?} + +As a convenience, the plain \MP\ format provides a "label" macro that automatically +turns strings into pictures for you using whatever font name is the current value of +"defaultfont" and scaled to the current value of "defaultscale".\vadjust{\moveright5.5in\vbox to 0pt{\vss\hsize4in +\noindent +The "label" macro is defined (essentially) to do this: +\begin{code} + def *label(expr s, z) = + draw s if string s: infont defaultfont + scaled defaultscale fi shifted z + enddef; +\end{code} +plus some clever code to align the label for you. +}} + +\newpage +\subsubsection{Bounding boxes and alignment with \texttt{infont}}\label{textsize} +\label{infontbbox} + +To allow you to align a text label on a specific point, \MP\ provides five unary +operators to measure the bounding box of a picture; they are shown in +\textcolor{red!67!black}{red} in the diagram, and you can use them to measure the +width, depth, and height of a textual picture. You can also work out the location +of the baseline of the text or the x-height, provided you know how much your picture +has been shifted. +The easiest way to do this is to +measure the picture \textit{before} you shift it.\vadjust{\moveright5.5in\vbox to 0pt{\vss\hsize4in +$$\includegraphics[width=4in]{infont-annotated}$$ +\vss}} +\begin{code} + picture pp; pp = "proof" infont "pplri8r"; +\end{code} +Here the picture \id{pp} +is created with the origin of the text sitting at coordinates $(0,0)$; +then you can get the dimensions like this +\begin{code} + wd = xpart urcorner pp; + ht = ypart urcorner pp; + dp = ypart lrcorner pp; +\end{code} +In this particular case you will find that you have $wd=20.47292$, $ht=7.19798$, and +$dp=-2.60017$. The depth is negative because the descenders on the +\textdemo{\textit{p}} and the \textdemo{\textit{f}} +in the chosen font stick down below the base line. The height is greater than the +x-height, because the \textdemo{\textit{f}} also sticks up, so you need to make another measurement: +\begin{code} +numeric xheight; xheight = ypart urcorner ("x" infont "pplri8r"); +\end{code} +Armed with these measurements you can align your text labels neatly so that they are +all positioned on the base line or vertically centred on the lower case letters +regardless of any ascenders or descenders. To draw your label left-aligned with its +origin at position $(x,y)$ you just need to use: \kw{draw} \id{pp} \kw{shifted} $(x,y)$. +To draw it right-aligned, you subtract +\id{wd} from the $x$-coordinate: \kw{draw} \id{pp} \kw{shifted} $(x-wd,y)$. Or to +centre it, subtract $1/2wd$. To center it vertically on the lowercase letters, +subtract $1/2\id{xheight}$ from the $y$-coordinate. You might of course like to +wrap these adjustments up in your own convenient macro to help you maintain +consistency in a diagram with many labels. + +Alternatively you can adjust the bounding box of your textual picture and then use +it with "label" as normal. Assuming \id{wd} is set to the width of your picture +and \id{xheight} is set correctly for the current font, then +\begin{code} + setbounds pp to unitsquare xscaled wd yscaled xheight; +\end{code} +will make the "label" alignment routines ignore any ascenders or descenders. +\vadjust{\moveright5.5in\vbox to 0pt{\hsize 4in \vss \small +\noindent\llap{\nb}Beware that if the resulting label is right at the edge of your +drawing then any parts of the text that stick out of the adjusted bounding box will +be clipped. See also §\ref{sec:rotated-boxes} for more on what happens if you +rotate the text.}} + +\newpage +\subsubsection{Setting Greek letters with \texttt{infont}} + +\leavevmode\hbox{} +$$\includegraphics[width=0.5\textwidth]{greek-homer}$$ +While it's technically possible to set the whole of Homer's \textsl{Iliad} using the Greek +fonts available to \mpl{infont}, it's probably not a great use of time; on the other hand you +might want to label parts of a diagram with Greek letters, and for single Greek +letters \mpl{infont} is more than adequate. + +The Greek letters for Computer Modern are in the maths-italic font "cmmi10", which +uses the encoding shown on p.\thinspace 430 of \textsl{The \TeX{}book}. For historical +reasons there's no omicron available, so you are supposed to use the $o$~character +instead. Fortunately you are unlikely to need more than the first few, and it's +quite easy to remember that $"char 11"=\alpha$, $"char 12"=\beta$, and so on. +Producing the upper case letters is a bit more of a fiddle with this encoding as you +need to know which ones use a Roman letter form; for details examine the program on +the right, or check the table in §\ref{euler} that shows +Herman Zapf's elegant Euler font, available as "eurm10". This +makes a refreshing change for some diagrams.\vadjust{\moveright5.5in\vbox to +0pt{\hsize4in\kern-164pt +\begin{code} +beginfig(1); +string ab, AB; +ab = ("" for i=11 upto 23: & char i endfor + & "o" for i=24 upto 33: & char i endfor); +AB = ("AB" & char 0 & char 1 & "EZH" & char 2 & "IK" + & char 3 & "MNO" & char 4 & char 5 & "P" + & char 6 & "T" & char 7 & char 8 & "X" + & char 9 & char 10); + +draw ab infont "cmmi10"; +draw AB infont "cmmi10" shifted 12 down; +draw ab infont "eurm10" shifted 32 down; +draw AB infont "eurm10" shifted 44 down; +endfig; +\end{code} +\centerline{\includegraphics{greek-default-encoding}} + +\begin{code} +string ab, AB; +ab = "abgdezhjiklmnoxprstufqyw"; +AB = "ABGDEZHJIKLMNOXPRSTUFQYW"; +\end{code} +\centerline{\includegraphics{greek-gfs-encoding}} +\vss}} + +\medskip\noindent +If you have fonts installed from the Greek Font Society, then you get a wider +choice, and a slightly more modern encoding. All of the plain letters are available +in the normal ASCII positions, so you do not have to muck about with "char xx" +so much. However in recent versions there is no character you +can use as a word space, so if you want to set Greek text rather than individual +letters, see §\ref{sec:neo-otf}. + +\vbox to 0pt{\centerline{\includegraphics{porson}}\vss} + + + +\newpage +\subsection{Setting text with \texttt{btex ... etex}}\label{btex} + +As soon as you need anything complicated in a label, like multiple fonts, multiple +lines, or mathematics, you will find it easier to switch from "infont" to the +"btex ... etex" mechanism that calls \TeX\ to create your textual picture. In fact +you might prefer to use \TeX\ for all your labels, even simple strings, for the sake +of consistency. The only downside is that this mechanism is a little bit slower. + +The "btex" mechanism produces a textual picture just as "infont" does with a height, +width, and depth that you can measure, and adjust, as discussed in +section~\ref{infontbbox}. And again, just like "infont" you can either use \mpl{draw} +to place the resulting picture directly, or pass it to the "label" macro. + +What you need to be aware of is that \MP\ places everything you put between the +"btex" and "etex" into an "\hbox{...}" and processes it with plain \TeX. This has +several implications: on the positive side you have easy access to italics and bold +letters, mathematical formulae, symbols like "$\alpha$", and anything else you can +normally put in an "\hbox{}"; but there are some restrictions especially if you want +to do anything more than produce a simple single-line label in the default Computer +Modern type face. The next few sections deal with some of the things you might want +to do. + +\subsubsection{Producing display maths} + +One of the obvious restrictions that \TeX\ imposes in +restricted horizontal mode +is that you can't use "$$ ... $$" to produce display maths. This means that +the various mode-sensitive constructs like $\sum$ and $\int$ will come out in their +smaller forms. And your fractions will look like they are $3\over4$ size. +If you want them big, then the solution is simple: just add +"\displaystyle" at the beginning of your formula $\longrightarrow$\vadjust{\moveright5.5in\vbox to 0pt{\hsize 4in\vss\noindent +\begin{code} +... +label(btex $\displaystyle \int_0^t 3x^2\, dx$ etex, (x,y)); +... +\end{code} +}} + +\subsubsection{Getting consistent baselines for your labels} + +As already discussed §\ref{infontbbox}, you can fiddle with the +bounding box of a text picture to make +the \mpl{label} macro line things up on a common baseline, but there is a +much easier way with "btex" and "etex". Plain \TeX\ provides a "\strut" +command that inserts an invisible rule that sticks up 8.5pt above the baseline +and 3.5pt below. If you put one of these in each of the your labels, +then they will all have the same vertical size and will all line up +neatly:\quad\mpl{label(btex \\strut a etex, origin);} + + +\subsubsection{Multi-line text labels} + +Another consequence of the "hbox" feature is that there is no automatic text +wrapping done for you, but again you can work round this easily because \TeX\ lets +you nest a "\vbox" inside an "\hbox". This gives you proper paragraph-like wrapping +but you will almost certainly need to adjust the line length, justification, and +indentation in order to get a satisfactory result \rightarrow \vadjust{\moveright5.5in\vbox to 0pt{\hsize 4in\vskip -42pt\noindent +\begin{code} +... +label(btex \vbox{\hsize 2in\parindent 0pt\raggedright + An extended caption or label that will be set as a + small paragraph with automatic hyphenation and + line-wrapping. +} etex, z0); +... +\end{code}\vss +}} + +\medskip\noindent +You may only need the full power of \TeX's paragraph making system occasionally +though: more usually you will just have one or two lines in each label, and you +might be quite happy to control the line breaks manually. In this case it's helpful +to wrap a little tabular structure around your text. Here's how to define something +suitable in plain \TeX. First you need to define a suitable macro at the start of +your figure +\begin{code} +verbatimtex +\def\s#1{\let\\\cr\vbox{\halign{\hfil\strut ##\hfil\cr#1\crcr}}} +etex +\end{code} +then you can write labels like this: +\begin{code} +... +label(btex \s{Single line} etex, z1); +label(btex \s{Longer text split\\onto a new line} etex, z2); +... +\end{code} +Notice how you can still use the macro with single lines, you just get a one-line +table as it were. +Note also that the definition of "\s" as given will centre each line of the text under +the one above. If you want them left aligned or right aligned, omit one of the +"\hfil" commands. \mpic{-160pt}{split-labels} +The three examples from this page are typeset over here \rightarrow\break +The small red circle show the reference points and the pale blue lines the +bounding boxes of the pictures that \MP\ gets back from \TeX. + +You can of course achieve the same effects using \LaTeX\ tabular structures, but +then you have to use the "-tex=latex" option to run +\MP. + +\smallskip\noindent +\textbf{Note}: In case it's not obvious, if you want text wrapping or tabular +arrangements as discussed here, you need to use \texttt{btex ... etex} to set your +labels. There's no text wrapping with "infont". On the other hand if all of your +labels are in "infont" but you just want one extra that has two lines, you can split +the text into two separate labels and position them independently. + +\newpage +\subsubsection{Dynamic labels}\label{sec:old-and-dynamic} + +If you are a maven of programming language syntax you may have noticed that +"btex ... etex" fits into the type system that \MP\ inherits from \MF\ as a "picture" and +not as a "string". Effectively, "btex" and "etex" act as a special pair of quotation +marks that create a picture; however the contents are used verbatim, so that the +whole construction is a syntactical atom. This means that you \textbf{cannot} write +this sort of thing: +\begin{code} +for i=0 upto 4: % this won't work + label(btex "$p_" & decimal i & "$" etex, (10i,0)); +endfor +\end{code} +Given this input \MP\ would attempt to get \TeX\ +to typeset +\begin{code} +\hbox{"$p_" & decimal i & "$"} +\end{code} +which would probably result in a +`Misplaced alignment tab character' error. +To get round this problem, \MP\ provides a general mechanism to write out a +string to a file, and then read the file back in. This is the mechanism used by the +"TEX()" macro that is provided alongside "plain.mp". This allows you to write: +\begin{code} +input TEX +... +for i=0 upto 4: + label(TEX("$p_" & decimal i & "$"), (10i,0)); +endfor +\end{code} +This works because the TEX macro is expecting a "string" so the normal string +concatenation rules are applied. The macro wraps the result with "btex" and "etex", +writes them out to a file, and then reads the file in again so that \MP\ gets the +correct contents to pass to \TeX. + +The only trouble with this is that it makes \MP\ open a file, +write to it, close it, and then read it in again for each label one at a time; this +means that it's very slow. +The example on the right shows how to speed things up, by using the same file for +all the labels and only writing it once. +The "write" command is a \MP\ primitive, and "EOF" is defined in "plain.mp".% +\vadjust{\moveright5.5in\vbox to 0pt{\vss\hsize4in\noindent +\begin{code} +path c; c = fullcircle scaled 100; draw c withcolor .67 red; +for i=0 upto 7: + fill fullcircle scaled 3 shifted point i of c; + z[i] = point i of c scaled 1.15; + write "label(btex $p_" & decimal i & "$ etex,(" + & decimal x[i] & "," & decimal y[i] + & "));" to ".mplabels"; +endfor +write EOF to ".mplabels"; +input ".mplabels"; +\end{code} +$$ +\includegraphics{dynamic-labels} +$$ +Note that you can't use "decimal" on a "pair" variable, but you can save the pair +as a "z"-variable and then use the "x" and "y" syntax. The scaling trick used here +only works because "c" is centred on the origin. If "c" were drawn elsewhere, you +would have to write: +\begin{code} +... point i of c shifted -center c + scaled 1.15 + shifted center c ... +\end{code} +}} + +\newpage +\subsection{Matching fonts}\label{sec:fonts} + +Despite the apparent restriction of using plain \TeX\ it is almost always possible +to match the font and format of an enclosing \LaTeX\ document. +The simplest approach is to use the plain +\TeX\ font mechanism with the names from "psfonts.map". +\begin{code} +verbatimtex +\font\rm=ptmr8r\rm +etex +\end{code} +Adding this at the top of your \MP\ program will set your text in Times New Roman, +although any maths will still be set using Computer Modern. +To fix this, all you +have to do is to redefine all the maths fonts in all sizes you need; this is not really +that hard but it is a fiddle to get all the details right. Fortunately +the wonderful "font-change" package has done it all for you for a large range of +fonts; with this package installed you can use +\begin{code} +verbatimtex +\input font_times +etex +\end{code} +instead, and all of your \TeX\ labels, including bold letters, italics, small caps, +and mathematics will be set in Times New Roman.\vadjust{\moveright5.5in\vbox to +0pt{\hsize 4in\vss\noindent +Here are some samples of the fonts available in the "font-change" package. For full +details, and especially details about using AMS symbols, see the package +documentation. +$$\includegraphics{trad-font-changes}$$ +}} + +If you still can't get your labels to match, you can force \MP\ to use \LaTeX\ +instead of plain \TeX. You need to use the "-tex" command line switch: +\begin{code} + mpost -tex=latex +\end{code} +and also load the packages you need in a "verbatimtex" block at the top of your +file\rlap{\ $\longrightarrow$}\vadjust{\moveright 5.5in\vbox to 0pt{\hsize 4in\raggedright +\vskip -42pt +\noindent +\begin{smallcode} +verbatimtex + \documentclass{article} + \usepackage{mathpazo} + \usepackage{xcolor} + \begin{document} +etex +\end{smallcode} +Note that the "\documentclass" and the "\begin{document}" lines are required, but +\MP\ is smart enough to add an "\end{document}" for you.\vss}} + +Plain "mpost" needs an old-fashioned ".dvi" file to work with, so you can only use +an engine that still produces one, like "latex" or "elatex", and not any of the +more modern engines, like "pdflatex". Generating the labels takes a little bit +longer because you have to load rather more `infrastructure' for \LaTeX, and you +are limited to whatever font packages you have that work with the traditional +\LaTeX\ engine. For a more modern approach, read on into section +\ref{sec:neo-labels}. + +\newpage +\subsection{Setting verbatim listings}\label{sec:verbatim} + +\textsc{There is a good chance} that you will never need to set a verbatim +listing in a \MP\ drawing, but if you do there are a couple of things to think +about. The issue about setting text verbatim with \TeX\ is that turning off the +control characters can be tricky, so if you have text for a label with characters +that are special in \TeX\ like the backslash or the underscore, then the simplest +thing to do is to avoid \TeX\ completely and use \mpl{infont} instead. +\mpic{-36pt}{verbatim-with-infont} +\begin{code} +string s; s = "\TeX\ sets maths like this $e=mc^2$"; +draw ("1. " & s) infont defaultfont; +draw ("2. " & s) infont "texnansi-lmr10" shifted 20 down; +draw ("3. " & s) infont "cmtt10" shifted 40 down; +draw ("4. " & s) infont "texnansi-lmtt10" shifted 60 down; +\end{code} +But as you can see, (1) this is a bit of a disaster with the default font "cmr10" +because it does not have all the glyphs in the usual ASCII positions (as noted +above §\ref{sec:charsets}). The solution is to use the version of the font +with the "texnansi" encoding (2), but you probably want it in the monofont (3) and +as you can see "cmtt10" has the “visible space” character instead of a regular +space. If this is not what you want then use the alternative encoding (4). + +If you want more than this, then you really need to use \LaTeX\ to process the +label, as discussed in §\ref{sec:fonts}, and load the appropriate preamble. \rightarrowfill\break +\vadjust{\moveright5.5in\vbox to 0pt{\vskip -72pt\begin{smallcode} +prologues := 3; outputtemplate := "%j.eps"; +verbatimtex +\documentclass{article} +\usepackage{listings} +\newcommand\mpstyle{\lstset{language=Metapost, basicstyle=\ttfamily, +columns=fullflexible, keepspaces=true, showstringspaces=false}} +\lstnewenvironment{code}[1][]{\mpstyle\lstset{#1}}{} +\begin{document} +etex +beginfig(1); +picture P; +P = thelabel(btex \vbox{\begin{code} + % special operators + vardef incr suffix $ = $:=$+1; $ enddef; + vardef decr suffix $ = $:=$-1; $ enddef; + + def reflectedabout(expr w,z) = % reflects about the line w..z + transformed begingroup transform T_; + w transformed T_ = w; + z transformed T_ = z; + xxpart T_ = -yypart T_; + xypart T_ = yxpart T_; % T_ is a reflection + T_ endgroup enddef; +\end{code}} etex, origin); +fill bbox P withcolor (1,1,31/32); draw P; draw bbox P; +endfig; end. +\end{smallcode}\vss}} + +\bigskip +$$\includegraphics[width=0.9\textwidth]{verbatim-listing}$$ + + + +%% \subsubsection{Getting full access to your system fonts} +%% +%% If you want full access to all of your system fonts you can +%% approach the problem the other way round and use one of the various means to include +%% \MP\ graphics as part of your \LaTeX\ source code. These include "gmp" for +%% pdf\LaTeX, "luamplib" for lua\LaTeX, and the whole Context system. The great +%% advantage of these systems is that all of your \MP\ labels directly inherit the +%% environment of the parent document, and give you access to all your system fonts and +%% full Unicode support -- the only disadvantages are that it's not so fast or simple as plain \MP\ +%% and you have to compile every graphic everytime you compile the document. It is of course always possible to +%% use these systems to produce standalone PDF graphics that you can then include in a +%% more conventional \TeX\ document. The example on the right shows how; in this case +%% the text uses the fonts set in the \LaTeX\ preamble.\vadjust{\moveright5.5in\vbox to +%% 0pt{\hsize 4in\vss\noindent +%% Here is a version of the Lua logo, with a Unicode accent for show. +%% $$\includegraphics[scale=0.7]{lulu.pdf}$$ +%% produced with "luamplib": +%% \begin{smallcode} +%% \documentclass[margin=5mm]{standalone} +%% \usepackage{fontspec} +%% \setmainfont{TeX Gyre Heros} +%% \usepackage{luamplib} +%% \begin{document} +%% \begin{mplibcode} +%% beginfig(1); +%% color lemon, midnight; lemon = (1,1,1/2); midnight = (0,0,1/2); +%% +%% fill unitsquare shifted -(1/2,1/2) scaled 4cm withcolor lemon; +%% fill fullcircle scaled 3cm withcolor midnight; +%% draw fullcircle scaled 3.7cm dashed evenly scaled 2 withcolor .5 white; +%% +%% fill fullcircle scaled 8mm shifted (0.7cm,0.7cm) withcolor white; +%% fill fullcircle scaled 8mm shifted (1.4cm,1.4cm) withcolor midnight; +%% +%% label.bot(btex Luã etex scaled 2.8,origin) withcolor white; +%% endfig; +%% \end{mplibcode} +%% \end{document} +%% \end{smallcode} +%% }} + + +\newpage +\section{Modern labels and annotations}\label{sec:neo-labels} + +This section is a re-working of the previous section, that attempts to show how much +nicer it is to handle labels in the new world of "luamplib". If this is all new +to you, you probably should start by doing "texdoc luamplib" on your system +and reading the documentation provided with the package. In order to use these +newfangled facilities you need to create your \MP\ diagrams inside a \TeX-wrapper +as explained above in §\ref{sec:sa-lua-flow}. + +\medskip\noindent +The first thing to say is that everything in the preceding section +will continue to work more or less the same when you use "luamplib" with Lua\LaTeX. +It is designed to be backwards-compatible, so that existing \MP\ programs using "infont" and +"btex" \dots\ "etex" will continue to work without change. The only differences are: +that the "TEX()" macro is re-implemented with internal library functions +so that it no longer uses temporary files, and is therefore very much faster; and it +is easier to integrate your drawings into \LaTeX\ because you no longer need to muck +about with \mpl{verbatimtex} blocks. +So the example code shown on the right,\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip-160pt +\begin{texcode} +\documentclass[border=5mm]{standalone} +\usepackage{fontspec} +\setmainfont{TeX Gyre Pagella} % <-- note chosen font +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +\end{texcode} +\vskip-\baselineskip +\begin{smallcode} + beginfig(1); + for x = 0 upto 1: + draw (80x,16) -- (80x, -68) withcolor 3/4[red, white]; + endfor + for y = 0 upto 3: + draw (0, -20y) -- (160, -20y) withcolor 3/4[red, white]; + endfor + + string s; s = "Hand gloves"; + draw s infont defaultfont shifted (0, 0); + draw s infont "phvr8r" shifted (0, -20); + draw TEX(s) shifted (0, -40); + draw btex Hand gloves etex shifted (0, -60); + + dotlabel.urt(s, (80, 0)); + dotlabel.urt(s infont "phvr8r", (80, -20)); + dotlabel.urt(TEX(s), (80, -40)); + dotlabel.urt(btex Hand gloves etex, (80, -60)); + endfig; +\end{smallcode} +\vskip-\baselineskip +\begin{texcode} +\end{mplibcode} +\end{document} +\end{texcode}\vss}} +will produce this: +$$\includegraphics{neo-labels.pdf}$$ +Note that "defaultfont" is still "cmr10" with the encoding that has the small stroke +(that plain \TeX\ uses for the \L\ character) instead of a space, and that you can +still use PostScript fonts like "phvr8r". But also notice that the "TEX()" macro +and the "btex" \dots\ "etex" construction have picked up the font set by the \LaTeX\ +wrapper. As you can see they produce exactly the same output; "TEX()" is generally +more useful because you can pass a primary string variable as an argument, which +makes it easier to construct dynamic labels. "TEX()" also has the synonym +"textext()" for compatibility with ConTexT. You can use either name, as you prefer. + +\smallskip\noindent +But this isn't the clever bit\dots + +\clearpage +\subsection{The magic of the \texttt{textextlabel} option}\label{ttlabel} + +The clever bit is that "luamplib" allows us to turn on the "TEX()" behaviour by +default, so that you can just use plain strings with the \mpl{label()} macro, and +have them automatically processed through \LaTeX. All you have to do is add this +to the preamble: +\begin{texcode} +\mplibtextextlabel{enable} +\end{texcode} +If you add this line to the example from the previous +page,\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip-80pt +\begin{texcode} +\documentclass[border=5mm]{standalone} +\usepackage{fontspec} +\setmainfont{TeX Gyre Pagella} +\usepackage{luamplib} +\mplibtextextlabel{enable} % <--- added option +\begin{document} +\begin{mplibcode} +\end{texcode} +\vskip-\baselineskip +\begin{smallcode} + beginfig(1); + for x = 0 upto 1: + draw (80x,16) -- (80x, -68) withcolor 3/4[red, white]; + endfor + for y = 0 upto 3: + draw (0, -20y) -- (160, -20y) withcolor 3/4[red, white]; + endfor + + string s; s = "Hand gloves"; + draw s infont defaultfont shifted (0, 0)0 + draw s infont "phvr8r" shifted (0, -20); + draw TEX(s) shifted (0, -40); + draw btex Hand gloves etex shifted (0, -60); + + dotlabel.urt(s, (80, 0)); + dotlabel.urt(s infont "phvr8r", (80, -20)); + dotlabel.urt(TEX(s), (80, -40)); + dotlabel.urt(btex Hand gloves etex, (80, -60)); + endfig; +\end{smallcode} +\vskip-\baselineskip +\begin{texcode} +\end{mplibcode} +\end{document} +\end{texcode} + +\vskip 58pt +\centerline{\red{$\star$ All the examples in the rest of this section $\star$}} +\centerline{\red{assume that you have set \texttt{\textbackslash +mplibtextextlabel}}} +\vss}} +you get this output: +$$\includegraphics{neo-labels-tte.pdf}$$ +As you can see, they all come out the same. +When the magic option is enabled, "luamplib" redefines +the primitive binary operator \mpl{infont}. +Ordinarily, this command takes two strings (the text you want to show, and the name +of the font to use) and produces a picture object consisting of the text typeset in +the given font. +$$\<string> \mathbin{"infont"} \<string> \longrightarrow \<picture>$$ +With the option enabled, +the right hand \<string> argument (which names the font) is completely ignored, +and the left hand \<string> argument (the text to show) is passed to the "TEX()" +macro. The result is still a \<picture> of course, but instead +of a simple rendering in a single font, the string will have been passed through +\LaTeX, so it can include maths, bold text, or any arbitrary typesetting +constructions. + +Note that even with the option enabled, \MP\ will not let you pass a \<string> +to \mpl{draw}. You have to put \mpl{infont "somefont"} after the string to get the +magic to work; the nice thing is that the \mpl{label()} macros do this for you. + +If you experiment a bit, you will find that even though the font name argument is +completely ignored, you can't leave it out; you have to give at least an empty +string: \mpl{draw "my text" infont ""}. However if you find yourself writing this, +you probably should try \mpl{draw TEX("my text")} instead. + +\subsection{Using Unicode and matching style with OTF fonts}\label{sec:neo-otf} + +If you read "texdoc luamplib" carefully, you will see that you \textit{can} use all +these new facilities with plain Lua\TeX, but this chapter is about using them with +Lua\LaTeX, and in particular it assumes some familiarity with the packages +"fontspec" and "unicode-math" that provide complete support for Unicode and OTF +fonts; you need this familiarity in order to use "luamplib" properly. +\mpic{1cm}{unicode} + +You also need an editor that will handle Unicode. \MP\ still restricts you to using +printable ASCII in your source code, except within a string literal or a "btex" +\dots\ "etex" picture literal. So it becomes very easy to write this sort of label: +\begin{smallcode} +label("café noir £2.50", origin); +\end{smallcode} +or even whole paragraphs that use Unicode: +\begin{smallcode} +label(btex \vbox{\hsize 4in + Nous étions à l'Étude, quand le Proviseur entra, suivi d'un + \textit{nouveau} habillé en bourgeois et d'un garçon de classe + qui portait un grand pupitre. Ceux qui dormaient se réveillèrent, + et chacun se leva comme surpris dans son travail. +\par} etex, 40 down); +\end{smallcode} +But you also need a font that actually supports the Unicode characters you use. +The default Latin Modern font used by Lua\LaTeX\ has a good range for English and +most European languages, but is a bit lacking in (say) polytonic Greek. So you will +need to define a suitable font in your preamble, and turn it on in your labels. +\begin{texcode} +\usepackage{fontspec} \newfontface\polytonic{GFS Porson} +\end{texcode} +then a box like this with proper polytonic Homeric Greek source +\begin{smallcode} +label(btex \vbox{\polytonic\halign{#\hfil\cr + ... + + (polytonic greek source in UTF8 that won't show up in + the Latin Modern Typewriter font being used here) + + ... \cr}} etex, 120 down); +\end{smallcode} +will produce the first few lines of the Iliad (just in case you wanted them). +Essentially if you can produce something in \LaTeX, you can produce exactly the same +in \MP\ using "luamplib" (but see also §\ref{sec:verbatim}). + + +\newpage +\subsection{Multi-line labels} + +It is a rule of syntax in \MP\ that a string token has to be given all on one line. +So if you have very long labels, or paragraphs of text, then you have to split them +up into separate shorter string tokens: +\begin{code} +label("\vbox{\hsize 4in It is a truth universally acknowledged," + & " that a single man in possession of a good fortune," + & " must be in want of a wife.\par}", origin); +\end{code} +taking care to include the necessary spaces, which can get fiddly. +\mpic{-15mm}{multi-line-labels} + +But this is where the "btex" \dots\ "etex" construction comes into play, even with +"luamplib". As we saw in the preceding section the construction fits into the \MP\ +syntax scheme as a special pair of quotation marks that produces a \<picture>. +Unlike regular string token, a "btex" \dots\ "etex" picture token can span several +lines of source code, so you can (more easily) write long \TeX\ labels like this: +\begin{code} +label(btex \vbox{\hsize 4in + It is a truth universally acknowledged, + that a single man in possession of a good fortune, + must be in want of a wife. + \par} etex, 128 down); +\end{code} +Thanks to the backward compatibility of the implementation, this works very well +even when you have "mplibtextextlabel" enabled. + +You also have full access to your \LaTeX\ environment, so you can get tables in the +same way in \MP\ using environments like "tabular": +\begin{code} +label(btex + \begin{tabular}{c} + A way to get simple\\ + two line labels + \end{tabular} etex, 256 down); +\end{code} +But recall that whatever you ask the \mpl{TEX()} macro to typeset like this is going into +a restricted horizontal mode box; so don't try to use floating environments like +"table" or "figure". And if you want automatic paragraph wrapping, you will have +to wrap your text in a suitable "\vbox", as shown above. + +\subsection{Display maths} + +Because the \mpl{TEX()} macro typesets everything in restricted horizontal mode, you +cannot use "$$ .. $$" to create display maths directly. This is not a \TeX\ +\textit{v} \LaTeX\ issue, it is just that for compatibility with plain \MP\ (and +common sense), the designers of "luamplib" chose to typeset labels into +horizontal-mode boxes This is usually what you want. If you prefer +large integral +operators (etc) in your labels, +then you should either add +"\displaystyle" at the beginning of your formula $\longrightarrow$\vadjust{\moveright5.5in\vbox to 0pt{\hsize 4in\vss\noindent +\begin{code} +... +label("$\displaystyle \int_0^t 3x^2\, dx$", z0); +... +label("\vbox{\hsize 2in $$\int_0^t 3x^2\, dx$$}", z1); +... +\end{code} +}} +\\ +or wrap the formula in a "\vbox" with a suitable "\hsize". Using "\displaystyle" is +probably simpler. + + +\subsection{Typographical minus signs and other dynamic labels} + +This is really easy with "mplibtextextlabel" enabled, because we can assemble a string +on the fly using standard \MP\ syntax:\mpic{0pt}{simple-number-line} +\begin{code} +draw (left--right) scaled 2in withcolor 2/3 red; +for i=-4 upto 4: + dotlabel.top("$" & decimal i & "$", (32i, 0)); +endfor +\end{code} +The normal operator precedence rules ensure that the string argument to +\mpl{dotlabel} is assembled before it is passed to the \mpl{TEX()} macro. +The individual parts of the string you assemble do not have to be +valid bits of \TeX\ in themselves; they only have to make sense once they +are actually passed to the macro. With "luamplib" there are no slow +external files being used, so the complexities used above +[§\ref{sec:old-and-dynamic}] to label points around a +circle can be simplified without sacrificing speed:\mpic{-80pt}{simple-circle-labels} +\begin{code} +path C; C = fullcircle scaled 100; draw C withcolor 2/3 red; +for i=0 upto 11: + drawdot point 2/3 i of C withpen pencircle scaled dotlabeldiam; + label("$p_{" & decimal i & "}$", point 2/3 i of C scaled 1.17); +endfor +\end{code} + +\smallskip\noindent\nb But note that you can't do this string concatenation with "btex" \dots\ "etex"; +although they might appear to be special quotation marks, they produce a +\<picture>, and \mpl{&} only works with strings or paths. + +\newpage +\subsection{Drawing on an external image}\label{sec:extimage} + +One of the limitations of the way that plain "mpost" uses \TeX\ is that +any "\special" commands are removed from the intermediate file that \MP\ translates +into a \<picture> variable. Hence, in particular, you cannot use +"\includegraphics" in a \TeX\ label. Fortunately, "luamplib" removes this +limitation, so it is now possible to annotate images using the full array of \MP\ +tools. +$$\includegraphics[width=360pt]{marked-up-photo}$$ +The source is shown on the right. +\vadjust{\moveright5.5in\vbox to 0pt{\vskip-3.75in\hsize 4in\raggedright\noindent +\begin{texcode} +\documentclass[border=1mm]{standalone} +\usepackage{luamplib} +\usepackage{graphicx} +\usepackage{fontspec}\setmainfont[Scale=0.6]{Helvetica} +\mplibtextextlabel{enable} +\begin{document} +\begin{mplibcode} +\end{texcode} +\begin{smallcode} +beginfig(1); +draw btex \includegraphics[width=5in]{glenshiel.jpg} etex; +if false: +for i=1 upto 36: + draw (origin -- 180 up) shifted (10i, 0) + withcolor if i mod 10 = 0: red else: 1/2 fi; +endfor +for i=1 upto 18: + draw (origin -- 360 right) shifted (0, 10i) + withcolor if i mod 10 = 0: red else: 1/2 fi; +endfor +fi +vardef callout@#(expr t, p, o) = + save T; picture T; T = thelabel.@#(t, p+o); + draw T; drawarrow p+o -- p cutbefore bbox T; +enddef; +ahangle := 20; ahlength := 2; +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2 blue); +callout.top("Sgurr na Ciste Dubhe", (80, 96), (-10, 20)); +callout.top("Sgurr nan Spainteach", (100, 91), (6, 12)); +label.top("\tiny Cuillin Ridge, Isle of Skye", (140, 81)); +label.top("Sgurr na Carnach", (190, 90)); +label.top("Sgurr Fhuaran", (282, 94)); +label.bot("\itshape View looking west ...", + point 5/2 of bbox currentpicture shifted 4 down); +endfig; +\end{smallcode} +\begin{texcode} +\end{mplibcode} +\end{document} +\end{texcode} +\vss}} +The two loops commented out with \mpl{if false:} will add a grid +on top of the photo like this: +$$\includegraphics[width=200pt]{marked-up-photo-with-grid}$$ +This makes it easier to find the appropriate coordinates for your annotations. + +\newpage +\section{Working with pictures} + +\MP\ inherits the mechanism of \<picture> variables directly from \MF, except that +the contents of these variables are a bit more complex. The system keeps track +of the active picture in a variable called \mpl{currentpicture}, which can be copied +to your own variables, or manipulated in various useful ways. In \MF\ the contents +of the variable is a pattern of pixels for a font, in \MP\ the contents are vector +graphics commands. This section reviews some of the things you can do with a +\<picture> variable — including putting one in a frame (see §\ref{sec:pictureframe}) +like so\rlap{\ $\longrightarrow$} +\mpic{-96pt}{youth} + +\bigskip\noindent +Plain \MP\ provides two built-in \<picture> variables: \mpl{nullpicture}, which is +empty, and \mpl{currentpicture}, which accumulates the results of drawing +commands. When you compile your program, the \mpl{beginfig} macro will +set \mpl{currentpicture} to blank, and the \mpl{endfig} macro will make \MP\ write +the accumulated contents of the picture to an output file, usually as PostScript. +You can also do these things yourself at any point in a program, using these +macros from "plain.mp": +\begin{code} +def clearit = currentpicture := nullpicture enddef; +def shipit = shipout currentpicture enddef; +\end{code} + +\smallskip\noindent +When you are creating a diagram with several independent elements, it is often +helpful to save the \mpl{currentpicture} in a \<picture> variable and start again. +In fact it is so useful that plain \MP\ includes an \mpl{image} macro that uses the +magic of macro grouping to make the process a bit easier. + +\begin{code} +vardef image(text t) = + save currentpicture; + picture currentpicture; + currentpicture := nullpicture; + t; currentpicture +enddef; +\end{code} +The general idea is that you declare a variable and then save a drawing into it: +\begin{code} +picture P; P = image(...); +\end{code} +and then you have a picture element that can be manipulated or copied as needed. +The rows of decorative beads in the frame on the right were created like this. + + +\newpage +\subsection{Creating and transforming pictures} + +After you have declared a variable with \mpl{picture P;} you can give it some contents +in a number of ways: +\begin{itemize} + \item \mpl{P = nullpicture;} --- this makes $P$ empty. + \item \mpl{P = currentpicture;} --- save a copy of your current picture (if any). + \item \mpl{P = image(... MP tokens ...);} --- capture some drawing commands. + \item \mpl{P = "string" infont "font-name";} --- capture an image of "string" set in the given font. + \item \mpl{P = btex ... TeX tokens ... etex;} --- capture the result of passing some arbitrary + tokens through \TeX. + \item \mpl{P = TEX("string");} --- capture the result of passing some arbitrary + string of tokens through \TeX, using the "TEX" macro. +\end{itemize} +You can read more about the details of type setting in §\ref{sec:trad-labels} and +§\ref{sec:neo-labels}, but the point here is that the results are normal \<picture> +variables that you can manipulate and use like any other. You can apply any of the +normal \MP\ transformations to a picture, so it can be slanted, scaled, rotated, or +shifted like any \<pair> or \<path>. Each picture has a reference point that is the +position of the origin for pictures created with \mpl{image} or by saving +\mpl{currentpicture} directly, and is usually the bottom left-hand corner of a +typeset picture created by \TeX. So to add three copies of $P$ to your current +picture, you could do: +\begin{code} + for i=1 upto 3: draw P shifted (20i, 0); endfor +\end{code} +and \MP\ will add copies of $P$ with the reference points shifted to $(20,0)$, +$(40,0)$, and $(60,0)$. A selection of other transformations is shown on the right $\longrightarrow$ +\vadjust{\moveright5.5in\vbox to 0pt{\hsize4in\vskip-240pt\centerline{\includegraphics[width=4in]{twister}}\par +\centerline{\textsl{\small The reference point for each compass is the small dot in the middle.}} +\vss}} + +\smallskip\noindent +If you need to measure the size of your picture, you can get the coordinates of the +corners with the built-in corner commands, and do some arithmetic like this: +\begin{code} + (wd, ht) = urcorner P - llcorner P; +\end{code} +You also get \mpl{ulcorner}, \mpl{lrcorner}, and \mpl{center}; plus \mpl{bbox} which +returns the rectangular path round the four corners, expanded by the current value +of \mpl{bboxmargin}. (See also §\ref{sec:rotated-boxes}). + +\newpage +\subsection{Clipping and bounding boxes} + +Once you have got your \<picture> variable, and possibly transformed it, the main +thing you can do with it is to use \mpl{draw} to add it to the current picture. But +there are two other commands that are sometimes helpful that allow you to alter the +apparent size of the picture. +\begin{itemize} + \item $\kw{setbounds}\ \<picture> \kw{to}\ \<path expression>$ + \item $\kw{clip}\ \<picture> \kw{to}\ \<path expression>$ +\end{itemize} +Both commands set the boundary of your picture to the arbitrary path expression, and +then the \mpl{clip} command also erases all of the picture that lies outside the +boundary. +(Note that this is not the same as setting the bounding box. The arbitrary path does +not have to be a rectangle; after either of these commands the bounding box will be +the rectangle that fits around the arbitrary path). + +\MP\ inherits the \mpl{clip} command from PostScript; there is no equivalent in \MF. +It can be useful as an alternative to \mpl{buildcycle}, but it is most commonly used +for trimming a repeating pattern to a particular shape. The usual approach is to +define a particular shape, $s$, then draw your pattern over a large area that covers +the shape, and finally call \mpl{clip currentpicture to s} to trim the pattern to +the shape. This is best done after you define all your shapes, but before you draw +any others or add any labels. The example on the right shows this approach in +action. + +You can apply \mpl{clip} to any picture, so you might prefer to capture your +pattern in \<picture> variable with \mpl{image}, apply \mpl{clip} to that, and +then \mpl{draw}. This works nicely if you want to repeat the clipped image. +\vadjust{\moveright 5.5in\vbox to 0pt{\hsize 4.2in \vss +\noindent\includegraphics[width=4in]{shady-circles} + +\medskip +\begin{smallcode} +path c, C; numeric r; r = 60; +c = fullcircle scaled 2r shifted (-r, 0); +C = fullcircle scaled 4r shifted (2r, 0); + +numeric t, u; +(t, whatever) = C intersectiontimes + C shifted (center c - center C); +(u, whatever) = c intersectiontimes (point t of C -- center c); + +path s; +s = subpath (0, u) of c -- subpath (t, 4) of C -- cycle; +for i=0 upto r: + draw (left--right) scaled 2r rotated -42 shifted (3i, 0) + withpen pencircle scaled 1/4 withcolor 2/3 blue; +endfor +clip currentpicture to s; + +draw c; draw C; +draw center c -- center C -- point t of C -- cycle; +\end{smallcode} +\vskip -126pt}} + +You can also use this technique to fill with a gradient: just reduce the gap +between each line and use the index variable to blend between two colours. +Something like \mpl{withcolor (i/r)[blue, white]} in the example shown. + +\vfill\noindent +\hey Why would you ever want to use \mpl{setbounds}? Mainly to help with aligning type set labels, +as discussed in §\ref{infontbbox}, or if you want to make boundaries of different +picture elements consistent in order to line them up more easily. Or perhaps to set +a margin for the whole image by using something like this just before the +\mpl{endfig}. +\begin{code} +setbounds currentpicture to bbox currentpicture; +\end{code} +This has the effect of adding a \mpl{bboxmargin} wide strip all round. + +\newpage +\subsection{Bounding boxes of transformed pictures}\label{sec:rotated-boxes} + +When you rotate a text label, or otherwise transform a picture, the corner-points +also change, but not quite in the way you might think. It turns out that +the \mpl{bbox} is always a rectangle aligned to the edges of your page. +Effectively, the corners are determined \textit{after} any transformation, +and the \mpl{center} is strictly the intersection of the lines between opposite +corners.\mpic{-68pt}{corners} + +You will notice this if you use the technique given on p.\thinspace29 of the \MP\ manual to draw a +label on a coloured (or erased) background; if you have rotated the label, the +\mpl{bbox} may be larger than you want. One solution is to define your label +untransformed and then apply the transformation twice: first when you fill the bounding box +and again when you draw the label, for example: +\begin{code} +picture p; p = thelabel.top("Correctly", origin); +unfill bbox p rotated 30 shifted z0; +draw p rotated 30 shifted z0; +\end{code} + +\vfill +\subsection{Using pictures to assemble a complex diagram} + +If you have a diagram with several independent parts, like the comparison above then +there is a useful general technique: declare a subscripted \<picture> variable, and +then use \mpl{image} to draw each part separately. The advantage of this is that +you do not have to worry about where the \mpl{origin} is, which often makes a +drawing simpler (for example because you can use \mpl{rotated} rather than +\mpl{rotatedaround}). Once you have created all the parts you can then add them to +the final image using \mpl{draw} as shown on the right $\longrightarrow$ +\vadjust{\moveright 396pt\vbox to 0pt{\hsize 4in \vskip -81pt +\noindent The illustration above was drawn using this general sub-picture +technique, approximately like this: +\begin{code} +picture P[]; +P1 = image( + % first drawing... +); +P2 = image( + % second drawing... +); +draw P1 shifted 100 up; draw P2 shifted 100 down; +\end{code} +\vss}} + +Sometimes it is more convenient to use \mpl{label} to place the pictures, taking +advantage of the automatic alignment provided. Note also that, unless you have +explicitly filled them with \mpl{white} colour, the blank parts of each picture are +really transparent so you can overlap them when appropriate. + +\newpage +\subsection{Adding a caption to the current picture} + +When you have finished a complicated picture, you may want +to add a caption or some other label which would look neat if it +were exactly centred at the bottom of the everything else. You could +keep track of exactly how wide and deep you have made the picture to do this, but +there is an easier way, that will adjust itself automatically if you change the +contents of the picture later. +\begin{code} +beginfig(1); +% ... complete drawing that needs a caption ... +label.bot("This picture needs a label at the bottom", + point 1/2 of bbox currentpicture); +endfig; +\end{code} +Through the automatic alignment routines in \mpl{label}, this +will produce a label neatly centred at the bottom. If it is too close +you can either set a larger \id{bboxmargin} or use something like: +\begin{code} +label.bot("This picture needs a label at the bottom", + point 1/2 of bbox currentpicture shifted 42 down); +\end{code} +Note that in both cases the addition of the label will move +the corner points of the picture, so that the bounding box +will have been expanded to include the new label. You can use this feature to +add a series of centered labels. +But if this is not what you want, perhaps because you want to add two labels +side by side at bottom, then you can “freeze” the current bounding box like this: +\begin{code} + picture bb; bb = bbox currentpicture; + label.bot("Left label", point 1/4 of bb); + label.bot("Right label", point 3/4 of bb); +\end{code} + +\noindent +The path returned by \mpl{bbox} has four points starting at the lower left and +proceeding clockwise like a \mpl{unitsquare}. +So \mpl{point 1/2 of bbox currentpicture} is half way between lower left and lower +right, while \mpl{point 5/2 of bbox currentpicture} is half-way from upper right to +upper left. + +Note that the path is defined even if the current picture is empty. If you call +\mpl{bbox currentpicture} at the start of a picture you will get a square path +centered on the origin and scaled to $2 \id{bboxmargin}$. + +\moveright5.5in\vbox to 0pt{\hsize 4in \vss +\noindent +Here is an example. +$$\includegraphics[width=4in]{double-angle}$$ +The labels at the bottom were added like this: +\begin{smallcode} +label.bot("$\triangle ACD \sim \triangle ABC$", + point 1/2 of bbox currentpicture shifted 24 down); + +path bb; bb = bbox currentpicture shifted 12 down; +label.bot(btex \vbox{....} etex, point 1/4 of bb); +label.bot(btex \vbox{....} etex, point 3/4 of bb); +\end{smallcode} +The second call to \mpl{bbox currentpicture} gets the bounding box +that includes the first centered label.} + + +\newpage +\subsection{Drawing pictures with various colours and pens} + +Consider the \<picture> with different colours and pens in the example here +$\longrightarrow$ +\vadjust{\moveright5.5in\vbox to 0pt{\vskip +-1.5\baselineskip +\begin{smallcode} +numeric s; s = 21; +path alpha; +alpha = ((-2s, s) {right} + .. halfcircle rotated -90 scaled 2s shifted (2s, 0) + .. {left} (-2s, -s)) shifted (s*left); + +vardef overdraw(expr a, b, r, P, shade) = + linecap := butt; + undraw subpath (a+r, b-r) of P withpen pencircle scaled 2; + draw subpath (a, b) of P withcolor shade; +enddef; + +picture cb; cb = image( + draw alpha withcolor 2/3 red; + undraw alpha rotated 180 withpen pencircle scaled 2; + draw alpha rotated 180 withcolor 2/3 blue; + overdraw(0.21, 0.36, 0.02, alpha, 2/3 red); + overdraw(0.67, 0.86, 0.02, alpha, 2/3 red); + overdraw(3.4, 4.3, 0.1, alpha, 2/3 red); + overdraw(5.4, 5.6, 0.02, alpha, 2/3 red); + overdraw(5.4, 5.6, 0.02, alpha rotated 180, 2/3 blue); +); +\end{smallcode} + +\vskip 48pt\noindent +\begin{minipage}{4in} +\begin{itemize} + \item Example 1 shows that by default \mpl{draw} uses the colours and pens + defined in the picture + + \item Examples 2, 3, and 4 show what happens if you change the pen, or the + colour, or both. + + \item Example 5 shows you how to make a bagel in \MP. + + \item Example 6 shows you the slightly tricky syntax to extract the paths, pens, + and colours from the \<picture> and adjust them as needed. +\end{itemize} +\end{minipage} +\vss}} +With this captured in a \<picture> variable, you can \mpl{draw} it with +different colours and pens to obtain a variety of effects: +$$\includegraphics[width=\textwidth]{draw-picture.pdf}$$ +The picture is supposed to represent a fancy knot (a “Carrick bend”), and to show +the red and blue strands crossing each other. The \mpl{overdraw} macro tries to do +this by undrawing with a thick pen, then redrawing the upper strand on top. + +\newpage +\subsection{Simulating transparency} + +Filling with transparent colour can sometimes be a very effective graphic technique, but the +underlying technical implementation is often dauntingly complex. Plain \MP\ +provides no colour model that directly supports transparency for any output format, +so if you need it you will have to resort to layering and managing the colour +blending yourself. This page presents an example of the basic technique, that can +be adapted to more general purpose macros as required. The technique is included in +this section, because it involves more manipulation of \<picture> variables. +\vadjust{\moveright5.5in\vbox to 0pt{\vskip -128pt +$$\includegraphics[width=3in]{fake-transparency}$$ +Omitting the simple grid, this drawing was produced like this: +\begin{smallcode} +% Large A +label.urt("A" infont defaultfont scaled 8, origin) withcolor 1/4 green; +% An arbitrary shape +path shape; shape = (superellipse(right, up, left, down, 0.81)) + shifted 1/2 right scaled 30 rotated 30; +% Parameters +alpha = 5/8; % alpha: 0=invisible, 1=opaque +color filler; filler = .95[red,white]; +picture fg, bg; +bg = currentpicture; % capture the current drawing +fg = image( + for e within bg: % redraw everything in blended color + draw e withcolor alpha[colorpart e, filler]; + endfor % and add some decorations + draw shape withpen pencircle scaled 2 withcolor 3/4; + draw subpath (2.718, 3.1415) of shape + shifted - center shape scaled 7/8 shifted + center shape + withpen pencircle scaled 2 withcolor white; +); +clip fg to shape; % now clip the fg drawing to the shape +fill shape withcolor filler; % fill the shape +draw fg; % and put the fg drawing on top +draw shape withcolor 3/4 blue; % make a nice edge +\end{smallcode} +\vss}}% +The two useful tools in the plain \MP\ kit bag are: +\begin{itemize} + \item The ability to loop through all the elements of a picture + \item The ability to blend colours using the mediation syntax +\end{itemize} +The example drawing on the right consists of a regular grid and a text picture, with +a bubble drawn over the top. The bubble can be made to look transparent like this: +\begin{enumerate} + \item Define the shape that you want to be transparent, decide on how opaque + you want it, and the colour to use. + \item Capture the current drawing in a \<picture> variable. + \item Loop over all the elements in that picture, redrawing each one with + a blended color, and capture all this in another \<picture>. + \item Add some decoration; here there is an internal margin, and a hint of a + reflection line to make it look shiny. + \item Clip the new blended-colour picture to the shape. + \item Fill the shape with the filler colour. + \item Draw the blended-colour parts on top. + \item Finally, add a neat edge (if needed). +\end{enumerate} +As you may appreciate, with this approach, you need to do the transparent parts +after you have drawn everything else in your drawing. + +\newpage +\subsection{Adding a background and other post-processing}\label{backgrounds} + +The \<picture> capture technique provides a simple way to add a background or do +other post-processing on your drawing. The advantage is that you do not have to +work out the size of your drawing before you start. + +\moveright5.5in\vbox to 0pt{\vskip -43pt +Here is an example that adds graph paper behind +a drawing. +$$\includegraphics{graph-paper}$$ +\begin{smallcode} +input colorbrewer-rgb +path C; C = fullcircle scaled 120 shifted 12 up rotated 6; +for t=0,1.2,4: draw center C -- point t of C withcolor Blues 8 8; endfor +draw C withcolor Reds 8 8; dotlabel.urt("Start", point 0 of C); + +vardef grid(expr ll, ur, grid_unit) = + save llx, lly, urx, ury, G; numeric llx, lly, urx, ury; + (llx, lly) = ll; (urx, ury) = ur; + picture G; G = image( + for x = floor(llx / grid_unit) + 1 upto floor(urx / grid_unit): + draw (x * grid_unit, lly) -- (x * grid_unit, ury); + endfor + for y = floor(lly / grid_unit) + 1 upto floor(ury / grid_unit): + draw (llx, y * grid_unit) -- (urx, y * grid_unit); + endfor + fill fullcircle; % <-- show the origin + ); G enddef; + +picture P; P = currentpicture; currentpicture := nullpicture; +drawoptions(withpen pencircle scaled 1/4); +draw grid(llcorner P, urcorner P, 1mm) withcolor Blues 8 1; +draw grid(llcorner P, urcorner P, 10mm) withcolor Blues 8 2; +drawoptions(); +draw P; +\end{smallcode} +\vss} + +\smallskip\noindent +You could add a subtle off-white background fill like this: +\begin{smallcode} +picture P; P = currentpicture; fill bbox P withcolor (1,1,31/32); draw P; +\end{smallcode} +Or you can be more ambitious, as shown in the example on the right +\rightarrowfill\break +In general, you draw any background you want, like this: +\begin{smallcode} +picture P; P = currentpicture; currentpicture := nullpicture; +% do complex background drawing... +clip currentpicture to bbox P; draw P; +\end{smallcode} +Or you can do things like make automatic adjustments to the scale. If you wanted +to be sure that your drawing was not more than 5 inches wide, you could try this +just before the \mpl{endfig}: +\begin{smallcode} +numeric wd; wd = xpart (urcorner currentpicture + - llcorner currentpicture); +if wd > 360: currentpicture := currentpicture scaled (360/wd); fi +\end{smallcode} + +\bigskip\noindent +If you wanted to apply one of these changes to all the figures in your "mpost" input file then +you can use the hook provided by plain \MP: +\begin{smallcode} +extra_endfig := "picture P; P = currentpicture;" & + "fill bbox P withcolor (1,1,31/32); draw P;"; +\end{smallcode} +The definition of \mpl{endfig}, includes the line \mpl{scantokens extra_endfig;} so +that any contents of the string variable \mpl{extra_endfig} are automatically +processed before the figure is produced. +If you are using "luamplib" then you can use the alternative hook that it provides +so that you do not even have to type \mpl{endfig}: +\begin{smallcode} +\everyendmplib{picture P; P = currentpicture; +fill bbox P withcolor (1,1,31/32); draw P; endfig;} +\end{smallcode} + +\newpage +\subsection{Adding a ruler} + +\textsc{If you wish} to check the dimensions of your drawing, it can be useful to +add a temporary ruler that shows you the dimensions of the bounding box like this: +$$\includegraphics{icosahedron}$$ +The red rulers were added by putting \mpl{input ruler-cm} at the end of the figure. +\vadjust{\moveright5.5in\vbox to 0pt{\vss\hsize 4in\raggedright\noindent +Here is the implementation of "ruler.mp": +\begin{code} +% add a ruler along the left hand and lower edges +% of the bounding box of the currentpicture +path B; B = bbox currentpicture; +for s=0, 1: + path p; numeric a; pair o; + p = subpath (0, 1) of if s=0: reverse fi B; + a = arclength p; + o = if s=0: left else: down fi; + for i=0 upto 3: + exitif not known u[i]; + for j=0 upto floor(a/u[i]): + pair t; t = point arctime j*u[i] of p of p; + draw (origin -- (6 - 2i) * o) shifted t; + if i=0: label(decimal j, t shifted 12 o); fi + endfor + endfor + draw p; +endfor +\end{code} +The inner loop draws successively shorter lines at each of the +minor units, and numbers at the major units. +\vss}} +They are drawn round the bounding box, set here with the default +margin of 2\unit{bp}. +The "ruler-cm.mp" file looks like this: +\begin{code} +numeric u[]; u0 = 1 cm; u1 = 1 mm; +drawoptions(withcolor 0.54 red); +input ruler +drawoptions(); +\end{code} +and there is a companion "ruler-inch.mp" file that looks like this: +\begin{code} +numeric u[]; u0 = 1 in; u1 = 1/4 in; u2 = 1/12 in; +drawoptions(withcolor 3/4 blue); +input ruler +drawoptions(); +\end{code} +The idea is that you set a subscripted variable \mpl{u[]} to a number of unit sizes +where you want markers and then call \mpl{input ruler}. + + +\newpage +\subsection{Adding a border}\label{pics-border} + +\textsc{In most documents} the drawings look just fine without any decoration, but sometimes +you might want to add emphasis or pick out part of a drawing. The examples here can +be applied to \mpl{currentpicture} or any other \<picture> +variable.\vadjust{\moveright 396pt\vbox to 0pt{\hsize 4in\vskip -42pt +$$\includegraphics{border.pdf}$$ +\begin{smallcode} +% don't take this one too seriously... +vardef rope expr c = +save hemp, s, w, n, A, a, b; +color hemp; hemp = 1/256 (224, 202, 169); +numeric s, w, n, A; +A = arclength c; s = A/floor(A/2); w = -1; n = -1; +path a[]; for t=0 step s until A + 1: a[incr n] = + (0,+w) rotated angle direction arctime t-3/2s of c of c + shifted point arctime t-3/2s of c of c +.. (0,+w) rotated angle direction arctime t-1/2s of c of c + shifted point arctime t-1/2s of c of c +.. (0,-w) rotated angle direction arctime t+1/2s of c of c + shifted point arctime t+1/2s of c of c +.. (0,-w) rotated angle direction arctime t+3/2s of c of c + shifted point arctime t+3/2s of c of c; +endfor +image(for i=1 upto n: + path b; b = buildcycle(a[i-1], reverse a[i]); + fill b withcolor hemp; + draw b withpen pencircle scaled 1/8; +endfor) enddef; +interim bboxmargin := 16; +draw rope rounded_corners bbox currentpicture; +\end{smallcode}\vss}} + +$$\vbox{\halign{$\vcenter{\hbox{\includegraphics[width=150bp]{#}}}$&\quad$\vcenter{\hbox{#}}$\cr +border-shadow& +\begin{smallcode} +picture P; P = currentpicture; +fill bbox P shifted (3,-3) withcolor 3/4; +unfill bbox P; draw bbox P; +draw P; +\end{smallcode}\cr +border-dashed& +\begin{smallcode} +vardef rounded_corners expr p = + for i=1 upto length p: + subpath (i-15/16, i-1/16) of p .. + endfor cycle +enddef; + +interim bboxmargin := 16; +draw rounded_corners bbox currentpicture + dashed evenly scaled 3/4; +\end{smallcode}\cr +}}$$ + +\newpage +\subsection{Adding a frame}\label{sec:pictureframe} + +As promised at the start of this section, here is the code for the picture +frame drawn round Raphael's young man. +\begin{code} +input picture_frame +beginfig(1); + picture F; + F = thelabel(TEX("\includegraphics{youth.jpg}"), origin); + draw F; draw frame bbox F; +endfig; +\end{code} +All the heavy lifting is done by "frame" macro defined in +"picture_frame.mp" $\longrightarrow$ +\vadjust{\moveright5.2in\vbox to 0pt{\hsize 4in\vskip-184pt +\begin{smallcode} +vardef frame expr P = + save base, side, f, t, u, xx; + picture base, side; path f; numeric t, u, xx; + % work out some measurements + t = arclength subpath (0,1) of P; + u = arclength subpath (1,2) of P; + xx = max(t, u) + 2 pf_wd; + f = unitsquare xscaled xx yscaled pf_wd; + % convenience / nonce function + vardef paint_strip(expr y, wd, shade) = + draw subpath (0, 1) of f + shifted (0, if y < 0: pf_wd + fi y) + withpen pencircle scaled wd + withcolor shade + enddef; + base = image( + % background colour + fill f withcolor gold; + % grey strips + paint_strip(2, 3, 5/4 grey); + paint_strip(3.5, 1/4, grey); + paint_strip(5, 1/4, 1/2[gold, dark]); + paint_strip(-6.5, 1/4, 1/2[gold, dark]); + paint_strip(-6, 1/4, 1/2[gold, dark]); + paint_strip(-2, 2, 5/4 grey); + % spatter with random spots + for i=0 upto 4 * arclength(subpath (0,1) of f): + fill fullcircle scaled uniformdeviate 3/4 + shifted (uniformdeviate xx, uniformdeviate pf_wd) + withcolor dark; + endfor + % decorative balls + for x = 2 step 3 until xx: + draw ball shifted (x, 2); + endfor + ); + % make two trapezium shapes + side = base; + clip side to (pf_wd, 0) -- (pf_wd + u, 0) + -- (2 pf_wd + u, pf_wd) -- (0, pf_wd) -- cycle; + clip base to (pf_wd, 0) -- (pf_wd + t, 0) + -- (2 pf_wd + t, pf_wd) -- (0, pf_wd) -- cycle; + % arrange the pieces around path P + image( + draw base rotated 180 shifted point 1 of P shifted (+pf_wd, 0); + draw base rotated 0 shifted point 3 of P shifted (-pf_wd, 0); + draw side rotated 90 shifted point 0 of P shifted (0, -pf_wd); + draw side rotated 270 shifted point 2 of P shifted (0, +pf_wd); + ) +enddef; +\end{smallcode} +\vss}} + +\noindent This macro also needs some colours: +\begin{smallcode} +color gold, dark, grey; +gold = 1/256(243, 197, 127); +dark = 1/256(144, 87, 50); +grey = 1/256(156, 147, 138); +\end{smallcode} +a picture of a small silvery-gold ball: +\begin{smallcode} +picture ball; ball = image(for i=0 upto 16: + fill interpath(i/16, + fullcircle scaled 10, + fullcircle scaled 3 shifted (-2, 2) + ) withcolor (i/16)[gold, 15/16 white]; +endfor) scaled 1/4; +\end{smallcode} +and an internal variable that defines the width of the frame: +\begin{smallcode} +newinternal pf_wd; pf_wd := 21; +\end{smallcode} +The macro takes a rectangular path $P$ as an argument, and +makes a thin rectangle $f$ that is scaled to the desired width and +the longer of the two sides of the path. This thin rectangle is then +decorated with background colour, strips of colour to suggest depth, a random +spatter-pattern, and a row of little balls. The macro then makes two trapezium +shaped copies of the decorated rectangle, pieces them together around $P$, and returns the result as a +\<picture>. + +\newpage +\section{Annotations} + +\textsc{In some awkward corners}, you may find that you just can't get your label in +the right place with \mpl{dotlabel} even if you adjust \mpl{labeloffset}. In these +cases there are two simple techniques you can use. First, you could separate +drawing the dot from placing the label; given a point $P$ you can try:\mpic{0pt}{callout} +\begin{code} +drawdot P withpen pencircle scaled dotlabeldiam; +label("$P$", P shifted 10 dir 68); +\end{code} +Using \mpl{dotlabeldiam} ensures that your dots match any others done with +\mpl{dotlabel}. +Secondly, if that's not enough, use a temporary pair to create a call out line: +\begin{code} +z0 = P + 20 dir -20; +draw z0 -- P + cutafter fullcircle scaled 8 shifted P + withpen pencircle scaled 1/4; +label.rt("\textit{pole}", z0); +\end{code} +If you want to do this sort of thing often, then it might be worth making a macro. +It is hard to write anything completely general, but see §\ref{sec:extimage} for an +example. + +\medskip\noindent +\textsc{You might also want} to mark a straight line between two points. +\vadjust{\moveright5.3in\vbox to 0pt{\hsize 4.2in +\begin{smallcode} +vardef do_brace(expr a,b,m,r) = + save d, e, n, bb; numeric d, n; pair e; path bb; + n = 1/2 m; d = angle (b-a); + e = up scaled m rotated d shifted r[a,b]; + bb = ((origin {0,n} .. {right} (abs n,n)) + rotated d shifted a -- + ((-abs n,-n){right} .. {0,n} origin {0,-n} .. {right}(abs n,-n)) + rotated d shifted e -- + ((-abs n,n){right} .. {0,-n} origin) + rotated d shifted b + ) shifted (up scaled n rotated d); + draw bb withpen pencircle yscaled .6 xscaled .1666 rotated d; + point 3 of bb +enddef; + +label.lrt("Here",do_brace(z1, z2, -12, 3/4)); +\end{smallcode} +Note that, as well as drawing the braces, the macro uses the grouping provided by \mpl{vardef} to return the +mid point so that you can put a label next to it. +\vss}} +The simplest way to do this is just to use \mpl{drawdblarrow} on a copy of your straight +path shifted to one side, like so: +\begin{code} +drawdblarrow (z1--z2) shifted (12 up rotated angle (z2-z1)); +\end{code} +If you combine this with temporarily setting \mpl{ahangle:=180}, you get the simple +dimension line shown in blue. +$$\includegraphics{braces}$$ +The red braces are a more complex variation on this theme $\longrightarrow$ + +\newpage +\section{Line caps and line joins} + +The PostScript language defines parameters that affect how the ends of each line are +drawn and how lines are joined together. Plain \MP\ provides access to these +parameters through internal variables called "linecap" and "linejoin"; it sets both +of them to the value "rounded" at the start of each job. +\mpic{0pt}{line-caps-and-joins} + +The figure on the right shows the affect of the different settings, using an +exaggerated line width of 2 points (instead of the usual 0.5 points). Some +observations +to note: +\begin{itemize} + \item When $\id{linecap}=\id{squared}$ then \kw{drawdot} produces diamond-shaped + dots, even when you are drawing with the default circular pen. + + \item When $\id{linecap}=\id{butt}$ then \kw{drawdot} produces invisible dots. + They still count towards the bounding box of the picture but there's no mark + on the page. + + \item When $\id{linecap}=\id{squared}$ then \kw{drawarrow} produces some + unpleasant results; even when $\id{linejoin}=\id{mitered}$, you can still + see small jaggies on the slopes of the arrows. + + \item The arrows are nice and sharp when + $\id{linejoin}=\id{mitered}$, but they over shoot the mark slightly. + + \item If you zoom in, you can see the effect of $\id{linejoin}$ on the corners of the grey box + as well as on the arrow heads, but you might not notice the difference when + the picture is printed unless you have a very high resolution printer. + + \item + This drawing was done with \mpl{pencircle scaled 2}, so that the dots + would be easy to see. This does make the arrows drawn with the + default line modes (rounded caps and rounded joins) looks a bit fat; + they look better with the usual \mpl{pencircle scaled .5}. + +\end{itemize} + +There is one more PostScript parameter affecting line joins. \MP\ makes it available as "miterlimit" +and it affects how much a mitered join is allowed to stick out at each corner. Plain \MP\ sets +"miterlimit=10;" which is correct for nearly all drawings. If you set +"miterlimit:=0;" then the mitered line join mode becomes more or less the same as the +beveled mode. + +\newpage +\section{Plotting functions}\label{func} + +\textsc{A selection} of graphs of mathematical functions is presented in this +section, taken from real examples collected over several years. For data +visualizations, see §\ref{dviz}. As ever in this document, the focus is on plain +\MP; the plain format provides no built-in facilities for graphs so you have to do +everything from scratch; but on the other hand there are no new macros or commands +to learn, you get full control of what goes on the page, and you will not spend +hours scratching your head wondering how to adjust the axis labels. + +\subsection{Making axes} + +You can start by drawing a simple set of axes.\mpic{-124pt}{plain-axes} +\begin{smallcode} +path xx, yy; +xx = (left -- right) scaled 130; +yy = (down -- up) scaled 80; +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); +\end{smallcode} +Here the axes are scaled arbitrarily to 130\,pt and 80\,pt, but you will probably +find it useful to set consistent units, and express sizes in terms of them. Purely +from habit, I use $u$ for the horizontal unit and $v$ for the vertical unit. This +makes it more convenient when you want to add a grid and/or a number +scale.\mpic{-16pt}{numbered-axes} + +\vskip -12pt +\vbox to 0pt{ +\begin{smallcode} +numeric u, v; u = 40; v = 29; +path xx, yy; +xx = (3/2 left -- 5 right) scaled u; +yy = (3/2 down -- 4 up) scaled v; +for x=-1, 1, 2, 3, 4: + draw yy shifted (x * u, 0) withcolor 7/8; % grid + draw (down--up) shifted (x * u, 0); % ticks + label("$" & decimal x & "$", (x * u, -8)); +endfor +for y=-1, 1, 2, 3: + draw xx shifted (0, y * v) withcolor 7/8; % grid + draw (left--right) shifted (0, y * v); % ticks + label("$" & decimal y & "$", (-10, y * v)); +endfor +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); +\end{smallcode}\vss} + +\newpage +\subsection{Drawing linear functions} + +\textsc{For simple} linear graphs, you just need to define two points and draw a +line between them; it is tempting to try to make some generalized macro to do this, +but it is hard to make something completely general, so for most graphs it is easier +just to specify two points and use \mpl{draw}; often it is handy to making your line +longer than you need, then trim it using \mpl{cutbefore} and/or \mpl{cutafter} so +that it fits neatly. +\vadjust{\moveright 384pt\vbox to 0pt{\hsize 4in\vskip -100pt\begin{smallcode} +numeric u, m, m', b, b'; +u = 1.44cm; +b = 3.6u; b' = b + 1/2 u; +m = -1; m' = 3/4 m; + +path xx, yy; +xx = (left -- 5 right) scaled u; +yy = xx rotated 90; + +numeric minx, maxx; path ff, gg; +minx = xpart point 1/16 of xx; +maxx = xpart point 15/16 of xx; +ff = (minx, minx * m + b) -- (maxx, maxx * m + b); +gg = (minx, minx * m' + b') -- (maxx, maxx * m' + b'); + +z0 = point 0.4 of ff; +z1 = point 0.54 of ff; +z1 0 = whatever [point 0 of gg, point 1 of gg]; x1 0 = x0; +z1 1 = whatever [point 0 of gg, point 1 of gg]; x1 1 = x1; + +forsuffixes @=0, 1: + draw (x@, 0) -- z@ -- (0, y@) dashed evenly scaled 3/4; + draw z@ -- z1 @ -- (0, y1 @) dashed withdots scaled 1/2; + label.bot("$x_{" & decimal @ & "}$", (x@, 0)); + label.lft("$y_{" & decimal @ & "}$", (0, y@)); + label.lft("$y'_{" & decimal @ & "}$", (0, y1 @)); +endfor +draw ff withcolor 2/3 red; +draw gg withcolor 3/4 blue; +drawarrow xx; drawarrow yy; + +label.rt("$x$", point 1 of xx); +label.top("$y$", point 1 of yy); + +dotlabel.urt("$b$", (0, b)); +dotlabel.urt("$b'$", (0, b')); + +draw thelabel("slope: $m=" & decimal m & "$", 7 up) + rotated angle (1, m) shifted point 2/3 of ff; +draw thelabel("slope: $m'=" & decimal m' & "$", 7 up) + rotated angle (1, m') shifted point 2/3 of gg; +\end{smallcode}\vss}} +$$\includegraphics[width=0.84\textwidth]{linear-graph.pdf}$$ +But in this example it was easier to calculate them using $y=mx+b$. +Note also that spaces are allowed in suffixes, which makes the loop a bit simpler. + +\newpage +In this second example of a linear function, the emphasis of the diagram was on +the angles at the $x$-axis made by the two lines, so the lines were defined using +\mpl{rotated}, \mpl{shifted}, and \mpl{cutbefore} instead. +\vadjust{\moveright 384pt\vbox to 0pt{\hsize 4in\vskip -24pt\begin{smallcode} +path ell, tee, arc; +ell = (left--right) scaled 10u rotated theta shifted P cutbefore xx; +tee = (left--right) scaled 10u rotated psi shifted P cutbefore xx; +arc = subpath (1.6, 3.2) of halfcircle rotated -180 shifted 1/2 up + scaled 10u shifted P rotatedabout(P, psi); +% the rest of the points then follow... +Q = ell intersectionpoint subpath (1.5, 3) of arc; +H = P + whatever * dir psi; +xpart H = xpart Q = xpart M; ypart M = ypart N = 0; +xpart N = xpart P; +L = point 0 of ell; +T = point 0 of tee; +% now get on with the drawing +draw arc dashed withdots scaled 1/4 withcolor 1/2 red; +draw P--N dashed withdots scaled 1/2; +draw Q--M dashed withdots scaled 1/2; +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); +drawoptions(withcolor 2/3 blue); +draw fullcircle scaled 32 shifted T cutafter tee; +draw fullcircle scaled 28 shifted L cutafter ell; +label("$\psi$", 24 right rotated 1/2 psi shifted T); +label("$\theta$", 20 right rotated 1/2 theta shifted L); +drawoptions(); +% trim the function lines neatly +path boundary; +z1 = point .95 of xx; +z2 = point .95 of yy; +boundary = z1--(x1,y2)--z2; +draw ell cutafter boundary; +draw tee cutafter boundary; +% and finally label the points. +label.bot("$T$", T); +label.bot("$L$", L); +label.bot("$N$", N); +label.bot("$M$", M); +dotlabel.ulft("$P$", P); +dotlabel.lrt("$Q$", Q); +dotlabel.lrt("$H$", H); +\end{smallcode}\vss}} +$$\includegraphics[width=\textwidth]{angles-lines.pdf}$$ +The lines are both trimmed to a convenient path (\id{boundary}) when they are +drawn.\vadjust{\moveright296pt\vbox to 0pt{\vskip 80pt\begin{mplibcode}beginfig(0);drawarrow +origin {right} .. {dir 70} 42 dir 60; + label.lft("\small \dots continues", origin); +endfig;\end{mplibcode}\vss}} + +\begin{smallcode} +numeric u; u = 1cm; + +path xx, yy; +xx = (2 left -- 10 right) scaled u; +yy = (down -- 7 up) scaled u; + +numeric theta, psi; psi = 28; theta = 50; + +pair P, Q, H, N, M, T, L; +P = (4u, 3u); +\end{smallcode} + + +\newpage +\subsection{Making curves for functions with a loop} + +\textsc{To plot a function} you can construct a suitable path using an in-line +\kw{for} +loop like this: +\begin{code} +vardef f(expr x) = x ** 2 enddef; +path ff; +ff = (for x = minx step s until maxx - s: + (x, f(x)) .. + endfor (maxx, f(maxx))) xscaled u yscaled v; +\end{code} +provided you have first defined variables \id{minx} and \id{maxx} to represent the +domain of $x$, and worked out appropriate values for horizontal and vertical +units, $u$, and $v$ so that the range of $f(x)$ fits neatly on your graph. + +The loop +above also uses a variable $s$ to control the number of points used to define the +path.\mwpic{-180pt}{func-powers} +The figures on the right show that over the domain $-3$ to $3$ a step of $\frac12$ +gives enough points for \MP's Bezier curve fitting routines to draw the functions +$x^2$ and $x^3$ accurately, but that you need a step size of $\frac18$, and hence +four times as many points, +for $x^6$ and $x^7$. On modern machines it does not really hurt to calculate +dozens of points, but a step size that generates 1000s of points will be slow to +compile. + +\bigskip\noindent +There are two other techniques to improve the shape of the curve produced by these loops: you can increase +the tension between each point by using \mpl{...} or \mpl{--} instead of \mpl{..} in +the loop; and +if you know how to differentiate your function, you can add a direction at each +step using the $\{\<pair>\}$ syntax: +\begin{code} +vardef f(expr x) = x ** 2 enddef; +vardef fp(expr x) = 2x enddef; % NB "fp" because "f'" is illegal +path ff; +ff = (for x = minx step s until maxx - s: + (x, f(x)){1, fp(x)} .. + endfor (maxx, f(maxx))) xscaled u yscaled v; +\end{code} + +\vfill +\noindent +\hey However in general it is simpler just to increase the number of samples by making a +smaller step size. + +\newpage +\subsection{Making curves for functions from path pieces}\label{sec:pathpieces} + +\textsc{In some situations}, you might find it easier to stich together various +\<path> pieces to make your curve. This can be especially elegant if there is a +symmetry in the path. For example:\mwpic{-48pt}{func-reflection} +\begin{code} +path ff, negative_ff; + +ff = (1,1) for x = 3/2 step 1/2 until 6: ... (x, 1/x) endfor; +ff := reverse ff reflectedabout(origin, dir 45) & ff; +ff := ff scaled 24; +negative_ff = ff reflectedabout(origin, dir -45); + +draw ff withcolor 2/3 red; +draw negative_ff withcolor 2/3 red; +\end{code} + +\smallskip\centerline{\small (Omitting code for the axes, the grid, and the dots at +each \mpl{point} of the paths).} + +\smallskip +\noindent +\begin{itemize} + \item Notice that you can update the path using the assignment operator + “\mpl{:=}”. + \item You need to \mpl{reverse} the reflected portion so that the two ends coincide. + \item The two path segments are spliced together with \mpl{&}. You could use + a path join like \mpl{..} instead, but then the joined path would have an extra + \mpl{point} at \mpl{(1,1)}. + \item Notice also how the vertical part has the same equal spacing of points as + the more horizontal part. +\end{itemize} + +Reflection of a function in the line at 45° gives the inverse of the function, which +is especially useful for $y=1/x$, but it applies to functions generally. So if you want to +plot $y=\sqrt x$ it may be easier to define a path for $y=x^2$ and then reflect it. +This is particularly useful if you want to plot, say, $y=\sqrt[3]x$, over a domain +that includes negative numbers, because \MP\ will not calculate reciprocal powers of +negative numbers.\mwpic{-60pt}{func-cuberoot} +The curve in this chart was created by reflecting the line $y=x^3$. + +\newpage +\subsubsection{Exponential and logarithm functions by reflection} + +\textsc{A further example} of creating paths by +transformation.\mpic{-36pt}{func-exponential} +\begin{smallcode} +numeric u; u = 42; +path xx, yy; +xx = (-3u, 0) -- (5u, 0); +yy = xx rotated 90; +path ee, ll, nn; +numeric minx, maxx, s; +minx = -3; maxx = 1/256 mlog(4.5); s = 1/4; +ee = (for x = minx step s until maxx - s: + (x, mexp(256x)){1, mexp(256x)} ... +endfor (maxx, mexp(256 maxx)){1, mexp(256 maxx)}) scaled u; +ll = ee reflectedabout(origin, dir 45); +nn = (for x=1 step s until 4-s: (x, 1/x) ... endfor (4, 1/4)) scaled u; +nn := reverse nn reflectedabout(origin, dir 45) & nn; + +draw unitsquare xscaled mexp(256) scaled u withcolor 7/8; +draw unitsquare yscaled mexp(256) scaled u withcolor 7/8; +path T[]; +numeric p; p = directiontime dir 45 of ee; +T1 = (precontrol p of ee -- postcontrol p of ee) + shifted - point p of ee scaled 1/2 u shifted point p of ee; +T2 = (precontrol p of ll -- postcontrol p of ll) + shifted - point p of ll scaled 1/2 u shifted point p of ll; +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); +draw T1; draw T2; draw interpath(1/2, T1, T2) dashed evenly; +drawoptions(); + +draw nn; label.urt("$1/x$", point 0 of nn); +draw ee withcolor 2/3 blue; label.top("$e^x$", point infinity of ee); +draw ll withcolor 3/4 red; label.top("$\ln(x)$", point infinity of ll); +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); +dotlabel.lft("$e$", (0, mexp(256) * u)); +dotlabel.bot("$e$", (mexp(256) * u, 0)); +dotlabel.ulft("$1$", (0, u)); +dotlabel.lrt("$1$", (u, 0)); +\end{smallcode} +\moveright6in\vbox to 0pt{\vss\noindent +If you prefer more ‘normal’ functions, you can define: +\begin{smallcode} +vardef exp(expr x) = mexp(256x) enddef; +vardef log(expr x) = 1/256 mlog(x) enddef; +\end{smallcode}} + +\newpage +\subsection{Functions using trigonometric functions} + +\textsc{As noted} in §\ref{trig}, \MP’s built-in trigonometric functions work in +degrees, this example shows how you might use them in a graph.% +\vadjust{\moveright5in\vbox to 0pt{\vskip-24pt\begin{code} + numeric u, pi; u = 50; pi = 3.141592653589793; + + path xx, yy; + xx = (3.5 left -- 3.6 right) scaled u; + yy = (1.1 down -- 1.2 up) scaled u; + + path ss; + ss = origin for t=1 upto 360: -- (t, sind(t)) endfor; + ss := ss shifted 360 left & ss; + ss := ss xscaled (pi/180) scaled u; + + draw ss cutbefore yy shifted point 0 of xx + cutafter yy shifted point 1 of xx + withcolor 3/4 blue; + draw ss shifted (-1/2 pi * u ,0) + cutbefore yy shifted point 0 of xx + cutafter yy shifted point 1 of xx + withcolor 2/3 red; + + drawoptions(dashed withdots scaled 1/4); + draw ((1/4 pi, 0) .. (1/4 pi, sind(45))) scaled u; + draw ((1/2 pi, 0) .. (1/2 pi, sind(90))) scaled u; + draw ((3/4 pi, 0) .. (3/4 pi, sind(135))) scaled u; + drawoptions(); + + drawarrow xx; label.rt("$t$", point 1 of xx); + drawarrow yy; label.top("$u(t)$", point 1 of yy); + + for i=-4, -3, -2, -1, 1, 2, 3, 4: + draw (down--up) scaled 2 shifted (pi * i/4 * u, 0); + label.bot(pi_quarters(i), (pi * i/4 * u, -2)); + endfor +\end{code}\vss}} +$$\includegraphics[width=\textwidth]{func-sines}$$ + +For this diagram, the sine wave path ("ss", shown in \blue{blue}) +needs to have two complete cycles, so it is constructed in stages. First the +section from the origin to $2\pi$ is created in a loop; with 360 steps, you can use +\mpl{--} and still get a smooth path. Secondly the cycle is duplicated by splicing +itself to a shifted copy. Thirdly it is $x$-scaled to radians, and then scaled in both +directions to the chosen unit size, and drawn chopped off to the width of +the $x$-axis. The cosine path is the same path, shifted $\frac12\pi$ left, +drawn in \red{red}, and chopped off to fit the same width. + +The fancy fraction labels were produced with this subroutine: +\begin{smallcode} +vardef pi_quarters(expr n) = + save s, f, q; string s, f; numeric q; + s = if n < 0: "-" else: "" fi; q = abs(n); + if q mod 4 = 0: + f = if q > 4: decimal 1/4 q else: "" fi; + elseif q mod 2 = 0: + f = "\frac{" & decimal 1/2 q & "}{2}"; + else: + f = "\frac{" & decimal q & "}{4}"; + fi + "$\scriptstyle" & s & f & "\pi$" +enddef; +\end{smallcode} + +\newpage +\subsection{Manipulating functions} + +\textsc{This second example} with trigonometric functions shows one way to add +two functions, by combining the \MP\ paths themselves.% +\vadjust{\moveright5in\vbox to 0pt{\vskip-24pt\begin{code} + numeric u, pi; u = 50; pi = 3.141592653589793; + + path xx, yy; + xx = (3.5 left -- 4 right) scaled u; + yy = (1.2 down -- 1.3 up) scaled u; + + path ss, tt, uu; + ss = origin for x=1 upto 360: -- (x, sind(x)) endfor; + tt = origin for x=1 upto 360: -- (x, 1/2 sind(3x)) endfor; + uu = origin for x=1 upto 360: -- + (x, ypart point x of ss + ypart point x of tt) endfor; + + forsuffixes $=ss, tt, uu: + $ := $ shifted 360 left & $; + $ := $ xscaled (pi/180) scaled u; + $ := $ cutbefore yy shifted point 0 of xx + cutafter yy shifted point 1 of xx; + endfor + + draw ss withcolor 1/2[blue, white]; + draw tt withcolor 1/2[red, white]; + draw uu withcolor 1/4 green; + + label.top("$f(x)=sin(x)$", point 290 of ss); + label.bot("$g(x)=\frac12 sin(3x)$", point 295 of tt); + label.urt("$f(x) + g(x)$", point 350 of uu); + + drawarrow xx; label.rt("$x$", point 1 of xx); + drawarrow yy; label.top("$y$", point 1 of yy); + + for i=-6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 7: + draw (down--up) scaled 2 shifted (pi * i/6 * u, 0); + label.bot(pi_sixths(i), (pi * i/6 * u, -2)); + endfor +\end{code}\vss}} +$$\includegraphics[width=\textwidth]{func-addition-of-sines}$$ +Notice how the same extension, scaling, and trimming operations +can be applied to all three paths using a \mpl{forsuffixes} loop. +Note that you can use any regular variable name for the loop index; +you don't have to use \mpl{$}, but like \mpl{@} it is a +valid variable name in \MP, and it looks a bit like a placeholder marker in other +languages. + +Note that you need to start with the first point of the path outside the loop +so that you don't end up with a dangling \mpl{--} path connector. Using +\mpl{origin} is just a short cut for writing \mpl{(0, sind(0))}. If you were +plotting a different function this would not work. For example, \mpl{(0, cosd(0))} +is \mpl{(0,1)}. + +\bigskip\noindent +\hey The missing \mpl{pi_sixths} macro is left as an exercise for the reader. +Hint: you can adapt the \mpl{pi_quarters} on the previous page, allowing for halves, +thirds, and sixths instead of halves and quarters. +\newpage +\subsection{Focus on a specific region of a function} + +\textsc{This visual proof} required a large $y$-axis scale.\mwpic{-24pt}{func-epi-v-pie} +The axes are separated to show the +discontinuity in scales, and that the origin is not on the chart.\enlargethispage\baselineskip + +\begin{smallcode}[xleftmargin=0pt] +numeric minx, maxx, s, u, v; +minx = 13/8; s = 1/16; maxx = 19/4; u = 89; v = 3072; + +def f(expr x) = 1/256 mlog(x) / x enddef; +path ff, xx, yy; +ff = for x=minx step s until maxx-s: (x, f(x)) .. endfor (maxx, f(maxx)); +ff := ff xscaled u yscaled v; +xx = origin -- right scaled (maxx-minx) scaled u; +yy = origin -- up scaled 0.09v; +xx := xx shifted point 0 of ff shifted 20 down; +yy := yy shifted point 0 of ff shifted 20 left; + +numeric pi, e, fpi, fe; +pi = 3.141592653589793 u; fpi = f(3.141592653589793) * v; +e = 2.718281828459045 u; fe = f(2.718281828459045) * v; +path ee, pp; +ee = (e, ypart point 0 of xx) -- (e, fe) -- (xpart point 0 of yy, fe); +pp = (pi, ypart point 0 of xx) -- (pi, fpi) -- (xpart point 0 of yy, fpi); + +draw ee dashed withdots scaled 1/4 withcolor 2/3 red; +draw pp dashed withdots scaled 1/2 withcolor 2/3 red; +draw ff withcolor 3/4 blue; + +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); +for x=2 upto 4: + draw (down--up) scaled 2 shifted (x * u, ypart point 0 of xx); + label.bot("$" & decimal x & "$", (x * u, ypart point 0 of xx - 2)); +endfor +for y=31 upto 38: + draw (left--right) scaled 2 shifted (xpart point 0 of yy, y/100*v); +endfor +for y=32, 35, 38: + label.lft("$" & decimal (y/100) & "$", (xpart point 0 of yy-2, y/100*v)); +endfor +\end{smallcode} + +\moveright384pt\vbox to 0pt{\vss\hsize 4in\begin{smallcode} +drawoptions(withcolor 1/2 red); +label.bot("$e$", point 0 of ee shifted 4 down); +label.lft("$1/e$", point 2 of ee shifted 2 left); + +label.bot("$\pi$", point 0 of pp shifted 4 down); +label.lft("$\ln\pi/\pi$", point 2 of pp shifted 2 left); + +drawoptions(withcolor 2/3 blue); +label.urt("$\displaystyle y={\ln x\over x}$", point 42 of ff); + +drawoptions(); +label("\dots\ hence\enspace $e^\pi > \pi^e$.", (4u, 0.38v)); +\end{smallcode}} + +\newpage +\subsection{Approximate function diagrams} + +\textsc{Sometimes} you may need to plot a function that does not have a simple +mathematical definition. You can use \MP’s normal path definitions to make a likely +looking approximation.\mwpic{-48pt}{func-stress} +\begin{smallcode}[xleftmargin=0pt, xrightmargin=-140pt] +z1 = 377 right; z2 = 233 up; +path ff; ff = origin .. (72, 144){1,2} .. (84, 144) .. (96, 144){1,1} + .. (220, 220){right} .. (370, 160){2,-1.3}; + +for t=2, 4, 4.9: + draw point t of ff -- (xpart point t of ff, y2 + 6) + dashed evenly scaled 1/2; +endfor +label.top("Strain hardening", + (1/2 (xpart point 2 of ff + xpart point 4 of ff), y2)); +label.top("Necking", + (1/2 (xpart point 4 of ff + xpart point 4.9 of ff), y2)); + +path rr; +rr = point 0.4 of ff -- (xpart point 0.8 of ff, ypart point 0.4 of ff) -- point 0.8 of ff; +draw rr; label.bot("Run", point 1/2 of rr); label.rt("Rise", point 3/2 of rr); + +vardef pin_label@#(expr p, a, b)= + draw a -- b cutbefore fullcircle scaled 8 shifted a withpen pencircle scaled 1/4 withcolor 1/2 white; + label@#(p, b); +enddef; +pin_label.lrt("Yield strength", point 1.2 of ff, point 2 of ff + (8, -18)); +pin_label.bot("Ultimate strength", point 4 of ff, point 4 of ff + (4, -24)); +pin_label.bot("Fracture", point 5 of ff, point 5 of ff + (-8, -18)); + +draw ff withpen pencircle scaled 1 withcolor 2/3 blue; +clip currentpicture to unitsquare scaled 400; % clip thick pen at origin + +drawdblarrow z1 -- origin -- z2; +label.ulft("Strain, $\epsilon$", z1); +label.urt("Stress, $\sigma$", z2); + +label("$\displaystyle\hbox{Young's modulus} = \hbox{Slope} = {\hbox{Rise}\over\hbox{Run}}$", + 1/2 z1 shifted 36 up) withcolor 2/3 blue; +\end{smallcode} + +\newpage +\subsubsection{Taming Bezier paths with controls} + +\textsc{It takes some} practice to translate a sketch of a curve into a smooth +path. Plain \MP\ inherits from \MF\ a useful \id{flex} macro, that takes a list of \<pair>s +(any number of them) and produces a pleasing path through them. As Knuth says in the +\mfbook: +“The idea is to specify two endpoints, $z_1$ and $z_n$, together with +one or more intermediate points where the path is traveling in the +same direction as the straight line from $z_1$ to~$z_n$; these +intermediate points are easy to see on a typical curve, so they +are natural candidates for key points.” For example: +\begin{code} +draw flex(z1,z2,z3) & flex(z3,z4,z5) +flex(z5,z6,z7) & flex(z7,z8,z9,z1) & cycle; +\end{code} +(with appropriate definitions of the points), produces this: +\lower 12pt\hbox{\smash{\begin{mplibcode} +beginfig(0); +z1=(0,509); +z2=(-14,492); +z3=(-32,481); +z4=(-42,455); +z5=(-62,430); +z6=(-20,450); +z7=(42,448); +z8=(38,465); +z9=(4,493); + draw flex(z1,z2,z3) & flex(z3,z4,z5) & +flex(z5,z6,z7) & flex(z7,z8,z9,z1) & cycle; + for i=1 upto 9: draw z[i] withpen pencircle scaled 2 withcolor red; endfor + currentpicture := currentpicture scaled 0.7071 rotatedabout(z3, -20); +endfig; +\end{mplibcode}}} + +\bigskip\noindent +\textsc{Another approach} is just to define the end-points and some control points, +and then define a path that is shaped by the control points but does not actually go +through them. Consider this program:\mwpic{-20pt}{func-pulse} +\begin{code} +vardef pulse(expr w, h, d) = +for i=0 upto 4: x[i] = w/4 * i; endfor + y0 = y1 = 0; + y3 = y4 = d; + y2 = h; + z0 .. 1/2[z0, z1] .. controls z1 + .. 1/2[z1, z2] .. controls z2 + .. 1/2[z2, z3] .. controls z3 + .. 1/2[z3, z4] .. z4 +enddef; +draw pulse(300, 100, -40); +\end{code} +This produces the smooth blue line shown on the right. A second copy of the line is +shown below, decorated with the three control points $z_1$, $z_2$, and $z_3$ in red, +and showing the six points of the path as small black circles. You can tweak this +curve by adjusting the controls left or right, or changing the mediation parameters +so that the points on the path are closer to one control point than the other. + +\newpage +\subsection{Parametric plots} + +\textsc{If you want to plot} one function against another, then you can make each +coordinate a function of an independent variable. All functions can be converted, +trivially, to this form: +\begin{code} +vardef f(expr x) = x enddef; +vardef g(expr x) = sind(x) enddef; % or whatever function ... +path ff; ff = for t = mint step s until maxt - s: + (f(t), g(t)) .. +endfor (f(maxt), g(maxt)); +\end{code} +But you can make more complicated curves, for example curves +that can have more than one value for $y$ for a given $x$, if you change $f(x)$ and +$g(x)$ appropriately. +The first example\mwpic{-24pt}{func-lemniscate} shows the lemniscate of Bernoulli +and was drawn like this: +\begin{smallcode} +numeric a, c; c = 128; a = sqrt(2) * c; +vardef f(expr x) = a * cosd(x) / (1 + sind(x) ** 2) enddef; +vardef g(expr x) = f(x) * sind(x) enddef; +numeric mint, maxt, s; mint = 0; s = 30; maxt = 360; +path p; +p = for t = mint step s until maxt - s: + (f(t), g(t)) ... +endfor cycle; +draw p withcolor 3/4 blue; +path xx, yy; +xx = (left -- right) scaled 200; +yy = (down -- up) scaled 64; +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); +dotlabel.bot("$F_1$", c * left); +dotlabel.bot("$F_2$", c * right); +\end{smallcode} +Although sometimes, especially when you know the domain is 0° to 360° and that the +path is cyclic, it is simpler to write the two expressions directly in the +loop:\mwpic{-108pt}{func-parametric} +\begin{smallcode} +path p; +p = for t = 0 upto 360: (144 cosd(3t), 89 sind(2t)) ... endfor cycle; +draw p withcolor 2/3 red; +\end{smallcode} +which produces this Lissajous curve $\longrightarrow$ +\newpage +\subsubsection{Parametric plots with polar coordinates: Maurer roses} + +\textsc{Instead of} defining separate functions for the $x$ and $y$ coordinates in a +parametric plot, it is sometimes convenient to use \MP’s polar coordinate notation (discussed in +§\ref{polar}). The family of “rose” plots, based on $r=\cos(n\theta)$, is easy to +do in this way. Here is a Maurer rose, based on $r=\cos(2\theta)$ and connecting +every 29th point on the curve.\mpic{-24pt}{func-maurer-rose} + +\begin{code} + numeric n; n = 2; + path r; r = (for t=0 upto 360: + cosd(n * t) * dir t ... + endfor cycle) scaled 150; + + numeric d; d = 29; + path k; k = for t = 0 upto 360: + point d * t mod 360 of r -- + endfor cycle; + + draw k dashed withdots scaled 1/8 + withpen pencircle scaled 1/4 + withcolor 1/2[blue, white]; + + draw r withcolor 2/3 red; + + path xx, yy; % you might not need the axes... + xx = (left -- right) scaled 160; + yy = (down -- up) scaled 160; + drawarrow xx; label.rt("$x$", point 1 of xx); + drawarrow yy; label.top("$y$", point 1 of yy); +\end{code} +Different values of $n$ and $d$ give an endless variety of patterns. But note that if +your choice of $d$ exceeds $2^{15}/360 \simeq 91$ you will need to use +"-numbersystem=double" to avoid arithmetic overflow. + +\newpage +\section{Drawing plane curves} + +{\textsc{“Plane curves offer} a rich \dots\ field of study which may be approached +from a quite elementary level. Anyone who can draw a circle with a given centre and +a given radius can draw a cardioid or a limaçon. Anyone who can use a set square +can draw a parabola or a strophoid” \hfill — \textsl{A Book of Curves}, E.\@ H.\@ +Lockwood\parfillskip0pt\par} + +\subsection{Parabola} + +\textsc{The simplest way} to get a parabola curve is to plot $y=x^2$ over $-1 \le x +\le 1$ and then transform as required (see next page), but it can be illuminating to follow more +traditional constructions, such as that shown on the +right.\mwpic{-180pt}{curves-parabola} +\vadjust{\moveright7.2in\vbox to 0pt{\hsize 200pt\noindent +The idea here is that you put the right angle of your set square on the vertical +axis with the short side touching $S$, and then draw the long side. If you do this +in enough places, the edges form a parabola. In the \MP\ code here, the intersection +of each ray with the one before is captured as variable $t$ and then added one at a +time to the \<path> \id{parabola}, (using a neat trick at the beginning). +\vss}} +\begin{smallcode}[xleftmargin=0pt] +pair A, S; A = origin; S = 66 right; +path parabola, last; +for q = -144 step 8 until 144: + pair Q; Q = (0, q); + path ray; + ray = (origin -- unitvector(S - Q)) scaled 300 + rotated if q < 0: - fi 90 shifted Q; + draw S -- ray withcolor 7/8; + if known last: + pair t; + t = whatever[point 0 of ray, point 1 of ray] + = whatever[point 0 of last, point 1 of last]; + parabola := if known parabola: parabola .. fi t; + fi + last := ray; +endfor + +draw parabola withcolor 3/4[red, white]; +for t=0 upto length parabola: + draw point t of parabola withpen pencircle scaled 3/2 withcolor red; +endfor +draw (up--down) scaled 300; + +dotlabel.lft("$A$", A); +dotlabel.rt("$S$", S); +\end{smallcode} + +\newpage +\subsubsection{Parabola from directrix and focus} + +\textsc{The classical definition} of the parabola is the locus of points that are +equidistant from a given line (the \textit{directrix}, shown as $A\to B$ on the +right) to a given focus point (shown as $S$).\mpic{-24pt}{curves-parabola-directrix} +Each point on the parabola path is related to each point, and you can construct an +equilateral parallelogram at each point as shown. This leads to a macro that +generates a parabola given two \<pair> variables to define the directrix and another +to define the focus: +\begin{code} +vardef parabola(expr A, B, S) = + save m, q, n, parabola; + pair n; % n = nearest point to S on A--B + n = whatever[A, B]; n - S = whatever * (A-B) rotated 90; + path parabola; + for t=0 step 1/64 until 1: + pair m, q; + m = 1/2[S, t[A, B]]; + q = whatever[S, n]; + q - m = whatever * (S - m) rotated 90; + parabola := if known parabola: parabola -- fi + q reflectedabout(S, m); + endfor + parabola +enddef; + +\end{code} + +\subsubsection{Parabola from $y=x^2$ and $dy/dx=2x$} + +Alternatively you could define a “unit parabola” like this: +\begin{code} +path ff; ff = (-1, 1){1, -2} .. (-1/2, 1/4){1, -1} .. + (0, 0){right} .. (1/2, 1/4){1, 1} .. (1, 1){1, 2}; +\end{code} +and then — using the points defined above, where $o$ is the mid-point of $n\to S$ — scale it and place it like this: +\begin{code} +draw ff scaled 4 abs(S-o) rotated angle (B-A) shifted o; +\end{code} + +\newpage +\subsection{Hyperbola} + +\textsc{The traditional construction} for the hyperbola is identical to the +construction for the parabola given above, except that the base line is a circle +rather than a straight line.\mwpic{-48pt}{curves-hyperbola-construction} +As a result, the shape of the curve changes depending on the radius of the base +circle, unlike the parabola. The curve is bounded by the two asymptotes, which are +the lines from the centre of the circle $O$ through the tangent points from the focus +$S$. When the ratio of $OS/OA = \sqrt2$, the asymptotes are at right angles. + +You can also draw the hyperbola as the function $y=1/x$ (as shown in +§\ref{sec:pathpieces}), which can be transformed to any desired shape. +The untransformed function is shown on the bottom left, with the focus $S$ +at the point $(\sqrt2, \sqrt2)$. If the desired angle between the asymptotes is +$2\alpha$, the transformation can be created like this: +\begin{code} +numeric alpha; alpha = 34; transform t; +origin transformed t = origin; +right transformed t = dir -alpha; +up transformed t = dir alpha; +\end{code} +This can be applied to the hyperbola curve itself and to the axes. But the focus +will remain at the same distance from the origin, as shown below right. +$$\includegraphics[width=0.9\textwidth]{curves-hyperbola-function}$$ + +\newpage +\subsection{Ellipse} + +\textsc{You can draw an ellipse} in \MP\ by scaling the standard \mpl{fullcircle} +path by a different amount in each direction. By convention, the $y$-axis is the +minor axis of an ellipse.\mwpic{-24pt}{curves-ellipse} +The ellipse shown on the right, was defined like this: +\begin{code} +path ellipse; +ellipse = fullcircle scaled 320 yscaled 5/8 rotated 13; +\end{code} +Then the lengths of the semi-major axes, $a$ and $b$, were extracted like this: +\begin{code} +numeric a, b; +2a = abs (point 4 of ellipse - point 0 of ellipse); +2b = abs (point 6 of ellipse - point 2 of ellipse); +\end{code} +If you already had $a$ and $b$, then you could use them directly to scale your +ellipse; the following snippet would produce the same elliptical path: +\begin{code} +numeric a, b; a = 160; b = 100; path ellipse; +ellipse = fullcircle xscaled 2a yscaled 2b rotated 13; +\end{code} + +\bigskip\noindent +The “eccentricity”, $e$, of the ellipse is the ratio between the distance from the centre +to each focus and the semi-major axis, $a$. By definition, the distance from $F_1$ +to $T$ to $F_2$ is constant as $T$ moves round the ellipse and is equal to $2a$. +Hence when $T$ lies on the minor axis, you have $TF_1=a$, and so $a^2 = b^2 + +a^2e^2$, and $e^2=1-b^2/a^2$: +\begin{code} +numeric e; e = 1 +-+ b/a; % the Pythagorean difference operator +\end{code} +The focus points can then be found like this: +\begin{code} +z0 = 1/2[point 0 of ellipse, point 4 of ellipse]; +z1 = e[z0, point 0 of ellipse]; +z2 = e[z0, point 4 of ellipse]; +\end{code} +\vfill +\moveright 384pt\vbox to 0pt{\vss +The tangent and the normal at $T$ were added like this: +\begin{code} +numeric t; t = 1.732; +draw (left--right) scaled 21 + rotated angle direction t of ellipse + shifted point t of ellipse; +draw (1/2 down--up) scaled 21 + rotated angle direction t of ellipse + shifted point t of ellipse; +\end{code}} +\newpage +\subsubsection{Tangent from external point to ellipse} + +$$\includegraphics[width=\textwidth]{curves-ellipse-tangents}$$ +To find the tangent points from an external point $A$ to an ellipse, the classical +construction is to draw an arc centred at $A$ through one focus point, then draw a +second arc centred at the other focus with radius $2a$. The intersection points, +$P$ and $Q$, of these two arcs are the images of the first focus point in the +required tangents (because $F_2TF_1=2a$ by definition, and $F_2P=2a$ by +construction), and so the tangent points $T$ and $T'$ are the intersections of +$F_2P$ and $F_2Q$ with the ellipse. + +\vfill\noindent +There is no such direct construction for the nearest point on an ellipse to a given +point, but you can use the macro \id{solve} to find it numerically. The blue arrow +shows the shortest distance from $A$ to the ellipse. +\vadjust{\moveright 384pt\vbox to 0pt{\vss +\begin{smallcode} +secondarydef a through b = + begingroup; save d; numeric d; d = abs(a-b); + (1+12/d)[b,a]--(1+12/d)[a,b] endgroup +enddef; +path ellipse; ellipse = fullcircle scaled 300 yscaled 5/8 rotated 13; +z0 = 1/2[point 4 of ellipse, point 0 of ellipse]; +numeric a, b, e; +2a = abs (point 4 of ellipse - point 0 of ellipse); +2b = abs (point 6 of ellipse - point 2 of ellipse); +e = 1 +-+ b/a; +z1 = e[z0, point 0 of ellipse]; +z2 = e[z0, point 4 of ellipse]; +z3 = 240 dir 25; +path pp, qq; +pp = fullcircle scaled 2 abs (z1 - z3) shifted z3; +qq = fullcircle scaled 4a shifted z2; +z4 = pp intersectionpoint qq; +z5 = reverse pp intersectionpoint qq; +numeric t, u; +(t, whatever) = ellipse intersectiontimes (z2 -- z4); +(u, whatever) = ellipse intersectiontimes (z2 -- z5); +vardef f(expr x) = + angle (z3 - point x of ellipse) + 90 > angle direction x of ellipse +enddef; +drawarrow z3 -- point solve f(0, t) of ellipse withcolor 2/3 blue; +drawoptions(dashed evenly withcolor 7/8); +draw point 0 of ellipse through point 4 of ellipse; +draw point 2 of ellipse through point 6 of ellipse; +drawoptions(withcolor 3/4); +draw z4 -- z1 -- z5; +draw z2 -- z4 -- z3 -- z5 -- cycle; +draw point t of ellipse -- z1 -- point u of ellipse; +drawoptions(); +draw point t of ellipse -- z3 -- point u of ellipse withcolor 2/3 red; +dotlabel.llft("$F_1$", z1); dotlabel.llft("$F_2$", z2); +dotlabel.urt("$A$", z3); +dotlabel.ulft("$P$", z4); dotlabel.lrt ("$Q$", z5); +dotlabel.ulft("$T$", point t of ellipse); +dotlabel.lrt("$T'$", point u of ellipse); +draw ellipse; +picture P; P = currentpicture; currentpicture := nullpicture; +draw pp dashed withdots scaled 1/2; +draw qq dashed withdots scaled 1/2; +bboxmargin := 24; clip currentpicture to bbox P; draw P; +\end{smallcode} +\vskip -48pt}} + +\newpage +\subsection{Cardioid} + +\textsc{To draw a cardioid by hand}, you can draw a base circle, mark a fixed point $A$ on it, and +then draw a circle centred at any point $Q$ on the circle that passes through point +$A$. If you then repeat this for many different positions of Q, the cardioid is the +curve that encloses all the circles.\mpic{-24pt}{curves-cardioid-simple} +But for \MP, you want only a single point $P$ from the circumference of each circle; +this turns out to be the image of $A$ reflected in the tangent at each point $Q$, like so: +$$\includegraphics[width=0.7\textwidth]{curves-cardioid-construction}$$ +\vskip -84pt\noindent +With a small step size $s$ and a base circle \id{base}, this suggests: \begin{smallcode} +pair A; A = point 0 of base; +path cardioid; cardioid = for t = 0 step s until length base: + A reflectedabout(precontrol t of base, postcontrol t of base) .. +endfor cycle; +\end{smallcode} +You can also show that $AP=2a(1+\cos\theta)$, where $a$ is the radius of the +base circle and $\theta$ is the angle that $AP$ makes with the diameter through +$A$, so you might use: +\begin{smallcode} +cardioid = for t=0 upto 360: 2a * (1+cosd(t)) * dir t .. endfor cycle; +\end{smallcode} + +\newpage +\subsection{Limaçon} + +\textsc{The limaçon can be seen} as a generalization of the cardioid, obtained by +moving point $A$ off the base circle. +\vadjust{\moveright356pt\vbox to 0pt{\vskip -20pt\noindent +\rlap{\includegraphics[scale=0.833]{curves-limacon-simple}}\vss}} +Here $A$ has been moved to the left, but each $P$ on the curve is still $A$ +reflected in the tangent at each $Q$. The “hole” gets larger as $A$ moves away from +the \id{base} circle; when $A$ touches the \id{base} the hole disappears and the +curve becomes the cardioid, as before. + +Following this ruler-and-compasses approach, the red limaçon path in the figure here +was generated from the \id{base} circle shown in blue. +\begin{smallcode} +pair A; A = 2[center base, point 0 of base]; +path limacon; limacon = for t = 0 step s until length base: + A reflectedabout(precontrol t of base, postcontrol t of base) .. +endfor cycle; +\end{smallcode} +Or if you prefer a more trigonometrical approach: +\begin{smallcode} +limacon = for t=0 upto 359: 2a*(1+2cosd(t))*dir t .. endfor cycle; +\end{smallcode} +Here $2a$ is the diameter of the blue \id{base} circle. Note that if you use +\mpl{sind}\\ instead of \mpl{cosd} you get the same curve rotated $90^\circ$. + +\vfill +\noindent +An alternative approach (due to Albrecht Dürer) is shown below. In this dia- +\par\kern 2pt +\vbox{\halign{#&\quad\vbox{\hsize=2.4in\noindent #}\cr +$$\includegraphics{curves-limacon-durer}$$ +&gram the base circle is divided into 12 parts like a +clock face. At 1 o’clock, you draw a line segment of a given +length parallel to the radius to 2 o’clock; at 2 you draw the same length +segment parallel to 4 o’clock, and so on. The limaçon is the curve through +the far ends of each segment (plus any intermediate points required). +This approach of doubling the angles makes it more obvious that the limaçon goes round twice, as it +were. The path was generated like this: \cr}} +\smallskip +\begin{smallcode} +limacon = for t=0 upto length base-1: + 42 dir angle point 2t of base shifted point t of base .. +endfor cycle; +\end{smallcode} + +\newpage +\subsection{Astroid} + +\textsc{Readers of a certain age} may recall threading strings between +pegs on a board to make the astroid. It is the envelope of a line of a given +length drawn from $x$-axis to $y$-axis at all possible points.\mpic{-24pt}{curves-astroid} +In \MP\ the simplest way to draw the astroid curve (and the “strings”) is to use a +base circle and the points $A$ and $B$ at the ends of each line that have the $x$ +part and $y$ part of each point $T$ round the base circle. +$$\includegraphics[scale=0.8]{curves-astroid-construction}$$ +Then the point $P$ on $A \to B$ that is closest to $T$ will lie on the astroid, so you +can make the path with: +\begin{smallcode} +path astroid; astroid = for t=0 step 1/16 until 8: + hide(pair a, b, p; + a = (xpart point t of base, 0); b = (0, ypart point t of base); + p = whatever[a, b]; p-point t of base = whatever * (a-b) rotated 90; + ) p -- endfor cycle; +\end{smallcode} +Note that you need to use “\mpl{--}” so that the cusps stay neatly +pointed.\marginpar{\vskip-38pt\small\noindent +\llap{\hey\,}The geometry of the subtended angles shows that the length of the arc +$T\to S$ equals the length of the arc from $T\to P$ on the quarter-sized circle +through $T$ and $M$. So the astroid is also the path of a point on the smaller +circle rolling around the inside of the base circle.} + +\newpage +\subsubsection{Astroid and cousins} + +The geometry of the astroid also allows us to define a simple parametric equation +for the point $P$. +$$\includegraphics[scale=0.8]{curves-astroid-construction}$$ +If the distance $OT = a$, then $OA = BT = a\cos\theta$. +But then $BP = BT\,\cos\theta = a\cos^2\theta$, and the +$x$-coordinate of $P = BP\,\cos\theta=a\cos^3\theta$. By a similar +argument the $y$-coordinate is $a\sin^3\theta$, so the parametric +equations for $P = (x, y)$: +$$\centerline{$x=a\cos^3\theta$ \quad and \quad $y=a\sin^3\theta$}$$ +This is used to make this rather psychedelic family of astroid cousins $\longrightarrow$ + +\moveright384pt\vbox to 36pt{\vss +$$\includegraphics[scale=0.9]{curves-astroid-family}$$ +\begin{smallcode} +numeric a; a = 144; input colorbrewer-rgb +for n=1 upto 7: + path p; p = a * right for t=6 step 6 until 90: + .. a * (cosd(t) ** n, sind(t) ** n) + endfor; + p := for i=0 upto 3: p rotated 90i & endfor cycle; + fill p withcolor Oranges[9][n]; draw p; +endfor +\end{smallcode}} + + +\newpage +\subsection{Cycloid} + +\textsc{Cycloids} are the curves made by points on the circumference of a +rolling wheel. In the first diagram the cycloid is drawn in red and the +corresponding rolling wheel in blue. The main idea in this diagram is to make the +whole drawing depend on just a few parameters; here there are two: the radius $r$ +and the amount of rotation $θ$. If we make $r$ bigger, the drawing will be +scaled up; if we change $θ$, the wheel will appear to have rolled along. +\mpic{-66pt}{cycloids} + + +\marginpar{\par\kern1in +\begin{itemize} + \item Near the beginning we define $\pi=3.14159265$, as there’s no such constant + built in, but it makes the source more understandable to write "pi/180" + instead of "0.017453". It would be nice to use the Greek letters themselves + in the source, but \MP\ only lets you use plain ASCII characters, to you + have to write "pi" instead. Later on "t" is used instead of $\theta$. + + \item The path of the cycloid $c$ is defined using an inline "for" loop. There’s a + slight awkwardness to doing this as you have to repeat yourself either at + the beginning or the end, because you can’t have a dangling "--" or ".." at + the end of the path. With a closed path it’s easier because you can just + put "--cycle" after the "endfor". The strange numbers here are because + we are going from a rotation of $-100°$ to $+460°$; $360°$ corresponds to + one hop of the cycloid. + + \item The axes are done in the usual way, except that we use "xpart" and the + "point .. of .." notation to make the $x$-axis neatly line up with the ends + of the cycloid path. + + \item To label points with dots but no text it’s convenient just to fill a + circle scaled to "dotlabeldiam"; this internal parameter is the current size + to be used for the dots in "dotlabel". + +\end{itemize}} + +\mpexternal[xleftmargin=0pt]{cycloids-code.mp} + +\noindent +You can generalize the picture to make cycloids where the point tracing the +cycloid is not on the circumference doing the rolling; the classic example is the +wheel of the train with a flange. Here I have added $R$ to define the radius of an +outer rim, while the wheel still rolls along a circle of radius $r$. +\mpic{-40pt}{cycloids-extra} +You might like to experiment with making $R<r$. Note also that variable names are +case sensitive in \MP. + +\mpexternal[xleftmargin=0pt]{cycloids-extra-code.mp} + +\noindent The output\mpic{-122pt}{cycloids-extra} is repeated at the right to save you flicking pages. + +\newpage +\subsubsection{The cycloid compared to other curves} + +\noindent\textsc{You can't easily} draw a cycloid through two arbitrary points, but +if you get the size of the generating circle right, you can make it pass through +points at each end of a quarter circle, and then it's easy to draw other curves +between the two points. \vadjust{\moveright5.25in\vbox to 0pt{\vskip-48pt\begin{smallcode} +numeric r; r = 164; +path Y, L, C, P, S; + +Y = origin for t=5 step 5 until 140: + -- (0, r) rotated t shifted (t/57.29577951308232*r, -r) +endfor cutafter (origin -- (4r, 0) rotated -45); + +z0 = point 0 of Y; +z1 = point infinity of Y; + +L = z0 -- z1; +C = quartercircle rotated 180 scaled 2x1 shifted (x1, y0); +% The idea here is to use the derivative as the direction at each point. +% If you treat A as x=-1, and B as x=0 and used x=-1/2 in the middle, +% then three points are enough to make the curves look realistic. +% parabola f = x^2, f' = 2x +P = z0{1,-2} + ... (xpart 1/2[z1, z0], ypart 1/4[z1, z0]){1,-1} ... z1 {1, 0}; +% sixth degree f = x^6, f' = 6x^5 +S = z0{1,-6} + ... (xpart 1/2[z1, z0], ypart 1/64[z1, z0]){1, -6/32} ... z1 {1, 0}; + +draw z0 -- (x0,y1) -- z1 withcolor 3/4; + +drawoptions(withcolor 2/3 red); + draw L; dotlabel.urt("Line", point 1/4 of L); +drawoptions(withcolor 1/2 green); + draw C; dotlabel.urt("Circle", point 1 of C); +drawoptions(withcolor 1/4[red, green]); + draw P; dotlabel.urt("Parabola", point 1/2 of P); +drawoptions(withcolor 3/4[red, green]); + draw S; dotlabel.llft("Sixth degree", point 3/4 of S); +drawoptions(withcolor 1/2 blue); + draw Y; dotlabel.urt("Cycloid", point 22 of Y); +drawoptions(); +dotlabel.ulft(btex $A$ etex, z0); +dotlabel.urt(btex $B$ etex, z1); +\end{smallcode}\vss}} +Here is an example with an inverted cycloid, the +so-called \textit{brachistochrone} (the curve joining two points such that a body +travelling along it under gravity takes a shorter time than is possible along any +other curve between the points). +$$\includegraphics{brachisto}$$ + +\newpage +\subsection{Spirals} + +\textsc{D'Arcy Thompson} tells us, in \textsl{On Growth and Form}, that the spiral +of Archimedes “may be roughly illustrated by the way a sailor coils a rope upon the +deck; as the rope is of uniform thickness, so in the whole spiral coil is each whorl +of the same breadth as that which precedes and as that which follows it”. In +mathematical terms the radius of the spiral is proportional to the angle turned, so that +$r=a\theta$.\mpic{-36pt}{curves-spiral-archimedes} +This is very simple to program in \MP. +\begin{code} +numeric a; a = 1/8; path S; +S = origin for t=1 upto 360: .. a * t * dir t endfor; +\end{code} +except that you are unlikely to need one point for every degree of turn in your spiral, so you are +more likely to code: +\begin{code} +S = origin for t=1 upto 360: .. a * t * dir 8t endfor; +\end{code} +which spreads the points out and gives you eight full turns, or perhaps +\begin{code} +S = origin for t=1 upto 90: .. 1/12 t * dir 16t endfor; +\end{code} +which would give you four complete turns with a tighter spacing.\marginpar{\small +The rope was drawn (very slowly) with the \mpl{rope} macro from §\ref{pics-border}.} + +\bigskip\noindent +The next simplest is the logarithmic spiral where you have +$r=a^\theta$.\mwpic{10pt}{curves-spiral-equiangular} This is also very simple to +program in \MP, provided you are careful about the scaling. The complete program +for the spiral shown on the right was: +\begin{code} +numeric a; a = 2.6; path S; +S = right for t=1 upto 360: .. a ** (t/64) * dir 4t endfor; +drawarrow S; +\end{code} +Note that $a$ was carefully chosen to get a curve that would fit the page, and that +$t$ has been divided by 64 to bring it into a suitable range to work with the +default number system. + +\newpage +\subsubsection{Logarithmic spiral and the golden rectangle} + +\textsc{The logarithmic spiral} is connected to growth in nature. If you start with +a small square and keep adding squares scaled to the longer side of the resulting +rectangle, you get the golden rectangle and the logarithmic spiral emerges from +it.\mpic{-48pt}{curves-spiral-gnomon-sq} +\begin{code}[xleftmargin=0pt, xrightmargin=-20pt] +drawoptions(withpen pencircle scaled 1/4 withcolor 1/4); +path s[]; s0 = unitsquare; +fill s0 withcolor 1/2[2/3 blue, white]; draw s0; +numeric a, b, t, n; a = 1; b = 1; n = 11; +for i = 1 upto n: + t := b; b := b + a; a := t; % Fibonacci sequence + s[i] = unitsquare scaled a; + s[i] := s[i] shifted (point i of s[i-1] - point i-1 of s[i]); + fill s[i] withcolor (1/2 + i/32)[2/3 blue, white]; + draw s[i]; +endfor +% cross hairs +for i = n-1 upto n: + draw point i-2 of s[i-2] -- point i of s[i]; +endfor +drawoptions(); +% Draw the spiral as a red arrow +drawarrow origin for i=0 upto n: .. point i of s[i] endfor + withcolor 2/3 red; +% Show the Fibonacci sizes for the bigger boxes +for i = 5 upto n: + label(TEX(decimal arclength subpath (0, 1) of s[i]) + scaled 0.8 rotated angle point n of s[n], center s[i]); +endfor +% Rotate whole picture to show off the spiral +currentpicture := currentpicture rotated - angle point n of s[n]; +\end{code} +\moveright 396pt \vbox to 0pt{\vss\hsize 4in\small +\noindent\llap{• }You can't assign to a pair literal in \MP, so you cannot write + \mpl{(a,b) := (b,a+b)}; use a temporary numeric instead. + +\noindent\llap{• }The uses of \mpl{point} with the square paths \mpl{s[]} +exploit the fact the the \mpl{unitsquare} path is cyclic, so point 4 is the same +as point 0, and so on. + +\noindent\llap{• }Note also the rotation of the labels, so that they are horizontal +when the whole picture is rotated at the end to show the spiral better. +\par} + + +\newpage +\subsubsection{Logarithmic spiral and similar triangles} + +\textsc{This drawing} starts with a large triangle and transforms it to make +smaller similar copies. The spiral is drawn through the apex of successive +transformed triangles.\mpic{0pt}{curves-spiral-gnomon-trig} +\begin{code}[xleftmargin=0pt, xrightmargin=-80pt] +path t[], base; pair apex; +base = (left--right) scaled 100; +apex = whatever * dir +72 shifted point 0 of base + = whatever * dir -72 shifted point 1 of base; +t0 = (base -- apex -- cycle); + +transform S; +r = arclength subpath (0, 1) of t0 / arclength subpath (2, 3) of t0; +point 0 of t0 transformed S = (r*r)[point 0 of t0, point 2 of t0]; +point 1 of t0 transformed S = point 0 of t0; +point 2 of t0 transformed S = point 1 of t0; + +n = 16; +for i=1 upto n: + t[i] = t[i-1] transformed S; + draw subpath (2,3) of t[i] withpen pencircle scaled 1/4; +endfor + +drawoptions(withpen pencircle scaled 1/8 withcolor blue); +draw point 0 of t0 -- point 3/2 of t0; +draw point 0 of t1 -- point 3/2 of t1; +drawoptions(); +draw t0; +drawarrow point 2 of t[n] for i=n-1 downto 0: + .. point 2 of t[i] endfor withcolor 2/3 red; + +z0 = whatever[point 0 of t0, point 3/2 of t0] + = whatever[point 0 of t1, point 3/2 of t1]; +currentpicture := currentpicture rotated - angle (point 2 of t0 - z0); +\end{code} +\moveright 396pt \vbox to 0pt{\vss\hsize 4in\small +\noindent The \<transform> $S$ is defined implicitly. It is sufficient +to give equations for three non-collinear points, and \MP\ will work out +the rest. In order to do this, you need to know the ratio of $x/a$, but by +definition $a/b=b/x$, so $x=b^2/a$, hence $x/a=(b/a)^2$. The code shown +sets $r=b/a$, and then uses $r^2$ as the required fraction along $a$. + +\smallskip\noindent\hey +This construction also works for other triangles, but it looks more elegant with +an isosceles with base angles of $72^\circ$ as shown. +\par} + + + +\newpage +\section{Eggs}\label{eggs} + +\textsc{Drawing bird's eggs} excites a curious fascination in some people. This +section shows some possible ways to make eggs with \MP. The first few +compass-and-ruler constructions are taken from Robert Dixon's +\textsl{Mathographics}. They all follow the same basic idea of constructing the +egg-shaped path from a series of circular arcs.\mnote{-16pt}{In these \MP\ implementations +each egg path has 8 points starting with point 0 at “3 o’clock” like \mpl{fullcircle}.} + +\subsection{Euclidean egg} + +The first is made of four circular arcs, defined here as parts of +circles $a$, $b$, $c$, \& $d$.\mwpic{0pt}{eggs-moss}\marginpar{\par\kern 3in +\begin{itemize} + \item "eggs-common.mp" defines the colours, and the \mpl{numbered_points} + routine that is used to show the points of the "egg" path. + \item Note that it is not necessary that the parts of the arcs touch; + in fact it is better to join them with the \mpl{..} connector in case the + ends are not close enough for you to use \mpl{&}. + \item The rigmarole with saving the current picture, is to show a copy + of the egg path with and without the construction lines. +\end{itemize} +} +\begin{code}[xleftmargin=0pt] +input eggs-common +path a, b, c, d, egg; numeric r; r = 100; +a = fullcircle scaled 2r; +b = fullcircle scaled 4r shifted point 4 of a; +c = fullcircle scaled 4r shifted point 0 of a; +d = fullcircle scaled 2 abs (point 2 of a - point 1 of b) + shifted point 2 of a; +egg = subpath (0, 1) of b .. point 2 of d .. + subpath (3, 4) of c .. subpath (5, 7) of a .. cycle; +beginfig(1); +fill egg withpen pencircle scaled 2 withcolor eggshell; +picture P; P = currentpicture; +drawoptions(withpen pencircle scaled 1/4 withcolor dark_eggshell); +draw a; draw d; +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); +draw point 1 of egg -- point 4 of egg -- + point 0 of egg -- point 3 of egg; +draw egg; +drawoptions(withpen pencircle scaled 2 withcolor 7/16); +draw center a; +draw center d; +draw numbered_points(egg); +drawoptions(); +draw P shifted 240 right; +endfig; +\end{code} +\newpage +\subsection{Pythagorean egg} + +The centres of the arcs are determined by the 3-4-5 triangle at the +origin.\mwpic{0pt}{eggs-thom} +\begin{code} +numeric r, a, b, t; a = 60; 4b = 3a; r = a ++ b; +path base, cap, egg; +base = subpath (4, 8) of fullcircle scaled 2(2r-b); +cap = subpath (0, 4) of fullcircle scaled 2r shifted (0, a) + cutbefore ((b, 0) -- (b, 2r)) + cutafter ((-b, 0) -- (-b, 2r)); +egg = point 4 of base {up} .. cap .. {down} base & cycle; +% more naturally "base {up} .. cap .. {down} cycle" +% but then point 0 would not be at 3 o'clock +\end{code} +Note that you can use $\to$ to create reasonably large circular arcs. +The parts of the drawing for filling the egg, +and showing the construction are similar to the first example. + +\subsection{A taller Pythagorean egg} + +A slightly different approach using a $\sqrt3$-$\sqrt4$-$\sqrt7$ +triangle.\mwpic{20pt}{eggs-357} +\begin{code} +path base, cup, cap, egg; +base = fullcircle scaled 200; +z1 = point -2/3 of base; +z2 = point 2/3 of base; +z3 = point 10/3 of base; +z4 = point 14/3 of base; +z5 = 1/2[z2, z3]; +z6 = 1/2[z4, z1]; +numeric a, b; +a = abs(z3 - z1) - abs(z4 - z6); +b = abs(z3 - z1) - abs(z4 - z5); +cup = subpath (4, 8) of fullcircle scaled 2a shifted z6; +cap = fullcircle scaled 2b shifted z5 + cutbefore (z5 -- 2[z4, z5]) + cutafter (z5 -- 2[z1, z5]); +egg = point 4 of cup {up} .. cap .. {down} cup & cycle; +\end{code} + +\newpage +\subsection{Golden section egg} + +An alternative construction.\mwpic{-10pt}{eggs-gold} +\begin{code}[xleftmargin=0pt] +path base, aa, bb; pair m, n, n'; +base = fullcircle scaled 200; m = 1/2 point 0 of base; +aa = halfcircle scaled 2 abs (point 2 of base - m) + shifted m cutbefore (origin -- 1000 up); +n = point infinity of aa; n' = n reflectedabout(up, down); +bb = subpath (0, 2) of base shifted n cutafter (origin -- 1000 up); + +path dome, cap, cup, egg; +dome = fullcircle + scaled 2 (abs(n - point 0 of base) - abs(n - point 0 of bb)) + shifted point infinity of bb; +cap = dome cutbefore (point 4 of bb -- 2[n, point 4 of bb]) + cutafter (point 4 of bb -- 2[n', point 4 of bb]); +cup = subpath (4, 8) of base; +egg = point 4 of cup {up} .. cap .. {down} cup & cycle; +\end{code} + +\subsection{Four point egg} +% eggs-four-point.mp +So far all the eggs have been drawn with semi-circular big end, but this +can be improved. +To get a smoother curve, you can use four different sized arcs with four different +centres of rotation to make up +each side of the egg.\mwpic{-36pt}{eggs-four-point} +\begin{code}[xleftmargin=0pt] +path egg, a, b, c, d; +a = fullcircle scaled 80; +b = a scaled 2 shifted point 6 of a; +c = halfcircle + scaled 2 (abs(point 0 of a - point 5 of b) - abs(point 0 of a)); +d = fullcircle + scaled 2 abs(point 2 of a - point 2 of c) shifted point 2 of c; + +egg = point 0 of c {up} .. subpath (1,3) of d .. {down} point 4 of c + .. subpath (5, 7) of b .. cycle; +\end{code} + +\newpage +\subsection{Five point egg} + +The next level of sophistication is to use five different arcs, +but this is more complex and you lose the +points at exactly E, N, W, and S.\mwpic{0pt}{eggs-five-better}\marginpar{\par\vskip 3in +\noindent\hey Instead of defining and joining circular arcs, this construction defines +the points for the egg and the desired directions at each point; +all the work of making the circular arcs is left to the $\to$ +connector. The six symmetrically-arranged rings $r_1$ to $r_6$ are used to define eight centres of +rotation $o_1$ to $o_8$ which are either points on the rings or intersections of +lines between them. Then eight directions $u_1$ to $u_8$ are defined at right +angles to lines between pairs of centre points. Finally the handy +\mpl{directionpoint} macro is used to find the points where the relevant circle is +moving in that direction. To make the egg, these points are joined up with $\to$ +constrained by the matching direction to make circular arcs.\\ +\llap{$\longleftarrow$\ }{\small Like so.} +} +\begin{smallcode}[xleftmargin=0pt] +numeric a; a = 56; +path r[]; % the rings +r1 = fullcircle scaled 2a shifted (0, -3/2 a); +r2 = fullcircle scaled 2a shifted (0, -1/2 a); +r3 = fullcircle scaled 2a shifted (0, +1/2 a); +r4 = fullcircle scaled 2a shifted (0, +3/2 a); +r5 = r2 rotatedabout(point 2 of r2, -60); +r6 = r2 rotatedabout(point 2 of r2, +60); +pair o[]; % the centres of rotation for each arc +o1 = point 6 of r5; +o2 = point 2 of r3; +o3 = point 6 of r6; +o4 = whatever[o3, point 2+4/3 of r2] = whatever[o2, point 2-4/3 of r1]; +o8 = whatever[o1, point 2-4/3 of r2] = whatever[o2, point 2+4/3 of r1]; +o6 = 1/2[point 2-4/3 of r1, point 2+4/3 of r1]; +o5 = whatever[o6, point 2-4/3 of r3] = whatever[o4, point 2+4/3 of r1]; +o7 = whatever[o6, point 2+4/3 of r3] = whatever[o8, point 2-4/3 of r1]; +pair u[], t[]; % directions and points for the egg path +u0 = (o8 - o1) rotated 90; t0 = directionpoint u0 of r6; +u1 = (o2 - o1) rotated 90; t1 = directionpoint u1 of r4; +u2 = (o2 - o3) rotated 90; t2 = directionpoint u2 of r4; +u3 = (o4 - o3) rotated 90; t3 = directionpoint u3 of r5; +u4 = (o5 - o4) rotated 90; +u5 = (o6 - o5) rotated 90; +u6 = (o6 - o7) rotated 90; +u7 = (o7 - o8) rotated 90; +t4 = directionpoint u4 of fullcircle scaled 2 abs (t3 - o4) shifted o4; +t5 = directionpoint u5 of fullcircle scaled 2 abs (t4 - o5) shifted o5; +t6 = directionpoint u6 of fullcircle scaled 2 abs (t5 - o6) shifted o6; +t7 = directionpoint u7 of fullcircle scaled 2 abs (t6 - o7) shifted o7; + +path egg; egg = for i=0 upto 7: t[i] {u[i]} .. endfor cycle; +\end{smallcode} + +\newpage +\subsection{A superellipse egg} + +All this construction is a good exercise in ingenuity, but if you just want +a simple quick egg path, then the \mpl{superellipse} macro gives you a \MP-specific +option. All you need is this:\mwpic{-36pt}{eggs-super} +\begin{code}[xleftmargin=0pt] +path egg; +egg = superellipse(right, 1.6 up, left, 1.2 down, 0.69); +egg := egg scaled 100; +\end{code} + + +\vfill \centerline{\tiny [This space intentionally left blank]} +\vfill + +\subsection{The perfect egg} + +The last word in egg curve perfection is the algebraic solution provided by the TDCC +Laboratory in Japan [{\small \mpl{https://nyjp07.com/index_E.html}}].\mwpic{0pt}{eggs-perfect} +\begin{code} +path egg; +egg = for t=-180 step 15 until 180 - eps: + (0.78 cosd(1/4 t) * sind(t), -cosd(t)) .. + endfor cycle; +egg := egg scaled 128; +\end{code} +Note that, unlike all the others, this path has 24 points and the path starts at the +top. You can draw it starting at 3 o’clock, and with only 8 points but the egg is slightly less perfect. +Just replace the loop above with: +\begin{code} +egg = for t=90, 135, 180, -135, -90, -45, 0, 45: + (0.78 cosd(1/4 t) * sind(t), -cosd(t)) .. + endfor cycle; +\end{code} + +\newpage +\subsection{Egg kitsch} + +If you want eggs that look solid, then you can use \mpl{interpath}:\mpic{-24pt}{eggs-shaded} +\begin{smallcode}[xleftmargin=0pt] +path egg, spot; +egg = (for t=-180 step 15 until 180 - eps: + (0.78 cosd(1/4 t) * sind(t), -cosd(t)) .. +endfor cycle) scaled 100; +spot = fullcircle scaled 4 shifted 3/4 point 3 of egg; +vardef fade_filled(expr egg, spot, dark, light, n) = image( + for i = 0 upto n: + fill interpath(i/n, egg, spot) withcolor ((i/n)**1/3)[dark,light]; + endfor) +enddef; +beginfig(1); + color a, b; + a = 1/256(150, 100, 60); + b = 1/256(256, 220, 180); + draw fade_filled(egg, spot, a, b, 256) rotated -30; +endfig; +\end{smallcode} +This works nicely with any of the "egg" paths defined in this section. +And finally, if all that has made you feel peckish, then how about +these?\mwpic{12pt}{eggs-fried} +\begin{smallcode}[xleftmargin=0pt, xrightmargin=-24pt] +path yolk, base; +color cooked_egg_yolk, cooked_egg_white; +cooked_egg_yolk = 1/256(216, 136, 49); +cooked_egg_white = 1/256(235, 237, 233); +vardef fried_egg(expr r) = + save base, yolk; path base, yolk; + yolk = for i=0 upto 17: (r + 1/8 normaldeviate) * dir 20i .. endfor cycle; + base = for i=0 upto 17: (2r + 1/8r * normaldeviate) * dir 20i .. endfor cycle; + base := base shifted (uniformdeviate r/2, uniformdeviate r/2); + image( + fill base withcolor cooked_egg_white; + fill yolk withcolor cooked_egg_yolk; + fill subpath (6.7, 9.6) of yolk scaled 0.8 -- + subpath (9.6, 6.7) of yolk scaled 0.66 -- cycle + withcolor 1/2[cooked_egg_yolk, white]; + ) enddef; +for i=0 upto 1: draw fried_egg(40) shifted 120 dir 120i; endfor +\end{smallcode} + +%=============================================================== +\newpage +\section{Data visualizations}\label{dviz} + +\textsc{Graphs and other displays} that show data, rather than a mathematical +function, are presented in this section, with a focus on illustrations for books and +technical papers. Most of the examples here follow those developed in Edward +Tufte's \textsl{The Visual Display of Quantitative Information}. + +\bigskip\noindent +Most of them have a pale manilla background, which was added using the technique +from §\ref{backgrounds}, like this: +\begin{code} +picture p; p = currentpicture; currentpicture := nullpicture; +bboxmargin := 12; +fill bbox p withcolor 1/32(32, 32, 31); +draw p; +\end{code} + +\newpage +\subsection{Simple time lines} + +\noindent\mpic{144pt}{tufte-mpg}\marginpar{ +\begin{itemize} + \item The complete Lua\TeX\ document shows the +use of "fontspec" to get the Palatino-like font, used in E.\@ Tufte's books. + \item The data is stored as \<pair> values in a \<path> variable. This only + works if both data values are numeric. If the values are larger than 4096 + you either need to scale them or use the "double" number system. + \item Notice how the horizontal unit $u$ is set by measuring the width of a label using +the technique discussed in §\ref{textsize}. And that you have to shift the data + path left before applying the $x$-scaling to avoid overflow. +\end{itemize} +} +\begin{texcode}[xleftmargin=0pt] +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{fontspec} +\setmainfont[Numbers=OldStyle]{TeX Gyre Pagella} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +\end{texcode}\vskip -8pt +\begin{smallcode}[xleftmargin=2pt] +beginfig(1); +path data, p; +data = (1978, 18) -- (1979, 19) -- (1980, 20) -- (1981, 22) + -- (1982, 24) -- (1983, 26) -- (1984, 27) -- (1985, 27.5); +numeric u, v; +u = xpart urcorner textext("1980\kern 0.75em"); v = 8; +p = data shifted -(xpart point 0 of data, 0) xscaled u yscaled v; +draw (xpart point 0 of p, 20) -- p -- (xpart point infinity of p, 20); +draw (xpart point 0 of p, 0) -- (xpart point infinity of p, 0); +for t=0 upto length p: + numeric x, y; (x, y) = point t of p; + undraw (x, y) withpen pencircle scaled 2 dotlabeldiam; + draw (x, y) withpen pencircle scaled dotlabeldiam; + label("\strut" & decimal ypart point t of data, (x, y + 8)); + draw (x, 0) -- (x, 4); + label("\strut" & decimal xpart point t of data, (x, 12)); +endfor +dotlabel.rt(btex \vbox to 6pt{\halign{\small #\hss\cr +13.7 mpg, average\cr for all cars on\cr road, 1978\cr}\vss} etex, +(xpart point 0 of p, 13.7v)); +dotlabel.lft(btex \vbox to 6pt{\halign{\small #\hss\cr +19.5 mpg, expected\cr average for all cars\cr on road, 1985\cr}\vss} etex, +(xpart point infinity of p, 19.5v)); +label.top(btex \vbox{\halign{\hss\textsc{#}\hss\cr +required fuel economy standards:\cr +new cars built from 1978 to 1985\cr}} etex, +point 5/2 of bbox currentpicture shifted 21 up); +endfig; +\end{smallcode}\vskip -10pt +\vbox to 0pt{\begin{texcode}[xleftmargin=0pt] +\end{mplibcode} +\end{document} +\end{texcode}\vss} + +\newpage +\subsection{Time line with minimal annotation} +\noindent\mpic{120pt}{tufte-budget}\marginpar{ +\begin{itemize} + \item This one has an even sparser frame, and + uses the \mpl{ahangle} trick to make the neat + I-beam annotation mark. + \item It is suggested that you resist the temptation to make very many + special macros to do charts like this; the ideas here are mainly + to show that \MP\ makes a good environment for following Tufte’s advice + about charts: to maximize data ink, and minimize chart junk. +\end{itemize}} +\begin{smallcode}[xleftmargin=0pt] +beginfig(1); +path data, p; +data = (1967, 311) -- (1968, 332) -- (1969, 372) -- (1970, 385) + -- (1971, 385) -- (1972, 393) -- (1973, 387) -- (1974, 381) + -- (1975, 387) -- (1976, 400) -- (1977, 380); +numeric u, v; +u = xpart urcorner textext("1980\kern 0.75em"); v = 1.414; +p = data shifted -(xpart point 0 of data, 300) xscaled u yscaled v; +draw p; +for d = 300 step 20 until 400: + numeric y; y = (d - 300) * v; + label.lft("\strut\scriptsize\$\,\small" & decimal d, (-12, y)); + draw (-8, y) -- (-12, y); + if d > 370: + draw (xpart point 3 of p, y) -- (xpart point infinity of p + 8, y) + dashed evenly scaled 1/4 withpen pencircle scaled 1/4; + fi +endfor +path a; a = (xpart point infinity of p + 21, (380 - 300) * v) + -- (xpart point infinity of p + 21, (400 - 300) * v); +interim ahangle := 180; +interim ahlength := 2; +drawdblarrow a withpen pencircle scaled 1/4; +label("\small 5\%", point 1/2 of a shifted 12 right); +for t=0 upto length p: + numeric x, y; (x, y) = point t of p; + undraw (x, y) withpen pencircle scaled 2 dotlabeldiam; + draw (x, y) withpen pencircle scaled dotlabeldiam; + draw (x, -8) -- (x, -12); + label("\strut\small" & decimal xpart point t of data, (x, -20)); +endfor +label.urt(btex \vbox{\halign{\small #\hfill\cr +Per capita\cr budget expenditure\cr in constant dollars\cr}} etex, +(0, (410 - 300) * v)); +endfig; +\end{smallcode} + +\newpage +\subsection{Time line with more complex dates} + +If the dates in your time line are more granular than years, then you need a +better way to deal with them. This chart shows the £/€ exchange rate by month. +$$\includegraphics[width=\textwidth]{tufte-currency}$$ +The dates on the horizontal axis where transformed from calendar dates to a +serial number using this routine: +\begin{code} +vardef base(expr Y, M, d) = + save m, y; numeric m, y; + if M < 3: m = M + 9; y = Y - 1; + else: m = M - 3; y = Y; fi + 365/1024 y + (floor(y/4) - floor(y/100) + floor(y/400) + + floor((2+3m)/5) + 30m + d - 307) / 1024 +enddef; +\end{code} +which allows you to add events like this: +\begin{code} +dotlabel.bot("Brexit vote", (base(2016, 6, 24) * u, 78 v)); +\end{code} +for suitable values of $u$ and $v$. + +\newpage +\section{Commutative diagrams} + +\textsc{If you want lots} of complex bells and whistles on your commutative diagrams, then +you probably want to use specialist tools like "tikz-cd" or "xypic", but if your +needs are simpler, plain \MP\ is more than capable, and if you are already +using it for other illustrations, there is one less thing to learn. + +Here are two examples to illustrate some general techniques. They may be familiar to +readers of the manuals for the tools referred to above. +$$ +\includegraphics{tikzcd-example} +\qquad +\includegraphics{xypic-example} +$$ +The complete code used to generate the right-hand picture is shown on the +right.\rlap{\rightarrow}% +\vadjust{\moveright5.5in\vbox to 0pt{\kern-218pt +\begin{smallcode} +picture U, XY, X, Y, Z; +z1 = -z2 = (-61, 42); + +U = thelabel("$U$", z1); +XY = thelabel("$X\times_ZY$", origin); +X = thelabel("$X$", (x2, 0)); +Y = thelabel("$Y$", (0, y2)); +Z = thelabel("$Z$", z2); + +forsuffixes @=U, XY, X, Y, Z: draw @; endfor + +ahangle := 20; +vardef curved_connect@#(expr s, a, b, d) = + save line, mark; + + path line; + line = center a {d} .. center b; + interim bboxmargin := 4; + drawarrow line cutbefore bbox a cutafter bbox b; + + picture mark; + mark = thelabel@#("$\scriptstyle " & s & "$", point 1/2 of line); + interim bboxmargin := 1; + unfill bbox mark; draw mark; +enddef; + +vardef connect@#(expr s, a, b) = + curved_connect@#(s, a, b, center b - center a) +enddef; + +connect.bot("p", XY, X); +connect.rt ("q", XY, Y); +connect.top("g", Y, Z); +connect.lft("f", X, Z); + +curved_connect.urt("x", U, X, right); +curved_connect.llft("y", U, Y, dir -80); + +drawoptions(dashed withdots scaled 1/2); +connect("(x,y)", U, XY); +drawoptions(); +\end{smallcode} +\noindent\textit{\small This example assumes you are using "mplibtextextlabel" – §\ref{ttlabel}} +\vss}} +The approach taken for both examples is first to define the node labels as pictures +placed as needed, and then write a special-purpose "connect" macro to make +consistent arrows between the nodes, using \mpl{center}, \mpl{bbox}, +\mpl{cutbefore}, and \mpl{cutafter} as appropriate. + +The $X\times_ZY$ example needs two variations. The "curved_connect" macro +takes a string for the label, the two nodes to be connected, and the initial +direction for the curved path to connect them. You can't have optional macro +arguments in \MP\ but you can call macros from inside another macro, so a second +simpler macro "connect" is defined to do straight connections. To avoid repetition, +this macro simply calls "curved_connect" with the required direction from $a$ to +$b$. + +For the $f^*E$ example, "connect" is simpler, because all the arrows are +straight and there are no labels, but needs to allow for arrows crossing: +\begin{smallcode} +vardef connect(expr a, b) = + save line; path line; interim bboxmargin := 4; + line = center a .. center b cutbefore bbox a cutafter bbox b; + cutdraw line withpen pencircle scaled 4 withcolor background; + drawarrow line +enddef; +\end{smallcode} + +\newpage +\section{Tilings and tessellations} + +\textsc{In mathematical terms}, a “tiling” is a countable set of tiles +that cover the plane without gaps or overlaps.\footnote{Adapted from \textsl{Tilings +and Patterns}, Branko Grünbaum \& G.\@ C.\@ Shephard, Freeman, 1987} +This section loosely follows that idea, and presents some ideas and general +techniques for creating tilings and other patterns or textures. + +You can make an effective grid by drawing repeated lines and then clipping to the +size you want:\mpic{-42pt}{tiling-simple} +\begin{code} +for i = -10 upto 10: + draw (left--right) scaled 200 shifted (0, 20i); + draw (down--up) scaled 200 shifted (20i, 0); +endfor +clip currentpicture to fullcircle scaled 200; +\end{code} +but this is a bit limited. If you want to produce more interesting tilings, you +need to define a unit shape or picture, and a pair of vectors to repeat it. +\begin{code} +path unit; pair u, v; color a, b; +unit = unitsquare scaled 24; +u = point 1 of unit - point 0 of unit; +v = point 3 of unit - point 0 of unit; +a = 3/4[red, white]; b = 3/4[blue, white]; +for i=-5 upto 5: + for j=-5 upto 5: + fill unit shifted (i*u + j*v) + withcolor if odd (i+j): a else: b fi; + draw unit shifted (i*u + j*v); + endfor +endfor +clip currentpicture to fullcircle scaled 200; +\end{code} +In tilings with more complex shapes you may find that using \mpl{fill} and +\mpl{draw} in the same loop causes uneven lines because the fill overlaps part of +the line. In these cases it is a good idea to duplicate the loops; use the first +set for filling, the second for drawing. + +\newpage +\subsection{Tiling with regular polygons}\label{sec:regtiling} + +\textsc{After tiling with squares}, the two simplest tilings are with triangles and +hexagons\mwpic{-24pt}{tiling-hex-trig} +(using the regular polygons from §\ref{polygons}). The basic loop is the same as the +previous page except +that the vectors $u$ and $v$ are now at 60° to each other (as shown in blue and +red in the examples to the right). All of these examples where drawn with the same +basic loop as before: +\begin{code} +for i = -n upto n: + for j = -n upto n: + draw P shifted (i * u + j * v); + endfor +endfor; +\end{code} +In the first row, $P$ was set to a simple polygon path: +\begin{code} +triangle = for i=0 upto 2: (0, 16) rotated 120i -- endfor cycle; +hexagon = for i=0 upto 5: (0, 16) rotated 60i -- endfor cycle; +\end{code} +The vectors $u$ (in red) and $v$ (in blue) were defined as (for both tilings): +\begin{code} +u = point 0 of triangle - point 1 of triangle; +v = u rotated -60; +\end{code} +To make the coloured versions, $P$ was defined as an appropriate \<picture>. +For the triangular tiling, it looked like this: $\vcenter{ +\begin{mplibcode} +input colorbrewer-rgb +path t, tt; +t = for i=0 upto 2: (0,8) rotated 120i --endfor cycle; +tt = t reflectedabout(point 2 of t, point 0 of t); + beginfig(0); +fill t withcolor Reds 8 2; fill tt withcolor Blues 8 2; draw t; draw tt; endfig; +\end{mplibcode}}$ so that the tiling actually filled the plane. In the hexagonal +tiling there are no gaps to fill, but in order to get a non-adjacent colouring, the +unit picture was defined as three shifted copies of the hexagon each filled with a +different color. The unit vectors were therefore scaled by $\sqrt3$ and rotated by +30° (as shown). + +The bottom row the unit pictures were replaced with drawings that connect the centre +of each shape to the midpoint of each side (in red), like this: +$$ +\begin{mplibcode} +path p[]; input colorbrewer-rgb +p3 = for i=0 upto 2: (0, 14) rotated 120i -- endfor cycle; +p6 = for i=0 upto 5: (0, 14) rotated 60i -- endfor cycle; +picture P[]; +for i=3,6: + P[i] = image( + for j=1 upto length p[i]: + draw origin -- point j+1/2 of p[i] withcolor Reds 7 6; + endfor + draw p[i]); +endfor + beginfig(0); draw P3; draw P3 reflectedabout(point 0 of p3, point 2 of p3); + draw P6 shifted (60, 3); endfig; +\end{mplibcode} +$$ +This has the effect of +connecting the centres of adjacent shapes in the tiling, which reveals that each +tiling is the dual of the other. + +\newpage +\subsection{Separating filling and drawing} + +\textsc{Repeating a unit image} can sometimes cause unwanted overlaps, so as noted +above, the solution is to make a filler unit and a drawing unit and do the filling +first and the drawing second. In this example\mwpic{-32pt}{arch-4-8-8} the drawing +unit (the octagon) is simple so you can just draw that path instead of making another +\<picture> for it. +\begin{smallcode} +input colorbrewer-rgb +path o, r[]; % o is the octagon, r[] are the "corners" +o = (for i=0 upto 7: 21 dir 45i -- endfor cycle) rotated -90/4; +pair t; t = whatever[point 0 of o, point 1 of o] + = whatever[point 2 of o, point 3 of o]; +r1 = subpath (1,2) of o -- t -- cycle; +r2 = r1 rotated 90; +r3 = r2 rotated 90; +r4 = r3 rotated 90; +picture filler; filler = image( + filldraw r1 withcolor Reds 8 3; + filldraw r3 withcolor Reds 8 3; + filldraw r2 withcolor Blues 8 3; + filldraw r4 withcolor Blues 8 3; + filldraw o withcolor Purples 8 2; +); +pair u, v; +u = point 0 of o - point 5 of o; v = u rotated 90; +beginfig(1); +numeric n; n = 5; +for i=-n upto n: + for j=-n upto n: + draw filler rotated ((i+j) mod 2 * 90) shifted (i*u + j * v); + endfor +endfor +for i=-n upto n: + for j=-n upto n: + draw o shifted (i*u + j * v); + endfor +endfor +\end{smallcode} +Rotating every other \mpl{filler} allows you to get the alternate colours in the squares. +\mpic{-60pt}{arch-4-8-8-parts}% +Using \mpl{filldraw} ensures that there are no gaps between adjacent segments. + +\newpage +\subsection{Tilings with more complex patterns} + +\textsc{The next example} also uses the square lattice,\mwpic{0pt}{arch-3-4-3-4} +but the unit is more complicated, so the drawing needs two \<picture> +variables, one for the colour fill and a second for the grid. +\begin{smallcode} +input colorbrewer-rgb +path s[], t[]; +s1 = unitsquare scaled 21 rotated 15; +s2 = s1 rotated 150; +t1 = subpath (4, 3) of s1 -- point 1 of s2 -- cycle; +t2 = t1 reflectedabout(point 1 of t1, point 2 of t1); +t3 = t1 rotated 150; +t4 = t1 rotated 210; +picture color_unit, grid_unit; +color_unit = image( + fill s1 withcolor Oranges 8 1; + fill s2 withcolor Oranges 8 2; + fill t1 withcolor Blues 8 1; + fill t2 withcolor Blues 8 2; + fill t3 withcolor Blues 8 3; + fill t4 withcolor Blues 8 4; +); +grid_unit = image( + draw s1; draw s2; + draw t2; draw t3; draw t4; +); +pair u, v; +u = point 1 of s1 - point 1 of s2; +v = u rotated 90; +numeric n; n = 4; +forsuffixes $=color_unit, grid_unit: + for i=-n upto n: + for j=-n upto n: + draw $ shifted (i * u + j * v); + endfor + endfor +endfor +\end{smallcode} + + + +\newpage +\subsection{Showing the dual tiling} + +\textsc{These tilings can be classified} by the configuration of the polygons that meet +at each vertex. This one is $(3^4, 6)$ because each vertex has four triangles and one +hexagon. It exists in two enantiomorph forms.\mxpic{-80pt}{3.6in}{arch-snub-hexagon} +The unit pictures look like this: +\par\bigskip +\vbox{\halign{#&\qquad\qquad\vbox to 48pt{\hsize=2.7in\noindent #\par\vss}\cr +\includegraphics[scale=0.75]{arch-snub-hexagon-unit} +&\hey\itshape To reveal the dual of the tiling, you can draw a line from +the median (or centroid if you prefer) of each polygon to the centre of each +edge.\cr}} + +\bigskip +\noindent They are drawn like this, where $h$ is the hexagon, $t_1 \dots\ t_6$ are the blue +triangles surrounding it, and $t_7$ \& $t_8$ are the two “connecting” triangles, +which swap sides to make the enantiomorphs. +\begin{code} +unit[k] = image( + for i=1 upto 6: + fill t[i] withcolor Blues 8 if odd i: 2 else: 3 fi; + endfor + for i=7 upto 8: + fill t[i] withcolor Oranges 8 if odd i: 3 else: 2 fi; + endfor + fill h withcolor Oranges 8 2; + forsuffixes S=h, t1, t2, t3, t4, t5, t6, t7, t8: + draw S withpen pencircle scaled 1/4 withcolor 3/4; + pair m; m = median(S); + for i=1 upto length S: + draw m -- point i - 1/2 of S withcolor 3/4; + endfor + endfor); +\end{code} +The $\id{median}()$ routine is from §\ref{polygons-given-side} and the colours are from +§\ref{colorbrewer}. This tiling is generated using the loop-with-triangular-grid-vectors +from §\ref{sec:regtiling}. + +\newpage +\subsection{Tiling with a dynamic unit} + +\textsc{In order to reveal} patterns in a tiling, you might want to vary the colours +or line styles used in each repeated drawing unit. In this case, you can write a +macro that takes a parameter and returns a picture to draw.\mwpic{-36pt}{arch-3-4-6-4} + +\begin{smallcode} +\mplibsetformat{metafun} % <-- for the "transparent" macro +\begin{mplibcode} +input colorbrewer-rgb +beginfig(1); +numeric r; r = 5; +path d; d = for i=1 upto 12: + (0, r) shifted (r * (sqrt(3)+2), 0) rotated 30i -- +endfor cycle; +color shade[]; +shade0 = Oranges 8 3; shade2 = Greens 8 3; +shade1 = Blues 8 3; shade3 = Reds 8 3; +vardef unit(expr n) = image( + fill d withcolor transparent (2, .9, shade[n mod 4]); + draw d; +) enddef; +pair u, v; +u = point 0 of d - point 3 of d; +v = u rotated 60; +numeric n; n = 6; +for i=-n upto n: + for j=-n upto n: + draw unit(i*j) shifted ((i - floor(j / 2)) * u + j * v); + endfor +endfor +clip currentpicture to unitsquare shifted -(1/2, 1/2) scaled (n * 9 r); +endfig; +\end{mplibcode} +\end{smallcode} +Instead of defining six triangles, six squares, and a hexagon, you can just define +the dodecagon and overlap each one. Using a macro to create the unit, allows you to +choose a different colour for each filler. Using the \mpl{transparent} macro from +Metafun automatically mixes the colours for the overlaps. But the edges don't look +so good, so you need to clip the whole picture to a neat square. + + +\newpage +\section{Recursion and iteration} + +\textsc{This chapter is not a tutorial} on recursion or iteration \textit{per se}, but rather +more of an exploration of the \MP\ techniques you can use to create some particular types +of drawing such as trees, plane-filling curves, non-periodic tilings, and fractals.\mwpic{-48pt}{rec-bush} + +\medskip\noindent +Even so, +it is perhaps useful to review some of the basic programming ideas +involved. +Consider for example the greatest common divisor algorithm presented in +§\ref{sec:gcd}. +\begin{smallcode} +vardef gcd(expr a, b) = if b = 0: a else: gcd(b, a mod b) fi enddef; +\end{smallcode} +The reason this works is that we know (mathematically) that $0 \le a \bmod b < b$, +and therefore that the arguments to the recursive call must get smaller each time +and eventually we must get to $b=0$ when the answer will be $a$. This is the basic +recursive approach: ensure at least one argument gets smaller each time and stop +when you get to a given limit. +The examples in this chapter use one of two simple approaches: +\begin{itemize} + \item Explicitly pass a \mpl{level} argument that is decremented on each + recursive call, and stop the recursion when the level gets to zero + \item Pass a \<path> or two \<pair> arguments, and stop the recursion when the + path is too short or the pairs are too close together. +\end{itemize} +If you find recursion confusing, you can nearly always use an iterative approach +instead. For example, you can implement the \mpl{gcd} function like this: +\begin{smallcode} +vardef gcd(expr A, B) = save r, a, b; numeric a, b, r; a := A; b := B; + forever: + r := a mod b; exitif r = 0; + a := b; b := r; + endfor b +enddef; +\end{smallcode} +Notice that you have to use assignment in the loop to update the variables, and +that you cannot assign to the arguments of a macro. Notice also that this version +requires both arguments to be positive integers. You need to use your judgement to +decide which is the better approach for a given problem. + +\newpage +\subsection{The Koch curve} + +\textsc{The Swedish mathematician} Helge von Koch\mpic{-30pt}{rec-koch-steps} originally devised the +Koch curve as an example of a non-differentiable curve that could be constructed with +elementary geometry. It makes a good introduction to recursive paths with \MP. The +construction is recursive: each straight line segment in the path is replaced with +four copies of itself, scaled down $\frac13$ and arranged as shown +at Level 1 \rightarrowfill\break +At each level of the construction, the number of points in the path increases +four-fold and the \mpl{arclength} of the path gets $\frac43$ longer. +\begin{code} +vardef koch(expr level, a, b) = + if level = 0: + a -- b + else: + save p, q, r; pair p, q, r; + p = 1/3[a,b]; r = 2/3[a,b]; q = r rotatedabout(p, 60); + koch(level-1, a, p) & + koch(level-1, p, q) & + koch(level-1, q, r) & + koch(level-1, r, b) + fi +enddef; +\end{code} +The five levels were drawing using this function in a loop like this: +\begin{code} +for n=0 upto 4: + draw koch(n, origin, 300 right) shifted (0, -100n) +endfor +\end{code} +The annotations in the middle show the \mpl{length} of the path and the +\mpl{arclength}. The curve will always fit in the same bounding box but in theory it +can become infinitely long. In practice it looks much the same past level 6, and +you will start running into \MP\ limitations. The length of a path is not limited by +the default scaled number system, but the numbers actually returned by \mpl{length} +and \mpl{arclength} may not be reliable for very long paths. Also the call stack may +get very large and the single-threaded processing time can get very slow, and when +the length of each segment shrinks to the size of your pen, you will start losing +detail. + +\newpage +\subsection{Sierpinski's gaskets} + +\textsc{The second example} of recursive construction also dates from the early 20th +century, but unlike von Koch's infinite curve, the area of Sierpinski's gasket tends +to zero.\mwpic{-12pt}{rec-sierpinski-triangle} In the original specification, you +are supposed to remove the central quarter of each triangle, but this program does +it the other way round and delays drawing the triangles until they are small enough. +\begin{code}[xleftmargin=0pt, xrightmargin=-2in] +vardef gasket(expr t, s, limit) = + if length (point 1 of t - point 0 of t) < limit: + fill t; + else: + save little_t; path little_t; little_t = t scaled s; + for i=1 upto length t: + gasket(little_t shifted (point i of t - point i of little_t), s, limit); + endfor + fi +enddef; +\end{code} +Note the useful idiom \mpl{shifted (point i of t - point i of little_t)} – this +neatly tucks a copy of the small triangle into the appropriate corner of the big +triangle. You can make this even simpler by coding the scaling parameter $s$ and +\id{limit} as constants in the recursive routine, so that you do not have to pass +them down each time. The triangular gasket was generated using the macro like this: +\begin{code}[xleftmargin=0pt, xrightmargin=-2in] +beginfig(1); + path T; T = for i = 1 upto 3: 220 up rotated 120i -- endfor cycle; + gasket(T, 1/2, 20); +endfig; +\end{code} +You can generalize it, for example by making +the path \id{T} into a pentagon with scaling factor $s=(3-\sqrt5)/2$, +or a hexagon with $s=1/3$, and so on: + +\vbox to 0pt{\vskip-2pt\hsize10in\noindent\includegraphics[width=10in]{rec-sierpinski-pentagon.pdf}\par\vss} + +\newpage +\subsection{The Heighway dragon} + +\textsc{The Heighway dragon curve} dates from the 1960s, and is created in a similar +way to the Koch curve: each straight line segment is recursively replaced with two +copies of itself scaled down and arranged as shown.\mwpic{-36pt}{rec-dragon} + +\medskip +\centerline{\includegraphics{rec-heighway-stages}} + +\noindent +Note that every other segment is flipped left and right. At each stage, the number of points +increases two-fold and the path gets $\sqrt2$ times longer. Here is a recursive +routine to generate the curve as a single path. +\begin{smallcode} +numeric r, theta; r = sqrt 1/2; theta = 45; +vardef dragon(expr level, a, b) = + if level > 0: + save p; pair p; + p = r[a, b] rotatedabout(a, theta); + dragon(level - 1, a, p) & reverse dragon(level - 1, b, p) + else: + a .. b + fi +enddef; +\end{smallcode} +The blue dragon was created with: {\small\mpl{draw dragon(15, origin, 240 right)}}\rlap{\raise1ex\hbox{\ $\smash{\nearrow}$}} + +After the fourth level the corners of the curve start to touch each +other, but they never cross. You can see this if you draw the curve with rounded +corners. +\begin{smallcode} +vardef rounded_corners expr p = + save r, n; numeric r, n; r = 1/3; n = length p; + subpath (0, 1-r) of p + for t=1 upto n-1: + .. subpath (t+r, t+1-r) of p + endfor .. subpath (n-r, n) of p +enddef; +\end{smallcode} +The red dragon was: {\small\mpl{draw rounded_corners dragon(10, origin, 240 +right)}}\rlap{\hbox{\ $\smash{\longrightarrow}$}} + +\vfill +\noindent{\small\llap{\hey\ }The length of the dragon path at level $n$ is +$2^n$, so if you want to do arithmetic with the path length you need to use a big +number system when $n > 11$.} + +\newpage +\subsection{Iterative dragons} + +\textsc{The dragon curve} can also be created with iteration instead of recursion. +Another way of viewing the stages of developing the curve is that at each stage the +whole curve is replaced by two copies of itself arranged as shown: + +\medskip +\centerline{\includegraphics{ifs-heighway-stages}} + +\smallskip\noindent +We can do this in two steps in a loop, like this:\mpic{-160pt}{ifs-heigh} +\begin{smallcode} +path p; p = origin -- dir 30; +numeric n; +for i=1 upto 12: + n := length p; + p := p rotated 45; + p := p & reverse p rotatedabout(point n of p, 90); +endfor +draw p scaled (384 / xpart (urcorner p - llcorner p)); +\end{smallcode} +In this approach, instead of scaling down each time, the curve is just allowed to +grow $\sqrt2$-times bigger in each loop, then scaled to the desired width +(384\unit{pt}) when it is complete. The 12 stages in the loop produce the rotated blue +dragon as shown.\rlap{\raise1ex\hbox{\ $\smash{\nearrow}$}} + +\vfill\noindent +With a small adaptation, you can also use this to explore variations. +\begin{smallcode} +path p; p = origin -- dir 30; +numeric n, r; r = 3; +for i=1 upto 12: + n := length p; + p := p rotated (45 - r); + p := p & reverse p rotatedabout(point n of p, 90 + 2r); +endfor +\end{smallcode} +The extra parameter $r$ is used to open up the folds, making it more +“organic”.\mpic{-200pt}{ifs-heigh-open} + +\newpage +\subsection{The golden dragon} + +\textsc{You can also explore variations} with the recursive approach by altering +the scaling and rotation. Here $r\simeq 0.74274$ and +$\theta\simeq 32.893°$ in the initial triangle.\vadjust{\moveright136pt\vbox to +0pt{\vskip 3.16in \hsize 3.6in \noindent +Because the initial triangle shape is shorter on one side, it is better to adapt +the \mpl{dragon} routine to measure the gap between points instead of using a fixed +level parameter: +\begin{smallcode} +vardef dragon(expr a, b) = + if abs(a-b) > 1: + save p; pair p; + p = r[a, b] rotatedaround(a, theta); + dragon(a, p) & reverse dragon(b, p) + else: + a .. b + fi +enddef; +\end{smallcode} +\vss}} +$$\hbox to \textwidth{\includegraphics{rec-dragon-golden.pdf}\hss}$$ + +\newpage +\subsection{The Peano-Gosper curve, or flow-snake} + +\textsc{The Peano-Gosper curve} is a space-filling curve. It is +constructed in the same way as the dragons, but generating shape has seven sections +instead of two, and they are cunningly arranged to fill the space.\mpic{-48pt}{rec-flowsnake-construction} +The arrows indicate the reversals so that the scaled-down copies fill the larger +part of the hexagons that contain them.\rlap{\hbox{\ $\smash{\rightarrow}$}} +$$\includegraphics[width=\textwidth]{rec-flowsnake}$$ +This fourth-level curve is the boundary between the filled and unfilled branches. +\vadjust{\moveright 384pt \vbox to 0pt{\vss\hsize 4in +\noindent Given the \id{snake} path scaled to unit length as shown above in +\red{red}, the +flow-snake path can be created with: +\begin{smallcode} +vardef rattle(expr level, a, b) = + if level > 0: + save s; path s; s = snake zscaled (b-a) shifted a; + reverse rattle(level - 1, point 1 of s, a) & + rattle(level - 1, point 1 of s, point 2 of s) & + rattle(level - 1, point 2 of s, point 3 of s) & + rattle(level - 1, point 3 of s, point 4 of s) & + reverse rattle(level - 1, point 5 of s, point 4 of s) & + reverse rattle(level - 1, point 6 of s, point 5 of s) & + rattle(level - 1, point 6 of s, b) + else: + a -- b + fi +enddef; +path S; S = rattle(4, origin, 360 right); +\end{smallcode} +}} + +\newpage +\subsection{Fractal trees} + +\textsc{You can make a tree} with a vertical line segment, a scaling factor $0< r < +1$, and an angle $0° < \theta < 180°$. Start with the vertical line as the trunk, +then make the first branch by scaling the trunk by $r$, rotating it $+\theta$, and +moving it to the top of the trunk; make the second in the same way but rotate it by +$-\theta$. Then repeat using each branch as a new trunk.\mpic{-114pt}{rec-simple-tree} +And stop after enough levels. + +This can be implemented easily in \MP, but requires a +slightly different technique. There is no simple way to represent the tree as a single +\<path> because a path cannot have branches, so +you need to draw each segment separately instead of trying to join them up. Compare +this to the \mpl{rattle} routine on the page before. +\begin{code} +vardef make_tree(expr level, bar) = + draw bar; + if level > 0: + for t=-theta, theta: + make_tree(level - 1, bar shifted - point 0 of bar + scaled r rotated t shifted point 1 of bar); + endfor + fi +enddef; +\end{code} +To combine several such trees in a drawing either move the initial \id{bar} path, or +capture the tree as a \<picture> using \mpl{image(make_tree(...))}, like so: +\begin{code} +picture T[]; numeric r, theta; +r = 0.58; theta = 60; T1 = image( + make_tree(3, origin -- 100 up); +); +% update r and theta... +r := 0.75; theta := 14; T2 = image( + make_tree(10, origin -- 100 up); +); +draw T1 shifted 32 up; +draw T2 shifted 128 right; +\end{code} +Notice that $r$ and $\theta$ are treated as global constants. You could pass them +to the \mpl{make_tree} routine instead. Notice also that the smaller tree only has +three levels, but the larger one has 10. And that the labels were omitted. + +\newpage +\subsubsection{More fractal vegetation} + +\textsc{If you are more interested} in the visual aspect of your tree than the +mathematical, you can tweak the \mpl{make_tree} routine a +bit.\mpic{-12pt}{rec-general-tree} +\begin{smallcode}[xleftmargin=0pt] +vardef make_tree(expr bar) = + save a; numeric a; a = abs(point 1 of bar - point 0 of bar); + cutdraw bar withpen pencircle scaled 1.2(1/8 a) withcolor background; + cutdraw bar withpen pencircle scaled (1/8 a) withcolor 1/256(57, 35, 32); + if a > leaf: + save s; pair s; + s = 1/32 a * r * unitvector(direction 1 of bar) rotated 90; + make_tree(bar shifted - point 0 of bar + shifted s scaled r rotated theta + shifted point 1 of bar); + make_tree(bar shifted - point 0 of bar + shifted -s scaled r rotated -theta + shifted point 1 of bar); + else: + draw point 1 of bar withpen pencircle scaled 1 withcolor 2/3 green; + fi +enddef; +beginfig(1); + numeric leaf, r, theta; + r = 0.75; theta = 14; leaf = 3; + make_tree(origin -- 100 up); +endfig; +\end{smallcode} +\begin{itemize} + \item The recursion is controlled by measuring the length of the branches instead + of using a \id{level} parameter. + \item Each branch is drawn with a pen scaled to $\frac18$ of its length using + \mpl{cutdraw}, and outlined so you can see the crossings. + \item Each branch is moved slightly left or right with $s$ to make the joins + neater. + \item Tiny green leaves are added on the ends of each branch. +\end{itemize} + +\newpage +\subsubsection{Randomized recursive plants} + +\vskip-\baselineskip +\noindent\hbox to \linewidth{\includegraphics{rec-general-tree-deviate}\hss} + +\noindent +\textsc{To get plants that look more natural} you can introduce some random factors. +Strictly these are not fractal because they are not +self-similar. The only change from the previous \mpl{make_tree} is to replace +\mpl{scaled r rotated theta} with +\begin{smallcode}[xleftmargin=0pt] +scaled (r + 1/16 normaldeviate) rotated (theta + 8 normaldeviate) +\end{smallcode} +The bush is similar, except that it splits into four branches at each step instead of +two and only the lengths randomized.\mwpic{-96pt}{rec-bush} +The colouring was a happy accident. +\begin{smallcode}[xleftmargin=0pt, xrightmargin=-100pt] +input colorbrewer-rgb +vardef bush(expr start, aim, level, limit) = + save s, target; numeric s; pair target; s = level / limit; + for a = -32, -8, 8, 16: + target = aim scaled ((32 + 16 normaldeviate) * s) rotated a shifted start; + draw start -- target withpen pencircle scaled s withcolor BrBG[limit][limit-level]; + if level > 1: bush(target, aim rotated a, level - 1, limit); fi + endfor +enddef; +beginfig(1); bush(origin, dir 80, 6, 8); endfig; +\end{smallcode} + +\newpage +\subsection{Penrose tilings} + +\textsc{Recursion is useful} for generating non-periodic tilings,% +\vadjust{\moveright5.5in\vbox to 0pt{\kern-8pt +\begin{smallcode} +% golden ratio constant +numeric psi; psi = (sqrt 5 - 1) / 2; + +% inflating tall shape makes two others +vardef inflate_tall(expr level, a, b, c) = + if level = 0: + draw a--b--c withpen pencircle scaled 1/8; + else: + save d; pair d; d = psi[b,a]; + inflate_tall(level - 1, d, c, a); + inflate_wide(level - 1, c, d, b); + fi +enddef; + +% inflating wide makes three +vardef inflate_wide(expr level, a, b, c) = + if level = 0: + draw a--b--c withpen pencircle scaled 1/8; + else: + save d, e; pair d, e; d = psi[a,b]; e = psi[a,c]; + inflate_tall(level - 1, d, e, b); + inflate_wide(level - 1, e, d, a); + inflate_wide(level - 1, c, e, b); + fi +enddef; + +% start with a tall triangle with apex at origin +pair a, b, c; +b = origin; +c = (sind(18), sind(72)) scaled 800; +a = (-xpart c, ypart c); + +% make an inflated "wedge" +picture P; P = image(inflate_tall(7, a, b, c)); + +% then exploit the five-fold symmetry to make the picture +for t = 0 upto 9: + draw P if odd t: reflectedabout(b, c) fi rotated 72t; +endfor; +\end{smallcode} + +\qquad\hey \textit{\small Colouring the tiling is left as an exercise for the +reader.}\par\vss}} +like those discovered in the 1970s by the British polymath Sir Roger Penrose. +$$\includegraphics[width=\textwidth]{penrose-P3.pdf}$$ +By definition, there is no periodic unit to repeat; but there are well-known +expansion rules to split each shape in the tiling into smaller congruent copies. So +you can start with a single large shape and apply the rules recursively to make the +tiling. The tiling above is made from two rhombus shapes, but each of these shapes can be +split in half into two triangles, and the expansion process actually works on these: +$$\includegraphics{penrose-stages.pdf}$$ + +\vbox to 0pt{\noindent +Each expansion of half the thick rhombus, produces two more thick halves and one +thin half; each expansion of half the thin rhombus produces a thick half and a thin +half. No other shapes are produced, so the recursive process can be repeated +indefinitely. At the desired lowest level the triangles are patched back into +rhombus shapes simply by \textit{not} drawing the edges shown with dots.\par\vss} + +\newpage +\subsection{Pinwheel tiling} + +\textsc{Another non-periodic tiling} is the so-called pinwheel tiling, devised in 1994 +by Charles Radin, based on +this dissection by John Conway.\mpic{-22pt}{pinwheel} +$$\includegraphics{conway.pdf}$$ +Starting with a triangle of this shape, the tiling recursively divides into +five smaller copies of itself. The colouring is passed down each level but only used +on the lowest. +\begin{smallcode} +input colorbrewer-rgb +vardef pinwheel(expr level, a, b, c, s) = + if level = 0: + fill a--b--c--cycle withcolor s; + draw a--b--c--cycle withpen pencircle scaled 1/8 withcolor white; + else: + save d, e, f, g; pair d, e, f, g; + d = 2/5[b, c]; e = 4/5[b,c]; f = 1/2[e, a]; g = 1/2[a, b]; + pinwheel(level - 1, e, a, c, Blues 9 4); + pinwheel(level - 1, f, g, a, Blues 9 3); + pinwheel(level - 1, f, g, e, Blues 9 2); + pinwheel(level - 1, d, e, g, Blues 9 5); + pinwheel(level - 1, d, b, g, Blues 9 6); + fi +enddef; +beginfig(1); + pinwheel(5, origin, 200 right, 100 up, ""); + pinwheel(5, (200,100), 100 up, 200 right, ""); +endfig; +\end{smallcode} +Notice that to make the dissection work, it is important to pass the three +\<pair> arguments in the right order. + + + +\newpage +\tableofcontents + +\newpage +\section*{To do...} + +- angle marks, including curved angle marks + +- drawing knots, double lines, ropes + +- decorating lines + +- the eye, hand + +- physics diagrams, pendulum, indicating movement and vibration + +- examining a glyph + +- all sorts of arrow, arrows between arrows, arrows next to a path (handles) + +- faking 3d + +- functions, def, vardef, recursion, primarydef etc, of syntax, expr suffix text + +- four box model charts - Tufte charts - Venn diagrams + +- tour of the plain format + +\end{document} diff --git a/info/drawing-with-metapost/src/angles-lines.mp b/info/drawing-with-metapost/src/angles-lines.mp new file mode 100644 index 0000000000..5b2f3be14c --- /dev/null +++ b/info/drawing-with-metapost/src/angles-lines.mp @@ -0,0 +1,78 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + % define a unit + numeric u; + u = 1cm; + + % define the axes + path xx, yy; + xx = (2 left -- 10 right) scaled u; + yy = (down -- 7 up) scaled u; + + % and the angles + numeric theta, psi; + psi = 28; + theta = 50; + + % and some points + pair P, Q, H, N, M, T, L; + P = (4, 3) scaled 1cm; + + % now some lines + path ell, tee, arc; + ell = (left--right) scaled 10u rotated theta shifted P cutbefore xx; + tee = (left--right) scaled 10u rotated psi shifted P cutbefore xx; + + % this is a bit if a fiddle, but gets us a circular arc tangent to + % the line tee at point P, which appears to be what is wanted.... + arc = subpath (1.6, 3.2) of halfcircle rotated -180 shifted 1/2 up + scaled 10u shifted P rotatedabout(P, psi); + + % the rest of the points then follow... + Q = ell intersectionpoint subpath (1.5, 3) of arc; + H = P + whatever * dir psi; + xpart H = xpart Q = xpart M; ypart M = ypart N = 0; + xpart N = xpart P; + L = point 0 of ell; + T = point 0 of tee; + + % now we can get with the drawing + draw arc dashed withdots scaled 1/4 withcolor 1/2 red; + draw P--N dashed withdots scaled 1/2; + draw Q--M dashed withdots scaled 1/2; + + drawarrow xx; label.rt("$x$", point 1 of xx); + drawarrow yy; label.top("$y$", point 1 of yy); + + drawoptions(withcolor 2/3 blue); + draw fullcircle scaled 32 shifted T cutafter tee; + draw fullcircle scaled 28 shifted L cutafter ell; + label("$\psi$", 24 right rotated 1/2 psi shifted T); + label("$\theta$", 20 right rotated 1/2 theta shifted L); + drawoptions(); + + path boundary; + z1 = point .95 of xx; + z2 = point .95 of yy; + boundary = z1--(x1,y2)--z2; + + draw ell cutafter boundary; + draw tee cutafter boundary; + + % and finally label the points. + label.bot("$T$", T); + label.bot("$L$", L); + label.bot("$N$", N); + label.bot("$M$", M); + dotlabel.ulft("$P$", P); + dotlabel.lrt("$Q$", Q); + dotlabel.lrt("$H$", H); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/angles-lines.pdf b/info/drawing-with-metapost/src/angles-lines.pdf Binary files differnew file mode 100644 index 0000000000..e894f32f21 --- /dev/null +++ b/info/drawing-with-metapost/src/angles-lines.pdf diff --git a/info/drawing-with-metapost/src/apollonius.mp b/info/drawing-with-metapost/src/apollonius.mp new file mode 100644 index 0000000000..91ed22b600 --- /dev/null +++ b/info/drawing-with-metapost/src/apollonius.mp @@ -0,0 +1,158 @@ +\documentclass[border=1mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +vardef radical_axis(expr ca, cb) = + numeric t, d, ra, rb; + ra = abs(center ca - point 0 of ca); + rb = abs(center cb - point 0 of cb); + d = abs(center cb - center ca); + 2t = 1 + (ra+rb) / d * (ra-rb) / d; + (up -- down) scaled 89 + rotated angle (center cb - center ca) + shifted t[center ca, center cb] +enddef; + +input tangent-times + +vardef invert_point(expr P, o, r) = + save p, d; pair p; numeric d; + p = P - o; d = abs p; + if d > 0: + o + unitvector p scaled (r/d*r) + else: + errmessage("Inversion undefined at center.") + fi +enddef; + +vardef pole(expr Line, Circle) = + save p, o, r; pair o, p; numeric r; + o = center Circle; + r = 1/2 abs (point 4 of Circle - point 0 of Circle); + p = whatever [point 1 of Line, point 0 of Line]; + p - o = whatever * direction 0 of Line rotated 90; + invert_point(p, o, r) +enddef; + +vardef polex(expr a, b, o, r) = + save p; pair p; + p = whatever [a, b]; + p - o = whatever * (a-b) rotated 90; + invert_point(p, o, r) +enddef; + +vardef three_point_circle(expr a,b,c) = + save m; pair m; + m = whatever [a,b] rotatedaround(.5[a,b],90) + = whatever [b,c] rotatedaround(.5[b,c],90); + fullcircle scaled 2 length(m-a) shifted m + enddef; + +vardef through(expr a, b, o) = + save d; numeric d; d = abs(a-b); + (1+o/d)[b, a] -- (1+o/d)[a, b] +enddef; + +beginfig(1); + path c[]; numeric r[]; + z1 = origin; r1 = 101; + z2 = 233 right rotated 4; r2 = 53; + z3 = 209 right rotated -42; r3 = 31; + + forsuffixes $=1, 2, 3: + c$ = fullcircle scaled 2 r$ shifted z$; + endfor + + pair ecs[], ics[]; + + for i=1 upto 3: + numeric j, k; + j = i mod 3 + 1; + k = 10i + j; + ics[k] = (r[i]/(r[i]+r[j]))[z[i], z[j]]; + ecs[k] = (r[i]/(r[i]-r[j]))[z[i], z[j]]; + endfor + path a[]; + a1 = radical_axis(c1, c2); + a2 = radical_axis(c2, c3); + a3 = radical_axis(c3, c1); + + z0 = whatever [point 0 of a1, point 1 of a1] + = whatever [point 0 of a2, point 1 of a2]; + + z11 = polex(ecs31, ecs12, z1, r1); + z21 = polex(ecs31, ecs12, z2, r2); + z31 = polex(ecs31, ecs12, z3, r3); + + z12 = c1 intersectionpoint (z0 -- z11); + z22 = c2 intersectionpoint (z0 -- z21); + z32 = c3 intersectionpoint (z0 -- z31); + + z13 = c1 intersectionpoint (z11 -- 8[z0,z11]); + z23 = c2 intersectionpoint (z21 -- 8[z0,z21]); + z33 = c3 intersectionpoint (z31 -- 8[z0,z31]); + + z14 = whatever[ecs12, ecs31] = whatever[z1, z11]; + z24 = whatever[ecs12, ecs31] = whatever[z2, z21]; + z34 = whatever[ecs12, ecs31] = whatever[z3, z31]; + + drawoptions(withcolor 3/4[blue, white]); + draw c1; draw c2; draw c3; + drawoptions(withcolor 1/4[blue, white]); + label.urt(btex $C_1$ etex, point 1 of c1); + label.top(btex $C_2$ etex, point 2 of c2); + label.rt (btex $C_3$ etex, point 1/2 of c3); + draw ecs12 -- ecs31 -- ecs23; + + drawoptions(withpen pencircle scaled 1/4 withcolor 3/4[2/3 blue, white]); + draw a1; draw a2; draw a3; + draw through(z1, z14, 6); + draw through(z2, z24, 6); + draw through(z3, z34, 6); + + drawoptions(withpen pencircle scaled 1/4 withcolor 1/2 white); + draw through(z0, z13, 6); + draw through(z0, z23, 6); + draw through(z0, z33, 6); + + drawoptions(withcolor 1/256(203, 92, 13)); + drawdot z0 withpen pencircle scaled 3/2 dotlabeldiam; + z99 = z0 shifted 24 dir -6; + label.rt("\vbox{\openup-4pt\halign{\hss #\hss\cr Radical\cr centre\cr}}", z99); + interim ahangle := 20; drawarrow z99 -- z0 + cutafter fullcircle scaled 16 shifted z0 + withpen pencircle scaled 1/3; + + drawoptions(withcolor 1/256(239, 114, 21)); + dotlabel.urt("\small Pole", z11); + dotlabel.ulft("\small Pole", z21); + dotlabel.urt("\small Pole", z31); + + drawoptions(withcolor 2/3 red); + draw three_point_circle(z12, z22, z32); + draw three_point_circle(z13, z23, z33); + drawdot z12 withpen pencircle scaled 3/4 dotlabeldiam; + drawdot z22 withpen pencircle scaled 3/4 dotlabeldiam; + drawdot z32 withpen pencircle scaled 3/4 dotlabeldiam; + drawdot z13 withpen pencircle scaled 3/4 dotlabeldiam; + drawdot z23 withpen pencircle scaled 3/4 dotlabeldiam; + drawdot z33 withpen pencircle scaled 3/4 dotlabeldiam; + + drawoptions(withcolor 1/2[blue, white]); + drawdot z1 withpen pencircle scaled dotlabeldiam; + drawdot z2 withpen pencircle scaled dotlabeldiam; + drawdot z3 withpen pencircle scaled dotlabeldiam; + drawdot z14 withpen pencircle scaled dotlabeldiam; + drawdot z24 withpen pencircle scaled dotlabeldiam; + drawdot z34 withpen pencircle scaled dotlabeldiam; + draw thelabel.top(btex Axis of similitude, as polar etex, origin) + rotated angle (ecs12 - ecs31) + shifted 3/4[ecs31, ecs12]; + + drawoptions(); + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/apollonius.pdf b/info/drawing-with-metapost/src/apollonius.pdf Binary files differnew file mode 100644 index 0000000000..7de0b11b20 --- /dev/null +++ b/info/drawing-with-metapost/src/apollonius.pdf diff --git a/info/drawing-with-metapost/src/arbelos.mp b/info/drawing-with-metapost/src/arbelos.mp new file mode 100644 index 0000000000..7fe9bfc796 --- /dev/null +++ b/info/drawing-with-metapost/src/arbelos.mp @@ -0,0 +1,63 @@ +\RequirePackage{luatex85} +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +% invert path or pair P in circle C +vardef invert(expr P, C) = + save I, r; pair I; numeric r; + I = center C; + r = abs(point 0 of C shifted -I); + if pair P: if abs(P-I) > 0: unitvector(P-I) scaled (r/abs(P-I)*r) shifted fi I + elseif path P: + save T; numeric T; + T = length P; + for t=0 upto T-1: invert(point t of P, C) .. endfor if cycle P: cycle else: invert(point T of P, C) fi + fi +enddef; +beginfig(1); + pair A,B,C; + A = origin; C = 244 right; B = 7/8[A, C]; + + path c[]; + c1 = fullcircle scaled 2 abs(A-C); % large circle for the inversions + c2 = fullcircle scaled abs(A-C) shifted 1/2[A,C]; + c3 = fullcircle scaled abs(A-B) shifted 1/2[A,B]; + c4 = fullcircle scaled abs(B-C) shifted 1/2[B,C]; + c5 = invert(c4,c1); + + numeric d; d = abs(point 0 of c5 - point 4 of c5); % diameter of c5 + + drawoptions(withcolor 3/4 white); + draw c4; draw c5; + draw invert(subpath(0, 3/2) of c2, c1); + draw invert(subpath(0, 3/2) of c3, c1); + drawoptions(); + + draw subpath(-1/4,1) of c1 withcolor 3/4[1/2 red,white]; + label.bot("\textit{circle of inversion}", point -1/4 of c1) withcolor 1/4[1/2 red, white]; + + for i=1 upto 72: + path c, c'; + c = c5 shifted (0, i*d); + c' = invert(c, c1); + if i<5: + drawoptions(withcolor 3/4 white); + draw c; draw origin -- center c; + fill fullcircle scaled dotlabeldiam shifted center c; + fill fullcircle scaled dotlabeldiam shifted center c'; + drawoptions(); + fi + draw c' withpen pencircle scaled 1/4 withcolor 2/3 blue; + endfor + + forsuffixes $=2,3,4: draw subpath (0,4) of c$ + withpen pencircle scaled 1/4 withcolor 2/3 blue; + endfor + + draw A--C; + dotlabel.lft("$A$", A); dotlabel.llft("$B$", B); dotlabel.rt("$C$", C); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/arbelos.pdf b/info/drawing-with-metapost/src/arbelos.pdf Binary files differnew file mode 100644 index 0000000000..df5f9a957f --- /dev/null +++ b/info/drawing-with-metapost/src/arbelos.pdf diff --git a/info/drawing-with-metapost/src/arch-3-4-3-4.mp b/info/drawing-with-metapost/src/arch-3-4-3-4.mp new file mode 100644 index 0000000000..42b01cb7eb --- /dev/null +++ b/info/drawing-with-metapost/src/arch-3-4-3-4.mp @@ -0,0 +1,62 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +vardef median(expr p) = origin for i=1 upto length p: + point i of p / length p endfor enddef; +beginfig(1); + path s[], t[]; + s1 = unitsquare scaled 21 rotated 15; + s2 = s1 rotated 150; + t1 = subpath (4, 3) of s1 -- point 1 of s2 -- cycle; + t2 = t1 reflectedabout(point 1 of t1, point 2 of t1); + t3 = t1 rotated 150; + t4 = t1 rotated 210; + + picture color_unit, grid_unit; + color_unit = image( + fill s1 withcolor Oranges 8 1; + fill s2 withcolor Oranges 8 2; + fill t1 withcolor Blues 8 1; + fill t2 withcolor Blues 8 2; + fill t3 withcolor Blues 8 3; + fill t4 withcolor Blues 8 4; + ); + grid_unit = image( + forsuffixes $=s1, s2, t1, t2, t3, t4: + draw $; + % pair m; m = median($); + % for i=1 upto length $: + % draw m -- point i - 1/2 of $ + % withpen pencircle scaled 1/4; + % endfor + endfor + ); + + pair u, v; + u = point 1 of s1 - point 1 of s2; + v = u rotated 90; + + numeric n; n = 3; + forsuffixes $=color_unit, grid_unit: + for i=-n upto n: + for j=-n upto n: + draw $ shifted (i * u + j * v); + endfor + endfor + endfor + label.bot("The arrangement of polygons in the units was carefully", point 1/2 of bbox currentpicture + shifted 21 down); + label.bot("chosen to give the tiling neat edges." , point 1/2 of bbox currentpicture); + + path b; b = bbox currentpicture shifted 34 down; + draw color_unit shifted point 1/4 of b; + draw grid_unit shifted point 3/4 of b; + path b; b = bbox currentpicture shifted 13 down; + label("\texttt{color\_unit}", point 1/4 of b); + label("\texttt{grid\_unit}", point 3/4 of b); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/arch-3-4-3-4.pdf b/info/drawing-with-metapost/src/arch-3-4-3-4.pdf Binary files differnew file mode 100644 index 0000000000..1ceea33b78 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-3-4-3-4.pdf diff --git a/info/drawing-with-metapost/src/arch-3-4-6-4.mp b/info/drawing-with-metapost/src/arch-3-4-6-4.mp new file mode 100644 index 0000000000..0a7094be3b --- /dev/null +++ b/info/drawing-with-metapost/src/arch-3-4-6-4.mp @@ -0,0 +1,54 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\mplibsetformat{metafun} +\begin{mplibcode} +input colorbrewer-rgb +beginfig(1); + numeric r; r = 5; + path d; d = for i=1 upto 12: + up scaled r + shifted (r * (sqrt(3)+2), 0) + rotated 30i -- + endfor cycle; + + color shade[]; + shade0 = Oranges 8 3; + shade1 = Blues 8 3; + shade2 = Greens 8 3; + shade3 = Reds 8 3; + + vardef unit(expr n) = image( + fill d withcolor transparent (2, .9, shade[n mod 4]); + draw d; + ) enddef; + pair u, v; + u = point 0 of d - point 3 of d; + v = u rotated 60; + numeric n; n = 6; + for i=-n upto n: + for j=-n upto n: + draw unit(i*j) shifted ((i - floor(j / 2)) * u + j * v); + endfor + endfor + clip currentpicture to unitsquare shifted -(1/2, 1/2) scaled (n * 9 r); + picture p; p = image( + draw d; draw point 0 of d withpen pencircle scaled 2; + drawoptions(withcolor 3/4[red, white]); + draw d shifted u; draw point 0 of d shifted u withpen pencircle scaled 2; + drawoptions(withcolor 3/4[blue, white]); + draw d shifted v; draw point 0 of d shifted v withpen pencircle scaled 2; + drawoptions(); + drawarrow origin -- u withcolor red; + drawarrow origin -- v withcolor blue; + ); + label.bot(p, point 1/2 of bbox currentpicture shifted 13 down); + label.bot(btex \vbox{\hsize 3.4in\noindent + The vectors are chosen so that the dodecagons overlap to make + the required triangles, squares, and hexagons.} etex, + point 1/2 of bbox currentpicture shifted 13 down); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/arch-3-4-6-4.pdf b/info/drawing-with-metapost/src/arch-3-4-6-4.pdf Binary files differnew file mode 100644 index 0000000000..e145578609 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-3-4-6-4.pdf diff --git a/info/drawing-with-metapost/src/arch-4-6-12.mp b/info/drawing-with-metapost/src/arch-4-6-12.mp new file mode 100644 index 0000000000..8d8ddf2684 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-4-6-12.mp @@ -0,0 +1,65 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input archimedean-tools +input colorbrewer-rgb +beginfig(1); + path dodo, h[], s[]; + dodo = poly 12 of (origin -- 21 up); + for i=1 upto 3: + s[i] = poly 4 of subpath(2i, 2i-1) of dodo; + endfor + h1 = poly 6 of subpath (4, 3) of s1; + h2 = poly 6 of subpath (2, 1) of s3; + + picture cols, lines; + cols = image( + fill dodo withcolor Greens 9 2; + for i=1 upto 2: + fill h[i] withcolor Greens 9 3; + endfor + for i=1 upto 3: + fill s[i] withcolor Greens 9 4; + endfor + ); + lines = image( + forsuffixes $=dodo, s1, s2, s3, h1, h2: + if known $: + draw $ withpen pencircle scaled 1/4; + pair m; m = median($); + for i=1 upto length $: + cutdraw m -- point i - 1/2 of $ + if odd i: cutbefore fullcircle scaled (length $ / 4) shifted m fi + withpen pencircle scaled 1/4 + withcolor Reds 4 3; + ; + endfor + fi + endfor + ); + + pair u, v; + u = point 2 of s1 - point 8 of dodo; + v = u rotated 60; + + numeric n; + n = 2; + + forsuffixes @ = cols, lines: + for i=-n upto n+1: + for j=-n upto n: + draw @ shifted (i*u + j*v - floor(i/2) * v); + endfor + endfor + endfor + + clip currentpicture to unitsquare xscaled ((2n+1) * xpart u) yscaled (2n * ypart v) + shifted median(s2 shifted (-n*u - n*v + floor(n/2) * v)); + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/arch-4-6-12.pdf b/info/drawing-with-metapost/src/arch-4-6-12.pdf Binary files differnew file mode 100644 index 0000000000..4868f44da3 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-4-6-12.pdf diff --git a/info/drawing-with-metapost/src/arch-4-8-8-parts.mp b/info/drawing-with-metapost/src/arch-4-8-8-parts.mp new file mode 100644 index 0000000000..53aec91d14 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-4-8-8-parts.mp @@ -0,0 +1,34 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +input colorbrewer-rgb +path o, r[]; +o = (for i=0 upto 7: 21 dir 45i -- endfor cycle) rotated -90/4; +pair t; t = whatever[point 0 of o, point 1 of o] + = whatever[point 2 of o, point 3 of o]; +r1 = subpath (1,2) of o -- t -- cycle; +r2 = r1 rotated 90; +r3 = r2 rotated 90; +r4 = r3 rotated 90; + +picture filler; +filler = image( + filldraw r1 withcolor Reds 8 3; + filldraw r2 withcolor Blues 8 3; + filldraw r3 withcolor Reds 8 3; + filldraw r4 withcolor Blues 8 3; + filldraw o withcolor Purples 8 2; +); +beginfig(1); + draw filler; + label.bot("\begin{tabular}{c}The \texttt{filler} picture\\(unrotated)\end{tabular}", point 1/2 of bbox filler); + + draw o shifted 120 right; + label.bot("The octagon path \texttt{o}", point 1/2 of bbox o shifted 120 right); + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/arch-4-8-8-parts.pdf b/info/drawing-with-metapost/src/arch-4-8-8-parts.pdf Binary files differnew file mode 100644 index 0000000000..6466e01641 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-4-8-8-parts.pdf diff --git a/info/drawing-with-metapost/src/arch-4-8-8.mp b/info/drawing-with-metapost/src/arch-4-8-8.mp new file mode 100644 index 0000000000..117588ff02 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-4-8-8.mp @@ -0,0 +1,40 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); +input colorbrewer-rgb +path o, r[]; +o = (for i=0 upto 7: 21 dir 45i -- endfor cycle) rotated -90/4; +pair t; t = whatever[point 0 of o, point 1 of o] + = whatever[point 2 of o, point 3 of o]; +r1 = subpath (1,2) of o -- t -- cycle; +r2 = r1 rotated 90; +r3 = r2 rotated 90; +r4 = r3 rotated 90; + +picture filler; +filler = image( + filldraw r1 withcolor Reds 8 3; + filldraw r2 withcolor Blues 8 3; + filldraw r3 withcolor Reds 8 3; + filldraw r4 withcolor Blues 8 3; + filldraw o withcolor Purples 8 2; +); +pair u, v; +u = point 0 of o - point 5 of o; v = u rotated 90; +beginfig(1); +numeric n; n = 5; +for i=-n upto n: + for j=-n upto n: + draw filler rotated ((i+j) mod 2 * 90) shifted (i*u + j * v); + endfor +endfor +for i=-n upto n: + for j=-n upto n: + draw o shifted (i*u + j * v); + endfor +endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/arch-4-8-8.pdf b/info/drawing-with-metapost/src/arch-4-8-8.pdf Binary files differnew file mode 100644 index 0000000000..e5fa45dde0 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-4-8-8.pdf diff --git a/info/drawing-with-metapost/src/arch-hexagon.mp b/info/drawing-with-metapost/src/arch-hexagon.mp new file mode 100644 index 0000000000..0cb259d124 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-hexagon.mp @@ -0,0 +1,59 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +input archimedean-tools +beginfig(1); + + path h[]; + % 42/sqrt(3) \simeq 24.2487; + for i=0 upto 2: + h[i] = poly 6 of (origin -- 42 / sqrt(3) * dir (30 + 120i)); + endfor + + picture unit; + unit = image( + % fill h0 withcolor Purples 8 3; + % fill h1 withcolor Oranges 8 3; + % fill h2 withcolor Blues 8 2; + numeric i; i = -1; + forsuffixes $ = Purples, Oranges, Blues: + pair m; m = median(h[incr i]); + for j=1 upto 3: + fill subpath (-2i+2j, 2-2i+2j) of h[i] -- m -- cycle withcolor $[8][1+j]; + endfor + for j=1 upto 3: + draw m -- point 2j of h[i] withcolor $[8][1]; + endfor + endfor + + % for i=0 upto 2: + % pair m; m = median(h[i]); + % for j=0 upto 2: + % draw point 2j+1/2 of h[i] -- m -- point 2j+3/2 of h[i]; + % endfor + % endfor + ); + + pair u, v; + u = point 3 of h1 - point 1 of h0; + v = u rotated 60; + + numeric n; + n = 4; + + for i=-n upto n: + for j=-n upto n: + draw unit shifted (i*u + j*v); + endfor + endfor + + clip currentpicture to superellipse(168 right, 168 up, 168 left, 168 down, 0.78); + + + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/arch-hst.mp b/info/drawing-with-metapost/src/arch-hst.mp new file mode 100644 index 0000000000..ebcf5668d2 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-hst.mp @@ -0,0 +1,62 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +input archimedean-tools +beginfig(1); + path t[], s[], h; + h = poly 6 of (origin -- 42 dir 30); + for i=1 upto 3: + s[i] = poly 4 of subpath (2i, 2i-1) of h; + endfor + t1 = poly 3 of subpath (4, 3) of s1; + t2 = poly 3 of subpath (2, 1) of s2; + + picture unit; + unit = image( + for i=1 upto 2: + fill t[i] withcolor Blues 8 4; + endfor + for i=1 upto 3: + fill s[i] withcolor Greens 8 1; + endfor + fill h withcolor Oranges 9 4; + forsuffixes $=t1, t2: + pair m; m = median($); + for i=1 upto length $: + draw m -- point i - 1/2 of $ withcolor Blues 8 8; + endfor + endfor + forsuffixes $=s1, s2, s3: + draw point 1/2 of $ -- point 5/2 of $ withcolor Oranges 8 8; + draw point 3/2 of $ -- point 7/2 of $ withcolor Blues 8 8; + endfor + for i=1 upto 3: + draw point i-1/2 of h -- point i+5/2 of h withcolor Oranges 8 8; + endfor + ); + + pair u, v; + u = point 2 of s1 - point -1 of h; + v = u rotated 60; + + numeric n; n=4; + for i=-n upto n: + for j=-n upto n: + draw unit rotated 0 shifted (i*u + j*v - floor (j/2) * u); + endfor + endfor + %draw bbox unit withcolor red; + + clip currentpicture to fullcircle scaled (3/2n * abs(v)); + + % picture p; p = currentpicture; + % (wd, ht) = (urcorner p - llcorner p); + % currentpicture := nullpicture; + % p := p scaled (148.5mm / wd); + % draw p; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/arch-hst.pdf b/info/drawing-with-metapost/src/arch-hst.pdf Binary files differnew file mode 100644 index 0000000000..6d973adde9 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-hst.pdf diff --git a/info/drawing-with-metapost/src/arch-snub-hex-parts.mp b/info/drawing-with-metapost/src/arch-snub-hex-parts.mp new file mode 100644 index 0000000000..bf3ff14355 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-snub-hex-parts.mp @@ -0,0 +1,49 @@ +input colorbrewer-rgb +input archimedean-tools +path t[], h; +h = poly 6 of (origin -- 21 dir 30); +for i=1 upto 6: + t[i] = poly 3 of subpath (i, i-1) of h; +endfor +picture P[], unit[]; +for k=1 upto 2: + pair u, v; + if k=1: + u = point 2 of h - point 2 of t6; + t7 := poly3 of subpath (0, -1) of t6; + t8 := poly3 of subpath (0, -1) of t3; + else: + u = point 3 of h - point 2 of t6; + t7 := poly3 of subpath (2, 1) of t6; + t8 := poly3 of subpath (2, 1) of t3; + fi + + v = u rotated -60; + + unit[k] = image( + for i=1 upto 6: + fill t[i] withcolor Blues 8 if odd i: 2 else: 3 fi; + endfor + for i=7 upto 8: + fill t[i] withcolor Oranges 8 if odd i: 3 else: 2 fi; + endfor + fill h withcolor Oranges 8 2; + forsuffixes $=h, t1, t2, t3, t4, t5, t6, t7, t8: + draw $ withpen pencircle scaled 1/4 withcolor 3/4; + pair m; m = median($); + for i=1 upto length $: + draw m -- point i - 1/2 of $ withcolor 3/4; + endfor + endfor + ); + + numeric n; n=4; + P[k] = image( + for i=-n upto n: + for j=-n upto n: + draw unit[k] shifted (i*u + j*v - floor (j/2) * u); + endfor + endfor + clip currentpicture to superellipse(168 right, 168 up, 168 left, 168 down, 0.78); + ); +endfor diff --git a/info/drawing-with-metapost/src/arch-snub-hexagon-unit.mp b/info/drawing-with-metapost/src/arch-snub-hexagon-unit.mp new file mode 100644 index 0000000000..b5200f6a6f --- /dev/null +++ b/info/drawing-with-metapost/src/arch-snub-hexagon-unit.mp @@ -0,0 +1,12 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input arch-snub-hex-parts +beginfig(1); + draw unit1 rotated 210; + draw unit2 rotated 210 shifted 100 left; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/arch-snub-hexagon-unit.pdf b/info/drawing-with-metapost/src/arch-snub-hexagon-unit.pdf Binary files differnew file mode 100644 index 0000000000..1dd83c0b0d --- /dev/null +++ b/info/drawing-with-metapost/src/arch-snub-hexagon-unit.pdf diff --git a/info/drawing-with-metapost/src/arch-snub-hexagon.mp b/info/drawing-with-metapost/src/arch-snub-hexagon.mp new file mode 100644 index 0000000000..09310036f2 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-snub-hexagon.mp @@ -0,0 +1,13 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input arch-snub-hex-parts +beginfig(1); +draw P1 shifted 180 left; +draw P2 shifted 180 right; +currentpicture := currentpicture rotated 90; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/arch-snub-hexagon.pdf b/info/drawing-with-metapost/src/arch-snub-hexagon.pdf Binary files differnew file mode 100644 index 0000000000..aed6e4d102 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-snub-hexagon.pdf diff --git a/info/drawing-with-metapost/src/arch-snub-square.mp b/info/drawing-with-metapost/src/arch-snub-square.mp new file mode 100644 index 0000000000..c607b4e5c4 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-snub-square.mp @@ -0,0 +1,72 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-cmyk +% a polygon of n sides on a given path +vardef poly expr n of p = + save x, y; + z0 = point 0 of p; + z1 = point 1 of p; + for i=2 upto n-1: + z[i] = z[i-2] rotatedabout(z[i-1], 180(2/n-1)); + endfor + for i=0 upto n-1: z[i] -- endfor cycle +enddef; +vardef median(expr P) = + (origin for i=1 upto length P: + point i of P endfor) / length P +enddef; +beginfig(1); + path t[], s[]; + s1 = poly 4 of (origin -- 42 dir 15); + t1 = poly 3 of subpath (2, 1) of s1; + t2 = poly 3 of subpath (3, 2) of s1; + t3 = poly 3 of subpath (2, 1) of t1; + t4 = poly 3 of subpath (2, 1) of t2; + s2 = poly 4 of subpath (2, 1) of t4; + + picture unit; + unit = image( + fill s1 withcolor Oranges 8 5; + fill t1 withcolor Blues 8 4; + fill t4 withcolor Blues 8 3; + fill t3 withcolor Greens 8 4; + fill t2 withcolor Greens 8 3; + fill s2 withcolor Reds 8 5; + forsuffixes $=s1, t1, s2, t2, t3, t4: + pair m; m = median($); + for i=1 upto length $: + draw point i - 1/2 of $ -- m + withpen pencircle scaled 3/2 withcolor Spectral 3 2; + endfor + endfor + ); + + pair u, v; + u = point 2 of t2 - point 0 of s1; + v = point 2 of t3 - point 0 of s1; + + for i=0 upto 6: + for j=0 upto 9: + draw unit shifted (i*u + j*v); + endfor + endfor + + % picture p; p = currentpicture; + % (wd, ht) = (urcorner p - llcorner p); + % currentpicture := nullpicture; + % p := p scaled (145mm / wd); + % undraw unitsquare xscaled 148mm yscaled 210mm shifted llcorner p shifted -(1.5mm, 1.5mm) + % withpen pencircle scaled 1/8; + % z0 = 1/2[llcorner currentpicture, ulcorner currentpicture]; + % draw z0 -- z0 shifted 3 right withpen pencircle scaled 1/8; + % z1 = 1/2[lrcorner currentpicture, urcorner currentpicture]; + % draw z1 -- z1 shifted 3 left withpen pencircle scaled 1/8; + % draw p; + % picture r; r = currentpicture rotated 90; + % currentpicture := nullpicture; + % draw r; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/arch-snub-square.pdf b/info/drawing-with-metapost/src/arch-snub-square.pdf Binary files differnew file mode 100644 index 0000000000..4cc73ce6e5 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-snub-square.pdf diff --git a/info/drawing-with-metapost/src/arch-square.mp b/info/drawing-with-metapost/src/arch-square.mp new file mode 100644 index 0000000000..3511abf4e9 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-square.mp @@ -0,0 +1,39 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +input archimedean-tools +beginfig(1); + + path s; + s = poly 4 of (origin -- 42 right); + pair m; + m = median(s); + + picture unit[]; + unit0 = image( + fill s withcolor Oranges 8 5; + for i=1 upto length s: + draw m -- point i-1/2 of s; + endfor<D-`> + + ); + unit1 = image(fill s withcolor Blues 8 4); + + + pair u, v; + u = point 1 of s - point 0 of s; + v = point 3 of s - point 0 of s; + + + numeric n; + n = 4 + + + + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/arch-triangle.mp b/info/drawing-with-metapost/src/arch-triangle.mp new file mode 100644 index 0000000000..cc80674664 --- /dev/null +++ b/info/drawing-with-metapost/src/arch-triangle.mp @@ -0,0 +1,45 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +input archimedean-tools +beginfig(1); + + path s, t; + s = poly 3 of (origin -- 42 right); + t = poly 3 of subpath(2,1) of s; + pair ms, mt; + ms = median(s); + mt = median(t); + + picture unit; + unit = image( + fill s withcolor Oranges 8 3; + fill t withcolor Blues 8 2; + draw point 5/2 of s -- ms; + draw point 3/2 of t -- mt; + draw point 1/2 of s -- ms -- mt -- point 5/2 of t; + ); + + pair u, v; + u = point 1 of s - point 0 of s; + v = point 2 of s - point 0 of s; + + numeric n; + n = 0; + + for i=-n upto n: + for j=-n upto n: + draw unit shifted (i*u + j*v); + endfor + endfor + + clip currentpicture to superellipse(168 right, 168 up, 168 left, 168 down, 0.78); + + + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/archimedean-tools.mp b/info/drawing-with-metapost/src/archimedean-tools.mp new file mode 100644 index 0000000000..e10c8f447f --- /dev/null +++ b/info/drawing-with-metapost/src/archimedean-tools.mp @@ -0,0 +1,33 @@ +% a polygon of n sides on a given path +vardef poly expr n of p = + save x, y; + z0 = point 0 of p; + z1 = point 1 of p; + for i=2 upto n-1: + z[i] = z[i-2] rotatedabout(z[i-1], 180(2/n-1)); + endfor + for i=0 upto n-1: z[i] -- endfor cycle +enddef; +vardef polyc expr n of p = + save x, y; + z0 = point 0 of p; + z1 = point 1 of p; + for i=2 upto n-1: + z[i] = z[i-2] rotatedabout(z[i-1], 180(2/n-1)); + endfor + for i=0 upto n-1: z[i] ... endfor cycle +enddef; + +% a polygon of n sides of a given length +vardef polya(expr n, s) = + save r, x, y; numeric r, x, y; + (x, y) = dir (180/n); + 2r = s * x / y; + for i=0 upto n-1: + (left * s/2) shifted (r * down) rotated (360 / n * i) -- + endfor cycle +enddef; +vardef median(expr P) = + (origin for i=1 upto length P: + point i of P endfor) / length P +enddef; + diff --git a/info/drawing-with-metapost/src/archimedes.mp b/info/drawing-with-metapost/src/archimedes.mp new file mode 100644 index 0000000000..81608868a8 --- /dev/null +++ b/info/drawing-with-metapost/src/archimedes.mp @@ -0,0 +1,189 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +numeric u; u = 10; +path box; box = unitsquare shifted -(1/2,1/2) scaled 12u; +picture P[]; +% (4,6,12) +numeric dx; dx = u*(1+ cosd(15)/sind(15)); +picture unit; unit = image( + path s; s = unitsquare shifted -(1/2,1/2) scaled u shifted (dx/2,0); + for t=0 upto 2: + draw s rotated 120t; + for tt=1 upto 3: + draw subpath(3,4) of s rotated (120t+30tt); + endfor + endfor +); +P1 = image( + draw unit withpen pencircle scaled 4 withcolor .8[red,white]; + for x=-5dx step dx until 5dx: + for y=-5dx step dx until 5dx: + draw unit shifted (x,0) shifted ((y,0) rotated 60); + endfor + endfor + clip currentpicture to box; draw box dashed withdots scaled 1/4; + label.bot("$(4, 6, 12)$", point 1/2 of bbox currentpicture); +); +numeric dx; dx = u*(cosd(180/8)/sind(180/8)); +path octagon; octagon = for t = 0 upto 7: 1/2(dx,u) rotated 45t -- endfor cycle; +P2 = image( + draw octagon withpen pencircle scaled 4 withcolor .8[red,white]; + for x=-5dx step dx until 5dx: + for y=-5dx step dx until 5dx: + draw octagon shifted (x,y); + endfor + endfor + clip currentpicture to box; draw box dashed withdots scaled 1/4; + label.bot("$(4, 8^2)$", point 1/2 of bbox currentpicture); +); + +numeric dx; dx = u*(cosd(180/12)/sind(180/12)); +path unit; unit = for t=0 upto 11: 1/2(dx,u) rotated 30t -- endfor cycle; +P3 = image( + draw unit withpen pencircle scaled 4 withcolor .8[red,white]; + for x=-5dx step dx until 5dx: + for y=-5dx step dx until 5dx: + draw unit shifted (x,0) shifted ((y,0) rotated 60); + endfor + endfor + clip currentpicture to box; draw box dashed withdots scaled 1/4; + label.bot("$(3, 12^2)$", point 1/2 of bbox currentpicture); +); + +% path box; box = unitsquare shifted -(1/2,1/2) scaled 12u; +% clip currentpicture to box; draw box dashed evenly; + +% endfig; + +% beginfig(4); +% % (6^3) +% save dx; dx = 3u; +% picture unit; unit = image( +% for t=0 upto 2: +% draw origin -- (u,0) rotated 120t; +% endfor +% ); + +% draw unit withpen pencircle scaled 4 withcolor .8[red,white]; + +% for x=-5dx step dx until 5dx: +% for y=-5dx step dx/3 until 5dx: +% draw unit shifted (x,0) shifted (y,0) shifted ((y,0) rotated 60); +% endfor +% endfor + +% path box; box = unitsquare shifted -(1/2,1/2) scaled 12u; +% clip currentpicture to box; draw box dashed evenly; + +% endfig; + +% (3,4,6,4) +numeric dx; dx = u*(1+cosd(30)/sind(30)); +picture unit; unit = image( + path s; s = unitsquare shifted -(1/2,1/2) scaled u shifted (dx/2,0); + for t=0 upto 2: + draw s rotated 120t; + endfor +); +P4 = image( + draw unit withpen pencircle scaled 4 withcolor .8[red,white]; + for x=-5dx step dx until 5dx: + for y=-5dx step dx until 5dx: + draw unit shifted (x,0) shifted ((y,0) rotated 60); + endfor + endfor + clip currentpicture to box; draw box dashed withdots scaled 1/4; + label.bot("$(3, 4, 6, 4)$", point 1/2 of bbox currentpicture); +); + +% beginfig(6); +% % (3,6,3,6) +% save dx; dx = 2u; +% path unit; unit = for t=0 upto 5: (u,0) rotated 60t -- endfor cycle; + +% draw unit withpen pencircle scaled 4 withcolor .8[red,white]; + +% for x=-5dx step dx until 5dx: +% for y=-5dx step dx until 5dx: +% draw unit shifted (x,0) shifted ((y,0) rotated 60); +% endfor +% endfor + +% path box; box = unitsquare shifted -(1/2,1/2) scaled 12u; +% clip currentpicture to box; draw box dashed evenly; + +% endfig; + +% beginfig(7); +% % (4^4) +% save dx; dx = 4u; +% path unit; unit = unitsquare shifted -(1/2,1/2) scaled 2u; + +% draw unit withpen pencircle scaled 4 withcolor .8[red,white]; + +% for x=-5dx step dx until 5dx: +% for y=-5dx step dx until 5dx: +% draw unit shifted (x,0) shifted (y/2,y/2); +% endfor +% endfor + +% path box; box = unitsquare shifted -(1/2,1/2) scaled 12u; +% clip currentpicture to box; draw box dashed evenly; + +% endfig; + +% beginfig(8); +% % (3^4,6) +% save dx; dx = 2u; +% picture unit; unit = image( +% for t=0 upto 5: draw ((2u,0) -- (u,0) +% -- (u,0) rotated 60 +% -- (u,0) rotated 60 shifted (u,0) +% ) rotated 60t; endfor +% ); + +% draw unit withpen pencircle scaled 4 withcolor .8[red,white]; + +% for x=-5dx step dx until 5dx: +% for y=-5dx step dx until 5dx: +% draw unit shifted (x,0) shifted ((x/2,0) rotated -60) +% shifted ((y,0) rotated 60) shifted (y/2,0); +% endfor +% endfor + +% path box; box = unitsquare shifted -(1/2,1/2) scaled 12u; +% clip currentpicture to box; draw box dashed evenly; + +% endfig; +% beginfig(9); +% % (3^2,4,3,4) +% save dx; dx = u; +% picture unit; unit = image( +% for t=0 upto 3: draw ( (u/2,-u/2) -- (u/2+u*sqrt(3)/2,0) -- (u/2,u/2) -- cycle) rotated 90t; endfor +% ); + +% draw unit withpen pencircle scaled 4 withcolor .8[red,white]; + +% for x=-5dx step dx until 5dx: +% for y=-5dx step dx until 5dx: +% draw unit shifted (x,0) shifted ((x,0) rotated 30) +% shifted (0,y) shifted ((0,y) rotated 30); +% endfor +% endfor + +% path box; box = unitsquare shifted -(1/2,1/2) scaled 12u; +% clip currentpicture to box; draw box dashed evenly; + +% endfig; +beginfig(1); +draw P1; +draw P2 shifted (12.5u, 0); +draw P3 shifted (0, -14.4u); +draw P4 shifted (12.5u, -14.4u); + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/archimedes.pdf b/info/drawing-with-metapost/src/archimedes.pdf Binary files differnew file mode 100644 index 0000000000..ead562d6f9 --- /dev/null +++ b/info/drawing-with-metapost/src/archimedes.pdf diff --git a/info/drawing-with-metapost/src/area-under-graph.mp b/info/drawing-with-metapost/src/area-under-graph.mp new file mode 100644 index 0000000000..e9dc5f5886 --- /dev/null +++ b/info/drawing-with-metapost/src/area-under-graph.mp @@ -0,0 +1,37 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +ahangle := 30; +beginfig(1); + numeric u; + u = 1cm; + path xx, yy, f; + xx = (1/4 left -- 17/4 right) scaled u; + yy = (1/4 down -- 9/4 up) scaled u; + + f = ((15/32,32/15) for x = 1/2 step 1/8 until 33/8: .. (x,1/x) endfor) scaled u; + + path A; + A = buildcycle(yy shifted (u,0), f, yy shifted (2.71828u,0), xx); + fill A withcolor Blues 8 2; + + drawoptions(withcolor 3/4); + for x=1 upto 4: + draw yy shifted (x*u,0) cutafter (xx shifted (0, ypart point 0 of f)); + endfor + for y=1 upto 2: + draw xx shifted (0,y*u) cutafter (yy shifted (xpart point infinity of f,0)); + endfor + + drawoptions(); + draw f withcolor Reds 8 7; + drawarrow xx; + drawarrow yy; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/area-under-graph.pdf b/info/drawing-with-metapost/src/area-under-graph.pdf Binary files differnew file mode 100644 index 0000000000..8230d85b40 --- /dev/null +++ b/info/drawing-with-metapost/src/area-under-graph.pdf diff --git a/info/drawing-with-metapost/src/axis-of-similitude.mp b/info/drawing-with-metapost/src/axis-of-similitude.mp new file mode 100644 index 0000000000..94ad49f695 --- /dev/null +++ b/info/drawing-with-metapost/src/axis-of-similitude.mp @@ -0,0 +1,113 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +vardef straight_through(expr Prefix, Path, Suffix) = + reverse (left scaled Prefix rotated angle direction 0 of Path shifted point 0 of Path) + .. Path .. (right scaled Suffix rotated angle direction infinity of Path shifted point infinity of Path) +enddef; + +vardef tangent_times(expr C, p) = + save m, a, b, G, H; + pair m; numeric a, b; path G, H; + m = 1/2[p, center C]; + H = halfcircle scaled abs (p - center C) + rotated angle (p - center C) shifted m; + G = H rotatedabout(m, 180); + (a, whatever) = C intersectiontimes H; + (b, whatever) = C intersectiontimes G; + (a, b if b < a: + 8 fi) +enddef; +vardef arc_of_tangents(expr C, p) = + subpath tangent_times(C, p) of C +enddef; + +beginfig(1); + + path c[]; numeric r[]; + z1 = origin; r1 = 101; + z2 = 233 right rotated 4; r2 = 53; + z3 = 209 right rotated -42; r3 = 31; + + forsuffixes $=1, 2, 3: + c$ = fullcircle scaled 2 r$ shifted z$; + endfor + + pair ecs[], ics[]; + + for i=1 upto 3: + numeric j, k; + j = i mod 3 + 1; + k = 10i + j; + ics[k] = (r[i]/(r[i]+r[j]))[z[i], z[j]]; + ecs[k] = (r[i]/(r[i]-r[j]))[z[i], z[j]]; + endfor + + show unitvector(ecs31 - ics12) dotprod unitvector(ecs12 - ics31); + + fill ecs12 -- arc_of_tangents(c1, ecs12) -- cycle withcolor 15/16[blue, white]; + fill ics12 -- arc_of_tangents(c1, ics12) -- cycle withcolor 7/8 [blue, white]; + fill ics12 -- arc_of_tangents(c2, ics12) -- cycle withcolor 7/8 [blue, white]; + + unfill c1; unfill c2; unfill c3; + + drawoptions(withcolor 3/4[blue, white]); + draw c1 withpen pencircle scaled 1; + draw c2 withpen pencircle scaled 1; + draw c3 withpen pencircle scaled 1; + + numeric t, u, v; + (t, whatever) = tangent_times(c1, ecs12); + (u, whatever) = tangent_times(c1, ics12); + (whatever, v) = tangent_times(c2, ics12); + draw point t of c1 -- z1 -- point u of c1 dashed evenly; + draw point t of c2 -- z2 -- point v of c2 dashed evenly; + + drawoptions(withcolor 1/4[blue, white]); + drawdot z1 withpen pencircle scaled dotlabeldiam; + drawdot z2 withpen pencircle scaled dotlabeldiam; + drawdot z3 withpen pencircle scaled dotlabeldiam; + label(btex $C_1$ etex, z1 shifted 10 dir 254); + label(btex $C_2$ etex, z2 shifted 10 dir 299); + label(btex $C_3$ etex, z3 shifted 10 dir 6); + label.lft(btex $r_1$ etex, 1/2[z1, point t of c1]); + label.lft(btex $r_2$ etex, 1/2[z2, point t of c2]); + + drawoptions(); + draw ecs31 -- ics12 -- ecs23 -- ecs12 -- ics31 dashed withdots scaled 1/2; + + drawoptions(withcolor 1/2 white); + draw straight_through(r1+24, z1 -- ecs12, 24); + draw straight_through(r1+24, z1 -- ecs31, 24); + draw straight_through(r2+24, z2 -- ecs23, 24); + + drawoptions(withcolor 2/3 red); + drawdot ics12 withpen pencircle scaled dotlabeldiam; + drawdot ics23 withpen pencircle scaled dotlabeldiam; + drawdot ics31 withpen pencircle scaled dotlabeldiam; + label(btex $I_{12}$ etex, ics12 shifted 12 dir 100); + label(btex $I_{23}$ etex, ics23 shifted 12 dir 340); + label(btex $I_{31}$ etex, ics31 shifted 10 dir 200); + + drawoptions(withcolor 5/8 blue); + drawdot ecs12 withpen pencircle scaled dotlabeldiam; + drawdot ecs23 withpen pencircle scaled dotlabeldiam; + drawdot ecs31 withpen pencircle scaled dotlabeldiam; + label(btex $E_{12}$ etex, ecs12 shifted 8 dir 100); + label(btex $E_{23}$ etex, ecs23 shifted 11 dir 345); + label(btex $E_{31}$ etex, ecs31 shifted 10 dir 277); + + draw thelabel.top(btex Axis of Similitude etex, origin) + rotated angle (ecs12 - ecs31) + shifted 1/4[ecs31, ecs12]; + + label(btex $\displaystyle {C_1E_{12} \over C_2E_{12}} = {r_1 \over r_2} + = {C_1I_{12} \over C_2I_{12}}$ etex, (x1, y3)); + + label(btex \it \& cetera \dots etex, (x1, y3-28)); + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/axis-of-similitude.pdf b/info/drawing-with-metapost/src/axis-of-similitude.pdf Binary files differnew file mode 100644 index 0000000000..422e03717c --- /dev/null +++ b/info/drawing-with-metapost/src/axis-of-similitude.pdf diff --git a/info/drawing-with-metapost/src/basedate.mp b/info/drawing-with-metapost/src/basedate.mp new file mode 100644 index 0000000000..f91c360894 --- /dev/null +++ b/info/drawing-with-metapost/src/basedate.mp @@ -0,0 +1,11 @@ +vardef base(expr Y, M, d) = + save m, y; numeric m, y; + if M < 3: + m = M + 9; + y = Y - 1; + else: + m = M - 3; + y = Y; + fi + 365/1024 y + (floor(y/4) - floor(y/100) + floor(y/400) + floor((2+3m)/5) + 30m + d - 307) / 1024 +enddef; diff --git a/info/drawing-with-metapost/src/bisection-euclidean.mp b/info/drawing-with-metapost/src/bisection-euclidean.mp new file mode 100644 index 0000000000..742d762011 --- /dev/null +++ b/info/drawing-with-metapost/src/bisection-euclidean.mp @@ -0,0 +1,33 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{dwmpcode} +\mplibtextextlabel{enable} +\begin{document} +\vbox{\openup 1pt \halign to 4in{#\cr +\begin{code} +vardef euclidean_bisector(expr a,b,c,r) = + save arc,p,q,e; + path arc; pair p,q,e; + arc = fullcircle scaled r shifted b; + p = (a--b) intersectionpoint arc; + q = (b--c) intersectionpoint arc ; + e = equilateral_triangle_point(p,q); + e +enddef; +\end{code} +\cr\kern 64pt +\begin{mplibcode} +input bisection +beginfig(1); +z0 = euclidean_bisector(A,B,C,100); +draw B -- 1.2[B,z0] withcolor .67 red; +draw A--B--C; +dotlabel.top(btex $A$ etex, A); +dotlabel.lft(btex $B$ etex, B); +dotlabel.lrt(btex $C$ etex, C); +dotlabel.lrt(btex $E$ etex, z0); +undraw bbox currentpicture; +endfig; +\end{mplibcode} +\cr}} +\end{document} diff --git a/info/drawing-with-metapost/src/bisection-euclidean.pdf b/info/drawing-with-metapost/src/bisection-euclidean.pdf Binary files differnew file mode 100644 index 0000000000..c5d2971ec7 --- /dev/null +++ b/info/drawing-with-metapost/src/bisection-euclidean.pdf diff --git a/info/drawing-with-metapost/src/bisection-interior.mp b/info/drawing-with-metapost/src/bisection-interior.mp new file mode 100644 index 0000000000..c39c6fd5a9 --- /dev/null +++ b/info/drawing-with-metapost/src/bisection-interior.mp @@ -0,0 +1,28 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{dwmpcode} +\mplibtextextlabel{enable} +\begin{document} +\vbox{\openup 1pt \halign to 4in{#\cr +\begin{code} +vardef interior_bisector(expr a,b,c) = + (1/(1+abs(c-b)/abs(a-b)))[a,c] +enddef; +\end{code} +\cr\kern 64pt +\begin{mplibcode} +input bisection +beginfig(1); +z0 = interior_bisector(A,B,C); +draw A--B--C--cycle; +draw B -- 1.2[B,z0] withcolor .67 red; +dotlabel.top(btex $A$ etex, A); +dotlabel.lft(btex $B$ etex, B); +dotlabel.lrt(btex $C$ etex, C); +draw z0 withpen pencircle scaled dotlabeldiam; +label(btex $E$ etex, z0 + (3,9)); +undraw bbox currentpicture; +endfig; +\end{mplibcode} +\cr}} +\end{document} diff --git a/info/drawing-with-metapost/src/bisection-interior.pdf b/info/drawing-with-metapost/src/bisection-interior.pdf Binary files differnew file mode 100644 index 0000000000..f58a457e9f --- /dev/null +++ b/info/drawing-with-metapost/src/bisection-interior.pdf diff --git a/info/drawing-with-metapost/src/bisection-vector.mp b/info/drawing-with-metapost/src/bisection-vector.mp new file mode 100644 index 0000000000..55a8166df9 --- /dev/null +++ b/info/drawing-with-metapost/src/bisection-vector.mp @@ -0,0 +1,30 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{dwmpcode} +\mplibtextextlabel{enable} +\begin{document} +\vbox{\openup 1pt \halign to 4in{#\cr +\begin{code} +vardef vector_bisector(expr a,b,c,r) = + b + unitvector (a-b) scaled r + + unitvector (c-b) scaled r +enddef; +\end{code} +\cr\kern 64pt +\begin{mplibcode} +input bisection +beginfig(1); +z0 = vector_bisector(A,B,C,50); +drawarrow B--A cutafter fullcircle scaled 5 shifted A; +drawarrow B--C cutafter fullcircle scaled 5 shifted C; +drawarrow B--z0 cutafter fullcircle scaled 5 shifted z0 +withcolor .67 red; +dotlabel.top(btex $A$ etex, A); +dotlabel.lft(btex $B$ etex, B); +dotlabel.lrt(btex $C$ etex, C); +dotlabel.lrt(btex $E$ etex, z0); +undraw bbox currentpicture; +endfig; +\end{mplibcode} +\cr}} +\end{document} diff --git a/info/drawing-with-metapost/src/bisection-vector.pdf b/info/drawing-with-metapost/src/bisection-vector.pdf Binary files differnew file mode 100644 index 0000000000..a3afc5aee0 --- /dev/null +++ b/info/drawing-with-metapost/src/bisection-vector.pdf diff --git a/info/drawing-with-metapost/src/bisection.mp b/info/drawing-with-metapost/src/bisection.mp new file mode 100644 index 0000000000..895fdefb20 --- /dev/null +++ b/info/drawing-with-metapost/src/bisection.mp @@ -0,0 +1,33 @@ +vardef equilateral_triangle_point(expr a, b) = + b shifted -a rotated 60 shifted a +enddef; + +vardef euclidean_bisector(expr a,b,c,r) = + save arc,p,q,e; + path arc; numeric p,q; pair e; + arc = fullcircle scaled r shifted b; + p = ypart ( (a--b) intersectiontimes arc ); + q = ypart ( (b--c) intersectiontimes arc ); + e = equilateral_triangle_point(point p of arc, point q of arc); + hide( + draw subpath(p-1/4,p+1/4) of arc withcolor .7[red,white]; + draw subpath(q-1/4,q+1/4) of arc withcolor .7[red,white]; + draw point p of arc -- point q of arc -- e -- cycle + withcolor .7[blue,white]; + ) + e +enddef; + +vardef vector_bisector(expr a,b,c,r) = + b + unitvector (a-b) scaled r + + unitvector (c-b) scaled r +enddef; + +vardef interior_bisector(expr a,b,c) = + (1/(1+abs(c-b)/abs(a-b)))[a,c] +enddef; + +pair A,B,C; +A = 120 right rotated 52; +B = origin; +C = 144 right rotated -4; diff --git a/info/drawing-with-metapost/src/blended-color-circles.mp b/info/drawing-with-metapost/src/blended-color-circles.mp new file mode 100644 index 0000000000..4da586d64b --- /dev/null +++ b/info/drawing-with-metapost/src/blended-color-circles.mp @@ -0,0 +1,44 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\usepackage{fontspec} +\setmainfont{Helvetica} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + +path C[], B[]; + +% arrange each circle so that point 0 is outside the others +C1 = fullcircle scaled 120 rotated 90 shifted 40 up; +C2 = C1 rotated 120; +C3 = C2 rotated 120; + +B0 = buildcycle(C1, C2, C3); +B1 = buildcycle(C1, C2); +B2 = buildcycle(C2, C3); +B3 = buildcycle(C3, C1); + +picture P; +for x=0 upto 1: + for y=0 upto 1: + P := image( + s := 1/4 + x/2; + k := 0 + y/4; + fill C1 withcolor s*(1,0,0,k); + fill C2 withcolor s*(0,1,0,k); + fill C3 withcolor s*(0,0,1,k); + fill B3 withcolor s*(1,0,1,k); + fill B2 withcolor s*(0,1,1,k); + fill B1 withcolor s*(1,1,0,k); + fill B0 withcolor s*(1,1,1,k); + undraw C1; undraw C2; undraw C3; + ) shifted -(200x, 200y); + draw P; + label.bot(("shade: " & decimal s & ", k: " & decimal k) + infont "phvr8r", point 1/2 of bbox P); + endfor +endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/blended-color-circles.pdf b/info/drawing-with-metapost/src/blended-color-circles.pdf Binary files differnew file mode 100644 index 0000000000..d5b4b4c631 --- /dev/null +++ b/info/drawing-with-metapost/src/blended-color-circles.pdf diff --git a/info/drawing-with-metapost/src/border-dashed.pdf b/info/drawing-with-metapost/src/border-dashed.pdf Binary files differnew file mode 100644 index 0000000000..0f46ba0340 --- /dev/null +++ b/info/drawing-with-metapost/src/border-dashed.pdf diff --git a/info/drawing-with-metapost/src/border-shadow.pdf b/info/drawing-with-metapost/src/border-shadow.pdf Binary files differnew file mode 100644 index 0000000000..5ba2540848 --- /dev/null +++ b/info/drawing-with-metapost/src/border-shadow.pdf diff --git a/info/drawing-with-metapost/src/border.mp b/info/drawing-with-metapost/src/border.mp new file mode 100644 index 0000000000..71dd27309b --- /dev/null +++ b/info/drawing-with-metapost/src/border.mp @@ -0,0 +1,50 @@ +\documentclass[border=2mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +input rope +beginfig(1); + + interim ahangle := 30; + + z0 = 42 left rotated normaldeviate; + z1 = 30 up rotated normaldeviate; + z2 = 31 right rotated normaldeviate; + z3 = 34 down rotated normaldeviate; + + path s[]; + for i=0 upto 3: + s[i] = unitsquare zscaled (z[(i+1) mod 4] - z[i]) shifted z[i]; + fill s[i] withcolor if odd i: Blues else: Oranges fi 8 1; + draw point 3 of s[i] -- subpath (1,4) of s[i] -- point 2 of s[i] withcolor 3/4; + drawarrow subpath (0, 1) of s[i]; + z[i+4] = 1/2[point 0 of s[i], point 2 of s[i]]; + endfor + + z8 = whatever[z4, z6] = whatever[z5, z7]; + + draw unitsquare scaled 5 rotated angle (z4-z6) shifted z8 withcolor 3/4; + + drawarrow z4--z6; + drawarrow z5--z7; + + vardef with_rounded_corners expr p = + for i=1 upto length p: + subpath (i-15/16, i-1/16) of p .. + endfor cycle + enddef; + + interim bboxmargin := 16; + draw rope with_rounded_corners bbox currentpicture; + % dashed evenly scaled 3/4; + %picture P; P = currentpicture; + %fill bbox P shifted (3,-3) withcolor 3/4; + %unfill bbox P; draw bbox P; + %draw P; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/border.pdf b/info/drawing-with-metapost/src/border.pdf Binary files differnew file mode 100644 index 0000000000..3b1e9fcd6f --- /dev/null +++ b/info/drawing-with-metapost/src/border.pdf diff --git a/info/drawing-with-metapost/src/braces.mp b/info/drawing-with-metapost/src/braces.mp new file mode 100644 index 0000000000..454adfb42a --- /dev/null +++ b/info/drawing-with-metapost/src/braces.mp @@ -0,0 +1,44 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef do_brace(expr a,b,m,r) = + save d, e, n, bb; + numeric d, n; pair e; path bb; + d = angle (b-a); + e = up scaled m rotated d shifted r[a,b]; + n = 1/2 m; + bb = ( + (origin {0,n} .. {right} (abs n,n)) + rotated d shifted a + -- + ((-abs n,-n) {right} .. {0,n} origin {0,-n} .. {right}(abs n,-n)) + rotated d shifted e + -- + ((-abs n,n){right}..{0,-n}origin) + rotated d shifted b + ) shifted (up scaled n rotated d); + draw bb withpen pencircle yscaled .6 xscaled .1666 rotated d; + point 3 of bb +enddef; +beginfig(1); + +z1 = origin; +z2 = 180 right rotated 13; +draw z1--z2 withcolor .7 white; +dotlabel.llft("$z_1$",z1); +dotlabel.urt("$z_2$",z2); + +begingroup; interim ahangle := 180; interim ahlength := 3; +drawdblarrow (z1--z2) shifted (12 up rotated angle (z2-z1)) withcolor 2/3 blue; +endgroup; + +drawoptions(withcolor 2/3 red); +label.lrt("Here",do_brace(z1,z2,-12,3/4)); +drawoptions(); + + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/braces.pdf b/info/drawing-with-metapost/src/braces.pdf Binary files differnew file mode 100644 index 0000000000..28064cabad --- /dev/null +++ b/info/drawing-with-metapost/src/braces.pdf diff --git a/info/drawing-with-metapost/src/brachisto.mp b/info/drawing-with-metapost/src/brachisto.mp new file mode 100644 index 0000000000..aab584db00 --- /dev/null +++ b/info/drawing-with-metapost/src/brachisto.mp @@ -0,0 +1,56 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + + path Y, L, C, P, S; + + % cycloid + numeric r, s; r = 160; + def sin(expr x) = sind( 57.29577951308232 x) enddef; + def cos(expr x) = cosd( 57.29577951308232 x) enddef; + vardef u(expr x) = x - sin(x) enddef; + vardef v(expr x) = 1 - cos(x) enddef; + vardef f(expr t) = u(t) < v(t) enddef; + tolerance := epsilon; + numeric a; a = solve f(2,3); + + Y = (origin for t = 1 upto 128: -- (u(t / 128 * a), -v(t / 128 * a)) endfor) scaled r; + + % Y = origin for t=5 step 5 until 140: + % -- (0, r) rotated t shifted (t / 57.29577951308232 * r, -r) + % endfor cutafter (origin -- 4r * dir -45); + + z0 = point 0 of Y; + z1 = point infinity of Y; + + % line + L = z0 -- z1; + + % circle + C = quartercircle rotated 180 scaled 2x1 shifted (x1, y0); + + % parabola f = x^2, f' = 2x + P = z0{1,-2} ... (xpart 1/2[z1, z0], ypart 1/4[z1, z0]){1,-1} ... z1 {1, 0}; + + % sixth degree f = x^6, f' = 6x^5 + S = z0{1,-6} ... (xpart 1/2[z1, z0], ypart 1/64[z1, z0]){1, -6/32} ... z1 {1, 0}; + + % axes + draw z0 -- (x0,y1) -- z1 withcolor 3/4; + + drawoptions(withcolor 2/3 red); draw L; dotlabel.urt("Line", point 1/4 of L); + drawoptions(withcolor 1/2 green); draw C; dotlabel.urt("Circle", point 1 of C); + drawoptions(withcolor 1/4[red, green]); draw P; dotlabel.urt("Parabola", point 1/2 of P); + drawoptions(withcolor 3/4[red, green]); draw S; dotlabel.llft("Sixth degree", point 3/4 of S); + drawoptions(withcolor 1/2 blue); draw Y; dotlabel.urt("Cycloid", point 100 of Y); + drawoptions(); + + dotlabel.ulft("$A$", z0); + dotlabel.urt("$B$", z1); + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/brachisto.pdf b/info/drawing-with-metapost/src/brachisto.pdf Binary files differnew file mode 100644 index 0000000000..9535832c97 --- /dev/null +++ b/info/drawing-with-metapost/src/brachisto.pdf diff --git a/info/drawing-with-metapost/src/brexit-map.mp b/info/drawing-with-metapost/src/brexit-map.mp new file mode 100644 index 0000000000..5f3f68d6b4 --- /dev/null +++ b/info/drawing-with-metapost/src/brexit-map.mp @@ -0,0 +1,99 @@ +\documentclass{standalone} +\usepackage{luamplib} +\mplibtextextlabel{enable} +\begin{document} +\begin{mplibcode} +input london-boroughs +input colorbrewer-rgb +vardef median(expr p) = + save n; numeric n; n = length p; + 1/n * point 0 of p for i=1 upto n: + 1/n * point i of p endfor +enddef; + +beginfig(1); + +fill Barking_and_Dagenham withcolor 0.638 [white, RdYlBu 9 3]; draw Barking_and_Dagenham; +fill Barnet withcolor 0.721 [white, RdYlBu 9 6]; draw Barnet; +fill Bexley withcolor 0.752 [white, RdYlBu 9 3]; draw Bexley; +fill Brent withcolor 0.650 [white, RdYlBu 9 5]; draw Brent; +fill Bromley withcolor 0.788 [white, RdYlBu 9 5]; draw Bromley; +fill Camden withcolor 0.654 [white, RdYlBu 9 7]; draw Camden; +fill City_of_London withcolor 0.735 [white, RdYlBu 9 7]; draw City_of_London; +fill Croydon withcolor 0.698 [white, RdYlBu 9 5]; draw Croydon; +fill Ealing withcolor 0.700 [white, RdYlBu 9 5]; draw Ealing; +fill Enfield withcolor 0.690 [white, RdYlBu 9 5]; draw Enfield; +fill Greenwich withcolor 0.695 [white, RdYlBu 9 5]; draw Greenwich; +fill Hackney withcolor 0.651 [white, RdYlBu 9 7]; draw Hackney; +fill Hammersmith_and_Fulham withcolor 0.699 [white, RdYlBu 9 6]; draw Hammersmith_and_Fulham; +fill Haringey withcolor 0.705 [white, RdYlBu 9 7]; draw Haringey; +fill Harrow withcolor 0.722 [white, RdYlBu 9 5]; draw Harrow; +fill Havering withcolor 0.760 [white, RdYlBu 9 3]; draw Havering; +fill Hillingdon withcolor 0.689 [white, RdYlBu 9 4]; draw Hillingdon; +fill Hounslow withcolor 0.697 [white, RdYlBu 9 5]; draw Hounslow; +fill Islington withcolor 0.703 [white, RdYlBu 9 7]; draw Islington; +fill Kensington_and_Chelsea withcolor 0.659 [white, RdYlBu 9 6]; draw Kensington_and_Chelsea; +fill Kingston_upon_Thames withcolor 0.783 [white, RdYlBu 9 6]; draw Kingston_upon_Thames; +fill Lambeth withcolor 0.673 [white, RdYlBu 9 7]; draw Lambeth; +fill Lewisham withcolor 0.630 [white, RdYlBu 9 6]; draw Lewisham; +fill Merton withcolor 0.734 [white, RdYlBu 9 6]; draw Merton; +fill Newham withcolor 0.592 [white, RdYlBu 9 5]; draw Newham; +fill Redbridge withcolor 0.675 [white, RdYlBu 9 5]; draw Redbridge; +fill Richmond_upon_Thames withcolor 0.820 [white, RdYlBu 9 6]; draw Richmond_upon_Thames; +fill Southwark withcolor 0.661 [white, RdYlBu 9 7]; draw Southwark; +fill Sutton withcolor 0.760 [white, RdYlBu 9 4]; draw Sutton; +fill Tower_Hamlets withcolor 0.645 [white, RdYlBu 9 6]; draw Tower_Hamlets; +fill Waltham_Forest withcolor 0.666 [white, RdYlBu 9 5]; draw Waltham_Forest; +fill Wandsworth withcolor 0.719 [white, RdYlBu 9 7]; draw Wandsworth; +fill Westminster withcolor 0.649 [white, RdYlBu 9 6]; draw Westminster; + +label("\tiny Barking \& Dagenham", median(Barking_and_Dagenham)); +label("\tiny Barnet", median(Barnet)); +label("\tiny Bexley", median(Bexley)); +label("\tiny Brent", median(Brent)); +label("\tiny Bromley", median(Bromley)); +label("\tiny Camden", median(Camden)); +label("\tiny City of London", median(City_of_London)); +label("\tiny Croydon", median(Croydon)); +label("\tiny Ealing", median(Ealing)); +label("\tiny Enfield", median(Enfield)); +label("\tiny Greenwich", median(Greenwich)); +label("\tiny Hackney", median(Hackney)); +label(TEX("\tiny Hammersmith \& Fulham") rotated -62, median(Hammersmith_and_Fulham)); +label("\tiny Haringey", median(Haringey)); +label("\tiny Harrow", median(Harrow)); +label("\tiny Havering", median(Havering)); +label("\tiny Hillingdon", median(Hillingdon)); +label("\tiny Hounslow", median(Hounslow) shifted 12 left); +label("\tiny Islington", median(Islington)); +label(TEX("\tiny Kensington \& Chelsea") rotated -54, median(Kensington_and_Chelsea)); +label(TEX("\tiny Kingston upon Thames") rotated 66, median(Kingston_upon_Thames)); +label("\tiny Lambeth", median(Lambeth)); +label("\tiny Lewisham", median(Lewisham)); +label("\tiny Merton", median(Merton)); +label("\tiny Newham", median(Newham)); +label("\tiny Redbridge", median(Redbridge)); +label("\tiny Richmond upon Thames", median(Richmond_upon_Thames)); +label("\tiny Southwark", median(Southwark)); +label("\tiny Sutton", median(Sutton)); +label("\tiny Tower Hamlets", median(Tower_Hamlets)); +label("\tiny Waltham Forest", median(Waltham_Forest)); +label("\tiny Wandsworth", median(Wandsworth)); +label("\tiny Westminster", median(Westminster)); + +label("\tiny Barking \& Dagenham", median(Barking_and_Dagenham)); +picture legend; legend = image( +numeric s; s = 10; +for i = 1 upto 9: + fill unitsquare scaled s shifted (i*s, 0) withcolor RdYlBu [9][i]; + if i > 1: draw (i*s, 0) -- (i*s, s); fi +endfor +draw unitsquare xscaled 9s yscaled s shifted (s,0); +label.top("Remain" infont "phvr8r", (10s, s)); +label.top("Leave" infont "phvr8r", (s, s));); +label.ulft(legend, point 1 of bbox currentpicture); +label.lrt("Brexit vote in London 2016" infont "phvr8r", point 3 of bbox currentpicture); + + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/brexit-map.pdf b/info/drawing-with-metapost/src/brexit-map.pdf Binary files differnew file mode 100644 index 0000000000..28b46ecadb --- /dev/null +++ b/info/drawing-with-metapost/src/brexit-map.pdf diff --git a/info/drawing-with-metapost/src/brown3a.pdf b/info/drawing-with-metapost/src/brown3a.pdf new file mode 100644 index 0000000000..6759b504a0 --- /dev/null +++ b/info/drawing-with-metapost/src/brown3a.pdf @@ -0,0 +1,8196 @@ +%PDF-1.3 +% +2 0 obj +<< +/Length 395073 +>> +stream +0.333 0.333 1 RG +1 J 1 j 0.48 w 10 M []0 d +/GS1 gs +1 i +397 312 m +397.509 311.707 398.102 311.594 398.683 311.68 c +402.325 312.217 403.189 317.651 399.197 320.387 c +397.182 321.769 394.498 321.883 392.769 320.244 c +390.668 318.251 390.753 313.916 387.558 313.678 c +383.95 313.409 383.851 318.247 383.328 322.265 c +383.227 323.038 382.977 323.838 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b/info/drawing-with-metapost/src/callout.mp new file mode 100644 index 0000000000..be171d03e5 --- /dev/null +++ b/info/drawing-with-metapost/src/callout.mp @@ -0,0 +1,19 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path a; a = (left .. right) scaled 42; +pair P; P = origin; r = 31; +draw a rotated r withcolor 2/3 red; draw a rotated (90+r) withcolor 2/3 red; +drawdot P withpen pencircle scaled dotlabeldiam; +label("$P$", P shifted 10 dir 68); + z0 = P + 20 dir -20; + draw z0 -- P + cutafter fullcircle scaled 8 shifted P + withpen pencircle scaled 1/4; + label.rt("\textit{pole}", z0); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/callout.pdf b/info/drawing-with-metapost/src/callout.pdf Binary files differnew file mode 100644 index 0000000000..07f6e21964 --- /dev/null +++ b/info/drawing-with-metapost/src/callout.pdf diff --git a/info/drawing-with-metapost/src/circumcircle.mp b/info/drawing-with-metapost/src/circumcircle.mp new file mode 100644 index 0000000000..b4cca18faa --- /dev/null +++ b/info/drawing-with-metapost/src/circumcircle.mp @@ -0,0 +1,49 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef through(expr a, b, o) = + save t; numeric t; t = 1+o/abs(a-b); + t[b,a] -- t[a,b] +enddef; + +beginfig(1); + path T; T = origin -- 377 dir 10 -- 233 dir 70 -- cycle; + pair m; + m = whatever * (point 0 of T - point 1 of T) rotated 90 shifted point 1/2 of T + = whatever * (point 1 of T - point 2 of T) rotated 90 shifted point 3/2 of T + = whatever * (point 2 of T - point 3 of T) rotated 90 shifted point 5/2 of T; + + drawoptions(withcolor 3/4); + draw unitsquare scaled 5 rotated angle (point 1 of T-point 0 of T) shifted point 1/2 of T; + draw unitsquare scaled 5 rotated angle (point 2 of T-point 1 of T) shifted point 3/2 of T; + draw unitsquare scaled 5 rotated angle (point 0 of T-point 2 of T) shifted point 5/2 of T; + drawoptions(dashed evenly scaled 1/4 withcolor 1/2); + draw through(point 1/2 of T, m, 10); + draw through(point 3/2 of T, m, 10); + draw through(point 5/2 of T, m, 10); + drawoptions(); + draw fullcircle scaled 2 abs (point 0 of T - m) shifted m withcolor 2/3 red; + draw T; + + draw m withpen pencircle scaled dotlabeldiam; label("$m$", m + 10 dir 12); + +vardef mark_equal(expr a, b, n) = + save m, s; picture m; m = image( + numeric s; 2s = n - 1; for t=-s upto s: + draw (down--up) scaled 2 rotated -13 shifted (t,0) withpen pencircle scaled 1/4; + endfor + ); + draw m rotated angle (b-a) shifted 1/4[a,b]; + draw m rotated angle (b-a) shifted 3/4[a,b]; +enddef; + + mark_equal(point 0 of T, point 1 of T, 1); + mark_equal(point 1 of T, point 2 of T, 2); + mark_equal(point 2 of T, point 0 of T, 3); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/circumcircle.pdf b/info/drawing-with-metapost/src/circumcircle.pdf Binary files differnew file mode 100644 index 0000000000..a1ce79e196 --- /dev/null +++ b/info/drawing-with-metapost/src/circumcircle.pdf diff --git a/info/drawing-with-metapost/src/clocks.mp b/info/drawing-with-metapost/src/clocks.mp new file mode 100644 index 0000000000..045c0f1682 --- /dev/null +++ b/info/drawing-with-metapost/src/clocks.mp @@ -0,0 +1,47 @@ +vardef clock_roman primary h = + if h<5: "" for i=1 upto h: & "I" endfor + elseif h<9: "V" for i=6 upto h: & "I" endfor + elseif h=9: "IX" + else: "X" for i=11 upto h: & "I" endfor + fi +enddef; + +path hand[]; +hand1 = origin .. (.257,1/50) .. (.377,1/60) & (.377,1/60) {up} .. (.40,3/50) .. (.60, 1/40) .. {right} (.75,0); +hand1 := (hand1 .. reverse hand1 reflectedabout(left,right) .. cycle) scaled 50; + +hand2 = origin .. (.60, 1/64) .. {right} (.925,0); +hand2 := (hand2 .. reverse hand2 reflectedabout(left,right) .. cycle) scaled 50; + +% hour of the day to degrees +vardef htod(expr hours) = 30*((15-hours) mod 12) enddef; +vardef mtod(expr minutes) = 6*((75-minutes) mod 60) enddef; + +vardef clock(expr hours, minutes) = image( + % face and outer ring + fill fullcircle scaled 100 withcolor 1/256(240, 240, 230); + draw fullcircle scaled 99 withcolor .8 white; + draw fullcircle scaled 100 withpen pencircle scaled 7/8; + % hour and minute marks + for t=0 step 6 until 359: + draw ((48,0)--(49,0)) rotated t; + endfor + for t=0 step 30 until 359: + draw ((47,0)--(49,0)) rotated t withpen pencircle scaled 7/8; + endfor + % numerals + for h=1 upto 12: + label( + if known roman_numerals_please: + clock_roman h infont "bchr8r" rotated ((12-h)*30) + else: + decimal h infont "bchr8r" + fi, (40,0) rotated htod(h)); + endfor + % hands rotated to the given time + filldraw hand1 rotated htod(hours+minutes/60) withpen pencircle scaled 7/8; + filldraw hand2 rotated mtod(minutes) withpen pencircle scaled 7/8; + % draw the center on top + fill fullcircle scaled 5; + fill fullcircle scaled 3 withcolor .4 white; +) enddef; diff --git a/info/drawing-with-metapost/src/closed-antireuleaux-set.mp b/info/drawing-with-metapost/src/closed-antireuleaux-set.mp new file mode 100644 index 0000000000..8d8da6b507 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-antireuleaux-set.mp @@ -0,0 +1,27 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +\mplibtextextlabel{enable} +vardef reuleaux(expr n, r) = + save a; numeric a; a = 90/n; + for t = 0 step 4a until 359: + (0, r) rotated t {left rotated (3a + t)} .. {left rotated (a + t)} + endfor cycle +enddef; +vardef median(expr p) = origin for i=1 upto length p: + point i of p / length p endfor enddef; +input colorbrewer-rgb +beginfig(1); + for n = 3 upto 7: + path p; p = reuleaux(n, 42) shifted (90n, 0); + pair m; m = point (n+1)/2 of p; + fill p withcolor if odd n: Oranges else: Blues fi 8 2; + % draw tensepath(p) withcolor Blues 8 6; + %draw p rotatedabout(median(p), 180/n) withcolor Blues 8 6; + draw p; + endfor + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/closed-antireuleaux-set.pdf b/info/drawing-with-metapost/src/closed-antireuleaux-set.pdf Binary files differnew file mode 100644 index 0000000000..9c0606cfd3 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-antireuleaux-set.pdf diff --git a/info/drawing-with-metapost/src/closed-fixed-polygon.mp b/info/drawing-with-metapost/src/closed-fixed-polygon.mp new file mode 100644 index 0000000000..793b4fac8b --- /dev/null +++ b/info/drawing-with-metapost/src/closed-fixed-polygon.mp @@ -0,0 +1,19 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +vardef polygon_with_side(expr n, s) = + save a, b, r; numeric a, b, r; + a * n = 360; a + 2b = 180; r = s * sind(b) / sind(a); + for i = 0 upto n-1: (0, r) rotated (a * i) -- endfor cycle +enddef; +for n = 11 downto 3: + path p; p = polygon_with_side(n, 72); + fill p withcolor (n/32)[white, 3/4 if odd n: red else: blue fi]; + draw p; +endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/closed-fixed-polygon.pdf b/info/drawing-with-metapost/src/closed-fixed-polygon.pdf Binary files differnew file mode 100644 index 0000000000..c53cf71886 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-fixed-polygon.pdf diff --git a/info/drawing-with-metapost/src/closed-points.mp b/info/drawing-with-metapost/src/closed-points.mp new file mode 100644 index 0000000000..9abf287338 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-points.mp @@ -0,0 +1,66 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +vardef median primary P = + (origin for i=1 upto length P: + point i of P endfor) / length P +enddef; + +primarydef p enlarged s = p shifted -median p scaled s shifted median p enddef; + +path u, c, s; +u = unitsquare shifted -(1/2, 1/2) scaled 72; +c = fullcircle scaled 72 shifted 108 right; +s = (superellipse(right, up, left, down, 0.8)) scaled 36 shifted 216 right; + +picture p[]; +p1 = image( +draw origin--left scaled 50 withcolor background; +ahangle := 30; numeric r, x; r = 9/16; x = -1; +forsuffixes $=u, c, s: + drawarrow subpath(r, r + length $) of $; + draw fullcircle scaled 2 shifted median $ withcolor .67 red; + for i=0 upto length $-1: + fill fullcircle scaled 2 shifted point i of $ withcolor .67 red; + label(decimal i, point i of ($ enlarged 1.15)); + endfor +endfor); + +p2 = image( +draw origin--left scaled 50 withcolor background; +forsuffixes $=u,c,s: + fill median $ -- subpath (1,2) of $ -- cycle withcolor .8[blue,white]; + draw median $ -- point 1 of $; + draw median $ -- point 2 of $; + draw $; +endfor +); + +p3 = image( +draw origin--left scaled 50 withcolor background; +forsuffixes $=u,c,s: + r := 1/5 length $; + fill median $ -- subpath (0,r) of $ -- cycle withcolor .8[red,white]; + draw median $ -- point 0 of $; + draw median $ -- point r of $; + draw $; + endfor +); + +p4 = image( +draw origin--left scaled 50 withcolor background; +forsuffixes $=u,c,s: + fill median $ -- subpath (-1,1) of $ -- cycle withcolor .8[black,white]; + draw median $ -- point -1 of $; + draw median $ -- point 1 of $; + draw $; + endfor +); + +beginfig(1); +for i=1 upto 4: draw p[i] shifted (0,-108i); endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/closed-points.pdf b/info/drawing-with-metapost/src/closed-points.pdf Binary files differnew file mode 100644 index 0000000000..f99b1bbf2f --- /dev/null +++ b/info/drawing-with-metapost/src/closed-points.pdf diff --git a/info/drawing-with-metapost/src/closed-polygon-chain.mp b/info/drawing-with-metapost/src/closed-polygon-chain.mp new file mode 100644 index 0000000000..14f9b28738 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-polygon-chain.mp @@ -0,0 +1,18 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input archimedean-tools +beginfig(1); + path P[]; P3 = for i=0 upto 2: 6 up rotated 120i -- endfor cycle; + fill P3 withcolor 3/4 red; draw P3; + for n = 4 upto 23: + numeric m; m = floor(n / 2); + P[n] = poly n of subpath (m, m-1) of P[n-1]; + fill P[n] withcolor (n/32)[3/4 if odd n: red else: blue fi, white]; + draw P[n]; label(decimal n, median(P[n])); + endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/closed-polygon-chain.pdf b/info/drawing-with-metapost/src/closed-polygon-chain.pdf Binary files differnew file mode 100644 index 0000000000..4034d6a359 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-polygon-chain.pdf diff --git a/info/drawing-with-metapost/src/closed-polygon-tops.mp b/info/drawing-with-metapost/src/closed-polygon-tops.mp new file mode 100644 index 0000000000..9943092ea8 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-polygon-tops.mp @@ -0,0 +1,35 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +vardef median primary P = + (origin for i=1 upto length P: + point i of P endfor) / length P +enddef; + +primarydef p enlarged s = p shifted -median p scaled s shifted median p enddef; + +vardef polygon(expr n, r) = + for i = 0 upto n - 1: (0, r) rotated (360 / n * i) -- endfor cycle +enddef; + +path u, c, s; +u = polygon(5, 36); +c = polygon(6, 36) shifted 108 right; +s = polygon(7, 36) shifted 216 right; + +beginfig(1); +draw origin--left scaled 50 withcolor background; +ahangle := 30; numeric r, x; r = 9/16; x = -1; +forsuffixes $=u, c, s: + drawarrow subpath(r, r + length $) of $; + draw fullcircle scaled 2 shifted median $ withcolor .67 red; + for i=0 upto length $-1: + fill fullcircle scaled 2 shifted point i of $ withcolor .67 red; + label(decimal i, point i of ($ enlarged 1.15)); + endfor +endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/closed-polygon-tops.pdf b/info/drawing-with-metapost/src/closed-polygon-tops.pdf Binary files differnew file mode 100644 index 0000000000..3c4a736834 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-polygon-tops.pdf diff --git a/info/drawing-with-metapost/src/closed-polygons.mp b/info/drawing-with-metapost/src/closed-polygons.mp new file mode 100644 index 0000000000..22d4e42f97 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-polygons.mp @@ -0,0 +1,70 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +vardef median primary P = + (origin for i=1 upto length P: + point i of P endfor) / length P +enddef; + +primarydef p enlarged s = p shifted -median p scaled s shifted median p enddef; + +vardef polygon(expr n, r) = + for i = 0 upto n - 1: (r, 0) rotated (360 / n * i) -- endfor cycle +enddef; + +path u, c, s; +u = polygon(5, 36); +c = polygon(6, 36) shifted 108 right; +s = polygon(7, 36) shifted 216 right; + +picture p[]; +p1 = image( +draw origin--left scaled 50 withcolor background; +ahangle := 30; numeric r, x; r = 9/16; x = -1; +forsuffixes $=u, c, s: + drawarrow subpath(r, r + length $) of $; + draw fullcircle scaled 2 shifted median $ withcolor .67 red; + for i=0 upto length $-1: + fill fullcircle scaled 2 shifted point i of $ withcolor .67 red; + label(decimal i, point i of ($ enlarged 1.15)); + endfor +endfor); + +p2 = image( +draw origin--left scaled 50 withcolor background; +forsuffixes $=u,c,s: + fill median $ -- subpath (1,2) of $ -- cycle withcolor .8[blue,white]; + draw median $ -- point 1 of $; + draw median $ -- point 2 of $; + draw $; +endfor +); + +p3 = image( +draw origin--left scaled 50 withcolor background; +forsuffixes $=u,c,s: + r := 1/5 length $; + fill median $ -- subpath (0,r) of $ -- cycle withcolor .8[red,white]; + draw median $ -- point 0 of $; + draw median $ -- point r of $; + draw $; + endfor +); + +p4 = image( +draw origin--left scaled 50 withcolor background; +forsuffixes $=u,c,s: + fill median $ -- subpath (-1,1) of $ -- cycle withcolor .8[black,white]; + draw median $ -- point -1 of $; + draw median $ -- point 1 of $; + draw $; + endfor +); + +beginfig(1); +for i=1 upto 4: draw p[i] shifted (0,-108i); endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/closed-polygons.pdf b/info/drawing-with-metapost/src/closed-polygons.pdf Binary files differnew file mode 100644 index 0000000000..4e827ad6d9 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-polygons.pdf diff --git a/info/drawing-with-metapost/src/closed-reuleaux-geometry.mp b/info/drawing-with-metapost/src/closed-reuleaux-geometry.mp new file mode 100644 index 0000000000..f4e69f3b51 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-reuleaux-geometry.mp @@ -0,0 +1,66 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef urp(expr n) = + save a; numeric a; a = 90/n; + for i=0 upto n-1: + 1/2 up rotated (a * 4i) {left rotated (a + a * 4i)} .. {left rotated (3a + a * 4i)} + endfor cycle +enddef; +vardef through@#(expr a, b) = save d; numeric d; d = abs(a-b); + (1+@#/d)[b, a] -- (1+@#/d)[a, b] enddef; +input colorbrewer-rgb +beginfig(1); + path p, c; p = urp(5) scaled 320; c = fullcircle scaled 320; + + pair A, B, C; + A = point 0 of p; + B = point 1 of p; + C = point 3 of p; + + drawoptions(withcolor 7/8); + draw c; + draw through 42(origin, A); + draw through 42(origin, B); + draw through 42(C, A); + draw through 42(C, B); + draw through 42(B, B shifted 64 up); + + forsuffixes $=origin, B: + draw (left--right) scaled 2 rotated 13 shifted $ shifted 42 up; + draw (left--right) scaled 2 rotated 13 shifted $ shifted 41 up; + endfor + ahangle := 30; + forsuffixes $=A, B: + drawarrow (reverse unitsquare scaled 5 -- 48 up) rotated angle ($ - C) shifted $; + endfor + + drawoptions(); + draw p withcolor Reds 8 7; + numeric a; a = 18; + label("$\alpha$", 32 dir 11/2 a shifted A); + label("$\alpha$", 32 dir 31/2 a shifted A); + label("$\alpha$", 32 dir 17/2 a shifted B); + label("$\alpha$", 32 dir 37/2 a shifted B); + label("$2\alpha$", 28 dir 14/2 a shifted C); + label("$3\alpha$", 24 dir 13/2 a shifted B); + label("$4\alpha$", 20 dir 7a); + + dotlabel.urt("$A$", A); + dotlabel.lrt("$B$", B); + dotlabel.lrt("$C$", C); + dotlabel.urt("$O$", origin); + + label.lft("$r$", 1/2 A); + +label.bot(btex \vbox{\hsize 4in \small\noindent +This proof only works for Reuleaux polygons with an odd number of sides, +because otherwise the point $C$ does not (quite) lie on the circle.\strut} etex, +point 1/2 of bbox currentpicture shifted 21 down); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/closed-reuleaux-geometry.pdf b/info/drawing-with-metapost/src/closed-reuleaux-geometry.pdf Binary files differnew file mode 100644 index 0000000000..190c7b6b22 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-reuleaux-geometry.pdf diff --git a/info/drawing-with-metapost/src/closed-reuleaux-set.mp b/info/drawing-with-metapost/src/closed-reuleaux-set.mp new file mode 100644 index 0000000000..4088ea8421 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-reuleaux-set.mp @@ -0,0 +1,31 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +\mplibtextextlabel{enable} +vardef reuleaux(expr n, r) = + save a; numeric a; a = 90/n; + for t = 0 step 4a until 359: + (0, r) rotated t {left rotated (a + t)} .. {left rotated (3a + t)} + endfor cycle +enddef; +vardef median(expr p) = origin for i=1 upto length p: + point i of p / length p endfor enddef; +input colorbrewer-rgb +beginfig(1); + for n = 3 upto 7: + path p; p = reuleaux(n, 42) shifted (90n, 0); + pair m; m = point (n+1)/2 of p; + fill p withcolor if odd n: Oranges else: Blues fi 8 2; + fill subpath (0, 1) of p -- m -- cycle + withcolor if odd n: Oranges else: Blues fi 8 3; + %draw fullcircle scaled 2 abs(point 0 of p - m) shifted m + % withpen pencircle scaled 2 withcolor 3/4; + %draw tensepath(p) withcolor Blues 8 6; + %draw p rotatedabout(median(p), 180/n) withcolor Blues 8 6; + draw p; + endfor + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/closed-reuleaux-set.pdf b/info/drawing-with-metapost/src/closed-reuleaux-set.pdf Binary files differnew file mode 100644 index 0000000000..9863d3d1cf --- /dev/null +++ b/info/drawing-with-metapost/src/closed-reuleaux-set.pdf diff --git a/info/drawing-with-metapost/src/closed-standards.mp b/info/drawing-with-metapost/src/closed-standards.mp new file mode 100644 index 0000000000..a3296e6af2 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-standards.mp @@ -0,0 +1,28 @@ +\documentclass{standalone} +\usepackage{luamplib} +\def \id#1{\mathord{\hbox{\it #1\/}}} +\def \kw#1{{\hbox{\bf #1\thinspace}}} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +path unitse; unitse = superellipse(1/2 right, 1/2 up, 1/2 left, 1/2 down,.8); +path xx, yy; xx = (left -- right) scaled 144; yy = (down -- up) scaled 89; +path u, c, s; +u = unitsquare scaled 72; +c = fullcircle scaled 72; +s = unitse scaled 72; + +beginfig(1); +drawarrow xx withcolor .67 white; +drawarrow yy withcolor .67 white; +draw u; label.top("$\id{unitsquare}$", point 5/2 of u); +drawoptions(withcolor .67 red); +draw c; label("$\id{fullcircle}$", 3/4[point 2 of c, point 6 of c]); +drawoptions(withcolor .67 blue); +draw s; label.lrt("$\id{superellipse}()$", point 7 of s); +drawoptions(); +dotlabel.bot("$\scriptstyle (1,0)$", point 1 of u); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/closed-standards.pdf b/info/drawing-with-metapost/src/closed-standards.pdf Binary files differnew file mode 100644 index 0000000000..7f3a42a63e --- /dev/null +++ b/info/drawing-with-metapost/src/closed-standards.pdf diff --git a/info/drawing-with-metapost/src/closed-triangles.mp b/info/drawing-with-metapost/src/closed-triangles.mp new file mode 100644 index 0000000000..b39a9bbf92 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-triangles.mp @@ -0,0 +1,29 @@ +\documentclass{standalone} +\usepackage{dwmpcode} +\usepackage{luamplib} +\mplibtextextlabel{enable} +\def \kw#1{\mathop{\hbox{\bf #1\thinspace}}} +\def \id#1{\mathord{\hbox{\it #1\/}}} +\begin{document} +\begin{mplibcode} +beginfig(1); +path xx, yy; xx = left scaled 8 -- right scaled 89; yy = down scaled 8 -- up scaled 55; +path t; t = origin -- (55,0) -- (55,34) -- cycle; +picture p[]; +p1 = image( +ahangle := 30; +drawarrow xx withcolor .67 white; +drawarrow yy withcolor .67 white; +fill t withcolor .8[blue,white]); + +p2 = image( +draw p1; +draw t; +dotlabel.rt(btex $(55,34)$ etex scaled 0.8, point 2 of t); +) shifted 144 right; +draw p1; draw p2; +label.bot(btex \mpl{fill t withcolor 0.8[blue,white];} etex scaled 0.8, point 1/2 of bbox p1); +label.bot(btex \mpl{draw t;} etex scaled 0.8, point 1/2 of bbox p2); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/closed-triangles.pdf b/info/drawing-with-metapost/src/closed-triangles.pdf Binary files differnew file mode 100644 index 0000000000..e42164b4b7 --- /dev/null +++ b/info/drawing-with-metapost/src/closed-triangles.pdf diff --git a/info/drawing-with-metapost/src/color-blend-toy.mp b/info/drawing-with-metapost/src/color-blend-toy.mp new file mode 100644 index 0000000000..d712791230 --- /dev/null +++ b/info/drawing-with-metapost/src/color-blend-toy.mp @@ -0,0 +1,21 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); + color brg, pbr; + brg = 1/256 (1, 66, 37); + pbr = 1/256 (223, 52, 57); + N = 5; n = 0; + for y=1 upto N: + for x=1 upto N: + fill fullcircle scaled 16 shifted 20(x,y) + withpen pencircle scaled 2 + withcolor (n/N/N)[pbr, brg]; + label(decimal incr n infont "phvr8r", 20(x,y)) + withcolor white; + endfor + endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/color-blend-toy.pdf b/info/drawing-with-metapost/src/color-blend-toy.pdf Binary files differnew file mode 100644 index 0000000000..9f1803d89c --- /dev/null +++ b/info/drawing-with-metapost/src/color-blend-toy.pdf diff --git a/info/drawing-with-metapost/src/color-hsv-bathymetric.mp b/info/drawing-with-metapost/src/color-hsv-bathymetric.mp new file mode 100644 index 0000000000..057e923edd --- /dev/null +++ b/info/drawing-with-metapost/src/color-hsv-bathymetric.mp @@ -0,0 +1,42 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +input color-hsv-macro +beginfig(2); + defaultfont := "phvr8r"; + defaultscale := 3/4; + path h,d,b; numeric n; + h = ((-2,0)--(0,0)--(-1,3)--(-2,3)--cycle) scaled 60; + d = h rotated 180; + n = 10; + b = subpath (0,1) of h -- point 1+1/n of d -- (xpart point 0 of h, ypart point 1+1/n of d) -- cycle; + fill b withcolor hsv_color(123, 1/8, 7/8); + draw subpath (2.13,4) of b; + + for i=1 upto n: + fill point 4-(i-1)/n of h -- point 1+(i-1)/n of h + -- point 1+i/n of h -- point 4-i/n of h -- cycle + withcolor hsv_color(42, 1/4 + 3/4 * i/n, 1 - i/3n); + fill point 4-(i-1)/n of d -- point 1+(i-1)/n of d + -- point 1+i/n of d -- point 4-i/n of d -- cycle + withcolor hsv_color(200, i/n - 1/n, 1 - i/3n); + endfor + string s; + for i=1 upto n-1: + draw point 4-i/n of h -- point 1+i/n of h; + draw point 4-i/n of d -- point 1+i/n of d; + s := decimal if i < 4: (i**2+1) else: (10 + (i-3)*10) fi & "00"; + label.rt(s, point 1+i/n of h); + label.lft(s, point 1+i/n of d); + endfor + label.rt("Metres", point 2 of h); + label.lft("Metres", point 2 of d); + label.lft("Hypsometric tints" infont defaultfont scaled defaultscale rotated 90, point 7/2 of h); + label.rt("Bathymetric tints" infont defaultfont scaled defaultscale rotated -90, point 7/2 of d); + label.lft("sea level", point 0 of h); + label("areas below sea level", center b); + draw h; draw d; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/color-hsv-bathymetric.pdf b/info/drawing-with-metapost/src/color-hsv-bathymetric.pdf Binary files differnew file mode 100644 index 0000000000..8f03ce2b73 --- /dev/null +++ b/info/drawing-with-metapost/src/color-hsv-bathymetric.pdf diff --git a/info/drawing-with-metapost/src/color-hsv-gamut.mp b/info/drawing-with-metapost/src/color-hsv-gamut.mp new file mode 100644 index 0000000000..392853e7bc --- /dev/null +++ b/info/drawing-with-metapost/src/color-hsv-gamut.mp @@ -0,0 +1,28 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +input color-hsv-macro +beginfig(1); + defaultfont := "phvr8r"; + numeric s[], v[]; + s0 = 1/2; v0 = 7/8; + s1 = 7/8; v1 = 7/8; + s2 = 7/8; v2 = 1/2; + for y=0 upto 2: + for h=0 step 15 until 360: + fill fullcircle scaled 24 shifted (h, -32y) withcolor hsv_color(h, s[y], v[y]); + draw fullcircle scaled 24 shifted (h, -32y) withcolor white; + if y=1: label(decimal h infont defaultfont scaled 1/2, (h,-16)); fi + endfor + endfor + + label.urt("Less saturation", (-20,12)); + label.lrt("Lower value", (-20,-76)); + + drawarrow (-15, -12) -- (-15,12); + drawarrow (-15, -52) -- (-15,-76); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/color-hsv-gamut.pdf b/info/drawing-with-metapost/src/color-hsv-gamut.pdf Binary files differnew file mode 100644 index 0000000000..a2a2d68a08 --- /dev/null +++ b/info/drawing-with-metapost/src/color-hsv-gamut.pdf diff --git a/info/drawing-with-metapost/src/color-hsv-macro.mp b/info/drawing-with-metapost/src/color-hsv-macro.mp new file mode 100644 index 0000000000..1ec265a055 --- /dev/null +++ b/info/drawing-with-metapost/src/color-hsv-macro.mp @@ -0,0 +1,15 @@ +vardef hsv_color(expr h,s,v) = + % following wikipedia article on "HSL and HSV" + save chroma, hh, x, m; + chroma = v*s; + hh = h/60; + x = chroma * (1-abs(hh mod 2 - 1)); + m = v - chroma; + if hh < 1: (chroma,x,0)+(m,m,m) + elseif hh < 2: (x,chroma,0)+(m,m,m) + elseif hh < 3: (0,chroma,x)+(m,m,m) + elseif hh < 4: (0,x,chroma)+(m,m,m) + elseif hh < 5: (x,0,chroma)+(m,m,m) + else: (chroma,0,x)+(m,m,m) + fi +enddef; diff --git a/info/drawing-with-metapost/src/compass-rose.mp b/info/drawing-with-metapost/src/compass-rose.mp new file mode 100644 index 0000000000..26952e26e2 --- /dev/null +++ b/info/drawing-with-metapost/src/compass-rose.mp @@ -0,0 +1,35 @@ +vardef compass_rose(expr shade, highlight) = + save currentpicture, s; + picture currentpicture; currentpicture := nullpicture; + drawoptions(withcolor shade); + draw fullcircle scaled 144 withpen pencircle scaled 2; + draw fullcircle scaled 138; + draw thelabel("NW" infont defaultfont scaled 3/4, 64 up) rotated +45; + draw thelabel("NE" infont defaultfont scaled 3/4, 64 up) rotated -45; + draw thelabel("SE" infont defaultfont scaled 3/4, 64 down) rotated +45; + draw thelabel("SW" infont defaultfont scaled 3/4, 64 down) rotated -45; + draw fullcircle scaled 120; + draw fullcircle scaled 118; + draw fullcircle scaled 108; + draw fullcircle scaled 106; + for t=0 step 15 until 350: draw ((53,0) -- (60,0)) rotated t; endfor + for t=0 step 45 until 350: draw ((69,0) -- (80,0)) rotated t; endfor + path s; s = 8 right -- 60 up -- 8 left -- cycle; + for t=0 upto 3: filldraw s rotated (45 + 90t); endfor + s := s scaled 1.414; + for t=0 upto 3: + s := s rotated 90t; + fill s withcolor background; + fill subpath (-1/2, 1) of s -- cycle if t=0: withcolor highlight fi; + draw s; + endfor + fill fullcircle scaled 28 withcolor background; + draw fullcircle scaled 28; + fill fullcircle scaled dotlabeldiam; + label.top("N" infont "cmbx12", 84 up); + label.bot("S" infont "cmbx12", 84 down); + label.rt("E" infont "cmbx12", 84 right); + label.lft("W" infont "cmbx12", 84 left); + drawoptions(); + currentpicture +enddef; diff --git a/info/drawing-with-metapost/src/complex-inverse-and-sqrt.mp b/info/drawing-with-metapost/src/complex-inverse-and-sqrt.mp new file mode 100644 index 0000000000..3b187fad26 --- /dev/null +++ b/info/drawing-with-metapost/src/complex-inverse-and-sqrt.mp @@ -0,0 +1,48 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +picture axes, dot, opendot, P[]; +numeric u; u = 42; +interim ahangle := 30; +axes = image( + path xx, yy; xx = (left--right) scaled 1.2 u; yy = xx rotated 90; + draw fullcircle scaled 2u; + drawarrow xx; label.rt("$\Re$", point 1 of xx); + drawarrow yy; label.top("$\Im$", point 1 of yy); +); +dot = image(fill fullcircle scaled dotlabeldiam); +opendot = image(draw dot; unfill fullcircle scaled 3/4 dotlabeldiam); +z1 = 0.7 dir 76; +z2 = 1.5 dir 60; +z3 = 1 dir 42; + +vardef connect(expr a, b, shade) = + path c; c = (a -- origin -- b) scaled u; + drawarrow c cutafter fullcircle scaled 5 shifted point 2 of c withcolor shade; + draw dot shifted point 0 of c; draw opendot shifted point 2 of c; +enddef; + +vardef zinverse(expr z) = 1/abs z * dir - angle z enddef; +vardef zsqrt(expr z) = if abs z > 0: sqrt(abs z) * dir 1/2 angle fi z enddef; + +input colorbrewer-rgb +P1 = image( +draw axes withcolor 3/4; +label.lrt("\small\texttt{zinverse()}", ulcorner axes shifted 10 left); +for i=1 upto 3: connect(z[i], zinverse(z[i]), SetTwo[7][2+i]); endfor +); +P2 = image( +draw axes withcolor 3/4; +label.lrt("\small\texttt{zsqrt()}", ulcorner axes); +for i=1 upto 3: connect(z[i], zsqrt(z[i]), SetTwo[7][2+i]); endfor +); +beginfig(1); + interim labeloffset := 12; + label.lft(P1, origin); + label.rt(P2, origin); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/complex-inverse-and-sqrt.pdf b/info/drawing-with-metapost/src/complex-inverse-and-sqrt.pdf Binary files differnew file mode 100644 index 0000000000..b1dae07134 --- /dev/null +++ b/info/drawing-with-metapost/src/complex-inverse-and-sqrt.pdf diff --git a/info/drawing-with-metapost/src/complex-operators.mp b/info/drawing-with-metapost/src/complex-operators.mp new file mode 100644 index 0000000000..a1510bc64b --- /dev/null +++ b/info/drawing-with-metapost/src/complex-operators.mp @@ -0,0 +1,32 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + numeric u; u = 1cm; + z1 = 2 dir 15; z2 = 1.2 dir 60; + z3 = z1+z2; z4 = z1 zscaled z2; z5 = (x1,-y1); + drawoptions(withcolor 2/3 white); + draw (1/2 left -- 3 right) scaled u ; + draw (1/2 down -- 3 up ) scaled u ; + draw subpath (0,3) of fullcircle scaled 2u rotated -22.5; + drawoptions(); + dotlabel.lrt (btex $\scriptstyle 1$ etex, (u,0)); + dotlabel.ulft(btex $\scriptstyle i$ etex, (0,u)); + interim ahangle := 30; + forsuffixes @=1,2,3,4,5: + x@ := x@ * u; y@ := y@ * u; + drawarrow origin -- z@ + cutafter fullcircle scaled 5 shifted z@ + withcolor 2/3 if @ < 3: blue else: red fi; + endfor + fill fullcircle scaled dotlabeldiam; + dotlabel.rt (btex $A$ etex, z1); + dotlabel.urt(btex $B$ etex, z2); + dotlabel.top(btex $A+B$ etex, z3); + dotlabel.top(btex $A \times B$ etex, z4); + dotlabel.rt (btex $\bar{A}$ etex, z5); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/complex-operators.pdf b/info/drawing-with-metapost/src/complex-operators.pdf Binary files differnew file mode 100644 index 0000000000..11370d7ce7 --- /dev/null +++ b/info/drawing-with-metapost/src/complex-operators.pdf diff --git a/info/drawing-with-metapost/src/conway.mp b/info/drawing-with-metapost/src/conway.mp new file mode 100644 index 0000000000..41d94bae34 --- /dev/null +++ b/info/drawing-with-metapost/src/conway.mp @@ -0,0 +1,22 @@ +\documentclass[margin=1mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +pair a, b, c, d, e, f, g; +a = origin; b = 180 right; c = 90 up; +d = 2/5[b, c]; e = 4/5[b,c]; f = 1/2[e, a]; g = 1/2[a, b]; + +fill a--b--c--cycle withcolor 15/16[blue, white]; +draw a--b--c--a--e--g--f; draw g--d; +dotlabel.llft("$a$", a); +dotlabel.lrt ("$b$", b); +dotlabel.ulft("$c$", c); +dotlabel.urt ("$d$", d); +dotlabel.urt ("$e$", e); +dotlabel.ulft("$f$", f); +dotlabel.bot ("$g$", g); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/conway.pdf b/info/drawing-with-metapost/src/conway.pdf Binary files differnew file mode 100644 index 0000000000..4ca336f367 --- /dev/null +++ b/info/drawing-with-metapost/src/conway.pdf diff --git a/info/drawing-with-metapost/src/corners.mp b/info/drawing-with-metapost/src/corners.mp new file mode 100644 index 0000000000..cf4e3ad416 --- /dev/null +++ b/info/drawing-with-metapost/src/corners.mp @@ -0,0 +1,36 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\usepackage{fontspec} +\setmainfont{TeX Gyre Pagella} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + picture p, q; + p = thelabel.rt("\fontsize{48}{48}\selectfont plain", origin); + q = thelabel.rt("\fontsize{48}{48}\selectfont rotated", origin) rotated 30; + + picture P[]; + numeric i; i = 0; + forsuffixes @=p,q: + P[incr i] = image( + fill bbox @ withcolor 15/16[red, white]; + draw llcorner @ -- lrcorner @ -- urcorner @ -- ulcorner @ -- cycle + withcolor 7/8[red, white]; + draw llcorner @ -- urcorner @ withcolor 7/8[red, white]; + draw ulcorner @ -- lrcorner @ withcolor 7/8[red, white]; + draw @ withcolor 3/4; + dotlabel.lft("\texttt{llcorner}", llcorner @); + dotlabel.rt ("\texttt{lrcorner}", lrcorner @); + dotlabel.rt ("\texttt{urcorner}", urcorner @); + dotlabel.lft("\texttt{ulcorner}", ulcorner @); + draw fullcircle scaled 3 shifted center @; + ); + endfor + labeloffset := 20; + label.top(P1, origin); + label.bot(P2, origin); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/corners.pdf b/info/drawing-with-metapost/src/corners.pdf Binary files differnew file mode 100644 index 0000000000..2588267ec8 --- /dev/null +++ b/info/drawing-with-metapost/src/corners.pdf diff --git a/info/drawing-with-metapost/src/curves-astroid-construction.mp b/info/drawing-with-metapost/src/curves-astroid-construction.mp new file mode 100644 index 0000000000..e6893d03ba --- /dev/null +++ b/info/drawing-with-metapost/src/curves-astroid-construction.mp @@ -0,0 +1,71 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path xx, yy, base; + +pair A, M, N, O, P, Q, R; +numeric t; t = 0.6; + +base = fullcircle scaled 500; +xx = 12 left -- point 0 of base + 12 right; +yy = 12 down -- point 2 of base + 12 up; + +O = origin; +A = point 0 of base; +R = (0, ypart point t of base); +Q = (xpart point t of base, 0); +M = 1/2[R, Q]; +N = 1/2[M, point t of base]; + +P = whatever[R, Q]; P - point t of base = whatever * (R-Q) rotated 90; + +path qa; qa = point 0 of base {left} + for i=1/4 step 1/4 until 2: + hide(pair a, b, p; + a = (xpart point i of base, 0); + b = (0, ypart point i of base); + p = whatever[a, b]; p - point i of base = whatever * (a - b) rotated 90; + ) .. p endfor; + +draw qa withcolor 3/4[1/2 red, white]; + +draw R -- point t of base -- O withcolor 3/4; +draw N -- P -- point t of base -- Q withcolor 3/4; +draw fullcircle scaled abs(M - point t of base) + shifted N withcolor 3/4; +draw R -- Q; + +draw subpath (-1, 3) of base + cutbefore xx shifted 1/2 point 0 of yy + cutafter yy shifted 1/2 point 0 of xx + withcolor 3/4[2/3 blue, white]; + +ahangle := 30; +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); + +labeloffset := 5; +dotlabel.bot("$S$", A); +dotlabel.llft("$O$", O); +dotlabel.lft("$B$", R); +dotlabel.bot("$A$", Q); +dotlabel.bot("$P$", P); +dotlabel.bot("$M$", M); +dotlabel.top("$N$", N); +dotlabel.urt("$T$", point t of base); + +draw P withpen pencircle scaled 2 withcolor 3/4[1/2 red, white]; + +drawoptions(withcolor 5/8[1/2 blue, white]); +label("$\scriptstyle \theta$", 16 dir 1/2 angle point t of base shifted O); +label("$\scriptstyle 2\theta$", 12 dir 0 shifted M); +label("$\scriptstyle 4\theta$", 8 dir angle (Q-R) shifted N); +drawoptions(); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-astroid-construction.pdf b/info/drawing-with-metapost/src/curves-astroid-construction.pdf Binary files differnew file mode 100644 index 0000000000..da5f00a8e5 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-astroid-construction.pdf diff --git a/info/drawing-with-metapost/src/curves-astroid-family.mp b/info/drawing-with-metapost/src/curves-astroid-family.mp new file mode 100644 index 0000000000..340144f721 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-astroid-family.mp @@ -0,0 +1,19 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +beginfig(1); + numeric a; a = 144; + for n=1 upto 7: + path p; p = a * right for t=6 step 6 until 90: + .. a * (cosd(t) ** n, sind(t) ** n) + endfor; + p := for i=0 upto 3: p rotated 90i & endfor cycle; + fill p withcolor Oranges[9][n]; draw p; + endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-astroid-family.pdf b/info/drawing-with-metapost/src/curves-astroid-family.pdf Binary files differnew file mode 100644 index 0000000000..18eb3b64d9 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-astroid-family.pdf diff --git a/info/drawing-with-metapost/src/curves-astroid.mp b/info/drawing-with-metapost/src/curves-astroid.mp new file mode 100644 index 0000000000..b572205217 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-astroid.mp @@ -0,0 +1,21 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + path base; base = fullcircle scaled 300; + path astroid; astroid = + for t=0 step 1/16 until 8: + hide(pair a, b, p; + a = (xpart point t of base, 0); + b = (0, ypart point t of base); + draw a -- b withpen pencircle scaled 1/4 withcolor 3/4; + p = whatever[a, b]; p - point t of base = whatever * (a - b) rotated 90; + ) p -- + endfor cycle; + draw astroid withcolor 2/3 red; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-astroid.pdf b/info/drawing-with-metapost/src/curves-astroid.pdf Binary files differnew file mode 100644 index 0000000000..550b18858a --- /dev/null +++ b/info/drawing-with-metapost/src/curves-astroid.pdf diff --git a/info/drawing-with-metapost/src/curves-cardioid-construction.mp b/info/drawing-with-metapost/src/curves-cardioid-construction.mp new file mode 100644 index 0000000000..4336fa7ab7 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-cardioid-construction.mp @@ -0,0 +1,54 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + + numeric u; u = 144; + + path base; base = fullcircle rotated 180 scaled u; + + path cardioid; + cardioid = for t = 0 step 1/4 until length base: + point 0 of base reflectedabout(precontrol t of base, postcontrol t of base) .. + endfor cycle; + + numeric t; t = 5.4; + pair A, M, P, P', Q, Q', R; + A = point 0 of base; + Q = point t of base; + Q' = point t-4 of base; + P = A reflectedabout(precontrol t of base, postcontrol t of base); + P' = A reflectedabout(precontrol t-4 of base, postcontrol t-4 of base); + M = 1/2[P, P']; + R = whatever[P, Q] = whatever[P', Q']; + + %draw Q' -- R -- Q withcolor 3/4; + draw 3[postcontrol t of base, precontrol t of base] + -- 5[precontrol t of base, postcontrol t of base] withcolor 3/4; + %draw 5[postcontrol t-4 of base, precontrol t-4 of base] + % -- 3[precontrol t-4 of base, postcontrol t-4 of base] withcolor 3/4; + + draw fullcircle scaled 2 abs (Q-A) shifted Q withcolor 7/8; + + % + %draw P' -- Q' -- Q -- P -- cycle; + draw A -- Q -- P -- cycle; + + draw base withcolor 2/3 blue; + draw origin withpen pencircle scaled dotlabeldiam withcolor 2/3 blue; + draw cardioid withcolor 1/4[white, 1/2 red]; + + dotlabel.ulft("$A$", A); + %dotlabel.ulft("$M$", M); + dotlabel.top ("$P$", P); + %dotlabel.llft("$P'$", P'); + dotlabel.urt ("$Q$", Q); + %dotlabel.bot ("$Q'$", Q'); + %dotlabel.lrt ("$R$", R); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-cardioid-construction.pdf b/info/drawing-with-metapost/src/curves-cardioid-construction.pdf Binary files differnew file mode 100644 index 0000000000..cab8862bc7 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-cardioid-construction.pdf diff --git a/info/drawing-with-metapost/src/curves-cardioid-simple.mp b/info/drawing-with-metapost/src/curves-cardioid-simple.mp new file mode 100644 index 0000000000..30a69897f3 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-cardioid-simple.mp @@ -0,0 +1,49 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + + numeric a; 2a = 144; + + path base; base = fullcircle rotated 180 scaled 2a shifted (a, 0); + path c[]; numeric n; n = -1; + numeric s; s = 1/4; + for t = 0 step s until length base: + c[incr n] = fullcircle + scaled 2 abs (point t of base - point 0 of base) + shifted point t of base; + draw c[n] withpen pencircle scaled 1/4 withcolor 3/4; + endfor + + path cardoid; + cardoid = for t = 0 step s until length base: + point 0 of base reflectedabout(precontrol t of base, postcontrol t of base) .. + endfor cycle; + + path polar; + polar = for t=0 upto 360: 2a * (1 + cosd(t)) * dir t .. endfor cycle; + + path rolling, aux; + aux = base rotatedabout(point 0 of base, 180); + rolling = for t = 0 step 1/4 until length base: + point t of aux rotatedabout(center base, 45t) .. + endfor cycle; + + % draw rolling withpen pencircle scaled 2 withcolor 3/4[blue, white]; + % draw polar withpen pencircle scaled 5/4 withcolor 3/4[green, white]; + + draw base withcolor 2/3 blue; + draw cardoid withcolor 2/3 red; + + % for i=1 upto n-1: + % draw center c[i] withpen pencircle scaled 3/2; + % endfor + + dotlabel.urt("$A$", point 0 of base); + dotlabel.urt("$Q$", point 5.4 of base); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-cardioid-simple.pdf b/info/drawing-with-metapost/src/curves-cardioid-simple.pdf Binary files differnew file mode 100644 index 0000000000..3deefb3619 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-cardioid-simple.pdf diff --git a/info/drawing-with-metapost/src/curves-ellipse-tangents.mp b/info/drawing-with-metapost/src/curves-ellipse-tangents.mp new file mode 100644 index 0000000000..a94e38463f --- /dev/null +++ b/info/drawing-with-metapost/src/curves-ellipse-tangents.mp @@ -0,0 +1,56 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +secondarydef a through b = + begingroup; save d; numeric d; d = abs(a-b); (1+12/d)[b,a]--(1+12/d)[a,b] endgroup +enddef; +beginfig(1); +path ellipse; ellipse = fullcircle scaled 300 yscaled 5/8 rotated 13; +z0 = 1/2[point 4 of ellipse, point 0 of ellipse]; +numeric a, b, e; +2a = abs (point 4 of ellipse - point 0 of ellipse); +2b = abs (point 6 of ellipse - point 2 of ellipse); +e = 1 +-+ b/a; +z1 = e[z0, point 0 of ellipse]; +z2 = e[z0, point 4 of ellipse]; +z3 = 240 dir 25; +path pp, qq; +pp = fullcircle scaled 2 abs (z1 - z3) shifted z3; +qq = fullcircle scaled 4a shifted z2; +z4 = pp intersectionpoint qq; +z5 = reverse pp intersectionpoint qq; +numeric t, u; +(t, whatever) = ellipse intersectiontimes (z2 -- z4); +(u, whatever) = ellipse intersectiontimes (z2 -- z5); + +vardef f(expr x) = + angle (z3 - point x of ellipse) + 90 > angle direction x of ellipse +enddef; +drawarrow z3 -- point solve f(0, t) of ellipse withcolor 2/3 blue; + +drawoptions(dashed evenly withcolor 7/8); +draw point 0 of ellipse through point 4 of ellipse; +draw point 2 of ellipse through point 6 of ellipse; +drawoptions(withcolor 3/4); +draw z4 -- z1 -- z5; +draw z2 -- z4 -- z3 -- z5 -- cycle; +draw point t of ellipse -- z1 -- point u of ellipse; +drawoptions(); +draw point t of ellipse -- z3 -- point u of ellipse withcolor 2/3 red; +dotlabel.llft("$F_1$", z1); dotlabel.llft("$F_2$", z2); +dotlabel.urt("$A$", z3); +dotlabel.ulft("$P$", z4); dotlabel.lrt ("$Q$", z5); +dotlabel.ulft("$T$", point t of ellipse); +dotlabel.lrt("$T'$", point u of ellipse); +draw ellipse; +picture P; P = currentpicture; currentpicture := nullpicture; +draw pp dashed withdots scaled 1/2; +draw qq dashed withdots scaled 1/2; +bboxmargin := 24; clip currentpicture to bbox P; +draw P; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-ellipse-tangents.pdf b/info/drawing-with-metapost/src/curves-ellipse-tangents.pdf Binary files differnew file mode 100644 index 0000000000..7cb7034ead --- /dev/null +++ b/info/drawing-with-metapost/src/curves-ellipse-tangents.pdf diff --git a/info/drawing-with-metapost/src/curves-ellipse.mp b/info/drawing-with-metapost/src/curves-ellipse.mp new file mode 100644 index 0000000000..f08f6ee204 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-ellipse.mp @@ -0,0 +1,49 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +secondarydef a through b = + begingroup; save d; numeric d; d = abs(a-b); (1+12/d)[b,a] -- (1+12/d)[a,b] endgroup +enddef; +beginfig(1); + path ellipse; ellipse = fullcircle scaled 300 yscaled 5/8 rotated 13; + numeric a, b, e; + 2a = abs (point 4 of ellipse - point 0 of ellipse); + 2b = abs (point 6 of ellipse - point 2 of ellipse); + e = 1 +-+ b/a; + + z0 = 1/2[point 0 of ellipse, point 4 of ellipse]; + z1 = e[z0, point 0 of ellipse]; + z2 = e[z0, point 4 of ellipse]; + + drawoptions(withcolor 3/4); + draw point 0 of ellipse through point 4 of ellipse; + draw point 2 of ellipse through point 6 of ellipse; + drawoptions(withcolor 1/2); + label.top("$a$", 1/2[z0, point 0 of ellipse]); + label.bot("$ae$", 1/2[z0, z1]); + label.rt("$b$", 1/2[z0, point 2 of ellipse]); + drawoptions(); + + numeric t; t = 1.732; + draw z1 -- point t of ellipse -- z2; + drawoptions(withcolor 1/2); + draw (left--right) scaled 21 + rotated angle direction t of ellipse + shifted point t of ellipse; + draw (1/2 down--up) scaled 21 + rotated angle direction t of ellipse + shifted point t of ellipse; + drawoptions(); + + draw ellipse withcolor 3/4 red; + + dotlabel.urt("$T$", point t of ellipse); + dotlabel.lrt("$F_1$", z1); + dotlabel.lrt("$F_2$", z2); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-ellipse.pdf b/info/drawing-with-metapost/src/curves-ellipse.pdf Binary files differnew file mode 100644 index 0000000000..0da741f270 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-ellipse.pdf diff --git a/info/drawing-with-metapost/src/curves-hyperbola-construction.mp b/info/drawing-with-metapost/src/curves-hyperbola-construction.mp new file mode 100644 index 0000000000..f8ea6ef263 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-hyperbola-construction.mp @@ -0,0 +1,54 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + + pair O; O = origin; + path c; c = fullcircle scaled 288; + pair S; S = point 0 of c shifted 42 right; + path hyperbola, last; + + numeric t; + (t, whatever) = c intersectiontimes fullcircle scaled abs(O-S) shifted 1/2[O,S]; + + numeric s; s = 1/16; + for t=-t + 6s step s until t - 5s: + path ray; + ray = (origin -- unitvector(S - point t of c)) scaled 200 + rotated if t < 0: - fi 90 shifted point t of c; + draw S -- ray withcolor 7/8; + if known last: + pair p; + p = whatever[point 0 of ray, point 1 of ray] + = whatever[point 0 of last, point 1 of last]; + hyperbola := if known hyperbola: hyperbola .. fi p; + fi + last := ray; + endfor + + + drawoptions(withcolor 1/2); + drawdblarrow 2[O, point t of c] -- O -- 2[O, point -t of c]; + draw O -- 1.2[O, S]; + + drawoptions(withcolor 3/4[blue, white]); + draw subpath (-1.4, 1.4) of c; + draw 1.4[S, point t of c] -- S -- 1.4[S, point -t of c]; + drawoptions(); + + draw hyperbola withcolor 3/4[red, white]; + for t=0 upto length hyperbola: + draw point t of hyperbola withpen pencircle scaled 3/2 withcolor red; + endfor + + + dotlabel.lrt("$S$", S); + dotlabel.llft("$A$", point 0 of c); + dotlabel.lft("$O$", origin); + +endfig; +\end{mplibcode}\llap{\texttt{\tiny\jobname}\quad} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-hyperbola-construction.pdf b/info/drawing-with-metapost/src/curves-hyperbola-construction.pdf Binary files differnew file mode 100644 index 0000000000..d08764066b --- /dev/null +++ b/info/drawing-with-metapost/src/curves-hyperbola-construction.pdf diff --git a/info/drawing-with-metapost/src/curves-hyperbola-function.mp b/info/drawing-with-metapost/src/curves-hyperbola-function.mp new file mode 100644 index 0000000000..d13ae0c09f --- /dev/null +++ b/info/drawing-with-metapost/src/curves-hyperbola-function.mp @@ -0,0 +1,82 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + numeric alpha; alpha = 34; + transform t; + origin transformed t = origin; + right transformed t = dir -alpha; + up transformed t = dir alpha; + + path xx, yy; + xx = (origin--right) scaled 144; + yy = (origin--up) scaled 144; + + numeric a, s; s = 1/64; a = 3/2; + path hh; + for x = 8 * a * s step s until 8/a: + hh := if known hh: hh ... fi (a * x, a / x); + endfor; + hh := hh scaled 16; + path C; C = fullcircle scaled 2a scaled sqrt(2) scaled 16; + pair F; F = (xpart point 0 of C, ypart point 2 of C); + + interim ahangle := 30; + picture P[]; + P1 = image( + draw point 0 of C -- F -- point 2 of C withcolor 3/4[blue, white]; + draw subpath (-1/2, 5/2) of C withcolor 3/4[blue, white]; + drawarrow xx withcolor 1/2; + drawarrow yy withcolor 1/2; + draw hh withcolor 2/3 red; dotlabel.urt("$S$", F); + ); + P2 = image( + hh := hh transformed t; + C := fullcircle scaled 2 abs(point directiontime down of hh of hh); + F := F rotated -45; + (p, whatever) = C intersectiontimes (fullcircle scaled abs(F) shifted 1/2 F); + draw point p of C -- F -- point -p of C withcolor 3/4[blue, white]; + draw subpath (-alpha/30, alpha/30) of C withcolor 3/4[blue, white]; + drawarrow xx transformed t withcolor 1/2; + drawarrow yy transformed t withcolor 1/2; + draw hh withcolor 2/3 red; + dotlabel.urt("$S$", F); + ); + + label.lft(P1, 10 left); + label.rt(P2, 10 right); + + + + %path C; + %C = fullcircle scaled 4 cosd(alpha) scaled 32a ; + %draw C withcolor 1/2; + + %pair S; + %S = whatever[precontrol +alpha/45 of C, postcontrol +alpha/45 of C] + % = whatever[precontrol -alpha/45 of C, postcontrol -alpha/45 of C]; + %draw (64a * right) withpen pencircle scaled 3 withcolor 3/4[blue, white]; + %draw S withpen pencircle scaled 3/2 withcolor red; + + %draw hh withcolor 2/3 red; + %%draw hh rotated 180 withcolor 2/3 red; + + %% path t; + %% t = point 4 of hh -- point 80 of hh -- point 400 of hh -- cycle; + + %% pair a, b, c; + %% a = whatever[point 0 of t, point 1 of t]; point 2 of t - a = whatever * (point 0 of t-point 1 of t) rotated 90; + %% b = whatever[point 2 of t, point 1 of t]; point 0 of t - b = whatever * (point 2 of t-point 1 of t) rotated 90; + + %% draw t; c = whatever[point 2 of t, a] = whatever[point 0 of t, b]; + + %% draw c withpen pencircle scaled 3; + + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-hyperbola-function.pdf b/info/drawing-with-metapost/src/curves-hyperbola-function.pdf Binary files differnew file mode 100644 index 0000000000..2129104b88 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-hyperbola-function.pdf diff --git a/info/drawing-with-metapost/src/curves-limacon-durer.mp b/info/drawing-with-metapost/src/curves-limacon-durer.mp new file mode 100644 index 0000000000..6066360a2a --- /dev/null +++ b/info/drawing-with-metapost/src/curves-limacon-durer.mp @@ -0,0 +1,36 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path base; +base = for t=0 upto 11: 42 up rotated -30t .. endfor cycle; + +path limacon; limacon = for t=0 upto length base - 1: + 42 dir angle point 2t of base + shifted point t of base .. +endfor cycle; + +draw base withcolor 2/3 blue; + +for i=1 upto 9: + draw point i/10 of base -- point i/10 of limacon + withpen pencircle scaled 1/4 + withcolor 7/8; +endfor +for i=1 step 1 until 12: + draw center base -- point i of base -- point i of limacon + withpen pencircle scaled 1/4 + withcolor 3/4; + draw point i of base withpen pencircle scaled 2; + %draw point i of limacon withpen pencircle scaled 1; +endfor +draw limacon withcolor 2/3 red; + +draw origin withpen pencircle scaled 2; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-limacon-durer.pdf b/info/drawing-with-metapost/src/curves-limacon-durer.pdf Binary files differnew file mode 100644 index 0000000000..22d08a83bc --- /dev/null +++ b/info/drawing-with-metapost/src/curves-limacon-durer.pdf diff --git a/info/drawing-with-metapost/src/curves-limacon-simple.mp b/info/drawing-with-metapost/src/curves-limacon-simple.mp new file mode 100644 index 0000000000..9647c805a4 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-limacon-simple.mp @@ -0,0 +1,39 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + + numeric a; 2a = 144; + + path base; base = fullcircle rotated 180 scaled 2a shifted (2a, 0); + path c[]; numeric n; n = -1; + numeric s; s = 1/4; + pair A; A = 2[center base, point 0 of base]; + for t = 0 step s until length base: + c[incr n] = fullcircle + scaled 2 abs (point t of base - A) + shifted point t of base; + draw c[n] withpen pencircle scaled 1/4 withcolor 3/4; + endfor + + path limacon; + limacon = for t = 0 step s until length base: + A reflectedabout(precontrol t of base, postcontrol t of base) .. + endfor cycle; + + % path polar; + % polar = for t=0 upto 359: 2a * (1 + 2 cosd(t)) * dir t .. endfor cycle; + % draw polar withpen pencircle scaled 5/4 withcolor 3/4[green, white]; + + draw base withcolor 2/3 blue; draw center base withpen pencircle scaled 2 withcolor 2/3 blue; + draw limacon withcolor 2/3 red; + + dotlabel.lft("$A$", A); + dotlabel.urt("$Q$", point 5.4 of base); + dotlabel.ulft("$P$", A reflectedabout(precontrol 5.4 of base, postcontrol 5.4 of base)); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-limacon-simple.pdf b/info/drawing-with-metapost/src/curves-limacon-simple.pdf Binary files differnew file mode 100644 index 0000000000..8ceb70b520 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-limacon-simple.pdf diff --git a/info/drawing-with-metapost/src/curves-parabola-directrix.mp b/info/drawing-with-metapost/src/curves-parabola-directrix.mp new file mode 100644 index 0000000000..52f3f8c3cc --- /dev/null +++ b/info/drawing-with-metapost/src/curves-parabola-directrix.mp @@ -0,0 +1,77 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\def\kw#1{\begingroup\def\_{\kern.04em + \vbox{\hrule width.3em height .6pt}\kern.08em}% +\ifmmode\mathop{\textbf{#1}}\else\hbox{\bf#1\/}\fi\endgroup} +\def\op#1{\begingroup\def\_{\kern.04em + \vbox{\hrule width.3em height .6pt}\kern.08em}% +\ifmmode\mathop{\textrm{#1}}\else\hbox{\rm#1\/}\fi\endgroup} +\def\id#1{\begingroup\def\_{\kern.04em + \vbox{\hrule width.3em height .6pt}\kern.08em}% +\ifmmode\mathop{\textit{#1}}\else\hbox{\it#1\/}\fi\endgroup} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +% A--B = directrix, S = focus +vardef parabola(expr A, B, S) = + save m, q, n, parabola; + pair n; + n = whatever[A, B]; + n - S = whatever * (A-B) rotated 90; + path parabola; + for t=0 step 1/64 until 1: + pair m, q; + m = 1/2[S, t[A, B]]; + q = whatever[S, n]; q - m = whatever * (S - m) rotated 90; + parabola := if known parabola: parabola -- fi q reflectedabout(S, m); + endfor + parabola +enddef; +input colorbrewer-rgb +beginfig(1); + + numeric i; i = 16; + pair a, b, s; a = 160 up rotated 5; b = 160 down rotated 5; s = 60 * right; + draw a--b; + + path ff; ff = (-1, 1){1, -2} .. (-1/2, 1/4){1, -1} .. (0, 0){right} .. (1/2, 1/4){1, 1} .. (1, 1){1, 2}; + + path p; p = parabola(a, b, s); + pair tab; tab = whatever[a, b]; point i of p - tab = whatever * (a-b) rotated 90; + pair q; q = point i of p reflectedabout(s, tab); + pair n; n = whatever[a,b] = whatever[q, s]; + pair o; o = 1/2[n, s]; + + % draw ff scaled 4 abs(s-o) + % rotated angle (b-a) + % shifted o + % withpen pencircle scaled 2 withcolor 7/8[red, white]; + + interim ahangle := 24; + drawarrow (b--a) shifted (o-n) withcolor 3/4; + drawarrow (1/2[a, n] --b) rotatedabout(n, 90) withcolor 3/4; + + draw tab -- point i of p -- q -- tab -- s withcolor 3/4; + + drawarrow s -- point i of p + -- s reflectedabout(precontrol i of p rotatedabout(point i of p, 90), + postcontrol i of p rotatedabout(point i of p, 90)) + withcolor Blues 8 4; + + draw p withcolor Blues 8 6; + dotlabel.lrt("$S$", s); + dotlabel.urt("$A$", a); + dotlabel.lrt("$B$", b); + dotlabel.ulft("\kw{point} $i$ \kw{of} $p$", point i of p); + interim dotlabeldiam := 2; + dotlabel.bot("$\scriptstyle q$", q); + dotlabel.llft("$\scriptstyle n$", n); + dotlabel.llft("$\scriptstyle o$", o); + dotlabel.lft("$\scriptstyle m\:$", 1/2[tab, s]); + dotlabel.lft("$\scriptstyle t[A, B]$", tab); + +endfig; +\end{mplibcode}\llap{\tiny\texttt{\jobname}} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-parabola-directrix.pdf b/info/drawing-with-metapost/src/curves-parabola-directrix.pdf Binary files differnew file mode 100644 index 0000000000..ab70182803 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-parabola-directrix.pdf diff --git a/info/drawing-with-metapost/src/curves-parabola.mp b/info/drawing-with-metapost/src/curves-parabola.mp new file mode 100644 index 0000000000..98c166560b --- /dev/null +++ b/info/drawing-with-metapost/src/curves-parabola.mp @@ -0,0 +1,41 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + pair O, S; O = origin; S = 66 right; + path parabola, last; + for q = -144 step 8 until 144: + pair Q; Q = (0, q); + path ray; + ray = (origin -- unitvector(S - Q)) scaled 300 + rotated if q < 0: - fi 90 shifted Q; + draw S -- ray withcolor 7/8; + if known last: + pair t; + t = whatever[point 0 of ray, point 1 of ray] + = whatever[point 0 of last, point 1 of last]; + parabola := if known parabola: parabola .. fi t; + fi + last := ray; + endfor + + draw parabola withcolor 3/4[red, white]; + for t=0 upto length parabola: + draw point t of parabola withpen pencircle scaled 3/2 withcolor red; + endfor + draw (up--down) scaled 300; + + dotlabel.lft("$A$", O); + dotlabel.rt("$S$", S); + + % vardef signum(expr x) = if x=0: 1 else: abs(x) / x fi enddef; + % path cart; + % cart = subpath (1, infinity) of (origin for x=-304 step 8 until 304: + % .. (x, 16 sqrt abs(x)) scaled signum(x) endfor); + + % drawarrow cart withcolor 1/3 green; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/curves-parabola.pdf b/info/drawing-with-metapost/src/curves-parabola.pdf Binary files differnew file mode 100644 index 0000000000..2889b2a82b --- /dev/null +++ b/info/drawing-with-metapost/src/curves-parabola.pdf diff --git a/info/drawing-with-metapost/src/curves-spiral-archimedes.mp b/info/drawing-with-metapost/src/curves-spiral-archimedes.mp new file mode 100644 index 0000000000..ba9b37122c --- /dev/null +++ b/info/drawing-with-metapost/src/curves-spiral-archimedes.mp @@ -0,0 +1,16 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\mplibnumbersystem{decimal} +\begin{mplibcode} +beginfig(1); + path S; S = origin for t=1 upto 360: .. 1/12 t * dir 8t endfor; + drawarrow S; + input rope + draw rope S shifted 64 right; + z0 = urcorner currentpicture - llcorner currentpicture; + currentpicture := currentpicture scaled (300 / x0); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/curves-spiral-archimedes.pdf b/info/drawing-with-metapost/src/curves-spiral-archimedes.pdf Binary files differnew file mode 100644 index 0000000000..a74df7785a --- /dev/null +++ b/info/drawing-with-metapost/src/curves-spiral-archimedes.pdf diff --git a/info/drawing-with-metapost/src/curves-spiral-equiangular.mp b/info/drawing-with-metapost/src/curves-spiral-equiangular.mp new file mode 100644 index 0000000000..81e45147c7 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-spiral-equiangular.mp @@ -0,0 +1,14 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + numeric a; a = 2.6; path S; + S = right for t=1 upto 360: .. a ** (t/64) * dir 4t endfor; + drawarrow S; +% input ruler-cm +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-spiral-equiangular.pdf b/info/drawing-with-metapost/src/curves-spiral-equiangular.pdf Binary files differnew file mode 100644 index 0000000000..0226fe100c --- /dev/null +++ b/info/drawing-with-metapost/src/curves-spiral-equiangular.pdf diff --git a/info/drawing-with-metapost/src/curves-spiral-gnomon-sq.mp b/info/drawing-with-metapost/src/curves-spiral-gnomon-sq.mp new file mode 100644 index 0000000000..5347da9484 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-spiral-gnomon-sq.mp @@ -0,0 +1,32 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + drawoptions(withpen pencircle scaled 1/4 withcolor 1/4); + path s[]; s0 = unitsquare; + fill s0 withcolor 1/2[2/3 blue, white]; draw s0; + numeric a, b, t, n; a = 1; b = 1; n = 11; + for i = 1 upto n: + t := b; b := b + a; a := t; + s[i] = unitsquare scaled a; + s[i] := s[i] shifted (point i of s[i-1] - point i-1 of s[i]); + fill s[i] withcolor (1/2 + i/32)[2/3 blue, white]; + draw s[i]; + endfor + for i = n-1 upto n: + draw point i-2 of s[i-2] -- point i of s[i]; + endfor + drawoptions(); + drawarrow origin for i=0 upto n: .. point i of s[i] endfor withcolor 2/3 red; + for i = 5 upto n: + label(TEX(decimal arclength subpath (0, 1) of s[i]) + scaled 0.8 rotated angle point n of s[n], center s[i]); + endfor + currentpicture := currentpicture rotated - angle point n of s[n]; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/curves-spiral-gnomon-sq.pdf b/info/drawing-with-metapost/src/curves-spiral-gnomon-sq.pdf Binary files differnew file mode 100644 index 0000000000..0863722fc3 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-spiral-gnomon-sq.pdf diff --git a/info/drawing-with-metapost/src/curves-spiral-gnomon-trig.mp b/info/drawing-with-metapost/src/curves-spiral-gnomon-trig.mp new file mode 100644 index 0000000000..5bf37d00b7 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-spiral-gnomon-trig.mp @@ -0,0 +1,39 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + path t[], base; pair apex; + base = (left--right) scaled 100; + apex = whatever * dir +72 shifted point 0 of base + = whatever * dir -72 shifted point 1 of base; + t0 = (base -- apex -- cycle); + transform S; + r = arclength subpath (0, 1) of t0 / arclength subpath (2, 3) of t0; + point 0 of t0 transformed S = (r*r)[point 0 of t0, point 2 of t0]; + point 1 of t0 transformed S = point 0 of t0; + point 2 of t0 transformed S = point 1 of t0; + + n = 16; + for i=1 upto n: + t[i] = t[i-1] transformed S; + draw subpath (2,3) of t[i] withpen pencircle scaled 1/4; + endfor + draw point 0 of t0 -- point 3/2 of t0 withpen pencircle scaled 1/8 withcolor blue; + draw point 0 of t1 -- point 3/2 of t1 withpen pencircle scaled 1/8 withcolor blue; + draw t0; + drawarrow point 2 of t[n] for i=n-1 downto 0: .. point 2 of t[i] endfor withcolor 2/3 red; + + z0 = whatever[point 0 of t0, point 3/2 of t0] + = whatever[point 0 of t1, point 3/2 of t1]; + numeric a; a = angle (point 2 of t0 - z0); + drawoptions(withcolor 1/2 green); + label.lft(TEX("$a$") rotated a, point 5/2 of t0); + label.bot(TEX("$b$") rotated a, point 1/2 of t0); + label.rt(TEX("$x$") rotated a, point 1/2 of t1); + drawoptions(); + currentpicture := currentpicture rotated -a; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/curves-spiral-gnomon-trig.pdf b/info/drawing-with-metapost/src/curves-spiral-gnomon-trig.pdf Binary files differnew file mode 100644 index 0000000000..24cf426568 --- /dev/null +++ b/info/drawing-with-metapost/src/curves-spiral-gnomon-trig.pdf diff --git a/info/drawing-with-metapost/src/cycloids-code.mp b/info/drawing-with-metapost/src/cycloids-code.mp new file mode 100644 index 0000000000..f81c74ae6e --- /dev/null +++ b/info/drawing-with-metapost/src/cycloids-code.mp @@ -0,0 +1,63 @@ +numeric r, pi; +r = 1.25cm; % radius of the wheel +pi = 3.14159265; % constant +% +% define the cycloid +path c; +c = (0,-r) rotated 100 shifted (r*-100/180*pi,r) + for t=-99 upto 460: + -- (0,-r) rotated -t shifted (r*t/180*pi,r) + endfor; +% and the axes +drawoptions(withcolor .5 white); +path xx, yy; +yy = (down -- 5 up) scaled 1/2 r; +xx = (xpart point 0 of c, 0) -- (xpart point infinity of c,0); +draw fullcircle scaled 1/4r; drawarrow xx; drawarrow yy; +drawoptions(); +label.rt (btex $x$ etex, point 1 of xx); +label.top(btex $y$ etex, point 1 of yy); +% draw the cycloid on top of the axes +draw c withcolor .67 red; + +% define a couple of related points: +% z1 center of the blue wheel +% z2 intersection of rim and cycloid +t = 84; % if you change t then the wheel will "roll" along... +z1 = (r*t/180*pi,r); +z2 = (0,-r) rotated -t shifted z1; + +% draw the auxiliary lines +draw (0,y2) -- z2 -- (x2,0) dashed withdots scaled .6; +draw z2 -- z1 -- (x1,0); + +% draw the rolling circle +draw fullcircle scaled 2r shifted z1 withcolor .67 blue; +% mark the centre and intersection with cycloid +fill fullcircle scaled dotlabeldiam shifted z1; +fill fullcircle scaled dotlabeldiam shifted z2; + +% some arc arrows and labels +path a[]; +z3 = (x1,5/12y1); +a1 = z3 {left} .. {left rotatedabout(z1,-t)} z3 rotatedabout(z1,-t); +drawarrow subpath (.05,.95) of a1; +label.llft(btex $\theta$ etex, point .5 of a1); + +a2 = subpath (0,1) of reverse quartercircle scaled 2.2r shifted z1; +drawarrow a2 rotatedabout(z1,-100); +drawarrow a2 rotatedabout(z1,80); + +% finally all the other labels +label.top(btex $r$ etex, .5[z1,z2]); +label.lft(btex $y$ etex, (0,y2)); +% give all the x-axis labels a common baseline with mathstrut +label.bot(btex $\mathstrut x$ etex, (x2,0)); +label.bot(btex $\mathstrut r\theta$ etex, (x1,0)); +label.bot(btex $\mathstrut 2\pi r$ etex, (r*2pi,0)); +% notice how nicely the coordinates work... +dotlabel.top(btex $(\pi r,2r)$ etex, (pi*r,2r)); +% and a little alignment to finish +label(btex $\vcenter{\halign{&$#$\hfil\cr +x=r(\theta-\sin\theta)\cr +y=r(1-\cos\theta)\cr}}$ etex,(4.2r,r)); diff --git a/info/drawing-with-metapost/src/cycloids-extra-code.mp b/info/drawing-with-metapost/src/cycloids-extra-code.mp new file mode 100644 index 0000000000..15ea28144f --- /dev/null +++ b/info/drawing-with-metapost/src/cycloids-extra-code.mp @@ -0,0 +1,61 @@ +numeric R, r, pi; +R = 1.8cm; % radius of the outer part +r = 1.3cm; % radius of the inner part +pi = 3.14159265; % constant +% define the cycloid +path c; +c = (0,-R) rotated 100 shifted (r*-100/180*pi,r) + for t=-99 upto 460: + -- (0,-R) rotated -t shifted (r*t/180*pi,r) + endfor; +% axes +drawoptions(withcolor .5 white); +path xx, yy; +yy = (down -- 5 up) scaled 1/2 r; +xx = (xpart point 0 of c, 0) -- (xpart point infinity of c,0); +draw fullcircle scaled 1/4r; drawarrow xx; drawarrow yy; +drawoptions(); +label.rt (btex $x$ etex, point 1 of xx); +label.top(btex $y$ etex, point 1 of yy); + +% draw the cycloid on top of the axes +draw c withcolor .67 red; + +% define a couple of related points: +% z1 center of the blue wheel +% z2 intersection of rim and cycloid +t = 124; % if you change t then the wheel will "roll" along... +z1 = (r*t/180*pi,r); +z2 = (0,-R) rotated -t shifted z1; + +% draw the auxiliary lines +draw (0,y2) -- z2 -- (x2,0) dashed withdots scaled .6; +draw z2 -- z1 -- (x1,0); +% draw the rolling circle and mark centre and intersection +draw fullcircle scaled 2r shifted z1 withcolor 2/3 blue; +draw fullcircle scaled 2R shifted z1 withcolor 1/2[2/3 blue, white]; +fill fullcircle scaled dotlabeldiam shifted z1; +fill fullcircle scaled dotlabeldiam shifted z2; +% some arc arrows and labels +path a[]; +z3 = (x1,5/12y1); +a1 = z3 {left} .. {left rotatedabout(z1,-t)} z3 rotatedabout(z1,-t); +drawarrow subpath (.05,.95) of a1; +label.llft(btex $\theta$ etex, point .5 of a1); +a2 = subpath (0,1) of reverse quartercircle scaled 2.2r shifted z1; +drawarrow a2 rotatedabout(z1,-100); +% finally all the other labels +label.rt (btex $r$ etex, (x1,.5y1)); +label.urt(btex $R$ etex, .6[z1,z2]); +label.lft(btex $y$ etex, (0,y2)); +% give all the x-axis labels a common baseline with mathstrut +label.bot(btex $\mathstrut x$ etex, (x2,0)); +label.bot(btex $\mathstrut r\theta$ etex, (x1,0)); +label.bot(btex $\mathstrut 2\pi r$ etex, (r*2pi,0)); +draw (down--up) scaled 2 shifted (r*2pi,0) withcolor .5 white; +% notice how nicely the coordinates work... +dotlabel.top(btex $(\pi r,R+r)$ etex, (pi*r,R+r)); +% and a little alignment to finish +label(btex $\vcenter{\halign{&$#$\hfil\cr +x=r\theta-R\sin\theta\cr +y=r-R\cos\theta\cr}}$ etex,(4.75r,r)); diff --git a/info/drawing-with-metapost/src/cycloids-extra.mp b/info/drawing-with-metapost/src/cycloids-extra.mp new file mode 100644 index 0000000000..af41bcdf46 --- /dev/null +++ b/info/drawing-with-metapost/src/cycloids-extra.mp @@ -0,0 +1,10 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +input cycloids-extra-code +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/cycloids-extra.pdf b/info/drawing-with-metapost/src/cycloids-extra.pdf Binary files differnew file mode 100644 index 0000000000..80daf06271 --- /dev/null +++ b/info/drawing-with-metapost/src/cycloids-extra.pdf diff --git a/info/drawing-with-metapost/src/cycloids.mp b/info/drawing-with-metapost/src/cycloids.mp new file mode 100644 index 0000000000..86788b5c46 --- /dev/null +++ b/info/drawing-with-metapost/src/cycloids.mp @@ -0,0 +1,10 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + input cycloids-code +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/cycloids.pdf b/info/drawing-with-metapost/src/cycloids.pdf Binary files differnew file mode 100644 index 0000000000..6334dd9ed2 --- /dev/null +++ b/info/drawing-with-metapost/src/cycloids.pdf diff --git a/info/drawing-with-metapost/src/desargues.mp b/info/drawing-with-metapost/src/desargues.mp new file mode 100644 index 0000000000..27203d4961 --- /dev/null +++ b/info/drawing-with-metapost/src/desargues.mp @@ -0,0 +1,56 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +beginfig(1); + z.P = 200 up rotated 21; z.A = 100 left rotated -21; + z.B = origin; z.C = 90 right rotated 42; + + z.A' = 3/8[z.P, z.A]; + z.B' = 1/2[z.P, z.B]; + z.C' = 5/8[z.P, z.C]; + + z.R = whatever [z.A, z.B] = whatever [z.A', z.B']; + z.S = whatever [z.B, z.C] = whatever [z.B', z.C']; + z.T = whatever [z.C, z.A] = whatever [z.C', z.A']; + + path t[]; + t1 = z.A--z.B--z.C--cycle; + t2 = z.A'--z.B'--z.C'--cycle; + + fill t1 withcolor 7/8[red, white]; + fill t2 withcolor 7/8[blue, white]; + draw t1 withcolor 1/2 white; + draw t2 withcolor 1/2 white; + + drawoptions(dashed withdots scaled 1/2); + draw z.P--z.A; + draw z.P--z.B; + draw z.P--z.C; + + drawoptions(dashed evenly scaled 1/2); + draw z.B--z.R--z.B'; + draw z.C--z.S--z.C'; + undraw subpath (1/4, 3/4) of (z.C'--z.T) withpen + pencircle scaled 5; + draw z.C--z.T--z.C'; + drawoptions(withcolor 2/3 red); + draw 9/8[z.S,z.R] -- 9/8[z.R,z.S]; + picture pp; pp = thelabel("perspectrix", origin); + draw pp shifted 7 down rotated angle (z.S-z.R) + shifted 1/2[z.R, z.T]; + dotlabel.urt("perspector", z.P); + + drawoptions(); + dotlabel.lft (btex $A$ etex, z.A); + dotlabel.llft(btex $B$ etex, z.B); + dotlabel.lrt (btex $C$ etex, z.C); + dotlabel.lft (btex $A'$ etex, z.A'); + dotlabel.llft(btex $B'$ etex, z.B'); + dotlabel.bot (btex $C'$ etex, z.C'); + label.rt(btex Desargues' Theorem etex, (x.C', 1/2(y.P+y.C'))); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/desargues.pdf b/info/drawing-with-metapost/src/desargues.pdf Binary files differnew file mode 100644 index 0000000000..bbb38bb28b --- /dev/null +++ b/info/drawing-with-metapost/src/desargues.pdf diff --git a/info/drawing-with-metapost/src/dice.mp b/info/drawing-with-metapost/src/dice.mp new file mode 100644 index 0000000000..a68153004e --- /dev/null +++ b/info/drawing-with-metapost/src/dice.mp @@ -0,0 +1,41 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); +vardef dice(expr pip_count, pip_color) = + save d,r,p, ul, ur, lr, ll; + r=1/8; path d; picture p; + %d = (for i=0 upto 3: subpath (r+i,1+i-r) of unitsquare .. endfor cycle) scaled 30; + d = for i=0 upto 3: quartercircle scaled 3 shifted (15,15) rotated 90i -- endfor cycle; + p = image(draw fullcircle scaled 6; fill fullcircle scaled 6 withcolor pip_color); + pair ul, ur, ll, lr; + ul = 1/5[ulcorner d, lrcorner d]; + lr = 4/5[ulcorner d, lrcorner d]; + ur = 1/5[urcorner d, llcorner d]; + ll = 4/5[urcorner d, llcorner d]; + image(fill d withcolor background; draw d; + if odd(pip_count): + draw p shifted center d; + fi; + if pip_count > 1: + draw p shifted ul; + draw p shifted lr; + fi; + if pip_count > 3: + draw p shifted ur; + draw p shifted ll; + fi; + if pip_count = 6: + draw p shifted 1/2[ul,ur]; + draw p shifted 1/2[ll,lr]; + fi) +enddef; + +beginfig(1); +for i=0 upto 4: + draw dice(1+floor uniformdeviate 6, red) rotated (2 normaldeviate) shifted (36i,0); +endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/dice.pdf b/info/drawing-with-metapost/src/dice.pdf Binary files differnew file mode 100644 index 0000000000..e5fc396365 --- /dev/null +++ b/info/drawing-with-metapost/src/dice.pdf diff --git a/info/drawing-with-metapost/src/double-angle.mp b/info/drawing-with-metapost/src/double-angle.mp new file mode 100644 index 0000000000..f09885dfb5 --- /dev/null +++ b/info/drawing-with-metapost/src/double-angle.mp @@ -0,0 +1,69 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +ahangle := 30; +beginfig(1); + +path h; pair A, B, C, D, O; numeric theta; + +h = halfcircle scaled 320; + +O = origin; +A = point 4 of h; +B = point 0 of h; +C = point 5/4 of h; +D = (xpart C, ypart A); + +2theta = angle C; + +draw unitsquare scaled 8 rotated angle (C-D) shifted D withcolor 3/4; +draw unitsquare scaled 8 rotated angle (A-C) shifted C withcolor 3/4; + +draw A--C--B withcolor Reds 7 7; +draw O--C--D withcolor Reds 7 7; + +drawoptions(withcolor Blues 7 6); +draw h; +label.ulft("$x^2 + y^2 = 1$", point 3 of h); +drawoptions(); + +primarydef o through p = (1+o/arclength(p))[point 1 of p, point 0 of p] -- (1+o/arclength(p))[point 0 of p, point 1 of p] enddef; +drawarrow 16 through (A--B); +drawarrow 16 through (O--point 2 of h); + +dotlabel.bot("$A$", A); +dotlabel.bot("$B$", B); +dotlabel.urt("$C \smash{\;\bigl(\cos2\theta, \sin2\theta\bigr)}$", C); +dotlabel.bot("$D$", D); +dotlabel.llft("$O$", O); + +label("$\theta$", 28 dir 1/2 theta shifted A); +label("$2\theta$", 20 dir theta); + +label("$x$", B shifted 24 right); +label("$y$", point 2 of h shifted 24 up); + +draw thelabel.top("$2\cos\theta$", origin) rotated theta shifted 1/2[A, C]; +draw thelabel.top("$2\sin\theta$", origin) rotated (theta-90) shifted 1/2[B, C]; + +label.bot("$\triangle ACD \sim \triangle ABC$", point 1/2 of bbox currentpicture shifted 24 down); + +path bb; bb = bbox currentpicture shifted 12 down; + +label.bot(btex \vbox{\openup 4pt\halign{\hfil $#$\hfil\cr +CD \Big/ AC = BC \Big/ AB\cr +\sin 2\theta \big/ 2 \cos\theta = 2 \sin\theta \big/ 2\cr +\sin 2\theta = 2\sin\theta \cos\theta\cr}} etex, point 1/4 of bb); + +label.bot(btex \vbox{\openup 4pt\halign{\hfil $#$\hfil\cr +AD \Big/ AC = AC \Big/ AB\cr +\bigl(1 + \cos 2\theta \bigr) \big/ 2 \cos\theta = 2 \cos\theta \big/ 2\cr +\cos 2\theta = 2\cos^2\theta - 1\cr}} etex, point 3/4 of bb); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/double-angle.pdf b/info/drawing-with-metapost/src/double-angle.pdf Binary files differnew file mode 100644 index 0000000000..f0721c8b48 --- /dev/null +++ b/info/drawing-with-metapost/src/double-angle.pdf diff --git a/info/drawing-with-metapost/src/double-dragon.mp b/info/drawing-with-metapost/src/double-dragon.mp new file mode 100644 index 0000000000..a701cbe8a0 --- /dev/null +++ b/info/drawing-with-metapost/src/double-dragon.mp @@ -0,0 +1,57 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\mplibnumbersystem{double} +\begin{mplibcode} +beginfig(1); + +vardef zinverse(expr z) = 1/abs z * dir - angle z enddef; + +vardef fizz(expr X) = + pair m, n; + m = right; + n = origin; + numeric a, x; + x = X; + forever: + exitif x = 0; + m := m zscaled zinverse((-1, 1)); + a := x mod 2; + n := n + a * m; + x := x div 2; + endfor + n +enddef; +input colorbrewer-rgb +color shade[]; +shade0 = Reds 5 4; shade1 = Oranges 5 4; +shade2 = Greens 5 4; shade3 = Blues 5 4; + +beginfig(1); + numeric s, t; s = 256; t = 4; + for n=0 upto (s/t*s/t-1): + numeric h, v; + h = floor 1/8 (n mod 32); + v = n mod 4; + fill fullcircle scaled t shifted (fizz(n) scaled s) + withcolor (1/2 + 1/8 v)[white, shade[h]]; + endfor; + path xx, yy; + xx = (left--right) scaled (s+8); + yy = xx rotated 90; + for i=-1 upto 1: + draw xx shifted (0, s*i) withpen pencircle scaled 1/8; + draw yy shifted (s*i, 0) withpen pencircle scaled 1/8; + endfor + dotlabel.lrt(btex $-1-i$ etex, (-1, -1) scaled s); + dotlabel.lrt(btex $-1$ etex, (-1, 0) scaled s); + dotlabel.lrt(btex $-1+i$ etex, (-1, 1) scaled s); + dotlabel.lrt(btex $-i$ etex, (0, -1) scaled s); + dotlabel.lrt(btex $+i$ etex, (0, 1) scaled s); + dotlabel.lrt(btex $+1-i$ etex, (1, -1) scaled s); + dotlabel.lrt(btex $+1$ etex, (1, 0) scaled s); + dotlabel.lrt(btex $+1+i$ etex, (1, 1) scaled s); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/double-dragon.pdf b/info/drawing-with-metapost/src/double-dragon.pdf Binary files differnew file mode 100644 index 0000000000..ccede3d7c5 --- /dev/null +++ b/info/drawing-with-metapost/src/double-dragon.pdf diff --git a/info/drawing-with-metapost/src/draw-picture.mp b/info/drawing-with-metapost/src/draw-picture.mp new file mode 100644 index 0000000000..16c40d8a49 --- /dev/null +++ b/info/drawing-with-metapost/src/draw-picture.mp @@ -0,0 +1,73 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + numeric s; s = 21; + path alpha; + alpha = ((-2s, s) {right} + .. halfcircle rotated -90 scaled 2s shifted (2s, 0) + .. {left} (-2s, -s)) shifted (s*left); + + vardef overdraw(expr a, b, r, P, shade) = + linecap := butt; + undraw subpath (a+r, b-r) of P withpen pencircle scaled 2; + draw subpath (a, b) of P withcolor shade; + enddef; + + picture cb; cb = image( + draw alpha withcolor 2/3 red; + undraw alpha rotated 180 withpen pencircle scaled 2; + draw alpha rotated 180 withcolor 2/3 blue; + overdraw(0.21, 0.36, 0.02, alpha, 2/3 red); + overdraw(0.67, 0.86, 0.02, alpha, 2/3 red); + overdraw(3.4, 4.3, 0.1, alpha, 2/3 red); + overdraw(5.4, 5.6, 0.02, alpha, 2/3 red); + overdraw(5.4, 5.6, 0.02, alpha rotated 180, 2/3 blue); + ); + + picture P[]; + + P1 = image(draw cb; label.rt("\texttt{draw cb;}", point 3/2 of bbox currentpicture)); + P2 = image(draw cb withpen pencircle scaled 4; label.rt("\texttt{draw cb withpen pencircle scaled 4;}", point 3/2 of bbox currentpicture)); + P3 = image(draw cb withcolor 1/2; label.rt("\texttt{draw cb withcolor 1/2;}", point 3/2 of bbox currentpicture)); + P4 = image(draw cb withpen pencircle scaled 4 withcolor 1/2; label.rt("\texttt{draw cb withpen pencircle scaled 4 withcolor 1/2;}", point 3/2 of bbox currentpicture)); + P5 = image( + color bagel; bagel = 1/256(187, 146, 79); + for i=4 step -1/4 until 1/2: + draw cb withpen pencircle scaled i + withcolor (i/4)[white, bagel]; + endfor + label.rt(btex \vbox{\halign{\texttt{#}\cr + for i=4 step -1/4 until 1/2:\cr + \quad draw cb withpen pencircle scaled i \cr + \qquad withcolor sqrt(i/4)[white, bagel];\cr + endfor\cr}} etex, point 3/2 of bbox currentpicture); + ); + + P6 = image( + for e within cb: + draw pathpart e withpen penpart e scaled 4 + withcolor 7/8[colorpart e, background]; + endfor + label.rt(btex \vbox{\halign{\texttt{#}\cr + for e within cb:\cr + \quad draw pathpart e withpen penpart e scaled 4\cr + \qquad withcolor 7/8[colorpart e, background];\cr + endfor\cr}} etex, point 3/2 of bbox currentpicture); + ); + + for i=1 upto 6: + if known P[i]: + pair p; p = 60i * down; + label.rt(P[i], p); + fill superellipse(right, up, left, down, 0.8) scaled 6 shifted p + withcolor 2/3; + label("\textsf{" & decimal i & "}", p) withcolor white; + fi; + endfor + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/draw-picture.pdf b/info/drawing-with-metapost/src/draw-picture.pdf Binary files differnew file mode 100644 index 0000000000..6eb8f7dbff --- /dev/null +++ b/info/drawing-with-metapost/src/draw-picture.pdf diff --git a/info/drawing-with-metapost/src/drawing-circles.mp b/info/drawing-with-metapost/src/drawing-circles.mp new file mode 100644 index 0000000000..014541eb18 --- /dev/null +++ b/info/drawing-with-metapost/src/drawing-circles.mp @@ -0,0 +1,64 @@ +\documentclass[border=0mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +color TR, TB, TG; +TB = (.2,.2,.7); +TR = (0.54,0,0); +TG = (0, 0.44, 0); + +path xx, yy; +xx = (left--right) scaled 144; +yy = (down--up) scaled 89; + +ahangle := 30; +picture P[]; +P0 = image(drawarrow xx withcolor 1/2; drawarrow yy withcolor 1/2); +P1 = image(draw P0; + path c, a; + c = fullcircle scaled 4cm shifted (34, 21); + a = point 1 of c -- center c; + draw c withcolor TR; + draw fullcircle scaled 3 shifted (34,21); label.lrt("$(34,21)$", (34,21)); + drawdblarrow a; label.ulft("2 cm", point 1/2 of a); +); + +vardef circle_through(expr A, B, C) = + save o; pair o; + o = whatever * (A-B) rotated 90 shifted 1/2[A,B] + = whatever * (B-C) rotated 90 shifted 1/2[B,C]; + fullcircle scaled 2 abs (A-o) shifted o +enddef; + +P2 = image(draw P0; + pair A, B, C; + A = (-21, -13); + B = (58, 21); + C = (89, -34); + + path c[]; + c1 = fullcircle scaled 2 abs(A-B) shifted A; + c2 = fullcircle scaled abs(A-B) shifted 1/2[A, B]; + c3 = circle_through(A, B, C); + + draw c1 withcolor TR; label.ulft("I", point 3 of c1) withcolor TR; + + draw A -- B dashed withdots scaled 1/4 withpen pencircle scaled 1/4 withcolor TB; + draw c2 withcolor TB; label.ulft("II", point 3 of c2) withcolor TB; + draw fullcircle scaled 2 shifted 1/2[A, B] withcolor TB; + + draw c3 withcolor TG; label.urt("III", point 1 of c3) withcolor TG; + + dotlabel.llft("$A$", A); + dotlabel.urt("$B$", B); + dotlabel.lrt("$C$", C); +); + +beginfig(1); +labeloffset := 10; +label.top(P1, origin); +label.bot(P2, origin); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/drawing-circles.pdf b/info/drawing-with-metapost/src/drawing-circles.pdf Binary files differnew file mode 100644 index 0000000000..e7e8f22f0a --- /dev/null +++ b/info/drawing-with-metapost/src/drawing-circles.pdf diff --git a/info/drawing-with-metapost/src/dwmpcode.sty b/info/drawing-with-metapost/src/dwmpcode.sty new file mode 100644 index 0000000000..ccc13d95a8 --- /dev/null +++ b/info/drawing-with-metapost/src/dwmpcode.sty @@ -0,0 +1,49 @@ +\NeedsTeXFormat{LaTeX2e}[1994/06/01] +\ProvidesPackage{dwmpcode}[2021/12/30 v0.01 Code setting in DwMP] +\RequirePackage{listings} +\RequirePackage{xcolor} +\definecolor{textblue}{rgb}{.2,.2,.7} +\definecolor{textred}{rgb}{0.54,0,0} +\definecolor{textgreen}{rgb}{0,0.43,0} +\definecolor{squash}{rgb}{0.793, 0.359, 0.05} +\definecolor{carrot}{rgb}{0.934, 0.445, 0.082} +\def\red#1{\textcolor{textred}{#1}} +\def\blue#1{\textcolor{textblue}{#1}} +\def\green#1{\textcolor{textgreen}{#1}} +\newcommand\mpstyle{\lstset{ +language=Metapost, +basicstyle=\ttfamily, +keywordstyle=\color{textblue}, +keywordstyle=[5]\color{squash}, +keywordstyle=[6]\color{carrot}, +commentstyle=\color{textred}, +stringstyle=\color{textgreen}, +frame=none, +columns=fullflexible, +keepspaces=true, +xleftmargin=\parindent, +showstringspaces=false, +morekeywords={cmykcolor, step, eps, exitif, rotatedabout}, +morestring=[s]{btex}{etex} +}} +\lstnewenvironment{smallcode}[1][]{\mpstyle\lstset{basicstyle=\small\ttfamily,#1}}{} +\lstnewenvironment{code}[1][]{\mpstyle\lstset{#1}}{} +\newcommand\mpexternal[2][]{{\mpstyle\lstinputlisting[#1]{#2}}} +\newcommand\mpl[1]{{\mpstyle\lstinline!#1!}} +% +\newcommand\texstyle{\lstset{ +language=[LaTeX]Tex, +basicstyle=\ttfamily, +keywordstyle=\color{textblue}, +commentstyle=\color{textred}, +stringstyle=\color{textgreen}, +frame=none, +columns=fullflexible, +keepspaces=true, +xleftmargin=\parindent, +showstringspaces=false, +}} +\lstnewenvironment{texcode}[1][]{\texstyle\lstset{basicstyle=\small\ttfamily,#1}}{} +\newcommand\txl[1]{{\texstyle\lstinline!#1!}} +% +\endinput diff --git a/info/drawing-with-metapost/src/dynamic-labels.mp b/info/drawing-with-metapost/src/dynamic-labels.mp new file mode 100644 index 0000000000..d831d99a4b --- /dev/null +++ b/info/drawing-with-metapost/src/dynamic-labels.mp @@ -0,0 +1,19 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(2); +path c; c = fullcircle scaled 100; draw c withcolor .67 red; +for i=0 upto 7: + fill fullcircle scaled 3 shifted point i of c; + z[i] = point i of c scaled 1.15; + write "label(btex $p_" & decimal i & "$ etex,(" + & decimal x[i] & "," & decimal y[i] + & "));" to ".mplabels"; +endfor +write EOF to ".mplabels"; +input ".mplabels"; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/dynamic-labels.pdf b/info/drawing-with-metapost/src/dynamic-labels.pdf Binary files differnew file mode 100644 index 0000000000..94ed338990 --- /dev/null +++ b/info/drawing-with-metapost/src/dynamic-labels.pdf diff --git a/info/drawing-with-metapost/src/eggs-357.mp b/info/drawing-with-metapost/src/eggs-357.mp new file mode 100644 index 0000000000..746638a83c --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-357.mp @@ -0,0 +1,59 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input eggs-common +beginfig(1); + +path base, cup, cap, egg; + +base = fullcircle scaled 180; +z0 = origin; +z1 = point -2/3 of base; +z2 = point 2/3 of base; +z3 = point 10/3 of base; +z4 = point 14/3 of base; +z5 = 1/2[z2, z3]; +z6 = 1/2[z4, z1]; + +numeric a, b, c; +a = abs(z3 - z1); +b = a - abs(z4 - z6); +c = a - abs(z4 - z5); + +cup = subpath (4, 8) of fullcircle scaled 2b shifted z6; +cap = fullcircle scaled 2c shifted z5 + cutbefore (z5 -- 2[z4, z5]) + cutafter (z5 -- 2[z1, z5]); + +egg = point 4 of cup {up} .. cap .. {down} cup & cycle; + +fill egg withpen pencircle scaled 2 withcolor eggshell; +picture P; P = currentpicture; +fill z4 -- z5 -- z6 -- cycle withcolor light_eggshell; + +drawoptions(withpen pencircle scaled 1/4 withcolor dark_eggshell); +label.top("$\scriptstyle \sqrt 3$", 1/2[z4, z6]); +label.lft("$\scriptstyle \sqrt 4 $", 10 down); +label.ulft("$\scriptstyle \sqrt 7$", 7/16[z4, z5]); +draw base; +draw z1 -- z2 -- z3 -- z4 -- z6 -- z5; +draw z0 withpen pencircle scaled 2; +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); +draw point 0 of egg -- point 4 of egg; +draw z4 -- point 1 of egg; +draw z1 -- point 3 of egg; +draw egg; +drawoptions(withpen pencircle scaled 2 withcolor 7/16); +forsuffixes $=1,4,5,6: draw z$; endfor +draw numbered_points(egg); +drawoptions(); + +draw P shifted 240 right; + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/eggs-357.pdf b/info/drawing-with-metapost/src/eggs-357.pdf Binary files differnew file mode 100644 index 0000000000..1f445d9698 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-357.pdf diff --git a/info/drawing-with-metapost/src/eggs-common.mp b/info/drawing-with-metapost/src/eggs-common.mp new file mode 100644 index 0000000000..a9b909f388 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-common.mp @@ -0,0 +1,11 @@ +color eggshell, dark_eggshell, light_eggshell; +eggshell = 3/4[1/4[red, green], white]; +dark_eggshell = 1/2[1/4[red, green], white]; +light_eggshell = 7/8[3/16[red, green], white]; + +vardef numbered_points(expr egg) = image( + for i=1 upto length egg: + draw point i of egg; + label(decimal i, point i of egg scaled 1.06); + endfor; +) enddef; diff --git a/info/drawing-with-metapost/src/eggs-five-better.mp b/info/drawing-with-metapost/src/eggs-five-better.mp new file mode 100644 index 0000000000..472f3ab0a3 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-five-better.mp @@ -0,0 +1,80 @@ +\documentclass{standalone} +\usepackage{luamplib} +\mplibtextextlabel{enable} +\begin{document} +\begin{mplibcode} +numeric a; a = 56; +path r[]; % the rings +r1 = fullcircle scaled 2a shifted (0, -3/2 a); +r2 = fullcircle scaled 2a shifted (0, -1/2 a); +r3 = fullcircle scaled 2a shifted (0, +1/2 a); +r4 = fullcircle scaled 2a shifted (0, +3/2 a); +r5 = r2 rotatedabout(point 2 of r2, -60); +r6 = r2 rotatedabout(point 2 of r2, +60); + +pair o[]; % the centres of rotation for each arc +o1 = point 6 of r5; +o2 = point 2 of r3; +o3 = point 6 of r6; +o4 = whatever[o3, point 2+4/3 of r2] = whatever[o2, point 2-4/3 of r1]; +o8 = whatever[o1, point 2-4/3 of r2] = whatever[o2, point 2+4/3 of r1]; +o6 = 1/2[point 2-4/3 of r1, point 2+4/3 of r1]; +o5 = whatever[o6, point 2-4/3 of r3] = whatever[o4, point 2+4/3 of r1]; +o7 = whatever[o6, point 2+4/3 of r3] = whatever[o8, point 2-4/3 of r1]; + +pair u[], t[]; % directions and points for the egg +u0 = (o8 - o1) rotated 90; t0 = directionpoint u0 of r6; +u1 = (o2 - o1) rotated 90; t1 = directionpoint u1 of r4; +u2 = (o2 - o3) rotated 90; t2 = directionpoint u2 of r4; +u3 = (o4 - o3) rotated 90; t3 = directionpoint u3 of r5; + +u4 = (o5 - o4) rotated 90; +u5 = (o6 - o5) rotated 90; +u6 = (o6 - o7) rotated 90; +u7 = (o7 - o8) rotated 90; +t4 = directionpoint u4 of fullcircle scaled 2 abs (t3 - o4) shifted o4; +t5 = directionpoint u5 of fullcircle scaled 2 abs (t4 - o5) shifted o5; +t6 = directionpoint u6 of fullcircle scaled 2 abs (t5 - o6) shifted o6; +t7 = directionpoint u7 of fullcircle scaled 2 abs (t6 - o7) shifted o7; + +path egg; +egg = for i=0 upto 7: t[i] {u[i]} .. endfor cycle; + +input eggs-common + +beginfig(1); + fill egg withpen pencircle scaled 2 withcolor eggshell; + picture P; P = currentpicture; + + drawoptions(withpen pencircle scaled 1/4 withcolor dark_eggshell); + for i=1 upto 6: draw r[i]; endfor + draw point 2+4/3 of r1 -- point 2-4/3 of r1 -- o2 -- cycle; + draw point 2+4/3 of r3 -- o7; + draw point 2-4/3 of r3 -- o5; + + + drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); + draw t0 -- o1 -- t1; + draw t2 -- o3 -- t3; + draw o4 -- t4; + draw o5 -- t5; + draw o7 -- t6; + draw o8 -- t7; + draw egg; + + drawoptions(withpen pencircle scaled 2 withcolor 7/16); + t8 = t0; + for i=1 upto 8: + draw o[i]; + label("$\scriptscriptstyle " & decimal i &"$", + o[i] + 4.2 (unitvector(t[i-1]-o[i]) + unitvector (t[i]-o[i]))); + endfor + drawoptions(); + draw numbered_points(egg); + + draw P shifted 240 right; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/eggs-five-better.pdf b/info/drawing-with-metapost/src/eggs-five-better.pdf Binary files differnew file mode 100644 index 0000000000..bf092abbca --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-five-better.pdf diff --git a/info/drawing-with-metapost/src/eggs-five-point.mp b/info/drawing-with-metapost/src/eggs-five-point.mp new file mode 100644 index 0000000000..519fd82f1d --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-five-point.mp @@ -0,0 +1,84 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input eggs-common +beginfig(1); +numeric a; a = 56; +path r[], s[]; +for i=1 upto 4: + r[i] = fullcircle scaled 2a shifted (0, a * i - 5/2 a); % -2.5a so the origin is in centre +endfor +r5 = r2 rotatedabout(point 2 of r2, 60); +r6 = r2 rotatedabout(point 2 of r2, -60); + + +z1 = subpath (4, 8) of r5 intersectionpoint (point 6 of r6 -- 8[point 6 of r6, point 2/3 of r2]); +z2 = subpath (0, 4) of r4 intersectionpoint (point 6 of r6 -- 8[point 6 of r6, point 2 of r3]); +z3 = whatever [z1, point 6 of r6] = whatever [point 2 of r3, point 10/3 of r1]; +z4 = z3 reflectedabout(up, down); +z7 = 1/2[point 10/3 of r1, point 2/3 of r1]; +z5 = whatever [z3, point 2/3 of r1] = whatever [z7, point 10/3 of r3]; +z6 = z5 reflectedabout(up, down); + +s1 = fullcircle scaled 2 length (z2 - point 6 of r6) shifted point 6 of r6 cutbefore z1 cutafter z2; +s3 = fullcircle rotated angle (point 2/3 of r1 - z3) scaled 2 length (z1-z3) shifted z3 cutafter z1; +s5 = fullcircle rotated angle (z7 - z5) scaled 2 (length (z1-z3) - length (z3-z5)) shifted z5 cutafter point 0 of s3; + +s2 = reverse s1 reflectedabout(up, down); +s4 = reverse s3 reflectedabout(up, down); +s6 = reverse s5 reflectedabout(up, down); + +path egg; + +egg = point 0 of s1 {direction 0 of s1} .. + point 2 of s1 {direction 2 of s1} .. + point 0 of s2 {direction 0 of s2} .. + point 0 of s4 {direction 0 of s4} .. + point 0 of s6 {direction 0 of s6} .. + point 1 of s6 {direction 1 of s6} .. + point 0 of s5 {direction 0 of s5} .. + point 0 of s3 {direction 0 of s3} .. cycle; + +fill egg withpen pencircle scaled 2 withcolor eggshell; +picture P; P = currentpicture; + +drawoptions(withpen pencircle scaled 1/4 withcolor dark_eggshell); +for i=1 upto 6: + draw r[i]; +endfor +draw point 2 of r3 -- point 2/3 of r1 -- point 10/3 of r1 -- cycle; +draw z5 -- point 10/3 of r3; +draw z6 -- point 2/3 of r3; +draw point 2/3 of r1 withpen pencircle scaled 2; +draw point 10/3 of r1 withpen pencircle scaled 2; +draw point 2/3 of r3 withpen pencircle scaled 2; +draw point 10/3 of r3 withpen pencircle scaled 2; + +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); +draw point 6 of r6 -- point 8 of s1; +draw point 6 of r6 -- point 0 of s1; +draw point 6 of r5 -- point 8 of s2; +draw point 6 of r5 -- point 0 of s2; + +draw z3 -- point 0 of s3; +draw z4 -- point 8 of s4; +draw z5 -- point 0 of s5; +draw z6 -- point 8 of s6; + +for $=z3, z4, z5, z6, z7, point 2 of r3, point 6 of r5, point 6 of r6: + draw $ withpen pencircle scaled 2; +endfor + +draw egg; +draw numbered_points(egg); + +drawoptions(); + +draw P shifted 240 right; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/eggs-five-point.pdf b/info/drawing-with-metapost/src/eggs-five-point.pdf Binary files differnew file mode 100644 index 0000000000..8e9e1dd3c8 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-five-point.pdf diff --git a/info/drawing-with-metapost/src/eggs-four-point.mp b/info/drawing-with-metapost/src/eggs-four-point.mp new file mode 100644 index 0000000000..e168217aad --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-four-point.mp @@ -0,0 +1,45 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input eggs-common +beginfig(1); + +path egg, a, b, c, d; +a = fullcircle scaled 80; +b = a scaled 2 shifted point 6 of a; +c = halfcircle scaled 2 (abs(point 0 of a - point 5 of b) - abs(point 0 of a)); +d = fullcircle scaled 2 abs(point 2 of a - point 2 of c) shifted point 2 of c; +egg = point 0 of c {up} .. subpath (1,3) of d .. {down} point 4 of c .. subpath (5, 7) of b .. cycle; + +fill egg withpen pencircle scaled 2 withcolor 3/4[1/4[red, green], white]; +picture P; P = currentpicture; +drawoptions(withpen pencircle scaled 1/4 withcolor darker_eggshell); +draw a; draw b; draw c; draw d; +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); +draw point 0 of c -- point 3 of egg; +draw point 4 of c -- point 1 of egg; +draw point 0 of a -- point 5 of egg; +draw point 4 of a -- point 7 of egg; +draw point 0 of c -- point 4 of c; +draw point 2 of d -- point 6 of b; +draw egg; +drawoptions(withpen pencircle scaled 2 withcolor 7/16); +for i=1 upto length egg: + draw point i of egg; + label(decimal i, point i of egg scaled 1.06); +endfor; +draw point 0 of a; +draw point 4 of a; +draw point 6 of a; +draw point 2 of c; +drawoptions(); + +draw P shifted 240 right; + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/eggs-four-point.pdf b/info/drawing-with-metapost/src/eggs-four-point.pdf Binary files differnew file mode 100644 index 0000000000..ffbf1a3438 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-four-point.pdf diff --git a/info/drawing-with-metapost/src/eggs-fried.mp b/info/drawing-with-metapost/src/eggs-fried.mp new file mode 100644 index 0000000000..2c8c6d7226 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-fried.mp @@ -0,0 +1,33 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); +path yolk, base; +color cooked_egg_yolk, cooked_egg_white; +cooked_egg_yolk = 1/256(216, 136, 49); +cooked_egg_white = 1/256(235, 237, 233); + +vardef fried_egg(expr r) = + save base, yolk; path base, yolk; + yolk = for i=0 upto 17: (r + 1/8 normaldeviate) * dir 20i .. endfor cycle; + base = (for i=0 upto 17: (2r + 1/8r * normaldeviate) * dir 20i .. endfor cycle) + shifted (uniformdeviate r/2, uniformdeviate r/2); + image( + fill base withcolor cooked_egg_white; + fill yolk withcolor cooked_egg_yolk; + fill subpath (6.7, 9.6) of yolk scaled 0.8 -- + subpath (9.6, 6.7) of yolk scaled 0.66 -- cycle + withcolor 1/2[cooked_egg_yolk, white]; + ) +enddef; + +for i=0 upto 1: + draw fried_egg(40) shifted 120 dir 120i; +endfor + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/eggs-fried.pdf b/info/drawing-with-metapost/src/eggs-fried.pdf Binary files differnew file mode 100644 index 0000000000..b75e01b5bb --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-fried.pdf diff --git a/info/drawing-with-metapost/src/eggs-gold.mp b/info/drawing-with-metapost/src/eggs-gold.mp new file mode 100644 index 0000000000..6c13a6afb3 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-gold.mp @@ -0,0 +1,63 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input eggs-common +beginfig(1); + +path base; +base = fullcircle scaled 200; + +pair m, n, n', p; +path aa, bb; + +m = 1/2 point 0 of base; +aa = halfcircle scaled 2 abs (point 2 of base - m) shifted m cutbefore (origin -- 1000 up); +n = point infinity of aa; +n' = n reflectedabout(up, down); +bb = subpath (0, 2) of base shifted n cutafter (origin -- 1000 up); +p = point infinity of bb; + +path dome; +path cap, cup, egg; +dome = fullcircle + scaled 2 (abs(n - point 0 of base) - abs(n - point 0 of bb)) + shifted p; +cap = dome + cutbefore (point 4 of bb -- 2[n, point 4 of bb]) + cutafter (point 4 of bb -- 2[n', point 4 of bb]); +cup = subpath (4, 8) of base; +path egg; egg = point 4 of cup {up} .. cap .. {down} cup & cycle; + +fill egg withpen pencircle scaled 2 withcolor eggshell; +picture P; P = currentpicture; +drawoptions(withpen pencircle scaled 1/4 withcolor dark_eggshell); +draw m -- aa; +draw bb; +draw origin -- point 2 of base; +draw base; draw dome; +label.bot("$m$", m); +label.ulft("$n$", n); +label.urt("$n'$", n'); +% label.lft("$p$", p); + +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); +draw n -- point 1 of egg; +draw n' -- point 3 of egg; +draw point 0 of egg -- point 4 of egg; +draw egg; + +forsuffixes $=n, n', p, origin: draw $ withpen pencircle scaled 2; endfor +draw m withpen pencircle scaled 2 withcolor dark_eggshell; +draw numbered_points(egg); + +drawoptions(); + +draw P shifted 240 right; + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/eggs-gold.pdf b/info/drawing-with-metapost/src/eggs-gold.pdf Binary files differnew file mode 100644 index 0000000000..171112387e --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-gold.pdf diff --git a/info/drawing-with-metapost/src/eggs-moss.mp b/info/drawing-with-metapost/src/eggs-moss.mp new file mode 100644 index 0000000000..6bd228dd4f --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-moss.mp @@ -0,0 +1,31 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +input eggs-common +path a, b, c, d, egg; numeric r; r = 100; +a = fullcircle scaled 2r; +b = fullcircle scaled 4r shifted point 4 of a; +c = fullcircle scaled 4r shifted point 0 of a; +d = fullcircle scaled 2 abs (point 2 of a - point 1 of b) + shifted point 2 of a; +egg = subpath (0, 1) of b .. point 2 of d .. + subpath (3, 4) of c .. subpath (5, 7) of a .. cycle; +beginfig(1); +fill egg withpen pencircle scaled 2 withcolor eggshell; +picture P; P = currentpicture; +drawoptions(withpen pencircle scaled 1/4 withcolor dark_eggshell); +draw a; draw d; +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); +draw point 1 of egg -- point 4 of egg -- + point 0 of egg -- point 3 of egg; +draw egg; +drawoptions(withpen pencircle scaled 2 withcolor 7/16); +draw center a; +draw center d; +draw numbered_points(egg); +drawoptions(); +draw P shifted 240 right; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/eggs-moss.pdf b/info/drawing-with-metapost/src/eggs-moss.pdf Binary files differnew file mode 100644 index 0000000000..3cea5ce5d2 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-moss.pdf diff --git a/info/drawing-with-metapost/src/eggs-pentagons.mp b/info/drawing-with-metapost/src/eggs-pentagons.mp new file mode 100644 index 0000000000..adbd86c3bc --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-pentagons.mp @@ -0,0 +1,35 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef pentapath(expr a, b) = + save c, d, e; + pair c, d, e; + c = a rotatedabout(b, -108); + d = b rotatedabout(c, -108); + e = b rotatedabout(a, 108); + a -- b -- c -- d -- e -- cycle +enddef; +input colorbrewer-rgb +beginfig(1); + + path p[]; + p0 = for i=1 upto 5: 20 right rotated -18 rotated 72i -- endfor cycle; + draw p0 dashed withdots scaled 1/4; + for i=1 upto 3: + p[i] = pentapath(point 4 of p[i-1], point 2 of p[i-1]); + draw p[i] dashed withdots scaled 1/4; + draw point 0 of p0 {point 2 of p0 - point 3 of p0} + .. point 1 of p0 {point 3 of p0 - point 4 of p0} + .. point 2 of p[i] {point 4 of p[i] - point 0 of p[i]} + .. point 3 of p[i] {point 0 of p[i] - point 1 of p[i]} + .. point 4 of p[i] {point 1 of p[i] - point 2 of p[i]} + .. cycle withcolor Spectral[4][i]; + endfor + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/eggs-pentagons.pdf b/info/drawing-with-metapost/src/eggs-pentagons.pdf Binary files differnew file mode 100644 index 0000000000..18fa3c1fe7 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-pentagons.pdf diff --git a/info/drawing-with-metapost/src/eggs-perfect.mp b/info/drawing-with-metapost/src/eggs-perfect.mp new file mode 100644 index 0000000000..8f63cd6e8b --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-perfect.mp @@ -0,0 +1,30 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input eggs-common +beginfig(1); +path egg; +%egg = for t=90,105,120,135,150,165,-180,-165,-150,-135,-120,-105,-90,-75,-60,-45,-30,-15,0,15,30,45,60,75: +%egg = for t=90, 135, 180, -135, -90, -45, 0, 45: +egg = for t=-180 step 15 until 180 - eps: + (0.78 cosd(1/4 t) * sind(t), -cosd(t)) .. + endfor cycle; +egg := egg scaled 128; + +fill egg withpen pencircle scaled 2 withcolor eggshell; +picture P; P = currentpicture; +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); +draw egg; +drawoptions(withpen pencircle scaled 2 withcolor 7/16); +draw origin; +for i=1 upto length egg: draw point i of egg; endfor +drawoptions(); + +draw P shifted 240 right; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/eggs-perfect.pdf b/info/drawing-with-metapost/src/eggs-perfect.pdf Binary files differnew file mode 100644 index 0000000000..9a291f9059 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-perfect.pdf diff --git a/info/drawing-with-metapost/src/eggs-shaded.mp b/info/drawing-with-metapost/src/eggs-shaded.mp new file mode 100644 index 0000000000..3eeca4babe --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-shaded.mp @@ -0,0 +1,27 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +path egg, spot; +egg = (for t=-180 step 15 until 180 - eps: + (0.78 cosd(1/4 t) * sind(t), -cosd(t)) .. +endfor cycle) scaled 100; + +spot = fullcircle scaled 4 shifted 3/4 point 3 of egg; + +vardef fade_filled(expr egg, spot, dark, light, n) = image( + for i = 0 upto n: + fill interpath(i/n, egg, spot) withcolor ((i/n)**1/3)[dark,light]; + endfor) +enddef; + +beginfig(1); + color a, b; + a = 1/256(150, 100, 60); + b = 1/256(256, 220, 180); + draw fade_filled(egg, spot, a, b, 256) rotated -30; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/eggs-shaded.pdf b/info/drawing-with-metapost/src/eggs-shaded.pdf Binary files differnew file mode 100644 index 0000000000..ec2143cdc2 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-shaded.pdf diff --git a/info/drawing-with-metapost/src/eggs-super.mp b/info/drawing-with-metapost/src/eggs-super.mp new file mode 100644 index 0000000000..89f8dda5b4 --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-super.mp @@ -0,0 +1,20 @@ +\documentclass{standalone} +\usepackage{luamplib} +\mplibtextextlabel{enable} +\begin{document} +\begin{mplibcode} +path egg; +egg = (superellipse(right, 1.6 up, left, 1.2 down, 0.69)) scaled 100; +input eggs-common +beginfig(1); + fill egg withpen pencircle scaled 2 withcolor eggshell; + picture P; P = currentpicture; + drawoptions(withcolor 1/2); + draw origin withpen pencircle scaled 2; + draw egg withpen pencircle scaled 1/4; + drawoptions(); + draw numbered_points(egg) withpen pencircle scaled 2; + draw P shifted 240 right; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/eggs-super.pdf b/info/drawing-with-metapost/src/eggs-super.pdf Binary files differnew file mode 100644 index 0000000000..2022f1ed2d --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-super.pdf diff --git a/info/drawing-with-metapost/src/eggs-thom.mp b/info/drawing-with-metapost/src/eggs-thom.mp new file mode 100644 index 0000000000..f38d86fcad --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-thom.mp @@ -0,0 +1,45 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input eggs-common + +numeric r, a, b, t; a = 60; b = 45; r = a ++ b; +pair p, q; p = -q = (b, 0); +path base, cap, egg; +base = subpath (4, 8) of fullcircle scaled 2(2r-b); +cap = subpath (0, 4) of fullcircle scaled 2r shifted (0, a) + cutbefore ((b, 0) -- (b, 2r)) + cutafter ((-b, 0) -- (-b, 2r)); +egg = point 4 of base {up} .. cap .. {down} base & cycle; +% more naturally +% base {up} .. cap .. {down} & cycle +% but then point 0 would not be at 3 o'clock + +beginfig(1); +fill egg withpen pencircle scaled 2 withcolor eggshell; +picture P; P = currentpicture; +fill origin -- (-b, 0) -- (0, a) -- cycle withcolor light_eggshell; +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); +label.top("$\scriptstyle 3$", (-1/2b, 0)); +label.lft("$\scriptstyle 4$", (0, 1/2a)); +label.ulft("$\scriptstyle 5$", 1/2[(-b, 0), (0, a)]); + +draw fullcircle scaled 2(2r-b); draw origin withpen pencircle scaled 2; +draw fullcircle scaled 2r shifted (0, a); draw (0, a) withpen pencircle scaled 2; + +draw p -- subpath (3, 4) of egg -- cycle; draw p withpen pencircle scaled 2; +draw q -- subpath (0, 1) of egg -- cycle; draw q withpen pencircle scaled 2; + +draw numbered_points(egg); + +drawoptions(); + +draw P shifted 240 right; + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/eggs-thom.pdf b/info/drawing-with-metapost/src/eggs-thom.pdf Binary files differnew file mode 100644 index 0000000000..2e4a9089cf --- /dev/null +++ b/info/drawing-with-metapost/src/eggs-thom.pdf diff --git a/info/drawing-with-metapost/src/escher.mp b/info/drawing-with-metapost/src/escher.mp new file mode 100644 index 0000000000..eac172d4f3 --- /dev/null +++ b/info/drawing-with-metapost/src/escher.mp @@ -0,0 +1,38 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); + +numeric s; s = 13; +path atom; +atom = origin + -- (2s,0) rotated -30 -- (2s,0) rotated -30 + (0,s) + -- ( s,0) rotated 30 -- ( s,0) rotated 30 + (0,s) + -- (0,2s) -- cycle; + +picture p[]; +for i=0 upto 2: + p[i] = image( + fill atom rotated -120i withcolor (7/8 - 1/8i) ; + draw atom rotated -120i; + ); +endfor + +n = 13; +for i=-n upto n: + for j=-n upto n: + forsuffixes $=0,1,2: + draw p$ shifted ((3i*s,0) rotated -30 + + (0,floor(1/2i)*3s + 3j*s)); + endfor + endfor +endfor + +clip currentpicture to (unitsquare shifted -(1/2,1/2) + xscaled 55.425s yscaled 30s); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/escher.pdf b/info/drawing-with-metapost/src/escher.pdf Binary files differnew file mode 100644 index 0000000000..391fc26d5d --- /dev/null +++ b/info/drawing-with-metapost/src/escher.pdf diff --git a/info/drawing-with-metapost/src/euler-sampler.mp b/info/drawing-with-metapost/src/euler-sampler.mp new file mode 100644 index 0000000000..72101fb1da --- /dev/null +++ b/info/drawing-with-metapost/src/euler-sampler.mp @@ -0,0 +1,24 @@ +\documentclass[border=8pt]{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); + for i=0 upto 15: + label.top(decimal i infont "cmss10" scaled 0.7, (15i,108)) withcolor .67 red; + for j=0 upto 7: + draw char (i+16j) infont "eurm10" shifted (15i-3,97-14j); + if i=0: + label.lft(decimal 16j infont "cmss10" scaled 0.7, (-8,100-14j)) withcolor .67 red; + fi + % if (i + 16j) > 0: + % draw char (i + 16j) infont "cmmi10" + % scaled 0.5 + % shifted (15i - 6, 97-14j) + % withcolor .67 blue; + % fi + endfor + endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/euler-sampler.pdf b/info/drawing-with-metapost/src/euler-sampler.pdf Binary files differnew file mode 100644 index 0000000000..1da31b1984 --- /dev/null +++ b/info/drawing-with-metapost/src/euler-sampler.pdf diff --git a/info/drawing-with-metapost/src/excircle.mp b/info/drawing-with-metapost/src/excircle.mp new file mode 100644 index 0000000000..01307ce3d0 --- /dev/null +++ b/info/drawing-with-metapost/src/excircle.mp @@ -0,0 +1,39 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef through(expr a, b, o) = + save t; numeric t; t = 1+o/abs(a-b); + t[b,a] -- t[a,b] +enddef; + +beginfig(1); + pair A, B, C; + A = origin; B = 233 dir 10; C = 144 dir 70; + pair a, b, m, t; + a = A + 22 unitvector (C-A) - 22 unitvector (B-A); + b = B + 22 unitvector (A-B) + 22 unitvector (C-B); + m = whatever[A,a] = whatever [B,b]; t = whatever[A,B]; + t-m = whatever * (B-A) rotated 90; + + drawoptions(dashed evenly scaled 1/4 withcolor 1/2 white); + draw t -- m; + draw unitsquare scaled 5 rotated angle (B-A) shifted t; + draw through(A, m, 10); + draw through(A, t, 10); + draw through(B, m, 10); + drawoptions(); + draw fullcircle scaled 2 abs (t-m) shifted m withcolor 2/3 blue; + draw A--B--C--cycle; + dotlabel.urt(btex $m$ etex, m); + dotlabel.bot(btex $t$ etex, t); + dotlabel.bot(btex $A$ etex, A); + dotlabel.urt(btex $B$ etex, B); + dotlabel.top(btex $C$ etex, C); + dotlabel.urt(btex $a$ etex, a); + dotlabel.top(btex $b$ etex, b); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/excircle.pdf b/info/drawing-with-metapost/src/excircle.pdf Binary files differnew file mode 100644 index 0000000000..817cd59350 --- /dev/null +++ b/info/drawing-with-metapost/src/excircle.pdf diff --git a/info/drawing-with-metapost/src/explode.mp b/info/drawing-with-metapost/src/explode.mp new file mode 100644 index 0000000000..8d03d94846 --- /dev/null +++ b/info/drawing-with-metapost/src/explode.mp @@ -0,0 +1,31 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\usepackage{fontspec} +\setmainfont{PlayfairDisplay-Black} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +randomseed:=2128.5073; +beginfig(1); +n = 40; r = 7 ; s = 44; +path explosion, splash; +explosion = for i=1 upto n: + (s if odd(i): - else: + fi r + uniformdeviate r,0) rotated (i*360/n) -- +endfor cycle; + +splash = for i=1 upto n: + (s if odd(i): - else: + fi r + uniformdeviate r,0) rotated (i*360/n) .. +endfor cycle; +splash := splash shifted (3s,0); + +fill explosion withcolor Oranges 7 4; +draw explosion withpen pencircle scaled 2 withcolor Reds 7 7; +label(TEX("\textbf{BOOM!}") scaled 1.6, center explosion) withcolor Reds 7 7; + +fill splash withcolor Blues 7 2; +draw splash withpen pencircle scaled 2 withcolor Blues 7 7; +label(TEX("\bfseries SPLAT!") scaled 1.6, center splash) withcolor Blues 7 7; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/explode.pdf b/info/drawing-with-metapost/src/explode.pdf Binary files differnew file mode 100644 index 0000000000..d5c1c094d9 --- /dev/null +++ b/info/drawing-with-metapost/src/explode.pdf diff --git a/info/drawing-with-metapost/src/fake-transparency.mp b/info/drawing-with-metapost/src/fake-transparency.mp new file mode 100644 index 0000000000..dfe7086052 --- /dev/null +++ b/info/drawing-with-metapost/src/fake-transparency.mp @@ -0,0 +1,57 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +% parameters +u = 1cm; +ymax = xmax = 2.75; +xmin = ymin = -1.75; + +% make a plain grid +path xx, yy; +xx = ((xmin,0) -- (xmax,0)) scaled u; +yy = ((0,ymin) -- (0,ymax)) scaled u; + +drawoptions(dashed withdots scaled 1/4 withcolor 3/4 white); +for i = ceiling ymin upto floor ymax: draw xx shifted (0,i*u); endfor +for i = ceiling xmin upto floor xmax: draw yy shifted (i*u,0); endfor + +drawoptions(withpen pencircle scaled .7); +xx := xx scaled 1.05; +yy := yy scaled 1.05; +drawarrow xx; +drawarrow yy; +drawoptions(); + +label.urt("A" infont defaultfont scaled 8, origin) withcolor 1/4 green; + +% draw the "transparent" superellipse box +path se; +%se = unitsquare shifted 1/2 down shifted 1/4 left scaled 3cm; +se = (superellipse(right, up, left, down, 0.81)) +shifted 1/2 right scaled 30 rotated 30; + +alpha = 5/8; % alpha: 0=invisible, 1=opaque +color filler; filler = .95[red,white]; +picture fg, bg; +bg = currentpicture; +fg = image( + for e within bg: + draw e withcolor alpha[colorpart e, filler]; + endfor + draw se withpen pencircle scaled 2 withcolor 3/4; + draw subpath (2.718, 3.1415) of se + shifted - center se scaled 7/8 shifted + center se + withpen pencircle scaled 2 withcolor white; + +); +clip fg to se; + +fill se withcolor filler; +draw fg; +draw se withcolor 3/4 blue; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/fake-transparency.pdf b/info/drawing-with-metapost/src/fake-transparency.pdf Binary files differnew file mode 100644 index 0000000000..07f247a29d --- /dev/null +++ b/info/drawing-with-metapost/src/fake-transparency.pdf diff --git a/info/drawing-with-metapost/src/func-addition-of-sines.mp b/info/drawing-with-metapost/src/func-addition-of-sines.mp new file mode 100644 index 0000000000..d7f9a41db5 --- /dev/null +++ b/info/drawing-with-metapost/src/func-addition-of-sines.mp @@ -0,0 +1,58 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef pi_sixths(expr n) = + save s, f, q; string s, f; numeric q; + s = if n < 0: "-" else: "" fi; q = abs(n); + if q mod 6 = 0: + f = if q > 6: decimal 1/6 q else: "" fi; + elseif q mod 3 = 0: + f = "\frac{" & decimal 1/3 q & "}{2}"; + elseif q mod 2 = 0: + f = "\frac{" & decimal 1/2 q & "}{3}"; + else: + f = "\frac{" & decimal q & "}{6}"; + fi + "$\scriptstyle" & s & f & "\pi$" +enddef; + +beginfig(1); + numeric u, pi; u = 50; pi = 3.141592653589793; + + path xx, yy; + xx = (3.5 left -- 4 right) scaled u; + yy = (1.2 down -- 1.3 up) scaled u; + + path ss, tt, uu; + ss = origin for x=1 upto 360: -- (x, sind(x)) endfor; + tt = origin for x=1 upto 360: -- (x, 1/2 sind(3x)) endfor; + uu = origin for x=1 upto 360: -- (x, ypart point x of ss + ypart point x of tt) endfor; + + forsuffixes $=ss, tt, uu: + $ := $ shifted 360 left & $; + $ := $ xscaled (pi/180) scaled u; + $ := $ cutbefore yy shifted point 0 of xx + cutafter yy shifted point 1 of xx; + endfor + + draw ss withcolor 1/2[blue, white]; + draw tt withcolor 1/2[red, white]; + draw uu withcolor 1/4 green; + + label.top("$f(x)=sin(x)$", point 290 of ss) withcolor 1/2[blue, white]; + label.bot("$g(x)=\frac12 sin(3x)$", point 295 of tt) withcolor 1/2[red, white]; + label.urt("$f(x) + g(x)$", point 350 of uu) withcolor 1/4 green; + + drawarrow xx; label.rt("$x$", point 1 of xx); + drawarrow yy; label.top("$y$", point 1 of yy); + + for i=-6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 7: + draw (down--up) scaled 2 shifted (pi * i/6 * u, 0); + label.bot(pi_sixths(i), (pi * i/6 * u, -2)); + endfor + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/func-addition-of-sines.pdf b/info/drawing-with-metapost/src/func-addition-of-sines.pdf Binary files differnew file mode 100644 index 0000000000..29973d3b2a --- /dev/null +++ b/info/drawing-with-metapost/src/func-addition-of-sines.pdf diff --git a/info/drawing-with-metapost/src/func-cuberoot.mp b/info/drawing-with-metapost/src/func-cuberoot.mp new file mode 100644 index 0000000000..21e17dca8c --- /dev/null +++ b/info/drawing-with-metapost/src/func-cuberoot.mp @@ -0,0 +1,34 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path xx, yy; +xx = (left -- right) scaled 150; +yy = (down -- up) scaled 50; + +numeric n; n = 6; + +path ff; ff = origin {right} for x = 1 upto n: ... {1, 3x*x} (x, x*x*x) endfor; +ff := ff reflectedabout(origin, dir 45); +ff := reverse ff rotated 180 & ff; +ff := ff xscaled (150/n/n/n) yscaled (50/n); + +for i = 1-n upto n-1: + draw yy shifted (i * xpart directionpoint (1, 1/3) of ff, 0) withcolor 7/8; +endfor +for i = -1, 1: + draw xx shifted (0, i * ypart directionpoint (1, 1/3) of ff) withcolor 7/8; +endfor + +draw ff withcolor 2/3 blue; + +label("$y = \sqrt[3]x$", (-84, 14)); +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/func-cuberoot.pdf b/info/drawing-with-metapost/src/func-cuberoot.pdf Binary files differnew file mode 100644 index 0000000000..4f3dc1f744 --- /dev/null +++ b/info/drawing-with-metapost/src/func-cuberoot.pdf diff --git a/info/drawing-with-metapost/src/func-epi-v-pie.mp b/info/drawing-with-metapost/src/func-epi-v-pie.mp new file mode 100644 index 0000000000..cb36f02f6d --- /dev/null +++ b/info/drawing-with-metapost/src/func-epi-v-pie.mp @@ -0,0 +1,64 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + +numeric minx, maxx, s, u, v; +minx = 13/8; s = 1/16; maxx = 19/4; +u = 89; v = 3072; + +def f(expr x) = 1/256 mlog(x) / x enddef; + +path ff, xx, yy; +ff = for x=minx step s until maxx-s: (x, f(x)) .. endfor (maxx, f(maxx)); +ff := ff xscaled u yscaled v; +xx = origin -- right scaled (maxx-minx) scaled u; +yy = origin -- up scaled 0.09v; +xx := xx shifted point 0 of ff shifted 20 down; +yy := yy shifted point 0 of ff shifted 20 left; + +numeric pi, e, fpi, fe; +pi = 3.141592653589793 u; fpi = f(3.141592653589793) * v; +e = 2.718281828459045 u; fe = f(2.718281828459045) * v; + +path ee, pp; +ee = (e, ypart point 0 of xx) -- (e, fe) -- (xpart point 0 of yy, fe); +pp = (pi, ypart point 0 of xx) -- (pi, fpi) -- (xpart point 0 of yy, fpi); + +draw ee dashed withdots scaled 1/4 withcolor 2/3 red; +draw pp dashed withdots scaled 1/2 withcolor 2/3 red; +draw ff withcolor 3/4 blue; +drawarrow xx; +drawarrow yy; + +for x=2 upto 4: + draw (down--up) scaled 2 shifted (x * u, ypart point 0 of xx); + label.bot("$" & decimal x & "$", (x * u, ypart point 0 of xx - 2)); +endfor +for y=31 upto 38: + draw (left--right) scaled 2 shifted (xpart point 0 of yy, y/100*v); +endfor +for y=32, 35, 38: + label.lft("$" & decimal (y/100) & "$", (xpart point 0 of yy-2, y/100*v)); +endfor + +drawoptions(withcolor 1/2 red); +label.bot("$e$", point 0 of ee shifted 4 down); +label.lft("$1/e$", point 2 of ee shifted 2 left); + +label.bot("$\pi$", point 0 of pp shifted 4 down); +label.lft("$\ln\pi/\pi$", point 2 of pp shifted 2 left); + +drawoptions(withcolor 2/3 blue); +label.urt("$\displaystyle y={\ln x\over x}$", point 42 of ff); + +drawoptions(); +label.rt("$x$", point 1 of xx); +label.top("$y$", point 1 of yy); +label("\dots\ hence\enspace $e^\pi > \pi^e$.", (4u, 0.38v)); + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/func-epi-v-pie.pdf b/info/drawing-with-metapost/src/func-epi-v-pie.pdf Binary files differnew file mode 100644 index 0000000000..175bde4577 --- /dev/null +++ b/info/drawing-with-metapost/src/func-epi-v-pie.pdf diff --git a/info/drawing-with-metapost/src/func-exponential.mp b/info/drawing-with-metapost/src/func-exponential.mp new file mode 100644 index 0000000000..bfea46f174 --- /dev/null +++ b/info/drawing-with-metapost/src/func-exponential.mp @@ -0,0 +1,49 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +numeric u; u = 42; + +path xx, yy; +xx = (-3u, 0) -- (5u, 0); +yy = xx rotated 90; + +path ee, ll, nn; +numeric minx, maxx, s; +minx = -3; maxx = 1/256 mlog(4.5); s = 1/4; +ee = (for x = minx step s until maxx - s: + (x, mexp(256x)){1, mexp(256x)} ... +endfor (maxx, mexp(256 maxx)){1, mexp(256 maxx)}) scaled u; +ll = ee reflectedabout(origin, dir 45); +nn = (for x=1 step s until 4-s: (x, 1/x) ... endfor (4, 1/4)) scaled u; +nn := reverse nn reflectedabout(origin, dir 45) & nn; + +draw unitsquare xscaled mexp(256) scaled u withcolor 7/8; +draw unitsquare yscaled mexp(256) scaled u withcolor 7/8; + +path T[]; +numeric p; p = directiontime dir 45 of ee; +T1 = (precontrol p of ee -- postcontrol p of ee) shifted - point p of ee scaled 1/2 u shifted point p of ee; +T2 = (precontrol p of ll -- postcontrol p of ll) shifted - point p of ll scaled 1/2 u shifted point p of ll; +drawoptions(withpen pencircle scaled 1/4 withcolor 1/2); +draw T1; draw T2; draw interpath(1/2, T1, T2) dashed evenly; +drawoptions(); + +draw nn; label.urt("$1/x$", point 0 of nn); +draw ee withcolor 2/3 blue; label.top("$e^x$", point infinity of ee) withcolor 2/3 blue; +draw ll withcolor 3/4 red; label.top("$\ln(x)$", point infinity of ll) withcolor 3/4 red; +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); + +dotlabel.lft("$e$", (0, mexp(256) * u)); +dotlabel.bot("$e$", (mexp(256) * u, 0)); +dotlabel.ulft("$1$", (0, u)); +dotlabel.lrt("$1$", (u, 0)); + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/func-exponential.pdf b/info/drawing-with-metapost/src/func-exponential.pdf Binary files differnew file mode 100644 index 0000000000..eab1af76ed --- /dev/null +++ b/info/drawing-with-metapost/src/func-exponential.pdf diff --git a/info/drawing-with-metapost/src/func-lemniscate.mp b/info/drawing-with-metapost/src/func-lemniscate.mp new file mode 100644 index 0000000000..cf56cede6b --- /dev/null +++ b/info/drawing-with-metapost/src/func-lemniscate.mp @@ -0,0 +1,25 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +numeric a, c; c = 100; a = sqrt(2) * c; +vardef f(expr x) = a * cosd(x) / (1 + sind(x) ** 2) enddef; +vardef g(expr x) = f(x) * sind(x) enddef; +numeric mint, maxt, s; mint = 0; s = 30; maxt = 360; +path p; +p = for t = mint step s until maxt - s: + (f(t), g(t)) ... +endfor cycle; +draw p withcolor 3/4 blue; +path xx, yy; +xx = (left -- right) scaled 150; +yy = (down -- up) scaled 55; +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); +dotlabel.bot("$F_1$", c * left); +dotlabel.bot("$F_2$", c * right); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/func-lemniscate.pdf b/info/drawing-with-metapost/src/func-lemniscate.pdf Binary files differnew file mode 100644 index 0000000000..d9dd8653c7 --- /dev/null +++ b/info/drawing-with-metapost/src/func-lemniscate.pdf diff --git a/info/drawing-with-metapost/src/func-maurer-rose.mp b/info/drawing-with-metapost/src/func-maurer-rose.mp new file mode 100644 index 0000000000..afa0d85467 --- /dev/null +++ b/info/drawing-with-metapost/src/func-maurer-rose.mp @@ -0,0 +1,31 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + + numeric n; n = 2; + path r; r = (for t=0 upto 360: + cosd(n * t) * dir t ... + endfor cycle) scaled 150; + + numeric d; d = 29; + path k; k = for t = 0 upto 360: + point d * t mod 360 of r -- + endfor cycle; + + draw k dashed withdots scaled 1/8 + withpen pencircle scaled 1/4 + withcolor 1/2[blue, white]; + draw r withcolor 2/3 red; + + path xx, yy; + xx = (left -- right) scaled 160; + yy = (down -- up) scaled 160; + drawarrow xx; label.rt("$x$", point 1 of xx); + drawarrow yy; label.top("$y$", point 1 of yy); + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/func-maurer-rose.pdf b/info/drawing-with-metapost/src/func-maurer-rose.pdf Binary files differnew file mode 100644 index 0000000000..a4a94febe5 --- /dev/null +++ b/info/drawing-with-metapost/src/func-maurer-rose.pdf diff --git a/info/drawing-with-metapost/src/func-parametric.mp b/info/drawing-with-metapost/src/func-parametric.mp new file mode 100644 index 0000000000..7d86c3b2ab --- /dev/null +++ b/info/drawing-with-metapost/src/func-parametric.mp @@ -0,0 +1,24 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path p; +numeric a, b, j, k; a = 144; b = 89; j = 3; k = 2; +p = for t = 0 upto 360: + (a * cosd(j * t), b * sind(k * t)) ... +endfor cycle; + +draw p withcolor 2/3 red; + +path xx, yy; +xx = (left -- right) scaled 150; +yy = (down -- up) scaled 100; +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/func-parametric.pdf b/info/drawing-with-metapost/src/func-parametric.pdf Binary files differnew file mode 100644 index 0000000000..8f7bed75f4 --- /dev/null +++ b/info/drawing-with-metapost/src/func-parametric.pdf diff --git a/info/drawing-with-metapost/src/func-powers.mp b/info/drawing-with-metapost/src/func-powers.mp new file mode 100644 index 0000000000..b88aa49a6a --- /dev/null +++ b/info/drawing-with-metapost/src/func-powers.mp @@ -0,0 +1,42 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +input colorbrewer-rgb +numeric t, s, u, v; +t = 3; s = 1; +u = 50; v = 10; + +path xx, yy; +xx = (left -- right) scaled t scaled u; +yy = 50 down -- 100 up; + +picture P[]; +for k = 1 upto 3: + s := s * 1/2; + P[k] = image( + for n=2 upto 7: + path c; c = (for x = -t step s until t-s: (x, x**n) ... endfor (t, t**n)) xscaled u yscaled v + cutbefore xx shifted point if odd n: 0 else: 1 fi of yy + cutafter xx shifted point 1 of yy if n > 5: shifted (0, v*n - 5v) fi; + draw c withcolor Spectral[6][n-1]; + for i=0 upto length(c): + draw point i of c withpen pencircle scaled 1; + endfor + label("$\scriptstyle x^" & decimal n & "$", point infinity of c shifted 3 right shifted 6 unitvector(direction infinity of c)); + label("Step: $\frac1{" & decimal (1/s) & "}$", (30, 80)); + endfor + for i=1,2: + draw (i*u, 0) -- (i*u, -2); + label.bot(decimal i, (i*u, -2)); + endfor + drawarrow xx; + drawarrow yy; + ); + draw P[k] shifted (0, -160k); +endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/func-powers.pdf b/info/drawing-with-metapost/src/func-powers.pdf Binary files differnew file mode 100644 index 0000000000..4888cb89f4 --- /dev/null +++ b/info/drawing-with-metapost/src/func-powers.pdf diff --git a/info/drawing-with-metapost/src/func-pulse.mp b/info/drawing-with-metapost/src/func-pulse.mp new file mode 100644 index 0000000000..ae07cfd570 --- /dev/null +++ b/info/drawing-with-metapost/src/func-pulse.mp @@ -0,0 +1,35 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +vardef pulse(expr w, h, d) = +for i=0 upto 4: x[i] = w/4 * i; endfor + y0 = y1 = 0; + y3 = y4 = d; + y2 = h; + z0 .. 1/2[z0, z1] .. controls z1 + .. 1/2[z1, z2] .. controls z2 + .. 1/2[z2, z3] .. controls z3 + .. 1/2[z3, z4] .. z4 +enddef; + +beginfig(1); + path p; p = pulse(300, 100, -40); + + draw point 0 of p -- postcontrol 1 of p -- postcontrol 2 of p -- postcontrol 3 of p -- point 5 of p + dashed withdots scaled 1/2 withcolor 2/3 red; + for i=1 upto 3: draw postcontrol i of p withpen pencircle scaled 2 withcolor 2/3 red; endfor + label.lrt("$z_1$", postcontrol 1 of p) withcolor 2/3 red; + label.top("$z_2$", postcontrol 2 of p) withcolor 2/3 red; + label.llft("$z_3$", postcontrol 3 of p) withcolor 2/3 red; + + draw p withcolor 1/2 blue; + draw p shifted 120 up withcolor 1/2 blue; + for i=0 upto length p: draw point i of p withpen pencircle scaled 2; endfor + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/func-pulse.pdf b/info/drawing-with-metapost/src/func-pulse.pdf Binary files differnew file mode 100644 index 0000000000..6b60a628bd --- /dev/null +++ b/info/drawing-with-metapost/src/func-pulse.pdf diff --git a/info/drawing-with-metapost/src/func-reflection.mp b/info/drawing-with-metapost/src/func-reflection.mp new file mode 100644 index 0000000000..a9bbf7f85e --- /dev/null +++ b/info/drawing-with-metapost/src/func-reflection.mp @@ -0,0 +1,35 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path xx, yy; +xx = (left -- right) scaled 150; +yy = (down -- up) scaled 150; + +path ff, negative_ff; +ff = (1,1) for x = 3/2 step 1/2 until 6: ... (x, 1/x) endfor; +ff := reverse ff reflectedabout(origin, dir 45) & ff; +ff := ff scaled 24; +negative_ff = ff reflectedabout(origin, dir -45); + +for i=-6 upto 6: + draw xx shifted (0, 24i) withpen pencircle scaled 1/4 withcolor 7/8; + draw yy shifted (24i, 0) withpen pencircle scaled 1/4 withcolor 7/8; +endfor + +forsuffixes @ = ff, negative_ff: + draw @ withcolor 2/3 red; + for i=0 upto length @: + draw point i of @ withpen pencircle scaled 1; + endfor +endfor + +label("$y = 1/x$", (-84, 84)); +drawarrow xx; label.rt("$x$", point 1 of xx); +drawarrow yy; label.top("$y$", point 1 of yy); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/func-reflection.pdf b/info/drawing-with-metapost/src/func-reflection.pdf Binary files differnew file mode 100644 index 0000000000..34670c9605 --- /dev/null +++ b/info/drawing-with-metapost/src/func-reflection.pdf diff --git a/info/drawing-with-metapost/src/func-sines.mp b/info/drawing-with-metapost/src/func-sines.mp new file mode 100644 index 0000000000..2a854ede62 --- /dev/null +++ b/info/drawing-with-metapost/src/func-sines.mp @@ -0,0 +1,57 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef pi_quarters(expr n) = + save s, f, q; string s, f; numeric q; + s = if n < 0: "-" else: "" fi; q = abs(n); + if q mod 4 = 0: + f = if q > 4: decimal 1/4 q else: "" fi; + elseif q mod 2 = 0: + f = "\frac{" & decimal 1/2 q & "}{2}"; + else: + f = "\frac{" & decimal q & "}{4}"; + fi + "$\scriptstyle" & s & f & "\pi$" +enddef; + +beginfig(1); + numeric u, pi; u = 50; pi = 3.141592653589793; + + path xx, yy; + xx = (3.5 left -- 3.6 right) scaled u; + yy = (1.1 down -- 1.2 up) scaled u; + + path ss; + ss = origin for t=1 upto 360: -- (t, sind(t)) endfor; + ss := ss shifted 360 left & ss; + ss := ss xscaled (pi/180) scaled u; + + drawoptions(dashed withdots scaled 1/4); + draw ((1/4 pi, 0) .. (1/4 pi, sind(45))) scaled u; + draw ((1/2 pi, 0) .. (1/2 pi, sind(90))) scaled u; + draw ((3/4 pi, 0) .. (3/4 pi, sind(135))) scaled u; + drawoptions(); + + draw ss cutbefore yy shifted point 0 of xx + cutafter yy shifted point 1 of xx + withcolor 3/4 blue; + draw ss shifted (-1/2 pi * u ,0) + cutbefore yy shifted point 0 of xx + cutafter yy shifted point 1 of xx + withcolor 2/3 red; + + drawarrow xx; label.rt("$t$", point 1 of xx); + drawarrow yy; label.top("$u(t)$", point 1 of yy); + + for i=-4, -3, -2, -1, 1, 2, 3, 4: + draw (down--up) scaled 2 shifted (pi * i/4 * u, 0); + label.bot(pi_quarters(i), (pi * i/4 * u, -2)); + endfor + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/func-sines.pdf b/info/drawing-with-metapost/src/func-sines.pdf Binary files differnew file mode 100644 index 0000000000..cce783b5bd --- /dev/null +++ b/info/drawing-with-metapost/src/func-sines.pdf diff --git a/info/drawing-with-metapost/src/func-stress.mp b/info/drawing-with-metapost/src/func-stress.mp new file mode 100644 index 0000000000..a22b39a783 --- /dev/null +++ b/info/drawing-with-metapost/src/func-stress.mp @@ -0,0 +1,44 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + z1 = 377 right; z2 = 233 up; + + path ff; + ff = origin .. (72, 144){1,2} .. (84, 144) .. (96, 144){1,1} .. (220, 220){right} .. (370, 160){2,-1.3}; + + for t=2, 4, 4.9: + draw point t of ff -- (xpart point t of ff, y2 + 6) dashed evenly scaled 1/2; + endfor + label.top("Strain hardening", (1/2 (xpart point 2 of ff + xpart point 4 of ff), y2)); + label.top("Necking", (1/2 (xpart point 4 of ff + xpart point 4.9 of ff), y2)); + + path rr; + rr = point 0.4 of ff -- (xpart point 0.8 of ff, ypart point 0.4 of ff) -- point 0.8 of ff; + draw rr; label.bot("Run", point 1/2 of rr); label.rt("Rise", point 3/2 of rr); + + vardef pin_label@#(expr p, a, b)= + draw a -- b cutbefore fullcircle scaled 8 shifted a withpen pencircle scaled 1/4 withcolor 1/2 white; + label@#(p, b); + enddef; + + pin_label.lrt("Yield strength", point 1.2 of ff, point 2 of ff + (8, -18)); + pin_label.bot("Ultimate strength", point 4 of ff, point 4 of ff + (4, -24)); + pin_label.bot("Fracture", point 5 of ff, point 5 of ff + (-8, -18)); + + draw ff withpen pencircle scaled 1 withcolor 2/3 blue; + clip currentpicture to unitsquare scaled 400; % clip thick pen at origin + + drawdblarrow z1 -- origin -- z2; + label.ulft("Strain, $\epsilon$", z1); + label.urt("Stress, $\sigma$", z2); + + label("$\displaystyle\hbox{Young's modulus} = \hbox{Slope} = {\hbox{Rise}\over\hbox{Run}}$", + 1/2 z1 shifted 36 up) withcolor 2/3 blue; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/func-stress.pdf b/info/drawing-with-metapost/src/func-stress.pdf Binary files differnew file mode 100644 index 0000000000..c880df8f0f --- /dev/null +++ b/info/drawing-with-metapost/src/func-stress.pdf diff --git a/info/drawing-with-metapost/src/gaussian.mp b/info/drawing-with-metapost/src/gaussian.mp new file mode 100644 index 0000000000..540256bddb --- /dev/null +++ b/info/drawing-with-metapost/src/gaussian.mp @@ -0,0 +1,69 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +numeric _sqrtpp; +_sqrtpp = 2.50662827463; + +vardef gauss(expr mu, sigma, x) = + if abs(x - mu) < 4 sigma: + mexp(-128 * (((x - mu) / sigma) ** 2)) / _sqrtpp / sigma + else: + 0 + fi + enddef; + +vardef gauss_curve(expr mu, sigma, a, b, s) = + (a, gauss(mu, sigma, a)) for x = a + s step s until b: .. (x, gauss(mu, sigma, x)) endfor + enddef; + +vardef percent(expr n, N) = + save p, s; + numeric p; + string s; + p = 1 - n / N / N; + s = decimal floor(1000 p); + substring (0, length(s) - 1) of s & "." & substring (length(s)-1, infinity) of s & "\thinspace\%" +enddef; + +path Z; Z = gauss_curve(0, 1, -4, 4, 1/8) xscaled 50 yscaled 220; +path A, B; +A = subpath (16,48) of Z -- (xpart point 48 of Z, 0) -- (xpart point 16 of Z, 0) -- cycle; +B = subpath (24,40) of Z -- (xpart point 40 of Z, 0) -- (xpart point 24 of Z, 0) -- cycle; +beginfig(1); + draw Z withcolor Reds 8 5; + numeric N, stack_height[]; N = 100; + numeric plump, fat, gross; gross = plump = fat = 0; + for i=1 upto N: for j = 1 upto N: + numeric r, k; + r = 50 normaldeviate; + k = round(r); + if known stack_height[k]: + stack_height[k] := stack_height[k] + 1; + else: + stack_height[k] := 1; + fi + pair z; z = (r, uniformdeviate 1/4 + stack_height[k]); + color shade; shade = Greens 8 8; + if abs(r) > 50: plump := plump + 1; shade := Blues 8 8; fi + if abs(r) > 100: fat := fat + 1; shade := Oranges 8 6 ; fi + if abs(r) > 150: gross := gross + 1; shade := Reds 8 8; fi + draw z withpen pencircle scaled 1/2 withcolor shade; + undraw z withpen pencircle scaled 1/4; + endfor endfor + label("\textsc{Observed values}", (-180, 96)); + label("$abs(r) \le 1$: " & percent(plump, N), (-180, 80)); + label("$abs(r) \le 2$: " & percent(fat, N), (-180, 68)); + label("$abs(r) \le 3$: " & percent(gross, N), (-180, 56)); + draw (left--right) scaled 220; + for i = -4 upto 4: + draw (50i, 0) -- (50i, -5); + label.bot("\hbox to 5pt{\hss$" & decimal i & "$}", (50i, -5)); + endfor + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/gaussian.pdf b/info/drawing-with-metapost/src/gaussian.pdf Binary files differnew file mode 100644 index 0000000000..1be302a804 --- /dev/null +++ b/info/drawing-with-metapost/src/gaussian.pdf diff --git a/info/drawing-with-metapost/src/geometry-triangles-on-circle.mp b/info/drawing-with-metapost/src/geometry-triangles-on-circle.mp new file mode 100644 index 0000000000..16852ad54e --- /dev/null +++ b/info/drawing-with-metapost/src/geometry-triangles-on-circle.mp @@ -0,0 +1,23 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +vardef equilateral_triangle_point(expr a, b) = + b shifted -a rotated 60 shifted a +enddef; +beginfig(1); +path c; c = fullcircle scaled 144; +numeric n; n = 8; +for i=0 upto n-1: + pair a,b,p,q; + a = point 8/n * i of c; + b = point 8/n * (i + 1) of c; + p = equilateral_triangle_point(a,b); + q = equilateral_triangle_point(b,a); + draw a -- p -- b withcolor .67 green; + draw a -- q -- b withcolor .67 red; +endfor +draw c withcolor .53 blue; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/geometry-triangles-on-circle.pdf b/info/drawing-with-metapost/src/geometry-triangles-on-circle.pdf Binary files differnew file mode 100644 index 0000000000..fce9394170 --- /dev/null +++ b/info/drawing-with-metapost/src/geometry-triangles-on-circle.pdf diff --git a/info/drawing-with-metapost/src/glenshiel.jpg b/info/drawing-with-metapost/src/glenshiel.jpg Binary files differnew file mode 100644 index 0000000000..65c5fae866 --- /dev/null +++ b/info/drawing-with-metapost/src/glenshiel.jpg diff --git a/info/drawing-with-metapost/src/gons.mp b/info/drawing-with-metapost/src/gons.mp new file mode 100644 index 0000000000..417c598690 --- /dev/null +++ b/info/drawing-with-metapost/src/gons.mp @@ -0,0 +1,28 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +vardef gcd(expr a, b) = + if b = 0: a else: gcd(b, a mod b) fi +enddef; +beginfig(1); +for n=2 upto 24: + for s=1 upto floor n/2: + pair p; p = (12n - 24s, -24n); + path gon; gon = ( + for t=0 upto n/gcd(s,n) - 1: + 10 up rotated (360/n * s * t) -- + endfor cycle); + if (n mod s = 0): + fill gon shifted p withcolor PuBuGn[9][1+floor (n/s/6)]; + label("$" & decimal (n/s) & "$", p); + fi + draw gon shifted p withpen pencircle scaled 1/8; + endfor +endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/gons.pdf b/info/drawing-with-metapost/src/gons.pdf Binary files differnew file mode 100644 index 0000000000..acdbb350d8 --- /dev/null +++ b/info/drawing-with-metapost/src/gons.pdf diff --git a/info/drawing-with-metapost/src/graph-paper.mp b/info/drawing-with-metapost/src/graph-paper.mp new file mode 100644 index 0000000000..b4fc2b8f1b --- /dev/null +++ b/info/drawing-with-metapost/src/graph-paper.mp @@ -0,0 +1,34 @@ +\documentclass[border=1mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef grid(expr ll, ur, grid_unit) = + save llx, lly, urx, ury, G; numeric llx, lly, urx, ury; + (llx, lly) = ll; (urx, ury) = ur; + picture G; G = image( + for x = floor(llx / grid_unit) + 1 upto floor(urx / grid_unit): + draw (x * grid_unit, lly) -- (x * grid_unit, ury); + endfor + for y = floor(lly / grid_unit) + 1 upto floor(ury / grid_unit): + draw (llx, y * grid_unit) -- (urx, y * grid_unit); + endfor + fill fullcircle; % <-- you might not want this + ); G enddef; + +input colorbrewer-rgb +beginfig(1); + path C; + C = fullcircle scaled 120 shifted 12 up rotated 6; + for t=0, 1.23, 4: draw center C -- point t of C withcolor Blues 8 8; endfor + draw C withcolor Reds 8 8; + dotlabel.urt("Start", point 0 of C); + + picture P; P = currentpicture; currentpicture := nullpicture; + draw grid(llcorner P, urcorner P, 1mm) withpen pencircle scaled 1/4 withcolor Blues 8 1; + draw grid(llcorner P, urcorner P, 1cm) withpen pencircle scaled 1/4 withcolor Blues 8 2; + draw P; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/graph-paper.pdf b/info/drawing-with-metapost/src/graph-paper.pdf Binary files differnew file mode 100644 index 0000000000..d6cd20458e --- /dev/null +++ b/info/drawing-with-metapost/src/graph-paper.pdf diff --git a/info/drawing-with-metapost/src/greek-default-encoding.mp b/info/drawing-with-metapost/src/greek-default-encoding.mp new file mode 100644 index 0000000000..76bbe6b86c --- /dev/null +++ b/info/drawing-with-metapost/src/greek-default-encoding.mp @@ -0,0 +1,29 @@ +%\documentclass[border=5mm]{standalone} +%\usepackage{luamplib} +%\begin{document} +%\begin{mplibcode} +prologues := 3; +outputtemplate := "%j.%{outputformat}"; +beginfig(1); +string ab, AB; + +ab = ("" for i=11 upto 23: & char i endfor + & "o" for i=24 upto 33: & char i endfor); + +AB = ("AB" & char 0 & char 1 & "EZH" & char 2 & "IK" + & char 3 & "MNO" & char 4 & char 5 & "P" + & char 6 & "T" & char 7 & char 8 & "X" + & char 9 & char 10); + +draw ab infont "cmmi10"; +draw AB infont "cmmi10" shifted 12 down; +draw ab infont "eurm10" shifted 32 down; +draw AB infont "eurm10" shifted 44 down; + +undraw (6 left -- 180 right) shifted 12 up; +undraw (6 left -- 180 right) shifted 50 down; + +endfig; +%\end{mplibcode} +%\end{document} +end diff --git a/info/drawing-with-metapost/src/greek-default-encoding.pdf b/info/drawing-with-metapost/src/greek-default-encoding.pdf Binary files differnew file mode 100644 index 0000000000..a98ad701dc --- /dev/null +++ b/info/drawing-with-metapost/src/greek-default-encoding.pdf diff --git a/info/drawing-with-metapost/src/greek-gfs-encoding.mp b/info/drawing-with-metapost/src/greek-gfs-encoding.mp new file mode 100644 index 0000000000..fbdcdcc513 --- /dev/null +++ b/info/drawing-with-metapost/src/greek-gfs-encoding.mp @@ -0,0 +1,21 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(2); +string ab, AB; +ab = "abgdezhjiklmnoxprstufqyw"; +AB = "ABGDEZHJIKLMNOXPRSTUFQYW"; +y := 0; +for $="grmn1000", "gporsonrg6r", "gneohellenicrg6r": + draw $ infont "cmss10" scaled 0.8 shifted (0,y) withcolor .67 red; + draw ab infont $ shifted (0,y-10); + draw AB infont $ shifted (0,y-22); + y := y - 40; +endfor +undraw (6 left -- 180 right) shifted 12 up; +undraw (6 left -- 180 right) shifted 108 down; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/greek-gfs-encoding.pdf b/info/drawing-with-metapost/src/greek-gfs-encoding.pdf Binary files differnew file mode 100644 index 0000000000..383e3a63df --- /dev/null +++ b/info/drawing-with-metapost/src/greek-gfs-encoding.pdf diff --git a/info/drawing-with-metapost/src/greek-homer.mp b/info/drawing-with-metapost/src/greek-homer.mp new file mode 100644 index 0000000000..0a872fe16e --- /dev/null +++ b/info/drawing-with-metapost/src/greek-homer.mp @@ -0,0 +1,25 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +vardef setg(text t) = + save G; picture G; + save x; numeric x; x = 0; + G = image( + for w = t: + picture g; + g = w infont "gporsonrg6r"; + draw g shifted (x, 0); + x := x + xpart (urcorner g - llcorner g) + 3; + endfor + ); G enddef; +beginfig(1); +picture homer; +homer = setg("m" & char 168 & "nin", char 138 & "eide", "je" & char 128, + "Phlh" & char 240 & char 136 & "deiw", ">Aqil" & char 168 & "oc"); +undraw 10 left -- 272 right; +draw homer scaled 1.732 withcolor .54 red; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/greek-homer.pdf b/info/drawing-with-metapost/src/greek-homer.pdf Binary files differnew file mode 100644 index 0000000000..f455cf1a21 --- /dev/null +++ b/info/drawing-with-metapost/src/greek-homer.pdf diff --git a/info/drawing-with-metapost/src/icosahedron.mp b/info/drawing-with-metapost/src/icosahedron.mp new file mode 100644 index 0000000000..b336628462 --- /dev/null +++ b/info/drawing-with-metapost/src/icosahedron.mp @@ -0,0 +1,53 @@ +\documentclass[border=2mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +vardef thatch(expr a, b, c, penw, gap) = + picture H; + pair v; + v = gap * unitvector(c - 1/2[a, b]); + H = image( + for i=1 upto 120: + draw (a--b) + shifted (i * v) + withpen pencircle scaled penw; + endfor + ); + clip H to a--b--c--cycle; draw H; +enddef; + +beginfig(1); + + path h, t; + + h = for i=0 upto 5: 84 right rotated 60i -- endfor cycle; + t = (origin -- subpath (0, 1) of h -- cycle) + shifted -1/3(point 0 of h + point 1 of h) + scaled 1.08 + rotated 90; + + thatch(point 2 of t, point 0 of t, point 1 of t, 1/4, 3/2); + thatch(point 4 of h, point 0 of t, point 2 of t, 1/4, 5/4); + thatch(point 0 of t, point 0 of h, point 1 of t, 1/4, 3/4); + thatch(point 1 of h, point 1 of t, point 0 of h, 1/4, 1); + thatch(point 4 of h, point 2 of t, point 3 of h, 1/4, 2); + thatch(point 4 of h, point 0 of t, point 5 of h, 1/4, 1); + thatch(point 5 of h, point 0 of t, point 6 of h, 1/4, 3/4); + + draw h; + draw t; + + for i=0 upto 2: + draw point 2i of h -- point i of t; + draw point 2i of h -- point i+1 of t; + draw point 2i+1 of h -- point i+1 of t; + endfor + + input ruler-cm + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/icosahedron.pdf b/info/drawing-with-metapost/src/icosahedron.pdf Binary files differnew file mode 100644 index 0000000000..48a513c8ff --- /dev/null +++ b/info/drawing-with-metapost/src/icosahedron.pdf diff --git a/info/drawing-with-metapost/src/ifs-heigh-open.mp b/info/drawing-with-metapost/src/ifs-heigh-open.mp new file mode 100644 index 0000000000..337bef2e61 --- /dev/null +++ b/info/drawing-with-metapost/src/ifs-heigh-open.mp @@ -0,0 +1,17 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); +path p; p = origin -- dir 30; +numeric n, r; r = 3; +for i=1 upto 12: + n := length p; + p := p rotated (45 - r); + p := p & reverse p rotatedabout(point n of p, 90 + 2r); +endfor +draw p scaled (384 / xpart (urcorner p - llcorner p)) withcolor .54 red; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/ifs-heigh-open.pdf b/info/drawing-with-metapost/src/ifs-heigh-open.pdf Binary files differnew file mode 100644 index 0000000000..5f26760b2e --- /dev/null +++ b/info/drawing-with-metapost/src/ifs-heigh-open.pdf diff --git a/info/drawing-with-metapost/src/ifs-heigh.mp b/info/drawing-with-metapost/src/ifs-heigh.mp new file mode 100644 index 0000000000..6ff5b94df0 --- /dev/null +++ b/info/drawing-with-metapost/src/ifs-heigh.mp @@ -0,0 +1,18 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); +path p; p = origin -- right rotated 30; +numeric n; +for i=1 upto 12: + n := length p; + p := p rotated 45; + p := p & reverse p rotatedabout(point n of p, 90); +endfor +draw p scaled (384 / xpart (urcorner p - llcorner p)) + withcolor (.2, .2, .7); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/ifs-heigh.pdf b/info/drawing-with-metapost/src/ifs-heigh.pdf Binary files differnew file mode 100644 index 0000000000..d779719752 --- /dev/null +++ b/info/drawing-with-metapost/src/ifs-heigh.pdf diff --git a/info/drawing-with-metapost/src/ifs-heighway-stages.mp b/info/drawing-with-metapost/src/ifs-heighway-stages.mp new file mode 100644 index 0000000000..a2647356f8 --- /dev/null +++ b/info/drawing-with-metapost/src/ifs-heighway-stages.mp @@ -0,0 +1,36 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +def width(expr P) = xpart (urcorner P - llcorner P) enddef; +picture s[]; +path p; p = origin -- 16 right; +s1 = image(undraw (origin -- up) scaled 21; draw p; + for t=0, 1: + draw point t of p withpen pencircle scaled 3/2 withcolor 0.74 red; + endfor +); +for i=2 upto 5: + s[i] = image( + p := p rotated 45; + p := p & reverse p rotatedabout(point length p of p, 90); + draw p; + for t=0, 1/2 length p, length p: + draw point t of p withpen pencircle scaled 3/2 withcolor 0.74 red; + endfor + ) shifted (0 for j=i-1 downto 1: + width(s[j]) + 30 endfor, 0); +endfor + +beginfig(1); +draw s1; draw s2; draw s3; draw s4; draw s5; +ahangle := 30; +drawoptions(withcolor (.2,.2,.7)); +for i=1 upto 4: + drawarrow (origin -- 14 right) shifted (xpart lrcorner s[i] + 6, 4); +endfor +drawoptions(); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/ifs-heighway-stages.pdf b/info/drawing-with-metapost/src/ifs-heighway-stages.pdf Binary files differnew file mode 100644 index 0000000000..a58b4bbcfc --- /dev/null +++ b/info/drawing-with-metapost/src/ifs-heighway-stages.pdf diff --git a/info/drawing-with-metapost/src/illusions-cafe-wall.mp b/info/drawing-with-metapost/src/illusions-cafe-wall.mp new file mode 100644 index 0000000000..da81df2963 --- /dev/null +++ b/info/drawing-with-metapost/src/illusions-cafe-wall.mp @@ -0,0 +1,24 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); + numeric u; u = 21; +picture strip; +strip = image( + for i=-10 upto 10: + fill unitsquare shifted (2i,0) scaled u; + draw unitsquare shifted (2i,0) scaled u withcolor 1/2; + endfor + for i=0, 1: + draw ((-20, i) -- (20, i)) scaled u withcolor 1/2; + endfor +); +for i=0 upto 9: + draw strip shifted (4 normaldeviate, i*u); +endfor +clip currentpicture to unitsquare shifted 1/2 left xscaled 29.2 yscaled 10 scaled u; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/illusions-cubes.mp b/info/drawing-with-metapost/src/illusions-cubes.mp new file mode 100644 index 0000000000..2b083debb1 --- /dev/null +++ b/info/drawing-with-metapost/src/illusions-cubes.mp @@ -0,0 +1,46 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +beginfig(2); + numeric u, v; + u = 60; + v = 21 * sqrt(3); + path face[]; + for i=0 upto 2: + face[i] = unitsquare scaled u rotated 45 yscaled (1/sqrt(3)) rotated (120-120i); + endfor + picture cube, sides; + cube = image( + for i=0 upto 2: + fill face[i] withcolor Oranges[8][i+1]; + endfor + for i=0 upto 2: + draw subpath (1, 4) of face[i]; + endfor + ); + sides = image( + for i=0, 2: + fill face[i] withcolor Oranges[8][i+1]; + endfor + for i=0, 2: + draw face[i]; + endfor + ); + draw cube shifted (0u, 2v); + draw cube shifted (0u, 4v); + draw cube shifted (0u, 6v); + draw cube shifted (1u, 5v); + draw cube shifted (2u, 4v); + draw cube shifted (3u, 3v); + draw cube shifted (2u, 2v); + draw cube shifted (u, v); + draw cube; + draw sides shifted (0u, 2v); +endfig; + +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/illusions-fraser-spiral.mp b/info/drawing-with-metapost/src/illusions-fraser-spiral.mp new file mode 100644 index 0000000000..850d957e79 --- /dev/null +++ b/info/drawing-with-metapost/src/illusions-fraser-spiral.mp @@ -0,0 +1,26 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\mplibsetformat{metafun} +\begin{mplibcode} +beginfig(1); + path s; + s = origin for i=1 upto 90: .. (30i, 0) rotated 4i endfor; + + path e[], f[]; + for i = 0 upto 17: + e[i] = s rotated 20i; + f[i] = s reflectedabout(left, right) rotated 20i; + endfor + + for i=0 upto 8: + fill e[2i] -- reverse e[2i+1] .. cycle withcolor 1/2; + endfor + for i=0 upto 8: + fill f[2i] -- reverse f[2i+1] .. cycle withcolor transparent(2, 1/2, 0); + endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/illusions-impossible-ring.mp b/info/drawing-with-metapost/src/illusions-impossible-ring.mp new file mode 100644 index 0000000000..85aaa6492f --- /dev/null +++ b/info/drawing-with-metapost/src/illusions-impossible-ring.mp @@ -0,0 +1,42 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\usepackage{graphicx} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +%label("$\includegraphics{ring-11.png}$", origin); +path S; +S = (-117, 4) {dir 30} .. {right} 30 up .. {dir -36} (117, -4) .. {left} 88 down + .. (-200, -50) .. {up} (-266, 30) {down} -- (-266, -50) + .. (-200, -108) {dir -28} .. 148 down {right} + .. (200, -108) .. {up} (266, -30); +S := S & subpath (5, 2) of S rotated 180 & cycle; +input colorbrewer-rgb + +fill S withcolor Greens 8 3; +fill S rotated 180 withcolor Reds 8 3; + +picture rules; rules = image( + for t=0 step 1/32 until directiontime down of subpath(12, 13) of S: + draw point 12+t of S -- point 12+t of S shifted 100 down + cutafter S; + endfor + for t=3 step 1/16 until 5: + draw point t of S -- point t of S shifted 100 down cutafter subpath (6, 8) of S; + endfor + for t=8+1/16 step 1/16 until 12: + draw point t of S -- 1/2[point 1 of S, point 3 of S] cutafter subpath (1, 3) of S; + endfor +); + +for r = 0, 180: + draw rules rotated r; + draw S rotated r; +endfor + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/illusions-impossible-triangle.mp b/info/drawing-with-metapost/src/illusions-impossible-triangle.mp new file mode 100644 index 0000000000..370a4800af --- /dev/null +++ b/info/drawing-with-metapost/src/illusions-impossible-triangle.mp @@ -0,0 +1,41 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +beginfig(1); + + numeric u; + + u = 42; + + z0 = (0, 3u); + z1 = z0 rotated 120; + z2 = z1 rotated 120; + + z3 = z0 shifted (1/2u, 3/2u * sqrt(3)); + z4 = z3 shifted (-u,0); + + z5-z4 = whatever * (z0-z1); + z6 = whatever [z0, z2]; + y5 = y6 = y2 - sqrt(3) * 1/2 u; + + z7-z3 = whatever * (z0-z1); + z7 = whatever [z1, z2]; + + %dotlabels.top(0, 1, 2, 3, 4, 5, 6, 7); + + path vii; + vii = z3--z4--z5--z6--z2--z7--cycle; + + for i=0 upto 2: + fill vii rotated 120i withcolor Blues[9][i+2]; + endfor + for i=0 upto 2: + draw vii rotated 120i; + endfor + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/illusions-pinna.mp b/info/drawing-with-metapost/src/illusions-pinna.mp new file mode 100644 index 0000000000..070802a5d9 --- /dev/null +++ b/info/drawing-with-metapost/src/illusions-pinna.mp @@ -0,0 +1,23 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + path s; s = unitsquare shifted -(1/2, 1/2) scaled 23; + for n = 1 upto 4: + numeric N; N = 10n + 8; + numeric a; a = 30 normaldeviate; + for t=0 upto N-1: + draw s rotated if odd n: - fi 13 shifted (5N * right) rotated (360 / N * t + a) + withcolor t mod 2; + endfor + endfor + picture P; + P = currentpicture; currentpicture := nullpicture; + fill bbox P withcolor 0.42; + draw P; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/illusions-shaded-diamonds.mp b/info/drawing-with-metapost/src/illusions-shaded-diamonds.mp new file mode 100644 index 0000000000..adb4b003a6 --- /dev/null +++ b/info/drawing-with-metapost/src/illusions-shaded-diamonds.mp @@ -0,0 +1,33 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +path diamond; picture shaded; +diamond = (left -- 3 down -- right -- 3 up -- cycle) scaled 13; +shaded = image( +for i=0 upto 256: + draw (left--right) scaled 15 shifted (0, 78/256 i - 39) + withpen pencircle scaled 0.32 + withcolor (i/256)[white, Greens 8 8]; +endfor); clip shaded to diamond; + + pair u, v; + u = point 2 of diamond - point 0 of diamond; + v = point 3 of diamond - point 0 of diamond; + +beginfig(1); + for i=0 upto 8: + for j=0 upto min(8-i, 4): + draw shaded shifted (i * u + j * v); + endfor + endfor + clip currentpicture to -1/2u -- 9u -- 5u + 4v -- 4v - 1/2u -- cycle; + + draw currentpicture rotated 180; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/illusions-wonky-grid.mp b/info/drawing-with-metapost/src/illusions-wonky-grid.mp new file mode 100644 index 0000000000..74fe8ddf99 --- /dev/null +++ b/info/drawing-with-metapost/src/illusions-wonky-grid.mp @@ -0,0 +1,27 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +input colorbrewer-rgb +beginfig(1); + path o; numeric r; r = 1/4; + o = (for i=0 upto 3: subpath (i+r, i+1-r) of unitsquare shifted -(1/2, 1/2) -- endfor cycle) scaled 42; + picture unit; path s; s = unitsquare scaled arclength subpath (1, 3/2) of o + rotated -45 shifted point 3/2 of o; + unit = image( + fill o withcolor Blues 8 5; + for i=0 upto 3: fill s rotated 90i; endfor + ); + + pair u, v; u = 42 right; v = u rotated 90; numeric n; n = 3; + for i=-n upto n: + for j=-2n upto 2n: + draw unit if (i+j) mod 4 < 2: reflectedabout (dir 135, dir -45) fi + shifted ((2i + j mod 2)*u + j*v); + endfor + endfor + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/incircle.mp b/info/drawing-with-metapost/src/incircle.mp new file mode 100644 index 0000000000..050beebf22 --- /dev/null +++ b/info/drawing-with-metapost/src/incircle.mp @@ -0,0 +1,38 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef through(expr a, b, o) = + save t; numeric t; t = 1+o/abs(a-b); + t[b,a] -- t[a,b] +enddef; + +beginfig(1); + pair A, B, C; + A = origin; B = 377 dir 10; C = 233 dir 70; + pair a, b, m, t; + a = A + 22 unitvector (C-A) + 22 unitvector (B-A); + b = B + 22 unitvector (A-B) + 22 unitvector (C-B); + m = whatever[A,a] = whatever [B,b]; t = whatever[A,B]; + t-m = whatever * (B-A) rotated 90; + + drawoptions(dashed evenly scaled 1/4 withcolor 1/2 white); + draw t -- m; + draw unitsquare scaled 5 rotated angle (B-A) shifted t; + draw through(A, m, 10); + draw through(B, m, 10); + drawoptions(); + draw fullcircle scaled 2 abs (t-m) shifted m withcolor 2/3 red; + draw A--B--C--cycle; + dotlabel.top(btex $m$\thinspace\strut etex, m); + dotlabel.bot(btex $t$ etex, t); + dotlabel.bot(btex $A$ etex, A); + dotlabel.urt(btex $B$ etex, B); + dotlabel.top(btex $C$ etex, C); + dotlabel.top(btex $a$ etex, a); + dotlabel.top(btex $b$ etex, b); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/incircle.pdf b/info/drawing-with-metapost/src/incircle.pdf Binary files differnew file mode 100644 index 0000000000..fe062e227e --- /dev/null +++ b/info/drawing-with-metapost/src/incircle.pdf diff --git a/info/drawing-with-metapost/src/infont-annotated.mp b/info/drawing-with-metapost/src/infont-annotated.mp new file mode 100644 index 0000000000..670c1f12b5 --- /dev/null +++ b/info/drawing-with-metapost/src/infont-annotated.mp @@ -0,0 +1,48 @@ +\documentclass[border=2mm]{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +picture p, pp; +p = "proof" infont "pplri8r" scaled 4; +pp = "proof" infont "pplri8r" scaled 8; +(wd, ht) = urcorner pp; +(xx, dp) = llcorner pp; +xheight = ypart urcorner ("x" infont "pplri8r" scaled 8); + +beginfig(3); +drawoptions( withpen pencircle scaled .5 withcolor .7[.74 red, white]); +draw origin -- (0,ht) -- (wd, ht) -- (wd, dp) -- (0,dp) -- cycle; +for y=0,xheight: + draw (0,y) -- (wd,y); +endfor +draw bbox pp dashed withdots scaled 1/4; +drawoptions(); +draw pp withcolor .5 white; +fill fullcircle scaled 4 shifted center pp withcolor .7[3/4 red,white]; + +vardef east primary p = 1/2[urcorner p, lrcorner p] enddef; +vardef west primary p = 1/2[ulcorner p, llcorner p] enddef; +vardef eastx primary p = lrcorner p shifted (+1,3) enddef; +vardef westx primary p = llcorner p shifted (-1,3) enddef; + +picture c[]; ahangle := 30; string s[]; +c0 = "center" infont "pplr8r" shifted center pp shifted (21,27); +c1 = "urcorner" infont "pplr8r" shifted urcorner pp shifted (13,8); +c2 = "ulcorner" infont "pplr8r" shifted ulcorner pp; c2 := c2 shifted (-13-xpart urcorner c2,8); +c3 = "llcorner" infont "pplr8r" shifted llcorner pp; c3 := c3 shifted (-13-xpart urcorner c3,-13); +c4 = "lrcorner" infont "pplr8r" shifted lrcorner pp shifted (13,-13); +c5 = "bboxmargin" infont "pplr8r" scaled 0.8; +draw c0 withcolor .67 red; drawarrow center c0 .. {dir -120} center pp shifted ( 1, 3) cutbefore bbox c0; +draw c1 withcolor .67 red; drawarrow westx c1 .. {dir -120} urcorner pp shifted ( 1, 1); +draw c2 withcolor .67 red; drawarrow eastx c2 .. {dir -60} ulcorner pp shifted (-1, 1); +draw c3 withcolor .67 red; drawarrow eastx c3 .. {dir +60} llcorner pp shifted (-2,-2); +draw c4 withcolor .67 red; drawarrow westx c4 .. {dir +120} lrcorner pp shifted ( 1,-1); +label.bot(c5, point 1/2 of bbox pp) withcolor 1/4[.67 red, white]; + +c6 = ("x-height" infont "pplr8r"); c6 := c6 shifted (-13-xpart urcorner c6,xheight-2.2); +c7 = ("origin" infont "pplr8r"); c7 := c7 shifted (-13-xpart urcorner c7,-2.2); +draw c6 withcolor .67 blue; drawarrow (-12,xheight) -- (-1,xheight); +draw c7 withcolor .67 blue; drawarrow (-12,0) -- (-1,0); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/infont-annotated.pdf b/info/drawing-with-metapost/src/infont-annotated.pdf Binary files differnew file mode 100644 index 0000000000..acc76a9f1d --- /dev/null +++ b/info/drawing-with-metapost/src/infont-annotated.pdf diff --git a/info/drawing-with-metapost/src/infont-example.mp b/info/drawing-with-metapost/src/infont-example.mp new file mode 100644 index 0000000000..0b2a4d965b --- /dev/null +++ b/info/drawing-with-metapost/src/infont-example.mp @@ -0,0 +1,19 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +interim bboxmargin := 2; +picture p, pp; +p = "proof" infont "pplri8r" scaled 4; +pp = image( +draw (left -- 7 right) scaled 14 withcolor .7 white; +draw p; +draw bbox currentpicture withcolor 3/4[red, white]); +beginfig(1); +draw bbox p withcolor 3/4[red, white]; +draw p; +draw pp shifted 120 right; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/infont-example.pdf b/info/drawing-with-metapost/src/infont-example.pdf Binary files differnew file mode 100644 index 0000000000..1e772668cc --- /dev/null +++ b/info/drawing-with-metapost/src/infont-example.pdf diff --git a/info/drawing-with-metapost/src/intersection-AB-or-BA.mp b/info/drawing-with-metapost/src/intersection-AB-or-BA.mp new file mode 100644 index 0000000000..e7f5e902cc --- /dev/null +++ b/info/drawing-with-metapost/src/intersection-AB-or-BA.mp @@ -0,0 +1,44 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +label(btex \vbox{\halign{\hfil\itshape # \hfil\cr +The paths are drawn with\cr +arrows to show where\cr +they start \& stop.\cr}} etex, origin); + +ahangle := 30; +path A, B; +A = fullcircle scaled 200 shifted 40 left; +B = fullcircle scaled 200 rotated 144 shifted 40 right; + +forsuffixes $=A, B: + drawarrow $ cutafter fullcircle scaled 6 shifted point 0 of $; + for t=0 upto 7: + drawdot point t of $ withpen pencircle scaled 2; + endfor +endfor + +label.ulft("$A$", point 3.14 of A); +label.urt("$B$", point 6.28 of B); + +drawoptions(withcolor 0.54 red); +numeric t, u; (t,u) = A intersectiontimes B; +draw fullcircle scaled 6 shifted point t of A; +z0 = point t of A shifted (-3, 16); +draw z0 -- point t of A cutafter fullcircle scaled 8 shifted point t of A; +label.top("\texttt{A intersectiontimes B}", z0); +drawoptions(withcolor (.2, .2, .7)); +numeric t, u; (t,u) = B intersectiontimes A; +draw fullcircle scaled 6 shifted point t of B; +z1 = point t of B shifted (-3, -16); +draw z1 -- point t of B cutafter fullcircle scaled 8 shifted point t of B; +label.bot("\texttt{B intersectiontimes A}", z1); +drawoptions(); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/intersection-AB-or-BA.pdf b/info/drawing-with-metapost/src/intersection-AB-or-BA.pdf Binary files differnew file mode 100644 index 0000000000..8cf9d43814 --- /dev/null +++ b/info/drawing-with-metapost/src/intersection-AB-or-BA.pdf diff --git a/info/drawing-with-metapost/src/intersection-algorithm.mp b/info/drawing-with-metapost/src/intersection-algorithm.mp new file mode 100644 index 0000000000..2915dc737c --- /dev/null +++ b/info/drawing-with-metapost/src/intersection-algorithm.mp @@ -0,0 +1,93 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +color tr, tb; tr = 0.54 red; tb = (0.2, 0.2, 0.7); +ahangle := 30; +path A, B; +A = subpath (1, 2) of fullcircle scaled 240 shifted 48 left; +B = subpath (7, 8) of fullcircle scaled 240 rotated 144 shifted 48 right; + +picture P[]; +P1 = image( + drawarrow A; label.rt("$A$", point 0 of A); + drawarrow B; label.rt("$B$", point 0 of B); +); +vardef boxed(expr p) = + save d; pair d; d = point infinity of p - point 0 of p; + unitsquare xscaled xpart d yscaled ypart d shifted point 0 of p +enddef; + +P2 = image( + draw A withcolor 3/4; + draw B withcolor 3/4; + draw boxed(A) withcolor tr; + draw boxed(B) withcolor tb; +); + +P3 = image( + draw A withcolor 3/4; + draw B withcolor 3/4; + path s[]; + numeric m; m = arctime 1/2 arclength A of A; + s1 = boxed(subpath (0, m) of A); + s3 = boxed(subpath (m, infinity) of A); + numeric m; m = arctime 1/2 arclength B of B; + s2 = boxed(subpath (0, m) of B); + s4 = boxed(subpath (m, infinity) of B); + draw s3 withcolor 7/8; + draw s4 withcolor 7/8; + draw s2 withcolor tb; + draw s1 withcolor tr; + label.urt("$\scriptstyle 1$", point 1 of s1); + label.ulft("$\scriptstyle 2$", point 0 of s2); + label.urt("$\scriptstyle 3$", point 1 of s3); + label.ulft("$\scriptstyle 4$", point 3 of s4); +); + +draw P1; +draw P2 shifted 120 right; +draw P3 shifted 240 right; + +P4 = image( + draw A withcolor 3/4; + draw B withcolor 3/4; + numeric m[]; + m1 = arctime 1/4 arclength A of A; + m2 = arctime 1/2 arclength A of A; + draw boxed(subpath (0, m1) of A) withcolor 7/8; + draw boxed(subpath (m1, m2) of A) withcolor tr; + numeric m[]; + m1 = arctime 1/4 arclength B of B; + m2 = arctime 1/2 arclength B of B; + draw boxed(subpath (0, m1) of B) withcolor 7/8; + draw boxed(subpath (m1, m2) of B) withcolor tb; +); +P5 = image( + draw A withcolor 3/4; + draw B withcolor 3/4; + numeric m[]; + m1 = arctime 1/4 arclength A of A; + m2 = arctime 3/8 arclength A of A; + m3 = arctime 1/2 arclength A of A; + draw boxed(subpath (m1, m2) of A) withcolor 7/8; + draw boxed(subpath (m2, m3) of A) withcolor tr; + numeric m[]; + m1 = arctime 1/4 arclength B of B; + m2 = arctime 3/8 arclength B of B; + m3 = arctime 1/2 arclength B of B; + draw boxed(subpath (m2, m3) of B) withcolor 7/8; + draw boxed(subpath (m1, m2) of B) withcolor tb; +); + +draw P4 shifted 89 down; +draw P5 shifted 89 down shifted 120 right; +interim bboxmargin := 0; +label.top("and so on\dots", center bbox P5 shifted 89 down shifted 240 right); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/intersection-algorithm.pdf b/info/drawing-with-metapost/src/intersection-algorithm.pdf Binary files differnew file mode 100644 index 0000000000..4d70f7d5e6 --- /dev/null +++ b/info/drawing-with-metapost/src/intersection-algorithm.pdf diff --git a/info/drawing-with-metapost/src/intersection-all-three.mp b/info/drawing-with-metapost/src/intersection-all-three.mp new file mode 100644 index 0000000000..5350b3b20c --- /dev/null +++ b/info/drawing-with-metapost/src/intersection-all-three.mp @@ -0,0 +1,28 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path A, B; +A = (origin {dir 42} .. 120 right {dir 42}) rotated 6; +B = A reflectedabout (left, right) shifted 10 up; +drawarrow A; label.llft("$A$", point 0 of A); +drawarrow B; label.ulft("$B$", point 0 of B); + +pair P[]; numeric n; n=0; +path R; R := A; %take a copy of A +forever: + R := R cutbefore B; % snip where we cross B + exitif length cuttings = 0; % stop if nothing was cut + P[incr n] = point 0 of R; % capture the point + R := subpath (epsilon, infinity) of R; % nudge along +endfor + +draw fullcircle scaled 4 shifted P1 withcolor .54 red; +draw fullcircle scaled 4 shifted P2 withcolor .54 red; +draw fullcircle scaled 4 shifted P3 withcolor .54 red; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/intersection-all-three.pdf b/info/drawing-with-metapost/src/intersection-all-three.pdf Binary files differnew file mode 100644 index 0000000000..2448555380 --- /dev/null +++ b/info/drawing-with-metapost/src/intersection-all-three.pdf diff --git a/info/drawing-with-metapost/src/intersection-only-two.mp b/info/drawing-with-metapost/src/intersection-only-two.mp new file mode 100644 index 0000000000..9ed3a02c93 --- /dev/null +++ b/info/drawing-with-metapost/src/intersection-only-two.mp @@ -0,0 +1,21 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path A, B; +A = (origin {dir 42} .. 120 right {dir 42}) rotated 6; +B = A reflectedabout (left, right) shifted 10 up; +drawarrow A; label.llft("$A$", point 0 of A); +drawarrow B; label.ulft("$B$", point 0 of B); +draw fullcircle scaled 4 + shifted (A intersectionpoint B) + withcolor .54 red; +draw fullcircle scaled 4 + shifted (reverse A intersectionpoint B) + withcolor (.2, .2, .7); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/intersection-only-two.pdf b/info/drawing-with-metapost/src/intersection-only-two.pdf Binary files differnew file mode 100644 index 0000000000..cb3c6d5570 --- /dev/null +++ b/info/drawing-with-metapost/src/intersection-only-two.pdf diff --git a/info/drawing-with-metapost/src/lemniscate-as-function.mp b/info/drawing-with-metapost/src/lemniscate-as-function.mp new file mode 100644 index 0000000000..977e5ced89 --- /dev/null +++ b/info/drawing-with-metapost/src/lemniscate-as-function.mp @@ -0,0 +1,26 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +vardef f(expr x) = sqrt(-1/2 - x * x + sqrt(1/4 + 2 * x * x)) enddef; +numeric s; s = 1/8; +path limb; + +limb = origin for x = s step s until 1: .. (x, f(x)) endfor {down}; +limb := limb scaled 128; + +path lemniscate; +lemniscate = limb + & reverse limb reflectedabout(left, right) + & limb reflectedabout(up, down) + & reverse limb rotated 180 + & cycle; + +draw limb withpen pencircle scaled 2 withcolor 7/8[red, white]; +drawarrow lemniscate withcolor 2/3 blue; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/lemniscate-as-function.pdf b/info/drawing-with-metapost/src/lemniscate-as-function.pdf Binary files differnew file mode 100644 index 0000000000..ef3d5bdfa3 --- /dev/null +++ b/info/drawing-with-metapost/src/lemniscate-as-function.pdf diff --git a/info/drawing-with-metapost/src/line-caps-and-joins.mp b/info/drawing-with-metapost/src/line-caps-and-joins.mp new file mode 100644 index 0000000000..cf52919d6e --- /dev/null +++ b/info/drawing-with-metapost/src/line-caps-and-joins.mp @@ -0,0 +1,56 @@ +\documentclass[border=2mm]{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); +numeric u; u = 32; +z1 = right scaled u; +z2 = z1 rotated 90; +z3 = z2 rotated 90; +z4 = z3 rotated 90; +z5 = 1/2[z1,z2]; +z6 = 1/2[z2,z3]; +z7 = 1/2[z3,z4]; +z8 = 1/2[z4,z1]; + +color tb, tr; +tb = (.2,.2,.7); +tr = 0.54 red; + +drawoptions (withpen pencircle scaled 2); +picture p; +for i=0 upto 2: + for j=0 upto 2: + linecap := i; + linejoin := j; + p := image( + draw unitsquare shifted -(1/2,1/2) scaled 2u withcolor 7/8; + label(decimal linecap, z7) withcolor tb; + label(decimal linejoin, z8) withcolor tr; + drawarrow z1--z3; + draw z2--z4; + drawdot(z5); + drawdot(z6); + drawarrow halfcircle scaled u rotated 180 shifted z4; + ) shifted (5/2u*i, -11/4u*j); + draw p; + if (i=1) and (j=1): draw bbox p dashed withdots scaled 3/4 + withpen pencircle scaled 1/2 + ; fi + endfor + endfor + + label("linecap" infont "cmtt10" scaled 1.2, (5/2u, 7/4u)); + label("butt" infont "cmtt10" , ( 0, 5/4u)) withcolor tb; + label("rounded" infont "cmtt10" , (5/2u, 5/4u)) withcolor tb; + label("squared" infont "cmtt10" , (5/1u, 5/4u)) withcolor tb; + + label("linejoin" infont "cmtt10" scaled 1.2 rotated 90, (-7/4u, -11/4u)); + label("mitered" infont "cmtt10" rotated 90, (-5/4u,0)) withcolor tr; + label("rounded" infont "cmtt10" rotated 90, (-5/4u,-11/4u)) withcolor tr; + label("beveled" infont "cmtt10" rotated 90, (-5/4u,-11/2u)) withcolor tr; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/line-caps-and-joins.pdf b/info/drawing-with-metapost/src/line-caps-and-joins.pdf Binary files differnew file mode 100644 index 0000000000..c65cacdde4 --- /dev/null +++ b/info/drawing-with-metapost/src/line-caps-and-joins.pdf diff --git a/info/drawing-with-metapost/src/linear-graph.mp b/info/drawing-with-metapost/src/linear-graph.mp new file mode 100644 index 0000000000..396c9ccd4c --- /dev/null +++ b/info/drawing-with-metapost/src/linear-graph.mp @@ -0,0 +1,52 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + numeric u, m, m', b, b'; + u = 1.44cm; + b = 3.6u; b' = b + 1/2 u; + m = -1; m' = 3/4 m; + + path xx, yy; + xx = (left -- 5 right) scaled u; + yy = xx rotated 90; + + numeric minx, maxx; path ff, gg; + minx = xpart point 1/16 of xx; + maxx = xpart point 15/16 of xx; + ff = (minx, minx * m + b) -- (maxx, maxx * m + b); + gg = (minx, minx * m' + b') -- (maxx, maxx * m' + b'); + + z0 = point 0.4 of ff; + z1 = point 0.54 of ff; + z1 0 = whatever [point 0 of gg, point 1 of gg]; x1 0 = x0; + z1 1 = whatever [point 0 of gg, point 1 of gg]; x1 1 = x1; + + forsuffixes @=0, 1: + draw (x@, 0) -- z@ -- (0, y@) dashed evenly scaled 3/4; + draw z@ -- z1 @ -- (0, y1 @) dashed withdots scaled 1/2; + label.bot("$x_{" & decimal @ & "}$", (x@, 0)); + label.lft("$y_{" & decimal @ & "}$", (0, y@)); + label.lft("$y'_{" & decimal @ & "}$", (0, y1 @)); + endfor + + draw ff withcolor 2/3 red; + draw gg withcolor 3/4 blue; + drawarrow xx; drawarrow yy; + + label.rt("$x$", point 1 of xx); + label.top("$y$", point 1 of yy); + + dotlabel.urt("$b$", (0, b)); + dotlabel.urt("$b'$", (0, b')); + + draw thelabel("slope: $m=" & decimal m & "$", 7 up) + rotated angle (1, m) shifted point 2/3 of ff; + draw thelabel("slope: $m'=" & decimal m' & "$", 7 up) + rotated angle (1, m') shifted point 2/3 of gg; + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/linear-graph.pdf b/info/drawing-with-metapost/src/linear-graph.pdf Binary files differnew file mode 100644 index 0000000000..5e511e976c --- /dev/null +++ b/info/drawing-with-metapost/src/linear-graph.pdf diff --git a/info/drawing-with-metapost/src/little-hexagon.mp b/info/drawing-with-metapost/src/little-hexagon.mp new file mode 100644 index 0000000000..102117cfb0 --- /dev/null +++ b/info/drawing-with-metapost/src/little-hexagon.mp @@ -0,0 +1,9 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); + draw for i=0 upto 5: 20 dir 60i -- endfor cycle; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/little-hexagon.pdf b/info/drawing-with-metapost/src/little-hexagon.pdf Binary files differnew file mode 100644 index 0000000000..5a3ea39922 --- /dev/null +++ b/info/drawing-with-metapost/src/little-hexagon.pdf diff --git a/info/drawing-with-metapost/src/london-boroughs.mp b/info/drawing-with-metapost/src/london-boroughs.mp new file mode 100644 index 0000000000..a8053b25ac --- /dev/null +++ b/info/drawing-with-metapost/src/london-boroughs.mp @@ -0,0 +1,34 @@ +path Kingston_upon_Thames, Croydon, Bromley, Hounslow, Ealing, Havering, Hillingdon, Harrow, Brent, Barnet, Lambeth, Southwark, Lewisham, Greenwich, Bexley, Enfield, Waltham_Forest, Redbridge, Sutton, Richmond_upon_Thames, Merton, Wandsworth, Hammersmith_and_Fulham, Kensington_and_Chelsea, Westminster, Camden, Tower_Hamlets, Islington, Hackney, Haringey, Newham, Barking_and_Dagenham, City_of_London; +Barking_and_Dagenham = (436.2, 348.3)--(435.8, 347.9)--(435.7, 347.8)--(435.7, 347.6)--(435.7, 347.4)--(435.7, 347.2)--(435.8, 347.0)--(435.8, 346.9)--(435.8, 346.7)--(435.7, 346.6)--(435.6, 346.4)--(435.6, 346.3)--(435.5, 346.1)--(435.5, 345.8)--(435.9, 345.3)--(435.7, 345.1)--(435.5, 344.9)--(435.5, 344.7)--(435.5, 344.5)--(435.7, 344.4)--(435.9, 344.4)--(436.0, 344.3)--(436.1, 344.1)--(436.1, 343.9)--(436.0, 343.4)--(435.9, 343.0)--(435.9, 342.8)--(435.9, 342.7)--(435.9, 342.6)--(435.9, 342.5)--(436.0, 342.4)--(436.1, 342.2)--(436.4, 342.2)--(436.5, 342.3)--(436.6, 342.3)--(436.7, 342.3)--(436.7, 342.2)--(436.8, 342.2)--(436.9, 342.1)--(437.1, 341.7)--(437.1, 341.6)--(437.1, 341.4)--(437.1, 341.2)--(437.2, 340.9)--(437.3, 340.6)--(437.4, 340.5)--(437.6, 340.1)--(437.8, 339.5)--(438.0, 339.1)--(438.2, 338.7)--(438.3, 338.4)--(438.6, 338.0)--(438.7, 337.7)--(438.7, 337.4)--(438.6, 337.2)--(438.8, 336.8)--(438.7, 336.6)--(439.2, 336.1)--(439.3, 336.0)--(439.5, 335.8)--(439.7, 335.5)--(439.8, 335.4)--(440.3, 334.5)--(440.4, 334.5)--(440.5, 334.2)--(440.5, 334.0)--(440.7, 333.1)--(440.6, 332.8)--(440.7, 332.7)--(440.8, 332.5)--(440.8, 332.4)--(440.7, 332.4)--(440.7, 332.5)--(440.6, 332.5)--(440.7, 332.6)--(440.6, 332.6)--(440.5, 332.7)--(440.1, 332.4)--(440.0, 332.5)--(439.8, 332.4)--(439.1, 332.0)--(439.0, 332.0)--(439.1, 332.0)--(438.7, 332.2)--(438.7, 332.7)--(438.6, 333.2)--(438.6, 333.6)--(438.5, 334.0)--(438.4, 334.6)--(438.3, 335.2)--(438.1, 335.8)--(438.1, 335.9)--(437.9, 335.9)--(437.9, 336.0)--(437.7, 336.4)--(437.7, 336.5)--(437.6, 336.8)--(437.0, 338.1)--(436.8, 338.4)--(436.8, 338.5)--(436.5, 339.1)--(436.4, 339.3)--(436.3, 339.4)--(436.3, 339.6)--(436.2, 339.7)--(436.1, 339.7)--(435.9, 339.5)--(435.8, 339.5)--(435.7, 339.5)--(435.6, 339.5)--(435.5, 339.5)--(435.4, 339.5)--(435.3, 339.5)--(435.3, 339.4)--(435.2, 339.5)--(435.3, 339.5)--(435.3, 339.6)--(435.4, 339.8)--(435.4, 340.0)--(435.4, 340.2)--(435.4, 340.4)--(435.8, 340.6)--(435.8, 340.7)--(435.6, 341.0)--(435.5, 341.1)--(435.4, 341.4)--(435.1, 341.9)--(434.9, 342.3)--(434.8, 342.6)--(434.7, 342.7)--(434.6, 342.9)--(434.4, 343.3)--(434.3, 343.5)--(434.3, 343.8)--(434.3, 343.9)--(434.2, 344.0)--(434.2, 344.3)--(434.2, 344.6)--(434.2, 344.8)--(434.2, 345.0)--(434.2, 345.1)--(434.2, 345.2)--(434.2, 345.3)--(434.2, 345.4)--(434.3, 345.5)--(434.3, 345.6)--(434.4, 346.1)--(434.5, 346.5)--(434.6, 346.7)--(434.8, 347.3)--(435.0, 347.7)--(435.1, 348.1)--(435.2, 348.4)--(435.2, 348.5)--(435.2, 348.6)--(435.8, 348.5)--(435.8, 348.4)--(436.0, 348.3)--(435.9, 348.2)--(435.8, 348.0)--(436.6, 348.1)--(437.0, 348.0)--(437.1, 348.0)--(437.2, 348.0)--(437.5, 347.8)--(437.7, 347.8)--(437.7, 348.0)--(437.8, 348.0)--(438.0, 348.0)--(438.3, 348.0)--(438.4, 348.0)--(438.5, 348.0)--(438.6, 348.0)--(438.6, 348.1)--(438.7, 348.1)--(438.8, 348.1)--(439.0, 348.1)--(439.3, 348.1)--(439.5, 348.1)--(439.8, 348.2)--(440.2, 348.3)--(440.4, 348.5)--(440.7, 348.6)--(441.0, 348.9)--(441.1, 349.0)--(441.3, 348.8)--(441.5, 348.9)--(441.6, 349.0)--(441.7, 348.8)--(441.8, 348.6)--(441.9, 348.4)--(442.0, 348.5)--(442.0, 348.6)--(442.1, 348.7)--(442.2, 348.7)--(442.3, 348.8)--(442.5, 348.8)--(443.0, 349.1)--(443.4, 349.3)--(443.8, 349.5)--(444.1, 349.6)--(444.3, 349.7)--(444.4, 349.8)--(444.5, 349.9)--(444.4, 350.0)--(444.4, 350.1)--(444.4, 350.2)--(444.5, 350.2)--(444.6, 350.2)--(444.8, 350.3)--(444.9, 350.4)--(445.0, 350.5)--(445.2, 350.6)--(445.2, 350.7)--(445.3, 350.8)--(445.4, 350.9)--(445.5, 350.9)--(445.6, 351.0)--(446.0, 351.2)--(446.1, 351.3)--(446.2, 351.4)--(446.6, 351.6)--(446.8, 351.8)--(447.2, 352.2)--(447.4, 352.3)--(447.6, 352.4)--(447.8, 352.4)--(448.5, 352.6)--(448.8, 352.7)--(448.9, 352.7)--(449.1, 352.7)--(449.4, 352.8)--(449.8, 352.9)--(450.0, 352.9)--(450.1, 352.9)--(450.2, 353.0)--(450.3, 353.0)--(450.4, 353.0)--(450.5, 353.0)--(450.5, 353.1)--(450.6, 353.2)--(450.7, 353.3)--(451.1, 353.4)--(451.2, 353.4)--(451.2, 353.5)--(451.2, 353.6)--(451.1, 353.8)--(451.2, 354.0)--(451.2, 354.1)--(451.2, 354.2)--(451.3, 354.3)--(451.4, 354.5)--(451.5, 354.7)--(451.9, 354.9)--(451.8, 354.8)--(451.8, 354.5)--(451.7, 354.1)--(451.7, 354.0)--(451.7, 353.8)--(451.7, 353.6)--(451.7, 353.5)--(451.9, 352.8)--(452.1, 352.1)--(452.2, 352.0)--(452.3, 351.6)--(452.4, 351.2)--(452.4, 351.0)--(452.5, 351.0)--(452.5, 350.9)--(452.6, 350.7)--(452.9, 350.9)--(453.1, 351.0)--(453.2, 351.1)--(453.7, 351.4)--(454.1, 351.6)--(454.6, 351.9)--(454.7, 352.1)--(455.1, 352.3)--(455.4, 352.4)--(455.6, 352.6)--(455.7, 352.7)--(455.8, 352.8)--(456.0, 352.9)--(456.3, 353.2)--(456.7, 353.5)--(456.8, 353.6)--(456.9, 353.7)--(457.1, 353.8)--(457.2, 353.9)--(457.5, 354.1)--(458.0, 354.3)--(458.6, 354.6)--(458.8, 354.8)--(458.9, 354.9)--(459.2, 355.0)--(459.5, 355.1)--(459.8, 355.3)--(459.9, 355.3)--(460.1, 355.5)--(460.2, 355.5)--(460.3, 355.6)--(460.5, 355.8)--(460.6, 355.9)--(460.8, 356.1)--(461.0, 356.3)--(461.1, 356.5)--(461.3, 356.7)--(461.5, 357.0)--(461.9, 357.6)--(462.0, 357.7)--(462.1, 357.8)--(462.2, 357.9)--(462.3, 357.9)--(462.5, 357.9)--(462.6, 357.9)--(462.7, 357.9)--(462.7, 358.6)--(463.7, 358.5)--(464.6, 358.4)--(465.2, 358.7)--(465.5, 359.0)--(465.4, 359.6)--(465.5, 359.6)--(465.7, 359.9)--(467.0, 361.7)--(466.7, 362.0)--(466.6, 362.0)--(466.6, 362.2)--(465.6, 363.0)--(465.9, 363.3)--(466.0, 363.9)--(468.7, 363.5)--(468.6, 364.0)--(469.2, 364.0)--(469.3, 364.0)--(469.9, 364.0)--(470.9, 365.4)--(469.8, 365.6)--(469.4, 365.8)--(469.1, 365.9)--(468.8, 366.0)--(469.6, 368.0)--(470.3, 369.7)--(469.7, 370.0)--(469.8, 370.4)--(470.0, 370.5)--(470.1, 370.5)--(470.2, 370.6)--(470.4, 370.6)--(473.7, 371.7)--(473.9, 371.9)--(474.1, 372.1)--(474.2, 372.2)--(474.3, 372.4)--(474.5, 372.6)--(474.8, 372.8)--(475.0, 373.0)--(475.1, 373.2)--(475.3, 373.4)--(475.4, 373.7)--(475.5, 373.8)--(475.6, 374.0)--(475.9, 374.1)--(476.0, 374.2)--(476.2, 374.3)--(476.3, 374.3)--(476.4, 374.3)--(476.8, 374.4)--(476.8, 374.5)--(476.8, 374.6)--(476.9, 374.8)--(477.0, 375.4)--(477.0, 375.8)--(477.0, 375.9)--(477.0, 376.2)--(477.0, 376.3)--(476.9, 376.5)--(476.9, 376.8)--(477.0, 377.0)--(477.1, 377.2)--(477.2, 377.6)--(477.3, 377.9)--(477.4, 378.3)--(477.5, 378.6)--(477.5, 378.7)--(477.4, 378.9)--(477.4, 379.0)--(477.4, 379.1)--(477.3, 379.1)--(477.2, 379.1)--(477.1, 379.2)--(477.1, 379.3)--(477.1, 379.6)--(477.1, 379.7)--(477.1, 379.8)--(477.0, 379.9)--(477.0, 380.0)--(477.6, 380.2)--(478.2, 380.4)--(477.9, 381.2)--(477.8, 381.6)--(477.7, 381.9)--(477.6, 382.0)--(477.7, 382.1)--(477.6, 382.1)--(477.5, 381.9)--(477.1, 381.7)--(477.1, 381.8)--(476.9, 382.2)--(476.8, 382.4)--(476.6, 382.7)--(476.6, 382.9)--(476.6, 383.1)--(476.6, 383.3)--(476.5, 383.3)--(476.5, 383.6)--(476.5, 384.0)--(476.5, 384.5)--(476.4, 384.7)--(476.4, 385.0)--(476.4, 385.3)--(476.8, 385.5)--(478.0, 386.2)--(477.9, 386.5)--(477.9, 386.6)--(477.9, 386.8)--(477.8, 387.0)--(477.7, 387.3)--(477.6, 387.6)--(477.6, 387.9)--(477.6, 388.0)--(477.5, 388.4)--(477.4, 388.4)--(477.3, 388.4)--(477.4, 388.7)--(479.3, 389.5)--(479.2, 389.7)--(479.1, 390.0)--(479.0, 390.2)--(478.9, 390.2)--(478.8, 390.4)--(478.8, 390.5)--(478.7, 390.7)--(478.6, 390.9)--(478.5, 390.9)--(478.3, 390.9)--(478.0, 391.0)--(476.2, 391.1)--(475.2, 391.2)--(475.2, 391.5)--(475.2, 391.8)--(475.3, 391.9)--(475.4, 392.0)--(475.4, 392.6)--(475.4, 392.7)--(475.1, 393.7)--(474.8, 393.7)--(474.9, 394.3)--(474.1, 396.8)--(474.2, 396.9)--(475.6, 397.3)--(476.2, 397.5)--(476.4, 397.5)--(476.5, 397.6)--(476.7, 397.6)--(477.3, 397.8)--(477.0, 398.4)--(476.9, 398.7)--(476.7, 399.1)--(476.6, 399.4)--(476.4, 399.8)--(476.3, 400.1)--(476.2, 400.6)--(476.9, 401.1)--(477.9, 402.0)--(478.7, 402.6)--(480.4, 404.0)--(481.1, 404.6)--(481.5, 404.9)--(481.5, 405.0)--(481.6, 405.0)--(481.7, 405.1)--(482.5, 405.9)--(483.6, 407.1)--(484.1, 407.6)--(484.5, 408.1)--(484.6, 408.1)--(485.2, 408.7)--(487.0, 410.7)--(487.7, 411.4)--(488.4, 411.1)--(488.7, 410.9)--(488.8, 410.9)--(488.8, 410.6)--(488.8, 410.5)--(488.7, 410.0)--(488.7, 409.5)--(488.6, 408.9)--(488.5, 408.5)--(488.8, 408.4)--(488.9, 408.4)--(488.8, 408.0)--(490.0, 408.8)--(490.2, 408.4)--(490.5, 407.9)--(490.8, 407.4)--(491.0, 407.0)--(490.9, 406.2)--(490.8, 405.2)--(490.8, 404.7)--(490.8, 403.8)--(490.7, 402.4)--(490.8, 402.1)--(491.0, 401.5)--(491.1, 401.1)--(491.2, 400.7)--(491.2, 400.4)--(491.2, 400.0)--(491.2, 399.2)--(491.2, 398.4)--(491.2, 397.2)--(491.2, 395.7)--(491.2, 394.5)--(491.2, 394.0)--(491.1, 393.7)--(491.0, 393.2)--(490.9, 393.0)--(490.6, 392.7)--(490.4, 392.4)--(490.0, 391.6)--(489.6, 390.9)--(489.5, 390.6)--(488.9, 390.7)--(488.8, 389.2)--(488.8, 387.5)--(488.8, 387.0)--(488.8, 386.7)--(488.8, 385.7)--(488.7, 385.0)--(489.5, 385.1)--(489.5, 384.9)--(489.6, 384.6)--(489.6, 384.5)--(489.6, 384.3)--(489.5, 383.8)--(489.5, 383.5)--(489.5, 383.0)--(489.5, 382.8)--(489.5, 382.4)--(489.5, 382.2)--(489.5, 382.1)--(490.2, 382.3)--(490.4, 380.9)--(490.5, 380.3)--(490.4, 380.3)--(490.5, 379.2)--(490.5, 379.0)--(490.6, 378.9)--(490.8, 378.8)--(490.8, 378.6)--(490.9, 378.4)--(489.8, 378.2)--(488.7, 378.0)--(488.8, 377.8)--(488.8, 377.5)--(488.8, 377.4)--(488.8, 377.3)--(488.9, 377.3)--(489.1, 377.3)--(489.1, 377.2)--(489.2, 377.2)--(489.2, 377.1)--(490.2, 377.2)--(490.3, 377.0)--(490.6, 376.6)--(490.7, 376.6)--(490.8, 376.6)--(490.9, 376.6)--(491.0, 376.6)--(491.0, 376.5)--(491.1, 376.5)--(491.1, 376.4)--(491.3, 376.3)--(491.5, 376.3)--(491.6, 376.3)--(491.7, 376.3)--(491.8, 376.4)--(491.9, 376.5)--(492.0, 376.6)--(492.1, 376.5)--(492.3, 376.3)--(492.7, 376.0)--(492.9, 375.8)--(493.3, 375.4)--(493.6, 375.0)--(493.9, 374.7)--(494.1, 374.5)--(494.4, 374.2)--(494.5, 374.1)--(494.7, 373.9)--(494.8, 373.8)--(495.0, 373.6)--(495.2, 373.5)--(495.3, 373.5)--(495.5, 373.4)--(495.6, 373.3)--(495.7, 373.2)--(495.8, 373.0)--(496.1, 372.8)--(496.5, 372.4)--(496.7, 372.2)--(496.9, 372.2)--(497.1, 372.0)--(497.3, 371.8)--(497.4, 371.7)--(497.6, 371.6)--(497.8, 371.4)--(498.0, 371.2)--(498.1, 371.0)--(498.2, 370.9)--(498.2, 370.7)--(498.3, 370.6)--(498.5, 370.5)--(498.6, 370.4)--(498.6, 370.3)--(498.7, 370.1)--(498.9, 370.0)--(499.1, 369.9)--(499.2, 369.9)--(499.4, 369.8)--(499.5, 369.7)--(499.5, 369.6)--(499.9, 369.8)--(500.1, 370.0)--(500.3, 370.1)--(500.6, 370.2)--(500.8, 370.4)--(501.2, 370.7)--(501.6, 370.9)--(501.9, 371.1)--(502.1, 371.2)--(502.2, 371.3)--(502.4, 371.4)--(502.6, 371.6)--(502.7, 371.6)--(502.8, 371.7)--(503.1, 371.8)--(503.4, 371.8)--(503.6, 371.8)--(503.7, 371.8)--(503.9, 371.9)--(504.0, 371.9)--(504.1, 372.0)--(504.3, 372.1)--(504.4, 372.1)--(504.5, 372.3)--(504.7, 372.4)--(504.8, 372.5)--(505.0, 372.6)--(505.1, 372.6)--(505.2, 372.7)--(505.3, 372.8)--(505.5, 372.9)--(505.9, 373.2)--(506.0, 373.2)--(506.3, 373.3)--(506.5, 373.4)--(506.8, 373.4)--(507.0, 373.5)--(507.2, 373.6)--(507.4, 373.7)--(507.7, 373.7)--(508.2, 373.6)--(508.8, 373.5)--(508.9, 373.4)--(509.2, 373.4)--(509.4, 373.4)--(509.7, 373.5)--(509.9, 373.5)--(510.2, 373.5)--(510.6, 373.6)--(511.0, 373.7)--(511.3, 373.7)--(511.6, 373.6)--(512.1, 373.6)--(512.2, 373.6)--(512.3, 373.6)--(512.7, 373.5)--(512.9, 373.5)--(513.0, 373.5)--(513.1, 373.6)--(513.2, 373.6)--(513.3, 373.5)--(513.8, 373.5)--(514.1, 373.5)--(514.3, 373.4)--(514.4, 373.4)--(514.5, 373.5)--(514.6, 373.6)--(514.9, 373.9)--(515.5, 374.4)--(515.5, 374.1)--(515.5, 373.9)--(515.5, 373.7)--(515.5, 373.6)--(515.5, 373.5)--(515.3, 373.3)--(515.2, 373.1)--(515.1, 373.0)--(514.9, 372.8)--(514.8, 372.7)--(514.8, 372.6)--(514.7, 372.5)--(514.7, 372.2)--(514.6, 371.7)--(514.5, 371.5)--(514.5, 371.4)--(514.4, 371.3)--(514.4, 371.2)--(514.3, 370.8)--(514.0, 370.1)--(513.9, 369.7)--(514.0, 369.7)--(514.0, 369.6)--(514.1, 369.5)--(514.1, 369.4)--(514.3, 369.3)--(514.5, 369.1)--(514.6, 368.9)--(515.0, 368.7)--(515.3, 368.5)--(515.3, 368.4)--(515.4, 368.3)--(515.4, 368.1)--(515.4, 367.9)--(515.4, 367.8)--(515.4, 367.7)--(515.4, 367.6)--(515.5, 367.5)--(515.5, 367.4)--(515.5, 367.2)--(515.5, 367.0)--(515.4, 366.9)--(515.3, 366.8)--(515.4, 366.7)--(515.4, 366.6)--(515.3, 366.5)--(515.4, 366.4)--(515.3, 366.3)--(515.3, 366.2)--(515.3, 366.0)--(515.3, 365.9)--(515.3, 365.8)--(515.3, 365.7)--(515.3, 365.5)--(515.5, 365.4)--(515.5, 365.3)--(515.6, 365.2)--(515.5, 365.1)--(515.5, 365.0)--(515.6, 364.8)--(515.7, 364.8)--(515.7, 364.7)--(515.7, 364.6)--(515.6, 364.6)--(515.5, 364.6)--(515.5, 364.4)--(515.5, 364.3)--(515.7, 364.3)--(516.0, 364.3)--(516.2, 364.3)--(516.3, 364.3)--(516.4, 364.2)--(516.6, 364.0)--(516.7, 363.8)--(517.0, 363.7)--(517.1, 363.7)--(517.1, 363.6)--(517.1, 363.5)--(517.1, 363.4)--(517.3, 363.3)--(517.4, 363.2)--(517.5, 363.0)--(517.6, 362.9)--(517.6, 362.8)--(517.7, 362.6)--(517.6, 362.6)--(517.6, 362.5)--(517.5, 362.4)--(517.5, 362.3)--(517.6, 362.2)--(517.7, 362.2)--(517.8, 362.2)--(517.9, 362.1)--(517.9, 362.0)--(517.8, 361.9)--(517.8, 361.8)--(517.9, 361.7)--(518.0, 361.7)--(518.2, 361.5)--(518.3, 361.4)--(518.3, 361.3)--(518.4, 361.2)--(518.5, 361.1)--(518.6, 361.1)--(518.7, 361.0)--(518.8, 360.9)--(519.0, 360.8)--(519.1, 360.8)--(519.3, 360.4)--(519.4, 360.4)--(519.4, 360.3)--(519.4, 360.2)--(519.3, 360.1)--(519.3, 360.0)--(519.3, 359.8)--(519.2, 359.6)--(519.1, 359.5)--(519.1, 359.3)--(519.1, 359.1)--(519.1, 359.0)--(519.1, 358.9)--(518.9, 358.9)--(518.8, 358.8)--(518.8, 358.7)--(518.6, 358.5)--(518.3, 358.2)--(518.1, 358.0)--(517.6, 357.5)--(517.6, 357.3)--(517.6, 357.2)--(517.5, 356.9)--(517.5, 356.8)--(517.5, 356.7)--(517.5, 356.6)--(517.4, 356.5)--(517.3, 356.3)--(517.2, 356.0)--(517.2, 355.9)--(517.2, 355.8)--(517.1, 355.7)--(517.1, 355.4)--(517.0, 355.2)--(516.9, 354.9)--(516.8, 354.7)--(516.7, 354.5)--(516.6, 354.2)--(516.6, 354.1)--(516.4, 354.0)--(516.3, 354.0)--(516.1, 353.9)--(516.0, 353.8)--(515.7, 353.6)--(515.5, 353.4)--(515.3, 353.2)--(515.1, 353.1)--(514.8, 352.9)--(514.6, 352.7)--(514.5, 352.6)--(514.4, 352.5)--(514.3, 352.4)--(513.9, 351.9)--(513.8, 351.8)--(513.7, 351.6)--(513.6, 351.4)--(513.5, 351.2)--(513.4, 351.1)--(513.2, 350.9)--(513.0, 350.7)--(512.9, 350.5)--(512.8, 350.3)--(512.7, 350.1)--(512.7, 349.8)--(512.7, 349.7)--(512.6, 349.5)--(512.6, 349.3)--(512.6, 349.2)--(512.5, 349.1)--(512.5, 348.9)--(512.5, 348.8)--(512.4, 348.6)--(512.5, 348.2)--(512.5, 348.1)--(512.5, 348.0)--(512.6, 348.0)--(512.5, 347.9)--(512.5, 347.8)--(512.5, 347.7)--(512.5, 347.6)--(512.5, 347.5)--(512.5, 347.3)--(512.4, 347.2)--(512.4, 346.9)--(512.4, 346.7)--(512.3, 346.6)--(512.3, 346.5)--(512.3, 346.4)--(512.3, 346.3)--(512.2, 346.3)--(512.1, 346.2)--(512.0, 346.1)--(512.0, 346.0)--(511.9, 346.0)--(511.9, 345.9)--(511.7, 345.7)--(511.6, 345.5)--(511.5, 345.4)--(511.4, 345.4)--(511.3, 345.4)--(511.2, 345.4)--(511.0, 345.4)--(510.9, 345.3)--(510.7, 345.2)--(510.6, 345.1)--(510.4, 345.0)--(510.3, 344.9)--(510.2, 344.8)--(509.9, 344.6)--(509.7, 344.4)--(509.4, 344.2)--(509.3, 344.1)--(509.2, 343.9)--(509.1, 343.8)--(508.7, 343.2)--(508.6, 343.0)--(508.3, 342.3)--(508.1, 341.9)--(507.8, 341.3)--(507.7, 341.0)--(507.6, 340.9)--(507.6, 340.8)--(507.6, 340.7)--(507.6, 340.6)--(507.5, 340.4)--(507.5, 340.3)--(507.4, 340.3)--(507.4, 340.1)--(507.4, 340.0)--(507.3, 339.9)--(507.3, 339.7)--(507.4, 339.6)--(507.4, 339.5)--(507.4, 339.4)--(507.3, 339.3)--(507.3, 339.2)--(507.3, 339.0)--(507.2, 338.9)--(507.1, 338.8)--(507.1, 338.7)--(507.0, 338.6)--(507.0, 338.4)--(506.9, 338.3)--(506.9, 338.1)--(506.8, 338.0)--(506.7, 337.6)--(506.6, 337.6)--(506.5, 337.5)--(506.5, 337.4)--(506.4, 337.4)--(506.4, 337.3)--(506.3, 337.2)--(506.2, 337.1)--(506.0, 336.8)--(505.9, 336.7)--(505.8, 336.6)--(505.7, 336.5)--(505.6, 336.3)--(505.4, 336.1)--(505.3, 336.0)--(505.2, 335.9)--(505.2, 335.8)--(505.1, 335.7)--(505.0, 335.6)--(504.9, 335.5)--(504.8, 335.5)--(504.8, 335.4)--(504.7, 335.3)--(504.7, 335.2)--(504.7, 335.0)--(504.6, 334.9)--(504.5, 334.8)--(504.5, 334.7)--(504.5, 334.6)--(504.4, 334.6)--(504.4, 334.5)--(504.3, 334.5)--(504.3, 334.4)--(504.2, 334.4)--(504.1, 334.3)--(504.0, 334.2)--(503.9, 334.1)--(503.9, 334.0)--(503.8, 333.9)--(503.8, 333.8)--(503.8, 333.5)--(503.7, 333.5)--(503.6, 333.4)--(503.6, 333.3)--(503.5, 333.3)--(503.5, 333.2)--(503.4, 333.0)--(503.3, 333.0)--(503.2, 333.0)--(503.1, 332.9)--(503.1, 332.8)--(503.1, 332.6)--(503.0, 332.4)--(502.9, 332.2)--(502.9, 332.0)--(502.8, 331.9)--(502.8, 331.8)--(502.7, 331.7)--(502.6, 331.5)--(502.4, 331.3)--(502.4, 331.2)--(502.4, 331.1)--(502.5, 330.9)--(502.4, 330.7)--(502.2, 330.3)--(502.2, 330.2)--(502.2, 330.1)--(502.2, 330.0)--(502.2, 329.9)--(502.2, 329.8)--(502.2, 329.7)--(502.2, 329.6)--(502.2, 329.5)--(502.1, 329.4)--(502.0, 329.4)--(502.0, 329.3)--(502.0, 329.2)--(502.0, 329.1)--(502.0, 329.0)--(501.9, 328.7)--(501.8, 328.4)--(501.6, 328.0)--(501.5, 327.7)--(501.4, 327.3)--(501.4, 326.8)--(501.3, 326.8)--(501.3, 326.6)--(501.1, 326.0)--(501.1, 325.9)--(501.1, 325.8)--(501.1, 325.7)--(501.1, 325.4)--(501.2, 325.3)--(501.2, 325.2)--(501.1, 325.0)--(501.1, 324.8)--(500.8, 323.8)--(500.8, 323.5)--(500.6, 323.0)--(500.5, 322.6)--(500.5, 322.5)--(500.4, 322.1)--(500.4, 321.9)--(500.3, 321.8)--(500.3, 321.7)--(500.3, 321.6)--(500.2, 321.3)--(500.1, 321.0)--(500.0, 320.7)--(500.0, 320.5)--(499.9, 320.3)--(499.9, 320.1)--(499.8, 320.0)--(499.8, 319.9)--(499.6, 319.4)--(499.6, 319.1)--(499.6, 319.0)--(499.5, 318.9)--(499.5, 318.8)--(499.4, 318.3)--(499.2, 317.2)--(499.2, 317.1)--(499.2, 317.0)--(499.1, 316.7)--(499.1, 316.4)--(499.1, 316.2)--(499.0, 316.1)--(499.1, 316.0)--(499.1, 315.9)--(499.1, 315.7)--(499.1, 315.6)--(499.0, 315.3)--(499.0, 315.0)--(498.9, 314.7)--(498.9, 314.6)--(498.9, 314.5)--(498.9, 314.4)--(498.7, 314.4)--(498.6, 314.3)--(497.9, 314.5)--(497.2, 314.6)--(497.0, 314.6)--(497.0, 314.7)--(496.8, 314.7)--(496.8, 314.6)--(496.6, 314.7)--(496.3, 314.6)--(496.1, 314.7)--(495.8, 314.8)--(495.5, 314.8)--(495.4, 314.8)--(495.2, 314.8)--(494.9, 314.9)--(494.8, 314.9)--(494.6, 314.9)--(494.5, 315.0)--(494.3, 315.0)--(493.8, 315.1)--(493.2, 315.4)--(492.4, 315.7)--(492.3, 315.7)--(492.0, 315.9)--(491.5, 316.0)--(491.1, 316.2)--(490.7, 316.4)--(490.5, 316.5)--(490.4, 316.6)--(490.4, 316.7)--(490.0, 316.7)--(489.5, 317.0)--(489.5, 317.2)--(489.9, 317.0)--(489.9, 317.2)--(489.9, 317.4)--(489.7, 317.6)--(489.5, 317.7)--(489.3, 317.7)--(488.9, 317.9)--(488.7, 317.7)--(488.6, 317.5)--(488.7, 317.4)--(488.6, 317.2)--(488.3, 317.6)--(488.1, 317.8)--(487.7, 318.0)--(487.3, 318.4)--(485.9, 319.4)--(486.0, 319.5)--(485.4, 320.0)--(485.2, 320.2)--(485.0, 320.3)--(484.1, 320.7)--(484.0, 320.8)--(483.5, 321.0)--(483.2, 321.1)--(483.0, 321.2)--(482.0, 321.4)--(481.5, 321.4)--(481.4, 321.5)--(481.3, 321.6)--(481.1, 321.7)--(481.0, 321.7)--(480.9, 321.7)--(480.8, 321.7)--(480.8, 321.6)--(480.7, 321.6)--(480.7, 321.5)--(480.6, 321.4)--(480.5, 321.4)--(480.4, 321.4)--(480.3, 321.4)--(480.2, 321.4)--(480.1, 321.5)--(479.5, 321.7)--(478.9, 321.8)--(478.6, 321.8)--(478.2, 321.9)--(477.8, 321.9)--(476.8, 322.0)--(476.4, 322.0)--(476.2, 322.0)--(476.1, 321.9)--(475.9, 321.9)--(475.6, 321.8)--(474.7, 321.6)--(474.3, 321.5)--(474.2, 321.5)--(474.1, 321.5)--(473.8, 321.5)--(473.6, 321.4)--(473.3, 321.3)--(472.7, 321.0)--(472.3, 320.8)--(472.1, 320.7)--(471.6, 320.5)--(471.3, 320.4)--(471.2, 320.3)--(471.0, 320.2)--(470.9, 320.0)--(470.6, 319.8)--(470.6, 319.7)--(470.5, 319.6)--(470.4, 319.6)--(470.2, 319.4)--(469.9, 319.1)--(469.7, 319.2)--(469.5, 319.0)--(469.3, 318.9)--(469.2, 318.8)--(468.9, 318.6)--(468.7, 318.5)--(468.7, 318.4)--(468.5, 318.3)--(468.0, 318.0)--(468.0, 317.8)--(468.1, 317.7)--(468.1, 317.6)--(467.5, 317.3)--(467.4, 317.4)--(464.4, 315.9)--(464.3, 315.9)--(464.0, 315.8)--(463.9, 315.8)--(463.6, 315.7)--(463.5, 315.7)--(463.2, 315.6)--(462.8, 315.5)--(462.4, 315.5)--(462.1, 315.6)--(461.9, 315.6)--(461.3, 315.6)--(461.2, 315.7)--(460.1, 315.9)--(460.1, 316.0)--(459.6, 316.0)--(459.5, 316.1)--(459.3, 316.1)--(459.2, 316.0)--(459.1, 315.9)--(458.5, 316.1)--(457.7, 316.3)--(457.6, 316.5)--(457.3, 316.7)--(457.1, 316.8)--(456.5, 317.4)--(456.2, 317.6)--(455.9, 317.8)--(455.8, 317.9)--(455.7, 318.0)--(455.6, 318.0)--(455.6, 318.1)--(455.5, 318.1)--(455.5, 318.2)--(455.4, 318.3)--(455.3, 318.3)--(455.3, 318.4)--(455.2, 318.5)--(455.1, 318.5)--(455.0, 318.6)--(455.0, 318.7)--(454.9, 318.9)--(454.8, 319.2)--(454.8, 319.3)--(454.9, 319.4)--(454.8, 319.5)--(454.8, 319.6)--(454.8, 319.7)--(454.8, 319.8)--(454.8, 320.2)--(454.7, 320.4)--(454.7, 320.8)--(454.8, 320.8)--(454.9, 321.0)--(454.8, 321.2)--(454.8, 321.5)--(454.8, 321.6)--(454.9, 321.9)--(455.0, 322.1)--(454.9, 322.4)--(455.0, 322.4)--(454.9, 322.6)--(454.8, 322.6)--(454.7, 322.7)--(454.4, 323.1)--(454.1, 323.5)--(453.6, 324.0)--(453.3, 324.3)--(452.9, 324.9)--(452.8, 325.1)--(452.8, 325.2)--(452.9, 325.4)--(452.9, 325.7)--(452.9, 326.0)--(453.0, 326.3)--(453.0, 326.6)--(453.1, 327.1)--(453.1, 327.2)--(453.0, 327.4)--(453.0, 327.6)--(452.9, 327.9)--(452.8, 328.4)--(452.8, 328.5)--(452.7, 328.6)--(452.7, 328.8)--(452.6, 328.9)--(452.3, 329.2)--(452.2, 329.4)--(452.0, 329.4)--(452.0, 329.3)--(451.9, 328.9)--(451.8, 328.9)--(451.7, 328.8)--(451.4, 328.7)--(451.2, 328.7)--(451.1, 328.8)--(450.8, 329.1)--(450.6, 329.3)--(450.3, 329.4)--(450.0, 329.5)--(449.6, 329.3)--(448.9, 329.0)--(448.8, 329.0)--(448.7, 328.9)--(448.6, 328.9)--(448.5, 328.8)--(448.4, 328.7)--(448.3, 328.7)--(448.2, 328.7)--(448.1, 328.7)--(448.1, 328.6)--(447.9, 328.5)--(447.8, 328.6)--(447.7, 328.7)--(447.6, 328.7)--(447.5, 328.7)--(447.5, 328.8)--(447.4, 328.8)--(447.2, 328.9)--(447.1, 328.9)--(447.0, 329.0)--(447.0, 329.1)--(446.6, 329.4)--(446.4, 329.6)--(446.0, 329.8)--(445.8, 329.9)--(444.7, 330.3)--(444.5, 330.4)--(444.3, 330.5)--(444.1, 330.6)--(443.6, 331.2)--(443.4, 331.3)--(443.3, 331.5)--(443.0, 331.9)--(442.7, 332.2)--(442.5, 332.5)--(442.0, 332.5)--(441.9, 332.5)--(441.8, 332.6)--(441.6, 332.5)--(441.4, 332.5)--(441.3, 332.6)--(441.2, 332.7)--(441.1, 332.9)--(441.0, 333.4)--(440.9, 334.3)--(440.7, 334.6)--(440.6, 334.9)--(440.3, 335.4)--(440.2, 335.4)--(440.0, 335.7)--(439.5, 336.3)--(439.6, 336.7)--(439.8, 337.0)--(439.7, 337.1)--(439.5, 337.4)--(439.2, 337.2)--(439.2, 337.3)--(439.1, 337.4)--(439.0, 337.5)--(439.0, 337.7)--(438.9, 337.8)--(439.0, 337.9)--(438.9, 338.1)--(438.7, 338.3)--(438.5, 338.7)--(438.1, 339.4)--(438.0, 339.7)--(437.9, 340.1)--(437.6, 340.6)--(437.5, 340.7)--(437.5, 340.8)--(437.4, 341.0)--(437.4, 341.5)--(437.3, 341.7)--(437.2, 342.0)--(437.0, 342.3)--(436.9, 342.5)--(436.9, 342.7)--(436.8, 342.8)--(436.8, 342.7)--(436.8, 342.5)--(436.7, 342.5)--(436.5, 342.5)--(436.3, 342.4)--(436.2, 342.5)--(436.2, 342.7)--(436.2, 342.8)--(436.2, 343.1)--(436.2, 343.2)--(436.2, 343.4)--(436.3, 343.9)--(436.3, 344.1)--(436.4, 344.2)--(436.4, 344.4)--(436.4, 344.8)--(435.9, 345.6)--(435.8, 345.8)--(435.8, 345.9)--(435.7, 346.0)--(435.8, 346.2)--(435.9, 346.4)--(436.0, 346.6)--(436.1, 347.1)--(436.0, 347.3)--(436.0, 347.7)--(436.0, 347.9)--(436.2, 348.2)--cycle; +Barnet = (245.8, 483.6)--(245.9, 483.2)--(246.1, 482.7)--(246.2, 482.3)--(246.3, 482.0)--(246.4, 481.7)--(246.5, 481.4)--(246.6, 481.4)--(246.7, 481.4)--(247.6, 481.4)--(247.9, 481.4)--(248.1, 481.4)--(248.4, 481.8)--(248.8, 481.7)--(249.3, 481.6)--(249.0, 480.8)--(249.0, 480.7)--(248.8, 480.2)--(248.5, 479.4)--(248.2, 478.4)--(248.1, 478.4)--(248.8, 478.2)--(249.2, 478.1)--(250.1, 477.9)--(251.0, 477.6)--(251.2, 477.6)--(251.5, 477.4)--(251.7, 477.2)--(251.9, 477.0)--(252.1, 476.7)--(252.1, 476.8)--(252.5, 477.7)--(253.2, 478.9)--(253.1, 478.9)--(253.4, 479.6)--(254.2, 481.2)--(255.6, 481.9)--(256.2, 482.0)--(256.2, 481.9)--(258.1, 482.1)--(257.4, 480.2)--(256.9, 478.9)--(256.1, 476.8)--(255.8, 476.0)--(255.7, 475.6)--(257.1, 475.2)--(257.2, 475.2)--(258.0, 474.9)--(258.2, 474.8)--(259.4, 474.4)--(260.5, 474.1)--(262.2, 473.5)--(262.9, 473.3)--(265.0, 472.6)--(270.2, 470.9)--(271.3, 470.8)--(271.5, 470.7)--(271.8, 470.6)--(272.8, 469.8)--(273.1, 469.6)--(273.4, 469.3)--(275.0, 468.8)--(275.2, 468.9)--(275.3, 468.7)--(276.9, 468.2)--(277.3, 468.1)--(277.4, 468.1)--(277.5, 467.7)--(277.5, 467.5)--(277.7, 467.4)--(277.8, 467.3)--(278.1, 467.1)--(278.2, 466.9)--(278.3, 466.4)--(278.3, 466.0)--(278.3, 465.7)--(278.4, 465.6)--(278.1, 465.5)--(278.0, 465.5)--(277.9, 465.2)--(278.0, 464.8)--(278.2, 464.9)--(278.4, 464.9)--(278.4, 464.8)--(278.5, 464.7)--(278.7, 464.2)--(278.8, 464.0)--(279.1, 463.6)--(279.1, 463.5)--(279.1, 463.4)--(279.2, 462.8)--(279.2, 462.3)--(279.2, 462.1)--(279.3, 462.0)--(279.3, 461.8)--(279.6, 461.9)--(280.2, 462.1)--(281.0, 460.6)--(279.7, 459.9)--(279.4, 459.7)--(279.3, 459.3)--(280.0, 459.0)--(280.2, 458.9)--(280.1, 458.6)--(280.0, 458.5)--(280.1, 458.4)--(279.5, 457.3)--(279.9, 457.1)--(279.8, 456.8)--(280.1, 456.7)--(280.2, 456.7)--(280.4, 456.7)--(280.8, 456.7)--(281.5, 456.8)--(281.6, 456.7)--(281.6, 456.5)--(281.6, 456.3)--(281.6, 456.1)--(281.6, 456.0)--(281.7, 455.5)--(281.8, 455.2)--(282.0, 454.3)--(282.1, 454.2)--(282.4, 453.6)--(282.5, 453.2)--(284.5, 453.9)--(285.3, 452.1)--(286.0, 451.2)--(286.3, 450.8)--(286.6, 450.3)--(286.5, 450.2)--(286.6, 450.0)--(286.7, 449.9)--(286.9, 449.7)--(287.2, 449.3)--(287.4, 449.0)--(287.6, 448.8)--(287.8, 448.5)--(288.2, 448.1)--(289.2, 447.2)--(289.3, 447.1)--(289.5, 447.0)--(289.8, 446.8)--(290.8, 446.1)--(291.3, 445.8)--(291.5, 445.7)--(291.9, 445.4)--(292.0, 445.2)--(292.1, 445.1)--(292.2, 445.0)--(292.2, 444.9)--(292.0, 444.7)--(292.2, 444.6)--(292.1, 444.4)--(292.2, 444.3)--(292.4, 444.5)--(292.4, 444.4)--(292.5, 444.4)--(292.6, 444.2)--(292.5, 444.1)--(292.7, 443.9)--(292.9, 443.7)--(293.1, 443.5)--(293.4, 443.4)--(293.8, 443.2)--(294.3, 442.9)--(294.6, 442.9)--(295.0, 442.8)--(295.5, 442.7)--(295.8, 442.6)--(295.7, 442.1)--(295.5, 441.7)--(295.4, 441.2)--(295.1, 441.2)--(294.9, 440.9)--(294.8, 440.5)--(294.7, 440.2)--(294.5, 439.7)--(294.8, 439.5)--(295.0, 439.4)--(294.9, 439.3)--(294.7, 439.3)--(294.5, 439.0)--(294.5, 438.9)--(294.5, 438.8)--(294.5, 438.7)--(294.4, 438.5)--(294.4, 438.4)--(294.3, 438.2)--(294.1, 437.9)--(294.0, 437.7)--(293.9, 437.6)--(293.8, 437.4)--(293.8, 437.2)--(293.7, 437.0)--(293.6, 436.8)--(293.5, 436.7)--(293.4, 436.5)--(293.2, 436.4)--(293.2, 436.3)--(293.1, 436.3)--(293.1, 436.2)--(293.0, 436.2)--(293.0, 436.1)--(293.0, 436.0)--(292.9, 436.0)--(292.9, 435.9)--(292.9, 435.6)--(292.8, 435.1)--(292.6, 434.4)--(292.6, 434.2)--(292.6, 434.0)--(292.5, 434.0)--(292.5, 433.9)--(292.4, 433.9)--(292.4, 433.8)--(292.4, 433.7)--(292.5, 433.6)--(292.5, 433.5)--(292.5, 433.3)--(292.5, 433.2)--(292.3, 432.8)--(292.3, 432.7)--(292.2, 432.3)--(292.1, 432.0)--(292.0, 431.8)--(292.0, 431.5)--(291.8, 431.4)--(291.7, 431.5)--(291.6, 431.5)--(291.7, 430.8)--(291.3, 430.2)--(290.9, 429.8)--(290.5, 429.4)--(290.3, 429.1)--(289.8, 428.4)--(289.7, 428.3)--(289.1, 427.5)--(289.0, 427.4)--(288.8, 427.2)--(288.9, 427.1)--(288.7, 427.0)--(288.3, 426.3)--(288.2, 426.2)--(288.1, 426.2)--(288.0, 426.2)--(287.8, 426.0)--(287.8, 425.9)--(287.8, 425.8)--(287.7, 425.7)--(287.6, 425.6)--(287.6, 425.5)--(287.5, 425.4)--(287.5, 425.2)--(287.4, 425.0)--(287.4, 424.8)--(287.3, 424.8)--(287.3, 424.7)--(287.3, 424.6)--(287.3, 424.5)--(286.9, 424.3)--(286.8, 424.3)--(286.6, 424.1)--(286.2, 423.9)--(286.0, 423.9)--(285.9, 423.8)--(285.6, 423.7)--(286.1, 422.9)--(286.8, 422.0)--(286.8, 421.9)--(287.4, 421.0)--(287.9, 420.3)--(288.3, 419.7)--(288.7, 419.2)--(289.2, 418.6)--(289.8, 417.9)--(288.1, 417.1)--(287.0, 416.6)--(286.6, 416.3)--(286.6, 415.7)--(286.6, 415.3)--(286.6, 415.2)--(286.6, 415.1)--(286.7, 414.7)--(286.8, 413.8)--(286.9, 413.6)--(287.0, 413.1)--(287.0, 413.0)--(287.1, 412.9)--(287.2, 412.6)--(287.4, 412.2)--(287.5, 412.1)--(287.5, 412.0)--(287.6, 411.6)--(287.7, 411.3)--(287.7, 410.7)--(287.7, 410.6)--(287.6, 410.4)--(287.6, 410.2)--(287.6, 410.1)--(287.4, 409.8)--(287.0, 409.3)--(286.9, 409.2)--(286.9, 409.0)--(286.8, 408.7)--(286.7, 408.5)--(286.5, 408.2)--(286.5, 408.0)--(286.4, 407.9)--(286.3, 407.7)--(286.1, 407.1)--(286.1, 406.9)--(286.1, 406.7)--(286.1, 406.6)--(285.7, 406.4)--(284.6, 405.7)--(284.4, 405.5)--(284.0, 405.3)--(284.1, 405.2)--(283.9, 405.0)--(283.2, 404.6)--(281.4, 403.5)--(281.3, 403.7)--(281.2, 403.8)--(280.9, 404.4)--(280.3, 405.3)--(280.2, 405.5)--(280.1, 405.8)--(280.1, 406.1)--(279.9, 407.1)--(279.9, 407.3)--(279.8, 407.4)--(279.9, 407.8)--(279.9, 408.2)--(279.9, 408.6)--(279.9, 409.3)--(279.7, 409.7)--(279.4, 410.1)--(279.1, 410.7)--(279.0, 410.8)--(278.9, 410.9)--(278.8, 411.0)--(278.6, 411.4)--(278.5, 411.4)--(278.4, 411.5)--(277.9, 411.9)--(277.9, 412.0)--(277.9, 412.1)--(277.9, 412.2)--(278.0, 412.3)--(277.7, 412.2)--(277.6, 412.0)--(277.4, 411.7)--(277.1, 410.9)--(276.2, 408.6)--(276.2, 408.5)--(275.4, 406.5)--(274.2, 403.6)--(274.2, 403.5)--(274.2, 403.4)--(274.2, 403.3)--(274.2, 403.0)--(274.2, 402.9)--(274.3, 402.7)--(274.4, 402.1)--(274.7, 401.0)--(274.8, 400.8)--(274.8, 400.4)--(275.0, 400.0)--(275.2, 399.8)--(275.2, 399.6)--(275.3, 399.4)--(275.4, 399.2)--(275.4, 399.1)--(275.4, 398.9)--(275.7, 398.2)--(275.7, 398.0)--(275.7, 397.9)--(275.8, 397.6)--(276.4, 397.8)--(276.4, 397.6)--(276.5, 397.4)--(276.5, 397.3)--(276.4, 397.2)--(276.2, 397.0)--(276.0, 396.9)--(275.9, 396.9)--(275.9, 396.2)--(275.8, 395.7)--(275.8, 395.5)--(275.7, 395.3)--(275.7, 395.2)--(275.8, 394.8)--(275.5, 394.7)--(275.5, 394.4)--(275.6, 394.2)--(275.6, 394.1)--(275.6, 394.0)--(275.6, 393.6)--(275.6, 393.4)--(275.6, 393.1)--(275.6, 393.0)--(275.5, 392.8)--(276.6, 392.9)--(276.6, 392.7)--(276.7, 392.4)--(276.6, 392.4)--(276.6, 392.2)--(276.6, 392.0)--(276.6, 391.4)--(277.3, 391.3)--(277.5, 391.3)--(277.6, 391.1)--(277.7, 390.9)--(277.6, 390.9)--(277.2, 390.9)--(277.2, 390.4)--(276.3, 389.3)--(276.1, 389.0)--(276.0, 388.9)--(275.9, 389.1)--(275.7, 389.2)--(275.6, 389.3)--(275.4, 389.5)--(275.2, 389.7)--(275.0, 389.9)--(274.9, 389.9)--(274.8, 389.8)--(274.4, 389.3)--(274.4, 389.2)--(274.2, 389.1)--(274.0, 388.9)--(273.9, 388.9)--(273.6, 388.8)--(273.3, 388.6)--(273.3, 388.7)--(273.2, 388.7)--(272.8, 388.3)--(272.7, 388.3)--(272.6, 388.3)--(272.4, 388.1)--(272.3, 387.9)--(272.4, 387.8)--(272.2, 387.7)--(271.8, 387.5)--(271.8, 387.1)--(271.6, 386.6)--(271.4, 386.3)--(271.5, 386.1)--(271.1, 386.0)--(271.0, 386.0)--(271.0, 385.9)--(271.0, 385.8)--(270.9, 385.6)--(270.9, 385.3)--(270.9, 385.1)--(270.9, 384.6)--(270.9, 384.5)--(271.0, 384.5)--(271.0, 383.5)--(271.1, 383.5)--(271.2, 382.5)--(271.2, 382.4)--(271.1, 382.2)--(271.0, 382.1)--(270.9, 381.9)--(271.0, 381.9)--(271.1, 381.9)--(270.9, 381.5)--(270.8, 381.4)--(270.7, 381.0)--(270.6, 380.9)--(270.4, 380.4)--(270.3, 380.5)--(269.5, 380.9)--(269.4, 380.6)--(269.4, 380.4)--(269.3, 380.2)--(269.3, 380.0)--(269.3, 379.8)--(269.4, 379.7)--(269.4, 379.1)--(269.4, 378.7)--(269.5, 378.1)--(269.5, 377.9)--(269.5, 377.7)--(269.5, 377.4)--(269.5, 377.1)--(269.2, 376.9)--(268.8, 376.8)--(268.5, 376.7)--(268.3, 376.7)--(268.3, 376.0)--(268.3, 375.4)--(268.3, 375.3)--(268.0, 375.1)--(267.9, 375.0)--(267.9, 374.9)--(267.8, 374.8)--(267.6, 374.8)--(267.3, 374.5)--(267.1, 374.3)--(267.0, 374.2)--(267.0, 374.1)--(266.9, 374.0)--(266.9, 373.9)--(266.8, 373.9)--(266.8, 373.8)--(266.8, 373.7)--(266.7, 373.5)--(266.7, 373.1)--(266.7, 372.9)--(266.7, 372.5)--(266.6, 372.5)--(266.6, 372.3)--(266.5, 372.2)--(266.4, 372.1)--(266.3, 371.9)--(266.3, 371.8)--(266.2, 371.8)--(265.9, 371.9)--(265.7, 372.0)--(265.7, 372.1)--(265.2, 372.3)--(265.2, 372.4)--(265.1, 372.5)--(264.7, 372.6)--(264.4, 372.8)--(264.1, 372.8)--(263.9, 372.8)--(263.7, 373.0)--(263.6, 372.8)--(263.6, 372.7)--(263.6, 372.6)--(263.5, 372.6)--(263.5, 372.5)--(263.4, 372.5)--(263.3, 372.5)--(263.2, 372.5)--(263.1, 372.4)--(263.0, 372.4)--(262.5, 372.3)--(262.2, 372.2)--(262.0, 372.1)--(261.8, 372.1)--(261.7, 372.1)--(261.8, 371.9)--(261.7, 371.8)--(261.2, 371.4)--(261.1, 371.3)--(261.0, 371.2)--(260.9, 371.0)--(260.8, 371.0)--(260.0, 370.2)--(259.9, 370.2)--(259.8, 370.2)--(260.0, 370.0)--(259.8, 369.7)--(259.6, 369.6)--(259.5, 369.5)--(259.1, 369.4)--(258.8, 369.3)--(258.5, 369.1)--(258.2, 369.0)--(258.1, 369.0)--(258.1, 368.9)--(258.0, 368.9)--(258.0, 368.8)--(257.9, 368.8)--(257.8, 368.7)--(257.7, 368.6)--(257.6, 368.5)--(257.5, 368.2)--(257.4, 368.1)--(257.3, 368.1)--(257.3, 368.0)--(257.2, 368.0)--(257.1, 368.0)--(257.0, 368.0)--(257.0, 367.9)--(256.8, 367.8)--(256.6, 367.7)--(256.4, 367.4)--(256.1, 367.4)--(256.0, 367.4)--(256.0, 367.3)--(255.9, 367.2)--(255.9, 367.1)--(255.8, 367.1)--(255.6, 366.8)--(255.6, 366.7)--(255.6, 366.6)--(255.5, 366.4)--(255.5, 366.2)--(255.5, 365.6)--(255.5, 365.4)--(255.5, 365.1)--(255.3, 364.3)--(255.3, 364.1)--(255.3, 363.8)--(255.2, 363.5)--(255.2, 363.3)--(255.1, 363.2)--(255.0, 363.1)--(254.8, 363.0)--(254.7, 362.9)--(254.4, 362.9)--(254.0, 362.9)--(253.6, 362.5)--(253.1, 362.2)--(253.0, 362.1)--(252.9, 362.2)--(252.8, 362.1)--(252.7, 362.1)--(252.6, 362.0)--(252.5, 362.0)--(252.5, 361.9)--(252.5, 361.8)--(252.3, 361.7)--(252.1, 361.6)--(252.0, 361.7)--(251.8, 361.7)--(251.5, 361.8)--(251.4, 361.8)--(251.3, 361.8)--(251.1, 361.9)--(251.0, 361.8)--(251.0, 361.9)--(250.9, 361.8)--(250.9, 361.5)--(250.9, 361.1)--(250.9, 361.0)--(250.9, 360.9)--(251.0, 360.6)--(251.0, 360.4)--(251.1, 360.2)--(251.0, 360.2)--(250.8, 360.1)--(250.6, 360.0)--(250.5, 359.9)--(250.4, 359.8)--(250.2, 360.1)--(250.0, 359.9)--(249.8, 359.7)--(249.8, 359.3)--(249.8, 359.1)--(249.8, 359.0)--(249.8, 358.9)--(249.7, 358.9)--(249.6, 358.7)--(249.6, 358.6)--(249.5, 358.6)--(249.4, 358.5)--(249.2, 358.0)--(249.1, 357.8)--(249.1, 357.7)--(249.1, 357.6)--(249.1, 357.5)--(249.1, 357.3)--(249.2, 357.2)--(249.2, 357.1)--(249.2, 357.0)--(249.2, 356.9)--(249.3, 356.8)--(248.9, 356.8)--(248.7, 356.8)--(248.5, 356.7)--(248.2, 356.6)--(247.9, 356.5)--(247.7, 356.5)--(247.6, 356.5)--(246.9, 356.3)--(246.5, 356.2)--(246.3, 356.1)--(246.1, 356.1)--(246.0, 356.1)--(245.9, 356.0)--(245.6, 355.9)--(245.4, 355.8)--(245.3, 355.8)--(245.2, 355.8)--(244.9, 355.7)--(244.8, 355.7)--(244.8, 355.9)--(244.8, 356.0)--(244.8, 356.4)--(244.8, 356.6)--(244.8, 356.7)--(244.7, 357.0)--(244.6, 356.9)--(244.5, 356.9)--(244.3, 356.9)--(244.0, 356.7)--(243.6, 356.6)--(243.3, 356.4)--(243.1, 356.3)--(242.7, 356.1)--(242.4, 356.5)--(242.2, 356.8)--(241.9, 357.3)--(241.3, 356.9)--(241.0, 356.8)--(240.7, 356.6)--(240.4, 356.3)--(240.2, 356.0)--(239.9, 355.7)--(239.7, 355.6)--(239.5, 355.5)--(239.3, 355.7)--(239.0, 356.2)--(238.5, 356.7)--(238.2, 357.2)--(238.1, 357.3)--(237.6, 357.9)--(237.4, 358.1)--(237.2, 358.3)--(237.1, 358.5)--(236.9, 358.7)--(236.5, 359.3)--(236.1, 359.8)--(235.8, 360.1)--(235.6, 360.5)--(235.3, 360.8)--(234.6, 362.1)--(234.4, 362.5)--(234.0, 363.2)--(233.7, 363.5)--(233.4, 363.9)--(233.1, 364.3)--(232.9, 364.6)--(232.4, 365.2)--(231.9, 365.9)--(231.5, 366.4)--(231.2, 366.7)--(231.1, 366.8)--(230.9, 367.1)--(230.8, 367.3)--(230.6, 367.5)--(230.2, 367.9)--(230.0, 368.1)--(229.9, 368.3)--(229.7, 368.4)--(229.3, 368.8)--(229.1, 368.9)--(228.9, 369.1)--(228.7, 369.3)--(228.6, 369.4)--(228.5, 369.5)--(228.1, 370.0)--(227.8, 370.5)--(227.6, 370.8)--(227.3, 371.1)--(227.0, 371.4)--(226.8, 371.8)--(226.4, 372.3)--(226.1, 372.7)--(225.2, 373.8)--(225.0, 373.7)--(224.9, 373.6)--(224.8, 373.5)--(224.7, 373.6)--(224.6, 373.6)--(224.5, 373.6)--(224.5, 373.5)--(224.4, 373.6)--(224.4, 373.8)--(224.2, 373.9)--(224.1, 374.0)--(224.0, 374.1)--(223.9, 374.1)--(223.8, 374.1)--(223.7, 374.1)--(223.7, 374.2)--(223.9, 374.2)--(223.9, 374.3)--(223.9, 374.4)--(223.7, 374.4)--(223.6, 374.4)--(223.5, 374.3)--(223.6, 374.2)--(223.6, 374.1)--(223.5, 374.1)--(223.4, 374.2)--(223.2, 374.2)--(223.1, 374.3)--(223.0, 374.3)--(222.7, 374.2)--(222.6, 374.2)--(222.5, 374.2)--(222.4, 374.3)--(222.3, 374.3)--(222.3, 374.2)--(222.2, 374.2)--(222.2, 374.1)--(222.3, 374.1)--(222.3, 374.0)--(222.3, 373.9)--(222.2, 373.8)--(222.1, 373.9)--(222.0, 373.9)--(222.0, 373.8)--(222.0, 373.7)--(222.2, 373.6)--(222.2, 373.5)--(222.1, 373.5)--(222.0, 373.5)--(221.8, 373.6)--(221.8, 373.7)--(221.7, 373.7)--(221.6, 373.9)--(221.5, 374.0)--(221.3, 374.0)--(221.2, 373.8)--(221.1, 373.8)--(220.9, 373.8)--(220.8, 373.8)--(220.7, 373.8)--(220.6, 373.8)--(220.6, 373.9)--(220.7, 374.0)--(220.8, 374.1)--(220.7, 374.1)--(220.6, 374.1)--(220.5, 374.1)--(220.4, 374.0)--(220.3, 373.9)--(220.2, 373.9)--(220.3, 374.0)--(220.2, 374.1)--(220.1, 374.1)--(220.1, 374.0)--(220.0, 373.8)--(219.9, 373.8)--(219.9, 373.9)--(219.8, 373.9)--(219.8, 374.0)--(219.7, 374.1)--(219.7, 374.2)--(219.7, 374.3)--(219.7, 374.4)--(219.6, 374.5)--(219.5, 374.5)--(219.3, 374.6)--(219.2, 374.6)--(219.1, 374.5)--(219.0, 374.6)--(218.9, 374.6)--(218.9, 374.5)--(218.9, 374.4)--(219.0, 374.4)--(219.0, 374.3)--(219.0, 374.2)--(218.9, 374.2)--(218.7, 374.2)--(218.6, 374.2)--(218.5, 374.1)--(218.3, 374.1)--(218.2, 374.0)--(218.0, 374.0)--(217.9, 374.0)--(217.8, 374.0)--(217.8, 374.1)--(217.8, 374.2)--(217.8, 374.3)--(217.7, 374.3)--(217.7, 374.2)--(217.6, 374.2)--(217.6, 374.1)--(217.6, 374.0)--(217.5, 373.9)--(217.4, 373.8)--(217.3, 373.7)--(217.2, 373.6)--(217.1, 373.5)--(216.9, 373.5)--(216.8, 373.6)--(216.7, 373.8)--(216.6, 373.8)--(216.5, 373.8)--(216.5, 373.7)--(216.3, 373.3)--(216.1, 373.2)--(216.1, 373.1)--(216.0, 373.1)--(216.0, 373.0)--(216.1, 373.0)--(216.2, 372.8)--(216.4, 372.7)--(216.5, 372.6)--(216.6, 372.5)--(216.6, 372.4)--(216.3, 372.1)--(216.3, 371.9)--(216.2, 371.8)--(216.2, 371.6)--(216.2, 371.4)--(216.2, 371.2)--(216.3, 371.2)--(216.3, 371.3)--(216.4, 371.3)--(216.5, 371.3)--(216.6, 371.3)--(216.6, 371.2)--(216.6, 371.1)--(216.5, 371.1)--(216.5, 370.9)--(216.3, 370.7)--(216.2, 370.6)--(216.2, 370.5)--(216.3, 370.3)--(216.2, 370.3)--(216.2, 370.2)--(216.0, 370.2)--(215.9, 370.3)--(215.8, 370.3)--(215.7, 370.2)--(215.7, 370.1)--(215.7, 370.0)--(215.8, 370.0)--(215.9, 369.9)--(215.9, 369.8)--(215.8, 369.7)--(215.7, 369.7)--(215.5, 369.7)--(215.3, 369.7)--(215.2, 369.8)--(215.1, 369.9)--(214.9, 369.7)--(214.8, 369.7)--(214.7, 369.8)--(214.7, 369.9)--(214.5, 370.0)--(214.4, 370.1)--(214.3, 370.1)--(214.2, 370.1)--(214.2, 370.2)--(214.1, 370.2)--(214.1, 370.3)--(214.0, 370.4)--(213.8, 370.6)--(213.7, 370.8)--(213.5, 371.1)--(213.3, 371.4)--(213.0, 371.6)--(213.0, 371.7)--(212.9, 371.7)--(212.9, 371.8)--(212.9, 372.0)--(212.8, 372.1)--(212.7, 372.3)--(212.6, 372.5)--(212.5, 372.6)--(212.3, 372.7)--(212.0, 373.0)--(211.9, 373.2)--(211.8, 373.5)--(211.7, 373.8)--(211.6, 373.9)--(211.6, 374.0)--(211.6, 374.1)--(211.6, 374.2)--(211.7, 374.2)--(211.7, 374.3)--(211.9, 374.3)--(212.0, 374.3)--(212.1, 374.3)--(212.3, 374.4)--(212.5, 374.4)--(212.6, 374.5)--(212.7, 374.6)--(212.8, 374.6)--(212.8, 374.7)--(212.8, 374.8)--(212.7, 374.9)--(212.5, 375.4)--(212.4, 375.6)--(212.2, 375.9)--(212.1, 376.2)--(211.9, 376.7)--(211.8, 377.0)--(211.7, 377.3)--(211.7, 377.4)--(211.6, 377.6)--(211.4, 378.5)--(211.3, 378.9)--(211.3, 379.1)--(211.2, 379.3)--(211.1, 379.4)--(211.1, 379.5)--(210.9, 379.9)--(210.8, 380.2)--(210.7, 380.7)--(210.5, 381.1)--(210.5, 381.2)--(210.4, 381.8)--(210.4, 381.9)--(210.4, 382.0)--(210.4, 382.1)--(210.4, 382.3)--(210.5, 382.4)--(210.5, 382.6)--(210.5, 382.8)--(210.5, 383.0)--(210.5, 383.3)--(210.5, 383.4)--(210.5, 383.6)--(210.5, 383.7)--(210.5, 383.9)--(210.5, 384.0)--(210.5, 384.1)--(210.6, 384.4)--(210.7, 384.5)--(210.8, 384.6)--(210.9, 384.7)--(211.0, 384.8)--(211.1, 384.9)--(211.3, 384.9)--(211.5, 385.1)--(211.7, 385.3)--(211.8, 385.4)--(211.9, 385.5)--(212.1, 385.8)--(212.3, 386.3)--(212.5, 386.4)--(212.9, 386.5)--(213.2, 386.7)--(213.5, 386.8)--(214.5, 387.2)--(214.6, 387.2)--(214.6, 387.3)--(214.4, 387.5)--(213.4, 389.2)--(213.1, 389.5)--(212.0, 391.2)--(211.6, 391.9)--(211.1, 392.4)--(211.0, 392.6)--(210.6, 393.0)--(210.3, 393.3)--(210.0, 393.5)--(209.4, 394.2)--(208.4, 395.4)--(208.1, 395.7)--(207.9, 395.9)--(207.7, 396.3)--(207.5, 396.5)--(206.4, 398.0)--(205.8, 398.8)--(204.9, 400.0)--(204.5, 400.4)--(204.3, 400.7)--(203.8, 401.3)--(203.5, 401.6)--(203.2, 402.0)--(202.8, 402.4)--(202.5, 402.8)--(202.2, 403.3)--(201.9, 403.8)--(201.2, 404.7)--(201.1, 404.8)--(200.8, 405.2)--(200.2, 406.0)--(200.0, 406.3)--(199.0, 407.4)--(198.9, 407.6)--(197.0, 409.9)--(196.4, 410.6)--(195.6, 412.0)--(195.3, 412.4)--(195.1, 412.7)--(194.8, 413.2)--(194.5, 413.5)--(194.3, 413.6)--(194.0, 413.9)--(193.9, 414.1)--(193.4, 414.7)--(192.8, 415.5)--(192.4, 416.1)--(191.8, 416.9)--(190.9, 418.3)--(190.4, 419.0)--(189.5, 420.0)--(189.1, 420.4)--(189.0, 420.6)--(188.8, 420.7)--(188.2, 421.2)--(187.7, 421.8)--(187.1, 422.5)--(186.6, 423.1)--(186.5, 423.3)--(186.2, 423.8)--(186.0, 424.1)--(185.1, 425.3)--(183.6, 427.2)--(182.9, 428.2)--(182.6, 428.7)--(182.0, 429.3)--(181.7, 429.9)--(181.3, 430.4)--(181.1, 430.7)--(180.6, 431.4)--(179.6, 432.7)--(179.3, 433.1)--(179.1, 433.4)--(178.6, 433.9)--(178.2, 434.4)--(177.6, 435.2)--(177.3, 435.7)--(176.8, 436.4)--(176.4, 437.0)--(176.1, 437.3)--(175.9, 437.7)--(174.8, 438.8)--(174.5, 439.4)--(174.1, 440.2)--(173.8, 440.9)--(173.6, 441.7)--(173.6, 441.9)--(173.6, 442.0)--(174.0, 443.1)--(174.4, 444.2)--(174.5, 444.2)--(174.6, 444.3)--(174.7, 444.4)--(174.8, 444.4)--(175.0, 444.3)--(175.1, 444.3)--(175.2, 444.3)--(175.3, 444.2)--(175.4, 444.2)--(175.4, 444.1)--(175.6, 444.0)--(175.8, 444.2)--(176.2, 444.2)--(176.6, 444.0)--(176.7, 444.0)--(176.8, 444.0)--(177.0, 443.8)--(177.1, 443.8)--(177.2, 443.7)--(177.3, 443.7)--(177.4, 443.7)--(177.6, 443.6)--(177.7, 443.6)--(177.9, 443.6)--(178.3, 443.6)--(178.4, 443.7)--(178.5, 443.7)--(178.7, 443.7)--(179.0, 443.7)--(179.4, 443.6)--(180.1, 443.3)--(181.3, 443.6)--(181.5, 443.7)--(181.9, 443.9)--(182.1, 444.0)--(182.9, 444.4)--(183.0, 444.5)--(183.1, 444.6)--(183.2, 444.6)--(183.4, 444.6)--(184.0, 444.7)--(184.4, 444.8)--(184.8, 444.8)--(184.9, 444.7)--(185.0, 444.7)--(185.5, 444.6)--(185.7, 444.6)--(185.9, 444.7)--(186.3, 444.8)--(186.5, 444.9)--(186.8, 445.0)--(187.1, 445.1)--(187.5, 445.2)--(187.7, 445.3)--(187.8, 445.4)--(188.0, 445.7)--(188.3, 445.9)--(188.6, 446.2)--(189.2, 446.4)--(190.3, 446.8)--(190.9, 446.8)--(191.2, 446.8)--(191.6, 446.8)--(191.9, 446.8)--(192.2, 446.8)--(192.5, 446.9)--(192.6, 447.0)--(192.8, 447.0)--(193.1, 447.1)--(193.3, 447.2)--(193.5, 447.4)--(193.7, 447.5)--(194.1, 447.4)--(194.3, 447.4)--(194.6, 447.4)--(194.9, 447.4)--(195.4, 447.3)--(195.6, 447.3)--(195.6, 447.4)--(195.4, 447.5)--(195.3, 447.6)--(195.3, 447.7)--(195.2, 447.8)--(195.2, 448.0)--(195.4, 448.6)--(195.5, 449.1)--(195.6, 449.3)--(195.8, 449.7)--(195.9, 450.0)--(195.9, 450.1)--(195.9, 450.2)--(195.9, 450.4)--(196.1, 450.7)--(196.1, 451.0)--(196.1, 451.1)--(196.7, 451.1)--(197.0, 451.1)--(197.3, 451.1)--(197.6, 451.1)--(198.0, 451.1)--(198.2, 451.2)--(198.4, 451.4)--(198.4, 451.3)--(198.9, 451.4)--(199.3, 451.5)--(199.2, 451.9)--(199.1, 452.3)--(199.1, 452.5)--(199.1, 452.7)--(199.1, 452.8)--(199.1, 452.9)--(199.1, 453.1)--(200.2, 453.5)--(201.1, 453.4)--(201.3, 453.4)--(201.6, 453.3)--(201.7, 453.2)--(201.8, 453.1)--(201.8, 453.0)--(201.9, 453.0)--(202.0, 453.2)--(202.1, 453.1)--(202.6, 454.3)--(202.7, 454.3)--(202.9, 454.2)--(203.0, 454.1)--(203.1, 454.0)--(203.2, 453.9)--(203.4, 453.7)--(203.5, 453.6)--(203.7, 453.5)--(203.8, 453.4)--(203.9, 453.3)--(204.0, 453.2)--(204.0, 453.1)--(204.1, 453.0)--(204.1, 452.8)--(204.1, 452.7)--(204.1, 452.6)--(204.2, 452.4)--(204.3, 452.2)--(204.3, 452.1)--(204.4, 452.1)--(204.5, 452.0)--(204.6, 451.8)--(204.8, 451.7)--(204.9, 451.7)--(205.1, 451.6)--(205.5, 451.5)--(205.7, 451.4)--(205.8, 451.4)--(206.1, 451.3)--(206.1, 451.2)--(206.4, 451.2)--(206.6, 451.2)--(206.7, 451.1)--(206.8, 451.1)--(206.9, 451.2)--(207.1, 451.9)--(207.4, 452.5)--(207.6, 452.9)--(208.1, 452.7)--(208.1, 452.8)--(208.1, 452.9)--(208.6, 453.1)--(208.7, 453.2)--(209.0, 453.7)--(209.3, 454.3)--(209.7, 455.4)--(210.0, 455.9)--(210.2, 456.4)--(210.3, 456.8)--(210.4, 457.1)--(210.5, 457.5)--(210.7, 458.0)--(210.8, 458.4)--(210.8, 458.5)--(210.8, 458.8)--(210.9, 459.1)--(211.0, 459.7)--(211.3, 462.6)--(211.6, 465.4)--(211.3, 465.5)--(211.2, 465.5)--(211.0, 465.6)--(210.9, 465.7)--(210.8, 465.8)--(210.7, 465.9)--(210.6, 466.1)--(210.7, 466.1)--(210.7, 466.2)--(210.7, 466.4)--(210.7, 466.6)--(210.8, 466.7)--(211.0, 466.9)--(211.1, 467.0)--(211.2, 467.0)--(211.2, 467.1)--(211.3, 467.1)--(211.5, 467.0)--(211.6, 467.0)--(211.7, 467.1)--(212.0, 467.0)--(212.2, 466.9)--(212.3, 466.8)--(212.6, 466.6)--(212.7, 466.5)--(212.9, 466.3)--(213.0, 466.2)--(213.8, 466.5)--(214.0, 466.7)--(214.3, 466.9)--(214.4, 466.9)--(214.9, 467.2)--(215.2, 467.4)--(215.4, 467.6)--(215.7, 467.7)--(215.9, 467.8)--(216.1, 467.9)--(216.2, 467.9)--(216.3, 467.9)--(216.4, 467.9)--(216.6, 468.0)--(216.8, 468.1)--(216.9, 468.2)--(217.1, 468.3)--(217.5, 468.4)--(217.7, 468.0)--(218.4, 468.4)--(218.7, 468.5)--(218.9, 468.7)--(219.4, 469.0)--(219.8, 469.1)--(219.6, 469.6)--(220.0, 469.8)--(220.3, 470.0)--(220.4, 470.1)--(220.7, 470.3)--(220.8, 470.3)--(221.0, 470.4)--(221.3, 470.5)--(221.7, 470.6)--(221.9, 470.5)--(222.0, 470.5)--(222.2, 470.1)--(222.4, 469.7)--(222.5, 469.8)--(222.6, 469.9)--(222.8, 469.9)--(223.0, 470.0)--(223.6, 470.4)--(224.4, 470.8)--(224.5, 470.9)--(224.6, 471.0)--(224.8, 471.1)--(224.9, 471.1)--(225.2, 471.4)--(225.3, 471.4)--(225.4, 471.4)--(225.6, 471.6)--(226.0, 471.7)--(226.1, 471.7)--(226.3, 471.7)--(226.3, 471.3)--(226.4, 471.1)--(226.5, 470.9)--(226.3, 470.9)--(226.5, 470.5)--(226.6, 470.2)--(226.7, 470.1)--(227.2, 469.4)--(227.4, 469.1)--(227.7, 468.8)--(227.8, 468.6)--(228.2, 469.0)--(228.0, 469.2)--(228.4, 469.4)--(228.8, 469.7)--(229.5, 470.3)--(230.2, 470.9)--(231.1, 471.6)--(231.3, 471.9)--(232.0, 472.4)--(232.3, 472.6)--(232.5, 472.8)--(232.7, 472.8)--(232.9, 472.0)--(234.1, 472.6)--(235.7, 473.9)--(235.9, 474.0)--(236.6, 474.6)--(236.7, 474.7)--(237.1, 474.2)--(237.4, 473.8)--(237.6, 473.5)--(237.7, 473.7)--(238.0, 473.8)--(238.3, 474.1)--(239.0, 474.5)--(240.3, 475.2)--(240.5, 475.3)--(239.1, 478.0)--(238.6, 478.9)--(238.1, 479.8)--(239.0, 480.5)--(240.3, 479.4)--(240.7, 480.0)--(240.9, 480.4)--(241.8, 481.6)--(241.9, 481.7)--(242.5, 481.6)--(243.1, 481.4)--(243.3, 481.3)--(243.5, 481.3)--(243.6, 481.2)--(243.8, 481.1)--(243.9, 481.0)--(244.2, 480.9)--(244.1, 481.6)--(244.0, 482.2)--(243.9, 482.3)--(243.9, 482.6)--(243.4, 483.4)--(244.1, 483.4)--(244.5, 483.4)--(245.0, 483.4)--(245.6, 483.5)--cycle; +Bexley = (472.3, 313.0)--(473.2, 313.4)--(474.1, 313.9)--(475.2, 314.6)--(475.4, 314.8)--(475.5, 314.8)--(475.6, 314.8)--(475.6, 314.9)--(475.8, 314.9)--(476.0, 315.0)--(476.2, 315.0)--(476.4, 315.1)--(476.6, 315.1)--(476.8, 315.1)--(477.3, 315.0)--(477.6, 315.0)--(477.8, 315.1)--(477.8, 315.2)--(477.9, 315.2)--(478.0, 315.2)--(479.3, 314.9)--(479.5, 314.8)--(479.6, 314.7)--(479.7, 314.6)--(480.6, 314.2)--(480.8, 314.0)--(481.1, 313.8)--(481.2, 313.7)--(481.4, 313.6)--(482.1, 313.2)--(482.2, 313.1)--(484.2, 311.8)--(484.3, 312.0)--(485.3, 311.4)--(486.1, 310.9)--(488.6, 309.5)--(488.8, 309.3)--(489.4, 309.1)--(489.6, 309.0)--(490.0, 308.9)--(490.3, 308.7)--(490.6, 308.6)--(490.8, 308.5)--(491.1, 308.3)--(491.3, 308.2)--(491.4, 308.2)--(491.6, 308.4)--(491.7, 308.4)--(491.7, 308.2)--(491.7, 308.1)--(491.7, 308.0)--(491.8, 307.9)--(491.9, 307.9)--(492.1, 308.0)--(492.2, 307.9)--(492.2, 307.8)--(492.4, 307.6)--(492.7, 307.5)--(492.8, 307.4)--(493.0, 307.4)--(493.0, 307.5)--(493.0, 307.9)--(493.1, 308.0)--(493.2, 307.9)--(493.6, 307.9)--(493.7, 307.9)--(493.8, 307.8)--(493.9, 307.9)--(494.0, 307.9)--(494.2, 308.0)--(494.3, 308.1)--(494.9, 308.0)--(495.2, 308.0)--(495.3, 308.0)--(495.6, 307.8)--(495.8, 307.8)--(496.7, 307.7)--(497.5, 307.6)--(497.5, 307.5)--(499.3, 307.4)--(499.9, 307.3)--(500.0, 307.3)--(500.2, 307.3)--(500.5, 307.1)--(500.9, 307.0)--(501.4, 306.8)--(501.5, 306.7)--(501.7, 306.6)--(501.8, 306.6)--(501.9, 306.6)--(502.0, 306.6)--(502.4, 306.5)--(502.8, 306.3)--(503.0, 306.3)--(503.2, 306.2)--(503.6, 305.9)--(504.4, 305.3)--(504.7, 305.1)--(504.8, 304.9)--(504.9, 304.8)--(505.2, 304.3)--(505.4, 304.0)--(506.2, 302.5)--(506.4, 302.3)--(506.4, 302.2)--(506.5, 302.2)--(506.6, 302.1)--(506.7, 302.0)--(506.8, 302.0)--(506.8, 301.9)--(507.2, 301.3)--(507.3, 301.1)--(507.3, 301.0)--(507.3, 300.9)--(507.3, 300.8)--(507.4, 300.7)--(507.5, 300.4)--(507.6, 300.1)--(507.7, 300.0)--(507.9, 299.7)--(508.0, 299.7)--(508.4, 298.7)--(508.3, 298.7)--(508.4, 298.5)--(508.5, 298.3)--(508.6, 298.1)--(508.6, 298.0)--(508.7, 297.8)--(508.7, 297.6)--(508.7, 297.4)--(508.7, 297.0)--(508.8, 296.3)--(508.8, 295.5)--(508.8, 295.1)--(508.8, 294.9)--(508.8, 294.1)--(508.8, 293.9)--(508.9, 292.7)--(508.9, 292.6)--(508.9, 292.5)--(508.9, 292.4)--(509.0, 292.3)--(509.1, 292.1)--(509.1, 292.0)--(509.0, 291.7)--(509.0, 291.6)--(509.2, 290.7)--(509.2, 290.5)--(509.3, 290.2)--(509.3, 290.0)--(509.2, 289.9)--(509.2, 289.8)--(509.3, 289.5)--(509.3, 289.2)--(509.4, 288.8)--(509.4, 288.7)--(509.5, 288.6)--(509.7, 288.3)--(509.8, 288.2)--(509.8, 288.1)--(509.8, 288.0)--(509.9, 287.9)--(510.0, 287.6)--(510.0, 287.4)--(510.0, 287.3)--(510.1, 287.3)--(510.1, 287.2)--(510.2, 287.2)--(510.2, 287.1)--(510.3, 287.0)--(510.5, 286.7)--(510.7, 286.8)--(511.3, 285.9)--(511.0, 285.7)--(511.1, 285.6)--(511.0, 285.5)--(511.1, 285.4)--(511.2, 285.2)--(511.3, 285.0)--(511.4, 285.1)--(511.6, 284.9)--(511.8, 285.0)--(513.3, 283.1)--(513.0, 282.7)--(513.1, 282.7)--(513.1, 282.6)--(513.2, 282.6)--(513.5, 282.9)--(514.5, 282.2)--(515.7, 281.4)--(516.1, 281.1)--(517.0, 280.6)--(517.1, 280.6)--(517.2, 280.5)--(517.3, 280.5)--(517.5, 280.4)--(517.7, 280.3)--(518.1, 280.2)--(519.3, 279.9)--(520.0, 279.7)--(520.8, 279.5)--(521.2, 279.4)--(521.6, 279.3)--(521.8, 279.3)--(521.9, 279.2)--(522.0, 279.2)--(522.0, 279.1)--(521.9, 278.5)--(522.1, 278.4)--(522.2, 279.1)--(522.3, 279.2)--(523.6, 279.2)--(523.6, 279.0)--(523.8, 279.0)--(523.8, 278.9)--(523.9, 278.9)--(524.0, 278.9)--(524.0, 278.8)--(524.1, 278.8)--(524.3, 278.8)--(524.4, 278.8)--(524.5, 278.8)--(524.6, 278.8)--(524.6, 278.7)--(524.7, 278.7)--(524.7, 278.8)--(524.8, 278.9)--(524.9, 279.0)--(525.1, 279.1)--(525.2, 279.1)--(525.3, 279.1)--(525.4, 279.1)--(525.5, 279.2)--(525.6, 279.2)--(525.7, 279.1)--(525.8, 279.1)--(525.8, 279.0)--(525.9, 279.0)--(526.0, 279.0)--(526.1, 279.1)--(526.2, 279.1)--(526.2, 279.0)--(526.3, 279.0)--(526.3, 279.1)--(526.3, 279.0)--(526.4, 279.1)--(526.5, 279.1)--(526.6, 279.1)--(526.7, 279.1)--(526.8, 279.1)--(526.8, 279.2)--(526.8, 279.1)--(526.9, 279.1)--(526.9, 279.0)--(527.0, 279.1)--(527.0, 279.0)--(527.1, 279.1)--(527.2, 279.1)--(527.3, 279.2)--(527.4, 279.2)--(527.4, 279.1)--(527.5, 279.2)--(527.6, 279.2)--(527.7, 279.2)--(527.8, 279.1)--(527.8, 279.0)--(527.7, 278.9)--(527.8, 278.9)--(527.9, 278.9)--(528.0, 279.0)--(528.1, 279.1)--(528.1, 279.2)--(528.2, 279.2)--(528.2, 279.3)--(528.2, 279.5)--(528.4, 279.6)--(528.5, 279.6)--(528.6, 279.6)--(528.6, 279.5)--(528.6, 279.4)--(528.8, 279.3)--(528.8, 279.2)--(528.9, 279.1)--(529.0, 279.2)--(529.1, 279.4)--(529.2, 279.5)--(529.2, 279.6)--(529.2, 279.7)--(529.2, 279.9)--(529.4, 279.9)--(529.5, 279.9)--(529.6, 279.9)--(529.7, 279.9)--(529.8, 279.9)--(529.7, 279.8)--(529.8, 279.8)--(529.8, 279.7)--(529.8, 279.6)--(529.9, 279.4)--(529.9, 279.3)--(530.0, 279.2)--(530.1, 279.2)--(530.1, 279.1)--(530.1, 279.0)--(530.1, 278.9)--(530.1, 278.7)--(530.1, 278.6)--(530.0, 278.6)--(529.9, 278.6)--(529.8, 278.6)--(529.7, 278.4)--(529.6, 278.4)--(529.4, 278.4)--(529.3, 278.3)--(529.2, 278.3)--(529.2, 278.2)--(529.3, 278.2)--(529.5, 278.3)--(529.6, 278.3)--(529.7, 278.3)--(529.7, 278.4)--(529.8, 278.4)--(529.9, 278.5)--(530.0, 278.5)--(530.1, 278.5)--(530.2, 278.6)--(530.2, 278.7)--(530.2, 279.1)--(530.2, 279.2)--(530.1, 279.3)--(530.0, 279.4)--(529.9, 279.5)--(530.0, 279.6)--(530.0, 279.7)--(529.9, 280.0)--(530.0, 280.0)--(530.1, 280.0)--(530.1, 280.1)--(530.2, 280.2)--(530.2, 280.3)--(530.2, 280.4)--(530.3, 280.4)--(530.4, 280.4)--(530.5, 280.4)--(530.6, 280.4)--(530.8, 280.5)--(530.9, 280.5)--(531.0, 280.5)--(531.2, 280.5)--(531.3, 280.5)--(531.3, 280.4)--(531.4, 280.4)--(531.4, 280.5)--(531.5, 280.5)--(531.5, 280.6)--(531.6, 280.6)--(531.8, 280.7)--(531.9, 280.7)--(532.0, 280.7)--(532.0, 280.6)--(531.9, 280.5)--(532.1, 280.7)--(532.2, 280.7)--(532.3, 280.5)--(532.3, 280.4)--(532.4, 280.3)--(532.5, 280.3)--(532.7, 280.5)--(532.9, 280.5)--(533.0, 280.6)--(533.0, 280.7)--(533.0, 280.8)--(532.9, 281.0)--(533.0, 281.1)--(533.0, 281.2)--(533.1, 281.2)--(533.1, 281.3)--(533.2, 281.3)--(533.3, 281.3)--(533.4, 281.3)--(533.6, 281.3)--(533.6, 281.4)--(533.8, 281.4)--(533.9, 281.4)--(534.0, 281.4)--(534.1, 281.5)--(534.1, 281.4)--(534.2, 281.5)--(534.2, 281.6)--(534.3, 281.6)--(534.4, 281.6)--(534.5, 281.6)--(534.6, 281.7)--(534.7, 281.7)--(534.8, 281.8)--(535.0, 281.8)--(535.0, 281.9)--(535.1, 281.9)--(535.2, 281.8)--(535.2, 281.7)--(535.3, 281.7)--(535.4, 281.9)--(535.6, 282.1)--(535.6, 282.3)--(535.8, 282.6)--(535.8, 282.7)--(536.3, 282.9)--(536.5, 282.9)--(536.7, 282.9)--(536.8, 282.9)--(537.0, 282.8)--(537.1, 282.7)--(537.2, 282.6)--(537.3, 282.6)--(537.4, 282.6)--(537.5, 282.6)--(537.7, 282.4)--(538.6, 281.9)--(538.7, 281.9)--(538.8, 281.8)--(538.9, 281.8)--(539.0, 281.7)--(539.3, 281.5)--(539.7, 281.3)--(539.8, 281.2)--(539.9, 281.2)--(540.1, 281.0)--(540.9, 280.6)--(540.9, 280.5)--(540.6, 280.4)--(540.5, 280.4)--(540.4, 280.3)--(540.3, 280.3)--(540.3, 280.2)--(540.2, 280.2)--(540.1, 280.1)--(540.1, 280.0)--(540.1, 279.9)--(540.0, 279.8)--(539.9, 279.7)--(539.9, 279.6)--(539.8, 279.5)--(539.8, 279.4)--(539.7, 279.3)--(539.7, 279.2)--(539.8, 278.8)--(539.8, 278.7)--(539.7, 278.6)--(539.8, 278.2)--(539.9, 278.2)--(540.0, 278.1)--(540.0, 277.8)--(539.9, 277.8)--(539.9, 277.7)--(540.0, 277.6)--(540.0, 277.3)--(540.0, 277.2)--(540.0, 277.1)--(539.9, 277.0)--(539.9, 276.9)--(539.9, 276.8)--(539.8, 276.7)--(539.7, 276.5)--(539.5, 276.4)--(539.4, 276.4)--(539.4, 276.3)--(539.3, 276.3)--(539.2, 276.3)--(539.1, 276.3)--(538.7, 276.1)--(538.6, 276.1)--(538.5, 276.1)--(538.3, 276.0)--(538.2, 275.9)--(538.0, 275.9)--(537.8, 275.8)--(537.5, 275.7)--(537.5, 275.6)--(537.4, 275.6)--(537.1, 275.3)--(537.0, 275.2)--(536.9, 275.0)--(536.8, 274.8)--(536.6, 274.7)--(536.6, 274.5)--(536.6, 274.4)--(536.6, 274.2)--(536.6, 273.9)--(536.6, 273.7)--(536.6, 273.6)--(536.6, 273.5)--(536.7, 273.3)--(536.7, 273.2)--(536.6, 273.1)--(536.7, 273.0)--(536.7, 272.8)--(536.8, 272.7)--(536.8, 272.5)--(536.8, 272.4)--(536.8, 272.3)--(536.3, 271.8)--(536.2, 271.6)--(536.1, 271.5)--(535.5, 270.8)--(535.4, 270.8)--(535.2, 270.7)--(535.1, 270.7)--(535.1, 270.6)--(535.0, 270.4)--(534.8, 269.9)--(534.7, 269.8)--(534.6, 269.7)--(534.6, 269.6)--(534.6, 269.5)--(534.7, 269.4)--(534.8, 269.3)--(534.8, 269.2)--(534.9, 269.1)--(534.9, 269.0)--(535.0, 268.9)--(535.1, 268.9)--(535.1, 268.8)--(535.1, 268.7)--(535.2, 268.7)--(535.2, 268.6)--(535.5, 268.3)--(535.9, 267.9)--(536.0, 267.9)--(536.3, 267.7)--(536.4, 267.6)--(536.5, 267.4)--(536.6, 267.2)--(536.7, 267.1)--(536.7, 266.9)--(536.7, 266.7)--(536.7, 266.6)--(536.7, 266.5)--(536.7, 266.4)--(536.7, 266.2)--(536.6, 265.6)--(536.4, 265.1)--(536.2, 264.4)--(536.1, 264.1)--(536.1, 264.0)--(536.1, 263.6)--(536.1, 263.5)--(536.1, 263.0)--(536.2, 262.8)--(536.2, 262.7)--(536.3, 262.0)--(536.4, 261.2)--(536.4, 260.7)--(536.4, 260.6)--(536.4, 260.5)--(536.3, 260.3)--(536.2, 260.3)--(536.2, 260.2)--(536.1, 260.2)--(535.2, 259.7)--(534.8, 259.5)--(534.5, 259.4)--(534.5, 259.3)--(534.4, 259.3)--(534.3, 259.0)--(534.1, 258.8)--(534.0, 258.7)--(533.9, 258.6)--(533.8, 258.5)--(533.4, 258.3)--(533.4, 258.2)--(533.4, 258.1)--(533.3, 258.0)--(533.2, 258.0)--(533.1, 257.8)--(533.0, 257.8)--(532.9, 257.8)--(532.6, 257.6)--(532.4, 257.5)--(532.4, 257.4)--(532.3, 257.4)--(532.2, 257.1)--(532.2, 257.0)--(532.1, 256.9)--(531.8, 256.6)--(531.7, 256.3)--(531.3, 255.7)--(531.2, 255.6)--(531.1, 255.6)--(530.0, 255.7)--(529.9, 255.7)--(528.7, 255.8)--(528.6, 255.8)--(528.4, 255.7)--(528.3, 255.7)--(527.6, 254.9)--(527.5, 254.6)--(527.7, 254.6)--(528.3, 255.4)--(528.6, 255.2)--(528.6, 255.1)--(528.2, 254.4)--(528.3, 254.4)--(528.4, 254.4)--(528.4, 254.5)--(528.5, 254.6)--(528.5, 254.7)--(528.7, 255.0)--(528.8, 255.1)--(529.1, 255.6)--(531.0, 255.4)--(531.4, 255.3)--(531.5, 255.2)--(531.6, 255.2)--(531.6, 255.1)--(531.7, 255.0)--(531.7, 254.8)--(531.6, 254.6)--(531.4, 254.0)--(531.3, 253.3)--(531.2, 253.2)--(531.3, 253.2)--(531.6, 253.2)--(531.8, 253.1)--(532.1, 253.1)--(532.1, 253.0)--(532.2, 252.7)--(532.2, 252.6)--(532.1, 252.5)--(531.8, 252.0)--(531.8, 251.9)--(531.8, 251.8)--(531.8, 251.7)--(531.9, 251.4)--(531.9, 251.2)--(531.8, 251.2)--(531.6, 251.1)--(531.6, 251.0)--(531.5, 250.8)--(531.4, 250.7)--(531.3, 250.7)--(531.2, 250.7)--(530.9, 250.9)--(530.8, 250.8)--(530.8, 250.7)--(530.5, 250.6)--(530.3, 250.4)--(530.2, 250.3)--(530.2, 250.2)--(530.3, 250.2)--(530.4, 250.1)--(530.4, 250.0)--(530.3, 250.0)--(530.3, 249.9)--(530.2, 249.9)--(530.1, 249.9)--(530.0, 249.9)--(530.0, 249.8)--(530.0, 249.7)--(530.1, 249.5)--(530.0, 249.5)--(529.9, 249.5)--(529.8, 249.5)--(529.8, 249.4)--(529.8, 249.3)--(529.7, 249.2)--(529.5, 249.3)--(529.4, 249.3)--(529.3, 249.3)--(529.3, 249.2)--(529.3, 249.1)--(529.2, 249.0)--(529.1, 249.0)--(529.0, 249.0)--(529.0, 248.9)--(528.9, 248.9)--(528.9, 248.8)--(529.0, 248.7)--(528.8, 248.6)--(528.6, 248.6)--(528.5, 248.6)--(528.4, 248.5)--(528.3, 248.5)--(528.3, 248.4)--(528.2, 248.3)--(528.1, 248.3)--(528.0, 248.4)--(528.0, 248.5)--(527.9, 248.4)--(527.9, 248.3)--(527.8, 248.2)--(527.7, 248.2)--(527.6, 248.2)--(527.5, 248.2)--(527.4, 248.2)--(527.3, 248.2)--(527.2, 248.2)--(527.1, 248.1)--(527.0, 248.2)--(526.9, 248.2)--(526.8, 248.2)--(526.8, 248.3)--(526.7, 248.4)--(526.8, 248.4)--(526.8, 248.5)--(526.9, 248.6)--(526.9, 248.7)--(526.8, 248.8)--(526.7, 248.8)--(526.7, 248.9)--(526.7, 248.8)--(526.6, 248.8)--(526.6, 248.7)--(526.6, 248.6)--(526.6, 248.5)--(526.5, 248.5)--(526.5, 248.4)--(526.4, 248.4)--(526.3, 248.5)--(526.3, 248.6)--(526.2, 248.6)--(526.1, 248.6)--(526.1, 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273.7)--(470.2, 273.8)--(470.5, 274.4)--(470.8, 274.8)--(471.0, 275.2)--(471.2, 275.5)--(471.9, 276.5)--(472.1, 276.8)--(472.4, 276.7)--(472.5, 276.6)--(472.6, 276.6)--(472.7, 276.5)--(473.0, 276.5)--(473.1, 276.5)--(473.3, 276.5)--(473.6, 276.4)--(473.7, 276.4)--(473.9, 276.3)--(474.1, 276.1)--(474.4, 276.0)--(474.5, 275.9)--(474.7, 275.7)--(475.0, 275.6)--(475.2, 275.4)--(475.6, 275.1)--(475.8, 274.9)--(475.9, 274.8)--(475.9, 274.7)--(476.0, 274.6)--(476.1, 274.6)--(476.0, 274.9)--(476.0, 275.1)--(475.9, 275.3)--(475.8, 275.9)--(475.6, 276.6)--(475.5, 277.1)--(475.2, 278.0)--(474.9, 279.0)--(474.8, 279.5)--(474.6, 280.1)--(474.5, 280.5)--(474.5, 280.8)--(474.5, 281.1)--(474.5, 282.1)--(474.4, 283.0)--(474.4, 284.3)--(474.3, 284.6)--(474.3, 284.8)--(474.3, 284.9)--(473.9, 286.2)--(473.9, 286.4)--(473.9, 286.5)--(473.9, 286.7)--(473.9, 286.8)--(473.9, 287.0)--(473.9, 287.6)--(473.9, 287.7)--(473.9, 287.8)--(473.8, 288.1)--(473.7, 288.9)--(473.7, 289.8)--(473.7, 290.4)--(473.6, 290.7)--(473.6, 292.0)--(473.5, 292.8)--(473.5, 293.3)--(473.4, 295.0)--(473.4, 296.1)--(473.4, 296.2)--(473.3, 297.3)--(473.2, 299.0)--(473.2, 300.0)--(473.1, 301.1)--(473.1, 301.8)--(473.1, 302.1)--(473.1, 302.5)--(473.1, 303.0)--(473.0, 303.7)--(473.0, 304.5)--(472.9, 305.8)--(472.9, 307.0)--(472.8, 307.7)--(472.8, 308.1)--(472.7, 309.0)--(472.7, 309.7)--(472.7, 310.0)--(472.7, 310.2)--(472.6, 310.8)--(472.6, 311.1)--(472.6, 311.5)--(472.5, 312.6)--cycle; +Brent = (252.0, 325.1)--(251.8, 325.2)--(251.8, 325.3)--(251.8, 325.4)--(251.7, 325.4)--(251.6, 325.4)--(251.5, 325.4)--(251.4, 325.3)--(251.1, 325.4)--(251.0, 325.9)--(250.9, 326.2)--(251.0, 326.3)--(251.0, 326.4)--(251.0, 326.6)--(251.0, 326.7)--(251.0, 326.8)--(251.0, 327.3)--(251.0, 327.6)--(251.0, 327.8)--(250.8, 328.1)--(250.7, 328.2)--(250.5, 328.4)--(250.5, 328.5)--(250.4, 328.6)--(250.3, 328.6)--(250.2, 328.7)--(250.1, 328.8)--(250.0, 328.9)--(249.9, 329.0)--(249.8, 329.1)--(249.6, 329.2)--(249.2, 329.5)--(249.1, 329.6)--(248.9, 329.7)--(248.8, 329.7)--(248.7, 329.8)--(248.6, 329.9)--(248.5, 330.0)--(248.5, 330.1)--(248.5, 330.2)--(248.4, 330.3)--(248.3, 330.5)--(248.2, 330.5)--(248.1, 330.5)--(247.9, 330.6)--(247.7, 330.7)--(247.5, 330.8)--(247.4, 330.9)--(247.4, 330.8)--(247.2, 330.8)--(247.0, 330.8)--(246.7, 330.8)--(246.3, 330.9)--(245.8, 330.9)--(245.6, 330.9)--(245.4, 330.8)--(245.1, 330.8)--(244.9, 330.7)--(244.8, 330.7)--(244.6, 330.7)--(244.4, 330.7)--(244.2, 330.7)--(244.0, 330.6)--(243.5, 330.5)--(243.3, 330.4)--(243.0, 330.3)--(242.8, 330.2)--(242.5, 330.0)--(242.4, 330.0)--(242.0, 329.7)--(241.8, 329.7)--(241.6, 329.6)--(241.4, 329.5)--(241.0, 329.5)--(240.7, 329.5)--(240.6, 329.4)--(240.5, 329.4)--(240.3, 329.4)--(240.1, 329.3)--(239.9, 329.3)--(239.8, 329.2)--(239.7, 329.2)--(239.3, 329.1)--(239.0, 329.0)--(238.9, 328.9)--(238.9, 328.8)--(238.9, 328.7)--(238.9, 328.6)--(238.9, 328.5)--(239.0, 328.0)--(239.1, 327.9)--(239.1, 327.8)--(239.1, 327.7)--(239.1, 327.6)--(239.0, 327.5)--(239.0, 327.4)--(238.9, 327.3)--(238.8, 327.2)--(238.7, 327.2)--(238.6, 327.1)--(238.5, 326.9)--(238.5, 326.6)--(238.4, 326.4)--(238.4, 326.2)--(238.5, 325.1)--(238.2, 325.3)--(238.1, 325.4)--(237.8, 325.6)--(237.6, 325.7)--(237.1, 325.9)--(236.9, 325.9)--(236.6, 326.0)--(236.4, 326.1)--(236.1, 326.2)--(235.2, 326.4)--(235.0, 326.4)--(234.7, 326.5)--(234.4, 326.6)--(234.1, 326.7)--(233.9, 326.9)--(233.7, 326.9)--(233.6, 327.0)--(233.3, 327.1)--(233.2, 327.2)--(233.1, 327.2)--(232.7, 327.3)--(232.5, 327.4)--(232.2, 327.4)--(231.9, 327.4)--(231.9, 327.3)--(231.4, 327.2)--(231.0, 327.2)--(230.4, 327.1)--(230.4, 327.4)--(230.3, 327.4)--(230.2, 327.4)--(230.1, 327.4)--(230.0, 327.5)--(229.9, 327.5)--(229.8, 327.6)--(229.6, 327.7)--(229.3, 327.7)--(229.1, 327.8)--(228.8, 327.9)--(228.7, 328.0)--(228.4, 328.1)--(228.2, 328.1)--(228.0, 328.2)--(227.8, 328.4)--(227.6, 328.5)--(227.1, 328.8)--(226.7, 329.1)--(226.4, 329.3)--(226.0, 329.7)--(225.8, 329.9)--(225.6, 330.0)--(225.5, 330.0)--(225.2, 329.9)--(225.1, 329.9)--(225.0, 329.8)--(224.9, 329.8)--(224.2, 329.5)--(224.0, 329.5)--(223.6, 329.3)--(222.8, 329.2)--(222.3, 329.2)--(222.1, 329.2)--(221.7, 329.2)--(221.4, 329.1)--(221.1, 329.1)--(221.0, 329.1)--(220.3, 329.0)--(219.5, 328.9)--(219.4, 328.9)--(219.3, 328.9)--(219.2, 329.0)--(219.0, 329.1)--(218.6, 329.2)--(218.5, 329.3)--(217.6, 329.8)--(217.4, 329.9)--(217.4, 330.0)--(217.3, 330.0)--(217.2, 330.1)--(217.0, 330.4)--(216.8, 330.7)--(216.7, 330.7)--(216.6, 330.8)--(216.4, 330.9)--(216.2, 330.9)--(215.5, 331.1)--(214.2, 331.5)--(214.1, 331.2)--(214.0, 331.1)--(213.9, 331.1)--(213.8, 331.2)--(213.7, 331.2)--(213.5, 331.3)--(213.4, 331.4)--(213.3, 331.4)--(213.2, 331.4)--(212.8, 331.4)--(212.0, 331.3)--(211.1, 331.2)--(210.8, 331.2)--(210.6, 331.1)--(210.5, 331.6)--(210.5, 331.7)--(210.5, 331.8)--(210.4, 331.8)--(210.4, 331.9)--(210.4, 332.0)--(210.4, 332.1)--(210.3, 332.1)--(210.3, 332.2)--(210.2, 332.2)--(209.7, 332.4)--(209.6, 332.5)--(209.3, 332.6)--(209.2, 332.5)--(209.1, 332.5)--(208.9, 332.3)--(208.1, 331.9)--(207.7, 331.8)--(207.6, 331.7)--(207.5, 331.7)--(207.5, 331.5)--(207.4, 331.4)--(207.0, 331.0)--(206.4, 330.4)--(205.9, 329.9)--(205.5, 329.6)--(205.2, 329.4)--(204.8, 329.2)--(204.8, 329.1)--(204.5, 329.0)--(204.6, 328.8)--(204.6, 328.7)--(204.5, 328.5)--(204.5, 328.4)--(204.4, 328.2)--(204.3, 328.2)--(204.3, 328.1)--(204.2, 328.0)--(203.7, 327.5)--(203.4, 327.1)--(203.1, 327.0)--(202.7, 326.7)--(202.5, 326.6)--(202.3, 326.5)--(201.8, 326.5)--(201.9, 326.4)--(201.9, 326.3)--(201.7, 326.2)--(201.6, 326.2)--(201.5, 326.2)--(201.4, 326.3)--(201.2, 326.5)--(200.7, 326.2)--(200.6, 326.5)--(200.0, 326.4)--(199.4, 326.3)--(198.6, 326.1)--(197.3, 325.8)--(196.0, 325.5)--(194.9, 325.0)--(194.8, 324.9)--(194.7, 324.9)--(194.6, 324.9)--(194.4, 324.8)--(194.3, 324.7)--(194.0, 324.6)--(193.9, 324.6)--(193.8, 324.6)--(193.7, 324.6)--(192.6, 324.9)--(192.4, 324.9)--(192.0, 325.0)--(191.8, 325.1)--(191.2, 325.2)--(191.1, 325.2)--(191.2, 325.1)--(190.1, 325.3)--(189.8, 325.7)--(189.7, 325.9)--(189.2, 326.3)--(188.8, 326.7)--(188.4, 327.1)--(188.0, 327.5)--(187.7, 327.8)--(187.5, 328.0)--(188.2, 328.5)--(188.9, 328.8)--(189.5, 329.2)--(189.9, 329.4)--(190.0, 329.4)--(190.0, 329.5)--(190.1, 329.7)--(190.3, 329.7)--(191.1, 330.1)--(191.2, 330.2)--(191.5, 329.4)--(192.4, 329.7)--(192.9, 329.9)--(193.3, 329.3)--(193.3, 329.2)--(193.8, 329.4)--(194.2, 329.6)--(194.5, 329.8)--(194.7, 329.9)--(194.8, 330.1)--(194.8, 330.3)--(194.9, 330.5)--(194.9, 330.9)--(195.0, 331.4)--(195.2, 331.4)--(195.2, 331.5)--(195.3, 331.9)--(195.4, 331.9)--(195.5, 332.0)--(195.3, 332.1)--(195.2, 332.2)--(195.0, 332.3)--(194.9, 332.5)--(194.7, 332.6)--(194.5, 332.9)--(194.1, 333.4)--(193.5, 334.2)--(192.8, 335.1)--(192.2, 335.8)--(192.1, 335.9)--(191.9, 335.8)--(191.0, 335.0)--(190.3, 334.5)--(189.9, 334.2)--(189.2, 333.7)--(187.9, 333.0)--(186.9, 332.6)--(186.7, 332.6)--(185.6, 332.0)--(185.5, 332.0)--(185.0, 331.8)--(184.7, 331.7)--(184.6, 331.6)--(184.0, 331.7)--(183.5, 331.5)--(183.4, 331.5)--(183.4, 331.6)--(183.3, 331.6)--(183.1, 331.9)--(183.0, 332.0)--(182.9, 332.0)--(182.8, 332.0)--(182.7, 332.0)--(182.6, 332.0)--(182.4, 331.9)--(182.3, 331.8)--(182.2, 331.8)--(182.2, 331.7)--(182.1, 331.6)--(182.0, 331.6)--(181.9, 331.5)--(181.7, 331.0)--(181.4, 330.7)--(181.1, 330.4)--(180.4, 329.6)--(179.8, 329.2)--(179.1, 328.7)--(178.4, 328.4)--(178.3, 328.2)--(176.8, 328.7)--(175.9, 329.0)--(175.0, 329.3)--(174.9, 329.3)--(175.0, 329.4)--(175.1, 329.5)--(175.2, 329.7)--(175.2, 329.8)--(175.3, 329.9)--(175.4, 330.0)--(175.6, 330.1)--(175.7, 330.2)--(175.7, 330.3)--(175.8, 330.6)--(175.8, 331.0)--(175.8, 331.5)--(175.9, 332.0)--(175.9, 332.6)--(175.9, 332.7)--(175.9, 333.4)--(175.9, 334.1)--(175.9, 334.4)--(175.8, 336.1)--(175.8, 336.4)--(176.6, 336.9)--(176.7, 336.9)--(176.5, 337.2)--(176.1, 337.6)--(175.8, 337.9)--(175.5, 338.2)--(175.1, 338.9)--(175.0, 339.0)--(174.6, 339.7)--(173.9, 340.6)--(174.0, 340.6)--(174.0, 340.7)--(174.1, 340.9)--(174.2, 341.1)--(174.3, 341.5)--(174.3, 341.9)--(174.2, 342.0)--(173.9, 342.4)--(173.6, 342.5)--(173.6, 342.8)--(174.2, 342.8)--(174.8, 342.9)--(174.9, 342.9)--(174.8, 343.0)--(174.8, 343.1)--(174.5, 343.5)--(174.1, 343.9)--(173.8, 344.2)--(173.2, 344.6)--(173.1, 344.7)--(173.0, 344.7)--(172.0, 345.2)--(171.9, 345.2)--(171.9, 345.3)--(171.8, 345.4)--(171.6, 345.6)--(171.5, 345.8)--(171.3, 346.0)--(171.2, 346.2)--(171.1, 346.5)--(171.0, 346.8)--(171.0, 347.1)--(170.8, 347.3)--(170.1, 347.8)--(169.7, 348.2)--(169.6, 348.1)--(168.7, 348.7)--(168.1, 349.1)--(168.2, 349.2)--(167.6, 349.6)--(167.4, 349.7)--(167.3, 349.8)--(167.2, 349.8)--(166.9, 350.0)--(166.4, 350.2)--(166.0, 350.5)--(165.4, 350.8)--(164.9, 351.1)--(164.4, 351.4)--(164.2, 351.5)--(163.8, 351.7)--(163.0, 352.2)--(162.4, 352.5)--(161.8, 352.8)--(161.5, 353.1)--(160.8, 353.4)--(160.2, 353.7)--(159.7, 353.8)--(159.2, 353.9)--(158.5, 354.1)--(157.8, 354.2)--(157.2, 354.4)--(156.5, 354.5)--(155.9, 354.7)--(155.8, 354.7)--(155.8, 354.6)--(155.4, 354.8)--(154.8, 355.0)--(154.9, 355.1)--(155.3, 356.4)--(155.9, 356.3)--(155.9, 356.6)--(155.9, 357.0)--(155.9, 357.6)--(156.3, 358.2)--(156.8, 359.0)--(157.3, 359.8)--(157.2, 359.9)--(157.8, 360.5)--(157.6, 360.6)--(157.7, 360.6)--(158.6, 361.4)--(158.7, 361.5)--(158.8, 361.6)--(158.9, 361.6)--(159.1, 361.6)--(159.2, 361.7)--(159.4, 361.7)--(159.8, 361.7)--(159.7, 362.0)--(159.5, 362.3)--(159.7, 362.6)--(159.5, 363.5)--(159.5, 364.4)--(159.4, 365.5)--(159.1, 365.6)--(158.7, 365.8)--(158.7, 366.3)--(158.6, 366.8)--(158.9, 366.9)--(159.3, 367.0)--(160.2, 367.3)--(160.1, 367.6)--(160.7, 367.7)--(161.1, 367.8)--(161.3, 367.9)--(161.8, 368.2)--(161.8, 368.4)--(162.2, 368.6)--(162.7, 368.9)--(163.3, 369.4)--(163.8, 369.8)--(163.6, 370.3)--(163.3, 370.9)--(162.8, 371.8)--(162.4, 372.7)--(161.9, 373.6)--(161.9, 373.9)--(161.8, 374.0)--(161.8, 374.2)--(161.8, 374.4)--(161.7, 374.7)--(161.4, 376.6)--(161.4, 376.8)--(161.3, 377.1)--(161.0, 377.8)--(160.7, 378.6)--(160.6, 379.0)--(160.5, 379.2)--(160.5, 379.3)--(160.5, 379.5)--(160.4, 379.7)--(160.5, 379.8)--(161.4, 380.1)--(162.7, 380.6)--(164.3, 381.4)--(164.6, 381.6)--(164.9, 381.8)--(165.2, 382.0)--(165.5, 382.3)--(165.9, 382.5)--(167.0, 383.7)--(168.9, 385.5)--(169.4, 385.9)--(169.5, 386.0)--(169.7, 386.2)--(170.6, 386.8)--(171.4, 387.3)--(173.0, 388.2)--(174.0, 388.8)--(174.3, 388.9)--(174.5, 389.1)--(174.7, 389.1)--(174.7, 389.2)--(175.0, 389.3)--(175.4, 389.4)--(175.9, 389.5)--(176.2, 389.6)--(176.6, 389.6)--(176.9, 389.7)--(177.2, 389.7)--(177.3, 389.7)--(177.9, 389.7)--(178.2, 389.7)--(178.4, 389.6)--(178.6, 389.6)--(178.7, 389.5)--(179.2, 389.3)--(179.7, 389.2)--(179.9, 389.2)--(180.0, 389.2)--(180.2, 389.2)--(181.7, 389.1)--(182.7, 389.0)--(184.2, 388.8)--(186.4, 388.4)--(189.8, 387.8)--(190.9, 387.5)--(190.9, 387.6)--(191.0, 387.6)--(191.0, 387.7)--(191.0, 388.3)--(191.0, 388.4)--(191.0, 388.5)--(191.0, 388.6)--(190.9, 388.7)--(190.8, 388.9)--(190.6, 389.4)--(190.5, 390.2)--(190.4, 390.6)--(190.4, 390.8)--(190.3, 391.2)--(190.2, 391.8)--(189.8, 392.9)--(189.7, 393.3)--(189.5, 393.6)--(189.4, 393.8)--(189.3, 394.0)--(189.3, 394.1)--(189.1, 394.4)--(188.9, 394.6)--(188.8, 394.7)--(188.7, 394.8)--(188.5, 395.0)--(188.4, 395.1)--(188.2, 395.2)--(188.0, 395.4)--(187.9, 395.5)--(187.7, 395.6)--(187.6, 395.6)--(187.4, 395.7)--(187.2, 395.8)--(186.6, 396.0)--(186.4, 396.0)--(186.1, 396.2)--(186.0, 396.2)--(185.9, 396.3)--(185.7, 396.4)--(185.6, 396.5)--(185.2, 396.9)--(185.4, 397.1)--(185.7, 397.2)--(185.9, 397.3)--(186.1, 397.4)--(186.6, 397.6)--(186.9, 397.7)--(187.1, 397.7)--(187.5, 398.0)--(187.7, 398.2)--(188.0, 398.5)--(188.1, 398.4)--(188.4, 398.3)--(188.5, 398.8)--(188.6, 398.8)--(188.9, 399.0)--(189.2, 399.1)--(189.5, 399.2)--(190.0, 399.4)--(190.5, 399.5)--(191.1, 399.7)--(191.4, 399.7)--(192.8, 400.0)--(194.8, 400.4)--(194.4, 400.8)--(195.3, 401.4)--(196.4, 402.0)--(196.4, 402.1)--(196.5, 402.1)--(196.6, 402.2)--(196.7, 402.3)--(196.8, 402.3)--(196.9, 402.4)--(197.0, 402.4)--(197.1, 402.5)--(197.2, 402.5)--(197.3, 402.5)--(197.4, 402.5)--(197.5, 402.5)--(197.6, 402.5)--(197.7, 402.5)--(197.8, 402.5)--(198.1, 402.5)--(198.1, 402.4)--(198.2, 402.4)--(198.3, 402.4)--(198.4, 402.4)--(198.4, 402.5)--(198.5, 402.5)--(198.6, 402.6)--(198.8, 402.8)--(199.0, 403.0)--(199.3, 403.3)--(199.6, 403.5)--(199.8, 403.7)--(200.0, 403.9)--(200.1, 404.0)--(200.3, 404.1)--(200.6, 404.3)--(201.1, 404.8)--(201.2, 404.7)--(201.9, 403.8)--(202.2, 403.3)--(202.5, 402.8)--(202.8, 402.4)--(203.2, 402.0)--(203.5, 401.6)--(203.8, 401.3)--(204.3, 400.7)--(204.5, 400.4)--(204.9, 400.0)--(205.8, 398.8)--(206.4, 398.0)--(207.5, 396.5)--(207.7, 396.3)--(207.9, 395.9)--(208.1, 395.7)--(208.4, 395.4)--(209.4, 394.2)--(210.0, 393.5)--(210.3, 393.3)--(210.6, 393.0)--(211.0, 392.6)--(211.1, 392.4)--(211.6, 391.9)--(212.0, 391.2)--(213.1, 389.5)--(213.4, 389.2)--(214.4, 387.5)--(214.6, 387.3)--(214.6, 387.2)--(214.5, 387.2)--(213.5, 386.8)--(213.2, 386.7)--(212.9, 386.5)--(212.5, 386.4)--(212.3, 386.3)--(212.1, 385.8)--(211.9, 385.5)--(211.8, 385.4)--(211.7, 385.3)--(211.5, 385.1)--(211.3, 384.9)--(211.1, 384.9)--(211.0, 384.8)--(210.9, 384.7)--(210.8, 384.6)--(210.7, 384.5)--(210.6, 384.4)--(210.5, 384.1)--(210.5, 384.0)--(210.5, 383.9)--(210.5, 383.7)--(210.5, 383.6)--(210.5, 383.4)--(210.5, 383.3)--(210.5, 383.0)--(210.5, 382.8)--(210.5, 382.6)--(210.5, 382.4)--(210.4, 382.3)--(210.4, 382.1)--(210.4, 382.0)--(210.4, 381.9)--(210.4, 381.8)--(210.5, 381.2)--(210.5, 381.1)--(210.7, 380.7)--(210.8, 380.2)--(210.9, 379.9)--(211.1, 379.5)--(211.1, 379.4)--(211.2, 379.3)--(211.3, 379.1)--(211.3, 378.9)--(211.4, 378.5)--(211.6, 377.6)--(211.7, 377.4)--(211.7, 377.3)--(211.8, 377.0)--(211.9, 376.7)--(212.1, 376.2)--(212.2, 375.9)--(212.4, 375.6)--(212.5, 375.4)--(212.7, 374.9)--(212.8, 374.8)--(212.8, 374.7)--(212.8, 374.6)--(212.7, 374.6)--(212.6, 374.5)--(212.5, 374.4)--(212.3, 374.4)--(212.1, 374.3)--(212.0, 374.3)--(211.9, 374.3)--(211.7, 374.3)--(211.7, 374.2)--(211.6, 374.2)--(211.6, 374.1)--(211.6, 374.0)--(211.6, 373.9)--(211.7, 373.8)--(211.8, 373.5)--(211.9, 373.2)--(212.0, 373.0)--(212.3, 372.7)--(212.5, 372.6)--(212.6, 372.5)--(212.7, 372.3)--(212.8, 372.1)--(212.9, 372.0)--(212.9, 371.8)--(212.9, 371.7)--(213.0, 371.7)--(213.0, 371.6)--(213.3, 371.4)--(213.5, 371.1)--(213.7, 370.8)--(213.8, 370.6)--(214.0, 370.4)--(214.1, 370.3)--(214.1, 370.2)--(214.2, 370.2)--(214.2, 370.1)--(214.3, 370.1)--(214.4, 370.1)--(214.5, 370.0)--(214.7, 369.9)--(214.7, 369.8)--(214.8, 369.7)--(214.9, 369.7)--(215.1, 369.9)--(215.2, 369.8)--(215.3, 369.7)--(215.5, 369.7)--(215.7, 369.7)--(215.8, 369.7)--(215.9, 369.8)--(215.9, 369.9)--(215.8, 370.0)--(215.7, 370.0)--(215.7, 370.1)--(215.7, 370.2)--(215.8, 370.3)--(215.9, 370.3)--(216.0, 370.2)--(216.2, 370.2)--(216.2, 370.3)--(216.3, 370.3)--(216.2, 370.5)--(216.2, 370.6)--(216.3, 370.7)--(216.5, 370.9)--(216.5, 371.1)--(216.6, 371.1)--(216.6, 371.2)--(216.6, 371.3)--(216.5, 371.3)--(216.4, 371.3)--(216.3, 371.3)--(216.3, 371.2)--(216.2, 371.2)--(216.2, 371.4)--(216.2, 371.6)--(216.2, 371.8)--(216.3, 371.9)--(216.3, 372.1)--(216.6, 372.4)--(216.6, 372.5)--(216.5, 372.6)--(216.4, 372.7)--(216.2, 372.8)--(216.1, 373.0)--(216.0, 373.0)--(216.0, 373.1)--(216.1, 373.1)--(216.1, 373.2)--(216.3, 373.3)--(216.5, 373.7)--(216.5, 373.8)--(216.6, 373.8)--(216.7, 373.8)--(216.8, 373.6)--(216.9, 373.5)--(217.1, 373.5)--(217.2, 373.6)--(217.3, 373.7)--(217.4, 373.8)--(217.5, 373.9)--(217.6, 374.0)--(217.6, 374.1)--(217.6, 374.2)--(217.7, 374.2)--(217.7, 374.3)--(217.8, 374.3)--(217.8, 374.2)--(217.8, 374.1)--(217.8, 374.0)--(217.9, 374.0)--(218.0, 374.0)--(218.2, 374.0)--(218.3, 374.1)--(218.5, 374.1)--(218.6, 374.2)--(218.7, 374.2)--(218.9, 374.2)--(219.0, 374.2)--(219.0, 374.3)--(219.0, 374.4)--(218.9, 374.4)--(218.9, 374.5)--(218.9, 374.6)--(219.0, 374.6)--(219.1, 374.5)--(219.2, 374.6)--(219.3, 374.6)--(219.5, 374.5)--(219.6, 374.5)--(219.7, 374.4)--(219.7, 374.3)--(219.7, 374.2)--(219.7, 374.1)--(219.8, 374.0)--(219.8, 373.9)--(219.9, 373.9)--(219.9, 373.8)--(220.0, 373.8)--(220.1, 374.0)--(220.1, 374.1)--(220.2, 374.1)--(220.3, 374.0)--(220.2, 373.9)--(220.3, 373.9)--(220.4, 374.0)--(220.5, 374.1)--(220.6, 374.1)--(220.7, 374.1)--(220.8, 374.1)--(220.7, 374.0)--(220.6, 373.9)--(220.6, 373.8)--(220.7, 373.8)--(220.8, 373.8)--(220.9, 373.8)--(221.1, 373.8)--(221.2, 373.8)--(221.3, 374.0)--(221.5, 374.0)--(221.6, 373.9)--(221.7, 373.7)--(221.8, 373.7)--(221.8, 373.6)--(222.0, 373.5)--(222.1, 373.5)--(222.2, 373.5)--(222.2, 373.6)--(222.0, 373.7)--(222.0, 373.8)--(222.0, 373.9)--(222.1, 373.9)--(222.2, 373.8)--(222.3, 373.9)--(222.3, 374.0)--(222.3, 374.1)--(222.2, 374.1)--(222.2, 374.2)--(222.3, 374.2)--(222.3, 374.3)--(222.4, 374.3)--(222.5, 374.2)--(222.6, 374.2)--(222.7, 374.2)--(223.0, 374.3)--(223.1, 374.3)--(223.2, 374.2)--(223.4, 374.2)--(223.5, 374.1)--(223.6, 374.1)--(223.6, 374.2)--(223.5, 374.3)--(223.6, 374.4)--(223.7, 374.4)--(223.9, 374.4)--(223.9, 374.3)--(223.9, 374.2)--(223.7, 374.2)--(223.7, 374.1)--(223.8, 374.1)--(223.9, 374.1)--(224.0, 374.1)--(224.1, 374.0)--(224.2, 373.9)--(224.4, 373.8)--(224.4, 373.6)--(224.5, 373.5)--(224.5, 373.6)--(224.6, 373.6)--(224.7, 373.6)--(224.8, 373.5)--(224.9, 373.6)--(225.0, 373.7)--(225.2, 373.8)--(226.1, 372.7)--(226.4, 372.3)--(226.8, 371.8)--(227.0, 371.4)--(227.3, 371.1)--(227.6, 370.8)--(227.8, 370.5)--(228.1, 370.0)--(228.5, 369.5)--(228.6, 369.4)--(228.7, 369.3)--(228.9, 369.1)--(229.1, 368.9)--(229.3, 368.8)--(229.7, 368.4)--(229.9, 368.3)--(230.0, 368.1)--(230.2, 367.9)--(230.6, 367.5)--(230.8, 367.3)--(230.9, 367.1)--(231.1, 366.8)--(231.2, 366.7)--(231.5, 366.4)--(231.9, 365.9)--(232.4, 365.2)--(232.9, 364.6)--(233.1, 364.3)--(233.4, 363.9)--(233.7, 363.5)--(234.0, 363.2)--(234.4, 362.5)--(234.6, 362.1)--(235.3, 360.8)--(235.6, 360.5)--(235.8, 360.1)--(236.1, 359.8)--(236.5, 359.3)--(236.9, 358.7)--(237.1, 358.5)--(237.2, 358.3)--(237.4, 358.1)--(237.6, 357.9)--(238.1, 357.3)--(238.2, 357.2)--(238.5, 356.7)--(239.0, 356.2)--(239.3, 355.7)--(239.5, 355.5)--(239.5, 355.4)--(239.7, 355.2)--(239.9, 354.9)--(240.0, 354.8)--(240.2, 354.4)--(240.3, 354.3)--(240.6, 353.8)--(240.8, 353.6)--(241.0, 353.4)--(241.1, 353.3)--(241.5, 352.8)--(241.8, 352.5)--(242.2, 352.0)--(242.6, 351.6)--(243.0, 351.1)--(243.5, 350.6)--(243.8, 350.3)--(243.9, 350.1)--(244.0, 350.0)--(244.3, 349.6)--(244.6, 349.2)--(245.1, 348.6)--(245.5, 347.9)--(246.1, 347.1)--(246.5, 346.5)--(246.6, 346.3)--(246.9, 346.0)--(247.2, 345.4)--(247.8, 344.7)--(247.8, 344.6)--(248.3, 343.9)--(249.6, 342.2)--(250.0, 341.7)--(250.8, 340.8)--(251.4, 339.9)--(251.6, 339.8)--(251.7, 339.7)--(251.8, 339.5)--(252.5, 338.6)--(253.7, 337.0)--(254.5, 335.9)--(254.6, 335.7)--(255.3, 334.8)--(254.3, 332.3)--(254.3, 332.2)--(254.5, 331.9)--(254.7, 331.5)--(254.8, 331.4)--(254.8, 331.2)--(254.8, 331.0)--(254.7, 330.7)--(254.6, 330.5)--(254.4, 330.2)--(254.3, 329.8)--(254.0, 329.3)--(253.5, 328.3)--(252.6, 326.4)--(252.1, 325.4)--(252.0, 325.2)--cycle; +Bromley = (403.7, 75.3)--(403.6, 75.5)--(403.6, 75.7)--(403.5, 75.9)--(403.4, 76.0)--(403.4, 76.2)--(403.4, 76.3)--(403.3, 76.5)--(403.2, 76.7)--(403.2, 76.9)--(403.1, 77.2)--(403.1, 77.5)--(403.0, 77.7)--(402.9, 77.9)--(402.9, 78.4)--(402.9, 78.5)--(402.9, 78.6)--(402.9, 79.0)--(402.8, 79.1)--(402.8, 79.2)--(402.8, 79.3)--(402.7, 79.9)--(402.7, 80.0)--(402.7, 80.1)--(402.7, 80.2)--(402.5, 80.6)--(402.4, 80.8)--(402.4, 80.9)--(402.4, 81.0)--(402.4, 81.1)--(402.4, 81.2)--(402.4, 81.4)--(402.3, 81.9)--(402.2, 82.0)--(402.2, 82.2)--(402.1, 82.3)--(402.1, 82.5)--(402.0, 82.6)--(402.0, 82.7)--(401.9, 82.8)--(401.8, 82.9)--(401.8, 83.0)--(401.7, 83.1)--(401.7, 83.2)--(401.7, 83.3)--(401.6, 83.4)--(401.6, 83.5)--(401.6, 83.6)--(401.6, 83.7)--(401.6, 84.0)--(401.4, 84.4)--(401.3, 84.9)--(401.3, 85.1)--(401.3, 85.4)--(400.9, 86.9)--(400.8, 87.0)--(400.8, 87.2)--(400.8, 87.4)--(400.8, 87.5)--(400.8, 87.7)--(400.9, 87.9)--(401.0, 88.2)--(401.1, 88.6)--(401.2, 88.9)--(401.3, 89.2)--(401.3, 89.6)--(401.3, 90.0)--(401.4, 90.8)--(401.5, 91.1)--(401.5, 91.2)--(401.5, 91.3)--(401.6, 91.6)--(401.7, 92.2)--(401.8, 92.8)--(402.0, 93.7)--(401.8, 93.6)--(401.4, 93.6)--(401.3, 93.5)--(401.2, 93.5)--(401.1, 93.5)--(400.6, 93.4)--(400.5, 93.3)--(400.4, 93.3)--(400.3, 93.2)--(400.0, 93.1)--(399.8, 93.1)--(399.7, 93.0)--(399.4, 93.0)--(399.4, 93.1)--(399.3, 93.7)--(399.0, 94.7)--(398.9, 95.1)--(398.6, 95.8)--(398.6, 96.1)--(398.5, 96.5)--(398.5, 96.6)--(398.5, 96.8)--(398.4, 97.4)--(398.4, 97.7)--(398.3, 98.0)--(398.2, 98.3)--(398.2, 98.5)--(398.2, 98.7)--(398.0, 100.0)--(398.0, 100.2)--(397.7, 101.0)--(397.4, 102.1)--(397.2, 102.5)--(397.1, 102.8)--(397.1, 103.2)--(397.0, 103.6)--(396.9, 104.1)--(396.8, 104.7)--(396.7, 105.1)--(396.6, 105.6)--(396.5, 106.2)--(396.3, 106.8)--(396.1, 107.4)--(396.0, 107.7)--(396.0, 108.0)--(395.9, 108.2)--(395.9, 108.3)--(395.9, 108.7)--(396.0, 108.8)--(396.3, 109.2)--(396.3, 109.3)--(396.4, 109.3)--(396.4, 109.5)--(396.3, 110.0)--(396.3, 110.1)--(396.3, 110.2)--(396.3, 110.3)--(396.5, 110.5)--(396.5, 110.7)--(396.6, 111.3)--(396.3, 111.4)--(396.1, 111.5)--(395.9, 111.7)--(395.6, 111.9)--(395.4, 112.3)--(395.2, 112.6)--(394.9, 112.8)--(394.8, 112.9)--(394.9, 113.0)--(394.8, 113.3)--(394.6, 113.9)--(394.6, 114.3)--(394.5, 115.0)--(394.3, 115.7)--(394.2, 116.1)--(394.1, 116.4)--(393.8, 117.6)--(393.7, 117.9)--(393.6, 118.4)--(393.5, 118.9)--(393.4, 119.3)--(393.3, 119.7)--(393.1, 120.2)--(393.1, 120.4)--(393.0, 120.5)--(393.1, 120.5)--(393.0, 120.8)--(392.9, 120.8)--(392.8, 121.5)--(392.9, 121.6)--(392.8, 121.9)--(392.7, 122.3)--(392.6, 122.6)--(392.6, 122.7)--(392.5, 123.1)--(392.3, 123.8)--(392.1, 124.3)--(392.1, 124.4)--(392.1, 124.6)--(392.1, 124.7)--(392.1, 125.0)--(392.0, 125.3)--(391.9, 125.5)--(391.8, 126.1)--(391.9, 126.1)--(391.8, 126.3)--(391.6, 126.7)--(391.6, 127.0)--(391.4, 127.5)--(391.2, 128.4)--(391.1, 128.5)--(391.0, 129.1)--(390.9, 129.6)--(390.8, 129.7)--(390.8, 129.9)--(390.7, 130.1)--(390.6, 130.4)--(390.5, 130.5)--(390.5, 130.8)--(390.4, 131.3)--(390.4, 131.9)--(390.4, 132.0)--(390.3, 132.4)--(390.3, 132.6)--(390.2, 132.9)--(390.1, 133.3)--(390.1, 133.4)--(390.1, 133.6)--(390.0, 133.7)--(390.0, 133.8)--(389.9, 133.9)--(390.0, 133.9)--(390.0, 134.0)--(389.9, 134.1)--(389.8, 134.2)--(389.8, 134.3)--(389.5, 134.7)--(389.4, 134.9)--(389.3, 135.1)--(389.1, 135.2)--(389.0, 135.3)--(389.0, 135.4)--(388.7, 135.7)--(388.5, 136.0)--(388.3, 136.3)--(388.2, 136.5)--(388.0, 136.8)--(387.8, 136.9)--(387.7, 137.2)--(387.5, 137.4)--(387.3, 137.6)--(387.1, 137.7)--(387.0, 137.9)--(386.8, 138.0)--(386.5, 138.2)--(385.8, 138.7)--(385.5, 138.9)--(385.3, 139.0)--(385.2, 139.0)--(385.1, 139.0)--(385.1, 138.9)--(385.0, 138.8)--(384.9, 138.8)--(384.6, 138.8)--(384.4, 138.9)--(384.2, 139.1)--(384.0, 139.3)--(383.8, 139.6)--(383.6, 139.8)--(383.4, 139.9)--(383.0, 140.3)--(382.6, 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69.5)--(427.0, 69.2)--(426.9, 68.9)--(426.7, 68.7)--(426.3, 68.5)--(426.2, 68.5)--(426.2, 68.4)--(426.1, 68.4)--(426.0, 68.4)--(426.0, 68.3)--(425.9, 68.3)--(425.8, 68.3)--(425.7, 68.3)--(425.7, 68.2)--(425.6, 68.2)--(425.5, 68.2)--(425.4, 68.2)--(425.3, 68.2)--(425.2, 68.2)--(425.1, 68.2)--(425.0, 68.2)--(425.1, 68.5)--(425.2, 69.2)--(425.2, 69.3)--(425.2, 69.5)--(425.2, 69.6)--(425.2, 69.8)--(425.3, 70.4)--(425.4, 70.5)--(425.4, 70.6)--(425.4, 70.7)--(425.4, 71.1)--(425.4, 71.8)--(425.5, 72.1)--(425.5, 72.5)--(425.6, 72.6)--(425.6, 72.7)--(425.6, 72.8)--(425.7, 73.0)--(425.7, 73.1)--(425.8, 73.4)--(425.8, 73.5)--(425.9, 73.9)--(425.9, 74.0)--(426.0, 74.3)--(426.0, 74.4)--(426.0, 74.6)--(426.1, 74.6)--(426.2, 74.8)--(426.1, 74.8)--(425.9, 74.9)--(425.8, 75.0)--(425.6, 75.1)--(425.5, 75.2)--(425.4, 75.2)--(425.3, 75.3)--(425.3, 75.8)--(425.3, 76.0)--(425.3, 76.2)--(425.2, 76.4)--(425.2, 76.5)--(425.1, 77.1)--(425.1, 77.4)--(425.0, 77.6)--(424.9, 77.9)--(424.8, 78.2)--(424.7, 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78.4)--(412.4, 78.1)--(412.3, 77.9)--(412.3, 77.8)--(412.2, 77.5)--(412.1, 77.2)--(412.0, 76.7)--(412.0, 76.5)--(412.0, 76.4)--(411.9, 76.3)--(411.9, 76.1)--(411.8, 75.9)--(411.8, 75.8)--(411.7, 75.6)--(411.5, 75.2)--(411.4, 75.0)--(411.3, 74.9)--(411.2, 74.7)--(410.8, 74.0)--(410.7, 73.7)--(410.2, 73.0)--(410.2, 72.9)--(410.1, 72.8)--(410.0, 72.3)--(409.9, 72.1)--(409.9, 71.8)--(409.9, 71.7)--(409.9, 71.5)--(409.9, 71.4)--(409.9, 71.1)--(409.9, 71.0)--(409.9, 70.8)--(409.9, 70.7)--(409.8, 70.6)--(409.6, 70.2)--(409.6, 70.1)--(409.6, 70.0)--(409.5, 69.8)--(409.5, 69.3)--(409.4, 69.2)--(409.4, 69.1)--(409.4, 68.8)--(409.4, 68.6)--(409.3, 68.3)--(409.0, 67.4)--(408.9, 66.9)--(408.8, 66.6)--(408.6, 66.6)--(408.1, 66.7)--(407.6, 66.7)--(406.9, 66.7)--(406.4, 66.7)--(406.0, 66.7)--(405.7, 67.3)--(405.6, 67.5)--(405.5, 67.8)--(405.5, 68.1)--(405.4, 68.3)--(405.4, 68.8)--(405.4, 68.9)--(405.2, 69.5)--(405.1, 69.7)--(405.0, 70.4)--(404.9, 70.8)--(404.7, 71.6)--(404.5, 72.0)--(404.4, 72.4)--(404.3, 72.8)--(404.2, 73.3)--(404.1, 73.9)--(403.9, 74.5)--(403.9, 74.8)--cycle; +Camden = (288.4, 372.2)--(288.3, 372.1)--(288.3, 372.0)--(288.5, 371.7)--(288.8, 371.3)--(289.0, 370.6)--(289.1, 370.1)--(289.3, 368.9)--(289.3, 368.4)--(289.4, 368.1)--(289.4, 368.0)--(289.5, 367.8)--(289.7, 367.2)--(289.7, 367.1)--(289.7, 367.0)--(289.7, 366.9)--(289.7, 366.8)--(289.7, 366.3)--(289.7, 366.1)--(289.7, 366.0)--(289.7, 365.8)--(289.7, 365.7)--(289.8, 365.5)--(289.8, 365.2)--(289.8, 364.9)--(289.8, 364.6)--(289.8, 364.5)--(289.8, 364.4)--(289.7, 364.2)--(289.6, 363.7)--(289.6, 363.6)--(289.6, 363.4)--(289.6, 363.2)--(289.6, 362.8)--(289.6, 362.6)--(289.6, 362.5)--(289.7, 362.3)--(289.7, 362.2)--(289.8, 362.1)--(289.8, 362.0)--(290.0, 361.8)--(290.6, 360.5)--(290.8, 360.1)--(290.9, 359.9)--(291.4, 358.8)--(291.5, 358.6)--(292.2, 357.3)--(292.3, 357.2)--(292.4, 357.1)--(292.4, 357.0)--(292.5, 356.9)--(292.9, 356.6)--(293.3, 356.4)--(293.6, 356.3)--(293.7, 356.2)--(293.8, 356.1)--(293.9, 356.0)--(294.0, 355.7)--(294.2, 355.4)--(294.2, 355.3)--(294.4, 355.1)--(294.7, 354.8)--(295.2, 354.2)--(296.0, 353.5)--(296.2, 353.3)--(296.6, 353.0)--(296.6, 352.9)--(296.8, 352.8)--(296.9, 352.7)--(296.9, 352.6)--(296.9, 352.5)--(297.0, 352.5)--(297.0, 352.1)--(297.0, 351.9)--(297.3, 351.0)--(297.4, 350.7)--(297.5, 350.6)--(297.7, 350.4)--(298.0, 350.0)--(298.2, 349.9)--(298.8, 349.4)--(299.0, 349.3)--(299.1, 349.1)--(299.2, 349.1)--(299.3, 349.0)--(299.5, 348.7)--(299.6, 348.6)--(299.6, 348.5)--(299.6, 348.4)--(299.8, 347.8)--(300.0, 346.9)--(300.1, 346.3)--(300.2, 345.8)--(300.2, 345.7)--(300.4, 344.7)--(300.5, 344.0)--(300.6, 343.6)--(300.6, 343.5)--(300.6, 343.3)--(300.6, 343.1)--(300.7, 342.8)--(300.8, 342.4)--(300.9, 342.0)--(301.1, 341.7)--(301.2, 341.4)--(301.2, 341.3)--(301.3, 341.1)--(301.4, 340.8)--(301.8, 339.9)--(302.0, 339.6)--(302.2, 339.3)--(302.5, 338.9)--(302.6, 338.6)--(302.7, 338.5)--(302.9, 337.8)--(303.0, 337.5)--(303.1, 337.2)--(303.2, 336.7)--(303.2, 336.0)--(303.3, 335.8)--(303.3, 335.5)--(303.3, 335.2)--(303.3, 334.1)--(303.3, 333.5)--(303.3, 332.1)--(303.2, 330.3)--(303.3, 329.9)--(303.3, 329.8)--(303.6, 329.9)--(304.2, 330.0)--(305.1, 330.1)--(305.7, 329.9)--(306.3, 329.7)--(306.7, 329.6)--(307.1, 329.4)--(307.5, 329.2)--(307.8, 329.0)--(307.9, 328.9)--(307.9, 328.8)--(307.9, 328.7)--(308.0, 328.6)--(308.0, 328.5)--(308.1, 327.7)--(308.1, 327.3)--(308.2, 326.9)--(308.2, 326.8)--(308.3, 326.7)--(308.3, 326.6)--(308.4, 326.5)--(308.8, 326.1)--(309.0, 325.9)--(309.4, 325.5)--(309.5, 325.4)--(309.6, 325.4)--(309.7, 325.3)--(310.0, 325.2)--(309.9, 325.0)--(309.5, 324.6)--(309.2, 324.2)--(309.2, 324.1)--(309.3, 323.9)--(309.4, 323.5)--(309.6, 323.2)--(309.6, 323.1)--(309.7, 323.0)--(309.9, 322.8)--(310.4, 322.3)--(310.5, 322.2)--(310.7, 322.1)--(310.9, 322.1)--(311.3, 321.9)--(311.8, 321.8)--(312.2, 321.6)--(312.3, 321.6)--(312.5, 321.6)--(312.6, 321.5)--(312.7, 321.4)--(312.9, 321.5)--(313.1, 321.1)--(313.2, 320.9)--(313.4, 320.5)--(314.3, 320.6)--(314.6, 320.1)--(314.7, 319.9)--(314.8, 319.6)--(314.9, 319.2)--(315.0, 319.0)--(315.1, 318.5)--(315.4, 317.3)--(315.5, 316.6)--(315.2, 316.4)--(315.0, 316.3)--(314.7, 316.2)--(314.4, 316.0)--(314.1, 315.8)--(314.0, 315.8)--(313.9, 315.7)--(313.6, 315.7)--(313.0, 315.8)--(312.7, 315.8)--(312.0, 316.0)--(311.2, 316.1)--(310.7, 316.2)--(310.2, 316.2)--(309.7, 316.1)--(309.7, 316.0)--(309.7, 315.8)--(309.8, 315.8)--(310.0, 315.4)--(310.4, 314.8)--(310.9, 313.9)--(311.1, 313.4)--(311.3, 312.9)--(311.2, 312.9)--(310.8, 312.8)--(310.8, 312.7)--(309.7, 312.3)--(309.1, 313.6)--(307.4, 312.9)--(307.1, 312.7)--(306.6, 312.4)--(306.4, 312.4)--(306.4, 312.2)--(306.4, 312.1)--(306.3, 312.0)--(306.0, 311.9)--(305.8, 311.8)--(305.3, 311.9)--(304.6, 312.1)--(304.2, 312.2)--(304.0, 311.9)--(303.5, 312.5)--(303.1, 312.1)--(303.0, 312.0)--(302.7, 311.7)--(302.4, 311.4)--(301.9, 310.9)--(301.2, 310.5)--(300.9, 310.2)--(300.8, 310.1)--(300.7, 310.0)--(300.7, 309.9)--(300.6, 309.7)--(300.5, 309.7)--(300.4, 309.7)--(300.3, 309.6)--(300.2, 309.7)--(300.0, 309.9)--(299.8, 310.1)--(299.7, 310.2)--(299.4, 310.4)--(299.3, 310.5)--(299.2, 310.5)--(299.0, 310.5)--(298.8, 311.2)--(298.7, 311.6)--(298.2, 313.2)--(298.2, 313.4)--(298.2, 313.5)--(298.2, 313.8)--(298.2, 313.9)--(298.2, 314.0)--(297.9, 314.5)--(297.7, 314.5)--(297.7, 314.4)--(297.4, 314.3)--(296.8, 314.1)--(296.7, 314.6)--(296.4, 314.8)--(296.0, 315.2)--(295.8, 315.0)--(295.7, 315.0)--(295.6, 315.0)--(295.4, 315.4)--(295.4, 315.5)--(294.8, 316.5)--(294.4, 316.2)--(294.0, 316.7)--(293.6, 316.4)--(292.8, 317.4)--(292.7, 317.6)--(292.7, 317.7)--(292.6, 317.8)--(292.3, 318.1)--(292.0, 318.5)--(291.6, 319.1)--(291.1, 319.7)--(290.4, 320.5)--(289.3, 321.9)--(289.2, 321.8)--(289.1, 321.9)--(288.9, 321.9)--(288.9, 322.0)--(289.0, 322.6)--(288.9, 322.7)--(288.6, 322.7)--(288.4, 322.7)--(288.5, 322.0)--(288.1, 322.0)--(288.0, 322.0)--(287.7, 321.9)--(287.3, 323.4)--(286.3, 323.1)--(284.3, 329.9)--(283.5, 332.2)--(283.1, 333.7)--(282.1, 336.9)--(281.7, 336.8)--(281.3, 336.7)--(281.1, 336.7)--(281.0, 336.7)--(280.7, 336.6)--(280.6, 336.6)--(280.4, 336.5)--(280.2, 336.5)--(280.0, 336.3)--(278.8, 335.9)--(278.1, 335.6)--(277.5, 335.5)--(276.8, 336.5)--(275.8, 336.0)--(274.8, 335.4)--(274.6, 335.3)--(274.5, 335.2)--(273.7, 334.7)--(272.6, 335.9)--(271.5, 337.0)--(271.4, 337.1)--(271.2, 337.2)--(270.8, 337.6)--(270.6, 337.7)--(270.5, 337.7)--(270.5, 337.8)--(269.5, 336.8)--(268.8, 337.5)--(267.7, 336.6)--(267.4, 337.1)--(267.3, 337.1)--(267.3, 337.2)--(267.3, 337.3)--(267.4, 338.7)--(266.1, 338.8)--(264.3, 338.9)--(264.2, 338.9)--(264.0, 338.9)--(263.8, 338.9)--(263.6, 338.8)--(263.3, 338.8)--(263.1, 338.7)--(263.0, 338.6)--(262.9, 338.6)--(262.1, 338.2)--(261.3, 337.7)--(260.5, 337.3)--(260.4, 337.1)--(260.3, 337.1)--(260.1, 336.9)--(260.0, 336.8)--(260.0, 336.7)--(259.5, 336.0)--(259.4, 335.9)--(257.9, 333.5)--(257.9, 333.4)--(257.8, 333.4)--(257.8, 333.3)--(257.4, 333.0)--(257.2, 332.9)--(257.0, 333.2)--(256.9, 333.3)--(256.8, 333.4)--(256.8, 333.5)--(256.7, 333.6)--(256.6, 333.6)--(256.5, 333.7)--(256.1, 334.0)--(255.8, 334.2)--(255.6, 334.4)--(255.4, 334.6)--(255.3, 334.8)--(254.6, 335.7)--(254.5, 335.9)--(253.7, 337.0)--(252.5, 338.6)--(251.8, 339.5)--(251.7, 339.7)--(251.6, 339.8)--(251.4, 339.9)--(250.8, 340.8)--(250.0, 341.7)--(249.6, 342.2)--(248.3, 343.9)--(247.8, 344.6)--(247.8, 344.7)--(247.2, 345.4)--(246.9, 346.0)--(246.6, 346.3)--(246.5, 346.5)--(246.1, 347.1)--(245.5, 347.9)--(245.1, 348.6)--(244.6, 349.2)--(244.3, 349.6)--(244.0, 350.0)--(243.9, 350.1)--(243.8, 350.3)--(243.5, 350.6)--(243.0, 351.1)--(242.6, 351.6)--(242.2, 352.0)--(241.8, 352.5)--(241.5, 352.8)--(241.1, 353.3)--(241.0, 353.4)--(240.8, 353.6)--(240.6, 353.8)--(240.3, 354.3)--(240.2, 354.4)--(240.0, 354.8)--(239.9, 354.9)--(239.7, 355.2)--(239.5, 355.4)--(239.5, 355.5)--(239.7, 355.6)--(239.9, 355.7)--(240.2, 356.0)--(240.4, 356.3)--(240.7, 356.6)--(241.0, 356.8)--(241.3, 356.9)--(241.9, 357.3)--(242.2, 356.8)--(242.4, 356.5)--(242.7, 356.1)--(243.1, 356.3)--(243.3, 356.4)--(243.6, 356.6)--(244.0, 356.7)--(244.3, 356.9)--(244.5, 356.9)--(244.6, 356.9)--(244.7, 357.0)--(244.8, 356.7)--(244.8, 356.6)--(244.8, 356.4)--(244.8, 356.0)--(244.8, 355.9)--(244.8, 355.7)--(244.9, 355.7)--(245.2, 355.8)--(245.3, 355.8)--(245.4, 355.8)--(245.6, 355.9)--(245.9, 356.0)--(246.0, 356.1)--(246.1, 356.1)--(246.3, 356.1)--(246.5, 356.2)--(246.9, 356.3)--(247.6, 356.5)--(247.7, 356.5)--(247.9, 356.5)--(248.2, 356.6)--(248.5, 356.7)--(248.7, 356.8)--(248.9, 356.8)--(249.3, 356.8)--(249.2, 356.9)--(249.2, 357.0)--(249.2, 357.1)--(249.2, 357.2)--(249.1, 357.3)--(249.1, 357.5)--(249.1, 357.6)--(249.1, 357.7)--(249.1, 357.8)--(249.2, 358.0)--(249.4, 358.5)--(249.5, 358.6)--(249.6, 358.6)--(249.6, 358.7)--(249.7, 358.9)--(249.8, 358.9)--(249.8, 359.0)--(249.8, 359.1)--(249.8, 359.3)--(249.8, 359.7)--(250.0, 359.9)--(250.2, 360.1)--(250.4, 359.8)--(250.5, 359.9)--(250.6, 360.0)--(250.8, 360.1)--(251.0, 360.2)--(251.1, 360.2)--(251.0, 360.4)--(251.0, 360.6)--(250.9, 360.9)--(250.9, 361.0)--(250.9, 361.1)--(250.9, 361.5)--(250.9, 361.8)--(251.0, 361.9)--(251.0, 361.8)--(251.1, 361.9)--(251.3, 361.8)--(251.4, 361.8)--(251.5, 361.8)--(251.8, 361.7)--(252.0, 361.7)--(252.1, 361.6)--(252.3, 361.7)--(252.5, 361.8)--(252.5, 361.9)--(252.5, 362.0)--(252.6, 362.0)--(252.7, 362.1)--(252.8, 362.1)--(252.9, 362.2)--(253.0, 362.1)--(253.1, 362.2)--(253.6, 362.5)--(254.0, 362.9)--(254.4, 362.9)--(254.7, 362.9)--(254.8, 363.0)--(255.0, 363.1)--(255.1, 363.2)--(255.2, 363.3)--(255.2, 363.5)--(255.3, 363.8)--(255.3, 364.1)--(255.3, 364.3)--(255.5, 365.1)--(255.5, 365.4)--(255.5, 365.6)--(255.5, 366.2)--(255.5, 366.4)--(255.6, 366.6)--(255.6, 366.7)--(255.6, 366.8)--(255.8, 367.1)--(255.9, 367.1)--(255.9, 367.2)--(256.0, 367.3)--(256.0, 367.4)--(256.1, 367.4)--(256.4, 367.4)--(256.6, 367.7)--(256.8, 367.8)--(257.0, 367.9)--(257.0, 368.0)--(257.1, 368.0)--(257.2, 368.0)--(257.3, 368.0)--(257.3, 368.1)--(257.4, 368.1)--(257.5, 368.2)--(257.6, 368.5)--(257.7, 368.6)--(257.8, 368.7)--(257.9, 368.8)--(258.0, 368.8)--(258.0, 368.9)--(258.1, 368.9)--(258.1, 369.0)--(258.2, 369.0)--(258.5, 369.1)--(258.8, 369.3)--(259.1, 369.4)--(259.5, 369.5)--(259.6, 369.6)--(259.8, 369.7)--(260.0, 370.0)--(259.8, 370.2)--(259.9, 370.2)--(260.0, 370.2)--(260.8, 371.0)--(260.9, 371.0)--(261.0, 371.2)--(261.1, 371.3)--(261.2, 371.4)--(261.7, 371.8)--(261.8, 371.9)--(261.7, 372.1)--(261.8, 372.1)--(262.0, 372.1)--(262.2, 372.2)--(262.5, 372.3)--(263.0, 372.4)--(263.1, 372.4)--(263.2, 372.5)--(263.3, 372.5)--(263.4, 372.5)--(263.5, 372.5)--(263.5, 372.6)--(263.6, 372.6)--(263.6, 372.7)--(263.6, 372.8)--(263.7, 373.0)--(263.9, 372.8)--(264.1, 372.8)--(264.4, 372.8)--(264.7, 372.6)--(265.1, 372.5)--(265.2, 372.4)--(265.2, 372.3)--(265.7, 372.1)--(265.7, 372.0)--(265.9, 371.9)--(266.2, 371.8)--(266.3, 371.8)--(266.3, 371.9)--(266.4, 372.1)--(266.5, 372.2)--(266.6, 372.3)--(266.6, 372.5)--(266.7, 372.5)--(266.7, 372.9)--(266.7, 373.1)--(266.7, 373.5)--(266.8, 373.7)--(266.8, 373.8)--(266.8, 373.9)--(266.9, 373.9)--(266.9, 374.0)--(267.0, 374.1)--(267.0, 374.2)--(267.1, 374.3)--(267.3, 374.5)--(267.6, 374.8)--(267.8, 374.8)--(267.9, 374.9)--(267.9, 375.0)--(268.0, 375.1)--(268.3, 375.3)--(268.3, 375.4)--(268.5, 375.5)--(268.8, 375.6)--(269.2, 375.8)--(269.8, 375.9)--(270.3, 375.9)--(271.0, 376.0)--(271.2, 376.0)--(271.4, 376.0)--(271.5, 376.0)--(271.7, 375.9)--(271.8, 375.8)--(272.0, 375.8)--(272.1, 375.8)--(272.2, 375.8)--(272.5, 375.7)--(272.7, 375.6)--(272.8, 375.6)--(272.9, 375.6)--(273.0, 375.6)--(273.1, 375.6)--(273.2, 375.6)--(273.3, 375.6)--(273.8, 375.6)--(274.3, 375.7)--(275.0, 375.8)--(275.1, 375.8)--(275.2, 375.9)--(275.3, 375.9)--(275.4, 375.9)--(275.5, 375.9)--(275.6, 375.9)--(275.7, 375.9)--(275.8, 375.9)--(275.9, 375.9)--(276.0, 375.9)--(276.2, 375.7)--(276.9, 375.3)--(278.2, 375.2)--(279.3, 375.1)--(280.3, 375.0)--(281.2, 374.9)--(282.3, 374.8)--(282.8, 374.8)--(283.0, 374.7)--(283.2, 374.5)--(283.4, 374.4)--(283.7, 374.2)--(284.4, 373.9)--(285.1, 373.8)--(285.8, 373.6)--(286.5, 373.3)--(286.9, 373.3)--(287.0, 373.2)--(287.2, 373.1)--(287.4, 373.0)--(288.4, 372.4)--cycle; +City_of_London = (311.5, 307.8)--(311.4, 308.0)--(311.4, 308.4)--(311.4, 308.6)--(311.5, 308.8)--(311.4, 308.9)--(311.3, 309.2)--(311.2, 309.2)--(311.0, 309.5)--(311.1, 309.6)--(311.0, 309.8)--(310.8, 310.0)--(310.9, 310.1)--(310.8, 310.1)--(310.9, 310.2)--(310.9, 310.3)--(311.3, 310.5)--(311.2, 310.8)--(311.1, 311.1)--(311.5, 311.2)--(311.7, 311.2)--(311.7, 311.4)--(311.4, 312.3)--(311.3, 312.8)--(311.3, 312.9)--(311.1, 313.4)--(310.9, 313.9)--(310.4, 314.8)--(310.0, 315.4)--(309.8, 315.8)--(309.7, 315.8)--(309.7, 316.0)--(309.7, 316.1)--(310.2, 316.2)--(310.7, 316.2)--(311.2, 316.1)--(312.0, 316.0)--(312.7, 315.8)--(313.0, 315.8)--(313.6, 315.7)--(313.9, 315.7)--(314.0, 315.8)--(314.1, 315.8)--(314.4, 316.0)--(314.7, 316.2)--(315.0, 316.3)--(315.2, 316.4)--(315.5, 316.6)--(315.6, 316.6)--(316.0, 316.8)--(317.0, 317.4)--(317.8, 317.8)--(318.0, 317.9)--(318.1, 318.0)--(318.5, 318.2)--(318.7, 318.3)--(318.9, 318.4)--(319.0, 318.4)--(319.0, 318.5)--(319.1, 318.5)--(319.4, 318.6)--(319.7, 318.7)--(320.0, 318.9)--(320.3, 319.0)--(320.8, 319.2)--(320.8, 319.5)--(320.9, 319.8)--(320.9, 320.1)--(321.0, 320.1)--(321.0, 320.2)--(321.0, 320.3)--(321.0, 320.4)--(320.9, 320.8)--(320.8, 321.1)--(320.7, 321.2)--(320.7, 321.3)--(320.6, 321.5)--(320.5, 321.5)--(321.3, 322.1)--(321.4, 321.9)--(321.5, 321.9)--(321.6, 321.5)--(321.9, 321.7)--(321.9, 321.6)--(322.5, 321.9)--(322.9, 321.0)--(323.0, 320.4)--(323.1, 320.1)--(323.4, 320.1)--(323.6, 320.2)--(323.6, 320.1)--(323.7, 320.1)--(323.8, 319.9)--(324.4, 320.1)--(324.5, 319.6)--(324.6, 319.6)--(324.9, 319.5)--(325.2, 319.5)--(325.5, 319.5)--(325.9, 319.4)--(326.3, 319.3)--(326.2, 318.5)--(326.3, 318.4)--(326.6, 318.3)--(327.5, 317.9)--(327.9, 317.6)--(328.7, 317.3)--(328.8, 317.2)--(329.0, 317.8)--(329.1, 317.9)--(329.2, 318.4)--(329.4, 318.7)--(329.5, 318.9)--(330.8, 318.4)--(331.2, 318.7)--(331.5, 319.0)--(331.7, 319.3)--(331.8, 319.5)--(332.0, 319.8)--(332.2, 320.2)--(332.3, 320.8)--(332.5, 320.8)--(333.2, 320.5)--(334.1, 320.4)--(333.9, 319.9)--(333.9, 319.5)--(333.9, 319.4)--(333.8, 318.8)--(333.7, 318.1)--(333.5, 317.4)--(334.3, 317.5)--(334.5, 317.6)--(334.5, 317.0)--(334.4, 317.0)--(334.6, 316.8)--(334.7, 316.6)--(334.7, 316.5)--(334.8, 316.5)--(334.8, 316.3)--(335.0, 315.9)--(335.0, 315.8)--(335.3, 315.1)--(335.4, 315.0)--(335.4, 314.9)--(335.7, 314.6)--(336.6, 313.6)--(336.8, 313.3)--(337.1, 313.0)--(337.2, 312.9)--(337.4, 312.6)--(337.7, 312.3)--(337.8, 311.8)--(337.9, 311.4)--(338.0, 310.9)--(338.1, 310.3)--(338.1, 309.7)--(338.2, 309.5)--(338.2, 309.3)--(338.3, 309.1)--(338.3, 308.9)--(338.4, 308.7)--(338.4, 308.6)--(338.4, 308.4)--(338.4, 308.1)--(338.4, 308.0)--(338.3, 307.9)--(338.3, 307.8)--(338.2, 307.8)--(338.1, 307.8)--(337.9, 307.8)--(337.7, 307.8)--(337.5, 307.7)--(337.4, 307.7)--(337.3, 307.6)--(337.1, 307.6)--(337.0, 307.6)--(336.6, 307.4)--(336.5, 307.4)--(336.4, 307.4)--(336.4, 307.5)--(336.3, 307.5)--(336.3, 307.7)--(336.3, 307.8)--(336.1, 307.7)--(336.0, 308.1)--(335.9, 308.2)--(335.8, 308.2)--(335.5, 308.2)--(335.5, 308.1)--(335.5, 308.0)--(335.5, 307.9)--(335.5, 307.8)--(335.4, 307.8)--(335.1, 307.8)--(334.9, 307.7)--(334.8, 307.6)--(334.7, 307.5)--(334.5, 307.2)--(334.4, 307.0)--(334.2, 306.9)--(334.1, 306.6)--(334.0, 306.5)--(334.3, 306.4)--(334.3, 306.3)--(334.2, 306.1)--(334.1, 305.8)--(334.1, 305.7)--(334.1, 305.6)--(334.0, 305.2)--(333.9, 305.2)--(333.8, 305.1)--(333.3, 305.2)--(333.2, 305.3)--(333.1, 305.4)--(333.0, 305.4)--(332.6, 305.6)--(332.5, 305.6)--(332.5, 305.7)--(332.4, 305.7)--(332.3, 305.7)--(331.1, 305.9)--(331.0, 305.9)--(330.9, 305.9)--(330.9, 305.7)--(330.4, 305.8)--(329.7, 305.9)--(329.6, 305.9)--(329.2, 305.9)--(328.8, 306.0)--(328.6, 306.1)--(328.6, 306.2)--(328.1, 306.3)--(328.0, 306.3)--(327.9, 306.3)--(327.8, 306.3)--(327.8, 306.2)--(327.6, 306.2)--(327.4, 306.2)--(327.0, 306.2)--(326.8, 306.2)--(326.6, 306.2)--(326.4, 306.2)--(325.9, 306.4)--(325.6, 306.5)--(325.3, 306.6)--(325.1, 306.7)--(324.8, 306.7)--(324.7, 306.8)--(324.2, 307.0)--(323.9, 307.1)--(323.5, 307.2)--(323.2, 307.3)--(323.1, 307.3)--(322.8, 307.4)--(322.7, 307.4)--(322.8, 307.7)--(322.9, 307.9)--(322.8, 307.9)--(322.8, 307.8)--(322.5, 307.9)--(322.4, 307.5)--(322.2, 307.6)--(322.2, 307.5)--(321.7, 307.7)--(321.2, 307.8)--(321.0, 307.9)--(321.0, 307.8)--(320.9, 307.9)--(320.8, 307.9)--(320.7, 307.9)--(320.6, 307.9)--(320.4, 308.0)--(320.2, 308.0)--(320.1, 308.0)--(319.9, 308.0)--(319.7, 308.0)--(319.6, 308.0)--(317.5, 308.0)--(317.5, 307.9)--(317.3, 307.9)--(317.2, 307.9)--(317.1, 307.9)--(317.1, 308.0)--(316.7, 308.0)--(316.3, 308.0)--(315.7, 308.0)--(314.3, 308.0)--(314.2, 308.0)--(313.6, 307.9)--(311.8, 307.9)--(311.8, 307.8)--(311.7, 307.8)--cycle; +Croydon = (350.1, 95.0)--(350.0, 95.0)--(347.5, 95.2)--(345.9, 95.2)--(345.1, 97.8)--(344.7, 97.6)--(344.0, 97.3)--(343.9, 97.2)--(343.8, 97.1)--(343.5, 97.0)--(343.1, 96.9)--(342.9, 96.8)--(342.5, 96.7)--(342.3, 96.6)--(342.1, 96.5)--(341.6, 96.5)--(341.6, 96.4)--(341.1, 96.2)--(340.8, 96.2)--(340.0, 96.1)--(339.7, 96.0)--(339.7, 95.6)--(339.7, 94.7)--(339.8, 94.2)--(339.5, 94.1)--(339.4, 94.1)--(339.3, 94.1)--(339.2, 94.1)--(339.1, 94.0)--(339.0, 94.0)--(338.8, 93.8)--(338.7, 93.7)--(338.6, 93.6)--(338.4, 93.4)--(338.2, 93.1)--(338.3, 93.0)--(338.2, 93.0)--(338.1, 92.7)--(338.0, 92.7)--(337.9, 92.6)--(337.8, 92.6)--(337.6, 92.9)--(337.5, 92.9)--(337.4, 92.9)--(337.2, 92.7)--(336.9, 92.6)--(336.5, 92.5)--(336.4, 92.5)--(336.5, 92.2)--(336.1, 92.0)--(335.7, 91.6)--(336.1, 91.1)--(336.3, 90.8)--(336.6, 90.3)--(336.8, 89.9)--(337.1, 89.5)--(337.5, 88.7)--(337.6, 88.5)--(337.8, 88.0)--(337.8, 87.7)--(337.9, 87.2)--(337.9, 86.9)--(337.8, 86.5)--(337.8, 85.8)--(337.6, 85.7)--(337.5, 85.6)--(337.4, 85.6)--(337.3, 85.5)--(336.8, 85.1)--(336.6, 85.0)--(336.4, 84.9)--(335.4, 84.2)--(334.6, 83.8)--(334.6, 83.7)--(334.5, 83.6)--(334.0, 83.0)--(333.4, 82.3)--(333.3, 82.1)--(333.1, 81.9)--(333.2, 81.5)--(333.2, 81.4)--(333.2, 81.2)--(333.2, 81.0)--(333.2, 80.6)--(333.2, 80.4)--(333.3, 79.7)--(333.2, 79.5)--(333.2, 79.3)--(333.2, 79.2)--(333.0, 78.9)--(332.7, 78.8)--(332.7, 78.7)--(332.7, 78.6)--(332.6, 78.5)--(332.6, 78.4)--(332.6, 78.3)--(331.9, 78.2)--(331.8, 78.1)--(331.6, 77.6)--(331.7, 77.5)--(331.8, 77.4)--(331.8, 76.9)--(331.8, 76.8)--(331.8, 76.6)--(331.8, 76.4)--(331.8, 76.0)--(331.7, 75.5)--(329.9, 74.7)--(329.4, 74.5)--(329.2, 74.5)--(328.8, 75.2)--(328.5, 75.3)--(328.5, 75.4)--(328.3, 75.4)--(328.0, 75.3)--(327.3, 75.0)--(327.4, 74.8)--(327.4, 74.7)--(327.3, 74.6)--(327.4, 74.5)--(327.5, 74.1)--(327.5, 74.0)--(327.7, 73.8)--(328.1, 73.5)--(328.3, 73.4)--(328.5, 73.3)--(328.7, 73.2)--(328.8, 73.2)--(329.0, 73.2)--(329.4, 73.2)--(329.5, 73.1)--(327.1, 71.2)--(326.1, 70.5)--(326.1, 70.4)--(325.9, 70.3)--(325.5, 69.9)--(325.4, 69.8)--(325.3, 69.7)--(325.2, 69.6)--(325.1, 69.6)--(325.0, 69.6)--(324.9, 69.5)--(324.6, 69.4)--(324.2, 69.3)--(324.1, 69.3)--(323.9, 69.2)--(323.8, 69.2)--(323.6, 69.0)--(321.2, 67.3)--(320.0, 66.3)--(319.5, 66.0)--(319.2, 65.7)--(318.7, 65.3)--(318.1, 64.9)--(315.5, 65.0)--(315.4, 65.0)--(315.4, 64.9)--(315.4, 64.8)--(315.5, 64.3)--(315.6, 64.0)--(315.5, 63.9)--(315.4, 64.0)--(315.2, 63.9)--(315.3, 63.8)--(315.2, 63.8)--(315.1, 63.7)--(315.0, 63.5)--(315.0, 63.4)--(314.8, 63.1)--(314.8, 63.0)--(314.7, 62.8)--(314.5, 62.4)--(314.4, 62.2)--(314.4, 62.1)--(314.3, 62.0)--(314.2, 62.0)--(314.1, 61.7)--(314.0, 61.5)--(314.0, 61.3)--(313.9, 61.2)--(314.0, 61.1)--(314.0, 60.8)--(313.9, 60.6)--(313.7, 60.1)--(313.6, 59.7)--(313.5, 59.5)--(313.5, 59.2)--(313.5, 59.0)--(313.2, 59.0)--(312.9, 59.0)--(311.1, 58.8)--(309.0, 58.5)--(308.9, 58.6)--(308.9, 58.8)--(308.8, 58.8)--(308.7, 58.9)--(308.6, 59.0)--(308.1, 59.7)--(307.9, 60.1)--(307.8, 60.2)--(307.7, 60.3)--(307.7, 60.5)--(307.6, 61.0)--(307.6, 61.4)--(307.5, 61.5)--(307.5, 61.6)--(307.0, 62.4)--(306.8, 62.6)--(306.3, 63.2)--(306.2, 63.4)--(306.1, 63.4)--(305.8, 63.8)--(305.2, 64.5)--(304.8, 64.9)--(304.7, 64.9)--(304.7, 65.0)--(304.5, 65.5)--(304.4, 65.6)--(304.2, 66.2)--(304.0, 66.7)--(303.9, 66.9)--(303.8, 67.2)--(303.8, 67.3)--(303.8, 67.4)--(303.8, 67.5)--(303.8, 67.6)--(303.9, 67.7)--(304.0, 67.9)--(304.0, 68.0)--(304.0, 68.1)--(303.9, 68.2)--(303.8, 68.3)--(303.4, 68.6)--(303.3, 68.8)--(303.1, 69.0)--(302.8, 69.5)--(302.7, 69.6)--(302.7, 69.7)--(302.6, 69.7)--(302.5, 69.8)--(302.4, 69.9)--(302.0, 70.6)--(301.6, 71.2)--(301.6, 71.3)--(301.5, 71.3)--(300.8, 71.2)--(300.7, 71.2)--(300.5, 71.1)--(300.4, 71.0)--(300.3, 71.0)--(300.3, 71.1)--(300.1, 72.0)--(300.0, 72.5)--(300.0, 72.8)--(299.9, 73.0)--(299.5, 73.9)--(299.4, 73.9)--(299.2, 73.7)--(299.1, 73.7)--(299.0, 73.7)--(298.7, 73.5)--(298.6, 73.4)--(298.5, 73.4)--(298.4, 73.3)--(297.9, 72.9)--(297.7, 72.8)--(297.5, 72.7)--(297.3, 72.6)--(297.3, 72.5)--(297.2, 72.5)--(297.1, 72.4)--(297.0, 72.4)--(296.8, 72.5)--(296.7, 72.6)--(296.5, 72.7)--(296.2, 72.8)--(296.0, 72.8)--(295.9, 72.8)--(295.7, 72.9)--(295.1, 74.3)--(294.6, 73.5)--(294.3, 73.7)--(293.7, 72.7)--(293.1, 72.9)--(292.0, 73.5)--(291.7, 73.6)--(291.0, 73.6)--(290.8, 73.0)--(290.7, 73.1)--(290.6, 73.1)--(290.5, 73.2)--(290.0, 73.6)--(289.9, 73.6)--(289.7, 73.7)--(288.9, 74.3)--(288.8, 74.3)--(288.3, 74.6)--(286.9, 74.1)--(286.9, 74.2)--(285.9, 76.7)--(285.7, 76.7)--(285.7, 76.8)--(285.2, 77.6)--(285.2, 77.7)--(285.2, 77.8)--(287.0, 79.1)--(287.4, 79.9)--(286.9, 79.8)--(286.5, 79.8)--(286.0, 79.8)--(285.9, 79.9)--(285.8, 80.0)--(285.7, 80.1)--(285.6, 80.1)--(285.5, 80.1)--(286.0, 80.8)--(286.2, 81.2)--(286.6, 82.1)--(286.9, 83.0)--(287.4, 84.1)--(287.4, 84.2)--(286.7, 84.3)--(286.5, 84.2)--(286.4, 84.3)--(286.1, 84.3)--(285.9, 84.3)--(285.8, 84.4)--(285.5, 84.6)--(285.3, 84.7)--(285.1, 84.8)--(284.8, 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187.9)--(347.0, 187.9)--(347.3, 187.8)--(347.3, 187.6)--(347.2, 187.4)--(347.7, 187.3)--(347.8, 187.3)--(348.1, 187.3)--(348.4, 187.2)--(348.9, 187.3)--(350.0, 185.7)--(350.4, 185.1)--(350.0, 184.8)--(350.1, 184.6)--(350.3, 184.8)--(350.9, 184.8)--(351.0, 184.9)--(351.3, 185.2)--(351.7, 185.5)--(352.3, 185.9)--(352.7, 186.2)--(353.2, 186.4)--(353.4, 186.6)--(353.8, 186.7)--(354.2, 186.9)--(354.7, 187.0)--(354.9, 187.1)--(355.2, 187.2)--(355.3, 187.4)--(355.7, 186.8)--(356.1, 186.3)--(356.4, 185.7)--(356.7, 185.6)--(357.0, 185.3)--(356.0, 183.8)--(356.6, 183.4)--(356.9, 183.2)--(357.2, 183.6)--(357.3, 183.5)--(357.0, 183.1)--(356.9, 182.6)--(356.3, 181.8)--(355.2, 180.3)--(355.3, 180.3)--(355.2, 180.1)--(355.4, 179.9)--(355.4, 180.0)--(355.8, 179.8)--(355.8, 179.7)--(355.9, 179.5)--(355.9, 179.2)--(356.2, 179.2)--(356.4, 179.1)--(356.4, 178.7)--(356.5, 178.6)--(356.6, 178.7)--(356.7, 178.5)--(357.3, 178.3)--(357.4, 178.3)--(357.9, 178.3)--(358.3, 178.3)--(358.5, 178.3)--(358.7, 178.3)--(359.0, 178.2)--(359.2, 178.1)--(359.3, 177.9)--(359.6, 177.3)--(359.6, 177.2)--(359.8, 177.0)--(359.9, 176.8)--(360.1, 176.9)--(360.3, 177.0)--(360.4, 176.8)--(360.5, 176.5)--(360.7, 176.2)--(361.0, 175.8)--(361.4, 175.3)--(362.1, 174.6)--(362.5, 174.4)--(362.8, 174.2)--(363.1, 174.1)--(363.5, 174.0)--(363.6, 174.1)--(363.8, 174.0)--(364.0, 173.9)--(364.4, 173.8)--(364.8, 173.7)--(365.0, 173.6)--(365.2, 173.6)--(365.3, 173.6)--(365.7, 173.6)--(366.0, 173.5)--(366.3, 173.4)--(366.8, 173.3)--(367.2, 173.2)--(367.2, 173.1)--(367.1, 172.2)--(366.8, 170.5)--(366.8, 170.3)--(366.6, 169.2)--(366.4, 167.4)--(366.4, 166.6)--(366.6, 165.5)--(367.0, 163.6)--(367.2, 162.6)--(367.6, 162.7)--(367.2, 160.5)--(367.3, 160.4)--(367.3, 160.5)--(367.4, 160.5)--(367.4, 160.4)--(367.5, 160.4)--(367.7, 160.3)--(368.1, 160.3)--(368.8, 160.4)--(369.7, 160.6)--(370.2, 160.6)--(371.1, 160.7)--(371.2, 160.7)--(371.3, 160.7)--(371.4, 160.7)--(371.5, 160.8)--(371.6, 160.8)--(371.7, 160.8)--(371.8, 160.8)--(371.9, 160.8)--(371.9, 160.9)--(372.0, 160.9)--(372.1, 160.9)--(372.2, 160.9)--(372.2, 161.0)--(372.3, 161.0)--(372.4, 161.0)--(372.4, 161.1)--(372.5, 161.1)--(372.6, 161.2)--(372.7, 161.2)--(372.7, 161.3)--(372.8, 161.3)--(372.8, 161.4)--(372.9, 161.4)--(372.9, 161.5)--(373.0, 161.6)--(373.0, 161.7)--(373.1, 161.7)--(373.1, 161.8)--(373.2, 161.8)--(373.2, 161.9)--(373.2, 162.0)--(373.3, 162.0)--(373.3, 162.1)--(373.3, 162.2)--(373.4, 162.3)--(373.4, 162.4)--(373.4, 162.5)--(373.4, 162.6)--(373.4, 162.7)--(373.4, 162.8)--(373.4, 162.9)--(373.3, 163.2)--(374.2, 163.3)--(374.2, 163.0)--(374.3, 162.7)--(374.3, 162.4)--(374.4, 162.1)--(374.5, 161.6)--(374.7, 161.1)--(374.8, 161.0)--(374.8, 160.9)--(374.8, 160.6)--(374.9, 160.5)--(374.9, 160.1)--(374.7, 160.0)--(374.9, 159.6)--(375.0, 159.6)--(375.0, 159.5)--(375.1, 159.2)--(375.3, 158.9)--(375.3, 158.7)--(375.5, 158.5)--(375.5, 158.3)--(375.5, 158.2)--(375.6, 158.0)--(375.6, 157.9)--(376.0, 157.0)--(376.2, 156.5)--(376.2, 156.2)--(376.4, 155.8)--(376.7, 155.4)--(376.7, 154.9)--(376.6, 154.7)--(376.6, 154.5)--(376.5, 154.4)--(376.6, 154.4)--(376.6, 154.3)--(376.7, 154.3)--(376.7, 154.0)--(376.7, 153.5)--(376.7, 153.1)--(376.7, 152.9)--(376.7, 152.7)--(376.8, 152.6)--(376.8, 152.5)--(376.8, 152.4)--(376.8, 152.2)--(376.8, 151.9)--(376.8, 151.7)--(376.8, 151.4)--(376.9, 151.1)--(377.0, 151.1)--(377.0, 150.8)--(377.0, 150.5)--(377.1, 150.2)--(377.1, 150.0)--(377.1, 149.6)--(377.0, 149.6)--(376.5, 149.6)--(376.5, 149.3)--(376.2, 149.3)--(376.2, 148.9)--(376.8, 148.1)--(376.7, 147.8)--(376.9, 147.7)--(377.1, 147.8)--(377.3, 148.0)--(377.8, 148.4)--(377.9, 148.2)--(378.0, 147.9)--(378.1, 147.8)--(378.3, 147.7)--(378.4, 147.5)--(378.5, 147.3)--(378.6, 147.1)--(378.7, 147.0)--(378.8, 146.7)--(378.9, 146.6)--(379.1, 146.4)--(379.3, 146.3)--(379.4, 146.2)--(379.4, 146.0)--(379.5, 146.0)--(379.5, 145.9)--(379.6, 145.8)--(379.7, 145.7)--(379.8, 145.6)--(379.9, 145.5)--(380.0, 145.3)--(380.1, 145.1)--(380.2, 145.0)--(380.2, 144.9)--(380.3, 144.7)--(380.3, 144.5)--(380.4, 144.2)--(380.5, 144.0)--(380.6, 143.8)--(380.7, 143.7)--(380.7, 143.5)--(380.8, 143.5)--(380.6, 143.4)--(380.7, 143.3)--(380.8, 143.2)--(381.0, 143.0)--(381.1, 142.8)--(381.4, 142.6)--(381.9, 141.7)--(382.2, 141.2)--(382.6, 140.7)--(383.0, 140.3)--(383.4, 139.9)--(383.6, 139.8)--(383.8, 139.6)--(384.0, 139.3)--(384.2, 139.1)--(384.4, 138.9)--(384.6, 138.8)--(384.9, 138.8)--(385.0, 138.8)--(385.1, 138.9)--(385.1, 139.0)--(385.2, 139.0)--(385.3, 139.0)--(385.5, 138.9)--(385.8, 138.7)--(386.5, 138.2)--(386.8, 138.0)--(387.0, 137.9)--(387.1, 137.7)--(387.3, 137.6)--(387.5, 137.4)--(387.7, 137.2)--(387.8, 136.9)--(388.0, 136.8)--(388.2, 136.5)--(388.3, 136.3)--(388.5, 136.0)--(388.7, 135.7)--(389.0, 135.4)--(389.0, 135.3)--(389.1, 135.2)--(389.3, 135.1)--(389.4, 134.9)--(389.5, 134.7)--(389.8, 134.3)--(389.8, 134.2)--(389.9, 134.1)--(390.0, 134.0)--(390.0, 133.9)--(389.9, 133.9)--(390.0, 133.8)--(390.0, 133.7)--(390.1, 133.6)--(390.1, 133.4)--(390.1, 133.3)--(390.2, 132.9)--(390.3, 132.6)--(390.3, 132.4)--(390.4, 132.0)--(390.4, 131.9)--(390.4, 131.3)--(390.5, 130.8)--(390.5, 130.5)--(390.6, 130.4)--(390.7, 130.1)--(390.8, 129.9)--(390.8, 129.7)--(390.9, 129.6)--(391.0, 129.1)--(391.1, 128.5)--(391.2, 128.4)--(391.4, 127.5)--(391.6, 127.0)--(391.6, 126.7)--(391.8, 126.3)--(391.9, 126.1)--(391.8, 126.1)--(391.9, 125.5)--(392.0, 125.3)--(392.1, 125.0)--(392.1, 124.7)--(392.1, 124.6)--(392.1, 124.4)--(392.1, 124.3)--(392.3, 123.8)--(392.5, 123.1)--(392.6, 122.7)--(392.6, 122.6)--(392.7, 122.3)--(392.8, 121.9)--(392.9, 121.6)--(392.8, 121.5)--(392.9, 120.8)--(393.0, 120.8)--(393.1, 120.5)--(393.0, 120.5)--(393.1, 120.4)--(393.1, 120.2)--(393.3, 119.7)--(393.4, 119.3)--(393.5, 118.9)--(393.6, 118.4)--(393.7, 117.9)--(393.8, 117.6)--(394.1, 116.4)--(394.2, 116.1)--(394.3, 115.7)--(394.5, 115.0)--(394.6, 114.3)--(394.6, 113.9)--(394.8, 113.3)--(394.9, 113.0)--(394.8, 112.9)--(394.9, 112.8)--(395.2, 112.6)--(395.4, 112.3)--(395.6, 111.9)--(395.9, 111.7)--(396.1, 111.5)--(396.3, 111.4)--(396.6, 111.3)--(396.5, 110.7)--(396.5, 110.5)--(396.3, 110.3)--(396.3, 110.2)--(396.3, 110.1)--(396.3, 110.0)--(396.4, 109.5)--(396.4, 109.3)--(396.3, 109.3)--(396.3, 109.2)--(396.0, 108.8)--(395.9, 108.7)--(395.9, 108.3)--(395.9, 108.2)--(396.0, 108.0)--(395.1, 107.8)--(394.4, 107.7)--(394.0, 107.5)--(393.8, 108.1)--(393.5, 108.7)--(393.2, 109.2)--(392.7, 109.6)--(391.8, 110.4)--(391.0, 111.2)--(390.8, 111.4)--(390.3, 111.7)--(389.6, 112.2)--(389.2, 112.5)--(389.3, 112.9)--(388.6, 113.0)--(388.5, 113.0)--(388.4, 113.0)--(388.3, 113.0)--(388.2, 113.0)--(388.1, 113.0)--(388.0, 113.1)--(387.9, 113.0)--(387.8, 113.0)--(387.7, 113.0)--(387.6, 113.0)--(387.5, 113.0)--(387.4, 113.0)--(387.3, 113.0)--(387.2, 113.0)--(387.1, 112.9)--(387.0, 112.9)--(386.9, 112.9)--(386.8, 112.9)--(386.7, 112.7)--(386.6, 112.6)--(386.7, 112.5)--(386.8, 112.1)--(386.6, 111.9)--(386.0, 111.9)--(385.0, 110.3)--(384.8, 110.3)--(384.8, 110.1)--(384.8, 109.6)--(384.7, 109.4)--(384.6, 109.2)--(384.4, 109.0)--(384.4, 108.6)--(384.4, 108.4)--(384.2, 108.6)--(384.2, 108.8)--(384.0, 109.1)--(383.8, 109.3)--(383.8, 109.4)--(383.7, 109.7)--(383.6, 110.0)--(383.5, 110.3)--(383.2, 110.8)--(383.0, 111.0)--(382.7, 111.4)--(382.4, 111.8)--(382.3, 111.9)--(382.2, 111.9)--(381.9, 111.8)--(381.8, 111.9)--(381.7, 112.2)--(381.6, 112.8)--(381.5, 113.1)--(381.1, 113.6)--(381.0, 113.9)--(380.9, 114.2)--(380.8, 114.3)--(380.7, 114.9)--(380.6, 115.0)--(380.6, 115.1)--(380.4, 115.2)--(380.2, 115.4)--(379.9, 115.7)--(379.9, 115.8)--(379.9, 116.1)--(379.8, 116.4)--(379.3, 117.0)--(379.2, 117.2)--(378.7, 117.5)--(378.5, 117.6)--(378.2, 117.7)--(377.6, 117.7)--(377.4, 117.7)--(377.3, 117.7)--(377.2, 117.8)--(376.9, 117.8)--(376.6, 118.0)--(376.3, 117.9)--(376.0, 117.9)--(375.8, 117.9)--(375.7, 117.8)--(375.4, 117.7)--(375.0, 117.6)--(374.8, 117.5)--(374.6, 117.5)--(374.3, 117.3)--(374.0, 117.2)--(373.9, 117.1)--(373.8, 117.0)--(373.6, 116.9)--(373.1, 116.8)--(373.0, 116.7)--(372.9, 116.7)--(372.7, 116.7)--(372.6, 116.6)--(372.5, 116.6)--(372.0, 116.6)--(371.8, 116.6)--(371.6, 116.6)--(371.3, 116.5)--(370.9, 116.6)--(370.7, 116.7)--(370.2, 117.1)--(370.1, 117.2)--(370.0, 117.2)--(369.8, 117.2)--(369.5, 117.3)--(369.1, 117.4)--(368.8, 117.4)--(368.4, 117.6)--(368.3, 117.7)--(368.1, 117.7)--(367.9, 117.7)--(367.7, 117.8)--(367.6, 117.8)--(367.3, 117.4)--(366.7, 116.8)--(366.6, 116.6)--(366.4, 116.2)--(366.3, 115.9)--(366.2, 115.5)--(366.0, 115.2)--(365.9, 115.0)--(365.9, 114.8)--(365.8, 114.6)--(365.7, 114.0)--(365.6, 113.6)--(365.4, 113.2)--(365.0, 112.3)--(364.4, 112.1)--(364.0, 112.1)--(363.2, 112.2)--(362.9, 112.1)--(362.6, 112.1)--(362.3, 112.0)--(361.8, 111.9)--(361.6, 111.7)--(361.3, 111.5)--(361.0, 111.3)--(360.7, 111.2)--(360.5, 111.1)--(360.4, 111.0)--(360.1, 111.0)--(359.9, 111.0)--(359.7, 110.9)--(359.2, 110.9)--(359.4, 110.3)--(359.4, 110.2)--(359.0, 110.0)--(358.9, 109.9)--(358.7, 109.7)--(358.7, 109.4)--(358.7, 109.1)--(358.7, 108.9)--(359.0, 108.2)--(359.2, 107.9)--(359.4, 107.8)--(359.7, 107.5)--(359.9, 107.3)--(360.3, 106.8)--(360.4, 106.5)--(360.2, 106.4)--(360.6, 105.8)--(360.7, 105.6)--(360.9, 105.1)--(361.0, 104.9)--(361.0, 104.8)--(361.0, 104.7)--(361.0, 104.5)--(361.1, 104.1)--(361.1, 103.9)--(361.1, 103.0)--(361.1, 102.7)--(360.9, 102.6)--(360.9, 102.7)--(360.8, 102.7)--(360.5, 102.8)--(360.2, 102.8)--(360.0, 102.8)--(359.8, 102.7)--(359.7, 102.3)--(359.4, 101.9)--(359.2, 101.6)--(359.1, 101.4)--(359.1, 101.3)--(359.0, 100.9)--(359.0, 100.8)--(358.8, 100.6)--(358.8, 100.0)--(358.8, 99.7)--(358.8, 99.5)--(358.5, 99.5)--(358.2, 99.4)--(358.0, 99.4)--(357.9, 99.4)--(357.7, 99.2)--(357.6, 99.1)--(357.2, 98.8)--(356.8, 98.5)--(356.4, 98.1)--(356.3, 98.1)--(355.8, 98.2)--(355.6, 98.3)--(355.4, 98.6)--(355.2, 98.5)--(355.1, 98.4)--(354.9, 98.2)--(354.4, 97.7)--(354.3, 97.6)--(354.1, 97.3)--(353.9, 97.1)--(353.7, 96.9)--(353.6, 96.9)--(353.5, 96.9)--(353.1, 96.7)--(353.0, 96.7)--(352.9, 96.7)--(352.8, 96.7)--(352.7, 96.7)--(352.5, 96.7)--(352.4, 96.7)--(352.2, 96.6)--(352.2, 96.5)--(351.9, 96.2)--(351.8, 96.0)--(351.6, 95.9)--(351.5, 95.8)--(351.4, 95.7)--(351.2, 95.6)--(350.6, 95.3)--(350.2, 95.1)--cycle; +Ealing = (102.5, 328.8)--(102.5, 328.9)--(102.4, 329.0)--(102.4, 329.1)--(102.3, 329.2)--(102.2, 329.3)--(102.1, 329.3)--(102.0, 329.3)--(101.9, 329.3)--(101.7, 329.4)--(101.6, 329.4)--(101.5, 329.4)--(101.4, 329.4)--(101.4, 329.5)--(101.3, 329.5)--(101.3, 329.7)--(101.3, 329.8)--(101.2, 330.0)--(101.2, 330.4)--(101.1, 330.6)--(101.1, 330.7)--(101.0, 330.7)--(100.9, 330.8)--(100.6, 330.8)--(100.5, 330.8)--(100.4, 330.9)--(100.3, 331.0)--(100.3, 331.1)--(100.2, 331.2)--(100.2, 331.3)--(100.1, 331.5)--(100.0, 331.5)--(100.0, 331.6)--(99.8, 331.6)--(99.7, 331.6)--(99.6, 331.6)--(99.5, 331.5)--(99.2, 331.5)--(99.1, 331.4)--(99.0, 331.4)--(98.9, 331.4)--(98.8, 331.5)--(98.7, 331.5)--(98.6, 331.5)--(98.6, 331.6)--(98.5, 331.7)--(98.4, 331.7)--(98.4, 331.8)--(98.4, 332.0)--(98.5, 332.2)--(98.5, 332.4)--(98.5, 332.5)--(98.5, 332.6)--(98.6, 332.7)--(98.6, 332.8)--(98.5, 332.9)--(98.4, 332.9)--(98.4, 333.0)--(98.3, 333.0)--(98.1, 333.0)--(97.8, 333.1)--(97.6, 333.2)--(97.5, 333.2)--(97.4, 333.3)--(97.3, 333.3)--(97.2, 333.3)--(97.1, 333.5)--(97.0, 333.7)--(97.0, 333.8)--(97.1, 333.9)--(97.2, 334.0)--(97.4, 334.0)--(97.7, 334.1)--(97.9, 334.2)--(97.9, 334.3)--(98.0, 334.3)--(98.0, 334.4)--(97.9, 334.4)--(97.8, 334.5)--(97.7, 334.6)--(97.6, 334.6)--(97.6, 334.7)--(97.5, 334.8)--(97.5, 334.9)--(97.4, 334.9)--(97.3, 335.0)--(97.1, 335.2)--(97.1, 335.3)--(97.1, 335.4)--(97.1, 335.5)--(97.1, 335.6)--(97.1, 335.7)--(97.2, 335.8)--(97.3, 335.8)--(97.4, 335.7)--(97.5, 335.6)--(97.6, 335.6)--(97.7, 335.6)--(97.7, 335.5)--(97.8, 335.4)--(97.9, 335.4)--(98.1, 335.4)--(98.3, 335.4)--(98.4, 335.4)--(98.5, 335.4)--(98.6, 335.5)--(98.9, 335.5)--(99.0, 335.5)--(99.2, 335.5)--(99.4, 335.6)--(99.5, 335.6)--(99.6, 335.7)--(99.9, 335.8)--(99.9, 335.9)--(100.0, 335.9)--(100.3, 336.0)--(100.6, 336.1)--(100.9, 336.2)--(101.3, 336.3)--(101.8, 336.5)--(102.3, 336.6)--(103.1, 336.9)--(104.0, 337.1)--(104.5, 337.3)--(105.1, 337.5)--(105.0, 337.6)--(105.6, 337.8)--(106.8, 338.2)--(107.2, 338.3)--(107.5, 338.3)--(107.9, 338.3)--(108.3, 338.4)--(109.0, 338.4)--(109.0, 338.3)--(109.5, 338.4)--(109.6, 338.4)--(110.3, 338.3)--(110.5, 338.3)--(111.1, 338.2)--(112.2, 338.2)--(112.4, 338.2)--(113.8, 338.5)--(113.7, 339.0)--(113.6, 339.8)--(113.6, 340.1)--(113.6, 340.2)--(113.5, 340.2)--(113.4, 340.5)--(113.4, 340.8)--(113.3, 341.0)--(113.2, 341.4)--(113.1, 341.7)--(113.1, 341.8)--(113.0, 341.9)--(112.8, 342.3)--(112.5, 342.1)--(111.9, 341.9)--(111.9, 342.0)--(111.8, 342.4)--(111.5, 343.6)--(111.6, 343.6)--(111.6, 343.7)--(111.5, 344.0)--(111.4, 344.0)--(111.4, 344.1)--(111.3, 344.4)--(111.4, 344.4)--(111.5, 344.4)--(111.9, 344.3)--(112.1, 344.3)--(112.3, 344.3)--(112.5, 344.2)--(112.8, 344.2)--(113.4, 344.2)--(114.0, 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294.0)--(187.4, 294.2)--(186.9, 294.3)--(186.5, 294.4)--(185.8, 294.5)--(185.5, 294.5)--(185.4, 294.5)--(185.2, 294.5)--(185.1, 294.5)--(185.0, 294.5)--(184.9, 294.4)--(184.4, 294.1)--(184.0, 293.9)--(183.6, 293.7)--(183.2, 293.4)--(182.0, 292.7)--(181.1, 292.3)--(180.9, 292.2)--(180.5, 291.9)--(180.5, 291.8)--(180.6, 291.3)--(180.7, 291.1)--(180.8, 291.1)--(180.8, 291.0)--(180.9, 290.9)--(181.0, 290.8)--(181.0, 290.6)--(181.0, 290.3)--(181.1, 290.2)--(181.2, 290.1)--(181.3, 289.9)--(181.4, 289.8)--(181.5, 289.5)--(181.7, 289.1)--(181.8, 288.9)--(181.9, 288.7)--(182.2, 288.2)--(182.3, 288.0)--(182.5, 287.7)--(182.4, 287.7)--(182.8, 287.0)--(182.3, 286.9)--(181.5, 286.7)--(180.4, 286.4)--(180.2, 286.8)--(180.2, 287.0)--(180.2, 287.2)--(180.2, 287.3)--(179.5, 287.5)--(179.2, 287.6)--(178.8, 287.7)--(178.4, 287.8)--(177.9, 288.0)--(177.4, 288.1)--(177.3, 288.2)--(177.2, 288.3)--(177.0, 288.3)--(177.0, 288.2)--(177.0, 288.0)--(177.0, 287.7)--(177.1, 287.4)--(177.2, 287.0)--(177.3, 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287.8)--(157.0, 287.9)--(156.6, 287.9)--(156.5, 287.9)--(156.1, 287.9)--(155.5, 287.9)--(155.0, 287.9)--(154.3, 287.9)--(153.8, 287.8)--(153.1, 287.7)--(152.4, 287.6)--(152.3, 287.7)--(151.8, 287.6)--(150.9, 288.1)--(150.0, 288.6)--(149.0, 289.2)--(148.2, 289.6)--(147.5, 290.0)--(146.7, 290.6)--(146.5, 290.7)--(146.4, 290.8)--(146.3, 290.8)--(146.3, 290.9)--(146.2, 290.9)--(146.0, 290.9)--(145.9, 290.9)--(145.8, 290.9)--(145.6, 290.8)--(144.7, 290.5)--(144.0, 290.2)--(143.2, 289.9)--(143.1, 289.9)--(143.1, 289.8)--(143.0, 289.9)--(142.9, 289.9)--(142.7, 289.8)--(142.5, 289.7)--(142.4, 289.7)--(142.3, 289.7)--(142.1, 289.7)--(142.1, 289.6)--(142.0, 289.6)--(141.9, 289.5)--(141.7, 289.5)--(141.5, 289.4)--(141.3, 289.3)--(141.1, 289.3)--(140.8, 289.3)--(140.7, 289.2)--(140.5, 289.1)--(140.4, 289.1)--(140.3, 288.9)--(140.0, 288.6)--(139.7, 288.4)--(139.3, 288.1)--(139.0, 288.0)--(138.9, 287.9)--(138.8, 287.8)--(138.6, 287.8)--(138.4, 287.6)--(138.3, 287.6)--(138.3, 287.5)--(138.2, 287.5)--(138.2, 287.4)--(138.1, 287.3)--(137.9, 287.3)--(137.8, 287.2)--(137.7, 287.2)--(137.6, 287.2)--(137.5, 287.1)--(137.4, 287.2)--(137.4, 287.1)--(137.2, 287.0)--(137.2, 286.9)--(137.2, 286.8)--(137.0, 286.7)--(137.0, 286.6)--(136.9, 286.6)--(136.9, 286.1)--(136.9, 285.8)--(136.8, 285.6)--(136.6, 285.5)--(136.4, 285.3)--(136.3, 285.1)--(136.2, 284.9)--(136.0, 284.6)--(135.8, 284.2)--(135.5, 284.6)--(135.3, 284.4)--(135.0, 284.1)--(134.9, 283.9)--(134.8, 283.8)--(134.8, 283.7)--(134.7, 283.7)--(134.7, 283.6)--(134.6, 283.5)--(134.4, 283.4)--(134.1, 283.2)--(133.9, 283.1)--(133.8, 283.0)--(133.7, 283.0)--(133.8, 282.8)--(134.0, 282.5)--(133.5, 282.2)--(133.0, 281.9)--(132.7, 281.7)--(131.8, 281.2)--(131.3, 280.9)--(131.3, 281.0)--(131.4, 281.1)--(131.4, 281.3)--(131.3, 281.4)--(131.3, 281.5)--(131.1, 281.5)--(131.0, 281.5)--(130.9, 281.5)--(130.8, 281.6)--(130.7, 281.8)--(130.5, 282.0)--(130.4, 282.1)--(130.3, 282.2)--(130.2, 282.4)--(130.1, 282.4)--(130.0, 282.5)--(129.9, 282.5)--(129.8, 282.9)--(129.3, 284.0)--(129.2, 284.3)--(129.1, 284.7)--(129.0, 284.9)--(128.9, 285.1)--(128.7, 285.4)--(128.6, 285.8)--(128.5, 286.3)--(128.0, 286.3)--(127.8, 286.2)--(127.6, 286.2)--(127.5, 286.2)--(127.0, 286.4)--(127.1, 287.0)--(127.1, 287.2)--(127.1, 287.4)--(127.1, 287.5)--(127.2, 287.6)--(127.0, 287.6)--(127.0, 287.7)--(126.6, 287.4)--(126.3, 287.3)--(126.1, 287.2)--(126.1, 287.1)--(125.8, 286.9)--(125.5, 286.8)--(124.7, 286.3)--(124.3, 286.1)--(124.1, 286.0)--(123.9, 285.9)--(123.7, 285.8)--(123.5, 285.7)--(123.3, 285.6)--(123.1, 285.6)--(123.0, 285.5)--(122.9, 285.5)--(122.5, 285.4)--(122.3, 285.4)--(122.0, 285.4)--(121.8, 285.4)--(121.6, 285.3)--(121.4, 285.3)--(120.7, 285.4)--(120.3, 285.6)--(119.9, 285.6)--(119.1, 285.9)--(118.1, 286.2)--(117.0, 286.6)--(115.8, 287.1)--(114.6, 287.6)--(113.1, 288.2)--(112.6, 288.4)--(111.6, 288.8)--(111.5, 288.9)--(111.5, 288.8)--(110.5, 289.2)--(109.7, 289.5)--(109.2, 289.7)--(108.2, 290.1)--(107.0, 290.6)--(106.8, 290.6)--(107.2, 291.4)--(107.9, 292.8)--(108.0, 292.8)--(108.5, 293.9)--(108.8, 294.3)--(108.9, 294.5)--(109.0, 294.5)--(109.0, 294.6)--(109.1, 294.7)--(109.2, 294.8)--(109.3, 294.9)--(109.4, 295.0)--(109.5, 295.1)--(109.6, 295.1)--(109.7, 295.2)--(109.8, 295.2)--(109.9, 295.3)--(110.0, 295.3)--(110.1, 295.4)--(110.3, 295.4)--(110.4, 295.5)--(110.6, 295.5)--(110.6, 295.6)--(111.1, 295.8)--(111.6, 296.0)--(112.1, 296.4)--(113.0, 297.0)--(113.9, 297.6)--(114.4, 297.9)--(114.9, 298.2)--(115.3, 298.5)--(115.5, 298.6)--(115.5, 298.7)--(115.6, 298.7)--(115.6, 298.8)--(115.8, 298.8)--(115.9, 298.9)--(116.0, 299.1)--(116.0, 299.2)--(116.2, 299.4)--(116.2, 299.6)--(116.3, 299.7)--(116.3, 299.9)--(116.4, 300.1)--(116.5, 300.4)--(116.7, 301.4)--(117.2, 303.2)--(117.5, 304.4)--(117.6, 304.8)--(117.7, 305.2)--(117.8, 305.5)--(117.8, 305.7)--(117.9, 305.9)--(117.9, 306.1)--(118.1, 306.5)--(118.2, 306.7)--(118.5, 307.2)--(118.6, 307.4)--(118.8, 307.6)--(119.1, 307.9)--(120.4, 308.7)--(120.8, 309.0)--(121.4, 309.4)--(121.6, 309.5)--(121.7, 309.6)--(121.8, 309.7)--(121.9, 309.8)--(122.1, 310.0)--(122.3, 310.3)--(122.4, 310.5)--(122.6, 310.8)--(122.7, 310.8)--(123.0, 311.3)--(123.2, 311.3)--(123.2, 311.4)--(123.3, 311.5)--(123.5, 311.8)--(123.7, 312.0)--(123.8, 312.1)--(123.8, 312.2)--(124.0, 312.7)--(124.3, 313.3)--(124.4, 313.4)--(124.8, 314.0)--(125.0, 314.3)--(125.4, 315.1)--(125.5, 315.4)--(125.7, 315.7)--(125.9, 316.2)--(126.0, 316.4)--(126.5, 317.4)--(126.8, 318.1)--(126.9, 318.4)--(127.2, 318.6)--(127.2, 318.8)--(127.4, 318.9)--(127.5, 319.2)--(127.5, 319.3)--(127.6, 319.6)--(127.6, 319.9)--(127.6, 320.1)--(127.6, 320.3)--(127.5, 320.8)--(127.5, 321.1)--(127.4, 321.4)--(127.4, 321.9)--(127.4, 323.5)--(127.3, 323.5)--(127.3, 323.6)--(127.2, 323.6)--(126.8, 323.7)--(126.4, 323.7)--(125.4, 323.8)--(125.0, 323.9)--(124.0, 324.1)--(123.6, 324.2)--(123.0, 324.3)--(121.9, 324.6)--(121.6, 324.7)--(120.8, 325.0)--(120.6, 325.1)--(120.1, 325.1)--(120.0, 324.8)--(119.9, 324.6)--(119.8, 324.2)--(119.3, 324.6)--(118.9, 324.9)--(118.5, 325.1)--(118.3, 325.3)--(118.3, 325.5)--(118.0, 325.6)--(117.9, 325.6)--(117.4, 325.7)--(116.1, 326.1)--(115.3, 326.3)--(114.6, 326.5)--(114.0, 326.6)--(113.7, 326.7)--(113.4, 326.7)--(113.0, 326.8)--(112.9, 326.8)--(111.8, 327.0)--(110.7, 327.3)--(109.8, 327.5)--(109.0, 327.6)--(107.7, 327.8)--(107.6, 327.8)--(107.5, 327.9)--(107.4, 327.9)--(107.3, 327.9)--(107.1, 327.9)--(106.9, 328.0)--(106.8, 328.0)--(105.9, 328.3)--(105.6, 328.3)--(104.2, 328.8)--(104.1, 328.8)--(103.6, 328.9)--(103.3, 328.9)--(103.2, 328.9)--(103.1, 329.0)--(103.0, 329.1)--cycle; +Enfield = (310.2, 509.3)--(310.4, 509.3)--(310.9, 509.3)--(311.4, 509.3)--(311.6, 509.3)--(311.9, 509.2)--(312.3, 509.0)--(312.6, 509.0)--(313.1, 508.8)--(313.6, 508.8)--(314.1, 508.7)--(314.6, 508.6)--(314.7, 508.5)--(315.1, 508.4)--(315.3, 508.4)--(315.5, 508.3)--(316.2, 508.1)--(316.4, 508.0)--(316.6, 507.9)--(316.7, 508.0)--(317.1, 507.8)--(317.5, 507.6)--(317.9, 507.6)--(318.2, 507.5)--(319.1, 507.5)--(319.8, 507.5)--(320.2, 507.4)--(320.4, 507.4)--(321.1, 507.4)--(321.5, 507.5)--(321.9, 507.5)--(322.2, 507.5)--(322.7, 507.6)--(323.2, 507.7)--(323.5, 507.7)--(323.8, 507.8)--(324.1, 507.8)--(324.4, 507.8)--(324.8, 507.7)--(325.4, 507.6)--(325.8, 507.4)--(326.3, 507.3)--(326.7, 507.1)--(326.8, 507.0)--(326.9, 507.0)--(326.9, 506.9)--(327.0, 506.9)--(327.1, 506.9)--(327.2, 506.9)--(327.3, 506.8)--(327.4, 506.8)--(327.5, 506.8)--(327.6, 506.8)--(327.7, 506.8)--(327.8, 506.8)--(328.2, 506.7)--(328.3, 506.7)--(328.4, 506.7)--(328.4, 506.6)--(328.5, 506.6)--(328.6, 506.6)--(328.8, 506.5)--(329.4, 506.2)--(329.6, 506.0)--(329.9, 505.8)--(330.4, 505.5)--(330.7, 505.4)--(331.0, 505.2)--(331.3, 505.0)--(331.6, 504.8)--(332.2, 504.5)--(332.4, 504.4)--(333.0, 504.1)--(333.4, 503.8)--(333.8, 503.6)--(334.0, 503.4)--(334.6, 503.0)--(335.1, 502.6)--(335.5, 502.4)--(335.9, 502.1)--(336.2, 502.0)--(336.6, 501.7)--(336.9, 501.6)--(337.1, 501.4)--(337.7, 501.1)--(338.1, 501.0)--(338.3, 500.9)--(338.6, 500.8)--(339.0, 500.7)--(339.3, 500.6)--(339.5, 500.6)--(339.7, 500.6)--(339.9, 500.5)--(340.7, 500.3)--(341.1, 500.2)--(341.4, 500.2)--(341.5, 500.2)--(341.7, 500.2)--(341.8, 500.2)--(342.3, 500.1)--(342.6, 500.1)--(342.8, 500.3)--(343.0, 500.3)--(343.0, 500.2)--(343.6, 500.3)--(344.0, 500.3)--(344.2, 500.3)--(344.5, 500.3)--(345.0, 500.5)--(345.3, 500.6)--(345.5, 500.7)--(345.7, 500.8)--(346.0, 500.8)--(346.2, 500.8)--(346.7, 500.8)--(347.1, 500.9)--(347.5, 501.0)--(347.9, 501.0)--(348.0, 501.1)--(348.3, 501.2)--(348.6, 501.4)--(348.7, 501.5)--(348.9, 501.6)--(349.2, 501.7)--(349.6, 501.9)--(349.7, 501.7)--(349.8, 501.7)--(349.9, 501.6)--(350.0, 501.6)--(350.2, 501.7)--(350.4, 501.5)--(350.5, 501.3)--(350.6, 501.3)--(350.6, 501.2)--(350.7, 501.2)--(350.7, 501.1)--(350.8, 501.1)--(350.9, 501.1)--(351.0, 501.1)--(351.0, 501.0)--(351.1, 501.0)--(351.2, 501.0)--(351.3, 501.0)--(351.4, 501.0)--(351.5, 501.0)--(351.6, 501.0)--(351.7, 501.0)--(351.8, 501.0)--(351.9, 501.0)--(352.3, 501.1)--(352.7, 501.1)--(353.0, 501.1)--(353.1, 501.0)--(353.4, 501.0)--(353.4, 500.8)--(353.4, 500.7)--(354.2, 500.7)--(354.3, 500.7)--(354.9, 500.7)--(355.2, 500.7)--(356.2, 500.6)--(357.9, 500.5)--(358.0, 500.5)--(358.0, 500.4)--(358.0, 500.3)--(358.1, 500.3)--(359.8, 500.2)--(360.0, 500.2)--(360.0, 500.3)--(360.5, 500.2)--(360.8, 500.2)--(360.9, 500.3)--(362.4, 500.2)--(362.4, 500.3)--(363.3, 500.2)--(364.0, 500.1)--(364.5, 500.1)--(364.6, 500.0)--(364.9, 499.9)--(365.3, 499.9)--(365.7, 499.8)--(366.2, 499.7)--(366.6, 499.6)--(366.7, 499.5)--(367.0, 499.5)--(367.5, 499.3)--(367.6, 499.4)--(367.8, 499.4)--(368.4, 499.3)--(369.2, 499.1)--(370.0, 499.0)--(370.1, 499.0)--(371.1, 498.9)--(371.4, 498.9)--(371.6, 498.9)--(372.6, 498.9)--(373.3, 498.9)--(374.0, 498.9)--(375.3, 498.8)--(375.4, 498.8)--(375.9, 498.8)--(376.0, 498.8)--(376.0, 498.7)--(376.0, 498.6)--(376.0, 497.9)--(376.1, 497.4)--(376.1, 497.3)--(376.3, 496.7)--(376.4, 496.1)--(376.5, 495.6)--(376.6, 494.9)--(376.6, 494.4)--(376.7, 494.2)--(376.7, 494.1)--(376.7, 493.9)--(376.6, 493.7)--(376.6, 493.4)--(376.5, 493.3)--(376.5, 492.9)--(376.4, 492.6)--(376.4, 492.3)--(376.4, 492.1)--(376.3, 491.7)--(376.2, 490.5)--(376.1, 488.9)--(376.1, 488.1)--(376.1, 488.0)--(376.1, 487.9)--(376.1, 487.8)--(376.1, 487.7)--(376.1, 487.6)--(376.1, 487.5)--(376.1, 487.4)--(376.2, 487.4)--(376.2, 487.3)--(376.2, 487.2)--(376.3, 487.1)--(376.5, 486.5)--(376.7, 486.2)--(376.7, 486.1)--(376.7, 486.0)--(376.7, 485.9)--(376.7, 485.8)--(376.7, 485.7)--(376.7, 485.6)--(376.7, 485.5)--(376.7, 485.4)--(376.6, 485.3)--(376.6, 485.2)--(376.5, 485.1)--(376.5, 485.0)--(376.5, 484.9)--(376.4, 484.9)--(376.4, 484.8)--(376.4, 484.7)--(376.4, 484.6)--(376.4, 484.4)--(376.3, 484.3)--(376.3, 484.1)--(376.3, 483.9)--(376.3, 483.8)--(376.3, 483.6)--(376.3, 483.4)--(376.3, 483.2)--(376.3, 481.6)--(376.2, 480.0)--(376.2, 479.9)--(376.3, 479.8)--(376.3, 479.7)--(376.3, 479.6)--(376.3, 479.5)--(376.3, 479.4)--(376.4, 479.3)--(376.4, 479.2)--(376.5, 479.1)--(376.5, 479.0)--(376.6, 478.9)--(376.8, 478.6)--(376.9, 478.4)--(377.2, 477.9)--(377.5, 477.6)--(377.6, 477.5)--(377.6, 477.4)--(377.7, 477.4)--(377.7, 477.3)--(377.8, 477.2)--(377.8, 477.1)--(377.9, 477.1)--(377.9, 477.0)--(378.0, 476.9)--(378.0, 476.8)--(378.0, 476.7)--(378.1, 476.6)--(378.1, 476.5)--(378.1, 476.4)--(378.1, 476.3)--(378.1, 476.2)--(378.1, 476.1)--(378.1, 476.0)--(378.1, 475.9)--(378.1, 475.8)--(378.1, 475.7)--(378.1, 475.6)--(378.1, 475.5)--(378.0, 475.4)--(378.0, 475.3)--(377.6, 473.4)--(377.3, 471.8)--(376.8, 469.8)--(376.6, 468.8)--(376.4, 468.1)--(376.4, 468.0)--(376.4, 467.9)--(376.4, 467.8)--(376.4, 467.7)--(376.4, 467.6)--(376.4, 467.5)--(376.4, 467.4)--(376.4, 467.2)--(376.4, 467.0)--(376.4, 466.8)--(376.4, 466.6)--(376.5, 466.4)--(376.5, 466.2)--(376.6, 465.4)--(376.6, 464.9)--(376.6, 464.2)--(376.5, 463.2)--(376.4, 463.0)--(376.3, 461.5)--(376.3, 460.3)--(376.0, 457.3)--(375.8, 454.0)--(375.8, 453.2)--(375.7, 452.9)--(375.6, 452.4)--(375.3, 451.2)--(374.9, 450.0)--(374.8, 449.7)--(374.8, 449.4)--(374.8, 449.2)--(374.7, 448.8)--(374.5, 448.3)--(374.4, 448.0)--(374.0, 447.3)--(373.6, 446.4)--(373.5, 446.1)--(373.3, 445.8)--(372.8, 444.7)--(372.3, 443.4)--(371.5, 441.8)--(369.8, 438.1)--(369.5, 437.4)--(369.3, 437.1)--(369.2, 436.8)--(369.1, 436.7)--(368.9, 436.3)--(368.8, 436.1)--(368.6, 435.8)--(368.5, 435.5)--(368.4, 435.2)--(368.3, 435.0)--(368.3, 434.8)--(368.2, 434.4)--(368.1, 434.0)--(368.0, 433.7)--(367.9, 433.5)--(367.8, 433.2)--(367.5, 432.8)--(367.4, 432.7)--(367.3, 432.6)--(366.8, 431.9)--(366.6, 431.6)--(366.5, 431.5)--(366.3, 431.1)--(366.2, 431.0)--(365.7, 430.3)--(365.2, 429.5)--(365.0, 429.2)--(364.6, 428.8)--(364.0, 428.1)--(363.8, 427.9)--(363.7, 427.7)--(363.6, 427.5)--(363.4, 427.3)--(363.3, 427.1)--(363.2, 427.0)--(363.1, 426.9)--(363.1, 426.8)--(363.0, 426.6)--(363.0, 426.5)--(362.9, 426.4)--(362.6, 425.8)--(362.2, 425.0)--(361.8, 424.1)--(361.5, 423.6)--(361.4, 423.5)--(361.4, 423.4)--(361.4, 423.3)--(361.3, 423.2)--(361.3, 423.1)--(361.3, 423.0)--(361.3, 422.9)--(361.3, 422.8)--(361.2, 422.8)--(361.2, 422.7)--(361.2, 422.6)--(361.2, 422.5)--(361.2, 422.4)--(361.1, 422.3)--(361.1, 422.2)--(361.1, 422.1)--(361.1, 422.0)--(361.1, 421.9)--(361.2, 421.8)--(361.2, 421.7)--(361.2, 421.6)--(361.3, 421.6)--(361.3, 421.5)--(361.3, 421.4)--(361.4, 421.3)--(361.4, 421.2)--(361.5, 421.1)--(361.5, 421.0)--(361.6, 421.0)--(361.6, 420.9)--(361.7, 420.8)--(362.0, 420.2)--(362.1, 420.2)--(362.1, 420.1)--(362.1, 420.0)--(362.2, 420.0)--(362.2, 419.9)--(362.2, 419.8)--(362.2, 419.7)--(362.3, 419.7)--(362.3, 419.6)--(362.3, 419.5)--(362.3, 419.4)--(362.3, 419.3)--(362.3, 419.2)--(362.3, 419.1)--(362.3, 419.0)--(362.3, 418.9)--(362.3, 418.8)--(362.2, 418.7)--(362.2, 418.6)--(362.2, 418.5)--(362.1, 418.5)--(362.1, 418.4)--(362.0, 418.3)--(362.0, 418.2)--(361.9, 418.2)--(361.8, 418.0)--(361.8, 417.9)--(361.7, 417.8)--(361.6, 417.7)--(361.5, 417.7)--(361.5, 417.6)--(361.4, 417.6)--(361.4, 417.5)--(361.3, 417.5)--(361.2, 417.4)--(361.1, 417.4)--(361.1, 417.3)--(361.0, 417.2)--(360.9, 417.1)--(360.7, 417.0)--(359.7, 416.4)--(359.2, 416.1)--(357.9, 415.2)--(357.8, 415.2)--(357.8, 415.1)--(357.7, 415.1)--(357.6, 415.0)--(357.5, 414.9)--(357.4, 414.8)--(357.4, 414.7)--(357.3, 414.7)--(357.3, 414.6)--(357.0, 414.7)--(356.3, 414.9)--(355.3, 415.1)--(354.6, 415.3)--(353.0, 415.8)--(352.7, 415.9)--(352.4, 415.9)--(352.1, 416.0)--(352.0, 416.1)--(351.8, 416.2)--(351.6, 416.2)--(351.4, 416.3)--(351.2, 416.4)--(350.2, 416.8)--(350.3, 416.8)--(350.1, 416.9)--(349.8, 417.0)--(349.5, 417.1)--(349.3, 417.1)--(349.3, 417.4)--(349.2, 417.4)--(349.2, 417.5)--(349.3, 417.6)--(349.2, 417.6)--(349.1, 417.7)--(348.5, 417.9)--(348.2, 418.0)--(347.4, 418.0)--(347.0, 417.9)--(345.7, 417.8)--(345.0, 417.7)--(343.5, 417.6)--(342.7, 417.5)--(342.3, 417.5)--(340.9, 417.5)--(340.7, 417.5)--(340.6, 417.5)--(340.0, 417.5)--(339.4, 417.8)--(339.2, 417.5)--(339.1, 417.5)--(339.0, 417.5)--(338.9, 417.5)--(338.8, 417.7)--(338.6, 417.5)--(337.9, 417.5)--(337.7, 417.5)--(337.2, 417.5)--(336.9, 417.5)--(336.8, 417.5)--(336.8, 417.6)--(336.4, 417.5)--(336.2, 417.5)--(336.0, 417.4)--(336.0, 417.5)--(335.9, 417.5)--(335.4, 417.4)--(334.8, 417.3)--(334.1, 417.2)--(333.5, 417.1)--(333.2, 417.1)--(332.8, 417.0)--(332.5, 417.1)--(331.8, 417.1)--(331.4, 417.1)--(331.2, 417.1)--(331.1, 417.1)--(331.0, 417.1)--(330.8, 417.1)--(330.0, 417.1)--(329.6, 417.0)--(329.1, 417.0)--(329.0, 417.0)--(328.2, 417.0)--(327.5, 417.0)--(327.4, 417.0)--(327.2, 417.0)--(326.5, 416.9)--(325.9, 417.0)--(325.6, 417.0)--(325.3, 417.1)--(325.0, 417.1)--(324.7, 417.2)--(324.5, 417.2)--(323.8, 417.3)--(323.0, 417.3)--(322.9, 417.2)--(322.6, 417.2)--(322.4, 417.3)--(322.3, 417.3)--(322.3, 417.2)--(322.3, 417.3)--(322.1, 417.3)--(322.1, 417.4)--(322.1, 417.5)--(321.9, 417.5)--(321.9, 417.2)--(321.8, 417.3)--(321.5, 417.3)--(321.1, 417.4)--(320.8, 417.2)--(320.0, 417.0)--(319.9, 417.2)--(319.9, 417.3)--(319.7, 417.7)--(319.5, 417.7)--(319.3, 417.6)--(319.1, 417.6)--(318.8, 417.5)--(318.7, 417.4)--(318.5, 417.2)--(318.4, 417.2)--(318.3, 417.2)--(318.4, 416.8)--(318.1, 416.8)--(317.3, 416.7)--(316.7, 416.7)--(316.4, 416.7)--(316.1, 416.7)--(315.9, 416.7)--(315.8, 416.7)--(315.6, 416.7)--(315.4, 416.7)--(315.3, 416.6)--(315.1, 416.6)--(314.9, 416.2)--(314.6, 416.4)--(314.2, 416.6)--(314.1, 416.6)--(313.9, 416.6)--(313.8, 416.6)--(313.0, 416.6)--(312.7, 416.5)--(312.6, 416.5)--(312.3, 416.5)--(311.8, 416.4)--(311.7, 416.3)--(310.2, 416.3)--(309.8, 416.3)--(309.4, 416.4)--(309.4, 416.6)--(309.3, 416.6)--(309.2, 416.7)--(309.2, 416.6)--(309.1, 416.5)--(308.7, 416.5)--(307.8, 416.4)--(307.1, 416.4)--(307.1, 416.3)--(306.9, 416.3)--(306.8, 416.3)--(306.6, 416.4)--(306.5, 416.4)--(306.0, 416.4)--(305.9, 416.4)--(305.6, 416.4)--(305.6, 416.2)--(304.7, 416.3)--(304.5, 416.1)--(304.1, 416.3)--(303.7, 416.6)--(303.4, 416.2)--(303.1, 416.3)--(301.9, 416.4)--(301.7, 416.5)--(301.6, 416.5)--(301.4, 416.5)--(301.1, 416.5)--(300.6, 416.6)--(300.5, 416.6)--(300.4, 416.5)--(300.0, 417.0)--(300.1, 417.0)--(300.0, 417.1)--(299.9, 417.1)--(299.9, 417.2)--(299.3, 417.8)--(299.4, 417.9)--(299.1, 418.0)--(298.6, 417.5)--(298.7, 417.5)--(298.4, 417.1)--(298.4, 417.2)--(298.3, 417.2)--(297.7, 417.2)--(297.3, 417.3)--(296.8, 417.4)--(296.0, 417.6)--(295.2, 418.1)--(294.9, 418.2)--(294.6, 418.3)--(293.5, 418.7)--(292.4, 419.1)--(292.2, 419.1)--(292.0, 419.0)--(291.7, 418.7)--(291.5, 418.6)--(290.5, 418.2)--(289.8, 417.9)--(289.2, 418.6)--(288.7, 419.2)--(288.3, 419.7)--(287.9, 420.3)--(287.4, 421.0)--(286.8, 421.9)--(286.8, 422.0)--(286.1, 422.9)--(285.6, 423.7)--(285.9, 423.8)--(286.0, 423.9)--(286.2, 423.9)--(286.6, 424.1)--(286.8, 424.3)--(286.9, 424.3)--(287.3, 424.5)--(287.3, 424.6)--(287.3, 424.7)--(287.3, 424.8)--(287.4, 424.8)--(287.4, 425.0)--(287.5, 425.2)--(287.5, 425.4)--(287.6, 425.5)--(287.6, 425.6)--(287.7, 425.7)--(287.8, 425.8)--(287.8, 425.9)--(287.8, 426.0)--(288.0, 426.2)--(288.1, 426.2)--(288.2, 426.2)--(288.3, 426.3)--(288.7, 427.0)--(288.9, 427.1)--(288.8, 427.2)--(289.0, 427.4)--(289.1, 427.5)--(289.7, 428.3)--(289.8, 428.4)--(290.3, 429.1)--(290.5, 429.4)--(290.9, 429.8)--(291.3, 430.2)--(291.7, 430.8)--(291.6, 431.5)--(291.7, 431.5)--(291.8, 431.4)--(292.0, 431.5)--(292.0, 431.8)--(292.1, 432.0)--(292.2, 432.3)--(292.3, 432.7)--(292.3, 432.8)--(292.5, 433.2)--(292.5, 433.3)--(292.5, 433.5)--(292.5, 433.6)--(292.4, 433.7)--(292.4, 433.8)--(292.4, 433.9)--(292.5, 433.9)--(292.5, 434.0)--(292.6, 434.0)--(292.6, 434.2)--(292.6, 434.4)--(292.8, 435.1)--(292.9, 435.6)--(292.9, 435.9)--(292.9, 436.0)--(293.0, 436.0)--(293.0, 436.1)--(293.0, 436.2)--(293.1, 436.2)--(293.1, 436.3)--(293.2, 436.3)--(293.2, 436.4)--(293.4, 436.5)--(293.5, 436.7)--(293.6, 436.8)--(293.7, 437.0)--(293.8, 437.2)--(293.8, 437.4)--(293.9, 437.6)--(294.0, 437.7)--(294.1, 437.9)--(294.3, 438.2)--(294.4, 438.4)--(294.4, 438.5)--(294.5, 438.7)--(294.5, 438.8)--(294.5, 438.9)--(294.5, 439.0)--(294.7, 439.3)--(294.9, 439.3)--(295.0, 439.4)--(294.8, 439.5)--(294.5, 439.7)--(294.7, 440.2)--(294.8, 440.5)--(294.9, 440.9)--(295.1, 441.2)--(295.4, 441.2)--(295.5, 441.7)--(295.7, 442.1)--(295.8, 442.6)--(295.5, 442.7)--(295.0, 442.8)--(294.6, 442.9)--(294.3, 442.9)--(293.8, 443.2)--(293.4, 443.4)--(293.1, 443.5)--(292.9, 443.7)--(292.7, 443.9)--(292.5, 444.1)--(292.6, 444.2)--(292.5, 444.4)--(292.4, 444.4)--(292.4, 444.5)--(292.2, 444.3)--(292.1, 444.4)--(292.2, 444.6)--(292.0, 444.7)--(292.2, 444.9)--(292.2, 445.0)--(292.1, 445.1)--(292.0, 445.2)--(291.9, 445.4)--(291.5, 445.7)--(291.3, 445.8)--(290.8, 446.1)--(289.8, 446.8)--(289.5, 447.0)--(289.3, 447.1)--(289.2, 447.2)--(288.2, 448.1)--(287.8, 448.5)--(287.6, 448.8)--(287.4, 449.0)--(287.2, 449.3)--(286.9, 449.7)--(286.7, 449.9)--(286.6, 450.0)--(286.5, 450.2)--(286.6, 450.3)--(286.3, 450.8)--(286.0, 451.2)--(285.3, 452.1)--(284.5, 453.9)--(282.5, 453.2)--(282.4, 453.6)--(282.1, 454.2)--(282.0, 454.3)--(281.8, 455.2)--(281.7, 455.5)--(281.6, 456.0)--(281.6, 456.1)--(281.6, 456.3)--(281.6, 456.5)--(281.6, 456.7)--(281.5, 456.8)--(280.8, 456.7)--(280.4, 456.7)--(280.2, 456.7)--(280.1, 456.7)--(279.8, 456.8)--(279.9, 457.1)--(279.5, 457.3)--(280.1, 458.4)--(280.0, 458.5)--(280.1, 458.6)--(280.2, 458.9)--(280.0, 459.0)--(279.3, 459.3)--(279.4, 459.7)--(279.7, 459.9)--(281.0, 460.6)--(280.2, 462.1)--(279.6, 461.9)--(279.3, 461.8)--(279.3, 462.0)--(279.2, 462.1)--(279.2, 462.3)--(279.2, 462.8)--(279.1, 463.4)--(279.1, 463.5)--(279.1, 463.6)--(278.8, 464.0)--(278.7, 464.2)--(278.5, 464.7)--(278.4, 464.8)--(278.4, 464.9)--(278.2, 464.9)--(278.0, 464.8)--(277.9, 465.2)--(278.0, 465.5)--(278.1, 465.5)--(278.4, 465.6)--(278.3, 465.7)--(278.3, 466.0)--(278.3, 466.4)--(278.2, 466.9)--(278.1, 467.1)--(277.8, 467.3)--(277.7, 467.4)--(277.5, 467.5)--(277.5, 467.7)--(277.4, 468.1)--(277.3, 468.1)--(276.9, 468.2)--(275.3, 468.7)--(275.2, 468.9)--(275.0, 468.8)--(273.4, 469.3)--(273.1, 469.6)--(272.8, 469.8)--(271.8, 470.6)--(271.5, 470.7)--(271.3, 470.8)--(270.2, 470.9)--(265.0, 472.6)--(262.9, 473.3)--(262.2, 473.5)--(260.5, 474.1)--(259.4, 474.4)--(258.2, 474.8)--(258.0, 474.9)--(257.2, 475.2)--(257.1, 475.2)--(255.7, 475.6)--(255.8, 476.0)--(256.1, 476.8)--(256.9, 478.9)--(257.4, 480.2)--(258.1, 482.1)--(258.5, 482.5)--(259.6, 483.7)--(260.0, 484.1)--(260.7, 484.9)--(261.0, 485.2)--(261.8, 486.1)--(261.5, 486.7)--(262.0, 486.7)--(262.6, 486.7)--(262.5, 487.3)--(262.7, 487.6)--(262.9, 487.4)--(264.7, 487.3)--(265.3, 488.3)--(266.1, 489.8)--(267.5, 492.4)--(267.7, 492.9)--(268.3, 493.8)--(269.3, 495.7)--(269.4, 495.8)--(269.5, 496.0)--(269.8, 496.6)--(270.5, 497.8)--(270.5, 500.0)--(270.5, 500.1)--(270.5, 500.8)--(270.5, 501.4)--(270.5, 501.7)--(270.5, 502.1)--(270.4, 502.6)--(270.5, 503.4)--(270.5, 504.1)--(270.7, 504.1)--(271.0, 504.1)--(271.4, 504.0)--(271.8, 503.9)--(272.2, 503.8)--(272.5, 503.7)--(272.9, 503.6)--(273.1, 503.6)--(273.4, 503.5)--(274.3, 503.3)--(274.5, 503.2)--(274.5, 503.1)--(274.6, 503.1)--(274.7, 503.1)--(274.7, 503.0)--(274.8, 503.0)--(274.8, 502.9)--(274.9, 502.9)--(274.9, 502.8)--(275.1, 502.7)--(275.5, 502.5)--(276.0, 502.3)--(276.5, 502.2)--(276.6, 502.1)--(277.1, 502.0)--(277.6, 501.8)--(278.0, 501.7)--(278.5, 501.6)--(278.9, 501.6)--(279.2, 501.5)--(279.5, 501.5)--(280.0, 501.5)--(280.3, 501.5)--(280.7, 501.6)--(281.3, 501.8)--(281.7, 501.9)--(282.2, 502.0)--(282.5, 502.1)--(282.9, 502.2)--(283.0, 502.4)--(283.3, 502.4)--(283.5, 502.5)--(283.9, 502.7)--(284.5, 503.0)--(285.1, 503.4)--(285.6, 503.7)--(286.0, 503.9)--(286.5, 504.1)--(287.0, 504.3)--(287.4, 504.5)--(287.8, 504.7)--(288.3, 504.9)--(288.8, 505.0)--(289.3, 505.2)--(289.9, 505.3)--(290.4, 505.4)--(290.7, 505.3)--(291.0, 505.3)--(291.0, 505.4)--(291.0, 505.5)--(291.3, 505.5)--(291.7, 505.6)--(292.1, 505.6)--(292.7, 505.6)--(293.4, 505.6)--(294.0, 505.6)--(294.5, 505.6)--(295.0, 505.6)--(295.5, 505.5)--(295.9, 505.5)--(296.3, 505.5)--(297.0, 505.4)--(297.6, 505.4)--(298.4, 505.4)--(298.7, 505.3)--(298.8, 505.3)--(298.9, 505.3)--(299.0, 505.3)--(299.1, 505.3)--(299.2, 505.4)--(299.3, 505.4)--(299.4, 505.4)--(299.5, 505.4)--(299.6, 505.5)--(299.7, 505.5)--(299.9, 505.5)--(300.0, 505.6)--(300.4, 505.7)--(300.9, 505.8)--(301.2, 505.9)--(301.6, 506.0)--(301.7, 505.8)--(301.9, 505.9)--(301.9, 506.0)--(302.0, 506.2)--(302.7, 506.4)--(303.1, 506.7)--(303.4, 506.9)--(303.8, 507.1)--(304.1, 507.3)--(304.5, 507.4)--(304.8, 507.6)--(305.0, 507.8)--(305.3, 508.0)--(305.5, 508.1)--(305.8, 508.3)--(306.5, 508.5)--(307.0, 508.6)--(307.6, 508.8)--(307.5, 509.0)--(307.8, 509.1)--(308.6, 509.2)--(309.1, 509.3)--(309.6, 509.3)--(310.0, 509.3)--cycle; +Greenwich = (472.3, 313.0)--(472.5, 312.6)--(472.6, 311.5)--(472.6, 311.1)--(472.6, 310.8)--(472.7, 310.2)--(472.7, 310.0)--(472.7, 309.7)--(472.7, 309.0)--(472.8, 308.1)--(472.8, 307.7)--(472.9, 307.0)--(472.9, 305.8)--(473.0, 304.5)--(473.0, 303.7)--(473.1, 303.0)--(473.1, 302.5)--(473.1, 302.1)--(473.1, 301.8)--(473.1, 301.1)--(473.2, 300.0)--(473.2, 299.0)--(473.3, 297.3)--(473.4, 296.2)--(473.4, 296.1)--(473.4, 295.0)--(473.5, 293.3)--(473.5, 292.8)--(473.6, 292.0)--(473.6, 290.7)--(473.7, 290.4)--(473.7, 289.8)--(473.7, 288.9)--(473.8, 288.1)--(473.9, 287.8)--(473.9, 287.7)--(473.9, 287.6)--(473.9, 287.0)--(473.9, 286.8)--(473.9, 286.7)--(473.9, 286.5)--(473.9, 286.4)--(473.9, 286.2)--(474.3, 284.9)--(474.3, 284.8)--(474.3, 284.6)--(474.4, 284.3)--(474.4, 283.0)--(474.5, 282.1)--(474.5, 281.1)--(474.5, 280.8)--(474.5, 280.5)--(474.6, 280.1)--(474.8, 279.5)--(474.9, 279.0)--(475.2, 278.0)--(475.5, 277.1)--(475.6, 276.6)--(475.8, 275.9)--(475.9, 275.3)--(476.0, 275.1)--(476.0, 274.9)--(476.1, 274.6)--(476.0, 274.6)--(475.9, 274.7)--(475.9, 274.8)--(475.8, 274.9)--(475.6, 275.1)--(475.2, 275.4)--(475.0, 275.6)--(474.7, 275.7)--(474.5, 275.9)--(474.4, 276.0)--(474.1, 276.1)--(473.9, 276.3)--(473.7, 276.4)--(473.6, 276.4)--(473.3, 276.5)--(473.1, 276.5)--(473.0, 276.5)--(472.7, 276.5)--(472.6, 276.6)--(472.5, 276.6)--(472.4, 276.7)--(472.1, 276.8)--(471.9, 276.5)--(471.2, 275.5)--(471.0, 275.2)--(470.8, 274.8)--(470.5, 274.4)--(470.2, 273.8)--(470.1, 273.7)--(470.0, 273.5)--(469.8, 273.2)--(469.5, 273.0)--(469.4, 272.8)--(469.1, 272.6)--(469.0, 272.2)--(468.8, 271.8)--(468.6, 271.4)--(468.5, 271.2)--(468.5, 271.1)--(468.4, 270.9)--(468.2, 270.6)--(468.1, 270.6)--(468.0, 270.4)--(467.8, 270.2)--(467.6, 270.3)--(467.6, 270.4)--(467.2, 270.7)--(467.0, 270.9)--(466.9, 271.0)--(466.9, 271.1)--(466.8, 271.2)--(466.6, 271.4)--(466.4, 271.7)--(466.2, 272.0)--(465.9, 272.3)--(465.6, 272.5)--(465.4, 272.8)--(465.3, 273.1)--(465.1, 273.3)--(465.1, 273.5)--(465.0, 273.5)--(464.7, 273.3)--(464.7, 273.2)--(464.6, 273.1)--(464.5, 273.1)--(464.5, 273.0)--(464.4, 273.0)--(464.4, 272.8)--(464.4, 272.7)--(464.3, 272.6)--(464.2, 272.6)--(463.8, 272.3)--(463.7, 272.2)--(463.3, 272.7)--(463.0, 272.4)--(462.8, 272.7)--(462.7, 272.7)--(462.6, 272.6)--(462.5, 272.4)--(462.3, 272.2)--(462.1, 272.0)--(462.0, 271.9)--(461.8, 271.7)--(461.7, 271.7)--(461.6, 271.6)--(461.6, 271.7)--(461.3, 271.9)--(461.2, 272.0)--(461.1, 272.0)--(460.8, 271.5)--(460.7, 271.5)--(460.6, 271.5)--(460.4, 271.8)--(460.4, 271.9)--(460.1, 272.1)--(459.5, 272.3)--(459.2, 272.4)--(458.9, 272.5)--(458.5, 272.5)--(458.2, 272.5)--(458.1, 272.4)--(458.1, 272.3)--(458.0, 272.2)--(457.7, 271.6)--(457.6, 271.5)--(457.4, 271.1)--(457.3, 271.0)--(457.2, 270.8)--(457.1, 270.1)--(457.1, 270.0)--(456.4, 270.0)--(456.2, 270.1)--(456.0, 270.1)--(455.7, 270.0)--(455.6, 270.0)--(455.4, 270.0)--(455.3, 269.9)--(455.2, 269.9)--(455.0, 269.8)--(454.7, 269.6)--(454.5, 269.4)--(454.4, 269.3)--(454.3, 269.2)--(454.0, 269.1)--(453.9, 269.0)--(453.9, 268.9)--(453.8, 268.9)--(453.6, 268.8)--(453.5, 268.7)--(453.4, 268.6)--(453.5, 268.6)--(453.4, 268.5)--(453.4, 268.4)--(453.3, 268.4)--(453.3, 268.3)--(453.4, 268.3)--(453.2, 268.0)--(452.9, 267.8)--(452.7, 267.5)--(452.5, 267.3)--(452.3, 267.1)--(451.9, 266.8)--(451.8, 266.7)--(451.6, 266.6)--(451.3, 266.5)--(450.9, 266.3)--(450.7, 266.3)--(450.5, 266.2)--(450.3, 266.0)--(450.1, 265.9)--(449.9, 265.7)--(449.7, 265.6)--(449.5, 265.2)--(449.3, 264.9)--(449.3, 264.8)--(449.0, 264.8)--(448.9, 264.3)--(448.7, 264.0)--(448.5, 263.5)--(448.2, 262.8)--(448.0, 262.3)--(447.4, 262.4)--(447.3, 262.3)--(447.3, 262.2)--(447.3, 261.8)--(447.3, 261.4)--(447.3, 260.7)--(447.3, 260.3)--(447.4, 260.2)--(447.6, 259.6)--(447.7, 259.4)--(447.7, 259.1)--(447.7, 258.7)--(447.6, 258.4)--(447.6, 258.0)--(447.6, 257.9)--(447.5, 257.5)--(447.5, 257.2)--(447.5, 256.8)--(447.5, 256.7)--(447.5, 256.3)--(447.5, 256.2)--(447.5, 256.1)--(447.5, 256.0)--(447.5, 255.9)--(447.6, 255.8)--(447.6, 255.6)--(447.7, 255.4)--(447.8, 255.1)--(448.0, 254.9)--(446.6, 254.5)--(445.6, 254.3)--(445.1, 254.2)--(445.1, 254.1)--(445.2, 254.0)--(445.7, 253.5)--(445.8, 253.4)--(445.9, 253.5)--(446.3, 253.0)--(446.6, 252.8)--(446.7, 252.8)--(447.0, 252.5)--(447.6, 252.3)--(448.2, 252.0)--(448.8, 251.8)--(449.6, 251.5)--(450.0, 251.4)--(450.0, 251.3)--(450.2, 250.9)--(450.3, 250.4)--(450.3, 250.2)--(450.3, 250.1)--(450.3, 250.0)--(450.3, 249.8)--(450.4, 249.6)--(450.4, 249.4)--(450.5, 249.2)--(450.5, 249.1)--(450.5, 248.8)--(450.6, 248.2)--(450.7, 247.8)--(450.7, 247.3)--(450.8, 246.7)--(450.8, 246.4)--(450.9, 246.1)--(450.9, 246.0)--(450.9, 245.7)--(451.0, 245.1)--(451.0, 244.8)--(451.0, 244.5)--(451.2, 244.0)--(451.5, 243.5)--(451.8, 242.5)--(451.8, 242.3)--(451.9, 242.0)--(452.0, 241.3)--(452.0, 240.8)--(452.0, 240.5)--(451.9, 240.2)--(451.8, 240.0)--(451.7, 239.8)--(451.5, 239.7)--(451.5, 239.5)--(451.5, 239.4)--(451.5, 239.3)--(451.4, 239.1)--(451.2, 239.0)--(451.2, 238.9)--(451.1, 238.9)--(451.0, 238.9)--(450.9, 238.8)--(450.8, 238.8)--(450.9, 238.4)--(451.0, 238.1)--(451.1, 237.8)--(451.2, 237.5)--(451.3, 237.2)--(451.5, 236.9)--(451.6, 236.5)--(451.3, 236.5)--(451.0, 236.5)--(450.9, 236.5)--(450.5, 236.2)--(450.4, 236.1)--(450.2, 236.0)--(450.0, 235.9)--(449.8, 235.7)--(449.5, 236.0)--(449.1, 236.4)--(448.7, 236.7)--(448.6, 236.6)--(448.5, 236.5)--(448.6, 236.3)--(448.7, 236.2)--(448.6, 236.1)--(448.5, 236.0)--(448.6, 235.9)--(448.8, 235.8)--(449.6, 235.1)--(449.5, 234.8)--(448.8, 234.0)--(448.6, 233.8)--(448.5, 233.6)--(448.4, 233.4)--(447.9, 232.8)--(447.9, 232.7)--(447.8, 232.8)--(447.5, 233.0)--(447.2, 233.2)--(446.9, 232.8)--(447.0, 232.7)--(446.8, 232.6)--(446.7, 232.3)--(446.8, 232.2)--(446.7, 232.1)--(446.6, 232.0)--(446.6, 231.9)--(446.7, 231.9)--(446.5, 231.7)--(446.5, 231.6)--(446.4, 231.6)--(446.4, 231.5)--(446.3, 231.5)--(446.4, 231.3)--(446.3, 231.2)--(446.1, 231.0)--(446.0, 230.9)--(445.9, 230.7)--(445.8, 230.6)--(445.9, 230.6)--(446.0, 230.5)--(445.9, 230.3)--(445.9, 230.1)--(445.8, 229.8)--(445.7, 229.6)--(445.6, 229.5)--(445.5, 229.4)--(445.4, 229.1)--(445.3, 228.9)--(445.2, 228.8)--(444.9, 228.9)--(444.7, 229.0)--(444.6, 229.1)--(444.3, 229.2)--(443.9, 228.7)--(443.3, 228.1)--(443.2, 228.0)--(443.6, 227.7)--(443.7, 227.5)--(444.0, 227.2)--(444.3, 227.0)--(444.4, 227.0)--(443.9, 226.5)--(444.2, 226.1)--(444.3, 226.0)--(444.0, 225.7)--(443.8, 225.6)--(444.1, 225.2)--(444.0, 224.9)--(443.8, 224.7)--(443.7, 224.5)--(443.4, 224.2)--(443.3, 224.1)--(443.2, 224.0)--(443.2, 223.9)--(443.1, 223.9)--(443.6, 223.8)--(443.6, 223.7)--(443.5, 223.3)--(443.3, 222.4)--(442.9, 223.2)--(442.7, 223.3)--(442.2, 223.5)--(442.0, 223.0)--(442.4, 222.9)--(441.5, 221.4)--(441.3, 221.1)--(441.1, 220.8)--(440.9, 220.6)--(440.2, 220.0)--(440.1, 219.9)--(440.0, 219.7)--(439.0, 218.7)--(438.0, 217.7)--(437.6, 217.3)--(437.7, 216.9)--(436.4, 216.5)--(436.5, 216.3)--(436.1, 216.2)--(436.3, 215.8)--(436.0, 215.5)--(435.7, 215.1)--(435.6, 215.1)--(435.6, 215.0)--(435.5, 214.9)--(435.4, 214.8)--(435.3, 214.7)--(434.4, 214.3)--(434.1, 214.7)--(433.8, 215.8)--(433.6, 215.8)--(433.1, 215.5)--(432.7, 215.3)--(432.6, 215.3)--(432.5, 215.3)--(432.4, 215.3)--(432.3, 215.3)--(432.2, 215.2)--(431.9, 215.5)--(431.6, 215.8)--(430.7, 216.9)--(430.6, 216.9)--(430.4, 217.3)--(429.5, 218.4)--(429.4, 218.5)--(429.2, 218.8)--(428.8, 219.4)--(428.5, 220.0)--(428.1, 220.7)--(427.8, 221.3)--(427.6, 221.6)--(427.0, 222.5)--(426.8, 222.8)--(426.1, 223.4)--(425.7, 223.8)--(425.5, 224.0)--(424.8, 224.7)--(424.4, 225.2)--(423.9, 225.9)--(423.6, 226.4)--(423.4, 226.6)--(423.3, 226.8)--(423.0, 227.2)--(422.5, 227.9)--(422.5, 228.1)--(422.5, 228.3)--(422.3, 228.5)--(421.9, 228.9)--(421.4, 229.4)--(421.2, 229.5)--(421.1, 229.7)--(420.9, 230.1)--(420.9, 230.2)--(420.7, 230.5)--(420.6, 230.9)--(420.4, 231.2)--(420.4, 231.4)--(420.2, 231.6)--(419.8, 231.9)--(419.5, 232.2)--(418.9, 232.8)--(418.9, 232.9)--(418.8, 232.9)--(418.7, 233.1)--(418.6, 233.1)--(418.5, 233.1)--(418.3, 233.2)--(418.1, 233.4)--(418.0, 233.4)--(417.7, 233.5)--(417.6, 233.5)--(417.3, 233.8)--(417.1, 233.9)--(416.7, 234.0)--(416.6, 234.1)--(416.5, 234.1)--(416.5, 234.2)--(416.4, 234.3)--(416.1, 234.5)--(416.0, 234.5)--(415.9, 234.5)--(415.8, 234.6)--(415.6, 234.6)--(415.4, 234.8)--(415.1, 235.0)--(415.0, 235.1)--(415.0, 235.2)--(414.8, 235.3)--(414.6, 235.3)--(414.6, 235.4)--(414.5, 235.4)--(414.5, 235.5)--(414.4, 235.7)--(414.3, 235.7)--(414.3, 235.8)--(414.2, 235.8)--(414.1, 235.8)--(414.0, 235.9)--(413.9, 235.9)--(413.8, 235.9)--(413.6, 236.0)--(413.5, 236.1)--(413.3, 236.1)--(413.2, 236.2)--(412.9, 236.3)--(412.7, 236.4)--(412.6, 236.4)--(412.4, 236.5)--(412.3, 236.5)--(412.1, 236.6)--(412.0, 236.6)--(411.8, 235.4)--(411.5, 234.1)--(411.5, 234.0)--(411.5, 233.9)--(411.5, 233.7)--(411.4, 233.7)--(411.1, 233.6)--(411.0, 233.6)--(410.9, 233.5)--(410.8, 233.5)--(410.6, 233.4)--(410.3, 233.4)--(409.9, 233.4)--(409.6, 233.5)--(409.5, 233.5)--(409.2, 233.5)--(408.6, 233.6)--(408.3, 233.7)--(408.1, 233.7)--(407.9, 233.8)--(407.8, 233.8)--(407.4, 234.0)--(407.1, 234.1)--(406.8, 234.3)--(406.3, 234.5)--(405.9, 234.6)--(405.6, 235.4)--(405.2, 236.3)--(405.8, 236.5)--(405.8, 236.8)--(405.6, 237.2)--(405.6, 237.4)--(405.6, 237.9)--(405.6, 239.0)--(405.8, 239.1)--(406.6, 239.6)--(406.7, 239.6)--(406.7, 239.9)--(406.5, 240.1)--(406.4, 240.6)--(406.3, 240.7)--(406.3, 240.8)--(406.2, 241.0)--(406.2, 241.1)--(406.1, 241.3)--(406.0, 241.6)--(405.7, 241.6)--(405.0, 241.7)--(404.1, 241.9)--(403.2, 242.0)--(402.7, 242.1)--(401.2, 242.6)--(401.0, 243.3)--(401.1, 243.3)--(401.2, 242.9)--(404.5, 243.5)--(404.4, 243.9)--(404.4, 244.0)--(404.3, 244.2)--(404.2, 244.6)--(404.1, 244.6)--(403.4, 244.4)--(403.2, 246.0)--(403.1, 246.0)--(403.1, 246.1)--(403.0, 246.3)--(402.9, 246.7)--(402.8, 246.9)--(402.8, 247.2)--(402.8, 247.3)--(402.7, 247.4)--(402.7, 247.5)--(402.7, 247.6)--(402.7, 247.7)--(402.7, 247.8)--(402.7, 247.9)--(402.7, 248.0)--(402.7, 248.3)--(401.7, 248.6)--(401.2, 248.7)--(400.8, 248.8)--(400.5, 248.9)--(400.4, 248.9)--(400.3, 249.0)--(399.9, 249.1)--(399.7, 249.2)--(399.6, 249.3)--(399.3, 249.4)--(399.1, 249.6)--(398.8, 249.7)--(398.6, 249.8)--(398.5, 249.9)--(398.5, 250.0)--(398.5, 250.1)--(398.5, 250.2)--(398.4, 250.3)--(398.3, 250.5)--(398.1, 250.9)--(398.0, 251.3)--(397.9, 251.5)--(397.7, 251.9)--(397.6, 252.2)--(397.5, 252.6)--(397.3, 252.9)--(397.3, 253.1)--(397.3, 253.4)--(397.2, 254.0)--(397.2, 254.6)--(397.2, 254.9)--(397.2, 255.2)--(397.2, 255.5)--(397.2, 255.9)--(397.2, 256.2)--(397.1, 256.4)--(397.1, 256.7)--(397.0, 257.0)--(396.9, 257.3)--(396.8, 257.7)--(396.8, 257.9)--(396.7, 258.1)--(396.7, 258.2)--(396.6, 258.6)--(396.6, 258.7)--(396.5, 258.8)--(396.5, 258.9)--(396.5, 259.0)--(396.6, 259.3)--(396.6, 259.5)--(396.6, 259.8)--(396.6, 260.1)--(397.1, 260.3)--(397.5, 260.4)--(397.7, 260.4)--(397.8, 260.5)--(398.1, 260.5)--(398.2, 260.6)--(398.3, 260.6)--(398.4, 260.7)--(398.6, 260.9)--(398.9, 261.1)--(399.1, 261.1)--(399.1, 261.2)--(399.2, 261.2)--(399.3, 261.3)--(399.4, 261.3)--(399.3, 261.4)--(399.2, 261.5)--(399.1, 261.8)--(399.0, 261.9)--(399.0, 262.0)--(399.0, 262.2)--(399.1, 262.3)--(399.2, 262.5)--(399.2, 262.6)--(399.2, 262.7)--(399.2, 262.9)--(399.2, 263.1)--(399.1, 263.5)--(398.9, 264.6)--(399.1, 264.6)--(399.5, 264.7)--(399.5, 264.6)--(399.6, 264.6)--(399.7, 264.6)--(399.9, 264.6)--(400.1, 264.5)--(400.2, 264.5)--(400.5, 264.4)--(400.7, 264.2)--(400.9, 264.1)--(401.1, 264.0)--(401.2, 263.9)--(401.3, 263.9)--(401.5, 263.7)--(401.6, 263.9)--(401.7, 264.1)--(401.7, 264.3)--(401.7, 264.5)--(401.5, 264.7)--(401.5, 264.8)--(401.4, 264.8)--(401.4, 264.9)--(401.4, 265.0)--(401.3, 265.1)--(401.1, 265.5)--(400.8, 266.3)--(400.7, 266.6)--(400.6, 267.0)--(400.5, 267.4)--(400.4, 267.7)--(400.4, 268.0)--(400.4, 268.3)--(401.1, 268.5)--(401.3, 268.6)--(401.5, 268.7)--(402.3, 269.0)--(402.6, 269.2)--(402.5, 269.2)--(401.8, 269.0)--(400.6, 268.8)--(400.0, 268.6)--(399.4, 268.5)--(398.4, 268.3)--(397.1, 268.0)--(396.4, 267.9)--(395.2, 267.7)--(393.8, 267.4)--(393.2, 267.3)--(392.7, 267.1)--(392.0, 267.0)--(391.2, 266.8)--(390.7, 266.7)--(390.3, 266.7)--(390.1, 266.6)--(389.8, 266.6)--(389.4, 266.6)--(389.2, 266.6)--(389.1, 266.6)--(389.0, 266.6)--(388.4, 266.7)--(387.5, 266.8)--(386.1, 267.1)--(385.4, 267.3)--(385.2, 267.3)--(384.9, 267.3)--(384.4, 267.5)--(384.0, 267.5)--(383.8, 267.4)--(383.7, 267.4)--(383.0, 267.3)--(382.9, 267.2)--(381.9, 267.0)--(381.3, 266.9)--(381.1, 266.9)--(380.9, 266.9)--(380.8, 267.0)--(380.5, 267.1)--(380.3, 267.3)--(380.1, 267.1)--(379.9, 266.7)--(380.0, 266.4)--(379.8, 266.4)--(379.5, 266.2)--(379.7, 265.8)--(380.1, 265.0)--(380.3, 264.4)--(380.4, 264.3)--(380.5, 264.1)--(380.6, 263.9)--(380.8, 263.6)--(381.0, 263.1)--(380.8, 262.9)--(380.4, 262.6)--(380.1, 262.4)--(380.0, 262.3)--(379.7, 262.1)--(379.5, 262.2)--(379.4, 262.3)--(379.4, 262.4)--(379.2, 262.6)--(379.1, 262.6)--(378.8, 263.0)--(378.7, 263.1)--(378.7, 263.2)--(378.6, 263.4)--(378.4, 263.2)--(378.2, 263.4)--(378.2, 263.5)--(378.1, 263.6)--(378.0, 263.6)--(377.8, 263.6)--(377.7, 263.6)--(377.5, 263.7)--(377.3, 263.8)--(377.1, 263.9)--(376.8, 264.1)--(376.9, 264.3)--(376.8, 264.5)--(376.7, 264.8)--(376.8, 264.8)--(376.9, 264.9)--(376.9, 264.8)--(377.1, 264.9)--(377.0, 265.0)--(376.9, 265.2)--(376.9, 265.4)--(377.0, 265.4)--(377.0, 265.5)--(376.9, 265.5)--(376.8, 265.6)--(376.8, 265.8)--(376.7, 266.3)--(376.6, 266.4)--(376.6, 266.5)--(376.2, 266.4)--(376.2, 266.5)--(376.1, 266.5)--(376.0, 266.6)--(375.9, 266.7)--(375.9, 266.8)--(375.9, 266.9)--(376.0, 267.1)--(376.1, 267.2)--(376.1, 267.3)--(376.1, 267.4)--(376.1, 267.5)--(376.1, 267.6)--(376.0, 267.7)--(375.9, 267.7)--(375.7, 267.7)--(375.6, 267.9)--(375.6, 268.0)--(375.5, 268.1)--(375.5, 268.2)--(375.4, 268.2)--(375.3, 268.3)--(375.2, 268.3)--(375.1, 268.4)--(375.0, 268.5)--(374.9, 268.6)--(374.8, 268.5)--(374.7, 268.6)--(374.6, 268.6)--(374.6, 268.7)--(374.5, 268.7)--(374.4, 268.7)--(374.4, 268.5)--(374.2, 268.6)--(374.3, 268.7)--(374.3, 268.9)--(374.1, 268.9)--(374.1, 269.1)--(374.1, 269.2)--(374.1, 269.4)--(374.0, 269.8)--(374.0, 269.9)--(374.0, 270.0)--(374.1, 270.0)--(374.1, 270.1)--(374.1, 270.0)--(374.2, 270.0)--(374.4, 270.0)--(374.5, 270.0)--(374.7, 269.9)--(375.2, 270.1)--(375.3, 270.0)--(375.3, 270.1)--(375.5, 270.3)--(375.7, 270.5)--(375.8, 270.6)--(375.8, 270.7)--(375.7, 270.7)--(375.6, 271.1)--(375.6, 271.3)--(375.6, 271.4)--(375.7, 271.5)--(375.8, 271.5)--(376.0, 271.8)--(376.1, 271.8)--(376.2, 271.9)--(376.3, 271.9)--(376.3, 271.8)--(376.2, 271.7)--(376.3, 271.6)--(376.6, 272.0)--(376.7, 272.0)--(376.7, 272.1)--(376.8, 272.3)--(376.7, 272.9)--(376.7, 273.2)--(376.7, 273.3)--(376.9, 274.2)--(377.1, 274.6)--(377.4, 275.0)--(377.5, 275.0)--(377.5, 275.1)--(377.9, 275.3)--(378.0, 275.3)--(378.1, 275.4)--(378.3, 276.7)--(377.9, 275.8)--(377.5, 276.1)--(377.4, 276.3)--(377.2, 276.3)--(377.0, 276.3)--(376.9, 276.2)--(376.8, 276.2)--(376.6, 276.1)--(375.9, 275.7)--(375.6, 275.5)--(375.7, 275.2)--(374.2, 274.9)--(373.8, 274.8)--(373.7, 275.5)--(373.7, 276.1)--(372.6, 276.5)--(371.4, 276.9)--(371.7, 277.2)--(371.7, 277.3)--(371.8, 277.5)--(371.8, 277.6)--(371.9, 278.1)--(371.9, 279.0)--(372.1, 279.9)--(372.1, 280.0)--(372.2, 280.4)--(372.2, 280.7)--(372.2, 280.8)--(372.3, 281.1)--(372.4, 281.4)--(372.5, 281.5)--(373.0, 281.2)--(372.9, 281.1)--(373.5, 280.8)--(373.9, 280.5)--(373.9, 280.4)--(374.0, 280.3)--(374.1, 280.4)--(375.0, 279.9)--(375.2, 279.8)--(375.3, 279.8)--(375.4, 279.7)--(375.7, 279.6)--(376.3, 279.3)--(376.6, 279.2)--(376.7, 279.1)--(376.7, 278.8)--(376.8, 278.3)--(376.9, 278.1)--(376.9, 278.0)--(377.0, 277.8)--(377.1, 277.6)--(377.3, 277.4)--(378.3, 277.2)--(378.3, 276.9)--(378.7, 276.7)--(378.8, 276.9)--(378.8, 277.0)--(378.9, 277.3)--(379.3, 277.2)--(379.3, 277.4)--(378.8, 277.4)--(378.6, 277.5)--(377.9, 277.8)--(377.7, 277.9)--(377.3, 278.1)--(377.3, 278.2)--(377.2, 278.3)--(377.2, 278.5)--(377.2, 278.7)--(377.2, 278.8)--(377.2, 278.9)--(377.3, 278.9)--(377.4, 279.0)--(377.9, 279.0)--(378.4, 279.0)--(378.4, 278.9)--(378.4, 278.7)--(378.7, 278.6)--(378.9, 278.6)--(379.0, 278.6)--(379.0, 278.5)--(379.1, 278.5)--(379.1, 278.8)--(379.1, 278.9)--(379.6, 278.8)--(379.6, 278.9)--(380.0, 278.9)--(380.0, 278.7)--(380.3, 278.7)--(380.5, 278.7)--(381.1, 278.7)--(381.2, 278.7)--(381.3, 278.7)--(381.3, 278.8)--(381.4, 278.9)--(381.6, 278.9)--(382.0, 278.9)--(382.1, 278.9)--(382.0, 279.2)--(382.4, 279.2)--(382.7, 279.3)--(382.8, 279.3)--(382.8, 279.4)--(382.8, 279.5)--(383.6, 279.8)--(383.8, 279.8)--(383.8, 279.7)--(383.9, 279.6)--(383.9, 279.5)--(384.6, 279.8)--(386.1, 280.6)--(386.2, 280.7)--(386.4, 280.8)--(386.6, 280.7)--(387.1, 281.1)--(387.2, 281.0)--(387.2, 281.1)--(387.3, 281.0)--(387.4, 281.1)--(387.3, 281.2)--(388.2, 281.9)--(388.1, 282.1)--(388.0, 282.2)--(388.4, 282.6)--(388.5, 282.5)--(388.4, 282.3)--(388.4, 282.2)--(388.7, 282.4)--(388.7, 282.5)--(388.7, 282.6)--(388.8, 282.7)--(389.1, 283.0)--(389.2, 282.9)--(389.5, 283.1)--(389.7, 283.3)--(390.1, 283.9)--(390.1, 284.1)--(390.2, 284.1)--(390.3, 284.3)--(390.3, 284.4)--(390.5, 284.3)--(390.6, 284.6)--(390.5, 284.6)--(390.7, 285.2)--(390.9, 285.1)--(390.9, 285.2)--(390.8, 285.3)--(390.9, 285.8)--(391.0, 285.8)--(391.2, 286.8)--(391.1, 286.8)--(391.1, 287.4)--(391.2, 287.4)--(391.1, 288.4)--(391.1, 288.5)--(391.3, 288.8)--(391.3, 288.9)--(391.2, 289.1)--(391.1, 289.4)--(391.1, 289.7)--(391.0, 290.0)--(391.1, 290.2)--(390.8, 290.1)--(390.7, 290.3)--(390.6, 290.5)--(390.6, 290.6)--(390.5, 290.9)--(390.6, 290.9)--(390.4, 291.3)--(390.5, 291.3)--(390.4, 291.4)--(390.3, 291.4)--(390.3, 291.5)--(390.0, 291.7)--(390.0, 291.8)--(390.0, 291.9)--(390.1, 292.0)--(390.1, 292.1)--(390.2, 292.1)--(390.2, 292.2)--(390.2, 292.4)--(390.1, 292.5)--(390.1, 292.6)--(389.9, 292.9)--(389.1, 292.6)--(388.6, 293.9)--(388.6, 294.0)--(388.5, 294.1)--(388.3, 295.0)--(388.5, 295.1)--(388.1, 296.1)--(388.2, 296.1)--(388.1, 296.6)--(387.9, 296.7)--(387.9, 296.6)--(387.8, 297.2)--(387.7, 297.3)--(387.7, 297.5)--(387.7, 297.6)--(387.6, 298.0)--(387.6, 299.0)--(387.6, 299.3)--(387.9, 299.3)--(387.9, 299.6)--(387.9, 299.8)--(387.8, 299.8)--(387.8, 300.0)--(387.9, 300.1)--(388.0, 300.0)--(388.1, 300.1)--(388.1, 300.2)--(387.7, 300.3)--(387.7, 300.4)--(387.9, 300.9)--(387.9, 301.1)--(388.0, 301.3)--(388.1, 301.4)--(388.2, 301.6)--(388.4, 301.8)--(388.7, 302.1)--(388.8, 302.3)--(388.9, 302.4)--(389.2, 302.6)--(389.3, 302.7)--(389.6, 303.0)--(390.1, 303.2)--(390.4, 303.3)--(390.5, 303.3)--(390.9, 303.4)--(391.3, 303.4)--(391.5, 303.4)--(391.9, 303.2)--(392.5, 302.9)--(392.6, 302.8)--(392.9, 302.6)--(393.2, 302.2)--(393.6, 301.7)--(393.8, 301.4)--(394.8, 300.1)--(394.9, 300.0)--(395.0, 299.8)--(395.4, 299.3)--(396.7, 297.5)--(397.2, 296.9)--(397.4, 296.6)--(397.8, 296.1)--(397.9, 296.1)--(398.1, 295.9)--(398.3, 295.7)--(398.5, 295.5)--(398.8, 295.3)--(398.8, 295.2)--(398.9, 295.2)--(398.9, 294.9)--(399.0, 294.8)--(399.3, 294.4)--(399.4, 294.3)--(399.5, 294.2)--(399.6, 294.1)--(399.7, 294.0)--(400.0, 293.8)--(400.2, 293.7)--(400.3, 293.6)--(400.3, 293.7)--(401.1, 293.2)--(401.3, 293.1)--(401.4, 293.1)--(401.5, 293.0)--(401.7, 292.9)--(401.7, 292.8)--(401.8, 292.8)--(401.9, 292.9)--(402.2, 292.7)--(402.7, 292.5)--(402.7, 292.6)--(403.2, 292.5)--(403.5, 292.4)--(403.8, 292.3)--(404.4, 292.2)--(404.4, 291.7)--(404.6, 291.6)--(404.7, 291.8)--(404.9, 292.0)--(405.2, 292.0)--(405.9, 291.8)--(405.9, 291.6)--(406.3, 291.6)--(406.7, 291.6)--(406.9, 291.6)--(407.5, 291.6)--(408.4, 291.5)--(408.7, 291.6)--(409.0, 291.6)--(409.1, 291.5)--(409.3, 291.6)--(409.3, 291.7)--(409.5, 291.7)--(409.8, 291.7)--(409.9, 291.6)--(410.1, 291.7)--(410.2, 291.8)--(411.2, 292.1)--(411.4, 292.1)--(411.5, 292.1)--(411.6, 292.2)--(412.1, 292.3)--(412.4, 292.3)--(412.7, 292.4)--(413.3, 292.7)--(413.5, 292.7)--(413.7, 292.8)--(413.9, 292.8)--(414.0, 292.9)--(414.2, 292.9)--(414.5, 292.8)--(414.7, 292.9)--(414.8, 292.9)--(415.0, 292.9)--(415.0, 293.0)--(415.1, 293.1)--(415.4, 293.1)--(415.7, 293.2)--(416.6, 293.5)--(417.0, 293.6)--(417.3, 293.6)--(418.0, 293.6)--(418.2, 293.6)--(418.4, 293.6)--(418.9, 293.6)--(419.6, 293.5)--(421.1, 293.4)--(422.7, 293.4)--(423.5, 293.3)--(423.9, 293.3)--(424.3, 293.2)--(424.4, 293.2)--(425.7, 293.1)--(426.7, 293.0)--(427.9, 292.9)--(428.0, 292.8)--(428.1, 292.8)--(428.2, 292.9)--(428.4, 292.9)--(428.4, 292.8)--(428.6, 292.8)--(428.7, 292.9)--(429.4, 292.9)--(429.5, 292.8)--(429.5, 292.4)--(429.7, 292.4)--(429.6, 292.8)--(429.7, 292.9)--(430.2, 292.9)--(430.3, 292.6)--(430.4, 292.6)--(430.4, 292.9)--(431.6, 293.0)--(432.6, 293.1)--(432.7, 293.0)--(433.1, 293.1)--(433.2, 293.1)--(433.7, 293.2)--(433.8, 293.1)--(433.9, 293.1)--(433.9, 293.3)--(434.4, 293.3)--(434.5, 293.3)--(436.5, 293.6)--(437.1, 293.7)--(437.3, 293.7)--(438.2, 293.9)--(439.8, 294.3)--(441.3, 294.8)--(442.0, 295.0)--(442.1, 294.9)--(442.2, 294.8)--(442.4, 294.8)--(442.6, 294.9)--(442.5, 295.3)--(442.9, 295.5)--(443.2, 295.7)--(444.1, 296.5)--(445.3, 297.4)--(446.3, 298.3)--(446.4, 298.2)--(446.5, 298.3)--(446.4, 298.4)--(446.9, 299.3)--(447.3, 300.0)--(447.5, 300.3)--(447.8, 300.7)--(448.8, 302.7)--(449.1, 303.3)--(449.2, 303.5)--(449.3, 303.5)--(449.3, 303.4)--(449.4, 303.4)--(449.5, 303.7)--(449.6, 303.8)--(449.8, 303.9)--(449.9, 304.0)--(449.9, 304.1)--(449.9, 304.4)--(450.0, 305.2)--(450.0, 305.7)--(450.0, 305.8)--(450.2, 305.9)--(450.4, 306.1)--(450.9, 306.9)--(452.1, 308.5)--(452.4, 309.0)--(452.6, 309.4)--(452.7, 309.5)--(452.7, 309.6)--(452.8, 309.7)--(452.9, 309.8)--(453.3, 309.8)--(453.5, 309.8)--(454.3, 309.8)--(454.6, 309.7)--(455.4, 310.0)--(456.1, 310.3)--(456.4, 310.4)--(457.3, 310.5)--(457.8, 310.6)--(458.5, 310.6)--(459.1, 310.6)--(460.0, 310.6)--(460.8, 310.5)--(461.4, 310.5)--(461.6, 310.5)--(462.3, 310.5)--(462.3, 310.4)--(463.1, 310.5)--(463.6, 310.6)--(464.5, 310.7)--(465.3, 310.9)--(466.3, 311.1)--(467.6, 311.4)--(468.2, 311.6)--(469.3, 311.9)--(470.2, 312.2)--(471.5, 312.7)--cycle; +Hackney = (319.3, 378.0)--(319.4, 378.0)--(319.5, 378.1)--(319.6, 378.1)--(319.6, 378.2)--(319.7, 378.2)--(319.8, 378.2)--(319.9, 378.2)--(320.0, 378.2)--(320.1, 378.2)--(320.3, 378.1)--(320.7, 378.0)--(320.8, 378.0)--(320.9, 378.0)--(321.0, 378.0)--(321.1, 378.0)--(321.2, 378.0)--(321.3, 378.0)--(321.4, 378.0)--(321.6, 378.2)--(321.9, 378.4)--(322.0, 378.4)--(322.1, 378.4)--(322.1, 378.5)--(322.3, 378.5)--(322.4, 378.5)--(322.5, 378.5)--(322.6, 378.5)--(322.7, 378.5)--(322.8, 378.5)--(323.0, 378.5)--(323.1, 378.5)--(323.2, 378.5)--(323.3, 378.5)--(323.4, 378.5)--(323.5, 378.5)--(323.6, 378.5)--(323.7, 378.5)--(323.9, 378.6)--(324.1, 378.8)--(324.3, 378.9)--(324.4, 378.9)--(324.5, 378.9)--(324.5, 379.0)--(324.6, 379.0)--(324.7, 379.0)--(324.9, 379.0)--(325.0, 379.0)--(325.1, 379.0)--(325.2, 379.0)--(325.3, 378.9)--(325.4, 378.9)--(325.5, 378.9)--(325.7, 378.9)--(325.9, 379.0)--(326.0, 379.0)--(326.1, 379.0)--(326.2, 379.0)--(326.4, 379.0)--(326.7, 378.9)--(326.7, 379.0)--(326.7, 379.1)--(326.6, 379.2)--(326.5, 379.1)--(326.5, 379.2)--(326.5, 379.4)--(326.9, 379.4)--(327.3, 379.5)--(327.3, 379.6)--(327.4, 379.6)--(327.5, 379.8)--(328.1, 379.9)--(328.8, 380.0)--(329.0, 380.0)--(329.5, 380.1)--(329.9, 380.2)--(330.2, 380.3)--(330.9, 380.5)--(331.6, 380.4)--(331.9, 380.4)--(332.2, 380.3)--(332.5, 380.3)--(332.8, 380.2)--(333.2, 380.1)--(333.5, 380.0)--(334.1, 379.8)--(334.3, 380.2)--(334.3, 380.3)--(334.4, 380.5)--(334.4, 380.4)--(334.6, 380.3)--(334.7, 380.3)--(334.8, 380.3)--(334.8, 380.5)--(335.3, 380.6)--(335.3, 380.7)--(335.4, 380.7)--(335.4, 380.6)--(335.5, 380.6)--(335.6, 380.7)--(335.8, 380.7)--(335.9, 380.7)--(335.9, 380.6)--(336.3, 380.7)--(336.5, 380.7)--(336.7, 380.7)--(336.8, 380.7)--(337.1, 380.8)--(337.1, 380.7)--(337.7, 380.8)--(338.1, 380.9)--(338.4, 381.0)--(338.6, 381.0)--(338.7, 381.1)--(338.8, 381.1)--(339.2, 381.3)--(339.5, 381.3)--(339.8, 381.4)--(340.2, 381.5)--(340.3, 381.5)--(340.4, 381.5)--(340.6, 381.6)--(340.6, 381.9)--(341.0, 382.0)--(341.5, 382.2)--(342.1, 382.4)--(342.3, 382.5)--(342.7, 382.6)--(343.4, 382.9)--(344.4, 383.3)--(344.6, 382.9)--(344.8, 382.2)--(345.1, 381.5)--(345.3, 380.9)--(345.3, 380.7)--(345.5, 380.2)--(345.6, 380.1)--(345.8, 379.6)--(346.0, 379.2)--(346.2, 378.8)--(346.3, 378.6)--(346.4, 378.2)--(346.4, 377.9)--(346.5, 377.7)--(346.5, 377.5)--(346.7, 377.3)--(347.1, 376.8)--(347.4, 376.5)--(347.7, 376.2)--(347.8, 375.8)--(348.0, 375.6)--(348.1, 375.5)--(348.2, 375.4)--(348.3, 375.3)--(348.4, 375.3)--(348.5, 375.3)--(348.7, 375.3)--(349.0, 375.3)--(349.1, 375.2)--(349.2, 375.2)--(349.3, 375.1)--(349.4, 375.0)--(349.5, 374.8)--(350.0, 373.7)--(350.1, 373.4)--(350.2, 373.3)--(350.3, 373.2)--(350.5, 373.0)--(351.1, 372.6)--(351.7, 372.3)--(351.9, 372.1)--(352.0, 372.0)--(352.2, 371.7)--(352.6, 371.3)--(353.1, 370.8)--(353.2, 370.7)--(353.3, 370.6)--(353.5, 370.3)--(353.6, 370.2)--(353.7, 370.1)--(353.8, 370.0)--(354.2, 369.7)--(354.3, 369.6)--(354.3, 369.5)--(354.8, 368.8)--(354.8, 368.7)--(354.9, 368.6)--(354.9, 368.5)--(354.9, 368.4)--(354.9, 368.3)--(354.9, 368.2)--(354.9, 368.1)--(354.9, 368.0)--(354.9, 367.9)--(354.8, 367.9)--(354.8, 367.8)--(354.7, 367.6)--(354.6, 367.4)--(354.6, 367.3)--(354.5, 367.3)--(354.5, 367.2)--(354.5, 367.1)--(354.5, 367.0)--(354.5, 366.9)--(354.5, 366.8)--(354.5, 366.7)--(354.6, 366.7)--(354.6, 366.6)--(354.7, 366.6)--(354.8, 366.5)--(354.9, 366.5)--(354.9, 366.4)--(355.0, 366.4)--(355.4, 366.3)--(355.6, 366.3)--(355.9, 365.9)--(356.1, 365.7)--(356.3, 365.7)--(356.5, 365.7)--(356.6, 365.7)--(356.7, 365.7)--(357.0, 365.7)--(357.3, 365.6)--(357.7, 365.4)--(357.8, 365.4)--(358.0, 365.4)--(358.1, 365.4)--(358.3, 365.4)--(358.4, 365.5)--(358.9, 365.7)--(359.1, 365.8)--(359.2, 365.8)--(359.3, 365.8)--(359.4, 365.9)--(359.5, 365.9)--(359.7, 365.8)--(359.8, 365.7)--(360.0, 365.5)--(360.1, 365.4)--(360.2, 365.3)--(360.3, 365.2)--(360.4, 365.2)--(360.5, 365.2)--(360.7, 365.2)--(360.9, 365.2)--(361.0, 365.3)--(361.2, 365.3)--(361.5, 365.5)--(361.6, 365.6)--(361.7, 365.6)--(361.8, 365.6)--(361.9, 365.6)--(362.0, 365.6)--(362.1, 365.6)--(362.4, 365.6)--(362.9, 365.4)--(363.1, 365.3)--(363.3, 365.2)--(363.6, 365.0)--(363.9, 364.8)--(364.0, 364.7)--(364.1, 364.7)--(364.3, 364.7)--(364.6, 364.7)--(364.8, 364.8)--(365.0, 364.9)--(365.4, 365.1)--(365.7, 365.3)--(365.8, 365.3)--(365.8, 365.4)--(365.9, 365.4)--(366.0, 365.4)--(366.0, 365.5)--(366.1, 365.5)--(366.2, 365.5)--(366.3, 365.6)--(366.4, 365.6)--(366.5, 365.6)--(366.7, 365.6)--(366.8, 365.6)--(367.0, 365.6)--(367.1, 365.5)--(367.6, 365.3)--(367.9, 365.1)--(368.0, 365.0)--(368.3, 364.9)--(368.8, 364.5)--(369.0, 364.4)--(369.1, 364.3)--(369.2, 364.2)--(369.4, 364.1)--(369.7, 363.8)--(370.0, 363.5)--(370.1, 363.3)--(370.2, 363.1)--(370.3, 363.0)--(370.4, 362.9)--(370.5, 362.8)--(370.5, 362.7)--(370.6, 362.5)--(370.6, 362.3)--(370.8, 361.9)--(370.8, 361.5)--(370.8, 361.3)--(370.9, 361.3)--(371.1, 361.2)--(371.4, 361.2)--(371.5, 361.2)--(371.8, 360.9)--(372.3, 360.6)--(373.1, 360.1)--(374.2, 359.3)--(374.5, 359.1)--(374.7, 358.9)--(375.0, 357.7)--(375.1, 357.9)--(375.2, 357.6)--(375.3, 357.3)--(375.4, 357.0)--(375.6, 355.8)--(375.7, 355.1)--(375.7, 355.0)--(375.7, 354.9)--(375.1, 354.9)--(374.5, 354.9)--(374.5, 354.7)--(374.6, 354.6)--(374.6, 354.4)--(374.5, 354.3)--(374.5, 354.1)--(374.5, 353.9)--(374.5, 353.7)--(374.6, 353.5)--(374.7, 353.2)--(375.0, 352.5)--(375.1, 352.0)--(375.3, 351.4)--(375.5, 350.9)--(375.6, 350.6)--(375.6, 350.5)--(375.7, 350.0)--(375.8, 349.6)--(376.0, 349.0)--(376.0, 348.7)--(376.1, 348.2)--(376.1, 347.9)--(376.1, 347.7)--(376.2, 347.6)--(376.2, 347.5)--(376.2, 347.1)--(376.3, 346.9)--(376.4, 346.4)--(376.4, 346.1)--(376.4, 345.8)--(376.3, 345.9)--(376.2, 345.8)--(376.1, 345.9)--(375.8, 345.9)--(375.5, 345.9)--(375.0, 346.0)--(374.5, 346.0)--(374.1, 346.0)--(373.8, 346.0)--(373.0, 345.9)--(372.4, 345.9)--(372.4, 345.8)--(372.2, 345.8)--(372.1, 345.7)--(371.9, 345.7)--(371.3, 345.6)--(370.3, 345.5)--(370.2, 345.6)--(370.1, 345.7)--(370.1, 345.5)--(370.0, 345.4)--(369.5, 345.1)--(369.1, 344.9)--(368.9, 344.8)--(368.6, 344.7)--(368.2, 344.5)--(368.0, 344.5)--(367.8, 344.4)--(367.7, 344.7)--(367.5, 345.0)--(367.5, 345.2)--(367.3, 345.3)--(367.1, 345.4)--(366.7, 345.6)--(366.4, 345.7)--(366.3, 345.8)--(366.0, 346.0)--(365.7, 346.3)--(365.0, 346.9)--(364.9, 347.0)--(364.8, 347.0)--(364.7, 346.9)--(364.5, 346.7)--(364.4, 346.6)--(364.4, 346.4)--(364.2, 346.3)--(363.9, 345.8)--(363.5, 345.5)--(363.0, 345.1)--(362.4, 344.4)--(361.7, 343.7)--(362.2, 343.3)--(362.1, 343.2)--(362.0, 343.2)--(362.0, 343.1)--(361.9, 343.0)--(361.6, 342.5)--(361.2, 341.8)--(360.9, 341.3)--(360.2, 340.2)--(359.8, 339.5)--(359.5, 339.1)--(359.3, 338.8)--(358.7, 338.3)--(357.9, 337.7)--(357.9, 337.6)--(357.8, 337.7)--(357.7, 337.7)--(357.6, 337.6)--(357.4, 337.3)--(357.1, 337.0)--(357.0, 336.9)--(356.2, 336.6)--(355.6, 336.5)--(355.4, 336.5)--(355.3, 336.5)--(355.1, 336.5)--(354.9, 336.6)--(354.8, 336.6)--(354.7, 336.6)--(354.6, 336.7)--(354.5, 336.7)--(354.5, 336.8)--(354.4, 336.8)--(354.2, 337.0)--(353.9, 337.2)--(353.7, 337.4)--(353.5, 337.5)--(353.2, 337.5)--(353.1, 337.5)--(353.0, 337.5)--(352.9, 337.5)--(352.8, 337.5)--(352.4, 337.4)--(352.3, 337.4)--(352.2, 337.3)--(352.1, 337.2)--(351.9, 337.1)--(351.8, 336.9)--(351.6, 336.7)--(351.6, 336.4)--(351.5, 336.0)--(351.5, 335.8)--(351.4, 335.8)--(351.1, 335.9)--(350.8, 336.0)--(350.6, 336.0)--(350.4, 336.0)--(350.2, 336.0)--(350.1, 335.9)--(350.0, 335.9)--(349.8, 335.9)--(349.7, 335.8)--(349.5, 335.8)--(349.2, 335.6)--(348.7, 335.3)--(348.4, 335.1)--(348.2, 335.0)--(347.9, 334.9)--(347.4, 334.8)--(347.2, 334.7)--(347.0, 334.7)--(346.8, 334.8)--(346.6, 334.8)--(346.4, 334.9)--(346.3, 334.9)--(346.2, 335.0)--(346.1, 335.1)--(345.9, 335.3)--(345.6, 335.7)--(345.3, 336.0)--(345.1, 336.2)--(344.8, 336.2)--(344.7, 336.0)--(344.6, 335.8)--(344.5, 335.7)--(344.6, 335.3)--(344.6, 334.8)--(344.7, 334.7)--(344.8, 334.3)--(344.9, 334.1)--(345.0, 333.9)--(345.0, 333.7)--(344.7, 333.7)--(344.3, 333.5)--(343.1, 333.3)--(343.0, 333.6)--(342.9, 333.3)--(342.7, 332.8)--(342.5, 332.3)--(342.3, 331.7)--(342.3, 331.4)--(341.6, 331.3)--(341.1, 331.2)--(340.2, 331.1)--(339.8, 331.1)--(339.3, 331.1)--(338.9, 331.0)--(338.6, 331.0)--(338.4, 330.9)--(338.1, 330.9)--(337.8, 330.7)--(337.7, 330.6)--(337.5, 330.5)--(337.1, 330.3)--(336.9, 330.1)--(336.8, 330.1)--(336.7, 329.9)--(336.5, 329.7)--(336.3, 329.2)--(336.1, 328.5)--(336.0, 328.3)--(335.9, 328.1)--(335.8, 328.1)--(335.6, 327.8)--(335.4, 327.5)--(335.0, 327.1)--(334.8, 326.8)--(335.4, 326.7)--(335.4, 326.4)--(335.3, 326.0)--(335.2, 325.8)--(335.2, 325.7)--(335.2, 325.6)--(335.3, 325.4)--(335.3, 325.1)--(335.3, 324.7)--(335.4, 323.5)--(335.8, 323.6)--(335.9, 323.0)--(336.0, 322.7)--(336.0, 322.6)--(335.9, 322.3)--(335.9, 322.1)--(335.9, 322.0)--(335.8, 321.6)--(335.7, 321.4)--(335.7, 321.2)--(335.7, 321.1)--(335.5, 321.0)--(335.3, 321.2)--(335.0, 321.1)--(334.5, 320.9)--(334.3, 320.8)--(334.1, 320.4)--(333.2, 320.5)--(332.5, 320.8)--(332.3, 320.8)--(332.2, 320.2)--(332.0, 319.8)--(331.8, 319.5)--(331.7, 319.3)--(331.5, 319.0)--(331.2, 318.7)--(330.8, 318.4)--(329.5, 318.9)--(329.6, 319.3)--(329.7, 319.7)--(329.8, 320.4)--(329.9, 320.9)--(329.9, 321.4)--(330.0, 322.4)--(330.0, 322.6)--(330.0, 322.9)--(329.8, 322.9)--(329.4, 322.9)--(329.4, 323.5)--(329.2, 323.6)--(329.2, 323.7)--(329.3, 324.0)--(329.4, 324.4)--(329.6, 324.6)--(329.5, 325.1)--(329.4, 325.5)--(328.5, 325.1)--(328.4, 325.1)--(328.2, 325.0)--(328.0, 325.0)--(327.9, 325.1)--(327.8, 325.2)--(327.5, 325.3)--(327.5, 325.5)--(327.3, 326.1)--(327.1, 326.4)--(327.0, 326.6)--(326.9, 326.7)--(326.8, 326.8)--(326.6, 326.9)--(326.2, 327.1)--(322.1, 328.6)--(322.3, 329.1)--(322.3, 329.3)--(322.3, 329.4)--(322.3, 329.6)--(322.2, 330.0)--(322.0, 330.4)--(321.7, 331.0)--(321.5, 331.6)--(320.9, 332.8)--(321.2, 333.0)--(321.5, 333.1)--(321.6, 333.2)--(322.3, 333.6)--(322.6, 333.9)--(322.6, 333.8)--(323.0, 334.0)--(323.1, 334.1)--(323.5, 334.4)--(324.0, 334.8)--(324.3, 335.1)--(324.6, 335.3)--(324.8, 335.4)--(325.2, 335.7)--(325.5, 336.0)--(325.9, 336.3)--(326.3, 336.6)--(326.6, 336.9)--(326.9, 337.2)--(327.3, 337.5)--(327.5, 337.6)--(327.7, 337.7)--(327.9, 337.8)--(328.0, 337.7)--(328.0, 337.9)--(328.1, 338.1)--(328.1, 338.3)--(328.4, 339.4)--(328.6, 340.6)--(328.7, 341.2)--(329.2, 343.5)--(329.6, 345.2)--(329.8, 346.5)--(329.8, 346.7)--(329.7, 348.2)--(330.1, 348.2)--(332.5, 348.0)--(333.5, 347.9)--(334.5, 347.7)--(334.8, 348.0)--(334.8, 348.1)--(334.8, 348.4)--(334.8, 348.7)--(334.8, 349.1)--(334.6, 349.8)--(334.5, 350.0)--(334.4, 350.3)--(334.3, 350.6)--(333.8, 351.4)--(333.7, 351.7)--(333.1, 353.1)--(333.0, 353.2)--(333.0, 353.3)--(332.8, 353.5)--(332.6, 353.8)--(332.5, 353.9)--(332.5, 354.0)--(332.4, 354.1)--(332.2, 354.1)--(332.0, 354.2)--(331.4, 354.2)--(329.9, 354.4)--(329.6, 354.3)--(329.1, 354.3)--(329.0, 354.2)--(328.9, 354.2)--(328.6, 354.3)--(328.5, 354.3)--(328.4, 354.3)--(328.3, 354.3)--(328.2, 354.4)--(327.9, 354.6)--(327.8, 354.7)--(327.7, 354.7)--(327.6, 354.8)--(327.4, 354.8)--(327.3, 354.9)--(327.0, 355.0)--(326.9, 355.0)--(326.8, 355.1)--(326.8, 355.2)--(326.1, 356.3)--(325.8, 356.1)--(325.7, 356.0)--(325.5, 356.2)--(325.4, 356.3)--(325.3, 356.3)--(325.3, 356.4)--(325.1, 356.5)--(325.0, 356.5)--(324.9, 356.5)--(324.9, 356.8)--(324.9, 357.3)--(324.9, 357.8)--(324.9, 358.3)--(324.9, 359.3)--(324.8, 360.4)--(324.9, 360.5)--(324.8, 360.5)--(324.8, 360.7)--(324.7, 361.1)--(324.5, 361.6)--(324.4, 361.8)--(324.3, 362.1)--(324.1, 362.5)--(323.9, 362.9)--(323.9, 363.0)--(323.9, 363.1)--(323.8, 363.2)--(323.5, 363.2)--(323.4, 363.3)--(323.1, 363.4)--(322.9, 363.4)--(322.8, 363.4)--(322.7, 363.5)--(322.5, 363.5)--(322.4, 363.5)--(322.0, 363.8)--(321.8, 363.9)--(321.6, 364.0)--(321.0, 364.2)--(320.8, 364.2)--(319.0, 363.7)--(318.4, 364.5)--(317.4, 365.5)--(317.3, 365.5)--(317.3, 365.6)--(317.2, 365.6)--(317.1, 365.7)--(317.0, 365.8)--(316.9, 365.9)--(316.6, 366.2)--(316.5, 366.4)--(315.3, 367.5)--(314.8, 367.9)--(314.9, 368.0)--(314.9, 368.1)--(315.8, 368.9)--(316.9, 370.2)--(317.4, 370.7)--(318.5, 372.1)--(318.7, 372.4)--(320.3, 374.5)--(320.5, 374.7)--(320.4, 374.8)--(320.4, 374.9)--(320.3, 375.1)--(320.2, 375.2)--(320.1, 375.5)--(320.0, 375.6)--(320.0, 375.7)--(319.9, 376.0)--(319.9, 376.1)--(319.4, 377.3)--(319.2, 378.0)--cycle; +Hammersmith_and_Fulham = (219.8, 281.0)--(219.7, 281.1)--(219.6, 281.4)--(219.4, 281.4)--(219.2, 281.6)--(219.1, 281.8)--(219.2, 281.9)--(219.1, 282.3)--(219.0, 282.4)--(219.0, 282.6)--(219.0, 282.7)--(219.0, 283.3)--(219.1, 284.2)--(219.1, 284.3)--(219.1, 284.7)--(219.1, 285.5)--(219.1, 285.8)--(219.3, 287.2)--(219.3, 287.6)--(219.4, 288.0)--(219.4, 288.4)--(219.4, 288.6)--(219.5, 288.7)--(219.5, 288.9)--(219.6, 289.2)--(219.4, 289.2)--(219.4, 289.5)--(219.4, 289.7)--(219.5, 289.8)--(219.5, 290.0)--(219.4, 290.6)--(219.5, 290.9)--(219.6, 291.3)--(219.5, 291.3)--(219.2, 291.2)--(219.1, 291.5)--(219.0, 291.8)--(218.9, 291.7)--(218.5, 291.5)--(218.3, 291.3)--(217.9, 291.1)--(217.6, 291.0)--(217.5, 290.9)--(217.3, 290.7)--(217.2, 290.6)--(216.7, 290.3)--(216.5, 290.2)--(216.3, 290.1)--(216.2, 290.4)--(215.7, 291.2)--(215.2, 292.4)--(215.0, 292.7)--(214.9, 292.8)--(215.0, 292.9)--(214.8, 293.2)--(214.8, 293.3)--(214.7, 293.3)--(214.5, 293.7)--(214.4, 293.9)--(214.2, 294.2)--(214.0, 294.4)--(213.7, 294.7)--(213.7, 294.8)--(213.5, 295.0)--(213.5, 295.2)--(213.5, 295.3)--(213.5, 295.4)--(213.4, 295.8)--(213.4, 296.2)--(213.4, 296.3)--(213.3, 296.4)--(213.3, 296.6)--(213.2, 296.7)--(213.2, 296.8)--(213.1, 296.9)--(212.9, 297.1)--(212.7, 297.4)--(212.5, 297.6)--(212.5, 297.7)--(212.5, 297.8)--(212.4, 297.8)--(212.0, 298.2)--(212.1, 298.2)--(212.0, 298.2)--(212.2, 298.3)--(212.4, 298.1)--(212.4, 298.0)--(212.6, 298.1)--(213.0, 298.2)--(213.0, 298.3)--(212.9, 298.5)--(213.0, 298.5)--(213.6, 298.6)--(213.8, 298.7)--(213.8, 298.5)--(213.8, 298.4)--(214.7, 298.6)--(214.8, 298.6)--(215.0, 298.5)--(215.3, 298.4)--(215.5, 298.4)--(215.6, 298.4)--(215.6, 298.5)--(215.6, 298.6)--(215.7, 298.6)--(215.8, 298.6)--(215.9, 298.6)--(215.9, 298.7)--(216.0, 298.7)--(216.1, 298.4)--(216.2, 298.5)--(216.9, 298.6)--(216.9, 298.4)--(217.9, 298.5)--(218.1, 298.6)--(218.5, 298.7)--(218.3, 299.5)--(218.1, 300.0)--(218.1, 300.1)--(218.0, 300.3)--(218.0, 300.5)--(218.0, 300.6)--(217.9, 300.8)--(217.9, 301.0)--(218.9, 301.0)--(218.9, 301.1)--(218.7, 302.0)--(218.6, 302.2)--(218.4, 303.0)--(218.4, 303.1)--(218.4, 303.5)--(218.3, 303.8)--(218.2, 304.0)--(218.2, 304.6)--(218.0, 305.0)--(217.9, 305.5)--(217.9, 305.6)--(217.7, 306.0)--(217.3, 306.8)--(216.9, 307.4)--(216.4, 308.2)--(216.1, 308.7)--(216.0, 308.9)--(216.0, 309.0)--(216.0, 309.1)--(215.9, 309.1)--(215.9, 309.3)--(215.8, 309.9)--(215.7, 310.1)--(215.7, 310.4)--(215.4, 311.1)--(215.1, 312.8)--(214.8, 314.1)--(214.7, 314.7)--(214.7, 314.8)--(214.7, 314.9)--(214.7, 315.0)--(214.7, 315.1)--(214.7, 315.2)--(214.7, 315.3)--(214.7, 315.4)--(214.8, 315.5)--(214.8, 315.6)--(214.9, 315.9)--(214.7, 316.2)--(214.7, 316.4)--(213.8, 318.2)--(213.7, 318.8)--(213.6, 319.2)--(213.5, 319.7)--(213.3, 320.1)--(213.2, 320.5)--(213.2, 320.8)--(213.2, 321.0)--(213.3, 321.3)--(213.4, 321.5)--(213.4, 321.6)--(214.2, 323.0)--(214.5, 323.4)--(214.7, 323.6)--(214.8, 323.7)--(214.8, 323.8)--(214.8, 323.9)--(214.5, 324.4)--(214.6, 324.4)--(214.8, 324.6)--(214.9, 324.8)--(214.8, 324.9)--(214.6, 325.0)--(214.7, 325.1)--(215.2, 325.8)--(215.2, 325.9)--(215.3, 326.0)--(215.4, 326.0)--(215.4, 325.9)--(215.7, 326.3)--(215.9, 326.6)--(216.5, 327.2)--(217.0, 327.7)--(217.2, 327.8)--(217.4, 328.0)--(217.6, 328.0)--(217.6, 328.1)--(217.7, 328.1)--(217.9, 328.1)--(218.0, 328.1)--(218.1, 328.3)--(217.4, 328.6)--(216.9, 328.9)--(217.0, 328.9)--(217.2, 329.4)--(217.3, 329.6)--(217.3, 329.8)--(217.4, 330.0)--(217.4, 329.9)--(217.6, 329.8)--(218.5, 329.3)--(218.6, 329.2)--(219.0, 329.1)--(219.2, 329.0)--(219.3, 328.9)--(219.4, 328.9)--(219.5, 328.9)--(220.3, 329.0)--(221.0, 329.1)--(221.1, 329.1)--(221.4, 329.1)--(221.7, 329.2)--(222.1, 329.2)--(222.3, 329.2)--(222.8, 329.2)--(223.6, 329.3)--(224.0, 329.5)--(224.2, 329.5)--(224.9, 329.8)--(225.0, 329.8)--(225.1, 329.9)--(225.2, 329.9)--(225.5, 330.0)--(225.6, 330.0)--(225.8, 329.9)--(226.0, 329.7)--(226.4, 329.3)--(226.7, 329.1)--(227.1, 328.8)--(227.6, 328.5)--(227.8, 328.4)--(228.0, 328.2)--(228.2, 328.1)--(228.4, 328.1)--(228.7, 328.0)--(228.8, 327.9)--(229.1, 327.8)--(229.3, 327.7)--(229.6, 327.7)--(229.8, 327.6)--(229.6, 327.1)--(229.9, 326.3)--(230.0, 326.2)--(230.0, 326.1)--(230.2, 325.9)--(230.3, 325.7)--(230.4, 325.6)--(230.5, 325.4)--(230.5, 325.2)--(230.6, 325.0)--(230.6, 324.9)--(230.6, 324.7)--(230.7, 324.4)--(230.7, 324.2)--(230.7, 324.0)--(230.7, 323.1)--(230.6, 323.0)--(230.7, 322.7)--(230.7, 322.6)--(230.8, 322.0)--(230.8, 321.8)--(230.8, 321.4)--(231.2, 321.4)--(231.2, 318.7)--(231.2, 318.3)--(230.3, 317.6)--(230.3, 317.4)--(230.2, 317.3)--(231.3, 315.8)--(231.4, 315.7)--(231.4, 315.6)--(231.5, 315.6)--(231.5, 315.5)--(231.6, 315.6)--(231.9, 315.1)--(232.1, 314.9)--(232.3, 314.5)--(232.5, 314.2)--(232.6, 314.0)--(232.7, 313.9)--(232.9, 313.5)--(233.1, 313.4)--(233.0, 313.3)--(233.5, 312.5)--(234.0, 311.6)--(234.2, 311.4)--(234.3, 311.3)--(234.4, 310.9)--(234.5, 310.7)--(234.7, 310.3)--(234.9, 309.9)--(235.1, 309.4)--(235.1, 309.3)--(235.1, 309.1)--(235.2, 308.9)--(235.4, 308.5)--(235.8, 307.6)--(235.9, 307.4)--(235.9, 307.2)--(236.0, 307.1)--(236.0, 307.0)--(236.1, 306.8)--(236.2, 306.6)--(236.2, 306.4)--(236.3, 306.3)--(236.3, 306.2)--(236.4, 306.1)--(236.4, 306.0)--(236.5, 305.5)--(236.6, 305.3)--(236.7, 304.7)--(236.7, 304.6)--(237.0, 304.7)--(237.7, 304.9)--(237.6, 305.4)--(237.7, 305.4)--(237.8, 305.4)--(237.9, 305.4)--(237.9, 305.5)--(238.0, 305.5)--(238.1, 305.6)--(238.2, 305.6)--(238.3, 305.3)--(238.3, 305.2)--(238.3, 305.1)--(238.4, 304.9)--(238.4, 304.8)--(238.5, 304.7)--(238.4, 304.5)--(238.6, 304.4)--(238.7, 304.3)--(238.7, 304.2)--(239.0, 304.2)--(239.2, 304.2)--(239.3, 303.5)--(239.5, 303.0)--(239.6, 302.6)--(239.5, 302.5)--(239.2, 302.4)--(239.0, 302.4)--(239.0, 302.3)--(239.0, 302.1)--(238.9, 301.8)--(239.1, 301.4)--(239.1, 301.2)--(238.9, 301.1)--(238.9, 301.2)--(238.8, 301.2)--(238.3, 301.0)--(237.7, 300.8)--(237.9, 300.4)--(238.0, 300.1)--(238.0, 300.0)--(238.2, 299.7)--(238.4, 299.0)--(238.5, 299.0)--(238.8, 298.4)--(238.9, 298.2)--(238.9, 298.1)--(239.0, 298.2)--(239.2, 297.8)--(239.6, 297.1)--(239.9, 296.6)--(240.1, 295.9)--(240.4, 295.3)--(240.6, 294.9)--(241.3, 293.9)--(241.7, 293.6)--(242.1, 293.2)--(242.3, 293.0)--(242.4, 292.9)--(242.9, 292.5)--(243.3, 292.1)--(243.5, 291.8)--(243.6, 291.7)--(243.8, 291.3)--(244.2, 290.6)--(244.8, 289.9)--(244.9, 289.8)--(245.4, 289.9)--(245.4, 289.8)--(245.4, 289.7)--(245.5, 289.7)--(245.5, 289.6)--(245.6, 289.5)--(245.8, 289.3)--(245.8, 289.2)--(245.9, 289.2)--(246.1, 289.0)--(246.3, 288.9)--(246.6, 288.7)--(246.9, 288.5)--(247.2, 288.2)--(247.5, 287.9)--(247.7, 287.6)--(247.9, 287.4)--(248.0, 287.3)--(248.2, 287.2)--(248.4, 286.9)--(248.5, 286.8)--(248.4, 286.8)--(248.5, 286.7)--(248.6, 286.4)--(248.8, 286.1)--(248.9, 285.9)--(249.0, 285.7)--(249.3, 285.3)--(249.3, 285.2)--(249.0, 285.0)--(249.4, 284.4)--(250.0, 283.6)--(250.2, 283.3)--(250.4, 283.2)--(251.0, 282.4)--(251.1, 282.2)--(251.4, 281.9)--(251.4, 281.8)--(251.9, 281.1)--(252.1, 281.0)--(252.2, 280.8)--(252.3, 280.7)--(252.4, 280.6)--(252.7, 280.4)--(253.2, 280.1)--(253.5, 280.3)--(254.0, 279.9)--(254.1, 279.8)--(254.3, 279.7)--(254.4, 279.6)--(254.4, 279.5)--(254.6, 279.4)--(254.7, 279.2)--(254.8, 279.1)--(254.9, 279.0)--(255.2, 278.6)--(255.5, 278.3)--(255.5, 278.2)--(255.7, 278.0)--(255.8, 277.7)--(256.0, 277.6)--(256.4, 277.0)--(256.5, 276.7)--(256.6, 276.7)--(256.8, 276.4)--(256.9, 276.2)--(257.1, 276.0)--(257.2, 275.8)--(257.3, 275.6)--(257.5, 275.4)--(257.6, 275.2)--(257.7, 275.2)--(258.2, 274.6)--(258.5, 274.1)--(259.0, 273.4)--(259.0, 273.3)--(259.1, 273.3)--(259.5, 272.8)--(259.7, 272.6)--(259.8, 272.5)--(259.9, 272.3)--(260.0, 272.2)--(260.2, 272.0)--(260.3, 272.2)--(260.4, 272.0)--(260.5, 271.9)--(260.6, 271.8)--(260.6, 271.6)--(260.8, 271.1)--(260.9, 271.0)--(261.1, 270.8)--(261.6, 270.2)--(261.7, 270.1)--(261.9, 269.9)--(262.1, 269.6)--(262.2, 269.5)--(262.2, 269.4)--(262.1, 269.4)--(262.0, 269.4)--(262.1, 269.3)--(262.0, 269.3)--(262.0, 269.2)--(262.1, 269.2)--(262.6, 269.2)--(262.8, 269.2)--(262.9, 269.2)--(263.0, 269.1)--(263.1, 269.1)--(263.2, 269.1)--(263.4, 269.2)--(264.4, 269.8)--(264.5, 269.8)--(264.6, 269.8)--(264.7, 269.9)--(264.9, 269.9)--(265.0, 269.9)--(265.1, 269.8)--(265.3, 269.8)--(265.1, 268.0)--(265.0, 267.8)--(265.0, 267.6)--(264.9, 266.7)--(264.8, 266.6)--(264.7, 266.5)--(264.8, 266.5)--(264.8, 266.4)--(264.6, 265.4)--(264.5, 265.0)--(264.4, 264.6)--(264.2, 263.8)--(263.9, 262.9)--(263.5, 261.7)--(263.1, 260.8)--(262.8, 259.9)--(262.6, 259.6)--(262.6, 259.5)--(262.6, 259.4)--(262.5, 259.3)--(262.0, 258.6)--(261.8, 258.3)--(261.5, 257.9)--(261.1, 257.5)--(260.8, 257.2)--(260.3, 257.0)--(259.3, 256.5)--(259.1, 256.4)--(259.0, 256.3)--(258.8, 256.2)--(258.1, 255.9)--(258.0, 255.8)--(257.9, 255.8)--(257.8, 255.8)--(257.7, 255.8)--(255.9, 255.3)--(255.7, 255.3)--(255.6, 255.3)--(255.3, 255.3)--(254.5, 255.3)--(253.3, 255.4)--(253.2, 255.7)--(253.1, 255.7)--(253.1, 255.6)--(253.0, 255.5)--(252.7, 255.5)--(252.4, 255.4)--(252.3, 255.4)--(252.2, 255.4)--(252.1, 255.4)--(251.9, 255.4)--(251.7, 255.4)--(251.5, 255.4)--(251.4, 255.4)--(251.3, 255.4)--(251.2, 255.4)--(250.7, 255.5)--(250.3, 255.6)--(250.1, 255.6)--(250.0, 255.6)--(249.7, 255.7)--(248.9, 256.0)--(248.4, 256.1)--(248.0, 256.3)--(247.5, 256.5)--(247.3, 256.6)--(247.2, 256.7)--(247.1, 256.7)--(247.0, 256.8)--(246.0, 257.0)--(245.2, 257.3)--(245.2, 257.4)--(244.8, 257.5)--(244.7, 257.4)--(244.0, 257.6)--(243.8, 257.7)--(243.9, 257.8)--(243.9, 257.9)--(244.0, 258.2)--(243.9, 258.2)--(243.7, 258.0)--(243.8, 257.9)--(243.6, 257.8)--(242.4, 258.4)--(241.4, 259.0)--(240.5, 259.5)--(240.2, 259.7)--(239.9, 259.9)--(239.6, 260.2)--(239.5, 260.4)--(239.3, 260.5)--(238.8, 261.1)--(238.7, 261.2)--(238.0, 262.2)--(238.0, 262.3)--(237.9, 262.3)--(237.1, 263.3)--(235.9, 264.9)--(235.6, 265.3)--(235.6, 265.4)--(235.3, 266.0)--(235.2, 266.3)--(235.3, 266.4)--(235.2, 266.5)--(234.8, 267.8)--(234.8, 267.9)--(234.8, 268.1)--(234.7, 268.6)--(234.6, 269.0)--(234.5, 269.5)--(234.4, 271.6)--(234.2, 273.6)--(234.2, 273.7)--(234.3, 273.7)--(234.4, 273.8)--(234.4, 273.9)--(234.3, 274.0)--(234.1, 274.4)--(234.0, 275.0)--(233.9, 275.3)--(233.9, 275.5)--(233.4, 276.4)--(233.0, 277.3)--(232.2, 278.9)--(232.2, 279.0)--(231.9, 279.5)--(231.7, 279.7)--(231.6, 279.9)--(230.8, 280.9)--(230.8, 281.0)--(230.8, 281.1)--(230.8, 281.2)--(230.8, 281.4)--(230.7, 281.3)--(230.4, 281.5)--(230.3, 281.7)--(229.4, 282.0)--(229.3, 282.0)--(229.1, 282.1)--(227.6, 283.0)--(226.9, 283.1)--(226.8, 283.1)--(226.3, 283.0)--(226.3, 283.1)--(226.2, 283.1)--(226.1, 283.0)--(226.0, 283.0)--(226.0, 283.1)--(225.9, 283.1)--(225.7, 283.1)--(225.6, 283.0)--(225.5, 283.0)--(225.3, 283.1)--(224.6, 283.0)--(224.1, 282.9)--(223.5, 282.8)--(223.1, 282.7)--(223.0, 282.8)--(222.5, 282.6)--(222.3, 282.5)--(222.1, 282.5)--(221.8, 282.4)--(221.5, 282.2)--(220.8, 281.9)--(220.7, 281.7)--(220.6, 281.7)--(220.5, 281.6)--(220.4, 281.5)--(220.4, 281.4)--cycle; +Haringey = (319.3, 378.0)--(319.2, 378.0)--(319.4, 377.3)--(319.9, 376.1)--(319.9, 376.0)--(320.0, 375.7)--(320.0, 375.6)--(320.1, 375.5)--(320.2, 375.2)--(320.3, 375.1)--(320.4, 374.9)--(320.4, 374.8)--(320.5, 374.7)--(320.3, 374.5)--(318.7, 372.4)--(318.5, 372.1)--(317.4, 370.7)--(316.9, 370.2)--(315.8, 368.9)--(314.9, 368.1)--(314.9, 368.0)--(314.8, 368.1)--(314.7, 368.0)--(314.6, 367.9)--(313.6, 368.8)--(313.3, 369.2)--(313.1, 369.4)--(313.0, 369.5)--(312.8, 369.7)--(312.6, 369.9)--(312.5, 370.1)--(312.4, 370.2)--(312.1, 370.6)--(312.0, 370.7)--(311.9, 370.8)--(311.7, 371.0)--(311.6, 371.1)--(311.4, 371.3)--(310.7, 372.0)--(310.4, 372.2)--(309.7, 372.7)--(307.3, 374.3)--(307.3, 374.4)--(307.2, 374.4)--(307.2, 374.6)--(307.2, 375.1)--(307.2, 375.2)--(307.0, 375.4)--(306.9, 375.4)--(307.0, 375.7)--(306.8, 375.9)--(306.6, 376.0)--(306.5, 376.1)--(306.9, 376.5)--(306.7, 376.8)--(306.2, 377.4)--(305.8, 377.9)--(305.6, 378.1)--(305.5, 378.2)--(305.3, 378.2)--(305.0, 378.3)--(304.6, 378.4)--(304.4, 378.4)--(304.1, 378.4)--(304.1, 378.5)--(304.5, 378.8)--(304.4, 379.0)--(304.3, 379.1)--(304.2, 379.4)--(304.1, 379.7)--(304.0, 379.7)--(303.8, 379.6)--(302.5, 379.1)--(302.5, 379.2)--(302.5, 379.1)--(302.1, 379.0)--(301.5, 378.7)--(301.4, 378.7)--(301.3, 378.7)--(301.2, 378.7)--(301.1, 378.6)--(301.0, 378.5)--(300.9, 378.4)--(300.8, 378.3)--(300.7, 378.3)--(300.6, 378.3)--(300.6, 378.2)--(300.6, 378.1)--(300.5, 378.1)--(300.4, 378.0)--(300.3, 378.0)--(300.0, 378.0)--(299.5, 377.9)--(299.3, 378.0)--(299.2, 378.0)--(299.1, 378.0)--(298.9, 378.0)--(298.6, 378.0)--(297.7, 377.8)--(297.3, 377.7)--(297.0, 377.6)--(296.5, 377.5)--(296.2, 377.4)--(296.0, 377.4)--(295.9, 377.3)--(295.8, 377.2)--(295.7, 377.2)--(295.6, 377.1)--(295.3, 376.9)--(295.0, 376.8)--(294.8, 376.7)--(294.7, 376.7)--(294.5, 376.5)--(294.0, 376.0)--(293.3, 375.4)--(293.0, 375.2)--(292.8, 375.1)--(292.7, 375.0)--(292.4, 374.7)--(292.3, 374.7)--(292.2, 374.6)--(291.9, 374.4)--(291.7, 374.3)--(291.4, 374.0)--(291.2, 373.9)--(291.1, 373.9)--(291.1, 373.8)--(291.0, 373.7)--(290.8, 373.6)--(290.7, 373.4)--(290.4, 373.2)--(289.7, 372.6)--(289.6, 372.5)--(289.4, 372.5)--(289.0, 372.3)--(288.6, 372.2)--(288.4, 372.2)--(288.4, 372.4)--(287.4, 373.0)--(287.2, 373.1)--(287.0, 373.2)--(286.9, 373.3)--(286.5, 373.3)--(285.8, 373.6)--(285.1, 373.8)--(284.4, 373.9)--(283.7, 374.2)--(283.4, 374.4)--(283.2, 374.5)--(283.0, 374.7)--(282.8, 374.8)--(282.3, 374.8)--(281.2, 374.9)--(280.3, 375.0)--(279.3, 375.1)--(278.2, 375.2)--(276.9, 375.3)--(276.2, 375.7)--(276.0, 375.9)--(275.9, 375.9)--(275.8, 375.9)--(275.7, 375.9)--(275.6, 375.9)--(275.5, 375.9)--(275.4, 375.9)--(275.3, 375.9)--(275.2, 375.9)--(275.1, 375.8)--(275.0, 375.8)--(274.3, 375.7)--(273.8, 375.6)--(273.3, 375.6)--(273.2, 375.6)--(273.1, 375.6)--(273.0, 375.6)--(272.9, 375.6)--(272.8, 375.6)--(272.7, 375.6)--(272.5, 375.7)--(272.2, 375.8)--(272.1, 375.8)--(272.0, 375.8)--(271.8, 375.8)--(271.7, 375.9)--(271.5, 376.0)--(271.4, 376.0)--(271.2, 376.0)--(271.0, 376.0)--(270.3, 375.9)--(269.8, 375.9)--(269.2, 375.8)--(268.8, 375.6)--(268.5, 375.5)--(268.3, 375.4)--(268.3, 376.0)--(268.3, 376.7)--(268.5, 376.7)--(268.8, 376.8)--(269.2, 376.9)--(269.5, 377.1)--(269.5, 377.4)--(269.5, 377.7)--(269.5, 377.9)--(269.5, 378.1)--(269.4, 378.7)--(269.4, 379.1)--(269.4, 379.7)--(269.3, 379.8)--(269.3, 380.0)--(269.3, 380.2)--(269.4, 380.4)--(269.4, 380.6)--(269.5, 380.9)--(270.3, 380.5)--(270.4, 380.4)--(270.6, 380.9)--(270.7, 381.0)--(270.8, 381.4)--(270.9, 381.5)--(271.1, 381.9)--(271.0, 381.9)--(270.9, 381.9)--(271.0, 382.1)--(271.1, 382.2)--(271.2, 382.4)--(271.2, 382.5)--(271.1, 383.5)--(271.0, 383.5)--(271.0, 384.5)--(270.9, 384.5)--(270.9, 384.6)--(270.9, 385.1)--(270.9, 385.3)--(270.9, 385.6)--(271.0, 385.8)--(271.0, 385.9)--(271.0, 386.0)--(271.1, 386.0)--(271.5, 386.1)--(271.4, 386.3)--(271.6, 386.6)--(271.8, 387.1)--(271.8, 387.5)--(272.2, 387.7)--(272.4, 387.8)--(272.3, 387.9)--(272.4, 388.1)--(272.6, 388.3)--(272.7, 388.3)--(272.8, 388.3)--(273.2, 388.7)--(273.3, 388.7)--(273.3, 388.6)--(273.6, 388.8)--(273.9, 388.9)--(274.0, 388.9)--(274.2, 389.1)--(274.4, 389.2)--(274.4, 389.3)--(274.8, 389.8)--(274.9, 389.9)--(275.0, 389.9)--(275.2, 389.7)--(275.4, 389.5)--(275.6, 389.3)--(275.7, 389.2)--(275.9, 389.1)--(276.0, 388.9)--(276.1, 389.0)--(276.3, 389.3)--(277.2, 390.4)--(277.2, 390.9)--(277.6, 390.9)--(277.7, 390.9)--(277.6, 391.1)--(277.5, 391.3)--(277.3, 391.3)--(276.6, 391.4)--(276.6, 392.0)--(276.6, 392.2)--(276.6, 392.4)--(276.7, 392.4)--(276.6, 392.7)--(276.6, 392.9)--(275.5, 392.8)--(275.6, 393.0)--(275.6, 393.1)--(275.6, 393.4)--(275.6, 393.6)--(275.6, 394.0)--(275.6, 394.1)--(275.6, 394.2)--(275.5, 394.4)--(275.5, 394.7)--(275.8, 394.8)--(275.7, 395.2)--(275.7, 395.3)--(275.8, 395.5)--(275.8, 395.7)--(275.9, 396.2)--(275.9, 396.9)--(276.0, 396.9)--(276.2, 397.0)--(276.4, 397.2)--(276.5, 397.3)--(276.5, 397.4)--(276.4, 397.6)--(276.4, 397.8)--(275.8, 397.6)--(275.7, 397.9)--(275.7, 398.0)--(275.7, 398.2)--(275.4, 398.9)--(275.4, 399.1)--(275.4, 399.2)--(275.3, 399.4)--(275.2, 399.6)--(275.2, 399.8)--(275.0, 400.0)--(274.8, 400.4)--(274.8, 400.8)--(274.7, 401.0)--(274.4, 402.1)--(274.3, 402.7)--(274.2, 402.9)--(274.2, 403.0)--(274.2, 403.3)--(274.2, 403.4)--(274.2, 403.5)--(274.2, 403.6)--(275.4, 406.5)--(276.2, 408.5)--(276.2, 408.6)--(277.1, 410.9)--(277.4, 411.7)--(277.6, 412.0)--(277.7, 412.2)--(278.0, 412.3)--(277.9, 412.2)--(277.9, 412.1)--(277.9, 412.0)--(277.9, 411.9)--(278.4, 411.5)--(278.5, 411.4)--(278.6, 411.4)--(278.8, 411.0)--(278.9, 410.9)--(279.0, 410.8)--(279.1, 410.7)--(279.4, 410.1)--(279.7, 409.7)--(279.9, 409.3)--(279.9, 408.6)--(279.9, 408.2)--(279.9, 407.8)--(279.8, 407.4)--(279.9, 407.3)--(279.9, 407.1)--(280.1, 406.1)--(280.1, 405.8)--(280.2, 405.5)--(280.3, 405.3)--(280.9, 404.4)--(281.2, 403.8)--(281.3, 403.7)--(281.4, 403.5)--(283.2, 404.6)--(283.9, 405.0)--(284.1, 405.2)--(284.0, 405.3)--(284.4, 405.5)--(284.6, 405.7)--(285.7, 406.4)--(286.1, 406.6)--(286.1, 406.7)--(286.1, 406.9)--(286.1, 407.1)--(286.3, 407.7)--(286.4, 407.9)--(286.5, 408.0)--(286.5, 408.2)--(286.7, 408.5)--(286.8, 408.7)--(286.9, 409.0)--(286.9, 409.2)--(287.0, 409.3)--(287.4, 409.8)--(287.6, 410.1)--(287.6, 410.2)--(287.6, 410.4)--(287.7, 410.6)--(287.7, 410.7)--(287.7, 411.3)--(287.6, 411.6)--(287.5, 412.0)--(287.5, 412.1)--(287.4, 412.2)--(287.2, 412.6)--(287.1, 412.9)--(287.0, 413.0)--(287.0, 413.1)--(286.9, 413.6)--(286.8, 413.8)--(286.7, 414.7)--(286.6, 415.1)--(286.6, 415.2)--(286.6, 415.3)--(286.6, 415.7)--(286.6, 416.3)--(287.0, 416.6)--(288.1, 417.1)--(289.8, 417.9)--(290.5, 418.2)--(291.5, 418.6)--(291.7, 418.7)--(292.0, 419.0)--(292.2, 419.1)--(292.4, 419.1)--(293.5, 418.7)--(294.6, 418.3)--(294.9, 418.2)--(295.2, 418.1)--(296.0, 417.6)--(296.8, 417.4)--(297.3, 417.3)--(297.7, 417.2)--(298.3, 417.2)--(298.4, 417.2)--(298.4, 417.1)--(298.7, 417.5)--(298.6, 417.5)--(299.1, 418.0)--(299.4, 417.9)--(299.3, 417.8)--(299.9, 417.2)--(299.9, 417.1)--(300.0, 417.1)--(300.1, 417.0)--(300.0, 417.0)--(300.4, 416.5)--(300.5, 416.6)--(300.6, 416.6)--(301.1, 416.5)--(301.4, 416.5)--(301.6, 416.5)--(301.7, 416.5)--(301.9, 416.4)--(303.1, 416.3)--(303.4, 416.2)--(303.7, 416.6)--(304.1, 416.3)--(304.5, 416.1)--(304.7, 416.3)--(305.6, 416.2)--(305.6, 416.4)--(305.9, 416.4)--(306.0, 416.4)--(306.5, 416.4)--(306.6, 416.4)--(306.8, 416.3)--(306.9, 416.3)--(307.1, 416.3)--(307.1, 416.4)--(307.8, 416.4)--(308.7, 416.5)--(309.1, 416.5)--(309.2, 416.6)--(309.2, 416.7)--(309.3, 416.6)--(309.4, 416.6)--(309.4, 416.4)--(309.8, 416.3)--(310.2, 416.3)--(311.7, 416.3)--(311.8, 416.4)--(312.3, 416.5)--(312.6, 416.5)--(312.7, 416.5)--(313.0, 416.6)--(313.8, 416.6)--(313.9, 416.6)--(314.1, 416.6)--(314.2, 416.6)--(314.6, 416.4)--(314.9, 416.2)--(315.1, 416.6)--(315.3, 416.6)--(315.4, 416.7)--(315.6, 416.7)--(315.8, 416.7)--(315.9, 416.7)--(316.1, 416.7)--(316.4, 416.7)--(316.7, 416.7)--(317.3, 416.7)--(318.1, 416.8)--(318.4, 416.8)--(318.3, 417.2)--(318.4, 417.2)--(318.5, 417.2)--(318.7, 417.4)--(318.8, 417.5)--(319.1, 417.6)--(319.3, 417.6)--(319.5, 417.7)--(319.7, 417.7)--(319.9, 417.3)--(319.9, 417.2)--(320.0, 417.0)--(320.8, 417.2)--(321.1, 417.4)--(321.5, 417.3)--(321.8, 417.3)--(321.9, 417.2)--(321.9, 417.5)--(322.1, 417.5)--(322.1, 417.4)--(322.1, 417.3)--(322.3, 417.3)--(322.3, 417.2)--(322.3, 417.3)--(322.4, 417.3)--(322.6, 417.2)--(322.9, 417.2)--(323.0, 417.3)--(323.8, 417.3)--(324.5, 417.2)--(324.7, 417.2)--(325.0, 417.1)--(325.3, 417.1)--(325.6, 417.0)--(325.9, 417.0)--(326.5, 416.9)--(327.2, 417.0)--(327.4, 417.0)--(327.5, 417.0)--(328.2, 417.0)--(329.0, 417.0)--(329.1, 417.0)--(329.6, 417.0)--(330.0, 417.1)--(330.8, 417.1)--(331.0, 417.1)--(331.1, 417.1)--(331.2, 417.1)--(331.4, 417.1)--(331.8, 417.1)--(332.5, 417.1)--(332.8, 417.0)--(333.2, 417.1)--(333.5, 417.1)--(334.1, 417.2)--(334.8, 417.3)--(335.4, 417.4)--(335.9, 417.5)--(336.0, 417.5)--(336.0, 417.4)--(336.2, 417.5)--(336.4, 417.5)--(336.8, 417.6)--(336.8, 417.5)--(336.9, 417.5)--(337.2, 417.5)--(337.7, 417.5)--(337.9, 417.5)--(338.6, 417.5)--(338.8, 417.7)--(338.9, 417.5)--(339.0, 417.5)--(339.1, 417.5)--(339.2, 417.5)--(339.4, 417.8)--(340.0, 417.5)--(340.6, 417.5)--(340.7, 417.5)--(340.9, 417.5)--(342.3, 417.5)--(342.7, 417.5)--(343.5, 417.6)--(345.0, 417.7)--(345.7, 417.8)--(347.0, 417.9)--(347.4, 418.0)--(348.2, 418.0)--(348.5, 417.9)--(349.1, 417.7)--(349.2, 417.6)--(349.3, 417.6)--(349.2, 417.5)--(349.2, 417.4)--(349.3, 417.4)--(349.3, 417.1)--(349.5, 417.1)--(349.8, 417.0)--(350.1, 416.9)--(350.3, 416.8)--(350.2, 416.8)--(351.2, 416.4)--(351.4, 416.3)--(351.6, 416.2)--(351.8, 416.2)--(352.0, 416.1)--(352.1, 416.0)--(352.4, 415.9)--(352.7, 415.9)--(353.0, 415.8)--(354.6, 415.3)--(355.3, 415.1)--(356.3, 414.9)--(357.0, 414.7)--(357.3, 414.6)--(357.2, 414.5)--(357.2, 414.4)--(357.1, 414.3)--(357.1, 414.2)--(357.0, 414.1)--(357.0, 414.0)--(356.9, 413.9)--(356.9, 413.7)--(356.8, 413.6)--(356.8, 413.5)--(356.8, 413.4)--(356.5, 412.7)--(356.2, 412.0)--(355.9, 411.4)--(355.6, 410.8)--(355.1, 409.9)--(354.4, 408.8)--(354.2, 408.4)--(354.0, 408.0)--(353.7, 407.6)--(353.1, 406.8)--(352.4, 405.9)--(351.9, 405.3)--(351.8, 405.1)--(351.7, 405.0)--(351.6, 404.9)--(351.5, 404.8)--(351.4, 404.7)--(351.4, 404.6)--(351.3, 404.5)--(351.3, 404.4)--(351.2, 404.4)--(351.2, 404.3)--(351.2, 404.2)--(351.1, 404.1)--(351.1, 404.0)--(351.0, 403.9)--(351.0, 403.8)--(351.0, 403.7)--(350.9, 403.6)--(350.9, 403.4)--(350.9, 402.9)--(350.9, 402.8)--(350.9, 402.7)--(350.8, 402.7)--(350.8, 402.6)--(350.8, 402.5)--(350.7, 402.4)--(350.6, 402.3)--(350.5, 402.2)--(350.5, 402.1)--(350.4, 402.1)--(350.4, 402.0)--(350.3, 401.9)--(350.3, 401.8)--(350.2, 401.8)--(350.2, 401.7)--(350.2, 401.6)--(350.1, 401.5)--(350.1, 401.4)--(350.1, 401.3)--(350.1, 401.2)--(350.1, 400.6)--(350.0, 400.0)--(350.0, 399.7)--(349.7, 399.0)--(349.6, 398.5)--(349.5, 398.2)--(349.4, 397.9)--(349.4, 397.5)--(349.4, 397.2)--(349.4, 396.9)--(349.5, 396.6)--(349.6, 396.3)--(349.8, 395.9)--(350.0, 395.6)--(350.0, 395.5)--(350.1, 395.4)--(350.1, 395.2)--(350.2, 395.1)--(350.3, 394.9)--(350.3, 394.8)--(350.4, 394.8)--(350.4, 394.7)--(350.4, 394.6)--(350.4, 394.5)--(350.4, 394.4)--(350.4, 394.3)--(350.3, 394.3)--(350.3, 394.2)--(350.3, 394.1)--(350.2, 394.1)--(350.2, 394.0)--(350.1, 394.0)--(350.1, 393.9)--(350.0, 393.9)--(349.7, 393.6)--(349.6, 393.5)--(349.2, 393.1)--(348.9, 392.8)--(348.9, 392.7)--(348.8, 392.7)--(348.8, 392.6)--(348.7, 392.5)--(348.7, 392.4)--(348.6, 392.3)--(348.6, 392.2)--(348.5, 392.1)--(348.5, 392.0)--(348.4, 391.9)--(348.4, 391.8)--(348.4, 391.7)--(348.4, 391.6)--(348.3, 391.5)--(348.3, 391.4)--(348.3, 391.3)--(348.3, 391.2)--(348.3, 391.1)--(348.2, 391.1)--(348.2, 391.0)--(348.2, 390.9)--(348.1, 390.8)--(348.1, 390.7)--(348.0, 390.6)--(348.0, 390.5)--(347.9, 390.4)--(347.9, 390.3)--(347.8, 390.3)--(347.8, 390.2)--(347.7, 390.1)--(347.6, 390.0)--(347.5, 389.9)--(347.4, 389.8)--(347.3, 389.7)--(347.2, 389.6)--(347.1, 389.5)--(347.0, 389.5)--(347.0, 389.4)--(346.9, 389.3)--(346.9, 389.2)--(346.8, 389.2)--(346.8, 389.1)--(346.8, 389.0)--(346.7, 388.9)--(346.7, 388.7)--(346.6, 388.7)--(346.6, 388.6)--(346.6, 388.5)--(346.5, 388.5)--(346.5, 388.4)--(346.4, 388.4)--(346.3, 388.3)--(346.2, 388.2)--(345.6, 387.7)--(345.3, 387.4)--(345.1, 387.3)--(344.6, 386.8)--(344.5, 386.8)--(344.5, 386.7)--(344.4, 386.6)--(344.3, 386.5)--(344.2, 386.4)--(344.2, 386.3)--(344.1, 386.2)--(344.0, 386.1)--(344.0, 386.0)--(343.9, 385.9)--(343.9, 385.8)--(343.8, 385.7)--(343.8, 385.6)--(343.8, 385.5)--(343.7, 385.4)--(343.7, 385.3)--(343.7, 385.2)--(343.6, 385.2)--(343.6, 385.1)--(343.6, 385.0)--(343.6, 384.9)--(343.7, 384.9)--(343.7, 384.8)--(343.7, 384.7)--(343.7, 384.6)--(343.8, 384.5)--(343.8, 384.4)--(343.9, 384.4)--(343.9, 384.3)--(344.2, 383.8)--(344.4, 383.4)--(344.4, 383.3)--(343.4, 382.9)--(342.7, 382.6)--(342.3, 382.5)--(342.1, 382.4)--(341.5, 382.2)--(341.0, 382.0)--(340.6, 381.9)--(340.6, 381.6)--(340.4, 381.5)--(340.3, 381.5)--(340.2, 381.5)--(339.8, 381.4)--(339.5, 381.3)--(339.2, 381.3)--(338.8, 381.1)--(338.7, 381.1)--(338.6, 381.0)--(338.4, 381.0)--(338.1, 380.9)--(337.7, 380.8)--(337.1, 380.7)--(337.1, 380.8)--(336.8, 380.7)--(336.7, 380.7)--(336.5, 380.7)--(336.3, 380.7)--(335.9, 380.6)--(335.9, 380.7)--(335.8, 380.7)--(335.6, 380.7)--(335.5, 380.6)--(335.4, 380.6)--(335.4, 380.7)--(335.3, 380.7)--(335.3, 380.6)--(334.8, 380.5)--(334.8, 380.3)--(334.7, 380.3)--(334.6, 380.3)--(334.4, 380.4)--(334.4, 380.5)--(334.3, 380.3)--(334.3, 380.2)--(334.1, 379.8)--(333.5, 380.0)--(333.2, 380.1)--(332.8, 380.2)--(332.5, 380.3)--(332.2, 380.3)--(331.9, 380.4)--(331.6, 380.4)--(330.9, 380.5)--(330.2, 380.3)--(329.9, 380.2)--(329.5, 380.1)--(329.0, 380.0)--(328.8, 380.0)--(328.1, 379.9)--(327.5, 379.8)--(327.4, 379.6)--(327.3, 379.6)--(327.3, 379.5)--(326.9, 379.4)--(326.5, 379.4)--(326.5, 379.2)--(326.5, 379.1)--(326.6, 379.2)--(326.7, 379.1)--(326.7, 379.0)--(326.7, 378.9)--(326.4, 379.0)--(326.2, 379.0)--(326.1, 379.0)--(326.0, 379.0)--(325.9, 379.0)--(325.7, 378.9)--(325.5, 378.9)--(325.4, 378.9)--(325.3, 378.9)--(325.2, 379.0)--(325.1, 379.0)--(325.0, 379.0)--(324.9, 379.0)--(324.7, 379.0)--(324.6, 379.0)--(324.5, 379.0)--(324.5, 378.9)--(324.4, 378.9)--(324.3, 378.9)--(324.1, 378.8)--(323.9, 378.6)--(323.7, 378.5)--(323.6, 378.5)--(323.5, 378.5)--(323.4, 378.5)--(323.3, 378.5)--(323.2, 378.5)--(323.1, 378.5)--(323.0, 378.5)--(322.8, 378.5)--(322.7, 378.5)--(322.6, 378.5)--(322.5, 378.5)--(322.4, 378.5)--(322.3, 378.5)--(322.1, 378.5)--(322.1, 378.4)--(322.0, 378.4)--(321.9, 378.4)--(321.6, 378.2)--(321.4, 378.0)--(321.3, 378.0)--(321.2, 378.0)--(321.1, 378.0)--(321.0, 378.0)--(320.9, 378.0)--(320.8, 378.0)--(320.7, 378.0)--(320.3, 378.1)--(320.1, 378.2)--(320.0, 378.2)--(319.9, 378.2)--(319.8, 378.2)--(319.7, 378.2)--(319.6, 378.2)--(319.6, 378.1)--(319.5, 378.1)--(319.4, 378.0)--cycle; +Harrow = (106.0, 416.9)--(106.6, 416.9)--(106.9, 416.9)--(107.2, 416.9)--(107.7, 416.9)--(107.8, 416.9)--(107.9, 416.9)--(108.0, 417.0)--(108.3, 417.1)--(108.4, 417.1)--(108.5, 417.1)--(108.8, 417.1)--(109.0, 417.2)--(109.2, 417.2)--(109.9, 417.5)--(110.2, 417.8)--(110.7, 418.0)--(110.9, 418.1)--(111.2, 418.3)--(111.7, 418.5)--(112.4, 418.8)--(112.8, 418.9)--(113.4, 419.1)--(113.8, 419.3)--(114.6, 419.4)--(115.1, 419.5)--(115.8, 419.5)--(116.0, 419.6)--(116.2, 419.6)--(116.4, 419.7)--(116.8, 419.7)--(117.2, 419.7)--(117.5, 419.8)--(117.7, 419.8)--(117.8, 419.8)--(118.0, 419.9)--(118.2, 420.0)--(119.0, 420.3)--(119.4, 420.5)--(119.8, 420.7)--(120.1, 420.9)--(120.3, 421.1)--(121.1, 421.9)--(121.2, 422.0)--(121.3, 422.0)--(121.5, 422.0)--(121.9, 422.0)--(122.2, 422.0)--(122.6, 422.1)--(123.2, 422.3)--(123.6, 422.4)--(123.7, 422.4)--(124.2, 422.4)--(124.6, 422.6)--(124.9, 422.7)--(125.3, 422.9)--(125.4, 422.9)--(125.7, 423.0)--(126.1, 423.2)--(126.6, 423.6)--(127.2, 423.8)--(127.5, 423.9)--(127.7, 424.0)--(128.3, 424.3)--(128.5, 424.4)--(129.1, 424.8)--(129.5, 425.1)--(129.9, 425.4)--(130.2, 425.5)--(130.7, 425.7)--(131.3, 425.9)--(131.5, 425.9)--(131.6, 426.0)--(131.7, 426.0)--(132.0, 426.2)--(132.5, 426.5)--(132.7, 426.8)--(133.0, 427.1)--(133.9, 427.9)--(134.0, 428.0)--(134.4, 429.0)--(134.5, 429.1)--(134.6, 429.0)--(134.6, 429.1)--(135.0, 429.3)--(135.2, 429.6)--(135.3, 429.9)--(135.5, 430.1)--(135.6, 430.3)--(135.7, 430.4)--(136.1, 430.8)--(136.4, 431.0)--(136.6, 431.1)--(136.9, 431.1)--(137.2, 431.2)--(137.5, 431.2)--(137.7, 431.2)--(137.9, 431.3)--(138.3, 431.4)--(138.6, 431.5)--(138.8, 431.5)--(138.9, 431.6)--(139.1, 431.7)--(139.5, 431.9)--(139.6, 431.9)--(139.7, 431.9)--(139.8, 431.9)--(140.1, 431.9)--(140.2, 431.9)--(141.0, 432.2)--(141.3, 432.3)--(141.5, 432.3)--(142.0, 432.5)--(142.5, 432.6)--(142.8, 432.6)--(142.8, 432.7)--(143.2, 432.8)--(143.4, 432.9)--(143.6, 433.0)--(143.7, 433.1)--(143.8, 433.1)--(144.3, 433.5)--(144.8, 433.8)--(145.2, 434.0)--(145.5, 434.3)--(145.8, 434.4)--(146.1, 434.5)--(146.4, 434.6)--(146.7, 434.8)--(147.0, 435.0)--(147.1, 435.1)--(148.7, 436.4)--(149.6, 437.1)--(150.0, 437.4)--(150.0, 437.5)--(150.2, 437.2)--(150.4, 437.0)--(150.6, 436.7)--(150.7, 436.7)--(150.8, 436.7)--(150.7, 436.6)--(150.7, 436.5)--(150.9, 436.3)--(151.1, 436.0)--(151.7, 436.8)--(151.7, 436.9)--(151.8, 437.0)--(152.1, 437.4)--(152.4, 437.7)--(152.6, 438.0)--(152.9, 438.4)--(153.1, 438.6)--(153.4, 439.0)--(153.6, 439.2)--(153.6, 439.3)--(153.7, 439.4)--(153.9, 439.6)--(154.2, 440.0)--(154.3, 440.1)--(154.5, 440.3)--(154.6, 440.5)--(154.7, 440.5)--(154.6, 440.6)--(154.6, 440.7)--(154.6, 440.8)--(154.7, 441.0)--(154.8, 440.9)--(155.0, 440.9)--(155.0, 441.0)--(155.5, 441.5)--(156.1, 442.1)--(156.7, 442.7)--(157.2, 443.0)--(157.5, 443.2)--(157.8, 443.5)--(158.5, 443.9)--(158.7, 443.9)--(158.9, 444.1)--(159.1, 444.3)--(159.5, 444.5)--(159.7, 444.6)--(160.4, 444.8)--(161.4, 445.2)--(162.0, 445.4)--(162.5, 445.5)--(163.2, 445.8)--(163.7, 446.0)--(163.9, 446.2)--(164.2, 446.5)--(164.4, 446.9)--(164.5, 447.0)--(164.6, 447.3)--(164.6, 447.5)--(164.6, 447.6)--(164.7, 447.8)--(164.8, 448.0)--(165.0, 448.2)--(165.2, 448.3)--(165.5, 448.5)--(165.6, 448.6)--(165.8, 448.7)--(166.3, 448.3)--(167.0, 447.9)--(167.5, 447.7)--(167.7, 447.6)--(168.1, 447.4)--(168.2, 447.3)--(168.6, 447.1)--(168.8, 447.0)--(169.3, 446.8)--(169.6, 446.7)--(169.7, 446.7)--(169.9, 446.6)--(171.1, 446.1)--(171.2, 446.1)--(171.7, 445.9)--(172.1, 445.7)--(172.6, 445.5)--(173.0, 445.3)--(173.7, 445.0)--(173.9, 444.9)--(174.2, 444.6)--(174.3, 444.5)--(174.3, 444.4)--(174.3, 444.3)--(174.3, 444.2)--(174.4, 444.2)--(174.0, 443.1)--(173.6, 442.0)--(173.6, 441.9)--(173.6, 441.7)--(173.8, 440.9)--(174.1, 440.2)--(174.5, 439.4)--(174.8, 438.8)--(175.9, 437.7)--(176.1, 437.3)--(176.4, 437.0)--(176.8, 436.4)--(177.3, 435.7)--(177.6, 435.2)--(178.2, 434.4)--(178.6, 433.9)--(179.1, 433.4)--(179.3, 433.1)--(179.6, 432.7)--(180.6, 431.4)--(181.1, 430.7)--(181.3, 430.4)--(181.7, 429.9)--(182.0, 429.3)--(182.6, 428.7)--(182.9, 428.2)--(183.6, 427.2)--(185.1, 425.3)--(186.0, 424.1)--(186.2, 423.8)--(186.5, 423.3)--(186.6, 423.1)--(187.1, 422.5)--(187.7, 421.8)--(188.2, 421.2)--(188.8, 420.7)--(189.0, 420.6)--(189.1, 420.4)--(189.5, 420.0)--(190.4, 419.0)--(190.9, 418.3)--(191.8, 416.9)--(192.4, 416.1)--(192.8, 415.5)--(193.4, 414.7)--(193.9, 414.1)--(194.0, 413.9)--(194.3, 413.6)--(194.5, 413.5)--(194.8, 413.2)--(195.1, 412.7)--(195.3, 412.4)--(195.6, 412.0)--(196.4, 410.6)--(197.0, 409.9)--(198.9, 407.6)--(199.0, 407.4)--(200.0, 406.3)--(200.2, 406.0)--(200.8, 405.2)--(201.1, 404.8)--(200.6, 404.3)--(200.3, 404.1)--(200.1, 404.0)--(200.0, 403.9)--(199.8, 403.7)--(199.6, 403.5)--(199.3, 403.3)--(199.0, 403.0)--(198.8, 402.8)--(198.6, 402.6)--(198.5, 402.5)--(198.4, 402.5)--(198.4, 402.4)--(198.3, 402.4)--(198.2, 402.4)--(198.1, 402.4)--(198.1, 402.5)--(197.8, 402.5)--(197.7, 402.5)--(197.6, 402.5)--(197.5, 402.5)--(197.4, 402.5)--(197.3, 402.5)--(197.2, 402.5)--(197.1, 402.5)--(197.0, 402.4)--(196.9, 402.4)--(196.8, 402.3)--(196.7, 402.3)--(196.6, 402.2)--(196.5, 402.1)--(196.4, 402.1)--(196.4, 402.0)--(195.3, 401.4)--(194.4, 400.8)--(194.8, 400.4)--(192.8, 400.0)--(191.4, 399.7)--(191.1, 399.7)--(190.5, 399.5)--(190.0, 399.4)--(189.5, 399.2)--(189.2, 399.1)--(188.9, 399.0)--(188.6, 398.8)--(188.5, 398.8)--(188.4, 398.3)--(188.1, 398.4)--(188.0, 398.5)--(187.7, 398.2)--(187.5, 398.0)--(187.1, 397.7)--(186.9, 397.7)--(186.6, 397.6)--(186.1, 397.4)--(185.9, 397.3)--(185.7, 397.2)--(185.4, 397.1)--(185.2, 396.9)--(185.6, 396.5)--(185.7, 396.4)--(185.9, 396.3)--(186.0, 396.2)--(186.1, 396.2)--(186.4, 396.0)--(186.6, 396.0)--(187.2, 395.8)--(187.4, 395.7)--(187.6, 395.6)--(187.7, 395.6)--(187.9, 395.5)--(188.0, 395.4)--(188.2, 395.2)--(188.4, 395.1)--(188.5, 395.0)--(188.7, 394.8)--(188.8, 394.7)--(188.9, 394.6)--(189.1, 394.4)--(189.3, 394.1)--(189.3, 394.0)--(189.4, 393.8)--(189.5, 393.6)--(189.7, 393.3)--(189.8, 392.9)--(190.2, 391.8)--(190.3, 391.2)--(190.4, 390.8)--(190.4, 390.6)--(190.5, 390.2)--(190.6, 389.4)--(190.8, 388.9)--(190.9, 388.7)--(191.0, 388.6)--(191.0, 388.5)--(191.0, 388.4)--(191.0, 388.3)--(191.0, 387.7)--(191.0, 387.6)--(190.9, 387.6)--(190.9, 387.5)--(189.8, 387.8)--(186.4, 388.4)--(184.2, 388.8)--(182.7, 389.0)--(181.7, 389.1)--(180.2, 389.2)--(180.0, 389.2)--(179.9, 389.2)--(179.7, 389.2)--(179.2, 389.3)--(178.7, 389.5)--(178.6, 389.6)--(178.4, 389.6)--(178.2, 389.7)--(177.9, 389.7)--(177.3, 389.7)--(177.2, 389.7)--(176.9, 389.7)--(176.6, 389.6)--(176.2, 389.6)--(175.9, 389.5)--(175.4, 389.4)--(175.0, 389.3)--(174.7, 389.2)--(174.7, 389.1)--(174.5, 389.1)--(174.3, 388.9)--(174.0, 388.8)--(173.0, 388.2)--(171.4, 387.3)--(170.6, 386.8)--(169.7, 386.2)--(169.5, 386.0)--(169.4, 385.9)--(168.9, 385.5)--(167.0, 383.7)--(165.9, 382.5)--(165.5, 382.3)--(165.2, 382.0)--(164.9, 381.8)--(164.6, 381.6)--(164.3, 381.4)--(162.7, 380.6)--(161.4, 380.1)--(160.5, 379.8)--(160.4, 379.7)--(160.5, 379.5)--(160.5, 379.3)--(160.5, 379.2)--(160.6, 379.0)--(160.7, 378.6)--(161.0, 377.8)--(161.3, 377.1)--(161.4, 376.8)--(161.4, 376.6)--(161.7, 374.7)--(161.8, 374.4)--(161.8, 374.2)--(161.8, 374.0)--(161.9, 373.9)--(161.9, 373.6)--(162.4, 372.7)--(162.8, 371.8)--(163.3, 370.9)--(163.6, 370.3)--(163.8, 369.8)--(163.3, 369.4)--(162.7, 368.9)--(162.2, 368.6)--(161.8, 368.4)--(161.8, 368.2)--(161.3, 367.9)--(161.1, 367.8)--(160.7, 367.7)--(160.1, 367.6)--(160.2, 367.3)--(159.3, 367.0)--(158.9, 366.9)--(158.6, 366.8)--(158.7, 366.3)--(158.7, 365.8)--(159.1, 365.6)--(159.4, 365.5)--(159.5, 364.4)--(159.5, 363.5)--(159.7, 362.6)--(159.5, 362.3)--(159.7, 362.0)--(159.8, 361.7)--(159.4, 361.7)--(159.2, 361.7)--(159.1, 361.6)--(158.9, 361.6)--(158.8, 361.6)--(158.7, 361.5)--(158.6, 361.4)--(157.7, 360.6)--(157.6, 360.6)--(157.8, 360.5)--(157.2, 359.9)--(157.3, 359.8)--(156.8, 359.0)--(156.3, 358.2)--(155.9, 357.6)--(155.9, 357.0)--(155.9, 356.6)--(155.9, 356.3)--(155.3, 356.4)--(154.9, 355.1)--(154.8, 355.0)--(154.7, 355.0)--(154.6, 355.1)--(154.4, 355.1)--(154.3, 355.1)--(154.3, 355.2)--(153.7, 355.4)--(153.4, 355.5)--(153.3, 355.6)--(152.9, 355.7)--(152.8, 355.7)--(152.6, 355.8)--(152.4, 355.8)--(151.1, 355.8)--(150.8, 355.9)--(150.3, 355.9)--(150.0, 355.9)--(149.7, 356.0)--(149.0, 356.2)--(148.5, 356.5)--(148.0, 356.8)--(148.0, 356.9)--(147.9, 356.9)--(147.8, 357.0)--(147.6, 357.1)--(146.7, 357.9)--(146.7, 358.0)--(146.5, 358.3)--(145.6, 358.0)--(144.3, 357.6)--(143.7, 357.4)--(142.3, 357.1)--(141.1, 356.8)--(139.9, 356.4)--(138.4, 355.9)--(137.8, 355.8)--(137.7, 355.7)--(136.9, 355.5)--(136.3, 355.4)--(136.2, 355.4)--(136.3, 355.1)--(136.4, 354.8)--(136.4, 354.6)--(136.3, 354.5)--(136.2, 354.5)--(136.1, 354.5)--(136.2, 354.4)--(136.2, 354.3)--(136.1, 354.3)--(136.1, 354.2)--(136.2, 354.0)--(135.9, 353.9)--(135.8, 353.8)--(135.5, 353.7)--(135.2, 353.6)--(134.9, 353.5)--(134.8, 353.4)--(134.7, 353.4)--(134.5, 353.4)--(134.3, 353.4)--(134.3, 353.3)--(133.8, 353.2)--(133.7, 353.2)--(133.4, 353.1)--(133.5, 352.9)--(133.3, 352.8)--(133.1, 352.8)--(132.9, 352.7)--(132.7, 352.7)--(132.6, 352.7)--(132.5, 352.7)--(132.0, 352.5)--(131.8, 352.5)--(131.6, 352.4)--(131.4, 352.3)--(131.3, 352.3)--(131.1, 352.3)--(131.0, 352.2)--(130.8, 352.1)--(130.6, 352.0)--(130.5, 351.9)--(130.4, 351.8)--(130.1, 351.6)--(130.0, 351.6)--(129.9, 351.6)--(129.8, 351.5)--(129.7, 351.5)--(129.6, 351.5)--(129.3, 351.5)--(129.1, 351.2)--(128.7, 351.6)--(128.5, 351.3)--(128.3, 351.4)--(127.6, 351.0)--(127.0, 350.5)--(126.6, 351.0)--(126.5, 351.0)--(126.3, 351.3)--(126.0, 351.7)--(125.8, 352.1)--(125.7, 352.4)--(125.7, 352.6)--(125.6, 352.6)--(125.5, 352.8)--(125.5, 353.0)--(125.4, 353.1)--(125.3, 353.4)--(125.2, 353.6)--(125.0, 354.0)--(124.9, 354.2)--(124.8, 354.4)--(124.8, 354.7)--(124.7, 354.9)--(124.6, 355.1)--(124.5, 355.5)--(124.5, 355.7)--(124.3, 356.0)--(124.1, 356.4)--(123.9, 356.8)--(123.8, 357.1)--(123.9, 357.3)--(123.9, 357.4)--(124.0, 357.5)--(123.9, 357.6)--(123.9, 357.7)--(123.9, 357.8)--(123.7, 357.8)--(123.7, 358.2)--(123.8, 358.5)--(123.8, 358.8)--(123.7, 358.9)--(123.6, 359.2)--(123.5, 359.3)--(123.5, 359.5)--(123.3, 359.7)--(123.0, 360.2)--(122.7, 360.7)--(122.6, 360.8)--(122.5, 360.9)--(122.3, 361.3)--(122.4, 361.4)--(122.5, 361.5)--(122.5, 361.6)--(122.4, 361.7)--(122.2, 362.1)--(122.1, 362.2)--(122.1, 362.4)--(122.1, 362.5)--(122.2, 362.9)--(122.2, 363.0)--(122.2, 363.1)--(122.2, 363.3)--(122.1, 363.6)--(122.0, 364.2)--(121.8, 364.7)--(121.8, 365.0)--(121.4, 366.0)--(121.5, 366.1)--(120.8, 367.8)--(120.6, 368.1)--(120.6, 368.2)--(120.5, 368.4)--(120.3, 369.0)--(119.8, 369.9)--(119.6, 370.2)--(119.2, 370.9)--(119.5, 371.2)--(119.7, 371.5)--(119.5, 371.9)--(119.0, 372.8)--(118.9, 373.1)--(118.6, 373.6)--(118.5, 374.0)--(118.0, 374.6)--(118.4, 375.0)--(118.2, 375.3)--(118.2, 375.4)--(118.1, 375.4)--(117.8, 376.7)--(117.7, 377.2)--(117.6, 377.7)--(117.3, 378.1)--(116.8, 379.0)--(116.4, 379.8)--(116.0, 380.8)--(115.7, 381.4)--(115.8, 381.4)--(115.6, 381.6)--(115.6, 381.7)--(115.4, 381.9)--(115.5, 382.1)--(115.8, 382.1)--(115.8, 382.5)--(115.7, 383.4)--(115.7, 383.6)--(115.8, 383.6)--(115.7, 384.2)--(115.6, 384.8)--(115.5, 384.8)--(115.5, 385.2)--(115.4, 385.7)--(115.4, 386.2)--(115.3, 386.9)--(115.2, 387.9)--(115.2, 388.0)--(115.1, 388.2)--(115.1, 388.5)--(115.0, 388.8)--(114.9, 389.4)--(114.8, 389.6)--(114.6, 389.9)--(114.3, 390.0)--(114.2, 390.2)--(113.9, 390.6)--(113.8, 390.9)--(113.4, 391.5)--(113.2, 391.8)--(113.0, 391.7)--(113.1, 391.6)--(112.9, 391.5)--(112.7, 391.6)--(112.5, 391.5)--(112.5, 391.7)--(112.5, 391.8)--(112.4, 392.0)--(112.5, 392.2)--(112.5, 392.3)--(112.5, 392.7)--(112.5, 393.0)--(112.4, 393.4)--(112.4, 393.5)--(112.4, 393.6)--(112.3, 393.8)--(112.1, 394.3)--(112.0, 394.6)--(111.9, 394.9)--(111.9, 395.0)--(111.8, 395.2)--(111.8, 395.5)--(111.7, 395.8)--(111.7, 396.3)--(111.7, 396.5)--(111.7, 396.8)--(111.7, 396.9)--(111.6, 397.0)--(111.5, 397.2)--(111.4, 397.5)--(111.3, 397.7)--(111.2, 397.7)--(111.1, 398.0)--(110.9, 398.2)--(110.8, 398.5)--(110.8, 398.6)--(110.8, 398.7)--(110.8, 398.8)--(110.8, 398.9)--(110.8, 399.0)--(110.5, 399.1)--(110.0, 399.2)--(109.3, 399.3)--(109.0, 399.4)--(109.1, 400.0)--(109.1, 400.1)--(109.2, 400.3)--(109.0, 400.7)--(108.8, 401.0)--(108.5, 401.5)--(108.3, 401.8)--(108.3, 401.9)--(108.2, 402.2)--(108.0, 402.6)--(108.0, 403.0)--(107.9, 403.3)--(107.7, 403.7)--(107.6, 404.0)--(107.5, 404.4)--(107.3, 405.5)--(107.3, 405.9)--(107.4, 406.4)--(107.3, 406.6)--(107.3, 407.1)--(107.3, 407.6)--(107.3, 408.0)--(107.3, 408.4)--(107.3, 409.1)--(107.5, 409.2)--(107.6, 409.5)--(107.7, 409.6)--(107.6, 409.7)--(107.5, 409.9)--(107.3, 410.5)--(107.2, 411.0)--(107.2, 411.4)--(107.1, 411.6)--(107.0, 411.8)--(106.9, 412.0)--(106.8, 412.2)--(106.7, 412.4)--(106.6, 412.7)--(106.5, 413.1)--(106.5, 413.6)--(106.5, 414.1)--(106.4, 414.9)--(106.3, 415.4)--(106.2, 415.9)--cycle; +Havering = (498.9, 314.6)--(498.9, 314.7)--(499.0, 315.0)--(499.0, 315.3)--(499.1, 315.6)--(499.1, 315.7)--(499.1, 315.9)--(499.1, 316.0)--(499.0, 316.1)--(499.1, 316.2)--(499.1, 316.4)--(499.1, 316.7)--(499.2, 317.0)--(499.2, 317.1)--(499.2, 317.2)--(499.4, 318.3)--(499.5, 318.8)--(499.5, 318.9)--(499.6, 319.0)--(499.6, 319.1)--(499.6, 319.4)--(499.8, 319.9)--(499.8, 320.0)--(499.9, 320.1)--(499.9, 320.3)--(500.0, 320.5)--(500.0, 320.7)--(500.1, 321.0)--(500.2, 321.3)--(500.3, 321.6)--(500.3, 321.7)--(500.3, 321.8)--(500.4, 321.9)--(500.4, 322.1)--(500.5, 322.5)--(500.5, 322.6)--(500.6, 323.0)--(500.8, 323.5)--(500.8, 323.8)--(501.1, 324.8)--(501.1, 325.0)--(501.2, 325.2)--(501.2, 325.3)--(501.1, 325.4)--(501.1, 325.7)--(501.1, 325.8)--(501.1, 325.9)--(501.1, 326.0)--(501.3, 326.6)--(501.3, 326.8)--(501.4, 326.8)--(501.4, 327.3)--(501.5, 327.7)--(501.6, 328.0)--(501.8, 328.4)--(501.9, 328.7)--(502.0, 329.0)--(502.0, 329.1)--(502.0, 329.2)--(502.0, 329.3)--(502.0, 329.4)--(502.1, 329.4)--(502.2, 329.5)--(502.2, 329.6)--(502.2, 329.7)--(502.2, 329.8)--(502.2, 329.9)--(502.2, 330.0)--(502.2, 330.1)--(502.2, 330.2)--(502.2, 330.3)--(502.4, 330.7)--(502.5, 330.9)--(502.4, 331.1)--(502.4, 331.2)--(502.4, 331.3)--(502.6, 331.5)--(502.7, 331.7)--(502.8, 331.8)--(502.8, 331.9)--(502.9, 332.0)--(502.9, 332.2)--(503.0, 332.4)--(503.1, 332.6)--(503.1, 332.8)--(503.1, 332.9)--(503.2, 333.0)--(503.3, 333.0)--(503.4, 333.0)--(503.5, 333.2)--(503.5, 333.3)--(503.6, 333.3)--(503.6, 333.4)--(503.7, 333.5)--(503.8, 333.5)--(503.8, 333.8)--(503.8, 333.9)--(503.9, 334.0)--(503.9, 334.1)--(504.0, 334.2)--(504.1, 334.3)--(504.2, 334.4)--(504.3, 334.4)--(504.3, 334.5)--(504.4, 334.5)--(504.4, 334.6)--(504.5, 334.6)--(504.5, 334.7)--(504.5, 334.8)--(504.6, 334.9)--(504.7, 335.0)--(504.7, 335.2)--(504.7, 335.3)--(504.8, 335.4)--(504.8, 335.5)--(504.9, 335.5)--(505.0, 335.6)--(505.1, 335.7)--(505.2, 335.8)--(505.2, 335.9)--(505.3, 336.0)--(505.4, 336.1)--(505.6, 336.3)--(505.7, 336.5)--(505.8, 336.6)--(505.9, 336.7)--(506.0, 336.8)--(506.2, 337.1)--(506.3, 337.2)--(506.4, 337.3)--(506.4, 337.4)--(506.5, 337.4)--(506.5, 337.5)--(506.6, 337.6)--(506.7, 337.6)--(506.8, 338.0)--(506.9, 338.1)--(506.9, 338.3)--(507.0, 338.4)--(507.0, 338.6)--(507.1, 338.7)--(507.1, 338.8)--(507.2, 338.9)--(507.3, 339.0)--(507.3, 339.2)--(507.3, 339.3)--(507.4, 339.4)--(507.4, 339.5)--(507.4, 339.6)--(507.3, 339.7)--(507.3, 339.9)--(507.4, 340.0)--(507.4, 340.1)--(507.4, 340.3)--(507.5, 340.3)--(507.5, 340.4)--(507.6, 340.6)--(507.6, 340.7)--(507.6, 340.8)--(507.6, 340.9)--(507.7, 341.0)--(507.8, 341.3)--(508.1, 341.9)--(508.3, 342.3)--(508.6, 343.0)--(508.7, 343.2)--(509.1, 343.8)--(509.2, 343.9)--(509.3, 344.1)--(509.4, 344.2)--(509.7, 344.4)--(509.9, 344.6)--(510.2, 344.8)--(510.3, 344.9)--(510.4, 345.0)--(510.6, 345.1)--(510.7, 345.2)--(510.9, 345.3)--(511.0, 345.4)--(511.2, 345.4)--(511.3, 345.4)--(511.4, 345.4)--(511.5, 345.4)--(511.6, 345.5)--(511.7, 345.7)--(511.9, 345.9)--(511.9, 346.0)--(512.0, 346.0)--(512.0, 346.1)--(512.1, 346.2)--(512.2, 346.3)--(512.3, 346.3)--(512.3, 346.4)--(512.3, 346.5)--(512.3, 346.6)--(512.4, 346.7)--(512.4, 346.9)--(512.4, 347.2)--(512.5, 347.3)--(512.5, 347.5)--(512.5, 347.6)--(512.5, 347.7)--(512.5, 347.8)--(512.5, 347.9)--(512.6, 348.0)--(512.5, 348.0)--(512.5, 348.1)--(512.5, 348.2)--(512.4, 348.6)--(512.5, 348.8)--(512.5, 348.9)--(512.5, 349.1)--(512.6, 349.2)--(512.6, 349.3)--(512.6, 349.5)--(512.7, 349.7)--(512.7, 349.8)--(512.7, 350.1)--(512.8, 350.3)--(512.9, 350.5)--(513.0, 350.7)--(513.2, 350.9)--(513.4, 351.1)--(513.5, 351.2)--(513.6, 351.4)--(513.7, 351.6)--(513.8, 351.8)--(513.9, 351.9)--(514.3, 352.4)--(514.4, 352.5)--(514.5, 352.6)--(514.6, 352.7)--(514.8, 352.9)--(515.1, 353.1)--(515.3, 353.2)--(515.5, 353.4)--(515.7, 353.6)--(516.0, 353.8)--(516.1, 353.9)--(516.3, 354.0)--(516.4, 354.0)--(516.6, 354.1)--(516.6, 354.2)--(516.7, 354.5)--(516.8, 354.7)--(516.9, 354.9)--(517.0, 355.2)--(517.1, 355.4)--(517.1, 355.7)--(517.2, 355.8)--(517.2, 355.9)--(517.2, 356.0)--(517.3, 356.3)--(517.4, 356.5)--(517.5, 356.6)--(517.5, 356.7)--(517.5, 356.8)--(517.5, 356.9)--(517.6, 357.2)--(517.6, 357.3)--(517.6, 357.5)--(518.1, 358.0)--(518.3, 358.2)--(518.6, 358.5)--(518.8, 358.7)--(518.8, 358.8)--(518.9, 358.9)--(519.1, 358.9)--(519.1, 359.0)--(519.1, 359.1)--(519.1, 359.3)--(519.1, 359.5)--(519.2, 359.6)--(519.3, 359.8)--(519.3, 360.0)--(519.3, 360.1)--(519.4, 360.2)--(519.4, 360.3)--(519.4, 360.4)--(519.3, 360.4)--(519.1, 360.8)--(519.0, 360.8)--(518.8, 360.9)--(518.7, 361.0)--(518.6, 361.1)--(518.5, 361.1)--(518.4, 361.2)--(518.3, 361.3)--(518.3, 361.4)--(518.2, 361.5)--(518.0, 361.7)--(517.9, 361.7)--(517.8, 361.8)--(517.8, 361.9)--(517.9, 362.0)--(517.9, 362.1)--(517.8, 362.2)--(517.7, 362.2)--(517.6, 362.2)--(517.5, 362.3)--(517.5, 362.4)--(517.6, 362.5)--(517.6, 362.6)--(517.7, 362.6)--(517.6, 362.8)--(517.6, 362.9)--(517.5, 363.0)--(517.4, 363.2)--(517.3, 363.3)--(517.1, 363.4)--(517.1, 363.5)--(517.1, 363.6)--(517.1, 363.7)--(517.0, 363.7)--(516.7, 363.8)--(516.6, 364.0)--(516.4, 364.2)--(516.3, 364.3)--(516.2, 364.3)--(516.0, 364.3)--(515.7, 364.3)--(515.5, 364.3)--(515.5, 364.4)--(515.5, 364.6)--(515.6, 364.6)--(515.7, 364.6)--(515.7, 364.7)--(515.7, 364.8)--(515.6, 364.8)--(515.5, 365.0)--(515.5, 365.1)--(515.6, 365.2)--(515.5, 365.3)--(515.5, 365.4)--(515.3, 365.5)--(515.3, 365.7)--(515.3, 365.8)--(515.3, 365.9)--(515.3, 366.0)--(515.3, 366.2)--(515.3, 366.3)--(515.4, 366.4)--(515.3, 366.5)--(515.4, 366.6)--(515.4, 366.7)--(515.3, 366.8)--(515.4, 366.9)--(515.5, 367.0)--(515.5, 367.2)--(515.5, 367.4)--(515.5, 367.5)--(515.4, 367.6)--(515.4, 367.7)--(515.4, 367.8)--(515.4, 367.9)--(515.4, 368.1)--(515.4, 368.3)--(515.3, 368.4)--(515.3, 368.5)--(515.0, 368.7)--(514.6, 368.9)--(514.5, 369.1)--(514.3, 369.3)--(514.1, 369.4)--(514.1, 369.5)--(514.0, 369.6)--(514.0, 369.7)--(513.9, 369.7)--(514.0, 370.1)--(514.3, 370.8)--(514.4, 371.2)--(514.4, 371.3)--(514.5, 371.4)--(514.5, 371.5)--(514.6, 371.7)--(514.7, 372.2)--(514.7, 372.5)--(514.8, 372.6)--(514.8, 372.7)--(514.9, 372.8)--(515.1, 373.0)--(515.2, 373.1)--(515.3, 373.3)--(515.5, 373.5)--(515.5, 373.6)--(515.5, 373.7)--(515.5, 373.9)--(515.5, 374.1)--(515.5, 374.4)--(514.9, 373.9)--(514.6, 373.6)--(514.5, 373.5)--(514.4, 373.4)--(514.3, 373.4)--(514.1, 373.5)--(513.8, 373.5)--(513.3, 373.5)--(513.2, 373.6)--(513.1, 373.6)--(513.0, 373.5)--(512.9, 373.5)--(512.7, 373.5)--(512.3, 373.6)--(512.2, 373.6)--(512.1, 373.6)--(511.6, 373.6)--(511.3, 373.7)--(511.0, 373.7)--(510.6, 373.6)--(510.2, 373.5)--(509.9, 373.5)--(509.7, 373.5)--(509.4, 373.4)--(509.2, 373.4)--(508.9, 373.4)--(508.8, 373.5)--(508.2, 373.6)--(507.7, 373.7)--(507.4, 373.7)--(507.2, 373.6)--(507.0, 373.5)--(506.8, 373.4)--(506.5, 373.4)--(506.3, 373.3)--(506.0, 373.2)--(505.9, 373.2)--(505.5, 372.9)--(505.3, 372.8)--(505.2, 372.7)--(505.1, 372.6)--(505.0, 372.6)--(504.8, 372.5)--(504.7, 372.4)--(504.5, 372.3)--(504.4, 372.1)--(504.3, 372.1)--(504.1, 372.0)--(504.0, 371.9)--(503.9, 371.9)--(503.7, 371.8)--(503.6, 371.8)--(503.4, 371.8)--(503.1, 371.8)--(502.8, 371.7)--(502.7, 371.6)--(502.6, 371.6)--(502.4, 371.4)--(502.2, 371.3)--(502.1, 371.2)--(501.9, 371.1)--(501.6, 370.9)--(501.2, 370.7)--(500.8, 370.4)--(500.6, 370.2)--(500.3, 370.1)--(500.1, 370.0)--(499.9, 369.8)--(499.5, 369.6)--(499.5, 369.7)--(499.4, 369.8)--(499.2, 369.9)--(499.1, 369.9)--(498.9, 370.0)--(498.7, 370.1)--(498.6, 370.3)--(498.6, 370.4)--(498.5, 370.5)--(498.3, 370.6)--(498.2, 370.7)--(498.2, 370.9)--(498.1, 371.0)--(498.0, 371.2)--(497.8, 371.4)--(497.6, 371.6)--(497.4, 371.7)--(497.3, 371.8)--(497.1, 372.0)--(496.9, 372.2)--(496.7, 372.2)--(496.5, 372.4)--(496.1, 372.8)--(495.8, 373.0)--(495.7, 373.2)--(495.6, 373.3)--(495.5, 373.4)--(495.3, 373.5)--(495.2, 373.5)--(495.0, 373.6)--(494.8, 373.8)--(494.7, 373.9)--(494.5, 374.1)--(494.4, 374.2)--(494.1, 374.5)--(493.9, 374.7)--(493.6, 375.0)--(493.3, 375.4)--(492.9, 375.8)--(492.7, 376.0)--(492.3, 376.3)--(492.1, 376.5)--(492.0, 376.6)--(491.9, 376.5)--(491.8, 376.4)--(491.7, 376.3)--(491.6, 376.3)--(491.5, 376.3)--(491.3, 376.3)--(491.1, 376.4)--(491.1, 376.5)--(491.0, 376.5)--(491.0, 376.6)--(490.9, 376.6)--(490.8, 376.6)--(490.7, 376.6)--(490.6, 376.6)--(490.3, 377.0)--(490.2, 377.2)--(489.2, 377.1)--(489.2, 377.2)--(489.1, 377.2)--(489.1, 377.3)--(488.9, 377.3)--(488.8, 377.3)--(488.8, 377.4)--(488.8, 377.5)--(488.8, 377.8)--(488.7, 378.0)--(489.8, 378.2)--(490.9, 378.4)--(490.8, 378.6)--(490.8, 378.8)--(490.6, 378.9)--(490.5, 379.0)--(490.5, 379.2)--(490.4, 380.3)--(490.5, 380.3)--(490.4, 380.9)--(490.2, 382.3)--(489.5, 382.1)--(489.5, 382.2)--(489.5, 382.4)--(489.5, 382.8)--(489.5, 383.0)--(489.5, 383.5)--(489.5, 383.8)--(489.6, 384.3)--(489.6, 384.5)--(489.6, 384.6)--(489.5, 384.9)--(489.5, 385.1)--(488.7, 385.0)--(488.8, 385.7)--(488.8, 386.7)--(488.8, 387.0)--(488.8, 387.5)--(488.8, 389.2)--(488.9, 390.7)--(489.5, 390.6)--(489.6, 390.9)--(490.0, 391.6)--(490.4, 392.4)--(490.6, 392.7)--(490.9, 393.0)--(491.0, 393.2)--(491.1, 393.7)--(491.2, 394.0)--(491.2, 394.5)--(491.2, 395.7)--(491.2, 397.2)--(491.2, 398.4)--(491.2, 399.2)--(491.2, 400.0)--(491.2, 400.4)--(491.2, 400.7)--(491.1, 401.1)--(491.0, 401.5)--(490.8, 402.1)--(490.7, 402.4)--(490.8, 403.8)--(490.8, 404.7)--(490.8, 405.2)--(490.9, 406.2)--(491.0, 407.0)--(490.8, 407.4)--(490.5, 407.9)--(490.2, 408.4)--(490.0, 408.8)--(488.8, 408.0)--(488.9, 408.4)--(488.8, 408.4)--(488.5, 408.5)--(488.6, 408.9)--(488.7, 409.5)--(488.7, 410.0)--(488.8, 410.5)--(488.8, 410.6)--(488.8, 410.9)--(488.8, 411.0)--(488.9, 411.3)--(489.1, 412.3)--(489.2, 413.2)--(489.2, 413.3)--(489.3, 413.5)--(489.3, 413.7)--(489.3, 413.8)--(489.3, 413.9)--(489.3, 414.0)--(489.6, 415.3)--(489.4, 415.6)--(489.3, 415.8)--(489.2, 415.9)--(489.2, 416.0)--(489.1, 416.2)--(489.1, 416.3)--(489.0, 416.7)--(488.9, 416.9)--(488.9, 417.0)--(488.9, 417.4)--(488.9, 417.5)--(488.9, 417.8)--(488.8, 418.4)--(488.8, 418.6)--(488.8, 418.8)--(488.8, 419.0)--(488.9, 419.3)--(488.9, 419.9)--(488.9, 420.2)--(488.9, 420.4)--(488.9, 420.7)--(488.9, 420.9)--(488.9, 421.2)--(488.9, 421.3)--(488.9, 421.5)--(488.8, 421.7)--(488.8, 422.0)--(488.7, 422.2)--(488.6, 422.5)--(488.5, 422.9)--(488.4, 423.0)--(488.3, 423.3)--(488.3, 423.6)--(488.2, 423.9)--(488.1, 424.4)--(488.1, 424.8)--(488.0, 425.5)--(487.9, 425.8)--(487.8, 425.9)--(487.8, 426.0)--(487.7, 426.2)--(487.5, 426.6)--(487.4, 426.8)--(487.2, 427.1)--(487.1, 427.3)--(486.9, 427.6)--(486.8, 427.7)--(486.7, 427.8)--(486.6, 427.9)--(486.5, 428.1)--(486.4, 428.2)--(486.4, 428.3)--(486.3, 428.4)--(486.3, 428.6)--(486.3, 428.7)--(486.3, 428.8)--(486.2, 428.8)--(486.2, 428.9)--(486.1, 429.0)--(486.1, 429.3)--(486.0, 429.6)--(485.9, 429.8)--(485.8, 430.0)--(485.7, 430.5)--(485.6, 430.7)--(485.5, 430.8)--(485.3, 430.9)--(485.3, 431.1)--(485.1, 431.3)--(484.9, 431.8)--(484.8, 432.0)--(484.7, 432.1)--(484.7, 432.2)--(484.5, 432.5)--(484.3, 432.8)--(484.2, 433.0)--(484.0, 433.3)--(483.9, 433.4)--(483.8, 433.6)--(483.1, 434.8)--(483.0, 434.9)--(483.0, 435.0)--(482.9, 435.1)--(482.8, 435.2)--(482.6, 435.4)--(482.5, 435.6)--(481.8, 436.7)--(481.1, 438.0)--(482.2, 438.0)--(485.8, 437.9)--(488.4, 437.9)--(488.6, 437.9)--(489.2, 437.9)--(490.0, 437.9)--(490.9, 438.0)--(491.1, 438.0)--(491.2, 438.1)--(491.4, 438.1)--(491.6, 438.1)--(492.4, 438.1)--(492.7, 438.1)--(492.9, 438.1)--(493.1, 438.1)--(493.3, 438.1)--(493.8, 438.1)--(494.8, 437.9)--(496.7, 437.7)--(498.0, 437.4)--(498.1, 437.7)--(498.4, 437.6)--(498.6, 437.6)--(499.0, 437.5)--(499.3, 437.4)--(499.7, 437.4)--(500.0, 437.3)--(500.2, 437.3)--(500.5, 437.2)--(501.4, 436.8)--(501.6, 436.8)--(501.7, 436.7)--(501.8, 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340.1)--(578.9, 340.1)--(578.7, 340.1)--(578.6, 340.1)--(578.4, 340.1)--(578.3, 340.2)--(578.2, 340.2)--(577.9, 340.3)--(577.8, 340.3)--(577.6, 340.3)--(577.5, 340.3)--(577.3, 340.4)--(577.1, 340.4)--(576.9, 340.3)--(575.9, 340.2)--(575.2, 340.0)--(575.0, 339.9)--(574.9, 339.9)--(574.7, 339.8)--(573.8, 339.4)--(573.5, 339.3)--(573.0, 339.2)--(572.8, 339.1)--(572.7, 339.1)--(572.2, 339.0)--(572.4, 336.2)--(572.4, 335.1)--(572.5, 334.2)--(572.5, 333.8)--(572.5, 331.9)--(572.5, 331.1)--(572.5, 331.0)--(572.5, 330.9)--(572.6, 330.8)--(572.6, 330.7)--(572.7, 330.7)--(572.8, 330.6)--(573.2, 330.6)--(573.5, 330.6)--(573.5, 329.8)--(573.5, 329.2)--(573.4, 328.8)--(572.8, 328.8)--(572.3, 328.7)--(571.9, 327.5)--(571.8, 327.3)--(571.8, 327.1)--(571.7, 326.8)--(571.7, 326.7)--(571.7, 326.4)--(571.7, 326.0)--(571.7, 325.8)--(571.7, 324.0)--(571.6, 323.5)--(571.6, 323.1)--(570.9, 323.2)--(570.6, 323.2)--(570.4, 323.1)--(570.3, 323.2)--(570.2, 323.2)--(570.1, 323.2)--(570.0, 323.3)--(569.6, 323.5)--(569.4, 323.6)--(569.3, 323.6)--(569.2, 323.6)--(569.1, 323.6)--(569.1, 323.5)--(569.0, 323.5)--(568.9, 323.5)--(568.7, 323.3)--(568.6, 323.3)--(568.6, 323.2)--(568.6, 323.1)--(568.7, 323.1)--(568.7, 323.0)--(568.6, 323.0)--(568.5, 323.0)--(568.4, 322.9)--(568.3, 322.9)--(568.2, 322.9)--(568.1, 323.0)--(567.9, 323.0)--(567.7, 323.0)--(567.6, 323.0)--(567.4, 323.0)--(567.3, 323.0)--(567.2, 323.0)--(567.1, 322.9)--(566.8, 322.9)--(566.7, 322.9)--(566.6, 322.9)--(566.5, 322.9)--(566.4, 322.8)--(566.3, 322.8)--(566.2, 322.8)--(566.2, 322.7)--(566.1, 322.7)--(566.0, 322.7)--(565.8, 322.8)--(565.6, 322.8)--(565.5, 322.8)--(565.4, 322.8)--(565.3, 322.8)--(565.2, 322.9)--(565.1, 322.9)--(564.9, 322.9)--(564.7, 322.9)--(564.7, 323.6)--(564.6, 324.3)--(564.6, 324.6)--(564.5, 324.9)--(564.4, 325.1)--(564.4, 325.3)--(564.4, 325.5)--(564.4, 325.7)--(564.3, 326.0)--(564.3, 326.4)--(564.1, 326.8)--(564.0, 327.2)--(563.9, 327.6)--(563.8, 327.9)--(563.7, 328.3)--(563.7, 328.7)--(563.6, 329.0)--(563.6, 329.2)--(563.6, 329.5)--(563.6, 329.7)--(563.5, 330.3)--(563.5, 330.6)--(563.4, 330.8)--(563.4, 330.9)--(563.3, 331.0)--(563.1, 331.3)--(563.0, 331.6)--(562.9, 331.9)--(562.8, 332.1)--(562.8, 332.3)--(562.8, 332.5)--(562.8, 332.8)--(562.8, 333.2)--(562.7, 333.6)--(562.6, 334.1)--(562.5, 334.4)--(562.4, 334.6)--(562.3, 334.8)--(562.3, 335.0)--(562.2, 335.0)--(561.6, 334.9)--(561.2, 334.9)--(561.0, 334.8)--(561.0, 334.0)--(561.0, 333.4)--(560.6, 333.6)--(560.0, 333.0)--(559.6, 332.8)--(559.3, 332.5)--(559.2, 332.4)--(559.2, 332.3)--(559.2, 332.2)--(559.2, 332.1)--(559.3, 332.0)--(559.6, 331.4)--(559.7, 331.2)--(559.8, 331.0)--(559.8, 330.9)--(559.8, 330.8)--(559.8, 330.7)--(559.6, 329.9)--(559.5, 329.8)--(559.5, 329.7)--(559.4, 329.6)--(559.3, 329.6)--(559.1, 329.2)--(559.0, 329.0)--(558.2, 329.4)--(556.9, 329.9)--(556.9, 329.7)--(556.9, 329.6)--(556.8, 329.5)--(556.6, 329.0)--(556.3, 328.7)--(556.0, 328.3)--(555.9, 328.1)--(555.8, 328.0)--(555.7, 328.0)--(555.7, 327.9)--(555.6, 327.6)--(555.6, 327.4)--(555.5, 327.3)--(555.5, 327.2)--(555.5, 327.0)--(555.5, 326.6)--(555.5, 326.3)--(555.8, 326.3)--(556.3, 326.2)--(556.6, 326.1)--(556.8, 326.0)--(557.0, 326.0)--(557.1, 326.0)--(557.2, 326.0)--(557.3, 325.9)--(557.5, 325.9)--(557.6, 325.9)--(557.7, 325.8)--(558.0, 325.6)--(558.1, 325.5)--(558.2, 325.4)--(558.3, 325.2)--(558.1, 325.1)--(558.0, 324.8)--(558.2, 324.5)--(558.5, 324.0)--(558.6, 323.9)--(558.6, 323.8)--(558.5, 323.7)--(558.4, 323.6)--(558.1, 323.3)--(557.9, 323.2)--(557.4, 323.4)--(555.8, 324.0)--(555.4, 324.2)--(555.3, 324.2)--(555.2, 324.3)--(555.0, 324.3)--(553.6, 324.2)--(553.1, 324.2)--(553.9, 322.5)--(554.2, 321.9)--(554.8, 320.5)--(554.9, 320.2)--(555.3, 319.3)--(555.6, 319.0)--(555.7, 318.7)--(556.0, 318.3)--(556.1, 318.0)--(555.9, 317.9)--(555.9, 317.7)--(555.6, 317.6)--(555.5, 317.6)--(555.4, 317.6)--(555.3, 317.5)--(555.3, 317.3)--(555.3, 317.0)--(555.4, 316.8)--(555.4, 316.7)--(555.5, 316.4)--(555.5, 316.2)--(555.5, 316.0)--(555.6, 315.4)--(555.6, 315.2)--(555.6, 314.9)--(555.6, 314.7)--(555.7, 314.7)--(555.7, 314.6)--(555.7, 314.5)--(555.8, 314.5)--(555.8, 314.4)--(555.9, 314.4)--(555.9, 314.5)--(556.0, 314.5)--(556.2, 314.5)--(556.4, 314.6)--(556.4, 314.4)--(556.6, 313.9)--(556.7, 313.4)--(556.8, 312.9)--(556.9, 312.4)--(556.9, 312.1)--(556.9, 311.8)--(556.8, 311.8)--(556.8, 311.6)--(556.7, 311.6)--(556.5, 311.5)--(556.3, 311.5)--(556.0, 311.4)--(555.9, 311.4)--(555.5, 311.3)--(555.5, 311.2)--(555.5, 311.1)--(555.5, 311.0)--(555.4, 310.9)--(555.4, 310.8)--(554.9, 310.8)--(553.6, 310.6)--(552.7, 310.4)--(552.4, 310.4)--(552.1, 310.3)--(552.0, 310.3)--(551.9, 310.4)--(551.8, 310.3)--(551.6, 310.3)--(551.4, 310.2)--(551.2, 310.1)--(551.0, 310.0)--(550.8, 310.0)--(550.3, 309.9)--(550.1, 309.9)--(550.0, 310.0)--(549.9, 310.0)--(549.7, 310.0)--(549.3, 310.0)--(549.1, 310.0)--(548.9, 310.1)--(548.6, 310.1)--(548.2, 310.1)--(547.9, 310.1)--(547.5, 310.0)--(546.7, 309.9)--(546.6, 309.9)--(546.5, 309.9)--(546.4, 309.9)--(546.4, 309.8)--(546.3, 309.8)--(546.2, 309.8)--(546.2, 309.7)--(546.1, 309.0)--(545.8, 309.0)--(545.9, 308.9)--(545.9, 308.8)--(546.0, 308.5)--(546.1, 308.3)--(546.2, 308.0)--(546.3, 307.7)--(546.3, 307.5)--(546.4, 307.2)--(546.6, 306.8)--(546.9, 306.3)--(546.9, 306.2)--(547.0, 306.1)--(547.1, 305.8)--(547.1, 305.6)--(547.2, 305.2)--(547.4, 304.7)--(547.6, 304.2)--(547.9, 303.6)--(548.2, 303.0)--(548.5, 302.4)--(548.8, 301.8)--(548.7, 301.8)--(548.5, 301.7)--(548.1, 301.6)--(547.9, 301.5)--(547.6, 301.4)--(547.4, 301.2)--(547.3, 301.1)--(547.2, 301.1)--(547.0, 300.8)--(546.7, 300.6)--(546.3, 300.2)--(546.0, 300.0)--(545.9, 299.9)--(545.7, 299.7)--(545.4, 299.6)--(545.2, 299.5)--(545.0, 299.4)--(544.9, 299.4)--(544.7, 299.7)--(544.6, 299.8)--(544.5, 300.0)--(544.4, 300.1)--(544.4, 300.0)--(544.3, 300.0)--(544.0, 299.9)--(543.8, 299.9)--(543.6, 299.9)--(543.3, 299.9)--(543.2, 299.9)--(543.1, 299.9)--(543.0, 299.9)--(542.7, 299.8)--(542.0, 299.6)--(541.7, 299.5)--(541.7, 299.4)--(541.7, 299.3)--(541.6, 299.3)--(541.2, 299.2)--(540.8, 299.1)--(540.7, 299.0)--(540.6, 298.8)--(540.4, 298.8)--(540.3, 298.7)--(540.1, 298.6)--(540.0, 298.5)--(539.9, 298.5)--(539.4, 298.4)--(539.2, 298.3)--(539.0, 298.4)--(538.9, 298.5)--(538.8, 298.4)--(538.7, 298.4)--(538.6, 298.4)--(538.5, 298.3)--(538.4, 298.1)--(538.1, 297.9)--(538.1, 297.8)--(538.0, 297.8)--(537.9, 297.8)--(537.9, 297.5)--(537.8, 297.2)--(537.7, 297.0)--(537.6, 296.6)--(537.5, 296.6)--(537.4, 296.6)--(537.3, 296.6)--(537.2, 296.6)--(537.1, 296.1)--(537.1, 296.0)--(537.1, 295.9)--(537.1, 295.8)--(537.1, 295.6)--(537.1, 295.5)--(537.1, 295.4)--(537.1, 295.3)--(537.1, 294.8)--(537.1, 294.7)--(537.1, 294.6)--(537.1, 294.5)--(536.9, 294.3)--(536.9, 294.2)--(536.8, 294.2)--(536.8, 294.1)--(536.8, 294.0)--(536.8, 293.9)--(536.8, 293.8)--(536.7, 293.7)--(536.6, 293.4)--(536.6, 292.9)--(536.6, 292.6)--(536.6, 292.4)--(536.6, 292.2)--(536.5, 292.1)--(536.3, 292.1)--(536.3, 292.0)--(536.2, 292.0)--(536.2, 291.8)--(536.2, 291.6)--(536.2, 291.5)--(536.1, 291.5)--(535.9, 291.4)--(535.6, 291.3)--(535.6, 291.2)--(535.5, 291.1)--(535.5, 291.0)--(535.4, 291.0)--(535.3, 291.0)--(535.3, 290.9)--(535.2, 290.9)--(535.2, 290.8)--(535.1, 290.8)--(535.0, 290.7)--(534.9, 290.8)--(534.8, 290.8)--(534.7, 290.8)--(534.6, 290.8)--(534.5, 290.8)--(534.5, 290.9)--(534.4, 291.0)--(534.3, 290.8)--(534.3, 290.7)--(534.2, 290.6)--(534.1, 290.6)--(533.7, 290.6)--(533.6, 290.6)--(533.5, 290.5)--(533.4, 290.5)--(533.4, 290.4)--(533.0, 290.4)--(532.8, 290.4)--(532.8, 290.3)--(532.7, 290.4)--(532.6, 290.4)--(532.5, 290.4)--(532.4, 290.4)--(532.1, 290.4)--(531.8, 290.3)--(531.5, 290.3)--(531.3, 290.2)--(530.9, 290.1)--(530.4, 290.0)--(530.3, 290.0)--(530.2, 290.0)--(530.0, 290.0)--(529.9, 290.0)--(529.3, 289.9)--(529.2, 289.9)--(529.1, 289.9)--(529.1, 289.8)--(528.5, 289.6)--(528.2, 289.5)--(527.5, 289.4)--(527.1, 289.1)--(526.7, 289.0)--(526.6, 288.9)--(526.5, 288.9)--(526.0, 288.8)--(525.9, 288.8)--(525.8, 288.7)--(525.9, 288.6)--(525.0, 288.5)--(525.0, 288.6)--(524.9, 288.6)--(524.8, 288.6)--(524.7, 288.6)--(524.3, 288.5)--(524.1, 288.5)--(523.6, 288.5)--(523.5, 288.4)--(523.3, 288.4)--(523.1, 288.5)--(523.0, 288.5)--(522.9, 288.5)--(522.7, 288.5)--(522.6, 288.5)--(522.4, 288.5)--(522.3, 288.4)--(522.0, 288.3)--(521.9, 288.2)--(521.4, 288.1)--(521.1, 288.0)--(520.9, 288.0)--(520.5, 288.0)--(520.4, 288.0)--(520.2, 288.0)--(520.1, 288.1)--(520.0, 288.1)--(519.8, 288.1)--(519.8, 288.2)--(519.8, 288.3)--(519.8, 288.4)--(519.8, 288.6)--(519.8, 288.7)--(519.7, 288.8)--(519.7, 288.9)--(519.6, 288.9)--(519.6, 289.0)--(519.5, 289.0)--(519.4, 289.0)--(519.4, 289.1)--(519.3, 289.1)--(519.3, 289.2)--(519.2, 289.3)--(519.1, 289.4)--(519.0, 289.7)--(518.8, 289.9)--(518.7, 290.1)--(518.6, 290.1)--(518.6, 290.2)--(518.4, 290.5)--(518.3, 290.7)--(518.1, 290.9)--(517.7, 291.2)--(517.7, 291.3)--(517.6, 291.6)--(517.3, 292.4)--(517.2, 292.7)--(517.2, 292.8)--(517.1, 292.9)--(517.0, 293.1)--(516.8, 294.2)--(516.8, 294.3)--(516.8, 294.4)--(516.8, 294.6)--(516.8, 294.8)--(516.8, 295.0)--(516.8, 295.2)--(516.7, 295.4)--(516.7, 295.5)--(516.7, 295.7)--(516.6, 296.0)--(516.5, 296.6)--(516.4, 296.7)--(516.4, 296.8)--(516.3, 296.8)--(516.1, 296.9)--(516.0, 297.0)--(516.0, 297.6)--(515.9, 297.6)--(515.9, 297.7)--(516.0, 297.8)--(516.2, 297.9)--(516.3, 298.0)--(516.4, 298.1)--(516.5, 298.2)--(516.7, 298.2)--(516.7, 298.3)--(516.4, 298.3)--(516.3, 298.4)--(516.3, 298.5)--(516.4, 298.5)--(516.3, 298.5)--(516.3, 298.6)--(516.3, 298.7)--(516.3, 298.8)--(516.3, 298.9)--(516.5, 299.0)--(516.5, 299.1)--(516.4, 299.1)--(516.3, 299.1)--(516.2, 299.1)--(516.2, 299.2)--(516.2, 299.7)--(516.2, 300.0)--(516.1, 300.1)--(515.9, 300.2)--(515.8, 300.2)--(515.8, 300.3)--(515.7, 300.3)--(515.7, 300.4)--(515.8, 300.4)--(515.8, 300.5)--(515.7, 300.5)--(515.7, 300.6)--(515.8, 300.7)--(515.8, 300.8)--(515.8, 300.9)--(515.6, 301.1)--(515.4, 301.4)--(515.3, 301.4)--(515.3, 301.5)--(515.0, 301.7)--(514.9, 301.8)--(514.8, 301.9)--(514.7, 302.0)--(514.5, 302.5)--(514.5, 302.6)--(514.2, 302.9)--(514.2, 303.0)--(514.2, 303.1)--(514.2, 303.2)--(514.2, 303.3)--(514.2, 303.4)--(514.3, 303.5)--(514.3, 303.6)--(514.3, 303.7)--(514.2, 303.8)--(514.2, 304.0)--(514.1, 304.0)--(514.0, 304.3)--(513.9, 304.6)--(513.6, 305.3)--(513.4, 305.9)--(513.3, 306.1)--(513.2, 306.3)--(513.0, 306.6)--(513.0, 306.9)--(513.0, 307.1)--(512.9, 307.3)--(512.9, 307.4)--(512.6, 307.6)--(512.5, 307.7)--(512.3, 307.8)--(512.2, 307.9)--(512.1, 308.0)--(512.0, 308.1)--(511.9, 308.3)--(511.7, 308.5)--(511.3, 309.0)--(511.0, 309.2)--(510.5, 309.7)--(510.0, 310.1)--(509.6, 310.4)--(509.5, 310.5)--(509.4, 310.5)--(509.3, 310.5)--(509.1, 310.5)--(508.7, 310.5)--(508.6, 310.6)--(508.5, 310.6)--(508.4, 310.6)--(508.2, 310.7)--(508.0, 310.9)--(507.9, 310.9)--(507.7, 311.1)--(507.4, 311.3)--(507.2, 311.4)--(506.3, 311.9)--(505.3, 312.4)--(504.2, 313.1)--(504.1, 313.2)--(503.9, 313.3)--(503.8, 313.4)--(503.5, 313.5)--(503.5, 313.6)--(503.4, 313.6)--(503.4, 313.7)--(503.4, 313.8)--(503.5, 313.9)--(503.5, 314.0)--(503.4, 314.0)--(503.4, 314.1)--(503.3, 314.1)--(503.2, 314.2)--(503.1, 314.2)--(503.0, 314.2)--(503.0, 314.1)--(502.9, 314.1)--(502.7, 314.1)--(502.5, 314.1)--(502.5, 314.3)--(502.4, 314.3)--(502.4, 314.4)--(502.3, 314.4)--(502.2, 314.4)--(502.2, 314.3)--(502.2, 314.2)--(502.1, 314.2)--(502.1, 314.3)--(501.9, 314.3)--(501.8, 314.3)--(501.7, 314.3)--(501.7, 314.2)--(501.6, 314.1)--(501.5, 314.1)--(501.5, 314.2)--(501.4, 314.2)--(501.1, 314.2)--(500.8, 314.1)--(500.6, 314.1)--(500.5, 314.1)--(500.4, 314.1)--(500.3, 314.1)--(500.2, 314.2)--(500.0, 314.2)--(499.9, 314.2)--(499.8, 314.2)--(499.7, 314.3)--(499.5, 314.3)--(499.5, 314.8)--(499.2, 314.8)--cycle; +Hillingdon = (106.0, 416.9)--(106.2, 415.9)--(106.3, 415.4)--(106.4, 414.9)--(106.5, 414.1)--(106.5, 413.6)--(106.5, 413.1)--(106.6, 412.7)--(106.7, 412.4)--(106.8, 412.2)--(106.9, 412.0)--(107.0, 411.8)--(107.1, 411.6)--(107.2, 411.4)--(107.2, 411.0)--(107.3, 410.5)--(107.5, 409.9)--(107.6, 409.7)--(107.7, 409.6)--(107.6, 409.5)--(107.5, 409.2)--(107.3, 409.1)--(107.3, 408.4)--(107.3, 408.0)--(107.3, 407.6)--(107.3, 407.1)--(107.3, 406.6)--(107.4, 406.4)--(107.3, 405.9)--(107.3, 405.5)--(107.5, 404.4)--(107.6, 404.0)--(107.7, 403.7)--(107.9, 403.3)--(108.0, 403.0)--(108.0, 402.6)--(108.2, 402.2)--(108.3, 401.9)--(108.3, 401.8)--(108.5, 401.5)--(108.8, 401.0)--(109.0, 400.7)--(109.2, 400.3)--(109.1, 400.1)--(109.1, 400.0)--(109.0, 399.4)--(109.3, 399.3)--(110.0, 399.2)--(110.5, 399.1)--(110.8, 399.0)--(110.8, 398.9)--(110.8, 398.8)--(110.8, 398.7)--(110.8, 398.6)--(110.8, 398.5)--(110.9, 398.2)--(111.1, 398.0)--(111.2, 397.7)--(111.3, 397.7)--(111.4, 397.5)--(111.5, 397.2)--(111.6, 397.0)--(111.7, 396.9)--(111.7, 396.8)--(111.7, 396.5)--(111.7, 396.3)--(111.7, 395.8)--(111.8, 395.5)--(111.8, 395.2)--(111.9, 395.0)--(111.9, 394.9)--(112.0, 394.6)--(112.1, 394.3)--(112.3, 393.8)--(112.4, 393.6)--(112.4, 393.5)--(112.4, 393.4)--(112.5, 393.0)--(112.5, 392.7)--(112.5, 392.3)--(112.5, 392.2)--(112.4, 392.0)--(112.5, 391.8)--(112.5, 391.7)--(112.5, 391.5)--(112.7, 391.6)--(112.9, 391.5)--(113.1, 391.6)--(113.0, 391.7)--(113.2, 391.8)--(113.4, 391.5)--(113.8, 390.9)--(113.9, 390.6)--(114.2, 390.2)--(114.3, 390.0)--(114.6, 389.9)--(114.8, 389.6)--(114.9, 389.4)--(115.0, 388.8)--(115.1, 388.5)--(115.1, 388.2)--(115.2, 388.0)--(115.2, 387.9)--(115.3, 386.9)--(115.4, 386.2)--(115.4, 385.7)--(115.5, 385.2)--(115.5, 384.8)--(115.6, 384.8)--(115.7, 384.2)--(115.8, 383.6)--(115.7, 383.6)--(115.7, 383.4)--(115.8, 382.5)--(115.8, 382.1)--(115.5, 382.1)--(115.4, 381.9)--(115.6, 381.7)--(115.6, 381.6)--(115.8, 381.4)--(115.7, 381.4)--(116.0, 380.8)--(116.4, 379.8)--(116.8, 379.0)--(117.3, 378.1)--(117.6, 377.7)--(117.7, 377.2)--(117.8, 376.7)--(118.1, 375.4)--(118.2, 375.4)--(118.2, 375.3)--(118.4, 375.0)--(118.0, 374.6)--(118.5, 374.0)--(118.6, 373.6)--(118.9, 373.1)--(119.0, 372.8)--(119.5, 371.9)--(119.7, 371.5)--(119.5, 371.2)--(119.2, 370.9)--(119.6, 370.2)--(119.8, 369.9)--(120.3, 369.0)--(120.5, 368.4)--(120.6, 368.2)--(120.6, 368.1)--(120.8, 367.8)--(121.5, 366.1)--(121.4, 366.0)--(121.8, 365.0)--(121.8, 364.7)--(122.0, 364.2)--(122.1, 363.6)--(122.2, 363.3)--(122.2, 363.1)--(122.2, 363.0)--(122.2, 362.9)--(122.1, 362.5)--(122.1, 362.4)--(122.1, 362.2)--(122.2, 362.1)--(122.4, 361.7)--(122.5, 361.6)--(122.5, 361.5)--(122.4, 361.4)--(122.3, 361.3)--(122.5, 360.9)--(122.6, 360.8)--(122.7, 360.7)--(123.0, 360.2)--(123.3, 359.7)--(123.5, 359.5)--(123.5, 359.3)--(123.6, 359.2)--(123.7, 358.9)--(123.8, 358.8)--(123.8, 358.5)--(123.7, 358.2)--(123.7, 357.8)--(123.9, 357.8)--(123.9, 357.7)--(123.9, 357.6)--(124.0, 357.5)--(123.9, 357.4)--(123.9, 357.3)--(123.8, 357.1)--(123.9, 356.8)--(124.1, 356.4)--(124.3, 356.0)--(124.5, 355.7)--(124.5, 355.5)--(124.6, 355.1)--(124.7, 354.9)--(124.8, 354.7)--(124.8, 354.4)--(124.9, 354.2)--(125.0, 354.0)--(125.2, 353.6)--(125.3, 353.4)--(125.4, 353.1)--(125.5, 353.0)--(125.5, 352.8)--(125.6, 352.6)--(125.6, 352.5)--(125.7, 352.2)--(125.2, 351.9)--(124.0, 351.1)--(123.7, 350.8)--(123.5, 350.7)--(123.1, 350.3)--(122.9, 350.1)--(122.8, 350.0)--(122.1, 349.4)--(121.5, 348.9)--(120.9, 348.5)--(120.5, 348.2)--(119.9, 347.9)--(119.4, 347.6)--(119.0, 347.5)--(118.6, 347.3)--(118.3, 347.0)--(118.0, 346.9)--(117.9, 347.0)--(117.2, 346.6)--(115.9, 346.0)--(115.4, 345.7)--(114.9, 345.5)--(114.6, 345.3)--(114.4, 345.3)--(113.9, 345.4)--(113.8, 345.1)--(113.9, 344.6)--(114.0, 344.1)--(113.4, 344.2)--(112.8, 344.2)--(112.5, 344.2)--(112.3, 344.3)--(112.1, 344.3)--(111.9, 344.3)--(111.5, 344.4)--(111.4, 344.4)--(111.3, 344.4)--(111.4, 344.1)--(111.4, 344.0)--(111.5, 344.0)--(111.6, 343.7)--(111.6, 343.6)--(111.5, 343.6)--(111.8, 342.4)--(111.9, 342.0)--(111.9, 341.9)--(112.5, 342.1)--(112.8, 342.3)--(113.0, 341.9)--(113.1, 341.8)--(113.1, 341.7)--(113.2, 341.4)--(113.3, 341.0)--(113.4, 340.8)--(113.4, 340.5)--(113.5, 340.2)--(113.6, 340.2)--(113.6, 340.1)--(113.6, 339.8)--(113.7, 339.0)--(113.8, 338.5)--(112.4, 338.2)--(112.2, 338.2)--(111.1, 338.2)--(110.5, 338.3)--(110.3, 338.3)--(109.6, 338.4)--(109.5, 338.4)--(109.0, 338.3)--(109.0, 338.4)--(108.3, 338.4)--(107.9, 338.3)--(107.5, 338.3)--(107.2, 338.3)--(106.8, 338.2)--(105.6, 337.8)--(105.0, 337.6)--(105.1, 337.5)--(104.5, 337.3)--(104.0, 337.1)--(103.1, 336.9)--(102.3, 336.6)--(101.8, 336.5)--(101.3, 336.3)--(100.9, 336.2)--(100.6, 336.1)--(100.3, 336.0)--(100.0, 335.9)--(99.9, 335.9)--(99.9, 335.8)--(99.6, 335.7)--(99.5, 335.6)--(99.4, 335.6)--(99.2, 335.5)--(99.0, 335.5)--(98.9, 335.5)--(98.6, 335.5)--(98.5, 335.4)--(98.4, 335.4)--(98.3, 335.4)--(98.1, 335.4)--(97.9, 335.4)--(97.8, 335.4)--(97.7, 335.5)--(97.7, 335.6)--(97.6, 335.6)--(97.5, 335.6)--(97.4, 335.7)--(97.3, 335.8)--(97.2, 335.8)--(97.1, 335.7)--(97.1, 335.6)--(97.1, 335.5)--(97.1, 335.4)--(97.1, 335.3)--(97.1, 335.2)--(97.3, 335.0)--(97.4, 334.9)--(97.5, 334.9)--(97.5, 334.8)--(97.6, 334.7)--(97.6, 334.6)--(97.7, 334.6)--(97.8, 334.5)--(97.9, 334.4)--(98.0, 334.4)--(98.0, 334.3)--(97.9, 334.3)--(97.9, 334.2)--(97.7, 334.1)--(97.4, 334.0)--(97.2, 334.0)--(97.1, 333.9)--(97.0, 333.8)--(97.0, 333.7)--(97.1, 333.5)--(97.2, 333.3)--(97.3, 333.3)--(97.4, 333.3)--(97.5, 333.2)--(97.6, 333.2)--(97.8, 333.1)--(98.1, 333.0)--(98.3, 333.0)--(98.4, 333.0)--(98.4, 332.9)--(98.5, 332.9)--(98.6, 332.8)--(98.6, 332.7)--(98.5, 332.6)--(98.5, 332.5)--(98.5, 332.4)--(98.5, 332.2)--(98.4, 332.0)--(98.4, 331.8)--(98.4, 331.7)--(98.5, 331.7)--(98.6, 331.6)--(98.6, 331.5)--(98.7, 331.5)--(98.8, 331.5)--(98.9, 331.4)--(99.0, 331.4)--(99.1, 331.4)--(99.2, 331.5)--(99.5, 331.5)--(99.6, 331.6)--(99.7, 331.6)--(99.8, 331.6)--(100.0, 331.6)--(100.0, 331.5)--(100.1, 331.5)--(100.2, 331.3)--(100.2, 331.2)--(100.3, 331.1)--(100.3, 331.0)--(100.4, 330.9)--(100.5, 330.8)--(100.6, 330.8)--(100.9, 330.8)--(101.0, 330.7)--(101.1, 330.7)--(101.1, 330.6)--(101.2, 330.4)--(101.2, 330.0)--(101.3, 329.8)--(101.3, 329.7)--(101.3, 329.5)--(101.4, 329.5)--(101.4, 329.4)--(101.5, 329.4)--(101.6, 329.4)--(101.7, 329.4)--(101.9, 329.3)--(102.0, 329.3)--(102.1, 329.3)--(102.2, 329.3)--(102.3, 329.2)--(102.4, 329.1)--(102.4, 329.0)--(102.5, 328.9)--(102.5, 328.8)--(103.0, 329.1)--(103.1, 329.0)--(103.2, 328.9)--(103.3, 328.9)--(103.6, 328.9)--(104.1, 328.8)--(104.2, 328.8)--(105.6, 328.3)--(105.9, 328.3)--(106.8, 328.0)--(106.9, 328.0)--(107.1, 327.9)--(107.3, 327.9)--(107.4, 327.9)--(107.5, 327.9)--(107.6, 327.8)--(107.7, 327.8)--(109.0, 327.6)--(109.8, 327.5)--(110.7, 327.3)--(111.8, 327.0)--(112.9, 326.8)--(113.0, 326.8)--(113.4, 326.7)--(113.7, 326.7)--(114.0, 326.6)--(114.6, 326.5)--(115.3, 326.3)--(116.1, 326.1)--(117.4, 325.7)--(117.9, 325.6)--(118.0, 325.6)--(118.3, 325.5)--(118.3, 325.3)--(118.5, 325.1)--(118.9, 324.9)--(119.3, 324.6)--(119.8, 324.2)--(119.9, 324.6)--(120.0, 324.8)--(120.1, 325.1)--(120.6, 325.1)--(120.8, 325.0)--(121.6, 324.7)--(121.9, 324.6)--(123.0, 324.3)--(123.6, 324.2)--(124.0, 324.1)--(125.0, 323.9)--(125.4, 323.8)--(126.4, 323.7)--(126.8, 323.7)--(127.2, 323.6)--(127.3, 323.6)--(127.3, 323.5)--(127.4, 323.5)--(127.4, 321.9)--(127.4, 321.4)--(127.5, 321.1)--(127.5, 320.8)--(127.6, 320.3)--(127.6, 320.1)--(127.6, 319.9)--(127.6, 319.6)--(127.5, 319.3)--(127.5, 319.2)--(127.4, 318.9)--(127.2, 318.8)--(127.2, 318.6)--(126.9, 318.4)--(126.8, 318.1)--(126.5, 317.4)--(126.0, 316.4)--(125.9, 316.2)--(125.7, 315.7)--(125.5, 315.4)--(125.4, 315.1)--(125.0, 314.3)--(124.8, 314.0)--(124.4, 313.4)--(124.3, 313.3)--(124.0, 312.7)--(123.8, 312.2)--(123.8, 312.1)--(123.7, 312.0)--(123.5, 311.8)--(123.3, 311.5)--(123.2, 311.4)--(123.2, 311.3)--(123.0, 311.3)--(122.7, 310.8)--(122.6, 310.8)--(122.4, 310.5)--(122.3, 310.3)--(122.1, 310.0)--(121.9, 309.8)--(121.8, 309.7)--(121.7, 309.6)--(121.6, 309.5)--(121.4, 309.4)--(120.8, 309.0)--(120.4, 308.7)--(119.1, 307.9)--(118.8, 307.6)--(118.6, 307.4)--(118.5, 307.2)--(118.2, 306.7)--(118.1, 306.5)--(117.9, 306.1)--(117.9, 305.9)--(117.8, 305.7)--(117.8, 305.5)--(117.7, 305.2)--(117.6, 304.8)--(117.5, 304.4)--(117.2, 303.2)--(116.7, 301.4)--(116.5, 300.4)--(116.4, 300.1)--(116.3, 299.9)--(116.3, 299.7)--(116.2, 299.6)--(116.2, 299.4)--(116.0, 299.2)--(116.0, 299.1)--(115.9, 298.9)--(115.8, 298.8)--(115.6, 298.8)--(115.6, 298.7)--(115.5, 298.7)--(115.5, 298.6)--(115.3, 298.5)--(114.9, 298.2)--(114.4, 297.9)--(113.9, 297.6)--(113.0, 297.0)--(112.1, 296.4)--(111.6, 296.0)--(111.1, 295.8)--(110.6, 295.6)--(110.6, 295.5)--(110.4, 295.5)--(110.3, 295.4)--(110.1, 295.4)--(110.0, 295.3)--(109.9, 295.3)--(109.8, 295.2)--(109.7, 295.2)--(109.6, 295.1)--(109.5, 295.1)--(109.4, 295.0)--(109.3, 294.9)--(109.2, 294.8)--(109.1, 294.7)--(109.0, 294.6)--(109.0, 294.5)--(108.9, 294.5)--(108.8, 294.3)--(108.5, 293.9)--(108.0, 292.8)--(107.9, 292.8)--(107.2, 291.4)--(106.8, 290.6)--(106.7, 290.6)--(106.5, 290.7)--(106.2, 290.9)--(105.8, 291.1)--(105.1, 291.3)--(104.9, 290.9)--(104.2, 288.9)--(104.1, 288.5)--(104.1, 288.4)--(104.1, 288.3)--(104.1, 288.2)--(104.1, 288.1)--(104.3, 287.2)--(104.3, 287.0)--(104.3, 286.8)--(104.3, 286.7)--(104.3, 286.6)--(104.3, 286.5)--(104.3, 286.4)--(104.1, 286.1)--(104.0, 286.0)--(103.9, 285.8)--(103.8, 285.6)--(103.8, 285.3)--(103.7, 284.3)--(103.6, 284.0)--(103.5, 282.5)--(103.5, 282.2)--(103.5, 282.1)--(103.2, 279.0)--(103.1, 278.3)--(103.1, 278.2)--(103.1, 278.1)--(103.0, 278.1)--(102.9, 278.0)--(102.9, 277.9)--(102.8, 277.9)--(102.8, 277.8)--(102.8, 277.7)--(102.8, 277.6)--(102.8, 277.4)--(102.7, 277.3)--(102.7, 277.2)--(102.6, 277.1)--(102.6, 277.0)--(102.6, 276.9)--(102.6, 276.8)--(102.6, 276.7)--(102.5, 276.7)--(102.4, 276.7)--(102.3, 276.7)--(102.2, 276.7)--(102.2, 276.6)--(102.0, 276.2)--(101.8, 275.9)--(101.8, 275.8)--(101.7, 275.6)--(101.8, 275.5)--(101.9, 275.5)--(102.1, 275.5)--(102.2, 275.5)--(102.2, 275.4)--(101.7, 274.9)--(101.4, 274.0)--(101.1, 273.2)--(100.9, 272.9)--(100.9, 272.7)--(100.8, 272.6)--(100.8, 272.1)--(100.7, 272.0)--(100.7, 271.9)--(100.6, 271.8)--(100.4, 271.6)--(100.3, 271.5)--(100.3, 271.4)--(100.3, 271.3)--(100.3, 271.2)--(100.4, 271.2)--(100.4, 271.1)--(100.5, 271.1)--(100.5, 271.0)--(100.6, 271.0)--(100.6, 270.9)--(100.5, 270.8)--(100.5, 270.3)--(100.4, 269.9)--(100.4, 269.6)--(100.4, 269.3)--(100.4, 269.1)--(100.5, 268.9)--(100.6, 268.7)--(100.7, 268.6)--(100.7, 268.5)--(100.9, 267.9)--(100.9, 267.8)--(101.0, 267.5)--(101.1, 267.3)--(101.3, 267.0)--(101.5, 266.1)--(101.5, 266.0)--(101.5, 265.9)--(101.7, 265.5)--(101.7, 265.0)--(101.8, 264.2)--(101.4, 263.3)--(101.3, 262.9)--(101.3, 262.7)--(101.2, 262.4)--(101.2, 262.0)--(101.2, 261.9)--(101.2, 261.8)--(101.3, 261.7)--(101.4, 261.6)--(101.5, 261.5)--(101.8, 261.4)--(101.9, 261.4)--(101.9, 261.3)--(102.0, 261.3)--(102.1, 261.1)--(102.2, 260.8)--(102.2, 260.7)--(102.3, 260.5)--(102.5, 260.3)--(102.6, 260.3)--(102.7, 260.2)--(102.8, 260.1)--(102.9, 260.0)--(103.0, 259.7)--(103.1, 259.4)--(103.2, 259.3)--(103.3, 259.1)--(103.3, 259.0)--(103.9, 258.6)--(104.2, 258.3)--(104.4, 258.1)--(104.5, 257.8)--(104.6, 257.4)--(104.5, 257.4)--(103.4, 256.8)--(102.3, 256.3)--(101.1, 255.7)--(100.2, 255.2)--(100.0, 255.1)--(99.3, 254.7)--(98.7, 254.4)--(98.4, 254.2)--(98.3, 254.1)--(98.3, 253.9)--(98.3, 253.8)--(98.1, 253.8)--(97.9, 253.9)--(97.8, 253.9)--(97.5, 253.8)--(97.3, 253.7)--(97.1, 253.6)--(97.0, 253.6)--(96.8, 253.5)--(96.1, 252.9)--(95.5, 252.5)--(94.5, 251.8)--(93.6, 251.2)--(92.7, 250.6)--(91.9, 250.0)--(91.7, 249.8)--(91.2, 249.5)--(90.5, 249.0)--(89.8, 248.5)--(89.2, 248.0)--(89.0, 247.7)--(88.9, 247.5)--(88.7, 247.2)--(88.5, 246.9)--(88.3, 246.4)--(88.2, 246.1)--(87.9, 245.7)--(87.8, 245.4)--(87.8, 245.2)--(87.6, 244.9)--(87.4, 244.5)--(87.3, 244.3)--(87.3, 244.0)--(87.1, 243.5)--(87.8, 243.2)--(88.5, 242.9)--(88.3, 242.6)--(88.1, 242.6)--(87.9, 242.6)--(87.5, 242.6)--(87.1, 242.6)--(86.7, 242.5)--(85.9, 242.4)--(85.6, 242.3)--(85.1, 242.3)--(85.0, 242.3)--(84.6, 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401.2)--(39.4, 401.5)--(39.5, 401.9)--(39.5, 402.0)--(39.6, 402.1)--(39.6, 402.2)--(39.7, 402.3)--(39.8, 402.3)--(39.9, 402.3)--(40.2, 402.4)--(40.3, 402.4)--(40.4, 402.4)--(40.5, 402.4)--(40.6, 402.3)--(40.7, 402.2)--(40.9, 402.0)--(41.0, 401.9)--(41.1, 401.9)--(41.3, 401.8)--(41.4, 401.8)--(41.5, 401.8)--(41.6, 401.9)--(41.7, 402.0)--(41.9, 402.1)--(41.9, 402.2)--(42.1, 402.3)--(42.1, 402.4)--(42.2, 402.5)--(42.1, 402.6)--(42.1, 402.7)--(42.1, 402.8)--(42.2, 403.1)--(42.2, 403.3)--(42.3, 403.4)--(42.3, 403.5)--(42.4, 403.5)--(42.4, 403.6)--(42.4, 403.7)--(42.3, 403.7)--(42.3, 403.8)--(42.1, 403.9)--(42.0, 404.1)--(41.9, 404.2)--(41.8, 404.3)--(41.8, 404.4)--(41.8, 404.5)--(41.7, 404.6)--(41.7, 404.7)--(41.6, 405.4)--(41.5, 405.6)--(41.5, 405.8)--(41.6, 405.9)--(41.6, 406.0)--(41.6, 406.1)--(41.7, 406.2)--(41.7, 406.3)--(41.6, 406.3)--(41.5, 406.7)--(41.3, 407.3)--(41.2, 407.6)--(41.1, 407.8)--(41.1, 408.0)--(41.1, 408.4)--(41.1, 408.5)--(41.0, 408.8)--(40.8, 409.3)--(40.7, 409.5)--(40.7, 409.6)--(40.7, 409.8)--(40.7, 410.0)--(40.7, 410.1)--(40.6, 410.2)--(40.4, 410.3)--(40.3, 410.4)--(40.3, 410.5)--(40.2, 410.6)--(40.2, 410.7)--(40.2, 410.8)--(40.2, 410.9)--(40.3, 410.9)--(40.3, 411.0)--(40.4, 411.1)--(40.7, 411.3)--(40.8, 411.3)--(40.8, 411.4)--(40.8, 411.5)--(40.8, 411.6)--(40.6, 411.9)--(40.4, 412.0)--(40.4, 412.1)--(40.3, 412.1)--(40.4, 412.3)--(40.4, 412.6)--(40.4, 412.7)--(40.4, 413.0)--(40.4, 413.1)--(40.3, 413.2)--(40.4, 413.3)--(40.4, 413.4)--(40.4, 413.5)--(40.5, 413.5)--(40.6, 413.6)--(40.6, 413.7)--(40.7, 413.7)--(40.7, 413.8)--(40.7, 413.9)--(40.6, 414.1)--(40.6, 414.3)--(40.6, 414.4)--(40.6, 414.5)--(40.8, 414.9)--(40.8, 415.1)--(40.9, 415.3)--(41.0, 415.5)--(41.0, 415.6)--(41.0, 415.7)--(41.0, 415.8)--(41.0, 415.9)--(41.0, 416.1)--(41.0, 416.2)--(41.0, 416.4)--(41.0, 416.5)--(41.0, 416.6)--(41.0, 416.7)--(41.1, 417.0)--(41.1, 417.1)--(41.1, 417.2)--(41.2, 417.2)--(41.3, 417.3)--(41.5, 417.3)--(41.5, 417.4)--(41.6, 417.4)--(41.6, 417.5)--(41.6, 417.7)--(41.7, 417.9)--(41.7, 418.0)--(41.8, 418.5)--(41.8, 418.7)--(41.9, 418.9)--(42.2, 419.5)--(42.2, 419.6)--(42.2, 419.7)--(42.2, 419.8)--(42.2, 420.0)--(42.2, 420.1)--(42.3, 420.4)--(42.3, 420.6)--(42.3, 420.7)--(42.4, 420.7)--(42.5, 420.9)--(42.5, 421.0)--(42.6, 421.5)--(42.6, 421.7)--(42.6, 421.8)--(42.6, 421.9)--(42.6, 422.0)--(42.6, 422.1)--(42.5, 422.2)--(42.4, 422.4)--(42.2, 422.7)--(42.1, 422.8)--(42.0, 422.9)--(42.0, 423.0)--(41.9, 423.0)--(41.8, 423.1)--(41.7, 423.1)--(41.7, 423.2)--(41.6, 423.4)--(41.6, 423.5)--(41.6, 423.6)--(41.6, 423.7)--(41.5, 423.8)--(41.4, 423.8)--(41.2, 423.9)--(41.1, 423.9)--(40.9, 424.0)--(40.7, 424.1)--(40.3, 424.1)--(40.1, 424.2)--(40.0, 424.2)--(39.9, 424.3)--(39.8, 424.4)--(39.8, 424.5)--(39.7, 424.7)--(39.5, 424.9)--(39.5, 425.0)--(39.3, 425.2)--(39.3, 425.4)--(39.2, 425.4)--(39.2, 425.5)--(39.3, 425.6)--(39.3, 425.7)--(39.3, 426.0)--(39.3, 426.1)--(39.3, 426.3)--(39.2, 426.5)--(39.2, 426.7)--(39.2, 426.8)--(39.2, 426.9)--(39.3, 427.0)--(39.3, 427.2)--(39.4, 427.4)--(39.5, 427.5)--(39.5, 427.6)--(39.6, 427.7)--(39.7, 427.8)--(39.9, 428.0)--(40.2, 428.2)--(40.3, 428.2)--(40.4, 428.3)--(40.4, 428.4)--(40.5, 428.5)--(40.5, 428.7)--(40.4, 428.9)--(40.5, 429.0)--(40.6, 429.1)--(40.8, 429.5)--(40.8, 429.6)--(40.8, 429.7)--(40.8, 430.0)--(40.9, 430.2)--(40.9, 430.5)--(40.9, 430.6)--(40.9, 430.7)--(40.8, 430.8)--(40.5, 431.0)--(40.4, 431.1)--(40.2, 431.3)--(39.9, 431.5)--(39.9, 431.6)--(39.8, 431.7)--(39.6, 432.0)--(39.4, 432.2)--(39.2, 432.5)--(39.1, 432.6)--(39.1, 432.7)--(39.0, 432.7)--(39.0, 432.9)--(39.0, 433.0)--(39.0, 433.1)--(39.1, 433.1)--(39.1, 433.2)--(39.3, 433.3)--(39.4, 433.4)--(39.4, 433.5)--(39.4, 433.7)--(39.4, 433.8)--(39.4, 434.0)--(39.4, 434.1)--(39.5, 434.3)--(39.5, 434.4)--(39.6, 434.5)--(39.6, 434.6)--(39.7, 434.7)--(39.8, 434.9)--(40.2, 435.2)--(40.2, 435.3)--(40.4, 435.4)--(40.5, 435.4)--(40.6, 435.7)--(40.8, 435.8)--(40.9, 436.0)--(41.1, 436.1)--(41.2, 436.1)--(41.3, 436.2)--(41.4, 436.2)--(41.5, 436.2)--(41.6, 436.2)--(41.7, 436.2)--(41.8, 436.2)--(41.8, 436.1)--(41.9, 436.1)--(41.9, 436.0)--(42.0, 435.9)--(42.1, 435.8)--(42.2, 435.6)--(42.2, 435.5)--(42.4, 435.5)--(42.6, 435.5)--(42.8, 435.4)--(42.9, 435.4)--(43.1, 435.5)--(43.2, 435.5)--(43.2, 435.4)--(43.3, 435.4)--(43.5, 435.2)--(43.5, 435.0)--(43.6, 434.9)--(43.7, 434.9)--(43.8, 435.0)--(43.9, 435.0)--(44.0, 435.0)--(44.2, 434.9)--(44.3, 434.9)--(44.4, 434.9)--(44.5, 434.9)--(44.7, 434.9)--(44.7, 435.0)--(44.8, 435.0)--(45.0, 435.2)--(45.3, 434.6)--(45.7, 433.8)--(46.3, 432.8)--(46.5, 432.4)--(46.7, 432.1)--(46.7, 432.0)--(46.8, 431.8)--(46.8, 431.7)--(46.9, 431.5)--(47.0, 431.5)--(47.1, 431.3)--(47.2, 431.2)--(47.5, 430.9)--(47.7, 430.8)--(47.9, 430.6)--(48.2, 430.4)--(48.4, 430.2)--(48.6, 430.1)--(48.9, 429.9)--(49.2, 429.6)--(49.4, 429.5)--(49.7, 429.3)--(50.0, 429.0)--(50.1, 429.0)--(50.2, 428.9)--(50.4, 428.7)--(50.6, 428.6)--(50.7, 428.5)--(50.8, 428.5)--(51.0, 428.4)--(51.3, 428.3)--(51.5, 428.2)--(51.5, 428.1)--(51.7, 427.9)--(51.8, 427.9)--(52.0, 427.8)--(52.0, 427.7)--(52.2, 427.6)--(52.6, 427.3)--(52.7, 427.4)--(53.1, 427.0)--(53.6, 426.6)--(54.0, 426.3)--(54.4, 426.0)--(54.4, 426.1)--(54.5, 426.0)--(54.8, 425.9)--(55.0, 425.8)--(55.3, 425.7)--(55.4, 425.6)--(55.5, 425.5)--(55.7, 425.4)--(55.8, 425.2)--(55.9, 425.1)--(56.2, 424.7)--(56.3, 424.5)--(56.4, 424.5)--(56.8, 424.0)--(57.2, 423.6)--(57.5, 423.3)--(57.8, 423.0)--(58.2, 422.6)--(58.4, 422.3)--(59.2, 421.7)--(59.5, 421.5)--(59.7, 421.3)--(60.8, 420.0)--(61.0, 419.9)--(61.2, 419.7)--(61.7, 419.4)--(62.3, 419.0)--(62.8, 418.6)--(63.7, 418.1)--(64.1, 417.8)--(64.5, 417.6)--(65.0, 417.3)--(65.4, 417.0)--(65.6, 417.1)--(65.7, 417.0)--(65.8, 416.9)--(65.9, 416.8)--(66.8, 416.4)--(67.0, 416.3)--(67.4, 416.0)--(68.1, 415.7)--(68.8, 415.4)--(69.1, 415.2)--(69.3, 415.1)--(69.5, 415.2)--(69.7, 415.3)--(70.0, 415.4)--(70.1, 415.4)--(70.3, 415.5)--(70.4, 415.5)--(70.5, 415.6)--(70.8, 415.8)--(71.0, 416.0)--(71.6, 416.3)--(72.5, 417.0)--(72.9, 417.3)--(73.8, 417.8)--(74.1, 418.0)--(74.3, 418.2)--(74.4, 418.2)--(74.6, 418.3)--(74.7, 418.4)--(74.9, 418.6)--(75.0, 418.7)--(75.3, 419.1)--(75.4, 419.2)--(75.4, 419.3)--(75.5, 419.4)--(75.5, 419.5)--(75.6, 419.6)--(75.6, 420.0)--(75.7, 420.3)--(75.8, 420.5)--(75.9, 420.7)--(75.9, 420.8)--(76.0, 421.2)--(76.2, 421.6)--(76.5, 422.4)--(76.6, 422.7)--(76.7, 422.9)--(76.9, 423.1)--(77.0, 423.3)--(77.1, 423.4)--(77.2, 423.5)--(77.3, 423.5)--(77.4, 423.5)--(77.6, 423.5)--(77.9, 423.5)--(78.0, 423.5)--(78.1, 423.4)--(78.5, 423.3)--(78.6, 423.3)--(78.7, 423.3)--(78.9, 423.4)--(79.0, 423.4)--(79.2, 423.4)--(79.5, 423.5)--(79.7, 423.6)--(80.0, 423.8)--(80.2, 423.8)--(80.3, 423.9)--(80.5, 424.0)--(80.7, 424.0)--(81.0, 424.0)--(81.4, 423.9)--(81.5, 423.9)--(82.0, 423.9)--(82.1, 423.9)--(82.2, 423.8)--(82.3, 423.8)--(82.4, 423.7)--(82.5, 423.6)--(82.7, 423.5)--(83.0, 423.5)--(83.1, 423.5)--(83.2, 423.4)--(83.5, 423.3)--(83.6, 423.3)--(83.6, 423.2)--(83.7, 423.1)--(83.8, 423.1)--(83.9, 423.0)--(84.0, 423.0)--(84.2, 423.0)--(84.4, 422.9)--(84.6, 422.8)--(84.8, 422.7)--(85.0, 422.6)--(85.2, 422.5)--(85.3, 422.5)--(85.5, 422.4)--(85.6, 422.4)--(85.7, 422.3)--(85.8, 422.3)--(85.9, 422.3)--(86.0, 422.2)--(86.6, 422.1)--(86.9, 422.0)--(87.0, 422.1)--(87.1, 422.1)--(87.3, 422.1)--(87.7, 422.0)--(87.9, 421.9)--(88.3, 421.8)--(88.5, 421.7)--(88.6, 421.7)--(88.9, 421.6)--(89.0, 421.6)--(89.1, 421.6)--(89.3, 421.5)--(89.9, 421.1)--(90.0, 421.0)--(90.1, 420.9)--(90.2, 420.9)--(90.3, 420.8)--(90.4, 420.8)--(90.5, 420.8)--(90.6, 420.8)--(90.8, 420.8)--(90.9, 420.8)--(91.2, 420.8)--(91.8, 420.7)--(92.0, 420.7)--(92.2, 420.7)--(92.5, 420.8)--(92.6, 420.8)--(92.9, 420.8)--(93.3, 420.8)--(93.7, 420.8)--(94.1, 420.8)--(94.2, 420.8)--(94.5, 420.8)--(95.0, 420.7)--(95.1, 420.7)--(95.4, 420.7)--(95.7, 420.6)--(96.0, 420.5)--(96.1, 420.5)--(96.2, 420.5)--(96.5, 420.5)--(96.7, 420.4)--(97.0, 420.4)--(97.2, 420.3)--(97.4, 420.2)--(97.6, 420.1)--(97.7, 420.0)--(97.8, 420.0)--(97.9, 419.9)--(98.0, 419.8)--(98.1, 419.7)--(98.6, 419.4)--(99.5, 418.8)--(99.6, 418.7)--(99.9, 418.4)--(100.0, 418.3)--(100.5, 417.9)--(100.6, 417.8)--(100.7, 417.7)--(101.0, 417.5)--(101.1, 417.5)--(101.2, 417.5)--(101.6, 417.5)--(101.9, 417.5)--(102.6, 417.4)--(103.0, 417.3)--(103.2, 417.3)--(103.5, 417.3)--(103.8, 417.3)--(105.1, 417.1)--(105.3, 417.1)--cycle; +Hounslow = (219.8, 281.0)--(219.7, 281.0)--(219.6, 281.0)--(219.0, 280.6)--(217.9, 279.6)--(217.9, 279.5)--(217.8, 279.4)--(217.6, 279.4)--(217.6, 279.3)--(217.4, 279.1)--(217.2, 278.9)--(216.6, 278.0)--(216.5, 277.9)--(216.5, 277.8)--(216.4, 277.6)--(216.3, 277.6)--(216.2, 277.6)--(216.4, 277.4)--(216.1, 276.9)--(216.0, 276.4)--(215.6, 275.4)--(215.5, 275.1)--(215.3, 274.1)--(215.1, 273.3)--(215.1, 272.8)--(215.1, 272.3)--(215.2, 272.3)--(215.2, 272.2)--(215.1, 272.2)--(215.3, 271.1)--(215.4, 270.0)--(215.4, 269.6)--(215.4, 269.3)--(215.3, 269.1)--(215.3, 269.0)--(215.2, 268.3)--(215.0, 267.4)--(214.7, 266.5)--(214.6, 266.1)--(214.2, 265.2)--(213.9, 264.5)--(213.8, 264.4)--(213.4, 263.6)--(213.4, 263.5)--(213.1, 263.2)--(213.0, 263.3)--(212.8, 263.1)--(212.8, 263.0)--(212.8, 262.9)--(211.8, 262.2)--(211.4, 262.0)--(210.9, 261.8)--(210.7, 261.8)--(210.3, 261.6)--(210.0, 261.5)--(209.7, 261.5)--(209.5, 261.4)--(209.4, 261.4)--(209.1, 261.4)--(208.3, 261.4)--(207.8, 261.4)--(206.9, 261.5)--(206.3, 261.7)--(206.0, 261.7)--(205.9, 261.8)--(205.8, 261.8)--(205.6, 261.9)--(204.9, 262.3)--(204.3, 262.7)--(203.8, 263.1)--(203.5, 263.4)--(203.4, 263.5)--(203.3, 263.6)--(203.2, 263.6)--(203.1, 263.8)--(203.0, 263.8)--(202.9, 264.0)--(202.5, 264.4)--(202.0, 264.9)--(202.0, 265.0)--(201.7, 265.5)--(201.6, 265.4)--(201.6, 265.5)--(201.5, 265.6)--(201.6, 265.7)--(201.5, 265.8)--(201.4, 265.7)--(201.2, 265.9)--(201.0, 266.4)--(200.4, 267.2)--(200.0, 268.1)--(199.8, 268.4)--(199.8, 268.7)--(199.7, 268.7)--(199.5, 269.4)--(199.4, 269.8)--(199.3, 270.3)--(198.8, 271.4)--(198.4, 272.4)--(198.5, 272.5)--(198.3, 273.0)--(198.2, 273.0)--(198.0, 273.4)--(197.4, 274.4)--(197.4, 274.5)--(197.3, 274.7)--(197.2, 274.8)--(197.1, 274.8)--(197.0, 275.0)--(196.6, 275.5)--(196.1, 275.9)--(195.7, 276.4)--(195.0, 276.9)--(194.4, 277.5)--(194.2, 277.6)--(194.1, 277.7)--(194.0, 277.7)--(193.9, 277.8)--(193.8, 277.8)--(192.7, 278.4)--(192.6, 278.4)--(192.2, 278.6)--(191.9, 278.7)--(191.3, 278.8)--(190.7, 279.0)--(190.7, 279.1)--(190.3, 279.2)--(190.2, 279.3)--(190.2, 279.1)--(190.1, 279.1)--(190.0, 279.2)--(189.9, 279.2)--(189.4, 279.2)--(189.1, 279.2)--(188.9, 279.2)--(188.4, 279.1)--(188.0, 279.0)--(187.3, 279.0)--(187.1, 279.0)--(186.9, 278.9)--(186.5, 278.8)--(186.3, 278.8)--(185.5, 278.4)--(185.2, 278.2)--(184.4, 277.9)--(183.9, 277.5)--(182.5, 276.6)--(182.1, 276.3)--(181.8, 276.0)--(181.5, 275.5)--(181.2, 275.1)--(181.1, 274.8)--(181.1, 274.7)--(181.3, 275.0)--(181.4, 275.0)--(181.7, 274.4)--(181.6, 274.3)--(181.6, 274.1)--(181.0, 274.6)--(180.7, 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238.6)--(81.6, 238.5)--(82.1, 238.5)--(82.7, 239.8)--(83.1, 240.6)--(83.5, 241.7)--(83.7, 242.2)--(83.8, 242.4)--(84.3, 242.3)--(84.6, 242.3)--(85.0, 242.3)--(85.1, 242.3)--(85.6, 242.3)--(85.9, 242.4)--(86.7, 242.5)--(87.1, 242.6)--(87.5, 242.6)--(87.9, 242.6)--(88.1, 242.6)--(88.3, 242.6)--(88.5, 242.9)--(87.8, 243.2)--(87.1, 243.5)--(87.3, 244.0)--(87.3, 244.3)--(87.4, 244.5)--(87.6, 244.9)--(87.8, 245.2)--(87.8, 245.4)--(87.9, 245.7)--(88.2, 246.1)--(88.3, 246.4)--(88.5, 246.9)--(88.7, 247.2)--(88.9, 247.5)--(89.0, 247.7)--(89.2, 248.0)--(89.8, 248.5)--(90.5, 249.0)--(91.2, 249.5)--(91.7, 249.8)--(91.9, 250.0)--(92.7, 250.6)--(93.6, 251.2)--(94.5, 251.8)--(95.5, 252.5)--(96.1, 252.9)--(96.8, 253.5)--(97.0, 253.6)--(97.1, 253.6)--(97.3, 253.7)--(97.5, 253.8)--(97.8, 253.9)--(97.9, 253.9)--(98.1, 253.8)--(98.3, 253.8)--(98.3, 253.9)--(98.3, 254.1)--(98.4, 254.2)--(98.7, 254.4)--(99.3, 254.7)--(100.0, 255.1)--(100.2, 255.2)--(101.1, 255.7)--(102.3, 256.3)--(103.4, 256.8)--(104.5, 257.4)--(104.6, 257.4)--(104.5, 257.8)--(104.4, 258.1)--(104.2, 258.3)--(103.9, 258.6)--(103.3, 259.0)--(103.3, 259.1)--(103.2, 259.3)--(103.1, 259.4)--(103.0, 259.7)--(102.9, 260.0)--(102.8, 260.1)--(102.7, 260.2)--(102.6, 260.3)--(102.5, 260.3)--(102.3, 260.5)--(102.2, 260.7)--(102.2, 260.8)--(102.1, 261.1)--(102.0, 261.3)--(101.9, 261.3)--(101.9, 261.4)--(101.8, 261.4)--(101.5, 261.5)--(101.4, 261.6)--(101.3, 261.7)--(101.2, 261.8)--(101.2, 261.9)--(101.2, 262.0)--(101.2, 262.4)--(101.3, 262.7)--(101.3, 262.9)--(101.4, 263.3)--(101.8, 264.2)--(101.7, 265.0)--(101.7, 265.5)--(101.5, 265.9)--(101.5, 266.0)--(101.5, 266.1)--(101.3, 267.0)--(101.1, 267.3)--(101.0, 267.5)--(100.9, 267.8)--(100.9, 267.9)--(100.7, 268.5)--(100.7, 268.6)--(100.6, 268.7)--(100.5, 268.9)--(100.4, 269.1)--(100.4, 269.3)--(100.4, 269.6)--(100.4, 269.9)--(100.5, 270.3)--(100.5, 270.8)--(100.6, 270.9)--(100.6, 271.0)--(100.5, 271.0)--(100.5, 271.1)--(100.4, 271.1)--(100.4, 271.2)--(100.3, 271.2)--(100.3, 271.3)--(100.3, 271.4)--(100.3, 271.5)--(100.4, 271.6)--(100.6, 271.8)--(100.7, 271.9)--(100.7, 272.0)--(100.8, 272.1)--(100.8, 272.6)--(100.9, 272.7)--(100.9, 272.9)--(101.1, 273.2)--(101.4, 274.0)--(101.7, 274.9)--(102.2, 275.4)--(102.2, 275.5)--(102.1, 275.5)--(101.9, 275.5)--(101.8, 275.5)--(101.7, 275.6)--(101.8, 275.8)--(101.8, 275.9)--(102.0, 276.2)--(102.2, 276.6)--(102.2, 276.7)--(102.3, 276.7)--(102.4, 276.7)--(102.5, 276.7)--(102.6, 276.7)--(102.6, 276.8)--(102.6, 276.9)--(102.6, 277.0)--(102.6, 277.1)--(102.7, 277.2)--(102.7, 277.3)--(102.8, 277.4)--(102.8, 277.6)--(102.8, 277.7)--(102.8, 277.8)--(102.8, 277.9)--(102.9, 277.9)--(102.9, 278.0)--(103.0, 278.1)--(103.1, 278.1)--(103.1, 278.2)--(103.1, 278.3)--(103.2, 279.0)--(103.5, 282.1)--(103.5, 282.2)--(103.5, 282.5)--(103.6, 284.0)--(103.7, 284.3)--(103.8, 285.3)--(103.8, 285.6)--(103.9, 285.8)--(104.0, 286.0)--(104.1, 286.1)--(104.3, 286.4)--(104.3, 286.5)--(104.3, 286.6)--(104.3, 286.7)--(104.3, 286.8)--(104.3, 287.0)--(104.3, 287.2)--(104.1, 288.1)--(104.1, 288.2)--(104.1, 288.3)--(104.1, 288.4)--(104.1, 288.5)--(104.2, 288.9)--(104.9, 290.9)--(105.1, 291.3)--(105.8, 291.1)--(106.2, 290.9)--(106.5, 290.7)--(106.7, 290.6)--(106.8, 290.6)--(107.0, 290.6)--(108.2, 290.1)--(109.2, 289.7)--(109.7, 289.5)--(110.5, 289.2)--(111.5, 288.8)--(111.5, 288.9)--(111.6, 288.8)--(112.6, 288.4)--(113.1, 288.2)--(114.6, 287.6)--(115.8, 287.1)--(117.0, 286.6)--(118.1, 286.2)--(119.1, 285.9)--(119.9, 285.6)--(120.3, 285.6)--(120.7, 285.4)--(121.4, 285.3)--(121.6, 285.3)--(121.8, 285.4)--(122.0, 285.4)--(122.3, 285.4)--(122.5, 285.4)--(122.9, 285.5)--(123.0, 285.5)--(123.1, 285.6)--(123.3, 285.6)--(123.5, 285.7)--(123.7, 285.8)--(123.9, 285.9)--(124.1, 286.0)--(124.3, 286.1)--(124.7, 286.3)--(125.5, 286.8)--(125.8, 286.9)--(126.1, 287.1)--(126.1, 287.2)--(126.3, 287.3)--(126.6, 287.4)--(127.0, 287.7)--(127.0, 287.6)--(127.2, 287.6)--(127.1, 287.5)--(127.1, 287.4)--(127.1, 287.2)--(127.1, 287.0)--(127.0, 286.4)--(127.5, 286.2)--(127.6, 286.2)--(127.8, 286.2)--(128.0, 286.3)--(128.5, 286.3)--(128.6, 285.8)--(128.7, 285.4)--(128.9, 285.1)--(129.0, 284.9)--(129.1, 284.7)--(129.2, 284.3)--(129.3, 284.0)--(129.8, 282.9)--(129.9, 282.5)--(130.0, 282.5)--(130.1, 282.4)--(130.2, 282.4)--(130.3, 282.2)--(130.4, 282.1)--(130.5, 282.0)--(130.7, 281.8)--(130.8, 281.6)--(130.9, 281.5)--(131.0, 281.5)--(131.1, 281.5)--(131.3, 281.5)--(131.3, 281.4)--(131.4, 281.3)--(131.4, 281.1)--(131.3, 281.0)--(131.3, 280.9)--(131.8, 281.2)--(132.7, 281.7)--(133.0, 281.9)--(133.5, 282.2)--(134.0, 282.5)--(133.8, 282.8)--(133.7, 283.0)--(133.8, 283.0)--(133.9, 283.1)--(134.1, 283.2)--(134.4, 283.4)--(134.6, 283.5)--(134.7, 283.6)--(134.7, 283.7)--(134.8, 283.7)--(134.8, 283.8)--(134.9, 283.9)--(135.0, 284.1)--(135.3, 284.4)--(135.5, 284.6)--(135.8, 284.2)--(136.0, 284.6)--(136.2, 284.9)--(136.3, 285.1)--(136.4, 285.3)--(136.6, 285.5)--(136.8, 285.6)--(136.9, 285.8)--(136.9, 286.1)--(136.9, 286.6)--(137.0, 286.6)--(137.0, 286.7)--(137.2, 286.8)--(137.2, 286.9)--(137.2, 287.0)--(137.4, 287.1)--(137.4, 287.2)--(137.5, 287.1)--(137.6, 287.2)--(137.7, 287.2)--(137.8, 287.2)--(137.9, 287.3)--(138.1, 287.3)--(138.2, 287.4)--(138.2, 287.5)--(138.3, 287.5)--(138.3, 287.6)--(138.4, 287.6)--(138.6, 287.8)--(138.8, 287.8)--(138.9, 287.9)--(139.0, 288.0)--(139.3, 288.1)--(139.7, 288.4)--(140.0, 288.6)--(140.3, 288.9)--(140.4, 289.1)--(140.5, 289.1)--(140.7, 289.2)--(140.8, 289.3)--(141.1, 289.3)--(141.3, 289.3)--(141.5, 289.4)--(141.7, 289.5)--(141.9, 289.5)--(142.0, 289.6)--(142.1, 289.6)--(142.1, 289.7)--(142.3, 289.7)--(142.4, 289.7)--(142.5, 289.7)--(142.7, 289.8)--(142.9, 289.9)--(143.0, 289.9)--(143.1, 289.8)--(143.1, 289.9)--(143.2, 289.9)--(144.0, 290.2)--(144.7, 290.5)--(145.6, 290.8)--(145.8, 290.9)--(145.9, 290.9)--(146.0, 290.9)--(146.2, 290.9)--(146.3, 290.9)--(146.3, 290.8)--(146.4, 290.8)--(146.5, 290.7)--(146.7, 290.6)--(147.5, 290.0)--(148.2, 289.6)--(149.0, 289.2)--(150.0, 288.6)--(150.9, 288.1)--(151.8, 287.6)--(152.3, 287.7)--(152.4, 287.6)--(153.1, 287.7)--(153.8, 287.8)--(154.3, 287.9)--(155.0, 287.9)--(155.5, 287.9)--(156.1, 287.9)--(156.5, 287.9)--(156.6, 287.9)--(157.0, 287.9)--(157.0, 287.8)--(157.4, 287.8)--(158.0, 287.6)--(158.4, 287.5)--(158.8, 287.4)--(159.0, 287.3)--(159.0, 287.4)--(159.4, 287.2)--(159.5, 287.2)--(159.6, 287.1)--(159.9, 287.0)--(160.3, 286.7)--(162.0, 287.8)--(162.4, 287.4)--(162.9, 286.7)--(163.3, 287.1)--(163.8, 287.7)--(163.9, 287.8)--(164.4, 287.6)--(164.8, 287.7)--(165.5, 287.3)--(166.2, 287.0)--(166.6, 286.8)--(166.9, 286.6)--(167.3, 286.3)--(167.8, 285.9)--(168.2, 286.2)--(168.5, 286.5)--(169.1, 287.1)--(170.1, 287.8)--(170.4, 287.6)--(170.4, 287.5)--(170.5, 287.5)--(170.2, 287.3)--(170.2, 287.2)--(170.9, 286.4)--(171.0, 286.2)--(171.1, 286.2)--(170.8, 285.8)--(170.7, 285.7)--(170.6, 285.7)--(170.8, 285.5)--(170.9, 285.4)--(171.1, 285.1)--(171.3, 285.3)--(171.4, 285.3)--(171.4, 285.2)--(171.5, 285.3)--(171.7, 285.1)--(171.9, 284.8)--(172.3, 284.3)--(172.4, 284.2)--(172.5, 284.1)--(172.9, 283.6)--(172.9, 283.5)--(173.6, 284.1)--(174.4, 284.7)--(175.0, 285.2)--(175.3, 285.4)--(175.6, 285.5)--(175.9, 285.7)--(176.6, 286.1)--(177.3, 286.5)--(177.2, 287.0)--(177.1, 287.4)--(177.0, 287.7)--(177.0, 288.0)--(177.0, 288.2)--(177.0, 288.3)--(177.2, 288.3)--(177.3, 288.2)--(177.4, 288.1)--(177.9, 288.0)--(178.4, 287.8)--(178.8, 287.7)--(179.2, 287.6)--(179.5, 287.5)--(180.2, 287.3)--(180.2, 287.2)--(180.2, 287.0)--(180.2, 286.8)--(180.4, 286.4)--(181.5, 286.7)--(182.3, 286.9)--(182.8, 287.0)--(182.4, 287.7)--(182.5, 287.7)--(182.3, 288.0)--(182.2, 288.2)--(181.9, 288.7)--(181.8, 288.9)--(181.7, 289.1)--(181.5, 289.5)--(181.4, 289.8)--(181.3, 289.9)--(181.2, 290.1)--(181.1, 290.2)--(181.0, 290.3)--(181.0, 290.6)--(181.0, 290.8)--(180.9, 290.9)--(180.8, 291.0)--(180.8, 291.1)--(180.7, 291.1)--(180.6, 291.3)--(180.5, 291.8)--(180.5, 291.9)--(180.9, 292.2)--(181.1, 292.3)--(182.0, 292.7)--(183.2, 293.4)--(183.6, 293.7)--(184.0, 293.9)--(184.4, 294.1)--(184.9, 294.4)--(185.0, 294.5)--(185.1, 294.5)--(185.2, 294.5)--(185.4, 294.5)--(185.5, 294.5)--(185.8, 294.5)--(186.5, 294.4)--(186.9, 294.3)--(187.4, 294.2)--(187.9, 294.0)--(188.4, 293.9)--(188.8, 293.8)--(189.2, 293.8)--(189.4, 293.7)--(190.1, 293.6)--(190.6, 293.7)--(190.9, 293.8)--(191.2, 294.0)--(191.3, 294.0)--(191.5, 294.1)--(192.1, 294.4)--(192.6, 294.8)--(193.1, 295.1)--(193.2, 295.3)--(194.1, 296.0)--(194.2, 296.1)--(194.3, 296.1)--(194.6, 295.7)--(194.9, 295.3)--(195.0, 295.1)--(195.1, 295.0)--(195.2, 294.8)--(195.4, 294.4)--(195.6, 293.9)--(195.7, 293.7)--(195.8, 293.5)--(195.9, 293.2)--(196.0, 293.0)--(196.1, 292.8)--(196.2, 292.6)--(196.3, 292.4)--(196.8, 291.9)--(197.2, 291.4)--(197.5, 291.1)--(197.9, 290.8)--(198.9, 290.1)--(199.3, 289.7)--(199.6, 289.5)--(199.8, 289.2)--(200.0, 289.0)--(200.2, 288.8)--(200.3, 288.3)--(200.4, 288.0)--(200.5, 287.8)--(200.7, 287.6)--(200.9, 287.4)--(200.9, 287.3)--(201.0, 287.3)--(201.2, 287.1)--(201.4, 287.0)--(201.9, 286.7)--(202.4, 286.6)--(202.7, 286.5)--(203.1, 286.5)--(203.3, 286.5)--(203.5, 286.5)--(203.6, 286.4)--(204.3, 286.6)--(205.0, 286.7)--(205.8, 286.9)--(206.3, 287.0)--(207.0, 287.1)--(207.3, 287.2)--(207.6, 287.2)--(207.6, 287.3)--(207.7, 287.3)--(207.9, 287.3)--(208.0, 287.3)--(208.6, 287.4)--(208.8, 287.4)--(209.0, 287.5)--(209.9, 287.6)--(210.6, 287.7)--(211.4, 287.7)--(211.7, 287.7)--(211.9, 287.7)--(212.0, 287.7)--(212.0, 288.4)--(212.1, 288.6)--(212.0, 288.7)--(212.0, 288.9)--(211.9, 288.9)--(211.9, 289.0)--(212.0, 289.0)--(212.0, 289.2)--(212.1, 289.3)--(212.2, 289.3)--(212.2, 289.5)--(212.2, 289.6)--(212.3, 289.8)--(212.3, 289.9)--(212.3, 290.0)--(212.5, 290.5)--(212.5, 290.6)--(212.8, 291.1)--(213.0, 291.6)--(213.2, 292.1)--(213.4, 292.3)--(213.5, 292.8)--(213.7, 293.1)--(213.8, 293.3)--(213.8, 293.5)--(213.8, 293.7)--(213.7, 293.8)--(213.7, 294.0)--(213.6, 294.1)--(213.6, 294.3)--(213.6, 294.6)--(213.5, 294.9)--(213.5, 295.0)--(213.7, 294.8)--(213.7, 294.7)--(214.0, 294.4)--(214.2, 294.2)--(214.4, 293.9)--(214.5, 293.7)--(214.7, 293.3)--(214.8, 293.3)--(214.8, 293.2)--(215.0, 292.9)--(214.9, 292.8)--(215.0, 292.7)--(215.2, 292.4)--(215.7, 291.2)--(216.2, 290.4)--(216.3, 290.1)--(216.5, 290.2)--(216.7, 290.3)--(217.2, 290.6)--(217.3, 290.7)--(217.5, 290.9)--(217.6, 291.0)--(217.9, 291.1)--(218.3, 291.3)--(218.5, 291.5)--(218.9, 291.7)--(219.0, 291.8)--(219.1, 291.5)--(219.2, 291.2)--(219.5, 291.3)--(219.6, 291.3)--(219.5, 290.9)--(219.4, 290.6)--(219.5, 290.0)--(219.5, 289.8)--(219.4, 289.7)--(219.4, 289.5)--(219.4, 289.2)--(219.6, 289.2)--(219.5, 288.9)--(219.5, 288.7)--(219.4, 288.6)--(219.4, 288.4)--(219.4, 288.0)--(219.3, 287.6)--(219.3, 287.2)--(219.1, 285.8)--(219.1, 285.5)--(219.1, 284.7)--(219.1, 284.3)--(219.1, 284.2)--(219.0, 283.3)--(219.0, 282.7)--(219.0, 282.6)--(219.0, 282.4)--(219.1, 282.3)--(219.2, 281.9)--(219.1, 281.8)--(219.2, 281.6)--(219.4, 281.4)--(219.6, 281.4)--(219.7, 281.1)--cycle; +Islington = (291.5, 358.6)--(291.4, 358.8)--(290.9, 359.9)--(290.8, 360.1)--(290.6, 360.5)--(290.0, 361.8)--(289.8, 362.0)--(289.8, 362.1)--(289.7, 362.2)--(289.7, 362.3)--(289.6, 362.5)--(289.6, 362.6)--(289.6, 362.8)--(289.6, 363.2)--(289.6, 363.4)--(289.6, 363.6)--(289.6, 363.7)--(289.7, 364.2)--(289.8, 364.4)--(289.8, 364.5)--(289.8, 364.6)--(289.8, 364.9)--(289.8, 365.2)--(289.8, 365.5)--(289.7, 365.7)--(289.7, 365.8)--(289.7, 366.0)--(289.7, 366.1)--(289.7, 366.3)--(289.7, 366.8)--(289.7, 366.9)--(289.7, 367.0)--(289.7, 367.1)--(289.7, 367.2)--(289.5, 367.8)--(289.4, 368.0)--(289.4, 368.1)--(289.3, 368.4)--(289.3, 368.9)--(289.1, 370.1)--(289.0, 370.6)--(288.8, 371.3)--(288.5, 371.7)--(288.3, 372.0)--(288.3, 372.1)--(288.4, 372.2)--(288.6, 372.2)--(289.0, 372.3)--(289.4, 372.5)--(289.6, 372.5)--(289.7, 372.6)--(290.4, 373.2)--(290.7, 373.4)--(290.8, 373.6)--(291.0, 373.7)--(291.1, 373.8)--(291.1, 373.9)--(291.2, 373.9)--(291.4, 374.0)--(291.7, 374.3)--(291.9, 374.4)--(292.2, 374.6)--(292.3, 374.7)--(292.4, 374.7)--(292.7, 375.0)--(292.8, 375.1)--(293.0, 375.2)--(293.3, 375.4)--(294.0, 376.0)--(294.5, 376.5)--(294.7, 376.7)--(294.8, 376.7)--(295.0, 376.8)--(295.3, 376.9)--(295.6, 377.1)--(295.7, 377.2)--(295.8, 377.2)--(295.9, 377.3)--(296.0, 377.4)--(296.2, 377.4)--(296.5, 377.5)--(297.0, 377.6)--(297.3, 377.7)--(297.7, 377.8)--(298.6, 378.0)--(298.9, 378.0)--(299.1, 378.0)--(299.2, 378.0)--(299.3, 378.0)--(299.5, 377.9)--(300.0, 378.0)--(300.3, 378.0)--(300.4, 378.0)--(300.5, 378.1)--(300.6, 378.1)--(300.6, 378.2)--(300.6, 378.3)--(300.7, 378.3)--(300.8, 378.3)--(300.9, 378.4)--(301.0, 378.5)--(301.1, 378.6)--(301.2, 378.7)--(301.3, 378.7)--(301.4, 378.7)--(301.5, 378.7)--(302.1, 379.0)--(302.5, 379.1)--(302.5, 379.2)--(302.5, 379.1)--(303.8, 379.6)--(304.0, 379.7)--(304.1, 379.7)--(304.2, 379.4)--(304.3, 379.1)--(304.4, 379.0)--(304.5, 378.8)--(304.1, 378.5)--(304.1, 378.4)--(304.4, 378.4)--(304.6, 378.4)--(305.0, 378.3)--(305.3, 378.2)--(305.5, 378.2)--(305.6, 378.1)--(305.8, 377.9)--(306.2, 377.4)--(306.7, 376.8)--(306.9, 376.5)--(306.5, 376.1)--(306.6, 376.0)--(306.8, 375.9)--(307.0, 375.7)--(306.9, 375.4)--(307.0, 375.4)--(307.2, 375.2)--(307.2, 375.1)--(307.2, 374.6)--(307.2, 374.4)--(307.3, 374.4)--(307.3, 374.3)--(309.7, 372.7)--(310.4, 372.2)--(310.7, 372.0)--(311.4, 371.3)--(311.6, 371.1)--(311.7, 371.0)--(311.9, 370.8)--(312.0, 370.7)--(312.1, 370.6)--(312.4, 370.2)--(312.5, 370.1)--(312.6, 369.9)--(312.8, 369.7)--(313.0, 369.5)--(313.1, 369.4)--(313.3, 369.2)--(313.6, 368.8)--(314.6, 367.9)--(314.7, 368.0)--(314.8, 368.1)--(314.9, 368.0)--(314.8, 367.9)--(315.3, 367.5)--(316.5, 366.4)--(316.6, 366.2)--(316.9, 365.9)--(317.0, 365.8)--(317.1, 365.7)--(317.2, 365.6)--(317.3, 365.6)--(317.3, 365.5)--(317.4, 365.5)--(318.4, 364.5)--(319.0, 363.7)--(320.8, 364.2)--(321.0, 364.2)--(321.6, 364.0)--(321.8, 363.9)--(322.0, 363.8)--(322.4, 363.5)--(322.5, 363.5)--(322.7, 363.5)--(322.8, 363.4)--(322.9, 363.4)--(323.1, 363.4)--(323.4, 363.3)--(323.5, 363.2)--(323.8, 363.2)--(323.9, 363.1)--(323.9, 363.0)--(323.9, 362.9)--(324.1, 362.5)--(324.3, 362.1)--(324.4, 361.8)--(324.5, 361.6)--(324.7, 361.1)--(324.8, 360.7)--(324.8, 360.5)--(324.9, 360.5)--(324.8, 360.4)--(324.9, 359.3)--(324.9, 358.3)--(324.9, 357.8)--(324.9, 357.3)--(324.9, 356.8)--(324.9, 356.5)--(325.0, 356.5)--(325.1, 356.5)--(325.3, 356.4)--(325.3, 356.3)--(325.4, 356.3)--(325.5, 356.2)--(325.7, 356.0)--(325.8, 356.1)--(326.1, 356.3)--(326.8, 355.2)--(326.8, 355.1)--(326.9, 355.0)--(327.0, 355.0)--(327.3, 354.9)--(327.4, 354.8)--(327.6, 354.8)--(327.7, 354.7)--(327.8, 354.7)--(327.9, 354.6)--(328.2, 354.4)--(328.3, 354.3)--(328.4, 354.3)--(328.5, 354.3)--(328.6, 354.3)--(328.9, 354.2)--(329.0, 354.2)--(329.1, 354.3)--(329.6, 354.3)--(329.9, 354.4)--(331.4, 354.2)--(332.0, 354.2)--(332.2, 354.1)--(332.4, 354.1)--(332.5, 354.0)--(332.5, 353.9)--(332.6, 353.8)--(332.8, 353.5)--(333.0, 353.3)--(333.0, 353.2)--(333.1, 353.1)--(333.7, 351.7)--(333.8, 351.4)--(334.3, 350.6)--(334.4, 350.3)--(334.5, 350.0)--(334.6, 349.8)--(334.8, 349.1)--(334.8, 348.7)--(334.8, 348.4)--(334.8, 348.1)--(334.8, 348.0)--(334.5, 347.7)--(333.5, 347.9)--(332.5, 348.0)--(330.1, 348.2)--(329.7, 348.2)--(329.8, 346.7)--(329.8, 346.5)--(329.6, 345.2)--(329.2, 343.5)--(328.7, 341.2)--(328.6, 340.6)--(328.4, 339.4)--(328.1, 338.3)--(328.1, 338.1)--(328.0, 337.9)--(328.0, 337.7)--(327.9, 337.8)--(327.7, 337.7)--(327.5, 337.6)--(327.3, 337.5)--(326.9, 337.2)--(326.6, 336.9)--(326.3, 336.6)--(325.9, 336.3)--(325.5, 336.0)--(325.2, 335.7)--(324.8, 335.4)--(324.6, 335.3)--(324.3, 335.1)--(324.0, 334.8)--(323.5, 334.4)--(323.1, 334.1)--(323.0, 334.0)--(322.6, 333.8)--(322.6, 333.9)--(322.3, 333.6)--(321.6, 333.2)--(321.5, 333.1)--(321.2, 333.0)--(320.9, 332.8)--(321.5, 331.6)--(321.7, 331.0)--(322.0, 330.4)--(322.2, 330.0)--(322.3, 329.6)--(322.3, 329.4)--(322.3, 329.3)--(322.3, 329.1)--(322.1, 328.6)--(326.2, 327.1)--(326.6, 326.9)--(326.8, 326.8)--(326.9, 326.7)--(327.0, 326.6)--(327.1, 326.4)--(327.3, 326.1)--(327.5, 325.5)--(327.5, 325.3)--(327.8, 325.2)--(327.9, 325.1)--(328.0, 325.0)--(328.2, 325.0)--(328.4, 325.1)--(328.5, 325.1)--(329.4, 325.5)--(329.5, 325.1)--(329.6, 324.6)--(329.4, 324.4)--(329.3, 324.0)--(329.2, 323.7)--(329.2, 323.6)--(329.4, 323.5)--(329.4, 322.9)--(329.8, 322.9)--(330.0, 322.9)--(330.0, 322.6)--(330.0, 322.4)--(329.9, 321.4)--(329.9, 320.9)--(329.8, 320.4)--(329.7, 319.7)--(329.6, 319.3)--(329.5, 318.9)--(329.4, 318.7)--(329.2, 318.4)--(329.1, 317.9)--(329.0, 317.8)--(328.8, 317.2)--(328.7, 317.3)--(327.9, 317.6)--(327.5, 317.9)--(326.6, 318.3)--(326.3, 318.4)--(326.2, 318.5)--(326.3, 319.3)--(325.9, 319.4)--(325.5, 319.5)--(325.2, 319.5)--(324.9, 319.5)--(324.6, 319.6)--(324.5, 319.6)--(324.4, 320.1)--(323.8, 319.9)--(323.7, 320.1)--(323.6, 320.1)--(323.6, 320.2)--(323.4, 320.1)--(323.1, 320.1)--(323.0, 320.4)--(322.9, 321.0)--(322.5, 321.9)--(321.9, 321.6)--(321.9, 321.7)--(321.6, 321.5)--(321.5, 321.9)--(321.4, 321.9)--(321.3, 322.1)--(320.5, 321.5)--(320.6, 321.5)--(320.7, 321.3)--(320.7, 321.2)--(320.8, 321.1)--(320.9, 320.8)--(321.0, 320.4)--(321.0, 320.3)--(321.0, 320.2)--(321.0, 320.1)--(320.9, 320.1)--(320.9, 319.8)--(320.8, 319.5)--(320.8, 319.2)--(320.3, 319.0)--(320.0, 318.9)--(319.7, 318.7)--(319.4, 318.6)--(319.1, 318.5)--(319.0, 318.5)--(319.0, 318.4)--(318.9, 318.4)--(318.7, 318.3)--(318.5, 318.2)--(318.1, 318.0)--(318.0, 317.9)--(317.8, 317.8)--(317.0, 317.4)--(316.0, 316.8)--(315.6, 316.6)--(315.5, 316.6)--(315.4, 317.3)--(315.1, 318.5)--(315.0, 319.0)--(314.9, 319.2)--(314.8, 319.6)--(314.7, 319.9)--(314.6, 320.1)--(314.3, 320.6)--(313.4, 320.5)--(313.2, 320.9)--(313.1, 321.1)--(312.9, 321.5)--(312.7, 321.4)--(312.6, 321.5)--(312.5, 321.6)--(312.3, 321.6)--(312.2, 321.6)--(311.8, 321.8)--(311.3, 321.9)--(310.9, 322.1)--(310.7, 322.1)--(310.5, 322.2)--(310.4, 322.3)--(309.9, 322.8)--(309.7, 323.0)--(309.6, 323.1)--(309.6, 323.2)--(309.4, 323.5)--(309.3, 323.9)--(309.2, 324.1)--(309.2, 324.2)--(309.5, 324.6)--(309.9, 325.0)--(310.0, 325.2)--(309.7, 325.3)--(309.6, 325.4)--(309.5, 325.4)--(309.4, 325.5)--(309.0, 325.9)--(308.8, 326.1)--(308.4, 326.5)--(308.3, 326.6)--(308.3, 326.7)--(308.2, 326.8)--(308.2, 326.9)--(308.1, 327.3)--(308.1, 327.7)--(308.0, 328.5)--(308.0, 328.6)--(307.9, 328.7)--(307.9, 328.8)--(307.9, 328.9)--(307.8, 329.0)--(307.5, 329.2)--(307.1, 329.4)--(306.7, 329.6)--(306.3, 329.7)--(305.7, 329.9)--(305.1, 330.1)--(304.2, 330.0)--(303.6, 329.9)--(303.3, 329.8)--(303.3, 329.9)--(303.2, 330.3)--(303.3, 332.1)--(303.3, 333.5)--(303.3, 334.1)--(303.3, 335.2)--(303.3, 335.5)--(303.3, 335.8)--(303.2, 336.0)--(303.2, 336.7)--(303.1, 337.2)--(303.0, 337.5)--(302.9, 337.8)--(302.7, 338.5)--(302.6, 338.6)--(302.5, 338.9)--(302.2, 339.3)--(302.0, 339.6)--(301.8, 339.9)--(301.4, 340.8)--(301.3, 341.1)--(301.2, 341.3)--(301.2, 341.4)--(301.1, 341.7)--(300.9, 342.0)--(300.8, 342.4)--(300.7, 342.8)--(300.6, 343.1)--(300.6, 343.3)--(300.6, 343.5)--(300.6, 343.6)--(300.5, 344.0)--(300.4, 344.7)--(300.2, 345.7)--(300.2, 345.8)--(300.1, 346.3)--(300.0, 346.9)--(299.8, 347.8)--(299.6, 348.4)--(299.6, 348.5)--(299.6, 348.6)--(299.5, 348.7)--(299.3, 349.0)--(299.2, 349.1)--(299.1, 349.1)--(299.0, 349.3)--(298.8, 349.4)--(298.2, 349.9)--(298.0, 350.0)--(297.7, 350.4)--(297.5, 350.6)--(297.4, 350.7)--(297.3, 351.0)--(297.0, 351.9)--(297.0, 352.1)--(297.0, 352.5)--(296.9, 352.5)--(296.9, 352.6)--(296.9, 352.7)--(296.8, 352.8)--(296.6, 352.9)--(296.6, 353.0)--(296.2, 353.3)--(296.0, 353.5)--(295.2, 354.2)--(294.7, 354.8)--(294.4, 355.1)--(294.2, 355.3)--(294.2, 355.4)--(294.0, 355.7)--(293.9, 356.0)--(293.8, 356.1)--(293.7, 356.2)--(293.6, 356.3)--(293.3, 356.4)--(292.9, 356.6)--(292.5, 356.9)--(292.4, 357.0)--(292.4, 357.1)--(292.3, 357.2)--(292.2, 357.3)--cycle; +Kensington_and_Chelsea = (262.2, 269.5)--(262.1, 269.6)--(261.9, 269.9)--(261.7, 270.1)--(261.6, 270.2)--(261.1, 270.8)--(260.9, 271.0)--(260.8, 271.1)--(260.6, 271.6)--(260.6, 271.8)--(260.5, 271.9)--(260.4, 272.0)--(260.3, 272.2)--(260.2, 272.0)--(260.0, 272.2)--(259.9, 272.3)--(259.8, 272.5)--(259.7, 272.6)--(259.5, 272.8)--(259.1, 273.3)--(259.0, 273.3)--(259.0, 273.4)--(258.5, 274.1)--(258.2, 274.6)--(257.7, 275.2)--(257.6, 275.2)--(257.5, 275.4)--(257.3, 275.6)--(257.2, 275.8)--(257.1, 276.0)--(256.9, 276.2)--(256.8, 276.4)--(256.6, 276.7)--(256.5, 276.7)--(256.4, 277.0)--(256.0, 277.6)--(255.8, 277.7)--(255.7, 278.0)--(255.5, 278.2)--(255.5, 278.3)--(255.2, 278.6)--(254.9, 279.0)--(254.8, 279.1)--(254.7, 279.2)--(254.6, 279.4)--(254.4, 279.5)--(254.4, 279.6)--(254.3, 279.7)--(254.1, 279.8)--(254.0, 279.9)--(253.5, 280.3)--(253.2, 280.1)--(252.7, 280.4)--(252.4, 280.6)--(252.3, 280.7)--(252.2, 280.8)--(252.1, 281.0)--(251.9, 281.1)--(251.4, 281.8)--(251.4, 281.9)--(251.1, 282.2)--(251.0, 282.4)--(250.4, 283.2)--(250.2, 283.3)--(250.0, 283.6)--(249.4, 284.4)--(249.0, 285.0)--(249.3, 285.2)--(249.3, 285.3)--(249.0, 285.7)--(248.9, 285.9)--(248.8, 286.1)--(248.6, 286.4)--(248.5, 286.7)--(248.4, 286.8)--(248.5, 286.8)--(248.4, 286.9)--(248.2, 287.2)--(248.0, 287.3)--(247.9, 287.4)--(247.7, 287.6)--(247.5, 287.9)--(247.2, 288.2)--(246.9, 288.5)--(246.6, 288.7)--(246.3, 288.9)--(246.1, 289.0)--(245.9, 289.2)--(245.8, 289.2)--(245.8, 289.3)--(245.6, 289.5)--(245.5, 289.6)--(245.5, 289.7)--(245.4, 289.7)--(245.4, 289.8)--(245.4, 289.9)--(244.9, 289.8)--(244.8, 289.9)--(244.2, 290.6)--(243.8, 291.3)--(243.6, 291.7)--(243.5, 291.8)--(243.3, 292.1)--(242.9, 292.5)--(242.4, 292.9)--(242.3, 293.0)--(242.1, 293.2)--(241.7, 293.6)--(241.3, 293.9)--(240.6, 294.9)--(240.4, 295.3)--(240.1, 295.9)--(239.9, 296.6)--(239.6, 297.1)--(239.2, 297.8)--(239.0, 298.2)--(238.9, 298.1)--(238.9, 298.2)--(238.8, 298.4)--(238.5, 299.0)--(238.4, 299.0)--(238.2, 299.7)--(238.0, 300.0)--(238.0, 300.1)--(237.9, 300.4)--(237.7, 300.8)--(238.3, 301.0)--(238.8, 301.2)--(238.9, 301.2)--(238.9, 301.1)--(239.1, 301.2)--(239.1, 301.4)--(238.9, 301.8)--(239.0, 302.1)--(239.0, 302.3)--(239.0, 302.4)--(239.2, 302.4)--(239.5, 302.5)--(239.6, 302.6)--(239.5, 303.0)--(239.3, 303.5)--(239.2, 304.2)--(239.0, 304.2)--(238.7, 304.2)--(238.7, 304.3)--(238.6, 304.4)--(238.4, 304.5)--(238.5, 304.7)--(238.4, 304.8)--(238.4, 304.9)--(238.3, 305.1)--(238.3, 305.2)--(238.3, 305.3)--(238.2, 305.6)--(238.1, 305.6)--(238.0, 305.5)--(237.9, 305.5)--(237.9, 305.4)--(237.8, 305.4)--(237.7, 305.4)--(237.6, 305.4)--(237.7, 304.9)--(237.0, 304.7)--(236.7, 304.6)--(236.7, 304.7)--(236.6, 305.3)--(236.5, 305.5)--(236.4, 306.0)--(236.4, 306.1)--(236.3, 306.2)--(236.3, 306.3)--(236.2, 306.4)--(236.2, 306.6)--(236.1, 306.8)--(236.0, 307.0)--(236.0, 307.1)--(235.9, 307.2)--(235.9, 307.4)--(235.8, 307.6)--(235.4, 308.5)--(235.2, 308.9)--(235.1, 309.1)--(235.1, 309.3)--(235.1, 309.4)--(234.9, 309.9)--(234.7, 310.3)--(234.5, 310.7)--(234.4, 310.9)--(234.3, 311.3)--(234.2, 311.4)--(234.0, 311.6)--(233.5, 312.5)--(233.0, 313.3)--(233.1, 313.4)--(232.9, 313.5)--(232.7, 313.9)--(232.6, 314.0)--(232.5, 314.2)--(232.3, 314.5)--(232.1, 314.9)--(231.9, 315.1)--(231.6, 315.6)--(231.5, 315.5)--(231.5, 315.6)--(231.4, 315.6)--(231.4, 315.7)--(231.3, 315.8)--(230.2, 317.3)--(230.3, 317.4)--(230.3, 317.6)--(231.2, 318.3)--(231.2, 318.7)--(231.2, 321.4)--(230.8, 321.4)--(230.8, 321.8)--(230.8, 322.0)--(230.7, 322.6)--(230.7, 322.7)--(230.6, 323.0)--(230.7, 323.1)--(230.7, 324.0)--(230.7, 324.2)--(230.7, 324.4)--(230.6, 324.7)--(230.6, 324.9)--(230.6, 325.0)--(230.5, 325.2)--(230.5, 325.4)--(230.4, 325.6)--(230.3, 325.7)--(230.2, 325.9)--(230.0, 326.1)--(230.0, 326.2)--(229.9, 326.3)--(229.6, 327.1)--(229.8, 327.6)--(229.9, 327.5)--(230.0, 327.5)--(230.1, 327.4)--(230.2, 327.4)--(230.3, 327.4)--(230.4, 327.4)--(230.4, 327.1)--(231.0, 327.2)--(231.4, 327.2)--(231.9, 327.3)--(231.9, 327.4)--(232.2, 327.4)--(232.5, 327.4)--(232.7, 327.3)--(233.1, 327.2)--(233.2, 327.2)--(233.3, 327.1)--(233.6, 327.0)--(233.7, 326.9)--(233.9, 326.9)--(234.1, 326.7)--(234.4, 326.6)--(234.7, 326.5)--(235.0, 326.4)--(235.2, 326.4)--(236.1, 326.2)--(236.4, 326.1)--(236.6, 326.0)--(236.9, 325.9)--(237.1, 325.9)--(237.6, 325.7)--(237.8, 325.6)--(238.1, 325.4)--(238.2, 325.3)--(238.5, 325.1)--(238.7, 325.0)--(238.6, 324.8)--(238.6, 324.7)--(238.6, 324.6)--(238.6, 324.5)--(238.7, 324.4)--(238.8, 324.0)--(238.9, 323.8)--(239.2, 323.8)--(239.5, 323.9)--(240.1, 324.0)--(240.4, 324.0)--(240.8, 324.0)--(241.4, 323.9)--(241.8, 323.9)--(242.5, 323.8)--(242.7, 323.8)--(243.0, 323.7)--(243.3, 323.6)--(243.5, 323.6)--(244.0, 323.4)--(244.1, 323.4)--(244.2, 323.4)--(244.4, 323.4)--(244.6, 323.4)--(244.8, 323.3)--(245.0, 323.3)--(245.2, 323.2)--(245.4, 323.1)--(245.6, 322.8)--(245.7, 322.6)--(245.9, 322.4)--(246.1, 321.9)--(246.2, 321.7)--(246.6, 321.0)--(246.7, 320.8)--(246.9, 320.5)--(247.1, 320.3)--(247.3, 320.2)--(247.7, 320.0)--(247.8, 319.9)--(247.9, 319.9)--(248.0, 319.9)--(248.6, 319.6)--(248.6, 319.4)--(248.7, 319.1)--(248.7, 318.8)--(248.7, 318.7)--(248.8, 318.7)--(249.0, 318.2)--(249.4, 317.4)--(249.4, 317.3)--(249.3, 317.3)--(249.3, 317.5)--(249.0, 317.5)--(248.7, 317.5)--(248.6, 317.5)--(248.3, 317.5)--(248.0, 317.5)--(247.7, 317.4)--(247.4, 317.3)--(247.2, 317.2)--(247.2, 317.1)--(247.3, 317.0)--(247.4, 316.8)--(247.5, 316.8)--(247.5, 316.6)--(247.5, 316.4)--(247.6, 316.4)--(247.6, 316.2)--(247.8, 315.9)--(248.2, 315.2)--(248.9, 314.1)--(249.4, 314.3)--(250.0, 313.0)--(250.5, 310.5)--(252.9, 311.0)--(253.4, 311.1)--(253.9, 308.5)--(254.2, 307.5)--(254.5, 307.4)--(254.9, 307.6)--(255.6, 305.4)--(255.9, 305.5)--(256.4, 305.6)--(257.2, 305.7)--(258.0, 305.8)--(258.5, 305.8)--(259.3, 303.2)--(260.2, 300.0)--(261.1, 296.9)--(261.3, 296.6)--(262.1, 296.3)--(262.4, 296.3)--(262.6, 296.3)--(262.8, 296.2)--(263.1, 296.2)--(263.3, 296.2)--(263.6, 296.2)--(264.0, 296.3)--(264.5, 293.2)--(264.6, 292.2)--(268.8, 292.8)--(268.8, 292.9)--(269.5, 293.0)--(269.5, 293.1)--(270.0, 293.2)--(270.0, 293.3)--(270.2, 293.3)--(271.5, 293.6)--(272.3, 293.8)--(272.5, 293.7)--(272.6, 293.6)--(272.7, 293.2)--(272.9, 293.0)--(274.1, 294.1)--(274.4, 293.5)--(276.2, 295.1)--(276.3, 295.3)--(276.5, 295.5)--(277.0, 296.3)--(277.2, 296.5)--(277.3, 296.6)--(277.4, 296.8)--(277.5, 296.8)--(277.6, 296.9)--(277.7, 297.0)--(278.7, 297.4)--(279.1, 297.5)--(279.1, 296.7)--(279.1, 296.6)--(279.1, 296.5)--(279.1, 296.4)--(279.2, 296.4)--(279.2, 296.2)--(279.7, 294.0)--(280.1, 293.5)--(280.9, 292.6)--(281.0, 292.3)--(281.3, 290.2)--(281.3, 290.1)--(281.4, 289.9)--(281.5, 289.6)--(281.3, 289.4)--(281.2, 289.3)--(281.2, 289.2)--(281.1, 289.1)--(281.1, 289.0)--(281.8, 288.3)--(281.1, 287.5)--(282.0, 286.2)--(281.3, 285.6)--(281.4, 285.1)--(281.4, 285.0)--(281.5, 284.7)--(281.6, 284.5)--(281.8, 284.0)--(281.3, 283.5)--(282.0, 282.9)--(283.0, 282.0)--(284.3, 280.8)--(284.5, 280.6)--(285.2, 279.9)--(285.3, 279.7)--(285.4, 279.6)--(285.4, 279.5)--(285.5, 279.0)--(285.1, 279.0)--(284.9, 278.9)--(284.2, 278.8)--(283.9, 278.7)--(283.7, 278.7)--(283.5, 278.6)--(282.5, 278.4)--(282.0, 278.2)--(281.6, 278.1)--(281.3, 278.0)--(280.9, 277.9)--(279.0, 277.5)--(277.3, 277.1)--(274.1, 276.3)--(273.6, 276.2)--(272.8, 276.0)--(272.3, 275.9)--(271.6, 275.7)--(271.2, 275.5)--(270.9, 275.4)--(270.8, 275.4)--(270.6, 275.3)--(270.4, 275.2)--(270.1, 275.1)--(269.8, 274.9)--(269.5, 274.8)--(269.3, 274.7)--(269.0, 274.5)--(268.9, 274.7)--(268.8, 274.6)--(268.7, 274.6)--(268.6, 274.5)--(267.9, 274.4)--(267.8, 274.3)--(267.7, 274.3)--(267.6, 274.2)--(267.5, 274.1)--(267.4, 274.0)--(267.4, 273.9)--(267.3, 273.9)--(267.2, 273.8)--(266.7, 273.2)--(266.6, 273.1)--(266.4, 272.8)--(266.3, 272.8)--(266.3, 272.7)--(266.5, 272.5)--(266.3, 272.3)--(266.2, 272.1)--(265.7, 271.2)--(265.6, 271.0)--(265.4, 270.2)--(265.3, 270.2)--(265.0, 270.2)--(264.6, 270.1)--(264.4, 270.1)--(264.3, 270.1)--(263.4, 269.5)--(263.0, 269.4)--(262.9, 269.5)--(262.6, 269.4)--(262.5, 269.4)--(262.4, 269.5)--cycle; +Kingston_upon_Thames = (164.0, 102.0)--(164.1, 102.1)--(164.1, 102.2)--(164.2, 102.2)--(164.3, 102.3)--(164.5, 102.5)--(164.7, 102.8)--(164.9, 103.0)--(164.9, 103.1)--(165.0, 103.2)--(165.1, 103.4)--(165.1, 103.6)--(165.0, 104.0)--(165.0, 104.2)--(164.9, 104.8)--(164.9, 104.9)--(164.7, 105.2)--(164.7, 105.6)--(164.7, 105.7)--(164.6, 105.9)--(164.6, 106.1)--(164.6, 106.2)--(164.6, 106.3)--(164.6, 107.0)--(164.6, 107.4)--(164.7, 107.8)--(164.7, 108.0)--(164.7, 108.1)--(164.7, 108.5)--(164.7, 108.8)--(164.8, 109.0)--(164.9, 109.4)--(164.9, 109.7)--(164.9, 110.2)--(164.9, 110.6)--(164.9, 111.6)--(165.0, 112.2)--(165.0, 112.8)--(165.0, 113.2)--(165.0, 113.6)--(165.0, 113.8)--(165.0, 114.0)--(164.9, 114.4)--(164.9, 114.5)--(165.0, 115.1)--(165.0, 115.5)--(165.2, 116.2)--(165.3, 116.6)--(165.4, 116.9)--(165.5, 117.6)--(165.6, 118.0)--(165.6, 118.4)--(165.6, 118.6)--(165.9, 118.7)--(165.4, 118.8)--(165.2, 119.3)--(165.0, 119.7)--(164.8, 120.2)--(164.6, 120.8)--(164.5, 121.1)--(164.0, 122.3)--(163.8, 123.0)--(163.6, 123.6)--(163.7, 123.8)--(163.8, 124.0)--(163.9, 124.2)--(164.0, 124.5)--(164.1, 124.9)--(164.3, 125.3)--(164.3, 125.5)--(164.4, 125.8)--(164.6, 126.2)--(164.8, 126.6)--(165.1, 127.3)--(165.3, 127.8)--(165.5, 127.9)--(165.8, 128.2)--(166.1, 128.5)--(166.3, 128.8)--(166.8, 129.2)--(167.5, 129.6)--(167.7, 129.8)--(167.9, 130.0)--(168.2, 130.4)--(168.3, 130.7)--(168.6, 131.2)--(168.8, 131.6)--(169.0, 132.0)--(169.2, 132.4)--(169.4, 132.9)--(169.7, 133.5)--(169.9, 134.1)--(170.1, 134.5)--(170.2, 134.8)--(170.5, 135.6)--(170.6, 136.0)--(170.6, 136.1)--(170.6, 136.2)--(170.7, 136.2)--(170.8, 136.6)--(171.0, 137.2)--(171.1, 137.7)--(171.2, 138.5)--(171.3, 139.2)--(171.8, 140.6)--(171.8, 140.8)--(171.7, 141.9)--(171.6, 142.6)--(171.6, 142.7)--(171.5, 143.2)--(171.5, 143.6)--(171.6, 144.4)--(171.7, 145.5)--(171.7, 146.6)--(171.8, 147.3)--(171.9, 147.6)--(171.9, 147.7)--(171.8, 148.1)--(171.8, 148.5)--(171.9, 149.2)--(172.1, 150.0)--(172.1, 150.1)--(172.2, 150.3)--(172.2, 150.5)--(172.4, 150.8)--(172.5, 151.1)--(172.6, 151.3)--(172.8, 151.6)--(172.9, 151.8)--(173.1, 152.0)--(173.3, 152.2)--(173.6, 152.6)--(173.9, 152.8)--(174.2, 153.0)--(174.5, 153.1)--(174.9, 153.1)--(175.2, 153.1)--(175.4, 153.2)--(175.5, 153.2)--(175.7, 153.3)--(176.2, 153.4)--(176.5, 153.5)--(176.9, 153.6)--(177.5, 153.8)--(177.9, 153.9)--(178.1, 154.0)--(178.3, 154.5)--(178.4, 154.7)--(178.6, 155.4)--(178.6, 155.8)--(178.7, 156.3)--(178.6, 157.3)--(179.0, 157.3)--(178.9, 158.0)--(178.3, 159.1)--(178.3, 159.2)--(178.1, 159.6)--(177.6, 159.6)--(177.6, 160.3)--(177.6, 160.5)--(177.6, 160.6)--(177.6, 160.7)--(177.6, 160.9)--(177.6, 161.0)--(177.5, 161.1)--(177.4, 161.1)--(177.4, 161.8)--(177.3, 161.9)--(177.2, 162.0)--(177.2, 162.2)--(177.1, 162.2)--(177.1, 162.3)--(176.8, 163.1)--(176.8, 163.3)--(176.8, 163.4)--(176.7, 163.7)--(176.7, 163.8)--(176.6, 163.9)--(176.5, 164.1)--(176.6, 164.2)--(176.6, 164.3)--(176.5, 164.4)--(176.5, 164.5)--(176.3, 164.6)--(176.2, 164.8)--(175.7, 165.5)--(176.4, 165.9)--(176.0, 166.5)--(175.9, 166.6)--(176.0, 166.7)--(175.8, 166.9)--(175.7, 167.0)--(175.7, 167.1)--(175.6, 167.2)--(175.7, 167.3)--(174.9, 168.4)--(174.7, 168.8)--(174.6, 168.7)--(172.8, 167.4)--(172.4, 168.1)--(172.3, 168.3)--(172.3, 168.5)--(172.2, 168.5)--(172.2, 168.6)--(172.2, 168.7)--(172.3, 168.8)--(172.3, 168.9)--(172.4, 168.9)--(172.4, 169.0)--(172.5, 169.0)--(172.8, 169.3)--(173.1, 169.5)--(173.4, 169.8)--(174.0, 170.3)--(173.9, 170.5)--(173.6, 170.9)--(173.5, 171.1)--(173.4, 171.3)--(173.2, 171.6)--(173.2, 171.7)--(172.8, 172.3)--(172.4, 172.9)--(172.1, 173.3)--(171.9, 173.5)--(171.8, 173.6)--(171.7, 173.8)--(171.4, 174.1)--(171.5, 174.2)--(171.6, 174.4)--(172.1, 175.0)--(173.8, 177.0)--(174.6, 178.0)--(174.9, 178.4)--(175.1, 179.1)--(175.2, 179.2)--(175.2, 179.4)--(175.3, 179.5)--(175.3, 179.6)--(175.6, 179.9)--(176.1, 180.5)--(176.3, 180.7)--(176.5, 180.9)--(176.6, 181.1)--(176.8, 181.2)--(176.8, 181.3)--(177.0, 181.7)--(177.0, 181.8)--(177.1, 182.0)--(177.2, 182.5)--(177.3, 182.6)--(177.3, 183.1)--(177.4, 183.8)--(177.4, 184.6)--(177.4, 185.2)--(177.4, 185.8)--(177.3, 186.6)--(177.2, 187.5)--(177.2, 187.6)--(177.1, 188.0)--(177.1, 188.3)--(177.1, 188.5)--(177.1, 188.7)--(177.0, 190.4)--(177.0, 191.6)--(177.0, 191.7)--(177.0, 191.9)--(177.1, 193.6)--(177.2, 194.2)--(177.3, 195.0)--(177.4, 195.2)--(177.4, 195.6)--(177.4, 195.9)--(177.3, 196.3)--(177.3, 196.5)--(177.3, 196.6)--(177.3, 196.8)--(177.3, 197.2)--(177.3, 197.4)--(177.3, 197.6)--(177.3, 197.8)--(177.4, 198.0)--(177.4, 198.3)--(177.7, 199.7)--(177.8, 200.0)--(177.8, 200.1)--(177.8, 200.3)--(177.9, 200.4)--(177.9, 200.6)--(177.9, 200.7)--(178.1, 201.0)--(178.1, 201.2)--(178.3, 201.8)--(178.4, 202.2)--(178.4, 202.5)--(178.4, 202.9)--(178.4, 203.2)--(178.4, 203.6)--(178.5, 203.8)--(178.5, 204.0)--(178.6, 204.3)--(178.8, 204.8)--(178.8, 205.0)--(178.8, 205.2)--(178.8, 205.6)--(178.7, 205.9)--(178.6, 206.1)--(178.6, 206.2)--(178.4, 206.6)--(178.3, 206.9)--(178.2, 207.2)--(178.1, 207.4)--(177.7, 208.0)--(177.5, 208.4)--(177.4, 208.7)--(176.9, 209.3)--(176.5, 209.9)--(176.2, 210.2)--(175.9, 210.4)--(175.7, 210.6)--(175.1, 211.0)--(174.6, 211.4)--(174.3, 211.7)--(173.9, 211.9)--(173.5, 212.2)--(172.8, 212.6)--(172.9, 213.0)--(173.1, 213.3)--(173.2, 213.7)--(173.9, 215.0)--(174.2, 215.6)--(174.4, 215.9)--(175.1, 216.9)--(175.3, 217.1)--(175.5, 216.9)--(175.7, 217.1)--(175.8, 217.1)--(175.9, 217.0)--(176.0, 216.9)--(176.2, 216.7)--(177.1, 215.9)--(176.8, 215.6)--(176.8, 215.5)--(177.1, 215.3)--(177.4, 215.2)--(177.6, 215.0)--(177.7, 215.0)--(177.9, 215.0)--(177.9, 214.9)--(178.3, 214.9)--(178.3, 215.3)--(178.4, 215.3)--(178.4, 216.0)--(178.5, 216.1)--(178.6, 216.2)--(178.7, 216.2)--(179.8, 215.9)--(181.3, 215.5)--(181.5, 215.2)--(181.6, 215.2)--(181.7, 215.4)--(181.5, 215.8)--(181.3, 216.1)--(181.6, 216.4)--(181.9, 216.0)--(182.1, 215.8)--(182.3, 215.7)--(182.5, 215.5)--(182.7, 215.2)--(183.6, 214.4)--(184.0, 214.0)--(184.0, 213.9)--(184.2, 213.7)--(184.4, 213.4)--(184.5, 213.4)--(184.6, 213.3)--(184.7, 213.2)--(184.8, 213.0)--(185.2, 212.8)--(185.5, 212.5)--(185.6, 212.4)--(185.7, 212.4)--(185.7, 212.5)--(186.0, 212.3)--(186.1, 211.9)--(186.0, 211.6)--(186.4, 211.6)--(186.5, 211.6)--(186.5, 211.3)--(186.5, 211.2)--(186.5, 211.0)--(187.4, 211.0)--(187.5, 211.0)--(187.5, 211.3)--(187.5, 211.7)--(187.5, 211.8)--(187.5, 213.3)--(187.8, 213.4)--(187.7, 213.7)--(189.3, 213.8)--(189.8, 212.4)--(189.8, 212.2)--(189.9, 212.0)--(190.1, 211.2)--(190.2, 211.1)--(190.2, 210.8)--(191.7, 206.1)--(192.3, 204.1)--(192.6, 204.2)--(193.0, 204.4)--(193.5, 204.6)--(195.0, 205.3)--(195.0, 205.4)--(196.6, 206.1)--(199.0, 209.4)--(199.3, 209.7)--(199.7, 210.3)--(200.0, 210.6)--(200.8, 211.8)--(201.2, 212.4)--(201.6, 212.9)--(202.2, 213.7)--(202.7, 214.4)--(202.8, 214.6)--(202.9, 214.6)--(203.0, 214.8)--(203.2, 215.0)--(203.4, 215.2)--(204.1, 215.8)--(204.4, 216.0)--(204.5, 216.2)--(204.7, 216.4)--(205.0, 216.5)--(205.2, 216.7)--(205.5, 216.9)--(205.9, 217.3)--(206.2, 217.7)--(206.5, 218.0)--(206.8, 218.2)--(207.3, 218.6)--(207.6, 218.8)--(208.4, 219.4)--(208.6, 219.5)--(209.0, 219.8)--(209.2, 219.9)--(209.8, 220.4)--(210.2, 220.6)--(210.5, 220.8)--(211.3, 221.4)--(211.8, 221.7)--(212.4, 222.2)--(212.6, 222.4)--(212.9, 222.6)--(213.0, 222.6)--(212.9, 222.8)--(213.0, 222.8)--(212.9, 222.8)--(213.1, 222.9)--(213.4, 222.9)--(213.7, 223.1)--(213.8, 223.2)--(213.9, 223.2)--(214.2, 223.5)--(214.5, 223.7)--(214.5, 223.6)--(214.7, 223.4)--(214.8, 223.3)--(214.9, 223.1)--(214.9, 223.0)--(215.0, 222.9)--(215.0, 222.6)--(215.0, 222.3)--(215.0, 222.1)--(215.0, 221.9)--(215.1, 221.8)--(215.1, 221.6)--(215.2, 221.5)--(215.3, 221.4)--(215.4, 221.3)--(215.4, 221.2)--(215.5, 221.2)--(215.6, 221.1)--(216.0, 220.8)--(216.2, 220.7)--(216.3, 220.6)--(216.4, 220.5)--(216.4, 220.4)--(216.5, 220.3)--(216.6, 220.2)--(216.7, 219.8)--(216.7, 219.5)--(216.7, 219.4)--(216.7, 219.2)--(216.7, 218.8)--(216.7, 218.5)--(216.6, 218.2)--(216.4, 217.8)--(216.3, 217.7)--(216.2, 217.6)--(216.0, 217.3)--(216.0, 217.2)--(215.8, 217.0)--(215.7, 216.9)--(215.7, 216.8)--(215.5, 216.7)--(215.4, 216.7)--(215.3, 216.6)--(215.2, 216.5)--(215.1, 216.3)--(215.0, 216.1)--(214.9, 215.9)--(214.8, 215.7)--(214.7, 215.4)--(214.7, 215.3)--(214.6, 215.1)--(214.6, 215.0)--(214.6, 214.9)--(214.6, 214.8)--(214.6, 214.7)--(214.7, 214.5)--(214.8, 214.2)--(214.9, 214.0)--(215.0, 213.6)--(215.1, 213.4)--(215.2, 213.1)--(215.3, 212.9)--(215.4, 212.7)--(215.5, 212.4)--(215.5, 212.2)--(215.7, 212.0)--(215.9, 211.7)--(216.0, 211.5)--(216.0, 211.2)--(216.0, 211.1)--(216.1, 211.0)--(216.1, 210.9)--(216.1, 210.7)--(216.0, 210.4)--(216.0, 210.0)--(215.9, 209.9)--(215.9, 209.7)--(215.8, 209.4)--(215.7, 209.2)--(215.7, 209.1)--(215.7, 209.0)--(215.7, 208.6)--(215.7, 208.3)--(215.7, 208.2)--(215.7, 208.0)--(215.8, 207.9)--(216.0, 207.8)--(216.3, 207.8)--(216.5, 207.7)--(216.5, 207.6)--(216.7, 207.4)--(216.9, 207.2)--(216.9, 207.1)--(217.0, 207.1)--(217.1, 206.6)--(217.2, 206.2)--(217.3, 205.9)--(217.3, 205.7)--(217.4, 205.6)--(217.4, 205.4)--(217.4, 205.1)--(217.4, 204.9)--(217.4, 204.7)--(217.5, 204.5)--(217.5, 204.3)--(217.6, 204.1)--(217.7, 203.9)--(217.9, 203.5)--(217.9, 203.4)--(218.0, 203.2)--(218.1, 202.9)--(218.1, 202.7)--(218.1, 202.6)--(218.2, 202.4)--(218.1, 202.2)--(218.1, 202.1)--(218.1, 201.9)--(218.0, 201.7)--(218.0, 201.3)--(218.0, 201.2)--(218.0, 200.7)--(218.0, 200.6)--(218.1, 200.2)--(218.1, 200.0)--(218.2, 199.4)--(218.3, 199.1)--(218.3, 199.0)--(218.2, 198.9)--(218.2, 198.8)--(218.3, 198.7)--(218.4, 198.4)--(218.6, 197.7)--(218.7, 197.3)--(219.2, 195.8)--(219.8, 194.0)--(219.9, 193.9)--(220.0, 193.1)--(220.1, 193.0)--(220.1, 192.9)--(220.1, 192.8)--(220.0, 192.8)--(220.0, 192.7)--(219.9, 192.5)--(219.8, 192.4)--(219.8, 192.2)--(219.8, 192.0)--(219.8, 191.8)--(219.9, 191.7)--(219.9, 191.6)--(220.0, 191.6)--(220.0, 191.5)--(220.0, 191.4)--(220.0, 191.3)--(220.0, 191.2)--(220.1, 191.2)--(220.1, 191.1)--(220.2, 191.0)--(220.3, 191.0)--(220.4, 191.0)--(220.5, 190.9)--(220.7, 190.9)--(220.8, 190.8)--(220.9, 190.8)--(221.0, 190.8)--(221.1, 190.8)--(221.1, 190.7)--(221.1, 190.6)--(221.3, 190.4)--(221.3, 190.3)--(221.4, 190.2)--(221.4, 190.1)--(221.3, 189.9)--(221.3, 189.7)--(221.3, 189.6)--(221.4, 189.4)--(221.5, 189.2)--(221.5, 189.0)--(221.5, 188.9)--(221.5, 188.7)--(221.6, 188.4)--(221.7, 188.3)--(221.7, 188.0)--(221.8, 187.7)--(221.8, 187.6)--(221.8, 187.5)--(221.8, 187.3)--(221.9, 187.2)--(221.9, 187.1)--(221.9, 186.6)--(221.9, 186.5)--(221.9, 186.1)--(221.9, 185.9)--(221.8, 185.6)--(221.8, 185.5)--(221.8, 185.4)--(221.7, 184.9)--(221.7, 184.8)--(221.6, 184.2)--(221.6, 184.3)--(221.5, 184.2)--(221.4, 184.1)--(221.3, 183.9)--(221.0, 183.6)--(221.0, 183.5)--(220.8, 183.2)--(220.6, 183.1)--(220.6, 183.0)--(220.5, 182.9)--(220.5, 182.8)--(220.6, 182.5)--(220.7, 182.3)--(220.7, 182.0)--(220.7, 181.7)--(220.7, 181.5)--(220.7, 181.4)--(220.7, 181.3)--(220.6, 181.2)--(220.6, 181.0)--(220.6, 180.9)--(220.6, 180.8)--(220.6, 180.7)--(220.5, 180.4)--(220.5, 180.3)--(220.4, 180.1)--(220.3, 179.9)--(220.2, 179.7)--(220.2, 179.6)--(220.5, 179.1)--(220.7, 178.9)--(220.9, 178.7)--(220.9, 178.6)--(221.0, 178.5)--(221.1, 178.2)--(221.2, 178.1)--(221.3, 178.0)--(221.4, 177.8)--(221.4, 177.7)--(221.5, 177.6)--(221.6, 177.5)--(221.6, 177.4)--(221.9, 177.1)--(222.0, 177.0)--(222.1, 176.8)--(222.2, 176.6)--(222.2, 176.4)--(222.5, 176.3)--(222.6, 176.2)--(222.7, 176.2)--(222.8, 176.1)--(223.0, 176.1)--(223.2, 176.1)--(223.5, 176.0)--(223.7, 176.0)--(223.9, 175.9)--(224.0, 175.8)--(224.2, 175.7)--(224.2, 175.6)--(224.3, 175.6)--(224.5, 175.1)--(224.5, 175.0)--(224.6, 174.9)--(224.7, 174.8)--(224.8, 174.3)--(224.9, 174.1)--(224.9, 174.0)--(224.9, 173.9)--(224.8, 173.7)--(224.9, 173.7)--(225.0, 173.7)--(225.0, 173.6)--(225.1, 173.6)--(225.1, 173.5)--(225.2, 173.3)--(225.2, 173.2)--(225.4, 172.9)--(225.5, 172.6)--(225.5, 172.5)--(225.4, 172.3)--(225.5, 172.2)--(225.5, 172.0)--(225.5, 171.8)--(225.5, 171.7)--(225.6, 171.4)--(225.6, 171.0)--(225.7, 170.8)--(225.7, 170.7)--(225.8, 170.6)--(225.8, 170.5)--(225.8, 170.4)--(225.9, 170.2)--(225.9, 170.1)--(225.9, 169.9)--(225.9, 169.8)--(225.9, 169.7)--(226.0, 169.5)--(226.0, 169.4)--(226.0, 169.3)--(226.0, 169.0)--(226.1, 168.8)--(226.2, 168.5)--(226.3, 168.3)--(226.3, 168.1)--(226.3, 167.8)--(226.4, 167.7)--(226.4, 167.6)--(226.5, 167.0)--(226.5, 166.9)--(226.4, 166.9)--(226.3, 166.8)--(226.2, 166.8)--(226.1, 166.6)--(225.9, 166.4)--(225.7, 166.3)--(225.6, 166.2)--(225.2, 165.8)--(225.0, 165.6)--(224.7, 165.4)--(224.6, 165.2)--(224.4, 165.0)--(224.4, 164.8)--(224.3, 164.6)--(224.0, 163.9)--(223.8, 163.6)--(223.7, 163.4)--(222.8, 164.0)--(222.7, 163.4)--(222.6, 162.8)--(222.5, 161.9)--(222.5, 161.2)--(222.4, 161.1)--(222.3, 160.8)--(222.3, 160.2)--(222.0, 160.2)--(221.9, 160.1)--(221.7, 160.3)--(221.6, 160.3)--(221.3, 160.3)--(221.2, 160.3)--(221.1, 160.2)--(220.8, 160.1)--(220.5, 160.2)--(220.5, 160.3)--(220.4, 160.3)--(220.4, 160.2)--(220.3, 160.3)--(220.0, 160.1)--(219.8, 160.1)--(219.7, 160.0)--(219.5, 160.0)--(218.9, 159.6)--(218.5, 159.4)--(218.3, 159.2)--(218.2, 159.2)--(217.9, 159.0)--(217.7, 158.9)--(217.2, 158.8)--(217.0, 158.7)--(216.5, 158.6)--(216.4, 158.6)--(216.1, 158.6)--(215.7, 158.6)--(215.4, 158.6)--(215.0, 158.6)--(214.9, 158.6)--(214.5, 158.6)--(213.9, 158.6)--(213.8, 158.6)--(213.7, 158.7)--(213.0, 158.7)--(212.4, 159.0)--(212.1, 159.0)--(212.0, 159.1)--(211.9, 159.1)--(211.8, 159.1)--(211.3, 159.4)--(211.2, 159.4)--(211.1, 159.4)--(211.1, 159.3)--(211.0, 158.9)--(210.9, 158.8)--(210.9, 158.2)--(210.8, 158.1)--(210.6, 157.9)--(210.6, 157.8)--(210.6, 157.7)--(210.3, 156.9)--(210.2, 156.8)--(210.0, 156.5)--(209.7, 156.3)--(209.3, 155.8)--(209.1, 155.4)--(208.8, 154.9)--(208.2, 154.5)--(207.3, 153.8)--(206.5, 153.1)--(205.9, 152.6)--(205.6, 152.5)--(205.3, 152.3)--(204.8, 152.0)--(204.4, 151.8)--(204.3, 151.7)--(203.8, 151.4)--(203.6, 151.3)--(203.4, 151.2)--(203.4, 151.1)--(203.3, 151.0)--(203.2, 150.9)--(203.1, 150.7)--(202.9, 150.6)--(202.8, 150.5)--(202.8, 150.4)--(202.7, 150.2)--(202.6, 150.0)--(202.5, 149.8)--(202.4, 149.6)--(202.4, 149.4)--(202.4, 149.2)--(202.2, 149.2)--(202.1, 149.1)--(202.0, 149.1)--(201.7, 148.9)--(201.5, 148.8)--(201.4, 148.8)--(201.3, 148.6)--(201.1, 148.5)--(201.1, 148.4)--(201.0, 148.4)--(201.0, 148.2)--(200.9, 148.1)--(200.9, 147.9)--(200.8, 147.7)--(200.9, 147.6)--(200.9, 147.4)--(200.8, 147.3)--(200.7, 147.2)--(200.7, 147.1)--(200.6, 147.1)--(200.5, 147.1)--(200.4, 146.9)--(200.3, 146.9)--(200.2, 146.8)--(200.1, 146.7)--(200.0, 146.7)--(199.7, 146.6)--(199.6, 146.6)--(199.4, 146.5)--(199.2, 146.5)--(199.0, 146.4)--(198.9, 146.3)--(198.7, 146.2)--(198.4, 146.0)--(198.3, 145.9)--(198.2, 145.8)--(198.0, 145.6)--(197.9, 145.5)--(197.8, 145.4)--(197.7, 145.3)--(197.6, 145.2)--(197.5, 145.0)--(197.4, 144.9)--(197.4, 144.8)--(197.3, 144.6)--(197.2, 144.5)--(197.1, 144.3)--(197.0, 144.2)--(197.0, 144.1)--(196.8, 143.9)--(196.7, 143.8)--(196.5, 143.6)--(196.4, 143.5)--(196.2, 143.4)--(196.1, 143.3)--(196.0, 143.3)--(195.8, 143.2)--(195.7, 143.1)--(195.6, 143.1)--(195.6, 143.0)--(195.5, 143.0)--(195.5, 142.9)--(195.3, 142.7)--(195.2, 142.6)--(195.1, 142.4)--(195.0, 142.3)--(194.9, 142.2)--(194.8, 142.1)--(194.6, 141.9)--(194.5, 141.8)--(194.4, 141.7)--(194.3, 141.5)--(194.2, 141.4)--(194.0, 141.1)--(193.9, 140.8)--(193.8, 140.7)--(193.7, 140.5)--(193.7, 140.4)--(193.6, 140.3)--(193.5, 140.1)--(193.4, 140.0)--(193.3, 139.9)--(193.2, 139.9)--(193.1, 139.8)--(192.9, 139.7)--(192.8, 139.7)--(192.8, 139.6)--(192.7, 139.5)--(192.9, 139.3)--(193.2, 139.2)--(193.4, 139.1)--(193.6, 139.0)--(193.9, 138.9)--(194.2, 138.8)--(194.4, 138.7)--(194.7, 138.6)--(194.9, 138.6)--(194.8, 138.4)--(194.3, 137.5)--(194.1, 137.2)--(193.9, 137.0)--(193.9, 136.9)--(193.8, 136.7)--(193.7, 136.5)--(192.9, 136.1)--(192.3, 135.7)--(192.1, 135.4)--(192.0, 135.2)--(192.0, 135.0)--(191.9, 134.8)--(191.8, 134.6)--(191.7, 134.5)--(191.5, 134.1)--(191.2, 133.7)--(191.0, 133.4)--(190.9, 133.3)--(190.9, 133.2)--(190.8, 132.8)--(190.8, 132.7)--(190.8, 132.5)--(190.8, 132.3)--(190.8, 131.9)--(190.8, 131.8)--(190.7, 131.5)--(190.6, 131.2)--(190.5, 130.9)--(190.3, 130.6)--(190.1, 130.4)--(190.0, 130.3)--(190.0, 130.2)--(189.8, 130.0)--(189.8, 129.8)--(189.8, 129.2)--(189.8, 129.0)--(189.8, 128.9)--(189.8, 128.8)--(189.7, 128.6)--(189.5, 128.4)--(189.4, 128.3)--(189.3, 128.2)--(188.9, 127.8)--(188.7, 127.7)--(188.6, 127.5)--(188.4, 127.3)--(188.3, 127.2)--(188.1, 126.9)--(187.9, 126.5)--(187.6, 126.2)--(187.3, 125.9)--(186.8, 125.4)--(186.6, 125.2)--(186.4, 125.1)--(186.4, 125.0)--(186.2, 124.8)--(186.0, 124.6)--(185.8, 124.4)--(185.6, 124.2)--(185.4, 124.0)--(185.3, 123.8)--(184.9, 123.3)--(184.8, 123.1)--(184.8, 122.9)--(184.7, 122.8)--(184.6, 122.3)--(184.5, 122.0)--(184.4, 121.7)--(184.2, 121.1)--(184.1, 120.8)--(184.0, 120.6)--(183.9, 120.4)--(183.7, 120.1)--(183.6, 120.0)--(183.4, 119.8)--(183.2, 119.6)--(182.7, 118.9)--(182.6, 118.7)--(182.4, 118.5)--(182.4, 118.4)--(182.3, 118.3)--(182.2, 118.1)--(182.1, 117.8)--(182.1, 117.6)--(182.0, 117.3)--(182.0, 117.1)--(181.9, 116.9)--(181.9, 116.7)--(181.9, 116.5)--(181.9, 116.2)--(181.8, 115.9)--(181.8, 115.6)--(181.7, 115.4)--(181.7, 115.1)--(181.6, 115.0)--(181.6, 114.8)--(181.4, 114.3)--(181.3, 114.1)--(181.3, 113.8)--(181.4, 113.4)--(181.4, 113.2)--(181.4, 112.9)--(181.5, 112.7)--(181.5, 112.6)--(181.6, 112.3)--(181.7, 112.1)--(181.8, 111.9)--(181.7, 111.7)--(181.7, 111.4)--(181.7, 111.0)--(181.7, 110.7)--(181.6, 110.4)--(181.5, 110.2)--(181.2, 109.5)--(181.1, 109.4)--(181.0, 109.3)--(180.9, 109.2)--(180.6, 109.0)--(180.4, 108.8)--(180.0, 108.6)--(179.9, 108.6)--(179.9, 108.5)--(179.4, 108.0)--(179.2, 107.6)--(178.9, 107.3)--(178.6, 106.9)--(178.4, 106.7)--(178.0, 106.4)--(177.7, 106.1)--(177.2, 105.8)--(176.9, 105.7)--(176.7, 105.6)--(176.5, 105.5)--(176.4, 105.5)--(176.2, 105.3)--(175.8, 105.0)--(175.6, 104.8)--(175.4, 104.5)--(174.9, 103.9)--(174.8, 103.8)--(174.2, 103.0)--(174.1, 102.9)--(173.7, 102.4)--(173.6, 102.3)--(173.4, 102.1)--(173.0, 101.7)--(172.6, 101.4)--(172.2, 101.0)--(172.0, 100.9)--(171.8, 100.8)--(171.4, 100.7)--(170.6, 100.5)--(170.4, 100.4)--(170.3, 100.4)--(170.0, 100.3)--(169.7, 100.1)--(169.3, 100.0)--(169.0, 99.8)--(168.8, 99.6)--(168.7, 99.6)--(168.7, 99.5)--(168.5, 99.5)--(168.4, 99.5)--(168.3, 99.4)--(168.0, 99.4)--(167.6, 99.4)--(167.3, 99.4)--(166.8, 99.4)--(166.7, 99.4)--(166.5, 99.5)--(166.4, 99.5)--(166.2, 99.1)--(165.7, 99.2)--(165.3, 99.6)--(165.1, 99.8)--(164.8, 100.0)--(164.4, 100.4)--(164.3, 100.8)--(164.2, 100.9)--(164.2, 101.1)--(164.1, 101.3)--(164.0, 101.6)--(164.0, 101.8)--(164.0, 101.9)--cycle; +Lambeth = (300.5, 278.9)--(300.6, 279.0)--(300.7, 279.1)--(300.8, 279.2)--(301.2, 279.5)--(301.3, 279.7)--(301.5, 279.9)--(301.6, 280.0)--(301.9, 280.3)--(302.2, 280.7)--(302.4, 280.9)--(302.4, 281.1)--(302.5, 281.2)--(302.7, 281.4)--(302.8, 281.7)--(302.9, 281.9)--(303.0, 282.1)--(303.3, 282.0)--(303.3, 282.1)--(303.2, 282.2)--(303.4, 282.6)--(303.4, 282.7)--(303.5, 282.9)--(303.6, 283.6)--(303.7, 283.9)--(303.8, 284.1)--(304.1, 284.8)--(304.3, 285.4)--(304.5, 286.0)--(304.7, 286.7)--(304.8, 287.3)--(304.9, 287.9)--(305.1, 289.4)--(305.2, 289.5)--(305.2, 290.0)--(305.3, 291.0)--(305.4, 291.4)--(305.5, 292.8)--(305.7, 294.2)--(305.9, 296.7)--(306.0, 297.6)--(305.9, 297.7)--(306.0, 298.4)--(306.2, 300.0)--(306.3, 300.1)--(306.3, 300.2)--(306.3, 300.5)--(306.4, 300.7)--(306.4, 300.9)--(306.5, 301.1)--(306.6, 301.3)--(306.9, 302.1)--(307.0, 302.3)--(307.2, 302.7)--(307.5, 303.0)--(307.8, 303.3)--(307.8, 303.4)--(307.9, 303.5)--(308.4, 303.9)--(308.5, 304.0)--(308.7, 304.1)--(308.9, 304.2)--(309.2, 304.3)--(309.5, 304.5)--(309.9, 304.6)--(310.1, 304.7)--(310.5, 304.8)--(310.8, 304.9)--(311.6, 305.2)--(311.7, 305.2)--(311.8, 305.1)--(312.1, 304.9)--(312.4, 304.9)--(312.4, 305.0)--(312.7, 305.1)--(312.8, 305.1)--(312.9, 305.2)--(313.2, 305.2)--(313.4, 305.3)--(313.4, 304.9)--(313.5, 304.5)--(313.6, 304.5)--(313.7, 304.5)--(313.7, 304.2)--(313.8, 303.8)--(313.9, 303.7)--(313.9, 303.6)--(314.5, 303.8)--(315.1, 301.8)--(315.2, 301.5)--(315.2, 301.3)--(315.2, 301.2)--(315.2, 301.0)--(315.1, 300.7)--(315.1, 300.5)--(315.2, 300.0)--(315.3, 299.7)--(314.5, 299.3)--(315.0, 298.4)--(315.5, 298.8)--(315.5, 298.6)--(315.5, 298.5)--(315.5, 298.4)--(315.5, 298.3)--(315.6, 298.2)--(315.2, 297.7)--(315.1, 297.8)--(315.0, 297.7)--(314.3, 296.8)--(314.1, 296.5)--(314.0, 296.3)--(313.5, 294.9)--(313.3, 294.2)--(313.2, 294.1)--(312.7, 292.4)--(312.7, 292.3)--(312.0, 291.9)--(312.3, 291.0)--(312.3, 290.9)--(312.4, 290.8)--(312.5, 290.8)--(312.6, 290.8)--(312.6, 290.7)--(313.4, 290.3)--(315.0, 289.4)--(316.0, 289.0)--(316.3, 288.9)--(316.6, 288.8)--(316.7, 288.6)--(316.7, 288.3)--(316.9, 288.1)--(316.9, 287.9)--(317.1, 287.7)--(317.2, 287.4)--(317.3, 287.0)--(317.3, 287.1)--(317.4, 287.1)--(317.5, 287.0)--(317.7, 286.9)--(317.6, 286.8)--(317.8, 286.5)--(317.7, 286.4)--(317.9, 286.2)--(317.8, 286.1)--(317.7, 285.9)--(317.4, 285.4)--(317.2, 284.9)--(316.8, 284.4)--(316.3, 283.5)--(314.6, 280.5)--(314.3, 280.2)--(314.5, 280.1)--(314.6, 280.0)--(314.7, 279.9)--(314.8, 279.8)--(315.8, 279.2)--(315.7, 278.9)--(315.6, 278.7)--(315.4, 278.2)--(315.2, 277.8)--(315.5, 277.6)--(315.9, 277.4)--(316.4, 277.0)--(316.2, 276.8)--(316.1, 276.7)--(316.4, 276.4)--(316.5, 276.2)--(316.6, 276.1)--(316.7, 276.1)--(316.9, 275.8)--(317.1, 275.5)--(317.1, 275.4)--(317.1, 275.3)--(317.5, 274.8)--(316.9, 274.6)--(316.5, 274.5)--(314.7, 274.1)--(314.6, 274.1)--(314.6, 274.0)--(314.7, 274.0)--(314.8, 273.9)--(314.9, 273.8)--(315.4, 273.5)--(316.5, 272.8)--(317.9, 271.9)--(319.4, 271.1)--(320.5, 270.4)--(320.4, 270.2)--(320.3, 270.1)--(320.2, 269.7)--(320.1, 268.0)--(320.1, 267.7)--(320.1, 267.5)--(320.1, 267.4)--(320.3, 267.1)--(320.7, 266.5)--(320.9, 266.1)--(321.0, 265.6)--(321.1, 265.2)--(321.2, 265.2)--(321.4, 264.9)--(322.6, 263.5)--(323.4, 262.6)--(324.1, 263.0)--(324.3, 263.2)--(324.5, 263.4)--(324.6, 263.5)--(324.7, 263.6)--(324.8, 263.7)--(324.8, 263.8)--(325.0, 264.6)--(325.0, 264.7)--(325.0, 264.8)--(325.1, 264.9)--(325.4, 265.2)--(325.4, 265.1)--(325.5, 263.9)--(325.5, 263.7)--(325.6, 263.0)--(325.7, 262.8)--(325.7, 262.6)--(326.1, 261.8)--(326.9, 260.2)--(327.6, 258.8)--(327.6, 258.7)--(327.7, 258.6)--(327.7, 258.5)--(327.7, 258.4)--(327.7, 258.1)--(327.3, 255.4)--(327.2, 254.9)--(327.0, 254.6)--(326.8, 254.0)--(326.3, 253.0)--(326.2, 252.9)--(326.1, 252.7)--(326.0, 252.5)--(325.9, 252.5)--(325.8, 252.4)--(325.7, 252.2)--(325.4, 251.7)--(325.0, 251.2)--(324.7, 250.8)--(324.6, 250.6)--(324.5, 250.4)--(324.4, 250.2)--(324.4, 250.0)--(324.3, 249.9)--(324.1, 249.3)--(324.0, 249.0)--(323.9, 248.8)--(323.7, 248.5)--(323.4, 248.2)--(323.0, 247.7)--(322.7, 247.4)--(322.6, 247.2)--(322.1, 246.8)--(321.7, 246.5)--(321.1, 245.8)--(320.7, 245.3)--(320.5, 245.1)--(320.4, 245.0)--(320.3, 244.7)--(320.3, 244.6)--(320.4, 244.0)--(320.4, 243.9)--(320.4, 243.8)--(320.4, 243.7)--(320.2, 243.3)--(320.2, 243.2)--(320.1, 243.1)--(320.1, 243.0)--(320.1, 242.8)--(320.2, 242.7)--(320.2, 242.6)--(320.2, 242.5)--(320.3, 242.4)--(320.4, 242.2)--(320.5, 242.0)--(320.6, 241.8)--(320.6, 241.7)--(320.7, 241.6)--(320.7, 241.2)--(320.7, 240.9)--(320.7, 240.8)--(321.0, 240.8)--(321.1, 240.8)--(321.2, 240.8)--(321.3, 240.8)--(321.3, 240.7)--(321.4, 240.7)--(321.4, 240.6)--(321.5, 240.5)--(321.5, 240.3)--(321.6, 240.0)--(321.7, 239.8)--(321.7, 239.6)--(321.7, 239.5)--(321.8, 239.5)--(322.0, 239.1)--(322.4, 238.8)--(322.7, 238.4)--(323.1, 238.0)--(324.0, 236.9)--(324.2, 236.7)--(324.3, 236.4)--(324.7, 235.8)--(324.7, 235.6)--(324.8, 235.5)--(324.8, 235.3)--(324.8, 235.2)--(324.8, 235.1)--(324.8, 234.9)--(324.8, 234.7)--(324.8, 234.6)--(324.9, 234.5)--(324.9, 234.3)--(325.2, 233.2)--(325.5, 232.6)--(325.6, 232.4)--(325.8, 232.2)--(325.9, 232.0)--(326.1, 231.7)--(326.2, 231.4)--(326.3, 231.1)--(326.4, 230.7)--(326.5, 230.4)--(326.7, 229.8)--(326.9, 229.0)--(327.1, 228.5)--(327.1, 228.4)--(327.2, 228.2)--(327.2, 228.1)--(327.3, 228.0)--(327.5, 227.6)--(327.7, 227.1)--(327.9, 226.6)--(328.1, 225.8)--(328.2, 225.6)--(328.3, 225.2)--(328.5, 224.1)--(328.6, 223.3)--(329.0, 221.5)--(329.2, 220.4)--(329.5, 219.0)--(329.8, 218.1)--(329.9, 217.8)--(330.0, 217.7)--(330.1, 217.6)--(330.3, 217.4)--(330.4, 217.3)--(330.7, 217.2)--(331.2, 217.1)--(331.5, 216.9)--(331.6, 216.9)--(331.6, 216.8)--(331.7, 216.8)--(331.7, 216.7)--(331.7, 216.6)--(331.6, 216.5)--(331.6, 216.3)--(331.5, 216.2)--(331.5, 216.1)--(331.6, 216.1)--(331.6, 216.0)--(331.7, 215.9)--(331.8, 215.9)--(332.0, 215.7)--(332.1, 215.6)--(332.2, 215.5)--(332.4, 215.1)--(332.6, 214.6)--(332.8, 214.2)--(333.0, 213.8)--(333.2, 213.4)--(333.3, 213.0)--(333.3, 212.7)--(333.4, 212.4)--(333.4, 212.0)--(333.4, 211.5)--(333.5, 211.2)--(333.5, 210.9)--(333.5, 210.7)--(333.5, 210.8)--(333.9, 210.9)--(334.5, 211.1)--(334.7, 211.2)--(334.7, 211.1)--(335.1, 210.0)--(335.3, 209.5)--(335.4, 209.5)--(335.7, 209.7)--(335.9, 209.7)--(336.0, 209.9)--(336.1, 209.8)--(336.2, 209.6)--(336.3, 209.4)--(336.7, 209.2)--(336.6, 208.8)--(336.5, 208.6)--(336.5, 208.5)--(336.5, 208.4)--(336.6, 208.4)--(337.2, 208.2)--(337.2, 208.1)--(337.1, 208.1)--(337.1, 208.0)--(337.1, 207.8)--(337.1, 207.6)--(337.0, 207.5)--(337.0, 207.3)--(336.9, 207.3)--(336.6, 207.2)--(336.5, 207.2)--(336.2, 207.1)--(336.0, 207.1)--(335.8, 207.1)--(335.6, 207.1)--(334.8, 207.2)--(334.4, 207.2)--(334.0, 207.1)--(333.7, 207.0)--(333.3, 206.9)--(332.5, 206.7)--(332.2, 206.6)--(332.0, 206.5)--(331.8, 206.5)--(331.7, 206.6)--(331.6, 206.6)--(331.4, 206.6)--(331.3, 206.7)--(331.2, 206.8)--(330.6, 207.1)--(330.4, 207.2)--(330.3, 207.2)--(330.1, 207.4)--(329.7, 207.9)--(329.5, 208.2)--(329.3, 208.4)--(329.2, 208.4)--(329.2, 208.5)--(329.1, 208.5)--(329.0, 208.6)--(328.9, 208.6)--(328.4, 208.9)--(328.2, 209.0)--(328.0, 209.2)--(327.8, 209.4)--(327.5, 209.7)--(327.4, 209.9)--(327.1, 210.1)--(326.9, 210.1)--(326.7, 210.2)--(326.6, 210.3)--(326.4, 210.3)--(326.2, 210.3)--(326.1, 210.3)--(325.9, 210.3)--(325.8, 210.3)--(325.6, 210.3)--(324.7, 210.2)--(324.2, 210.1)--(323.9, 210.1)--(323.8, 210.0)--(323.5, 210.0)--(323.3, 210.0)--(323.1, 210.0)--(322.9, 210.0)--(321.9, 210.0)--(321.5, 210.0)--(319.1, 210.0)--(318.8, 210.0)--(318.5, 209.9)--(318.0, 209.9)--(317.9, 209.9)--(317.8, 209.9)--(317.7, 209.9)--(317.5, 209.9)--(317.1, 210.0)--(316.5, 210.2)--(316.0, 210.3)--(315.6, 210.4)--(315.5, 210.5)--(315.3, 210.5)--(315.2, 210.4)--(315.1, 210.4)--(315.0, 210.4)--(314.7, 210.4)--(314.4, 210.4)--(314.0, 210.4)--(313.6, 210.5)--(313.2, 210.5)--(313.1, 210.4)--(313.0, 210.1)--(312.8, 209.9)--(312.4, 209.6)--(312.3, 209.4)--(312.2, 209.2)--(312.2, 209.1)--(312.2, 209.0)--(312.0, 208.7)--(311.9, 208.6)--(311.9, 208.5)--(311.7, 208.2)--(311.4, 207.9)--(311.4, 207.8)--(311.3, 207.7)--(311.2, 207.6)--(311.2, 207.5)--(311.0, 207.3)--(310.9, 207.2)--(310.6, 207.1)--(310.3, 207.0)--(310.2, 206.9)--(309.8, 206.6)--(309.7, 206.6)--(309.5, 206.4)--(309.4, 206.3)--(309.3, 206.2)--(309.0, 206.0)--(308.8, 205.8)--(308.6, 205.6)--(308.3, 205.4)--(308.2, 205.2)--(308.2, 204.9)--(308.2, 204.7)--(308.1, 204.5)--(308.0, 204.3)--(307.9, 204.0)--(307.8, 203.7)--(307.7, 203.6)--(307.7, 203.5)--(307.7, 203.4)--(307.6, 203.4)--(307.6, 203.3)--(307.5, 203.1)--(307.4, 203.0)--(307.4, 202.9)--(307.3, 202.7)--(307.2, 202.5)--(307.1, 202.3)--(307.0, 202.0)--(306.8, 201.8)--(306.7, 201.7)--(306.6, 201.5)--(306.6, 201.4)--(306.6, 201.2)--(306.5, 201.2)--(306.3, 201.2)--(306.2, 201.2)--(306.2, 201.0)--(306.0, 200.8)--(305.8, 200.7)--(305.6, 200.7)--(305.5, 200.7)--(305.4, 200.4)--(305.4, 200.2)--(305.3, 200.0)--(305.2, 199.8)--(305.2, 199.6)--(305.2, 199.5)--(305.2, 199.4)--(305.3, 199.3)--(305.0, 199.2)--(304.5, 198.8)--(304.5, 198.6)--(304.4, 198.5)--(304.2, 198.6)--(304.1, 198.8)--(304.0, 198.8)--(303.8, 198.6)--(303.3, 198.3)--(303.2, 198.3)--(303.0, 198.1)--(303.0, 198.0)--(302.9, 197.7)--(300.0, 198.0)--(299.6, 198.0)--(299.2, 198.2)--(299.0, 198.3)--(299.0, 198.4)--(299.1, 198.6)--(299.0, 198.7)--(298.9, 198.4)--(298.8, 198.3)--(298.8, 198.2)--(298.8, 198.1)--(298.8, 197.8)--(298.8, 197.7)--(298.7, 197.4)--(298.6, 196.5)--(298.4, 196.6)--(298.2, 196.7)--(298.0, 196.7)--(297.9, 196.7)--(297.7, 196.7)--(297.4, 196.8)--(296.9, 197.0)--(296.2, 197.2)--(295.8, 197.3)--(295.7, 197.4)--(295.5, 197.5)--(295.4, 197.5)--(294.9, 197.7)--(294.8, 197.8)--(294.5, 197.9)--(293.7, 198.3)--(293.1, 198.7)--(292.7, 198.9)--(292.6, 199.0)--(292.4, 198.7)--(291.6, 198.1)--(291.4, 198.0)--(291.4, 197.9)--(291.3, 197.9)--(291.2, 197.9)--(291.1, 197.8)--(290.9, 198.0)--(290.7, 198.0)--(290.4, 198.0)--(289.8, 198.0)--(289.3, 198.2)--(288.9, 198.3)--(289.0, 198.5)--(289.2, 198.8)--(289.2, 198.9)--(289.4, 199.1)--(289.5, 199.4)--(289.7, 199.5)--(289.8, 199.6)--(290.0, 199.8)--(290.2, 200.0)--(290.5, 200.3)--(290.8, 200.6)--(291.4, 201.1)--(291.2, 201.4)--(290.9, 201.6)--(291.0, 201.7)--(291.8, 202.6)--(292.3, 203.1)--(292.1, 203.3)--(293.2, 204.6)--(294.0, 205.6)--(294.6, 206.3)--(294.9, 206.6)--(295.3, 207.1)--(295.5, 207.4)--(295.8, 207.7)--(295.4, 208.7)--(295.2, 209.3)--(295.2, 209.4)--(295.0, 210.5)--(295.0, 210.7)--(294.9, 211.5)--(294.9, 211.7)--(294.9, 212.3)--(294.9, 213.2)--(295.0, 214.0)--(295.1, 215.3)--(295.1, 215.7)--(295.1, 216.2)--(295.1, 216.5)--(295.1, 217.1)--(295.1, 217.5)--(295.1, 217.9)--(295.1, 218.0)--(295.1, 218.1)--(295.1, 218.2)--(295.6, 218.1)--(295.9, 218.1)--(296.3, 218.0)--(296.8, 217.9)--(297.1, 217.9)--(297.2, 217.9)--(297.3, 218.0)--(297.2, 218.3)--(297.0, 218.9)--(296.8, 219.8)--(296.6, 220.7)--(296.4, 221.8)--(296.3, 221.9)--(296.2, 221.9)--(296.0, 221.9)--(295.6, 221.8)--(295.4, 221.8)--(295.1, 222.1)--(294.5, 222.9)--(294.2, 223.3)--(294.1, 223.4)--(294.1, 223.6)--(294.0, 223.8)--(294.0, 224.1)--(293.9, 224.5)--(293.8, 225.2)--(293.9, 225.3)--(294.0, 225.6)--(294.2, 225.9)--(294.3, 226.0)--(294.4, 226.1)--(294.5, 226.2)--(294.8, 226.4)--(295.0, 226.5)--(295.2, 226.7)--(295.3, 226.9)--(295.4, 227.1)--(295.5, 227.4)--(295.6, 227.7)--(295.6, 228.0)--(295.6, 228.3)--(295.6, 228.4)--(295.6, 228.5)--(295.5, 228.7)--(295.4, 228.8)--(295.5, 228.8)--(295.5, 228.9)--(295.5, 229.1)--(295.8, 229.0)--(295.9, 229.4)--(296.0, 229.5)--(296.2, 230.0)--(296.4, 230.4)--(296.5, 230.7)--(296.5, 230.9)--(296.4, 230.9)--(296.2, 230.8)--(295.9, 230.7)--(295.5, 230.6)--(295.1, 230.6)--(294.3, 230.5)--(293.1, 230.6)--(292.6, 230.6)--(292.0, 230.6)--(291.8, 230.6)--(291.4, 230.6)--(291.2, 230.6)--(291.0, 231.1)--(290.9, 231.5)--(290.6, 232.2)--(290.5, 232.4)--(290.4, 232.7)--(290.3, 232.9)--(290.2, 233.1)--(290.0, 233.5)--(289.8, 234.0)--(289.7, 234.5)--(289.6, 235.0)--(289.6, 235.1)--(289.7, 235.2)--(289.7, 235.6)--(289.8, 236.2)--(289.8, 236.7)--(289.9, 237.1)--(290.1, 237.7)--(290.2, 238.0)--(290.2, 238.2)--(290.6, 238.6)--(290.9, 238.9)--(291.1, 239.1)--(291.3, 239.5)--(291.4, 239.6)--(291.5, 239.8)--(291.6, 240.1)--(291.7, 240.3)--(291.8, 240.5)--(291.8, 240.8)--(291.2, 240.9)--(290.9, 241.0)--(290.8, 241.0)--(290.2, 241.3)--(288.8, 241.9)--(288.3, 242.1)--(288.0, 242.2)--(288.2, 242.9)--(288.1, 242.9)--(288.0, 242.9)--(287.9, 243.9)--(287.8, 244.9)--(287.7, 245.0)--(287.8, 245.1)--(287.8, 245.3)--(287.7, 246.2)--(287.6, 246.7)--(287.4, 247.6)--(287.4, 247.7)--(287.4, 247.8)--(287.2, 248.4)--(287.0, 249.0)--(286.8, 249.5)--(286.7, 249.7)--(286.7, 250.0)--(286.4, 252.1)--(286.4, 252.2)--(286.3, 252.4)--(286.2, 252.6)--(286.1, 252.8)--(285.9, 253.2)--(285.8, 253.3)--(285.8, 253.4)--(285.7, 253.7)--(285.6, 253.9)--(285.6, 254.1)--(285.4, 254.9)--(285.4, 255.1)--(285.4, 255.4)--(285.2, 257.2)--(285.7, 257.3)--(285.6, 257.7)--(285.6, 258.8)--(285.6, 258.9)--(285.6, 259.2)--(285.8, 259.2)--(286.4, 259.6)--(287.8, 260.6)--(288.6, 261.1)--(290.6, 262.5)--(290.7, 262.5)--(291.0, 262.0)--(291.8, 262.7)--(291.6, 263.0)--(292.2, 263.4)--(292.6, 263.8)--(292.9, 263.5)--(293.0, 263.5)--(293.3, 263.7)--(293.2, 263.8)--(293.1, 263.9)--(293.2, 264.0)--(293.3, 264.1)--(293.4, 264.1)--(293.5, 264.1)--(293.7, 264.2)--(293.7, 264.1)--(293.9, 264.1)--(294.0, 264.2)--(294.2, 264.4)--(294.3, 264.6)--(294.5, 264.9)--(294.9, 265.5)--(295.1, 265.3)--(295.2, 265.2)--(295.3, 265.1)--(295.4, 265.1)--(295.5, 265.1)--(295.6, 265.1)--(295.7, 265.1)--(295.8, 265.1)--(295.9, 265.1)--(296.0, 265.3)--(296.1, 265.4)--(296.2, 265.5)--(296.4, 265.8)--(296.6, 265.9)--(296.6, 266.1)--(296.6, 266.3)--(296.6, 266.4)--(296.6, 266.5)--(296.4, 266.6)--(296.1, 266.8)--(296.3, 267.3)--(296.2, 267.3)--(296.4, 267.6)--(296.5, 267.5)--(296.7, 267.8)--(296.8, 268.0)--(296.8, 268.3)--(296.9, 268.4)--(297.0, 268.4)--(297.0, 268.6)--(297.0, 268.7)--(297.1, 269.0)--(297.3, 269.6)--(297.5, 269.9)--(297.8, 270.6)--(298.0, 271.2)--(298.1, 271.5)--(298.2, 271.5)--(298.2, 271.6)--(298.3, 271.7)--(298.3, 271.8)--(298.4, 272.1)--(298.4, 272.3)--(298.5, 272.4)--(298.6, 272.6)--(298.6, 272.7)--(298.7, 272.8)--(298.9, 273.0)--(298.9, 273.2)--(299.1, 273.5)--(299.2, 273.8)--(299.3, 273.9)--(299.3, 274.0)--(299.4, 274.0)--(299.5, 274.6)--(299.9, 274.8)--(300.1, 274.9)--(300.4, 275.1)--(300.9, 275.4)--(301.2, 275.6)--(301.3, 276.0)--(301.3, 276.5)--(301.4, 277.1)--(301.5, 277.4)--(301.6, 277.7)--(301.9, 278.3)--(301.8, 278.4)--(301.5, 278.5)--(301.0, 278.6)--(300.8, 278.6)--(300.7, 278.6)--(300.5, 278.8)--cycle; +Lewisham = (366.9, 289.6)--(367.0, 289.4)--(367.1, 289.4)--(367.0, 289.3)--(367.0, 289.2)--(367.1, 289.1)--(367.2, 288.9)--(367.3, 288.5)--(367.5, 288.2)--(367.6, 288.0)--(367.9, 287.2)--(368.2, 286.7)--(368.4, 286.3)--(368.4, 286.2)--(368.6, 285.9)--(368.9, 285.5)--(369.1, 285.1)--(369.2, 284.9)--(369.2, 284.8)--(369.3, 284.8)--(369.6, 284.4)--(369.5, 284.3)--(370.0, 283.7)--(370.1, 283.5)--(370.6, 283.1)--(371.0, 282.7)--(371.2, 282.5)--(371.3, 282.4)--(371.6, 282.2)--(371.8, 282.0)--(372.4, 281.5)--(372.3, 281.4)--(372.4, 281.4)--(372.3, 281.1)--(372.2, 280.8)--(372.2, 280.7)--(372.2, 280.4)--(372.1, 280.0)--(372.1, 279.9)--(371.9, 279.0)--(371.9, 278.1)--(371.8, 277.6)--(371.8, 277.5)--(371.7, 277.3)--(371.7, 277.2)--(371.4, 276.9)--(372.6, 276.5)--(373.7, 276.1)--(373.7, 275.5)--(373.8, 274.8)--(374.2, 274.9)--(375.7, 275.2)--(375.6, 275.5)--(375.9, 275.7)--(376.6, 276.1)--(376.8, 276.2)--(376.9, 276.2)--(377.0, 276.3)--(377.2, 276.3)--(377.4, 276.3)--(377.5, 276.1)--(377.9, 275.8)--(377.9, 275.7)--(377.7, 275.6)--(377.7, 275.5)--(376.6, 275.1)--(376.5, 275.1)--(376.5, 275.0)--(376.5, 274.9)--(376.5, 274.7)--(376.4, 274.4)--(376.5, 274.4)--(376.4, 273.4)--(376.4, 273.3)--(376.4, 273.1)--(376.2, 273.1)--(376.1, 273.1)--(376.1, 273.0)--(376.3, 273.0)--(376.4, 273.0)--(376.4, 272.7)--(376.5, 272.7)--(376.5, 272.3)--(376.5, 272.2)--(376.4, 272.2)--(376.1, 272.1)--(376.0, 272.1)--(375.6, 271.7)--(375.5, 271.6)--(375.4, 271.5)--(375.3, 271.3)--(375.4, 271.3)--(375.4, 271.2)--(375.3, 271.2)--(375.3, 271.1)--(375.2, 271.1)--(375.1, 271.1)--(375.0, 271.1)--(374.9, 271.1)--(374.2, 270.7)--(374.2, 270.8)--(374.1, 270.7)--(374.4, 270.6)--(374.5, 270.7)--(374.7, 270.7)--(374.8, 270.8)--(375.0, 270.8)--(375.1, 270.9)--(375.4, 270.9)--(375.5, 270.7)--(375.4, 270.6)--(375.4, 270.5)--(375.3, 270.4)--(375.2, 270.4)--(374.5, 270.3)--(374.4, 270.2)--(374.3, 270.2)--(374.3, 270.1)--(374.1, 270.1)--(374.1, 270.0)--(374.0, 270.0)--(374.0, 269.9)--(374.0, 269.8)--(374.1, 269.4)--(374.1, 269.2)--(374.1, 269.1)--(374.1, 268.9)--(374.3, 268.9)--(374.3, 268.7)--(374.2, 268.6)--(374.4, 268.5)--(374.4, 268.7)--(374.5, 268.7)--(374.6, 268.7)--(374.6, 268.6)--(374.7, 268.6)--(374.8, 268.5)--(374.9, 268.6)--(375.0, 268.5)--(375.1, 268.4)--(375.2, 268.3)--(375.3, 268.3)--(375.4, 268.2)--(375.5, 268.2)--(375.5, 268.1)--(375.6, 268.0)--(375.6, 267.9)--(375.7, 267.7)--(375.9, 267.7)--(376.0, 267.7)--(376.1, 267.6)--(376.1, 267.5)--(376.1, 267.4)--(376.1, 267.3)--(376.1, 267.2)--(376.0, 267.1)--(375.9, 266.9)--(375.9, 266.8)--(375.9, 266.7)--(376.0, 266.6)--(376.1, 266.5)--(376.2, 266.5)--(376.2, 266.4)--(376.6, 266.5)--(376.6, 266.4)--(376.7, 266.3)--(376.8, 265.8)--(376.8, 265.6)--(376.9, 265.5)--(377.0, 265.5)--(377.0, 265.4)--(376.9, 265.4)--(376.9, 265.2)--(377.0, 265.0)--(377.1, 264.9)--(376.9, 264.8)--(376.9, 264.9)--(376.8, 264.8)--(376.7, 264.8)--(376.8, 264.5)--(376.9, 264.3)--(376.8, 264.1)--(377.1, 263.9)--(377.3, 263.8)--(377.5, 263.7)--(377.7, 263.6)--(377.8, 263.6)--(378.0, 263.6)--(378.1, 263.6)--(378.2, 263.5)--(378.2, 263.4)--(378.4, 263.2)--(378.6, 263.4)--(378.7, 263.2)--(378.7, 263.1)--(378.8, 263.0)--(379.1, 262.6)--(379.2, 262.6)--(379.4, 262.4)--(379.4, 262.3)--(379.5, 262.2)--(379.7, 262.1)--(380.0, 262.3)--(380.1, 262.4)--(380.4, 262.6)--(380.8, 262.9)--(381.0, 263.1)--(380.8, 263.6)--(380.6, 263.9)--(380.5, 264.1)--(380.4, 264.3)--(380.3, 264.4)--(380.1, 265.0)--(379.7, 265.8)--(379.5, 266.2)--(379.8, 266.4)--(380.0, 266.4)--(379.9, 266.7)--(380.1, 267.1)--(380.3, 267.3)--(380.5, 267.1)--(380.8, 267.0)--(380.9, 266.9)--(381.1, 266.9)--(381.3, 266.9)--(381.9, 267.0)--(382.9, 267.2)--(383.0, 267.3)--(383.7, 267.4)--(383.8, 267.4)--(384.0, 267.5)--(384.4, 267.5)--(384.9, 267.3)--(385.2, 267.3)--(385.4, 267.3)--(386.1, 267.1)--(387.5, 266.8)--(388.4, 266.7)--(389.0, 266.6)--(389.1, 266.6)--(389.2, 266.6)--(389.4, 266.6)--(389.8, 266.6)--(390.1, 266.6)--(390.3, 266.7)--(390.7, 266.7)--(391.2, 266.8)--(392.0, 267.0)--(392.7, 267.1)--(393.2, 267.3)--(393.8, 267.4)--(395.2, 267.7)--(396.4, 267.9)--(397.1, 268.0)--(398.4, 268.3)--(399.4, 268.5)--(400.0, 268.6)--(400.6, 268.8)--(401.8, 269.0)--(402.5, 269.2)--(402.6, 269.2)--(402.3, 269.0)--(401.5, 268.7)--(401.3, 268.6)--(401.1, 268.5)--(400.4, 268.3)--(400.4, 268.0)--(400.4, 267.7)--(400.5, 267.4)--(400.6, 267.0)--(400.7, 266.6)--(400.8, 266.3)--(401.1, 265.5)--(401.3, 265.1)--(401.4, 265.0)--(401.4, 264.9)--(401.4, 264.8)--(401.5, 264.8)--(401.5, 264.7)--(401.7, 264.5)--(401.7, 264.3)--(401.7, 264.1)--(401.6, 263.9)--(401.5, 263.7)--(401.3, 263.9)--(401.2, 263.9)--(401.1, 264.0)--(400.9, 264.1)--(400.7, 264.2)--(400.5, 264.4)--(400.2, 264.5)--(400.1, 264.5)--(399.9, 264.6)--(399.7, 264.6)--(399.6, 264.6)--(399.5, 264.6)--(399.5, 264.7)--(399.1, 264.6)--(398.9, 264.6)--(399.1, 263.5)--(399.2, 263.1)--(399.2, 262.9)--(399.2, 262.7)--(399.2, 262.6)--(399.2, 262.5)--(399.1, 262.3)--(399.0, 262.2)--(399.0, 262.0)--(399.0, 261.9)--(399.1, 261.8)--(399.2, 261.5)--(399.3, 261.4)--(399.4, 261.3)--(399.3, 261.3)--(399.2, 261.2)--(399.1, 261.2)--(399.1, 261.1)--(398.9, 261.1)--(398.6, 260.9)--(398.4, 260.7)--(398.3, 260.6)--(398.2, 260.6)--(398.1, 260.5)--(397.8, 260.5)--(397.7, 260.4)--(397.5, 260.4)--(397.1, 260.3)--(396.6, 260.1)--(396.6, 259.8)--(396.6, 259.5)--(396.6, 259.3)--(396.5, 259.0)--(396.5, 258.9)--(396.5, 258.8)--(396.6, 258.7)--(396.6, 258.6)--(396.7, 258.2)--(396.7, 258.1)--(396.8, 257.9)--(396.8, 257.7)--(396.9, 257.3)--(397.0, 257.0)--(397.1, 256.7)--(397.1, 256.4)--(397.2, 256.2)--(397.2, 255.9)--(397.2, 255.5)--(397.2, 255.2)--(397.2, 254.9)--(397.2, 254.6)--(397.2, 254.0)--(397.3, 253.4)--(397.3, 253.1)--(397.3, 252.9)--(397.5, 252.6)--(397.6, 252.2)--(397.7, 251.9)--(397.9, 251.5)--(398.0, 251.3)--(398.1, 250.9)--(398.3, 250.5)--(398.4, 250.3)--(398.5, 250.2)--(398.5, 250.1)--(398.5, 250.0)--(398.5, 249.9)--(398.6, 249.8)--(398.8, 249.7)--(399.1, 249.6)--(399.3, 249.4)--(399.6, 249.3)--(399.7, 249.2)--(399.9, 249.1)--(400.3, 249.0)--(400.4, 248.9)--(400.5, 248.9)--(400.8, 248.8)--(401.2, 248.7)--(401.7, 248.6)--(402.7, 248.3)--(402.7, 248.0)--(402.7, 247.9)--(402.7, 247.8)--(402.7, 247.7)--(402.7, 247.6)--(402.7, 247.5)--(402.7, 247.4)--(402.8, 247.3)--(402.8, 247.2)--(402.8, 246.9)--(402.9, 246.7)--(403.0, 246.3)--(403.1, 246.1)--(403.1, 246.0)--(403.2, 246.0)--(403.4, 244.4)--(404.1, 244.6)--(404.2, 244.6)--(404.3, 244.2)--(404.4, 244.0)--(404.4, 243.9)--(404.5, 243.5)--(401.2, 242.9)--(401.1, 243.3)--(401.0, 243.3)--(401.2, 242.6)--(402.7, 242.1)--(403.2, 242.0)--(404.1, 241.9)--(405.0, 241.7)--(405.7, 241.6)--(406.0, 241.6)--(406.1, 241.3)--(406.2, 241.1)--(406.2, 241.0)--(406.3, 240.8)--(406.3, 240.7)--(406.4, 240.6)--(406.5, 240.1)--(406.7, 239.9)--(406.7, 239.6)--(406.6, 239.6)--(405.8, 239.1)--(405.6, 239.0)--(405.6, 237.9)--(405.6, 237.4)--(405.6, 237.2)--(405.8, 236.8)--(405.8, 236.5)--(405.2, 236.3)--(405.6, 235.4)--(405.9, 234.6)--(406.3, 234.5)--(406.8, 234.3)--(407.1, 234.1)--(407.4, 234.0)--(407.8, 233.8)--(407.9, 233.8)--(408.1, 233.7)--(408.3, 233.7)--(408.6, 233.6)--(409.2, 233.5)--(409.5, 233.5)--(409.6, 233.5)--(409.9, 233.4)--(410.3, 233.4)--(410.6, 233.4)--(410.8, 233.5)--(410.9, 233.5)--(411.0, 233.6)--(411.1, 233.6)--(411.4, 233.7)--(411.4, 233.6)--(411.2, 233.4)--(411.2, 233.3)--(411.1, 233.2)--(411.1, 233.0)--(411.0, 232.8)--(410.9, 232.5)--(410.9, 232.2)--(410.9, 232.0)--(411.0, 231.5)--(411.1, 231.1)--(411.1, 230.9)--(411.1, 230.7)--(411.1, 230.4)--(411.0, 230.4)--(409.1, 229.9)--(409.1, 230.0)--(408.3, 229.7)--(408.3, 229.6)--(408.3, 229.4)--(408.4, 229.0)--(408.4, 228.6)--(408.4, 228.4)--(408.5, 228.1)--(408.6, 228.1)--(408.7, 227.3)--(408.7, 227.2)--(408.8, 226.9)--(408.9, 226.9)--(409.0, 227.2)--(409.4, 227.5)--(410.1, 227.9)--(410.3, 227.9)--(410.4, 227.9)--(410.4, 227.8)--(410.6, 227.8)--(411.0, 227.8)--(411.1, 227.8)--(411.1, 227.7)--(411.2, 227.2)--(411.3, 226.7)--(411.3, 226.3)--(411.4, 225.8)--(411.4, 225.5)--(411.3, 225.3)--(411.1, 225.1)--(411.4, 224.8)--(411.8, 224.6)--(412.0, 224.4)--(412.1, 224.2)--(412.2, 224.1)--(412.3, 224.1)--(412.4, 224.0)--(412.9, 223.7)--(413.0, 223.6)--(413.1, 223.5)--(412.8, 223.1)--(412.9, 223.0)--(413.0, 223.2)--(413.1, 223.3)--(413.2, 223.5)--(413.6, 223.3)--(413.7, 223.0)--(413.8, 222.8)--(414.4, 223.2)--(414.7, 222.6)--(415.3, 223.1)--(417.1, 224.3)--(417.4, 224.5)--(417.4, 224.0)--(417.4, 223.9)--(417.4, 223.5)--(417.3, 223.4)--(417.3, 223.3)--(417.3, 223.1)--(417.3, 222.8)--(417.3, 222.6)--(417.4, 222.4)--(417.4, 222.3)--(417.4, 222.1)--(417.5, 222.0)--(417.5, 221.9)--(417.6, 221.5)--(417.6, 221.4)--(417.6, 221.3)--(417.7, 221.1)--(417.7, 220.8)--(417.9, 220.3)--(417.8, 220.2)--(417.9, 219.9)--(418.0, 219.7)--(418.0, 219.5)--(418.0, 218.6)--(417.9, 218.5)--(417.7, 218.3)--(417.7, 218.1)--(418.1, 218.1)--(418.1, 217.7)--(418.2, 217.1)--(418.3, 216.9)--(418.4, 216.9)--(418.4, 216.6)--(418.4, 216.5)--(418.5, 216.5)--(418.6, 216.0)--(418.6, 215.9)--(418.6, 215.8)--(418.6, 215.7)--(418.6, 215.4)--(418.6, 215.0)--(418.6, 214.9)--(418.5, 214.9)--(418.4, 214.9)--(417.6, 213.6)--(416.7, 212.3)--(415.8, 213.1)--(415.6, 213.4)--(415.4, 213.6)--(415.1, 213.9)--(415.0, 214.2)--(414.8, 214.3)--(414.6, 214.5)--(414.4, 214.7)--(414.1, 214.9)--(414.0, 215.0)--(413.8, 215.1)--(413.3, 215.5)--(413.2, 215.7)--(412.9, 215.9)--(412.5, 216.1)--(412.4, 216.2)--(412.3, 216.3)--(412.0, 216.7)--(411.7, 217.0)--(411.2, 217.5)--(410.3, 218.5)--(410.1, 218.6)--(409.5, 219.2)--(409.0, 219.4)--(408.8, 219.2)--(408.4, 218.9)--(408.2, 218.8)--(408.1, 218.7)--(407.9, 218.6)--(407.9, 218.5)--(407.7, 218.4)--(407.5, 218.2)--(407.3, 218.0)--(407.1, 217.8)--(406.9, 217.7)--(406.6, 217.5)--(406.5, 217.4)--(406.4, 217.3)--(406.4, 217.2)--(406.3, 217.2)--(406.1, 217.0)--(406.0, 216.9)--(405.5, 217.0)--(405.3, 217.1)--(405.1, 217.2)--(405.0, 217.1)--(404.5, 217.2)--(404.3, 217.1)--(404.2, 217.0)--(404.0, 217.0)--(403.9, 217.0)--(403.7, 217.0)--(403.5, 216.9)--(403.4, 216.8)--(403.2, 216.6)--(402.6, 216.2)--(402.3, 216.1)--(401.9, 216.1)--(401.6, 216.1)--(401.5, 216.1)--(401.2, 216.1)--(401.4, 215.5)--(401.5, 215.0)--(400.9, 214.9)--(400.8, 214.4)--(400.0, 213.8)--(399.7, 213.7)--(399.6, 213.3)--(399.3, 213.2)--(399.2, 213.0)--(399.0, 212.8)--(398.6, 212.1)--(398.5, 211.9)--(397.7, 212.3)--(397.6, 212.4)--(397.2, 212.5)--(397.0, 211.9)--(396.8, 211.1)--(396.7, 210.6)--(396.4, 210.7)--(396.3, 210.8)--(396.3, 210.6)--(396.3, 210.4)--(396.4, 210.4)--(396.3, 210.1)--(396.0, 210.2)--(395.9, 209.9)--(395.8, 209.8)--(395.8, 209.7)--(395.8, 209.6)--(395.8, 209.4)--(395.7, 209.3)--(395.7, 209.2)--(395.6, 209.0)--(395.5, 208.9)--(395.5, 208.7)--(395.3, 208.3)--(395.5, 208.1)--(395.4, 208.0)--(395.4, 207.9)--(395.5, 207.6)--(395.6, 207.4)--(395.5, 207.3)--(395.3, 207.1)--(394.1, 206.1)--(393.9, 206.3)--(392.9, 207.4)--(392.4, 207.1)--(392.3, 206.9)--(392.1, 206.8)--(392.1, 206.7)--(391.8, 206.5)--(391.6, 206.3)--(391.6, 206.2)--(391.6, 206.1)--(391.6, 206.0)--(391.5, 205.7)--(391.5, 205.1)--(391.5, 205.0)--(391.5, 204.9)--(391.4, 204.8)--(391.1, 204.6)--(391.2, 204.6)--(391.2, 204.4)--(389.4, 204.1)--(389.4, 204.0)--(389.3, 204.0)--(389.3, 203.7)--(389.5, 203.4)--(388.3, 202.8)--(388.3, 202.9)--(388.3, 203.0)--(388.2, 203.0)--(388.2, 203.1)--(388.2, 203.2)--(388.1, 203.3)--(388.0, 203.6)--(387.9, 203.8)--(387.9, 204.2)--(387.6, 204.4)--(387.5, 204.4)--(387.4, 204.4)--(387.2, 204.4)--(387.1, 203.8)--(387.1, 203.5)--(387.1, 203.2)--(387.2, 202.9)--(386.5, 202.7)--(386.4, 202.7)--(386.3, 202.6)--(386.1, 202.6)--(385.9, 202.5)--(385.7, 202.4)--(385.6, 202.3)--(385.5, 202.2)--(385.4, 202.1)--(385.4, 202.0)--(384.5, 201.6)--(383.6, 201.6)--(382.9, 201.6)--(382.4, 201.6)--(381.9, 201.6)--(381.9, 201.8)--(381.8, 201.8)--(381.8, 201.9)--(381.7, 201.9)--(381.8, 203.0)--(381.8, 203.1)--(381.9, 203.1)--(381.9, 203.2)--(381.7, 203.2)--(381.2, 203.2)--(380.8, 203.1)--(380.3, 203.1)--(380.0, 203.1)--(379.7, 203.1)--(379.3, 203.1)--(379.3, 203.9)--(377.7, 204.1)--(377.6, 204.1)--(377.6, 204.2)--(377.5, 204.2)--(377.4, 204.1)--(377.1, 204.1)--(377.1, 204.2)--(377.1, 204.3)--(376.8, 204.5)--(376.6, 204.6)--(376.5, 204.8)--(376.3, 204.9)--(376.6, 205.5)--(376.0, 205.8)--(376.4, 206.2)--(376.6, 206.4)--(376.7, 206.6)--(376.8, 206.8)--(376.9, 207.0)--(376.9, 207.3)--(376.9, 207.8)--(376.9, 207.9)--(376.9, 208.0)--(377.0, 208.1)--(377.0, 208.2)--(377.0, 208.3)--(377.1, 208.4)--(377.2, 208.5)--(377.0, 208.7)--(376.8, 208.8)--(376.6, 209.0)--(376.5, 209.0)--(376.6, 209.1)--(377.0, 209.6)--(376.8, 209.9)--(376.2, 210.4)--(376.1, 210.5)--(376.0, 210.4)--(375.9, 210.3)--(375.8, 210.4)--(375.6, 210.7)--(375.5, 210.9)--(375.5, 211.0)--(375.2, 211.3)--(375.1, 211.5)--(374.9, 211.7)--(374.7, 211.9)--(374.6, 212.0)--(374.3, 212.1)--(374.1, 212.2)--(373.8, 212.3)--(373.7, 212.4)--(373.7, 212.5)--(373.6, 212.5)--(373.6, 212.6)--(373.4, 212.5)--(373.3, 212.6)--(372.9, 212.9)--(372.7, 213.0)--(372.6, 213.1)--(372.4, 213.2)--(372.2, 213.4)--(372.2, 213.5)--(372.1, 213.6)--(372.1, 213.7)--(371.7, 213.9)--(371.0, 214.4)--(370.9, 214.5)--(370.7, 214.5)--(370.5, 214.7)--(370.4, 214.7)--(370.3, 214.7)--(370.2, 214.7)--(369.9, 214.7)--(369.2, 214.5)--(369.3, 214.2)--(369.4, 213.6)--(369.0, 213.6)--(368.5, 213.5)--(368.3, 213.5)--(368.1, 213.5)--(368.1, 213.6)--(367.7, 213.6)--(367.7, 214.0)--(367.5, 213.9)--(367.2, 213.9)--(366.7, 213.9)--(366.5, 213.8)--(366.3, 213.8)--(366.1, 213.8)--(365.4, 213.8)--(365.2, 213.5)--(365.0, 213.1)--(364.9, 212.8)--(364.8, 212.6)--(364.7, 212.7)--(364.4, 212.8)--(364.0, 212.9)--(363.8, 212.9)--(363.7, 212.9)--(363.5, 213.0)--(363.2, 213.0)--(362.9, 212.9)--(362.9, 212.8)--(362.8, 212.7)--(362.5, 212.5)--(362.5, 212.4)--(362.6, 212.2)--(361.7, 211.8)--(361.7, 211.7)--(361.7, 211.9)--(361.6, 211.9)--(361.4, 211.8)--(361.3, 211.7)--(361.1, 211.7)--(361.1, 211.6)--(361.2, 211.4)--(360.7, 211.2)--(359.4, 210.7)--(359.4, 211.0)--(359.6, 211.0)--(359.6, 211.2)--(358.9, 211.2)--(358.7, 211.2)--(358.7, 211.3)--(357.8, 211.2)--(357.7, 211.2)--(357.4, 211.1)--(357.2, 211.1)--(356.9, 211.1)--(356.8, 211.0)--(356.5, 211.0)--(356.4, 210.9)--(356.3, 210.9)--(355.9, 210.8)--(355.6, 210.8)--(355.6, 211.0)--(355.6, 211.3)--(355.5, 212.0)--(355.0, 212.0)--(353.4, 212.1)--(353.4, 211.8)--(353.2, 211.2)--(352.7, 211.3)--(352.7, 211.2)--(352.4, 211.2)--(352.3, 211.2)--(352.2, 211.2)--(352.1, 211.2)--(351.9, 211.2)--(351.4, 211.3)--(351.1, 211.4)--(351.1, 211.5)--(350.9, 211.7)--(350.3, 212.0)--(350.1, 212.1)--(349.9, 212.2)--(349.8, 212.2)--(348.8, 212.8)--(348.7, 212.8)--(348.7, 212.9)--(348.7, 213.0)--(348.8, 213.1)--(348.6, 213.2)--(348.4, 213.4)--(348.3, 213.4)--(348.1, 213.5)--(347.9, 213.6)--(347.9, 213.5)--(347.8, 213.6)--(347.7, 213.6)--(347.4, 213.6)--(347.4, 213.7)--(347.3, 213.7)--(346.7, 213.7)--(346.6, 213.4)--(346.5, 213.4)--(346.4, 213.5)--(346.4, 213.6)--(346.2, 213.6)--(346.1, 213.6)--(345.9, 213.6)--(345.7, 213.7)--(345.5, 213.4)--(345.1, 213.7)--(344.9, 213.7)--(344.6, 213.9)--(344.2, 214.2)--(344.2, 214.3)--(344.0, 214.3)--(343.7, 214.3)--(343.6, 214.2)--(343.6, 213.9)--(343.3, 214.1)--(343.1, 214.2)--(342.6, 214.4)--(342.5, 214.4)--(342.5, 214.5)--(342.4, 214.5)--(342.3, 214.5)--(342.2, 214.6)--(342.1, 214.6)--(342.0, 214.6)--(341.9, 214.6)--(341.8, 214.6)--(341.7, 214.6)--(341.4, 214.6)--(341.2, 214.6)--(341.1, 214.6)--(340.8, 214.6)--(340.6, 214.5)--(340.4, 214.5)--(340.2, 214.5)--(340.1, 214.4)--(340.0, 214.4)--(340.0, 214.5)--(340.0, 214.8)--(339.9, 215.4)--(339.7, 215.9)--(339.6, 216.1)--(339.4, 216.6)--(339.3, 216.9)--(339.2, 217.1)--(339.2, 217.3)--(339.2, 217.8)--(339.2, 218.2)--(339.3, 218.3)--(339.3, 218.4)--(340.1, 220.2)--(340.6, 221.0)--(340.7, 221.2)--(340.8, 221.3)--(340.9, 221.4)--(341.1, 221.5)--(341.5, 221.5)--(341.7, 221.5)--(341.8, 221.5)--(341.9, 221.5)--(342.1, 221.5)--(342.3, 221.5)--(342.5, 221.6)--(342.6, 221.6)--(342.9, 221.7)--(342.9, 221.8)--(343.3, 222.0)--(343.6, 222.1)--(344.1, 222.6)--(344.5, 222.9)--(344.8, 223.2)--(345.2, 223.6)--(345.3, 223.7)--(345.3, 223.8)--(345.4, 223.9)--(345.4, 224.0)--(345.5, 224.1)--(345.6, 224.5)--(345.7, 224.9)--(345.9, 225.4)--(346.1, 225.9)--(346.2, 226.3)--(346.2, 226.4)--(346.3, 226.5)--(346.3, 226.6)--(346.3, 226.7)--(346.4, 226.8)--(346.5, 226.9)--(346.6, 227.0)--(346.9, 227.2)--(347.0, 227.3)--(347.0, 227.4)--(346.9, 227.5)--(346.8, 227.7)--(346.8, 227.8)--(346.8, 228.0)--(346.7, 228.9)--(346.7, 229.5)--(346.5, 230.9)--(346.3, 231.0)--(346.0, 231.1)--(345.8, 231.3)--(346.1, 231.7)--(346.3, 232.1)--(346.4, 232.2)--(346.4, 232.3)--(346.6, 232.8)--(346.6, 233.2)--(346.7, 233.7)--(346.6, 234.6)--(346.6, 235.1)--(346.7, 235.4)--(346.7, 235.9)--(346.8, 236.2)--(346.9, 236.5)--(347.0, 236.9)--(347.1, 237.2)--(347.3, 237.8)--(347.5, 238.0)--(347.6, 238.2)--(347.7, 238.4)--(347.8, 238.7)--(348.1, 239.1)--(348.4, 239.4)--(349.3, 240.1)--(349.7, 240.4)--(350.0, 240.7)--(350.4, 241.2)--(350.7, 241.6)--(351.4, 241.2)--(352.2, 240.7)--(352.7, 240.4)--(353.0, 240.2)--(353.2, 240.0)--(353.2, 239.9)--(353.3, 239.9)--(353.4, 239.9)--(353.5, 239.9)--(353.6, 239.9)--(353.9, 240.0)--(355.3, 240.8)--(356.0, 241.2)--(356.4, 241.4)--(356.6, 241.5)--(356.9, 241.5)--(357.2, 241.5)--(357.5, 241.5)--(358.1, 241.4)--(358.4, 241.4)--(358.7, 241.3)--(358.8, 241.7)--(359.1, 242.2)--(359.3, 242.5)--(359.3, 242.6)--(359.3, 242.8)--(359.4, 242.9)--(359.5, 242.9)--(359.7, 243.4)--(360.2, 244.2)--(360.3, 244.5)--(360.6, 245.0)--(360.8, 245.1)--(361.1, 245.8)--(361.6, 247.1)--(361.5, 247.1)--(361.4, 247.2)--(361.3, 247.3)--(360.9, 247.6)--(360.4, 248.1)--(359.5, 248.8)--(359.3, 248.9)--(358.4, 249.2)--(358.5, 249.4)--(358.7, 250.0)--(358.7, 250.1)--(358.7, 250.2)--(358.8, 250.5)--(359.0, 251.5)--(359.0, 251.6)--(359.0, 251.9)--(359.0, 252.4)--(359.1, 252.9)--(359.1, 253.3)--(359.1, 253.6)--(359.0, 254.1)--(359.0, 254.6)--(358.9, 255.0)--(358.8, 255.5)--(358.7, 255.9)--(358.5, 256.5)--(358.3, 257.0)--(358.2, 257.3)--(358.1, 257.5)--(357.9, 257.7)--(357.8, 257.8)--(357.6, 258.0)--(357.4, 258.1)--(357.2, 258.3)--(357.1, 258.4)--(356.5, 258.8)--(356.8, 259.1)--(356.5, 259.3)--(356.2, 259.4)--(355.9, 259.6)--(355.8, 259.6)--(355.7, 259.6)--(355.6, 259.7)--(355.3, 259.8)--(355.1, 259.8)--(354.9, 259.9)--(355.0, 260.0)--(355.1, 260.0)--(354.8, 260.4)--(354.5, 260.7)--(354.7, 260.8)--(354.8, 260.9)--(354.8, 261.1)--(354.4, 261.8)--(354.3, 262.0)--(354.0, 262.4)--(353.8, 262.7)--(353.9, 262.8)--(354.2, 263.2)--(354.2, 263.3)--(354.1, 263.7)--(354.0, 264.3)--(353.8, 264.9)--(353.4, 266.8)--(353.4, 267.8)--(353.3, 270.6)--(353.3, 272.5)--(352.1, 273.1)--(352.4, 273.8)--(352.3, 273.8)--(352.5, 274.4)--(352.5, 274.5)--(352.6, 274.5)--(352.6, 274.4)--(352.6, 274.6)--(352.7, 274.8)--(352.7, 274.9)--(352.6, 274.9)--(352.6, 275.0)--(352.6, 275.1)--(352.7, 275.0)--(352.7, 275.1)--(352.6, 275.1)--(352.6, 275.2)--(352.7, 275.2)--(352.7, 275.3)--(352.8, 275.5)--(352.8, 275.6)--(352.8, 275.7)--(352.8, 275.8)--(352.9, 275.8)--(352.8, 275.8)--(352.8, 275.9)--(352.9, 276.3)--(352.9, 276.6)--(352.9, 276.9)--(353.0, 277.4)--(353.0, 277.6)--(353.1, 277.6)--(353.1, 277.7)--(353.0, 277.7)--(353.0, 277.8)--(353.0, 277.9)--(352.9, 278.5)--(353.0, 278.5)--(353.0, 278.7)--(352.9, 278.7)--(352.8, 279.5)--(352.9, 279.6)--(352.9, 279.8)--(352.8, 280.1)--(352.8, 280.4)--(352.7, 280.4)--(352.7, 280.5)--(352.6, 280.5)--(352.6, 280.6)--(352.5, 280.8)--(352.4, 281.1)--(352.7, 281.2)--(352.7, 281.3)--(352.8, 281.3)--(352.9, 281.4)--(352.9, 281.5)--(352.9, 281.7)--(352.9, 281.8)--(352.9, 281.9)--(352.9, 282.0)--(352.9, 282.1)--(352.9, 282.2)--(352.8, 282.3)--(352.7, 282.4)--(352.6, 282.4)--(352.6, 282.5)--(352.5, 282.5)--(352.4, 282.7)--(352.4, 282.8)--(352.3, 283.0)--(352.1, 283.5)--(352.9, 283.2)--(353.0, 283.1)--(353.1, 283.3)--(353.2, 283.3)--(353.3, 283.4)--(353.4, 283.4)--(353.6, 283.4)--(353.9, 283.4)--(354.1, 283.4)--(354.3, 283.5)--(354.4, 283.5)--(354.5, 283.6)--(354.6, 283.6)--(354.6, 283.7)--(354.6, 283.8)--(354.8, 284.0)--(354.7, 284.0)--(353.5, 284.9)--(353.8, 285.2)--(354.0, 285.3)--(354.2, 285.4)--(354.6, 285.6)--(355.0, 285.7)--(355.0, 285.8)--(355.2, 285.9)--(355.3, 285.9)--(356.5, 286.3)--(356.4, 286.4)--(357.2, 286.8)--(357.2, 286.5)--(357.2, 286.3)--(357.4, 286.3)--(357.6, 286.2)--(357.7, 286.1)--(357.9, 286.2)--(358.2, 286.3)--(358.2, 286.4)--(358.4, 286.5)--(358.4, 286.4)--(358.6, 286.1)--(359.8, 286.0)--(360.1, 286.0)--(360.2, 286.0)--(360.0, 286.3)--(361.5, 287.2)--(360.9, 288.1)--(361.6, 288.5)--(361.3, 289.1)--(361.3, 289.2)--(361.4, 289.2)--(361.7, 289.3)--(361.8, 289.4)--(361.9, 289.4)--(362.0, 289.4)--(362.1, 289.4)--(362.1, 289.3)--(362.2, 289.3)--(362.3, 289.3)--(362.4, 289.3)--(362.4, 289.2)--(362.5, 289.2)--(362.5, 289.1)--(362.6, 289.1)--(362.7, 289.0)--(362.8, 288.9)--(362.9, 288.8)--(363.0, 288.8)--(363.0, 288.7)--(363.1, 288.7)--(363.1, 288.6)--(363.2, 288.6)--(363.3, 288.6)--(363.4, 288.6)--(363.5, 288.6)--(365.0, 289.1)--(366.3, 289.5)--(366.3, 289.4)--(366.6, 289.4)--(366.8, 289.5)--(366.8, 289.6)--cycle; +Merton = (299.1, 174.2)--(299.0, 174.1)--(298.8, 174.2)--(298.7, 174.2)--(298.6, 174.1)--(298.5, 174.1)--(298.2, 173.9)--(298.0, 173.8)--(297.8, 173.7)--(297.6, 173.6)--(297.4, 173.5)--(297.3, 173.4)--(297.0, 173.4)--(296.9, 173.4)--(296.8, 173.4)--(296.6, 173.3)--(296.5, 173.3)--(295.8, 172.9)--(295.6, 172.9)--(295.4, 172.8)--(295.3, 172.8)--(295.2, 172.8)--(295.0, 172.8)--(294.8, 172.7)--(294.7, 172.7)--(294.4, 172.6)--(293.9, 172.3)--(293.8, 172.3)--(293.7, 172.5)--(293.6, 172.6)--(292.9, 173.4)--(292.1, 173.7)--(290.4, 173.4)--(290.3, 173.3)--(290.2, 173.2)--(290.1, 173.2)--(289.5, 173.1)--(288.0, 172.8)--(287.0, 172.6)--(286.5, 172.5)--(286.5, 172.6)--(286.3, 172.6)--(286.2, 172.5)--(285.3, 172.3)--(284.2, 172.1)--(284.0, 172.1)--(282.9, 171.8)--(282.8, 171.9)--(282.7, 171.9)--(282.6, 171.9)--(282.5, 171.8)--(282.5, 171.7)--(282.4, 171.7)--(282.4, 171.6)--(282.4, 171.4)--(282.3, 171.4)--(282.2, 171.4)--(282.2, 171.5)--(282.1, 171.5)--(281.9, 171.4)--(281.5, 171.1)--(281.2, 171.0)--(281.0, 170.8)--(280.8, 170.5)--(280.6, 170.2)--(280.4, 170.4)--(280.2, 170.5)--(280.0, 170.6)--(279.9, 170.6)--(279.3, 170.6)--(279.3, 170.5)--(279.2, 170.5)--(279.1, 170.5)--(279.0, 170.5)--(278.9, 170.4)--(278.9, 170.5)--(278.8, 170.5)--(278.7, 170.5)--(278.6, 170.5)--(278.5, 170.5)--(278.4, 170.5)--(278.3, 170.5)--(278.2, 170.5)--(278.1, 170.4)--(277.7, 170.4)--(277.6, 170.3)--(277.5, 170.4)--(277.4, 170.7)--(277.3, 170.9)--(277.2, 171.1)--(277.0, 171.3)--(277.0, 171.6)--(276.9, 171.9)--(276.8, 172.0)--(276.8, 172.1)--(276.7, 172.2)--(276.7, 172.3)--(276.6, 172.3)--(276.4, 172.3)--(276.3, 172.4)--(276.1, 172.5)--(275.9, 172.6)--(275.8, 172.8)--(275.6, 173.1)--(275.5, 173.2)--(275.4, 173.4)--(275.1, 173.6)--(275.0, 173.6)--(274.8, 173.7)--(274.3, 174.1)--(274.3, 174.2)--(274.2, 174.4)--(274.1, 174.5)--(274.1, 174.6)--(273.7, 174.9)--(273.5, 175.1)--(273.4, 175.2)--(273.3, 175.3)--(273.2, 175.2)--(273.1, 175.3)--(272.9, 175.3)--(272.7, 175.3)--(272.5, 175.4)--(272.2, 175.4)--(272.1, 175.5)--(271.8, 175.6)--(271.4, 175.8)--(271.2, 176.0)--(270.8, 176.2)--(270.3, 175.6)--(269.7, 174.7)--(268.9, 173.6)--(268.1, 172.5)--(267.9, 172.3)--(267.8, 172.2)--(267.7, 172.0)--(267.4, 171.8)--(267.3, 171.9)--(267.0, 171.6)--(266.4, 171.0)--(265.8, 170.4)--(265.6, 170.2)--(265.5, 170.1)--(265.4, 170.1)--(265.4, 170.0)--(265.2, 170.0)--(264.5, 169.7)--(264.4, 169.6)--(264.2, 169.5)--(263.6, 168.9)--(263.2, 168.6)--(262.5, 168.0)--(261.6, 167.1)--(261.2, 167.5)--(260.8, 167.6)--(260.8, 167.7)--(260.7, 167.8)--(260.6, 167.9)--(260.4, 167.8)--(260.1, 167.9)--(260.2, 168.1)--(260.1, 168.2)--(259.9, 168.4)--(259.8, 168.5)--(259.7, 168.6)--(259.3, 168.9)--(259.1, 168.7)--(258.3, 169.7)--(258.2, 169.7)--(257.9, 169.5)--(257.7, 169.7)--(256.8, 168.9)--(256.7, 168.8)--(256.6, 168.8)--(256.0, 168.5)--(255.9, 168.4)--(255.0, 167.7)--(253.8, 166.8)--(253.3, 166.4)--(253.2, 166.3)--(252.0, 167.7)--(251.0, 168.8)--(250.0, 167.6)--(249.3, 168.0)--(248.9, 168.3)--(248.4, 168.6)--(248.2, 168.8)--(248.1, 168.9)--(248.0, 169.0)--(247.4, 169.4)--(247.3, 169.4)--(247.3, 169.5)--(247.1, 169.5)--(247.0, 169.6)--(246.9, 169.6)--(246.8, 169.7)--(246.6, 169.9)--(246.6, 169.7)--(246.5, 169.7)--(246.4, 169.6)--(246.4, 169.5)--(246.4, 169.4)--(246.3, 169.2)--(246.2, 169.2)--(246.1, 169.1)--(245.9, 168.7)--(245.8, 168.5)--(245.6, 168.2)--(245.4, 167.9)--(245.3, 167.7)--(245.2, 167.5)--(245.1, 167.4)--(245.0, 167.2)--(244.9, 167.0)--(244.7, 166.8)--(244.6, 166.6)--(244.5, 166.5)--(244.4, 166.4)--(244.2, 166.2)--(244.4, 166.1)--(244.5, 166.0)--(244.2, 165.6)--(244.2, 165.7)--(244.1, 165.7)--(243.8, 165.2)--(243.9, 165.2)--(244.0, 165.1)--(243.3, 164.2)--(242.8, 163.5)--(242.5, 163.0)--(242.3, 162.6)--(242.1, 162.4)--(242.0, 162.2)--(241.7, 161.8)--(241.5, 161.5)--(241.4, 161.3)--(241.2, 161.0)--(241.1, 160.9)--(241.0, 160.7)--(240.9, 160.8)--(240.8, 160.8)--(240.6, 161.0)--(240.3, 161.2)--(240.2, 161.3)--(239.9, 161.3)--(239.8, 161.3)--(239.7, 161.4)--(239.6, 161.4)--(239.5, 161.5)--(239.3, 161.5)--(239.0, 161.7)--(238.3, 162.2)--(236.5, 163.4)--(234.3, 164.8)--(233.4, 165.5)--(232.3, 166.6)--(231.9, 167.0)--(231.2, 167.7)--(230.0, 168.7)--(229.8, 168.9)--(230.0, 169.0)--(229.8, 169.4)--(229.8, 169.6)--(229.7, 169.8)--(229.6, 170.0)--(229.6, 170.3)--(229.5, 170.7)--(229.4, 171.2)--(229.4, 171.3)--(229.4, 171.5)--(229.3, 172.0)--(228.1, 171.6)--(227.9, 171.5)--(227.8, 171.5)--(227.2, 171.3)--(226.8, 171.1)--(226.6, 171.0)--(226.5, 171.0)--(226.4, 170.9)--(226.3, 170.8)--(226.1, 170.7)--(225.8, 170.6)--(225.8, 170.5)--(225.8, 170.6)--(225.7, 170.7)--(225.7, 170.8)--(225.6, 171.0)--(225.6, 171.4)--(225.5, 171.7)--(225.5, 171.8)--(225.5, 172.0)--(225.5, 172.2)--(225.4, 172.3)--(225.5, 172.5)--(225.5, 172.6)--(225.4, 172.9)--(225.2, 173.2)--(225.2, 173.3)--(225.1, 173.5)--(225.1, 173.6)--(225.0, 173.6)--(225.0, 173.7)--(224.9, 173.7)--(224.8, 173.7)--(224.9, 173.9)--(224.9, 174.0)--(224.9, 174.1)--(224.8, 174.3)--(224.7, 174.8)--(224.6, 174.9)--(224.5, 175.0)--(224.5, 175.1)--(224.3, 175.6)--(224.2, 175.6)--(224.2, 175.7)--(224.0, 175.8)--(223.9, 175.9)--(223.7, 176.0)--(223.5, 176.0)--(223.2, 176.1)--(223.0, 176.1)--(222.8, 176.1)--(222.7, 176.2)--(222.6, 176.2)--(222.5, 176.3)--(222.2, 176.4)--(222.2, 176.6)--(222.1, 176.8)--(222.0, 177.0)--(221.9, 177.1)--(221.6, 177.4)--(221.6, 177.5)--(221.5, 177.6)--(221.4, 177.7)--(221.4, 177.8)--(221.3, 178.0)--(221.2, 178.1)--(221.1, 178.2)--(221.0, 178.5)--(220.9, 178.6)--(220.9, 178.7)--(220.7, 178.9)--(220.5, 179.1)--(220.2, 179.6)--(220.2, 179.7)--(220.3, 179.9)--(220.4, 180.1)--(220.5, 180.3)--(220.5, 180.4)--(220.6, 180.7)--(220.6, 180.8)--(220.6, 180.9)--(220.6, 181.0)--(220.6, 181.2)--(220.7, 181.3)--(220.7, 181.4)--(220.7, 181.5)--(220.7, 181.7)--(220.7, 182.0)--(220.7, 182.3)--(220.6, 182.5)--(220.5, 182.8)--(220.5, 182.9)--(220.6, 183.0)--(220.6, 183.1)--(220.8, 183.2)--(221.0, 183.5)--(221.0, 183.6)--(221.3, 183.9)--(221.4, 184.1)--(221.5, 184.2)--(221.6, 184.3)--(221.6, 184.2)--(221.7, 184.8)--(221.7, 184.9)--(221.8, 185.4)--(221.8, 185.5)--(221.8, 185.6)--(221.9, 185.9)--(221.9, 186.1)--(221.9, 186.5)--(221.9, 186.6)--(221.9, 187.1)--(221.9, 187.2)--(221.8, 187.3)--(221.8, 187.5)--(221.8, 187.6)--(221.8, 187.7)--(221.7, 188.0)--(221.7, 188.3)--(221.6, 188.4)--(221.5, 188.7)--(221.5, 188.9)--(221.5, 189.0)--(221.5, 189.2)--(221.4, 189.4)--(221.3, 189.6)--(221.3, 189.7)--(221.3, 189.9)--(221.4, 190.1)--(221.4, 190.2)--(221.3, 190.3)--(221.3, 190.4)--(221.1, 190.6)--(221.1, 190.7)--(221.1, 190.8)--(221.0, 190.8)--(220.9, 190.8)--(220.8, 190.8)--(220.7, 190.9)--(220.5, 190.9)--(220.4, 191.0)--(220.3, 191.0)--(220.2, 191.0)--(220.1, 191.1)--(220.1, 191.2)--(220.0, 191.2)--(220.0, 191.3)--(220.0, 191.4)--(220.0, 191.5)--(220.0, 191.6)--(219.9, 191.6)--(219.9, 191.7)--(219.8, 191.8)--(219.8, 192.0)--(219.8, 192.2)--(219.8, 192.4)--(219.9, 192.5)--(220.0, 192.7)--(220.0, 192.8)--(220.1, 192.8)--(220.1, 192.9)--(220.1, 193.0)--(220.0, 193.1)--(219.9, 193.9)--(219.8, 194.0)--(219.2, 195.8)--(218.7, 197.3)--(218.6, 197.7)--(218.4, 198.4)--(218.3, 198.7)--(218.2, 198.8)--(218.2, 198.9)--(218.3, 199.0)--(218.3, 199.1)--(218.2, 199.4)--(218.1, 200.0)--(218.1, 200.2)--(218.0, 200.6)--(218.0, 200.7)--(218.0, 201.2)--(218.0, 201.3)--(218.0, 201.7)--(218.1, 201.9)--(218.1, 202.1)--(218.1, 202.2)--(218.2, 202.4)--(218.1, 202.6)--(218.1, 202.7)--(218.1, 202.9)--(218.0, 203.2)--(217.9, 203.4)--(217.9, 203.5)--(217.7, 203.9)--(217.6, 204.1)--(217.5, 204.3)--(217.5, 204.5)--(217.4, 204.7)--(217.4, 204.9)--(217.4, 205.1)--(217.4, 205.4)--(217.4, 205.6)--(217.3, 205.7)--(217.3, 205.9)--(217.2, 206.2)--(217.1, 206.6)--(217.0, 207.1)--(216.9, 207.1)--(216.9, 207.2)--(216.7, 207.4)--(216.5, 207.6)--(216.5, 207.7)--(216.3, 207.8)--(216.0, 207.8)--(215.8, 207.9)--(215.7, 208.0)--(215.7, 208.2)--(215.7, 208.3)--(215.7, 208.6)--(215.7, 209.0)--(215.7, 209.1)--(215.7, 209.2)--(215.8, 209.4)--(215.9, 209.7)--(215.9, 209.9)--(216.0, 210.0)--(216.0, 210.4)--(216.1, 210.7)--(216.1, 210.9)--(216.1, 211.0)--(216.0, 211.1)--(216.0, 211.2)--(216.0, 211.5)--(215.9, 211.7)--(215.7, 212.0)--(215.5, 212.2)--(215.5, 212.4)--(215.4, 212.7)--(215.3, 212.9)--(215.2, 213.1)--(215.1, 213.4)--(215.0, 213.6)--(214.9, 214.0)--(214.8, 214.2)--(214.7, 214.5)--(214.6, 214.7)--(214.6, 214.8)--(214.6, 214.9)--(214.6, 215.0)--(214.6, 215.1)--(214.7, 215.3)--(214.7, 215.4)--(214.8, 215.7)--(214.9, 215.9)--(215.0, 216.1)--(215.1, 216.3)--(215.2, 216.5)--(215.3, 216.6)--(215.4, 216.7)--(215.5, 216.7)--(215.7, 216.8)--(215.7, 216.9)--(215.8, 217.0)--(216.0, 217.2)--(216.0, 217.3)--(216.2, 217.6)--(216.3, 217.7)--(216.4, 217.8)--(216.6, 218.2)--(216.7, 218.5)--(216.8, 218.5)--(216.9, 218.5)--(217.0, 218.6)--(217.3, 218.7)--(217.7, 218.9)--(218.3, 219.2)--(218.7, 219.3)--(219.2, 219.5)--(219.9, 219.8)--(220.2, 220.0)--(220.9, 220.3)--(221.5, 220.6)--(221.8, 220.7)--(222.1, 220.8)--(222.4, 220.9)--(222.8, 221.1)--(223.2, 221.3)--(223.5, 221.4)--(223.9, 221.6)--(224.0, 221.7)--(224.5, 221.9)--(224.9, 222.1)--(225.4, 222.4)--(226.0, 222.7)--(226.5, 223.0)--(227.0, 223.3)--(227.7, 223.6)--(229.0, 224.3)--(229.7, 224.7)--(230.1, 224.9)--(230.3, 225.0)--(230.5, 225.0)--(230.6, 225.1)--(230.9, 225.1)--(231.2, 225.1)--(231.7, 225.3)--(232.0, 225.3)--(232.9, 225.5)--(233.3, 225.6)--(234.1, 225.7)--(234.3, 225.8)--(234.6, 225.8)--(234.8, 225.7)--(235.0, 225.6)--(235.2, 225.9)--(235.9, 225.8)--(237.4, 225.6)--(237.9, 225.5)--(238.4, 225.6)--(238.7, 225.7)--(239.0, 225.8)--(239.1, 225.8)--(239.2, 225.8)--(239.3, 225.8)--(239.4, 225.8)--(239.5, 225.8)--(239.6, 225.8)--(239.7, 225.8)--(239.8, 225.8)--(239.9, 225.8)--(240.0, 225.8)--(240.0, 225.7)--(240.2, 225.7)--(240.3, 225.6)--(240.6, 225.5)--(240.8, 225.3)--(240.8, 225.2)--(240.9, 225.2)--(240.9, 225.1)--(241.0, 225.1)--(241.0, 225.0)--(241.2, 224.8)--(241.3, 224.8)--(241.6, 224.9)--(242.2, 224.9)--(242.5, 225.0)--(242.9, 225.0)--(243.1, 225.0)--(243.2, 225.0)--(243.4, 225.0)--(243.6, 225.0)--(243.7, 225.0)--(244.0, 225.1)--(244.4, 225.3)--(244.9, 225.4)--(245.2, 225.4)--(245.7, 225.6)--(246.0, 225.6)--(246.3, 225.7)--(246.5, 225.7)--(246.6, 225.7)--(246.9, 225.8)--(247.0, 225.8)--(247.4, 226.0)--(248.0, 226.1)--(248.3, 226.2)--(248.9, 226.4)--(249.1, 226.4)--(249.5, 226.5)--(250.0, 226.6)--(250.1, 226.7)--(250.9, 227.0)--(253.1, 227.6)--(254.0, 228.0)--(254.3, 228.0)--(254.5, 228.2)--(255.7, 228.6)--(255.8, 228.5)--(255.9, 228.5)--(256.6, 228.8)--(257.0, 228.9)--(257.2, 228.9)--(258.7, 229.3)--(258.9, 229.4)--(258.9, 229.3)--(258.9, 229.2)--(258.9, 229.1)--(258.9, 228.9)--(258.9, 228.6)--(259.0, 228.2)--(259.1, 227.8)--(259.0, 227.6)--(259.0, 227.5)--(258.8, 227.1)--(258.8, 227.0)--(258.7, 227.0)--(258.6, 226.8)--(258.6, 226.6)--(258.5, 226.6)--(258.6, 226.5)--(258.8, 226.5)--(259.1, 226.4)--(259.2, 226.2)--(259.3, 226.2)--(259.6, 226.2)--(259.5, 225.7)--(259.5, 225.1)--(259.4, 224.8)--(259.3, 224.0)--(259.2, 223.5)--(259.2, 223.1)--(259.1, 222.5)--(259.0, 222.2)--(258.9, 222.0)--(258.9, 221.8)--(258.7, 221.2)--(258.6, 221.0)--(258.5, 220.9)--(258.5, 220.8)--(258.4, 220.8)--(258.4, 220.7)--(258.3, 220.5)--(258.3, 220.4)--(258.4, 220.3)--(258.4, 220.0)--(258.6, 219.7)--(258.7, 219.5)--(258.8, 219.3)--(258.9, 219.1)--(258.9, 219.0)--(258.9, 218.8)--(259.0, 218.7)--(259.0, 218.6)--(259.2, 218.4)--(259.4, 218.2)--(259.8, 218.4)--(259.7, 218.7)--(260.3, 219.0)--(260.4, 219.0)--(261.0, 219.0)--(261.0, 219.9)--(261.1, 220.0)--(261.6, 220.0)--(261.7, 220.0)--(262.1, 220.2)--(262.4, 219.7)--(262.5, 219.5)--(262.7, 219.2)--(262.9, 219.1)--(263.0, 218.9)--(263.1, 218.7)--(263.1, 218.5)--(263.1, 218.4)--(263.1, 218.3)--(263.1, 218.2)--(263.2, 218.2)--(263.3, 218.1)--(263.3, 217.8)--(263.2, 217.3)--(263.1, 216.2)--(263.1, 215.8)--(263.1, 215.2)--(263.1, 214.1)--(263.3, 213.4)--(263.5, 212.7)--(263.6, 212.5)--(264.1, 211.8)--(265.0, 210.9)--(265.8, 210.7)--(267.0, 210.6)--(267.2, 210.5)--(267.8, 210.4)--(268.1, 210.3)--(268.8, 210.1)--(269.3, 209.9)--(269.4, 209.9)--(269.8, 209.7)--(270.0, 209.6)--(270.1, 209.6)--(270.1, 209.5)--(270.7, 209.2)--(270.8, 209.2)--(270.8, 209.3)--(271.0, 209.2)--(271.4, 209.0)--(271.7, 208.8)--(271.8, 208.7)--(271.9, 208.6)--(272.3, 208.3)--(272.4, 208.2)--(272.6, 208.0)--(272.7, 207.9)--(272.9, 207.8)--(273.6, 207.4)--(274.0, 207.2)--(274.5, 207.0)--(274.5, 206.9)--(275.1, 206.7)--(275.4, 206.6)--(275.5, 206.6)--(275.7, 206.6)--(275.8, 206.5)--(276.1, 206.5)--(276.8, 206.4)--(277.0, 206.3)--(277.0, 206.1)--(276.9, 206.2)--(276.8, 206.1)--(277.2, 206.0)--(277.5, 206.0)--(277.9, 205.9)--(278.5, 205.9)--(279.2, 205.9)--(279.3, 205.9)--(279.5, 206.7)--(279.7, 207.1)--(280.1, 208.1)--(280.1, 208.2)--(280.3, 208.1)--(280.3, 208.2)--(280.4, 208.2)--(280.3, 208.2)--(280.4, 208.5)--(280.3, 208.5)--(280.3, 208.6)--(280.4, 208.6)--(280.4, 209.0)--(280.5, 208.9)--(280.5, 208.8)--(280.5, 208.7)--(280.6, 208.9)--(280.8, 208.8)--(281.2, 208.6)--(281.6, 208.3)--(282.0, 208.0)--(282.5, 207.8)--(283.0, 207.6)--(283.3, 207.5)--(283.6, 207.4)--(285.0, 206.7)--(286.0, 206.2)--(286.4, 206.0)--(286.5, 206.0)--(286.6, 206.1)--(286.8, 206.2)--(287.0, 206.3)--(287.1, 206.3)--(287.2, 206.3)--(287.4, 206.2)--(287.5, 206.2)--(287.7, 206.1)--(287.8, 206.1)--(287.9, 206.0)--(288.3, 205.8)--(288.6, 205.5)--(288.8, 205.3)--(289.0, 205.1)--(289.1, 205.1)--(289.4, 205.1)--(289.5, 205.1)--(289.6, 205.1)--(289.8, 204.9)--(290.0, 204.7)--(290.1, 204.5)--(290.2, 204.5)--(290.2, 204.4)--(290.3, 204.4)--(290.3, 204.3)--(290.4, 204.2)--(290.7, 204.1)--(290.9, 204.0)--(291.1, 203.9)--(291.2, 203.8)--(291.6, 203.9)--(291.7, 204.0)--(291.9, 204.1)--(292.2, 204.2)--(292.4, 204.3)--(292.5, 204.4)--(292.9, 204.7)--(293.1, 204.8)--(293.2, 204.9)--(293.4, 205.0)--(293.5, 205.1)--(293.6, 205.2)--(293.7, 205.3)--(293.8, 205.4)--(294.0, 205.6)--(293.2, 204.6)--(292.1, 203.3)--(292.3, 203.1)--(291.8, 202.6)--(291.0, 201.7)--(290.9, 201.6)--(291.2, 201.4)--(291.4, 201.1)--(290.8, 200.6)--(290.5, 200.3)--(290.2, 200.0)--(290.0, 199.8)--(289.8, 199.6)--(289.7, 199.5)--(289.5, 199.4)--(289.4, 199.1)--(289.2, 198.9)--(289.2, 198.8)--(289.0, 198.5)--(288.9, 198.3)--(289.3, 198.2)--(289.8, 198.0)--(290.4, 198.0)--(290.7, 198.0)--(290.9, 198.0)--(291.1, 197.8)--(291.2, 197.9)--(291.3, 197.9)--(291.4, 197.9)--(291.4, 198.0)--(291.6, 198.1)--(292.4, 198.7)--(292.6, 199.0)--(292.7, 198.9)--(293.1, 198.7)--(293.7, 198.3)--(294.5, 197.9)--(294.8, 197.8)--(294.9, 197.7)--(295.4, 197.5)--(295.5, 197.5)--(295.7, 197.4)--(295.8, 197.3)--(296.2, 197.2)--(296.9, 197.0)--(297.4, 196.8)--(297.7, 196.7)--(297.9, 196.7)--(298.0, 196.7)--(298.2, 196.7)--(298.4, 196.6)--(298.6, 196.5)--(298.7, 197.4)--(298.8, 197.7)--(298.8, 197.8)--(298.8, 198.1)--(298.8, 198.2)--(298.8, 198.3)--(298.9, 198.4)--(299.0, 198.7)--(299.1, 198.6)--(299.0, 198.4)--(299.0, 198.3)--(299.2, 198.2)--(299.6, 198.0)--(300.0, 198.0)--(302.9, 197.7)--(303.0, 198.0)--(303.0, 198.1)--(303.2, 198.1)--(303.1, 197.9)--(303.1, 197.7)--(303.0, 197.5)--(302.9, 196.8)--(302.8, 196.5)--(302.7, 196.5)--(302.7, 196.6)--(302.6, 196.6)--(302.5, 196.5)--(302.4, 196.4)--(302.5, 196.3)--(302.4, 196.2)--(302.3, 195.9)--(302.1, 195.5)--(302.0, 195.3)--(301.9, 195.3)--(301.9, 195.0)--(301.2, 194.9)--(300.9, 194.9)--(300.6, 194.9)--(300.6, 194.6)--(300.1, 194.5)--(300.1, 194.3)--(300.1, 194.2)--(300.2, 194.1)--(300.2, 193.7)--(300.0, 193.7)--(299.8, 193.6)--(300.0, 192.0)--(300.5, 190.3)--(300.6, 190.2)--(300.6, 190.1)--(300.7, 190.1)--(300.7, 189.9)--(300.6, 189.8)--(300.8, 189.0)--(300.8, 188.9)--(300.9, 188.8)--(301.1, 188.5)--(301.2, 188.3)--(301.3, 188.3)--(301.3, 188.2)--(301.4, 188.0)--(301.4, 187.9)--(302.1, 187.3)--(302.3, 187.2)--(302.5, 187.0)--(302.7, 186.8)--(302.8, 186.7)--(302.9, 186.5)--(303.0, 186.4)--(303.1, 186.3)--(303.2, 186.3)--(303.3, 186.0)--(303.4, 185.9)--(303.5, 185.7)--(303.7, 185.3)--(303.9, 185.1)--(304.1, 184.8)--(304.1, 184.7)--(304.2, 184.6)--(304.4, 184.4)--(304.4, 184.3)--(304.5, 184.2)--(304.7, 183.9)--(304.9, 183.6)--(305.0, 183.3)--(305.1, 183.3)--(305.1, 183.2)--(305.0, 183.1)--(305.0, 183.0)--(304.9, 182.6)--(304.8, 182.5)--(305.1, 182.5)--(305.3, 182.4)--(305.5, 182.4)--(305.6, 182.3)--(305.7, 182.2)--(305.8, 182.1)--(305.7, 182.0)--(305.9, 181.8)--(305.8, 181.6)--(305.7, 181.5)--(305.2, 180.9)--(304.4, 179.8)--(304.0, 179.2)--(303.4, 178.5)--(302.9, 178.0)--(302.4, 177.5)--(302.2, 177.3)--(302.3, 177.2)--(302.6, 177.1)--(302.8, 177.1)--(302.9, 177.1)--(302.3, 176.6)--(301.8, 176.3)--(300.6, 175.4)--(299.7, 174.8)--(299.6, 174.7)--(299.4, 174.6)--(299.3, 174.5)--(299.2, 174.4)--cycle; +Newham = (383.0, 323.4)--(382.9, 323.7)--(382.8, 323.8)--(382.8, 324.0)--(382.8, 324.3)--(382.9, 324.6)--(382.9, 325.5)--(383.0, 325.8)--(383.0, 326.0)--(383.1, 326.1)--(383.2, 326.3)--(383.2, 326.4)--(383.2, 326.7)--(383.2, 326.9)--(383.1, 327.1)--(383.0, 327.2)--(382.8, 327.4)--(382.7, 327.6)--(382.6, 327.8)--(382.5, 328.0)--(382.4, 328.3)--(382.3, 328.4)--(382.2, 328.6)--(381.8, 328.8)--(381.4, 329.2)--(381.3, 329.3)--(381.1, 329.5)--(381.1, 329.6)--(381.0, 329.8)--(381.0, 330.0)--(380.9, 330.2)--(380.8, 330.3)--(380.7, 330.4)--(380.6, 330.5)--(380.1, 330.6)--(379.5, 330.6)--(379.2, 330.7)--(378.8, 330.8)--(378.6, 330.9)--(378.5, 331.0)--(378.5, 331.1)--(378.4, 331.2)--(378.4, 331.3)--(378.2, 331.4)--(377.9, 331.6)--(377.5, 331.7)--(376.8, 332.1)--(376.6, 332.2)--(376.3, 332.4)--(376.1, 332.6)--(376.1, 332.7)--(375.9, 333.1)--(375.7, 333.4)--(375.6, 333.6)--(375.6, 333.7)--(375.6, 333.8)--(375.6, 334.0)--(375.7, 334.2)--(375.7, 334.3)--(375.7, 334.4)--(375.7, 334.6)--(375.6, 334.8)--(375.6, 334.9)--(375.4, 335.0)--(375.2, 335.2)--(375.1, 335.4)--(375.0, 335.5)--(375.0, 335.6)--(375.0, 335.8)--(374.9, 336.0)--(374.9, 336.2)--(374.8, 336.4)--(374.7, 336.7)--(374.6, 337.0)--(374.4, 337.2)--(374.3, 337.4)--(374.1, 337.6)--(373.9, 337.7)--(373.8, 337.7)--(373.6, 337.8)--(373.5, 337.9)--(373.4, 338.1)--(373.4, 338.3)--(373.4, 338.5)--(373.3, 338.6)--(373.3, 338.7)--(373.3, 338.8)--(373.4, 339.0)--(373.6, 339.3)--(373.8, 339.5)--(374.0, 339.7)--(374.2, 339.8)--(374.3, 339.9)--(374.4, 339.9)--(374.5, 340.1)--(374.6, 340.1)--(374.7, 340.4)--(374.7, 340.7)--(374.8, 341.1)--(374.9, 341.3)--(375.0, 341.4)--(375.1, 341.6)--(375.3, 341.8)--(375.4, 342.0)--(375.5, 342.1)--(375.6, 342.3)--(375.6, 342.4)--(375.6, 342.6)--(375.6, 343.0)--(375.6, 343.1)--(375.7, 343.2)--(375.8, 343.3)--(376.0, 343.3)--(376.2, 343.4)--(376.3, 343.5)--(376.4, 343.6)--(376.5, 343.8)--(376.5, 344.0)--(376.6, 344.5)--(376.7, 344.7)--(376.7, 344.8)--(376.7, 344.9)--(376.6, 345.2)--(376.5, 345.4)--(376.4, 345.5)--(376.4, 345.6)--(376.4, 345.8)--(376.4, 346.1)--(376.4, 346.4)--(376.3, 346.9)--(376.2, 347.1)--(376.2, 347.5)--(376.2, 347.6)--(376.1, 347.7)--(376.1, 347.9)--(376.1, 348.2)--(376.0, 348.7)--(376.0, 349.0)--(375.8, 349.6)--(375.7, 350.0)--(375.6, 350.5)--(375.6, 350.6)--(375.5, 350.9)--(375.3, 351.4)--(375.1, 352.0)--(375.0, 352.5)--(374.7, 353.2)--(374.6, 353.5)--(374.5, 353.7)--(374.5, 353.9)--(374.5, 354.1)--(374.5, 354.3)--(374.6, 354.4)--(374.6, 354.6)--(374.5, 354.7)--(374.5, 354.9)--(375.1, 354.9)--(375.7, 354.9)--(375.7, 354.8)--(375.8, 354.7)--(375.9, 354.8)--(375.9, 355.0)--(376.7, 355.1)--(377.6, 355.3)--(378.2, 355.5)--(378.4, 355.5)--(378.5, 355.6)--(378.6, 355.6)--(379.4, 355.9)--(379.9, 356.1)--(380.1, 356.2)--(380.2, 356.2)--(380.3, 356.2)--(380.4, 356.2)--(380.5, 356.1)--(380.7, 356.0)--(380.8, 355.9)--(381.0, 355.8)--(381.2, 355.3)--(381.3, 355.2)--(381.3, 355.0)--(381.3, 354.8)--(381.3, 354.6)--(381.3, 354.3)--(381.4, 354.0)--(381.5, 353.7)--(381.6, 353.3)--(381.7, 353.3)--(381.8, 353.3)--(381.9, 353.3)--(382.1, 353.4)--(382.8, 353.7)--(383.3, 353.8)--(383.5, 353.9)--(383.7, 354.0)--(384.1, 354.3)--(385.0, 354.7)--(385.1, 354.5)--(385.8, 354.8)--(386.5, 355.1)--(386.9, 355.3)--(387.0, 355.3)--(387.4, 355.4)--(387.9, 355.5)--(388.5, 355.6)--(389.1, 355.7)--(389.3, 355.8)--(389.8, 355.9)--(389.8, 355.8)--(389.9, 355.7)--(390.1, 355.8)--(390.3, 355.8)--(390.3, 355.7)--(390.3, 355.6)--(391.1, 355.7)--(391.0, 356.0)--(391.2, 356.0)--(391.2, 355.6)--(391.3, 355.7)--(391.6, 355.7)--(391.7, 355.5)--(391.6, 355.3)--(391.6, 355.0)--(391.6, 354.8)--(391.5, 354.6)--(391.5, 354.4)--(391.7, 354.3)--(392.4, 354.6)--(392.9, 354.8)--(393.0, 354.8)--(393.3, 354.9)--(394.2, 355.1)--(394.2, 355.2)--(394.2, 355.3)--(394.3, 355.5)--(394.6, 355.7)--(394.8, 355.9)--(395.1, 356.2)--(394.2, 357.5)--(395.2, 358.6)--(395.6, 358.1)--(397.2, 359.0)--(397.6, 359.3)--(398.4, 358.2)--(398.4, 358.1)--(398.6, 357.8)--(398.8, 357.4)--(399.2, 357.6)--(399.3, 357.4)--(400.0, 357.9)--(400.4, 358.2)--(401.0, 358.4)--(401.0, 358.5)--(401.5, 358.8)--(402.5, 359.4)--(402.2, 359.6)--(402.3, 359.7)--(401.6, 360.9)--(401.7, 360.8)--(402.0, 360.9)--(402.3, 360.9)--(402.6, 360.9)--(402.9, 360.8)--(403.1, 360.8)--(403.1, 361.3)--(403.1, 361.4)--(403.3, 361.5)--(403.5, 361.6)--(403.6, 361.6)--(403.7, 361.5)--(403.8, 361.4)--(403.8, 361.3)--(403.8, 361.2)--(403.8, 360.5)--(403.9, 360.2)--(404.0, 360.1)--(404.6, 360.2)--(405.0, 360.2)--(405.1, 359.3)--(405.1, 358.5)--(406.1, 358.8)--(406.6, 358.9)--(407.1, 358.9)--(407.5, 359.0)--(407.6, 359.0)--(407.8, 358.9)--(408.0, 358.9)--(409.0, 359.1)--(409.6, 359.2)--(409.8, 359.2)--(409.9, 359.2)--(410.4, 359.4)--(410.7, 359.6)--(411.0, 359.7)--(411.1, 359.8)--(411.7, 360.0)--(411.8, 360.0)--(412.0, 360.0)--(412.1, 360.0)--(412.5, 360.0)--(413.3, 360.0)--(413.9, 360.0)--(414.7, 359.9)--(415.3, 359.9)--(416.0, 359.8)--(416.3, 359.8)--(416.5, 359.9)--(416.6, 359.9)--(417.8, 359.9)--(418.9, 359.8)--(419.5, 359.8)--(419.6, 359.8)--(419.6, 359.9)--(419.6, 360.1)--(419.5, 360.3)--(419.3, 360.6)--(419.0, 361.1)--(418.7, 361.5)--(418.6, 361.8)--(418.4, 362.0)--(418.3, 362.3)--(418.3, 362.6)--(418.2, 362.8)--(418.1, 362.9)--(418.0, 363.0)--(417.6, 363.3)--(417.1, 363.7)--(417.2, 363.7)--(417.1, 364.4)--(416.9, 365.2)--(416.6, 366.2)--(416.5, 366.7)--(416.4, 367.1)--(416.6, 367.2)--(416.9, 367.3)--(417.6, 367.7)--(417.7, 367.8)--(418.2, 368.0)--(418.9, 368.4)--(419.4, 368.6)--(419.8, 368.9)--(420.2, 369.0)--(420.6, 369.2)--(421.0, 369.5)--(422.1, 370.0)--(422.2, 369.9)--(422.4, 369.3)--(422.4, 369.2)--(422.6, 368.8)--(422.9, 368.2)--(423.0, 368.1)--(423.2, 368.0)--(423.4, 367.8)--(423.6, 367.7)--(423.7, 367.6)--(423.8, 367.5)--(423.9, 367.5)--(423.9, 367.4)--(424.0, 367.4)--(424.1, 367.4)--(424.2, 367.3)--(424.3, 367.0)--(424.4, 366.9)--(424.4, 366.8)--(424.5, 366.8)--(424.6, 366.7)--(424.7, 366.7)--(424.8, 366.6)--(424.9, 366.5)--(424.9, 366.4)--(424.9, 366.3)--(424.9, 366.2)--(425.0, 366.2)--(425.0, 366.1)--(425.1, 366.0)--(425.1, 365.8)--(425.0, 365.7)--(425.0, 365.6)--(425.0, 365.5)--(425.1, 365.5)--(425.1, 365.4)--(425.1, 365.3)--(425.2, 365.3)--(425.2, 365.2)--(425.3, 365.2)--(425.5, 365.2)--(425.7, 365.1)--(425.8, 365.1)--(425.9, 365.1)--(425.9, 365.0)--(426.0, 365.0)--(426.1, 364.9)--(426.3, 364.7)--(426.5, 364.5)--(426.5, 364.4)--(426.4, 364.4)--(426.4, 364.3)--(426.4, 364.2)--(426.5, 364.1)--(426.6, 364.2)--(426.7, 364.2)--(426.7, 364.3)--(426.8, 364.3)--(426.9, 364.3)--(426.9, 364.2)--(426.9, 364.1)--(426.9, 363.8)--(427.0, 363.6)--(427.1, 363.5)--(427.3, 363.5)--(427.5, 363.4)--(427.6, 363.4)--(427.7, 363.4)--(427.8, 363.4)--(427.9, 363.1)--(427.9, 363.0)--(428.0, 362.9)--(428.1, 362.9)--(428.1, 362.8)--(428.1, 362.7)--(428.2, 362.7)--(428.3, 362.7)--(428.4, 362.6)--(428.4, 362.5)--(428.4, 362.4)--(428.5, 362.3)--(428.6, 362.4)--(428.7, 362.4)--(428.8, 362.4)--(428.8, 362.3)--(428.8, 362.2)--(428.9, 362.1)--(429.1, 361.9)--(429.2, 361.8)--(429.4, 361.9)--(429.7, 362.0)--(429.8, 362.0)--(430.1, 362.1)--(430.7, 362.5)--(431.5, 362.9)--(431.6, 362.7)--(431.9, 362.5)--(432.0, 362.3)--(432.3, 361.9)--(432.4, 361.8)--(432.5, 361.9)--(432.8, 361.5)--(433.0, 361.2)--(433.1, 361.1)--(433.0, 361.0)--(433.3, 360.6)--(433.5, 360.4)--(433.6, 360.2)--(433.7, 360.1)--(433.8, 359.6)--(433.9, 359.0)--(434.0, 358.2)--(434.6, 358.2)--(434.7, 356.9)--(434.8, 356.1)--(434.9, 355.2)--(435.0, 354.8)--(435.1, 354.1)--(435.1, 354.0)--(435.2, 353.5)--(435.2, 353.1)--(435.3, 352.9)--(435.4, 352.4)--(435.4, 352.1)--(435.4, 351.9)--(435.3, 351.8)--(435.4, 351.2)--(435.4, 350.8)--(435.4, 350.4)--(435.4, 350.2)--(435.4, 350.0)--(435.4, 349.4)--(435.3, 349.0)--(435.2, 348.7)--(435.2, 348.6)--(435.2, 348.5)--(435.2, 348.4)--(435.1, 348.1)--(435.0, 347.7)--(434.8, 347.3)--(434.6, 346.7)--(434.5, 346.5)--(434.4, 346.1)--(434.3, 345.6)--(434.3, 345.5)--(434.2, 345.4)--(434.2, 345.3)--(434.2, 345.2)--(434.2, 345.1)--(434.2, 345.0)--(434.2, 344.8)--(434.2, 344.6)--(434.2, 344.3)--(434.2, 344.0)--(434.3, 343.9)--(434.3, 343.8)--(434.3, 343.5)--(434.4, 343.3)--(434.6, 342.9)--(434.7, 342.7)--(434.8, 342.6)--(434.9, 342.3)--(435.1, 341.9)--(435.4, 341.4)--(435.5, 341.1)--(435.6, 341.0)--(435.8, 340.7)--(435.8, 340.6)--(435.4, 340.4)--(435.4, 340.2)--(435.4, 340.0)--(435.4, 339.8)--(435.3, 339.6)--(435.3, 339.5)--(435.2, 339.5)--(435.3, 339.4)--(435.3, 339.5)--(435.4, 339.5)--(435.5, 339.5)--(435.6, 339.5)--(435.7, 339.5)--(435.8, 339.5)--(435.9, 339.5)--(436.1, 339.7)--(436.2, 339.7)--(436.3, 339.6)--(436.3, 339.4)--(436.4, 339.3)--(436.5, 339.1)--(436.8, 338.5)--(436.8, 338.4)--(437.0, 338.1)--(437.6, 336.8)--(437.7, 336.5)--(437.7, 336.4)--(437.9, 336.0)--(437.9, 335.9)--(438.1, 335.9)--(438.1, 335.8)--(438.3, 335.2)--(438.4, 334.6)--(438.5, 334.0)--(438.6, 333.6)--(438.6, 333.2)--(438.7, 332.7)--(438.7, 332.2)--(438.7, 331.9)--(438.8, 331.9)--(438.9, 331.9)--(439.0, 331.9)--(439.1, 332.0)--(439.1, 331.9)--(439.3, 331.9)--(439.6, 331.9)--(439.8, 332.1)--(440.1, 332.1)--(440.4, 332.2)--(440.6, 332.2)--(440.6, 332.1)--(441.2, 332.0)--(441.3, 331.9)--(441.5, 331.9)--(441.9, 331.9)--(442.2, 331.9)--(442.5, 331.7)--(442.8, 331.5)--(442.6, 331.4)--(442.5, 331.3)--(442.4, 331.3)--(442.1, 331.5)--(442.2, 331.4)--(442.4, 331.2)--(442.4, 331.1)--(442.8, 331.0)--(443.0, 331.1)--(443.2, 330.8)--(443.5, 330.6)--(443.6, 330.4)--(443.9, 330.1)--(444.1, 330.0)--(444.2, 330.0)--(444.5, 329.8)--(444.8, 329.8)--(445.1, 329.6)--(445.3, 329.4)--(445.4, 329.4)--(445.7, 329.3)--(446.1, 329.1)--(446.7, 328.6)--(447.1, 328.3)--(447.5, 328.0)--(448.1, 328.0)--(448.3, 328.1)--(448.6, 328.2)--(448.7, 328.3)--(448.9, 328.4)--(449.1, 328.4)--(449.2, 328.4)--(449.2, 328.5)--(449.4, 328.6)--(449.5, 328.7)--(449.6, 328.7)--(449.7, 328.8)--(449.8, 328.8)--(450.0, 328.8)--(450.1, 328.8)--(450.2, 328.8)--(450.3, 328.8)--(450.3, 328.7)--(450.4, 328.7)--(450.4, 328.6)--(450.5, 328.5)--(450.6, 328.3)--(450.7, 328.2)--(450.8, 328.1)--(451.2, 328.0)--(451.3, 328.1)--(451.5, 328.1)--(451.6, 328.2)--(451.7, 328.2)--(451.9, 328.2)--(452.1, 328.2)--(452.2, 328.2)--(452.3, 328.2)--(452.3, 328.1)--(452.4, 328.0)--(452.4, 327.8)--(452.4, 327.6)--(452.4, 327.4)--(452.5, 327.3)--(452.5, 327.0)--(452.4, 326.6)--(452.4, 326.5)--(452.4, 326.4)--(452.3, 326.3)--(452.1, 326.4)--(452.0, 326.3)--(451.7, 326.4)--(451.5, 326.4)--(451.4, 326.4)--(451.5, 326.3)--(451.7, 326.2)--(451.8, 326.2)--(451.9, 326.2)--(451.9, 326.1)--(452.0, 326.0)--(452.1, 326.0)--(452.1, 325.9)--(452.2, 325.8)--(452.3, 325.5)--(452.2, 325.2)--(452.1, 324.9)--(452.2, 324.8)--(453.1, 323.7)--(453.1, 323.5)--(453.2, 323.4)--(453.3, 323.1)--(453.5, 323.0)--(453.7, 322.8)--(453.7, 322.7)--(453.8, 322.7)--(454.0, 322.5)--(454.1, 322.4)--(454.1, 322.3)--(454.1, 322.0)--(453.9, 321.4)--(453.9, 321.2)--(453.9, 321.1)--(453.9, 321.0)--(453.9, 320.9)--(453.9, 320.7)--(453.9, 320.6)--(453.9, 320.0)--(453.9, 319.8)--(453.9, 319.7)--(454.0, 319.4)--(454.1, 319.0)--(454.4, 318.2)--(454.5, 317.9)--(454.6, 317.9)--(454.7, 317.7)--(454.8, 317.6)--(454.8, 317.5)--(454.9, 317.5)--(454.9, 317.4)--(454.9, 317.3)--(455.0, 317.2)--(455.1, 317.1)--(455.2, 317.1)--(455.4, 317.0)--(455.5, 317.0)--(455.7, 316.9)--(455.8, 316.7)--(455.9, 316.6)--(455.8, 316.5)--(455.7, 316.6)--(455.6, 316.7)--(455.5, 316.7)--(455.3, 316.9)--(455.2, 316.9)--(455.2, 317.0)--(455.1, 317.0)--(455.1, 316.9)--(455.0, 316.9)--(454.9, 316.8)--(454.7, 316.6)--(454.6, 316.6)--(454.5, 316.5)--(454.4, 316.5)--(454.3, 316.4)--(454.2, 316.4)--(454.0, 316.4)--(453.2, 316.2)--(452.3, 316.1)--(451.8, 316.0)--(451.7, 316.0)--(451.6, 316.0)--(451.4, 316.0)--(451.6, 315.6)--(451.2, 315.4)--(450.2, 315.0)--(450.0, 314.9)--(449.7, 314.7)--(449.2, 314.4)--(448.8, 314.2)--(448.2, 313.7)--(448.0, 313.6)--(447.6, 313.2)--(447.5, 313.1)--(447.0, 312.6)--(446.7, 312.4)--(446.7, 312.0)--(446.8, 311.8)--(447.0, 311.8)--(446.6, 311.2)--(446.2, 310.6)--(445.8, 310.0)--(445.2, 309.1)--(445.0, 308.9)--(444.7, 308.8)--(444.6, 308.7)--(444.3, 308.1)--(444.0, 307.4)--(443.8, 306.8)--(443.7, 306.6)--(443.6, 306.3)--(443.2, 305.2)--(443.0, 305.1)--(442.9, 305.0)--(442.8, 305.0)--(442.5, 305.0)--(442.5, 304.9)--(442.5, 304.8)--(442.7, 304.7)--(442.8, 304.7)--(443.0, 304.6)--(443.0, 304.5)--(443.0, 304.4)--(442.7, 304.1)--(442.5, 303.6)--(442.3, 303.7)--(442.1, 303.7)--(442.0, 303.6)--(442.1, 303.5)--(442.2, 303.4)--(442.4, 303.3)--(442.2, 303.2)--(442.1, 303.1)--(442.0, 303.1)--(442.0, 303.0)--(441.9, 302.9)--(441.8, 302.8)--(441.8, 302.6)--(441.6, 302.3)--(441.5, 302.1)--(441.4, 301.9)--(441.2, 301.8)--(440.9, 301.8)--(440.8, 301.8)--(440.6, 301.8)--(440.5, 301.7)--(440.5, 301.6)--(440.7, 301.5)--(440.8, 301.4)--(441.0, 301.2)--(441.0, 301.1)--(441.0, 301.0)--(440.9, 300.9)--(440.7, 300.4)--(440.5, 300.0)--(440.4, 299.8)--(440.3, 299.8)--(439.7, 299.5)--(438.6, 298.9)--(438.5, 298.8)--(438.4, 298.8)--(438.4, 298.7)--(438.3, 298.7)--(438.2, 299.0)--(438.1, 298.9)--(438.2, 298.7)--(437.9, 298.6)--(437.9, 298.7)--(437.8, 298.7)--(437.7, 298.7)--(437.8, 298.5)--(437.1, 298.3)--(437.0, 298.5)--(436.8, 298.5)--(436.9, 298.2)--(435.4, 298.0)--(434.3, 297.8)--(433.4, 297.7)--(433.3, 297.7)--(431.8, 297.5)--(431.4, 297.3)--(431.3, 297.3)--(431.1, 297.3)--(431.0, 297.3)--(430.8, 297.3)--(430.6, 297.3)--(430.1, 297.3)--(429.9, 297.3)--(429.7, 297.3)--(429.3, 297.3)--(427.0, 297.8)--(426.2, 298.0)--(425.0, 298.2)--(424.4, 298.3)--(423.5, 298.4)--(422.8, 298.5)--(421.8, 298.6)--(421.4, 298.6)--(421.0, 298.6)--(420.3, 298.7)--(419.8, 298.7)--(419.3, 298.7)--(418.3, 298.7)--(416.3, 298.6)--(415.7, 298.5)--(415.4, 298.5)--(415.2, 298.4)--(414.9, 298.4)--(414.8, 298.3)--(414.7, 298.3)--(414.6, 298.3)--(414.4, 298.3)--(414.1, 298.2)--(413.9, 298.2)--(413.5, 298.1)--(413.5, 298.2)--(413.0, 298.1)--(413.0, 298.0)--(412.4, 297.9)--(411.6, 297.7)--(411.1, 297.6)--(411.2, 297.5)--(411.1, 297.5)--(411.0, 297.5)--(411.0, 297.6)--(410.1, 297.3)--(409.7, 297.2)--(409.2, 297.1)--(408.6, 297.0)--(407.5, 296.9)--(406.6, 296.9)--(406.3, 297.0)--(406.2, 296.5)--(405.7, 296.6)--(405.7, 296.9)--(405.6, 296.9)--(405.6, 297.2)--(404.8, 297.4)--(404.7, 297.2)--(404.6, 297.2)--(404.5, 297.2)--(404.0, 297.4)--(403.5, 297.5)--(403.3, 297.7)--(403.1, 297.7)--(403.0, 297.8)--(403.0, 297.9)--(403.0, 298.0)--(402.9, 297.9)--(402.3, 298.2)--(402.2, 298.4)--(401.9, 298.6)--(401.6, 298.8)--(401.5, 298.9)--(401.1, 299.4)--(400.6, 299.9)--(400.5, 300.0)--(400.3, 300.2)--(400.1, 300.5)--(400.0, 300.7)--(400.0, 301.0)--(399.9, 301.1)--(399.7, 301.0)--(399.2, 301.5)--(399.5, 301.7)--(399.4, 301.8)--(399.2, 301.6)--(399.2, 301.7)--(399.1, 301.8)--(398.7, 302.3)--(398.6, 302.5)--(398.2, 303.1)--(398.3, 303.5)--(398.4, 303.5)--(398.4, 303.6)--(398.3, 303.7)--(398.2, 303.7)--(397.5, 303.9)--(397.4, 304.0)--(396.6, 304.8)--(396.3, 305.1)--(395.9, 305.4)--(396.0, 305.5)--(395.9, 305.7)--(395.8, 305.6)--(395.7, 305.6)--(395.7, 305.7)--(395.6, 305.9)--(395.7, 306.0)--(395.8, 306.4)--(395.9, 306.9)--(396.0, 307.1)--(396.0, 307.2)--(396.2, 307.6)--(396.1, 307.7)--(396.0, 307.9)--(395.9, 308.1)--(395.8, 308.3)--(395.7, 308.4)--(395.6, 308.4)--(395.5, 308.5)--(395.4, 308.6)--(395.3, 308.6)--(394.4, 308.8)--(393.9, 309.0)--(393.8, 309.0)--(393.6, 309.1)--(393.5, 309.1)--(393.5, 309.2)--(393.4, 309.2)--(393.4, 309.3)--(393.4, 309.4)--(393.6, 309.7)--(393.9, 310.3)--(394.0, 310.5)--(394.0, 310.6)--(394.1, 310.7)--(394.2, 311.0)--(394.3, 311.2)--(394.4, 311.3)--(394.4, 311.4)--(394.4, 311.7)--(394.5, 312.1)--(394.5, 312.3)--(394.5, 312.5)--(394.5, 312.6)--(394.5, 312.8)--(394.4, 313.0)--(394.2, 313.5)--(394.0, 313.8)--(393.9, 313.9)--(393.8, 314.0)--(393.7, 314.0)--(393.7, 314.1)--(393.6, 314.1)--(393.4, 314.2)--(393.3, 314.2)--(393.2, 314.3)--(393.2, 314.2)--(392.9, 314.2)--(392.8, 314.1)--(392.7, 314.1)--(392.6, 314.0)--(392.4, 313.8)--(392.0, 313.3)--(391.9, 313.2)--(391.8, 312.9)--(391.7, 312.8)--(391.7, 312.7)--(391.7, 312.3)--(391.7, 311.4)--(391.7, 311.0)--(391.6, 310.9)--(391.6, 310.5)--(391.4, 310.2)--(391.3, 310.2)--(391.3, 310.1)--(390.8, 310.0)--(390.7, 310.1)--(390.7, 310.2)--(390.6, 310.4)--(390.7, 310.5)--(390.7, 310.7)--(390.7, 310.8)--(390.8, 310.9)--(390.9, 311.2)--(391.3, 312.1)--(391.3, 312.3)--(391.4, 312.5)--(391.5, 313.0)--(391.6, 313.5)--(391.7, 313.6)--(391.7, 313.7)--(391.8, 314.1)--(391.9, 314.2)--(391.9, 314.4)--(391.8, 314.5)--(391.8, 314.7)--(391.5, 315.0)--(391.4, 315.2)--(391.2, 315.3)--(390.7, 315.7)--(389.7, 316.3)--(388.8, 316.9)--(388.3, 317.2)--(387.9, 317.6)--(387.5, 317.8)--(387.0, 318.3)--(386.8, 318.5)--(386.7, 318.4)--(386.8, 318.4)--(386.9, 318.3)--(386.9, 318.2)--(387.0, 318.2)--(387.0, 318.1)--(386.9, 318.1)--(386.7, 318.1)--(386.6, 318.1)--(386.5, 318.1)--(386.4, 318.1)--(386.3, 318.0)--(386.0, 317.9)--(385.9, 317.9)--(385.9, 317.8)--(385.8, 317.8)--(385.7, 317.7)--(385.7, 317.6)--(385.6, 317.5)--(385.6, 317.4)--(385.5, 317.1)--(385.4, 316.9)--(385.3, 316.7)--(385.3, 316.6)--(385.2, 316.6)--(385.1, 316.6)--(385.0, 316.7)--(385.0, 316.8)--(384.9, 317.3)--(384.9, 317.4)--(384.8, 317.5)--(384.7, 317.7)--(384.6, 318.0)--(384.5, 318.2)--(384.4, 318.4)--(384.2, 318.7)--(384.0, 318.9)--(383.8, 319.1)--(383.6, 319.3)--(383.2, 319.5)--(382.9, 319.8)--(382.7, 320.0)--(382.6, 320.1)--(382.6, 320.2)--(382.6, 320.5)--(382.6, 320.8)--(382.6, 320.9)--(382.6, 321.1)--(382.7, 321.2)--(382.8, 321.4)--(383.0, 321.9)--(383.2, 322.3)--(383.3, 322.6)--(383.5, 323.2)--(383.5, 323.4)--(383.5, 323.8)--(383.5, 324.2)--(383.5, 324.4)--(383.5, 324.6)--(383.5, 325.0)--(383.6, 325.1)--(383.6, 325.3)--(383.8, 325.6)--(383.9, 326.0)--(383.9, 326.4)--(384.0, 326.5)--(384.0, 326.6)--(385.3, 327.1)--(385.4, 327.2)--(385.5, 327.3)--(385.7, 327.4)--(386.4, 327.7)--(386.8, 327.9)--(386.9, 327.9)--(387.1, 328.0)--(387.1, 328.1)--(387.2, 328.2)--(387.3, 328.4)--(387.5, 328.5)--(387.7, 328.6)--(387.9, 328.7)--(388.5, 329.1)--(388.6, 329.2)--(388.8, 329.4)--(388.8, 329.5)--(388.8, 329.9)--(388.8, 330.0)--(388.9, 330.2)--(388.9, 330.4)--(389.0, 330.6)--(389.0, 330.7)--(389.2, 331.0)--(389.3, 331.1)--(389.4, 331.4)--(389.4, 331.6)--(389.5, 331.7)--(389.3, 331.8)--(389.3, 331.9)--(389.2, 332.1)--(389.1, 332.2)--(389.1, 332.4)--(389.0, 332.5)--(389.0, 332.3)--(388.9, 332.2)--(388.8, 332.1)--(388.8, 332.0)--(388.6, 331.6)--(388.5, 331.2)--(388.3, 330.7)--(388.3, 330.6)--(388.2, 330.5)--(388.2, 330.4)--(388.1, 330.1)--(388.0, 330.0)--(388.0, 329.8)--(387.9, 329.7)--(387.9, 329.4)--(387.8, 329.3)--(387.8, 329.2)--(387.7, 329.1)--(387.7, 329.0)--(387.6, 329.0)--(387.4, 329.0)--(387.4, 328.9)--(387.1, 328.6)--(387.0, 328.4)--(386.9, 328.2)--(386.8, 328.1)--(386.4, 327.9)--(386.3, 327.9)--(386.2, 327.8)--(386.0, 327.8)--(385.9, 327.8)--(385.8, 327.8)--(385.7, 327.8)--(385.7, 328.0)--(385.6, 328.0)--(385.5, 327.8)--(385.5, 327.6)--(385.4, 327.5)--(385.2, 327.4)--(385.2, 327.3)--(385.0, 327.2)--(384.9, 327.2)--(384.7, 327.1)--(384.4, 327.0)--(384.3, 327.0)--(384.2, 327.0)--(384.1, 327.0)--(383.9, 327.1)--(383.8, 327.1)--(383.8, 327.2)--(383.7, 327.4)--(383.7, 327.5)--(383.5, 327.9)--(383.3, 327.9)--(383.2, 328.0)--(383.0, 328.1)--(382.9, 328.0)--(383.0, 327.9)--(383.1, 327.6)--(383.2, 327.5)--(383.3, 327.4)--(383.3, 327.2)--(383.4, 327.2)--(383.6, 326.8)--(383.6, 326.7)--(383.6, 326.6)--(383.6, 326.4)--(383.5, 326.2)--(383.4, 326.0)--(383.2, 325.4)--(383.2, 325.1)--(383.1, 325.1)--(383.1, 324.9)--(383.2, 324.3)--(383.2, 324.0)--(383.2, 323.6)--(383.1, 323.6)--(383.1, 323.7)--(383.0, 323.7)--(383.0, 323.5)--cycle; +Redbridge = (436.0, 348.3)--(435.8, 348.4)--(435.8, 348.5)--(435.2, 348.6)--(435.2, 348.7)--(435.3, 349.0)--(435.4, 349.4)--(435.4, 350.0)--(435.4, 350.2)--(435.4, 350.4)--(435.4, 350.8)--(435.4, 351.2)--(435.3, 351.8)--(435.4, 351.9)--(435.4, 352.1)--(435.4, 352.4)--(435.3, 352.9)--(435.2, 353.1)--(435.2, 353.5)--(435.1, 354.0)--(435.1, 354.1)--(435.0, 354.8)--(434.9, 355.2)--(434.8, 356.1)--(434.7, 356.9)--(434.6, 358.2)--(434.0, 358.2)--(433.9, 359.0)--(433.8, 359.6)--(433.7, 360.1)--(433.6, 360.2)--(433.5, 360.4)--(433.3, 360.6)--(433.0, 361.0)--(433.1, 361.1)--(433.0, 361.2)--(432.8, 361.5)--(432.5, 361.9)--(432.4, 361.8)--(432.3, 361.9)--(432.0, 362.3)--(431.9, 362.5)--(431.6, 362.7)--(431.5, 362.9)--(430.7, 362.5)--(430.1, 362.1)--(429.8, 362.0)--(429.7, 362.0)--(429.4, 361.9)--(429.2, 361.8)--(429.1, 361.9)--(428.9, 362.1)--(428.8, 362.2)--(428.8, 362.3)--(428.8, 362.4)--(428.7, 362.4)--(428.6, 362.4)--(428.5, 362.3)--(428.4, 362.4)--(428.4, 362.5)--(428.4, 362.6)--(428.3, 362.7)--(428.2, 362.7)--(428.1, 362.7)--(428.1, 362.8)--(428.1, 362.9)--(428.0, 362.9)--(427.9, 363.0)--(427.9, 363.1)--(427.8, 363.4)--(427.7, 363.4)--(427.6, 363.4)--(427.5, 363.4)--(427.3, 363.5)--(427.1, 363.5)--(427.0, 363.6)--(426.9, 363.8)--(426.9, 364.1)--(426.9, 364.2)--(426.9, 364.3)--(426.8, 364.3)--(426.7, 364.3)--(426.7, 364.2)--(426.6, 364.2)--(426.5, 364.1)--(426.4, 364.2)--(426.4, 364.3)--(426.4, 364.4)--(426.5, 364.4)--(426.5, 364.5)--(426.3, 364.7)--(426.1, 364.9)--(426.0, 365.0)--(425.9, 365.0)--(425.9, 365.1)--(425.8, 365.1)--(425.7, 365.1)--(425.5, 365.2)--(425.3, 365.2)--(425.2, 365.2)--(425.2, 365.3)--(425.1, 365.3)--(425.1, 365.4)--(425.1, 365.5)--(425.0, 365.5)--(425.0, 365.6)--(425.0, 365.7)--(425.1, 365.8)--(425.1, 366.0)--(425.0, 366.1)--(425.0, 366.2)--(424.9, 366.2)--(424.9, 366.3)--(424.9, 366.4)--(424.9, 366.5)--(424.8, 366.6)--(424.7, 366.7)--(424.6, 366.7)--(424.5, 366.8)--(424.4, 366.8)--(424.4, 366.9)--(424.3, 367.0)--(424.2, 367.3)--(424.1, 367.4)--(424.0, 367.4)--(423.9, 367.4)--(423.9, 367.5)--(423.8, 367.5)--(423.7, 367.6)--(423.6, 367.7)--(423.4, 367.8)--(423.2, 368.0)--(423.0, 368.1)--(422.9, 368.2)--(422.6, 368.8)--(422.4, 369.2)--(422.4, 369.3)--(422.2, 369.9)--(422.1, 370.0)--(421.0, 369.5)--(420.6, 369.2)--(420.2, 369.0)--(419.8, 368.9)--(419.4, 368.6)--(418.9, 368.4)--(418.2, 368.0)--(417.7, 367.8)--(417.6, 367.7)--(416.9, 367.3)--(416.6, 367.2)--(416.4, 367.1)--(416.5, 366.7)--(416.6, 366.2)--(416.9, 365.2)--(417.1, 364.4)--(417.2, 363.7)--(417.1, 363.7)--(417.6, 363.3)--(418.0, 363.0)--(418.1, 362.9)--(418.2, 362.8)--(418.3, 362.6)--(418.3, 362.3)--(418.4, 362.0)--(418.6, 361.8)--(418.7, 361.5)--(419.0, 361.1)--(419.3, 360.6)--(419.5, 360.3)--(419.6, 360.1)--(419.6, 359.9)--(419.6, 359.8)--(419.5, 359.8)--(418.9, 359.8)--(417.8, 359.9)--(416.6, 359.9)--(416.5, 359.9)--(416.3, 359.8)--(416.0, 359.8)--(415.3, 359.9)--(414.7, 359.9)--(413.9, 360.0)--(413.3, 360.0)--(412.5, 360.0)--(412.1, 360.0)--(412.0, 360.0)--(411.8, 360.0)--(411.7, 360.0)--(411.1, 359.8)--(411.0, 359.7)--(410.7, 359.6)--(410.4, 359.4)--(409.9, 359.2)--(409.8, 359.2)--(409.6, 359.2)--(409.0, 359.1)--(408.0, 358.9)--(407.8, 358.9)--(407.6, 359.0)--(407.5, 359.0)--(407.1, 358.9)--(406.6, 358.9)--(406.1, 358.8)--(405.1, 358.5)--(405.1, 359.3)--(405.0, 360.2)--(404.6, 360.2)--(404.0, 360.1)--(403.9, 360.2)--(403.8, 360.5)--(403.8, 361.2)--(403.8, 361.3)--(403.8, 361.4)--(403.7, 361.5)--(403.6, 361.6)--(403.5, 361.6)--(403.3, 361.5)--(403.1, 361.4)--(403.1, 361.3)--(403.1, 360.8)--(402.9, 360.8)--(402.6, 360.9)--(402.3, 360.9)--(402.0, 360.9)--(401.7, 360.8)--(401.6, 360.9)--(401.4, 361.1)--(401.2, 361.4)--(400.6, 362.1)--(400.4, 362.5)--(400.1, 362.8)--(400.0, 362.9)--(399.9, 363.0)--(399.8, 363.1)--(399.7, 363.2)--(399.3, 363.6)--(399.0, 363.8)--(398.9, 363.9)--(398.7, 364.1)--(398.5, 364.2)--(398.3, 364.2)--(398.2, 364.3)--(398.1, 364.3)--(397.8, 364.3)--(397.6, 364.3)--(397.4, 365.3)--(397.2, 366.4)--(397.1, 366.9)--(397.1, 367.4)--(397.1, 368.2)--(397.1, 368.7)--(397.2, 369.1)--(397.2, 369.2)--(397.2, 369.8)--(397.2, 370.1)--(397.2, 370.7)--(397.2, 371.1)--(398.0, 371.2)--(398.6, 371.2)--(399.1, 371.2)--(399.3, 371.2)--(399.5, 371.3)--(399.7, 371.4)--(399.8, 371.5)--(399.9, 371.6)--(400.0, 371.7)--(400.1, 371.8)--(400.2, 371.9)--(400.2, 372.0)--(400.3, 372.1)--(400.3, 372.2)--(400.3, 372.3)--(400.3, 372.4)--(400.4, 372.5)--(400.4, 372.6)--(400.3, 372.7)--(400.3, 372.8)--(400.3, 372.9)--(400.3, 373.0)--(400.0, 373.7)--(399.9, 374.2)--(399.7, 374.9)--(399.5, 375.5)--(399.3, 376.0)--(399.3, 376.1)--(399.3, 376.2)--(399.4, 376.2)--(399.5, 376.2)--(399.6, 376.2)--(399.7, 376.3)--(399.8, 376.2)--(400.0, 376.1)--(400.1, 376.0)--(400.3, 375.9)--(400.3, 376.0)--(400.3, 376.5)--(400.2, 377.1)--(400.1, 378.1)--(400.1, 378.6)--(400.1, 378.7)--(400.0, 378.7)--(399.5, 378.3)--(399.4, 378.5)--(399.3, 378.7)--(399.2, 378.8)--(399.1, 379.0)--(399.0, 379.1)--(398.8, 379.1)--(398.5, 379.3)--(398.4, 379.3)--(398.3, 379.3)--(398.4, 379.5)--(398.6, 380.1)--(398.9, 380.7)--(398.9, 380.8)--(399.0, 381.0)--(399.0, 381.2)--(398.9, 381.6)--(398.9, 382.1)--(398.9, 382.6)--(398.9, 383.3)--(398.9, 384.4)--(398.9, 384.5)--(398.3, 386.3)--(398.1, 386.4)--(397.8, 387.3)--(397.3, 389.1)--(397.4, 389.1)--(397.6, 389.2)--(397.5, 389.4)--(397.4, 389.5)--(397.4, 389.6)--(397.3, 390.0)--(397.4, 390.0)--(397.3, 390.2)--(397.4, 390.5)--(397.4, 390.9)--(397.0, 390.9)--(396.4, 390.9)--(395.9, 391.3)--(395.9, 391.7)--(395.8, 392.2)--(395.7, 393.1)--(395.6, 393.8)--(395.5, 394.9)--(395.4, 395.6)--(395.3, 396.7)--(395.2, 397.1)--(395.2, 397.5)--(395.1, 398.0)--(395.1, 398.6)--(395.0, 399.1)--(395.0, 399.3)--(395.0, 399.6)--(394.9, 399.8)--(394.9, 400.0)--(394.9, 400.1)--(394.9, 400.5)--(394.8, 400.6)--(394.8, 401.2)--(394.8, 401.6)--(394.8, 402.0)--(394.8, 402.1)--(394.9, 402.8)--(394.9, 403.0)--(394.9, 403.1)--(394.9, 403.3)--(394.9, 403.4)--(394.9, 403.6)--(395.0, 404.2)--(395.0, 404.8)--(395.0, 405.4)--(395.0, 405.9)--(395.1, 406.8)--(395.1, 407.6)--(395.1, 408.3)--(395.2, 408.5)--(395.3, 408.9)--(395.4, 409.3)--(395.7, 409.8)--(395.9, 410.4)--(396.4, 411.4)--(397.0, 412.8)--(397.8, 414.8)--(397.9, 415.1)--(398.0, 415.4)--(398.1, 415.4)--(398.4, 415.9)--(398.4, 416.0)--(398.5, 416.1)--(398.4, 416.2)--(398.6, 416.7)--(398.9, 417.4)--(399.1, 418.1)--(399.2, 418.1)--(399.4, 418.0)--(399.5, 418.0)--(399.4, 418.2)--(399.3, 418.5)--(399.2, 418.6)--(399.3, 418.7)--(399.4, 418.7)--(399.4, 418.8)--(399.5, 419.0)--(399.5, 419.1)--(399.5, 419.2)--(399.4, 419.2)--(399.5, 419.4)--(399.6, 419.6)--(399.8, 420.4)--(400.0, 420.3)--(400.0, 420.4)--(400.2, 420.8)--(400.1, 420.9)--(400.3, 421.3)--(400.4, 421.2)--(400.5, 421.5)--(400.6, 421.9)--(400.5, 422.0)--(400.6, 422.2)--(400.6, 422.4)--(400.6, 422.6)--(400.6, 422.7)--(400.6, 422.9)--(400.5, 423.0)--(400.5, 423.1)--(400.5, 423.2)--(400.4, 423.4)--(400.0, 423.8)--(396.9, 428.2)--(396.8, 428.2)--(396.7, 428.3)--(396.6, 428.4)--(396.2, 428.6)--(395.8, 428.7)--(395.0, 429.1)--(392.9, 429.9)--(391.9, 430.4)--(391.8, 430.4)--(391.6, 430.5)--(391.8, 430.6)--(392.1, 430.6)--(392.3, 430.6)--(392.4, 430.6)--(392.4, 430.5)--(392.5, 430.5)--(392.6, 430.5)--(392.6, 430.6)--(392.7, 430.6)--(392.8, 430.5)--(393.0, 430.4)--(393.1, 430.5)--(393.3, 430.5)--(393.5, 430.6)--(393.8, 430.6)--(394.0, 430.6)--(394.3, 430.6)--(395.3, 430.4)--(395.4, 430.4)--(395.5, 430.4)--(395.6, 430.4)--(395.7, 431.0)--(395.8, 431.1)--(395.7, 431.4)--(395.7, 431.6)--(395.6, 431.7)--(395.5, 431.8)--(395.4, 432.0)--(395.5, 432.0)--(395.5, 432.1)--(395.6, 432.1)--(395.7, 432.1)--(395.8, 432.3)--(395.9, 432.4)--(396.0, 432.5)--(396.1, 432.6)--(396.2, 432.6)--(396.3, 432.6)--(396.4, 432.7)--(396.5, 432.7)--(396.6, 432.9)--(396.6, 433.0)--(396.7, 433.0)--(396.8, 433.1)--(396.8, 433.2)--(396.8, 433.3)--(396.7, 433.3)--(396.7, 433.5)--(396.7, 433.6)--(396.8, 433.7)--(397.0, 433.7)--(397.2, 433.8)--(397.2, 433.9)--(397.3, 433.9)--(397.3, 434.0)--(397.3, 434.1)--(397.3, 434.2)--(397.3, 434.3)--(397.4, 434.3)--(397.4, 434.4)--(397.5, 434.4)--(397.5, 434.3)--(397.6, 434.3)--(397.7, 434.3)--(397.9, 434.5)--(398.0, 434.5)--(398.1, 434.6)--(398.1, 434.7)--(398.2, 434.8)--(398.3, 434.9)--(398.3, 435.0)--(398.3, 435.1)--(398.4, 435.2)--(398.5, 435.3)--(398.5, 435.4)--(398.4, 435.4)--(398.4, 435.5)--(398.3, 435.5)--(398.2, 435.6)--(398.3, 435.7)--(398.4, 435.7)--(398.5, 435.7)--(398.6, 435.7)--(398.6, 435.8)--(398.6, 435.9)--(398.5, 435.8)--(398.5, 435.9)--(398.4, 435.9)--(398.4, 436.0)--(398.4, 436.1)--(398.4, 436.2)--(398.3, 436.2)--(398.3, 436.3)--(398.3, 436.4)--(398.5, 436.4)--(398.6, 436.5)--(398.6, 437.0)--(398.6, 437.1)--(398.6, 437.2)--(398.7, 437.2)--(398.6, 437.3)--(398.8, 437.5)--(398.9, 437.7)--(398.8, 437.8)--(398.8, 437.9)--(398.8, 438.0)--(398.8, 438.1)--(398.9, 438.2)--(399.1, 438.3)--(399.1, 438.4)--(399.0, 438.5)--(399.1, 438.5)--(399.2, 438.6)--(399.2, 438.7)--(399.2, 438.9)--(399.3, 438.9)--(399.3, 439.0)--(399.3, 439.1)--(399.1, 439.3)--(399.1, 439.4)--(399.1, 439.5)--(399.3, 439.8)--(399.4, 439.8)--(399.5, 439.8)--(399.5, 439.9)--(399.6, 439.9)--(399.6, 440.0)--(399.5, 440.1)--(399.5, 440.2)--(399.5, 440.3)--(399.5, 440.5)--(399.5, 440.6)--(399.6, 440.8)--(399.6, 440.9)--(399.6, 441.0)--(399.7, 441.0)--(399.8, 441.0)--(399.9, 441.0)--(399.9, 441.1)--(399.9, 441.2)--(399.8, 441.3)--(399.7, 441.4)--(399.7, 441.5)--(399.8, 441.6)--(399.8, 441.7)--(399.9, 441.6)--(400.0, 441.6)--(400.1, 441.6)--(400.1, 441.7)--(400.0, 441.7)--(400.0, 441.8)--(400.2, 441.7)--(400.4, 441.6)--(400.5, 441.5)--(401.2, 441.0)--(401.9, 440.5)--(402.3, 440.2)--(402.6, 440.0)--(403.2, 439.3)--(403.6, 438.9)--(403.7, 438.7)--(403.8, 438.6)--(403.8, 438.5)--(403.9, 438.4)--(404.0, 438.4)--(404.1, 438.3)--(404.3, 438.3)--(404.4, 438.3)--(404.6, 438.2)--(404.7, 438.2)--(404.8, 438.2)--(404.9, 438.1)--(405.1, 437.9)--(405.5, 437.6)--(405.4, 437.1)--(405.7, 437.1)--(405.7, 437.0)--(405.8, 436.8)--(405.8, 436.7)--(405.9, 436.6)--(406.1, 436.4)--(406.7, 436.0)--(406.6, 435.6)--(407.2, 435.4)--(407.4, 435.2)--(407.7, 434.9)--(407.9, 434.8)--(408.0, 434.6)--(408.2, 434.5)--(408.2, 434.4)--(408.3, 434.2)--(409.1, 433.2)--(409.2, 433.0)--(409.2, 432.9)--(409.3, 432.9)--(409.5, 432.6)--(409.5, 432.5)--(410.1, 432.0)--(410.2, 431.8)--(410.5, 431.5)--(410.6, 431.4)--(410.7, 431.2)--(411.0, 430.8)--(411.2, 430.5)--(411.5, 430.3)--(411.5, 430.2)--(411.6, 430.0)--(411.6, 429.7)--(411.7, 429.8)--(412.2, 429.3)--(413.5, 428.1)--(413.4, 428.0)--(413.4, 427.8)--(413.5, 427.7)--(413.6, 427.7)--(413.8, 427.6)--(413.8, 427.5)--(413.9, 427.4)--(414.0, 427.3)--(414.0, 427.5)--(414.1, 427.6)--(414.2, 427.7)--(414.4, 427.9)--(414.6, 428.1)--(414.8, 428.3)--(415.1, 428.4)--(415.3, 428.6)--(415.6, 428.7)--(415.9, 428.8)--(416.3, 428.9)--(416.7, 429.0)--(417.1, 429.2)--(417.6, 429.2)--(417.6, 429.3)--(417.7, 429.3)--(418.2, 429.3)--(418.7, 429.4)--(418.8, 429.5)--(419.6, 429.8)--(420.2, 430.0)--(420.6, 430.1)--(421.1, 430.3)--(421.8, 430.5)--(422.3, 430.6)--(422.5, 430.6)--(422.5, 430.5)--(422.6, 430.5)--(422.8, 430.5)--(423.0, 430.1)--(423.2, 429.9)--(423.2, 429.8)--(423.3, 429.7)--(423.3, 429.6)--(423.2, 429.5)--(423.2, 429.4)--(423.1, 429.3)--(423.1, 429.2)--(423.1, 429.1)--(423.1, 429.0)--(423.2, 429.0)--(423.2, 428.9)--(423.2, 428.8)--(423.1, 428.7)--(422.8, 428.5)--(422.8, 428.4)--(422.7, 428.4)--(422.7, 428.3)--(422.6, 428.3)--(422.5, 428.3)--(422.4, 428.3)--(422.3, 428.4)--(422.3, 428.5)--(422.2, 428.6)--(422.1, 428.7)--(422.0, 428.7)--(421.9, 428.8)--(421.7, 428.8)--(421.6, 428.8)--(421.5, 428.8)--(421.4, 428.8)--(421.3, 428.7)--(421.3, 428.6)--(421.2, 428.6)--(420.8, 428.5)--(420.6, 428.4)--(420.4, 428.3)--(420.3, 428.3)--(420.3, 428.2)--(420.2, 428.1)--(420.2, 427.9)--(420.1, 427.8)--(420.1, 427.7)--(420.0, 427.6)--(419.9, 427.5)--(419.8, 427.5)--(419.6, 427.4)--(419.6, 427.3)--(419.5, 427.2)--(419.4, 427.2)--(419.4, 427.1)--(419.2, 426.7)--(419.2, 426.5)--(419.2, 426.4)--(419.1, 426.3)--(419.1, 426.2)--(419.1, 426.1)--(419.2, 425.9)--(419.3, 425.8)--(419.3, 425.7)--(419.3, 425.6)--(419.3, 425.5)--(419.2, 425.5)--(419.1, 425.3)--(419.1, 425.2)--(419.0, 425.2)--(419.0, 425.1)--(419.0, 425.0)--(419.0, 424.9)--(419.1, 424.8)--(419.1, 424.7)--(419.3, 424.9)--(419.4, 425.1)--(419.4, 425.2)--(419.5, 425.2)--(419.6, 425.1)--(419.7, 425.1)--(420.4, 424.7)--(421.0, 424.3)--(421.5, 423.9)--(421.7, 423.8)--(422.2, 423.5)--(422.4, 423.4)--(422.6, 423.2)--(422.8, 423.0)--(423.2, 422.7)--(424.6, 421.6)--(424.9, 421.4)--(425.6, 421.0)--(426.3, 420.5)--(426.9, 420.1)--(427.4, 419.8)--(427.9, 419.5)--(428.2, 419.2)--(428.2, 419.1)--(428.6, 418.8)--(428.7, 418.8)--(428.9, 418.6)--(429.2, 418.2)--(429.4, 418.2)--(429.6, 418.0)--(429.8, 418.3)--(430.2, 418.4)--(430.7, 418.1)--(431.2, 417.8)--(431.3, 417.7)--(431.6, 417.4)--(432.1, 417.3)--(432.6, 417.2)--(432.8, 417.2)--(433.1, 417.1)--(433.7, 416.8)--(434.1, 416.6)--(434.7, 416.3)--(435.2, 416.2)--(435.6, 416.0)--(436.0, 415.9)--(436.4, 415.7)--(436.9, 416.0)--(437.1, 416.2)--(438.0, 416.8)--(438.8, 417.4)--(439.2, 417.7)--(439.9, 418.1)--(440.5, 418.4)--(440.6, 418.5)--(440.7, 418.5)--(440.9, 418.5)--(441.0, 418.5)--(441.3, 418.4)--(441.7, 418.4)--(442.0, 418.3)--(442.4, 418.2)--(442.7, 418.1)--(443.3, 417.9)--(443.6, 417.9)--(443.7, 417.9)--(444.0, 417.9)--(444.4, 417.8)--(444.7, 417.6)--(445.0, 417.3)--(445.2, 417.1)--(445.2, 416.9)--(445.3, 417.0)--(445.9, 416.2)--(446.0, 416.0)--(446.1, 416.0)--(446.2, 416.0)--(446.2, 415.9)--(446.3, 415.8)--(447.6, 416.2)--(447.6, 416.6)--(447.9, 416.8)--(448.0, 417.0)--(448.3, 417.1)--(448.5, 417.2)--(448.5, 417.1)--(448.6, 417.2)--(449.3, 417.5)--(449.9, 417.7)--(450.0, 417.8)--(450.1, 417.8)--(450.0, 417.9)--(449.4, 419.2)--(449.5, 420.0)--(450.0, 421.5)--(450.7, 421.8)--(450.9, 421.9)--(451.0, 422.1)--(451.0, 422.3)--(451.1, 422.5)--(451.1, 422.7)--(451.1, 422.8)--(451.4, 422.9)--(451.5, 422.9)--(451.5, 423.0)--(451.4, 423.1)--(451.3, 423.4)--(451.1, 423.7)--(450.9, 424.3)--(450.6, 424.7)--(450.6, 424.8)--(450.5, 424.9)--(450.4, 425.1)--(450.0, 425.6)--(449.6, 426.0)--(450.0, 426.2)--(450.5, 426.5)--(451.4, 427.0)--(452.2, 427.4)--(452.5, 427.6)--(452.9, 427.7)--(453.1, 427.8)--(453.3, 427.9)--(453.6, 428.2)--(454.6, 426.2)--(455.1, 425.2)--(456.1, 425.6)--(457.3, 426.1)--(457.6, 425.1)--(457.8, 425.2)--(458.4, 425.4)--(458.8, 425.6)--(459.1, 425.7)--(459.9, 426.0)--(460.3, 426.2)--(461.6, 426.8)--(461.8, 426.9)--(464.2, 428.6)--(465.8, 429.7)--(466.4, 430.2)--(467.8, 428.5)--(468.3, 428.5)--(468.7, 428.8)--(469.0, 429.0)--(469.3, 429.3)--(469.4, 429.4)--(469.5, 429.5)--(469.7, 429.6)--(470.1, 429.9)--(470.2, 430.0)--(470.3, 430.0)--(470.5, 430.1)--(471.0, 430.6)--(471.1, 430.6)--(471.4, 430.8)--(471.6, 431.0)--(471.6, 431.1)--(471.7, 431.2)--(472.0, 431.6)--(472.1, 431.8)--(472.1, 431.9)--(472.2, 432.4)--(472.3, 432.7)--(472.6, 433.0)--(472.8, 433.4)--(473.3, 433.7)--(473.5, 433.7)--(473.5, 433.8)--(473.6, 433.9)--(473.7, 433.9)--(473.8, 434.0)--(473.9, 434.0)--(474.1, 434.1)--(474.2, 434.2)--(474.3, 434.3)--(474.4, 434.5)--(474.5, 434.5)--(474.7, 434.7)--(474.9, 435.5)--(475.6, 435.7)--(475.9, 435.8)--(476.2, 436.0)--(476.3, 436.1)--(477.0, 436.4)--(477.1, 436.5)--(477.3, 436.7)--(477.6, 436.9)--(478.3, 437.5)--(479.2, 437.9)--(479.4, 438.0)--(479.6, 438.0)--(479.7, 438.0)--(480.5, 438.0)--(480.5, 438.1)--(480.6, 438.1)--(480.7, 438.1)--(480.7, 438.0)--(480.9, 438.0)--(481.1, 438.0)--(481.8, 436.7)--(482.5, 435.6)--(482.6, 435.4)--(482.8, 435.2)--(482.9, 435.1)--(483.0, 435.0)--(483.0, 434.9)--(483.1, 434.8)--(483.8, 433.6)--(483.9, 433.4)--(484.0, 433.3)--(484.2, 433.0)--(484.3, 432.8)--(484.5, 432.5)--(484.7, 432.2)--(484.7, 432.1)--(484.8, 432.0)--(484.9, 431.8)--(485.1, 431.3)--(485.3, 431.1)--(485.3, 430.9)--(485.5, 430.8)--(485.6, 430.7)--(485.7, 430.5)--(485.8, 430.0)--(485.9, 429.8)--(486.0, 429.6)--(486.1, 429.3)--(486.1, 429.0)--(486.2, 428.9)--(486.2, 428.8)--(486.3, 428.8)--(486.3, 428.7)--(486.3, 428.6)--(486.3, 428.4)--(486.4, 428.3)--(486.4, 428.2)--(486.5, 428.1)--(486.6, 427.9)--(486.7, 427.8)--(486.8, 427.7)--(486.9, 427.6)--(487.1, 427.3)--(487.2, 427.1)--(487.4, 426.8)--(487.5, 426.6)--(487.7, 426.2)--(487.8, 426.0)--(487.8, 425.9)--(487.9, 425.8)--(488.0, 425.5)--(488.1, 424.8)--(488.1, 424.4)--(488.2, 423.9)--(488.3, 423.6)--(488.3, 423.3)--(488.4, 423.0)--(488.5, 422.9)--(488.6, 422.5)--(488.7, 422.2)--(488.8, 422.0)--(488.8, 421.7)--(488.9, 421.5)--(488.9, 421.3)--(488.9, 421.2)--(488.9, 420.9)--(488.9, 420.7)--(488.9, 420.4)--(488.9, 420.2)--(488.9, 419.9)--(488.9, 419.3)--(488.8, 419.0)--(488.8, 418.8)--(488.8, 418.6)--(488.8, 418.4)--(488.9, 417.8)--(488.9, 417.5)--(488.9, 417.4)--(488.9, 417.0)--(488.9, 416.9)--(489.0, 416.7)--(489.1, 416.3)--(489.1, 416.2)--(489.2, 416.0)--(489.2, 415.9)--(489.3, 415.8)--(489.4, 415.6)--(489.6, 415.3)--(489.3, 414.0)--(489.3, 413.9)--(489.3, 413.8)--(489.3, 413.7)--(489.3, 413.5)--(489.2, 413.3)--(489.2, 413.2)--(489.1, 412.3)--(488.9, 411.3)--(488.8, 411.0)--(488.8, 410.9)--(488.7, 410.9)--(488.4, 411.1)--(487.7, 411.4)--(487.0, 410.7)--(485.2, 408.7)--(484.6, 408.1)--(484.5, 408.1)--(484.1, 407.6)--(483.6, 407.1)--(482.5, 405.9)--(481.7, 405.1)--(481.6, 405.0)--(481.5, 405.0)--(481.5, 404.9)--(481.1, 404.6)--(480.4, 404.0)--(478.7, 402.6)--(477.9, 402.0)--(476.9, 401.1)--(476.2, 400.6)--(476.3, 400.1)--(476.4, 399.8)--(476.6, 399.4)--(476.7, 399.1)--(476.9, 398.7)--(477.0, 398.4)--(477.3, 397.8)--(476.7, 397.6)--(476.5, 397.6)--(476.4, 397.5)--(476.2, 397.5)--(475.6, 397.3)--(474.2, 396.9)--(474.1, 396.8)--(474.9, 394.3)--(474.8, 393.7)--(475.1, 393.7)--(475.4, 392.7)--(475.4, 392.6)--(475.4, 392.0)--(475.3, 391.9)--(475.2, 391.8)--(475.2, 391.5)--(475.2, 391.2)--(476.2, 391.1)--(478.0, 391.0)--(478.3, 390.9)--(478.5, 390.9)--(478.6, 390.9)--(478.7, 390.7)--(478.8, 390.5)--(478.8, 390.4)--(478.9, 390.2)--(479.0, 390.2)--(479.1, 390.0)--(479.2, 389.7)--(479.3, 389.5)--(477.4, 388.7)--(477.3, 388.4)--(477.4, 388.4)--(477.5, 388.4)--(477.6, 388.0)--(477.6, 387.9)--(477.6, 387.6)--(477.7, 387.3)--(477.8, 387.0)--(477.9, 386.8)--(477.9, 386.6)--(477.9, 386.5)--(478.0, 386.2)--(476.8, 385.5)--(476.4, 385.3)--(476.4, 385.0)--(476.4, 384.7)--(476.5, 384.5)--(476.5, 384.0)--(476.5, 383.6)--(476.5, 383.3)--(476.6, 383.3)--(476.6, 383.1)--(476.6, 382.9)--(476.6, 382.7)--(476.8, 382.4)--(476.9, 382.2)--(477.1, 381.8)--(477.1, 381.7)--(477.5, 381.9)--(477.6, 382.1)--(477.7, 382.1)--(477.6, 382.0)--(477.7, 381.9)--(477.8, 381.6)--(477.9, 381.2)--(478.2, 380.4)--(477.6, 380.2)--(477.0, 380.0)--(477.0, 379.9)--(477.1, 379.8)--(477.1, 379.7)--(477.1, 379.6)--(477.1, 379.3)--(477.1, 379.2)--(477.2, 379.1)--(477.3, 379.1)--(477.4, 379.1)--(477.4, 379.0)--(477.4, 378.9)--(477.5, 378.7)--(477.5, 378.6)--(477.4, 378.3)--(477.3, 377.9)--(477.2, 377.6)--(477.1, 377.2)--(477.0, 377.0)--(476.9, 376.8)--(476.9, 376.5)--(477.0, 376.3)--(477.0, 376.2)--(477.0, 375.9)--(477.0, 375.8)--(477.0, 375.4)--(476.9, 374.8)--(476.8, 374.6)--(476.8, 374.5)--(476.8, 374.4)--(476.4, 374.3)--(476.3, 374.3)--(476.2, 374.3)--(476.0, 374.2)--(475.9, 374.1)--(475.6, 374.0)--(475.5, 373.8)--(475.4, 373.7)--(475.3, 373.4)--(475.1, 373.2)--(475.0, 373.0)--(474.8, 372.8)--(474.5, 372.6)--(474.3, 372.4)--(474.2, 372.2)--(474.1, 372.1)--(473.9, 371.9)--(473.7, 371.7)--(470.4, 370.6)--(470.2, 370.6)--(470.1, 370.5)--(470.0, 370.5)--(469.8, 370.4)--(469.7, 370.0)--(470.3, 369.7)--(469.6, 368.0)--(468.8, 366.0)--(469.1, 365.9)--(469.4, 365.8)--(469.8, 365.6)--(470.9, 365.4)--(469.9, 364.0)--(469.3, 364.0)--(469.2, 364.0)--(468.6, 364.0)--(468.7, 363.5)--(466.0, 363.9)--(465.9, 363.3)--(465.6, 363.0)--(466.6, 362.2)--(466.6, 362.0)--(466.7, 362.0)--(467.0, 361.7)--(465.7, 359.9)--(465.5, 359.6)--(465.4, 359.6)--(465.5, 359.0)--(465.2, 358.7)--(464.6, 358.4)--(463.7, 358.5)--(462.7, 358.6)--(462.7, 357.9)--(462.6, 357.9)--(462.5, 357.9)--(462.3, 357.9)--(462.2, 357.9)--(462.1, 357.8)--(462.0, 357.7)--(461.9, 357.6)--(461.5, 357.0)--(461.3, 356.7)--(461.1, 356.5)--(461.0, 356.3)--(460.8, 356.1)--(460.6, 355.9)--(460.5, 355.8)--(460.3, 355.6)--(460.2, 355.5)--(460.1, 355.5)--(459.9, 355.3)--(459.8, 355.3)--(459.5, 355.1)--(459.2, 355.0)--(458.9, 354.9)--(458.8, 354.8)--(458.6, 354.6)--(458.0, 354.3)--(457.5, 354.1)--(457.2, 353.9)--(457.1, 353.8)--(456.9, 353.7)--(456.8, 353.6)--(456.7, 353.5)--(456.3, 353.2)--(456.0, 352.9)--(455.8, 352.8)--(455.7, 352.7)--(455.6, 352.6)--(455.4, 352.4)--(455.1, 352.3)--(454.7, 352.1)--(454.6, 351.9)--(454.1, 351.6)--(453.7, 351.4)--(453.2, 351.1)--(453.1, 351.0)--(452.9, 350.9)--(452.6, 350.7)--(452.5, 350.9)--(452.5, 351.0)--(452.4, 351.0)--(452.4, 351.2)--(452.3, 351.6)--(452.2, 352.0)--(452.1, 352.1)--(451.9, 352.8)--(451.7, 353.5)--(451.7, 353.6)--(451.7, 353.8)--(451.7, 354.0)--(451.7, 354.1)--(451.8, 354.5)--(451.8, 354.8)--(451.9, 354.9)--(451.5, 354.7)--(451.4, 354.5)--(451.3, 354.3)--(451.2, 354.2)--(451.2, 354.1)--(451.2, 354.0)--(451.1, 353.8)--(451.2, 353.6)--(451.2, 353.5)--(451.2, 353.4)--(451.1, 353.4)--(450.7, 353.3)--(450.6, 353.2)--(450.5, 353.1)--(450.5, 353.0)--(450.4, 353.0)--(450.3, 353.0)--(450.2, 353.0)--(450.1, 352.9)--(450.0, 352.9)--(449.8, 352.9)--(449.4, 352.8)--(449.1, 352.7)--(448.9, 352.7)--(448.8, 352.7)--(448.5, 352.6)--(447.8, 352.4)--(447.6, 352.4)--(447.4, 352.3)--(447.2, 352.2)--(446.8, 351.8)--(446.6, 351.6)--(446.2, 351.4)--(446.1, 351.3)--(446.0, 351.2)--(445.6, 351.0)--(445.5, 350.9)--(445.4, 350.9)--(445.3, 350.8)--(445.2, 350.7)--(445.2, 350.6)--(445.0, 350.5)--(444.9, 350.4)--(444.8, 350.3)--(444.6, 350.2)--(444.5, 350.2)--(444.4, 350.2)--(444.4, 350.1)--(444.4, 350.0)--(444.5, 349.9)--(444.4, 349.8)--(444.3, 349.7)--(444.1, 349.6)--(443.8, 349.5)--(443.4, 349.3)--(443.0, 349.1)--(442.5, 348.8)--(442.3, 348.8)--(442.2, 348.7)--(442.1, 348.7)--(442.0, 348.6)--(442.0, 348.5)--(441.9, 348.4)--(441.8, 348.6)--(441.7, 348.8)--(441.6, 349.0)--(441.5, 348.9)--(441.3, 348.8)--(441.1, 349.0)--(441.0, 348.9)--(440.7, 348.6)--(440.4, 348.5)--(440.2, 348.3)--(439.8, 348.2)--(439.5, 348.1)--(439.3, 348.1)--(439.0, 348.1)--(438.8, 348.1)--(438.7, 348.1)--(438.6, 348.1)--(438.6, 348.0)--(438.5, 348.0)--(438.4, 348.0)--(438.3, 348.0)--(438.0, 348.0)--(437.8, 348.0)--(437.7, 348.0)--(437.7, 347.8)--(437.5, 347.8)--(437.2, 348.0)--(437.1, 348.0)--(437.0, 348.0)--(436.6, 348.1)--(436.2, 348.3)--(436.2, 348.4)--(436.2, 348.5)--(436.2, 348.7)--(436.1, 348.9)--(436.1, 349.2)--(436.0, 349.3)--(436.0, 349.4)--(436.0, 349.6)--(436.0, 349.8)--(436.0, 350.0)--(436.0, 350.1)--(436.0, 350.2)--(436.0, 350.3)--(436.1, 350.6)--(436.1, 350.7)--(436.1, 350.8)--(436.1, 350.9)--(436.0, 351.0)--(435.8, 351.3)--(435.8, 351.5)--(435.8, 351.6)--(435.8, 351.7)--(435.8, 351.8)--(435.8, 351.9)--(435.8, 352.0)--(435.8, 352.1)--(435.8, 352.2)--(435.8, 352.3)--(435.8, 352.4)--(435.7, 352.6)--(435.7, 352.7)--(435.6, 353.0)--(435.6, 353.2)--(435.5, 353.3)--(435.5, 353.8)--(435.5, 353.9)--(435.5, 354.0)--(435.6, 354.2)--(435.7, 354.6)--(435.7, 354.8)--(435.8, 354.9)--(435.8, 355.1)--(435.8, 355.2)--(435.8, 355.3)--(435.8, 355.4)--(435.7, 355.4)--(435.7, 355.5)--(435.6, 355.6)--(435.5, 355.7)--(435.4, 355.8)--(435.3, 355.9)--(435.2, 356.3)--(435.2, 356.4)--(435.1, 356.5)--(435.1, 356.7)--(435.1, 356.9)--(435.1, 357.0)--(435.1, 357.2)--(435.0, 357.4)--(435.0, 357.5)--(435.0, 357.7)--(435.0, 358.0)--(435.0, 358.1)--(435.0, 358.2)--(434.9, 358.5)--(434.8, 359.1)--(434.6, 359.6)--(434.5, 360.1)--(434.3, 360.6)--(434.2, 360.9)--(434.1, 361.1)--(433.9, 361.3)--(433.6, 361.9)--(433.4, 362.2)--(433.3, 362.1)--(433.4, 361.9)--(433.5, 361.8)--(433.6, 361.7)--(433.7, 361.4)--(433.9, 361.2)--(434.0, 360.9)--(434.2, 360.3)--(434.3, 360.1)--(434.4, 360.0)--(434.5, 359.4)--(434.7, 358.6)--(434.9, 358.0)--(434.9, 357.9)--(434.9, 357.8)--(434.9, 357.7)--(434.9, 357.5)--(434.9, 357.4)--(434.9, 356.9)--(435.0, 356.5)--(435.0, 356.4)--(435.0, 356.3)--(435.1, 356.2)--(435.2, 355.9)--(435.3, 355.7)--(435.4, 355.5)--(435.6, 355.4)--(435.6, 355.3)--(435.6, 355.2)--(435.7, 355.1)--(435.7, 355.0)--(435.6, 354.9)--(435.6, 354.8)--(435.6, 354.7)--(435.5, 354.4)--(435.4, 354.2)--(435.4, 354.0)--(435.4, 353.9)--(435.4, 353.7)--(435.4, 353.5)--(435.4, 353.2)--(435.5, 352.7)--(435.5, 352.6)--(435.6, 352.3)--(435.6, 352.2)--(435.7, 352.0)--(435.7, 351.9)--(435.6, 351.8)--(435.6, 351.7)--(435.6, 351.5)--(435.6, 351.4)--(435.7, 351.2)--(435.8, 351.1)--(435.8, 350.9)--(435.9, 350.9)--(435.9, 350.8)--(435.9, 350.7)--(435.9, 350.3)--(435.8, 350.0)--(435.8, 349.9)--(435.8, 349.7)--(435.8, 349.4)--(435.9, 349.0)--(436.0, 348.7)--(436.0, 348.6)--(436.0, 348.4)--cycle; +Richmond_upon_Thames = (166.8, 253.8)--(166.9, 253.5)--(167.0, 253.3)--(167.1, 253.1)--(167.2, 252.8)--(167.4, 252.6)--(167.5, 252.4)--(167.6, 252.2)--(167.8, 252.0)--(167.9, 251.9)--(168.5, 251.2)--(168.7, 250.9)--(169.0, 250.6)--(169.5, 250.1)--(169.6, 250.1)--(169.7, 250.0)--(169.8, 249.9)--(169.7, 249.9)--(169.8, 249.8)--(170.5, 249.2)--(170.8, 249.0)--(170.8, 248.9)--(171.7, 248.1)--(171.9, 248.0)--(172.2, 247.7)--(172.3, 247.7)--(172.4, 247.6)--(172.6, 247.5)--(173.1, 247.1)--(173.1, 247.0)--(173.2, 247.0)--(173.3, 247.1)--(173.5, 247.0)--(173.7, 246.8)--(174.1, 246.6)--(174.0, 246.5)--(174.1, 246.5)--(174.3, 246.6)--(174.5, 246.4)--(174.4, 246.3)--(174.5, 246.2)--(174.6, 246.3)--(175.0, 246.0)--(175.1, 245.9)--(175.2, 245.7)--(175.5, 245.6)--(175.9, 245.2)--(176.1, 245.2)--(176.3, 245.2)--(176.6, 245.1)--(176.8, 244.9)--(177.0, 244.8)--(176.5, 244.4)--(176.6, 244.3)--(177.1, 244.7)--(177.3, 244.6)--(177.4, 244.5)--(177.5, 244.4)--(177.7, 244.2)--(177.7, 244.1)--(178.0, 243.8)--(178.3, 243.5)--(178.3, 243.3)--(178.7, 242.8)--(178.9, 242.2)--(179.3, 241.4)--(179.4, 241.2)--(179.5, 240.7)--(179.5, 240.5)--(179.6, 240.3)--(179.5, 240.1)--(179.5, 239.8)--(179.5, 239.7)--(179.4, 239.3)--(179.4, 239.1)--(179.2, 238.8)--(178.9, 238.3)--(178.8, 238.1)--(178.4, 237.6)--(178.3, 237.4)--(178.2, 237.3)--(178.1, 237.2)--(177.9, 237.1)--(177.6, 236.9)--(177.3, 236.5)--(177.1, 236.3)--(176.9, 236.1)--(176.3, 235.6)--(176.1, 235.4)--(175.8, 235.2)--(175.4, 235.0)--(175.2, 234.9)--(174.6, 234.6)--(174.3, 234.4)--(174.1, 234.3)--(173.6, 234.1)--(173.2, 234.0)--(173.1, 233.9)--(172.5, 233.7)--(172.1, 233.6)--(171.8, 233.5)--(171.7, 233.4)--(171.5, 233.4)--(171.4, 233.3)--(171.4, 233.2)--(171.4, 233.1)--(171.2, 233.0)--(170.9, 233.0)--(170.5, 232.9)--(170.1, 232.8)--(170.0, 232.8)--(169.7, 232.8)--(169.0, 232.8)--(168.8, 232.8)--(168.6, 232.9)--(168.3, 232.9)--(168.1, 232.9)--(168.0, 233.1)--(167.9, 233.1)--(167.9, 233.0)--(167.8, 233.0)--(167.8, 233.1)--(167.4, 233.2)--(167.4, 233.1)--(167.3, 233.1)--(167.1, 233.1)--(166.7, 233.0)--(166.4, 233.0)--(165.9, 232.9)--(165.8, 232.9)--(165.3, 232.8)--(165.0, 232.7)--(165.0, 232.8)--(164.9, 232.8)--(164.9, 232.6)--(164.9, 232.5)--(164.5, 232.4)--(164.3, 232.3)--(164.0, 232.0)--(163.8, 231.9)--(163.7, 231.9)--(163.5, 231.8)--(163.6, 231.7)--(163.5, 231.7)--(163.5, 231.6)--(163.2, 231.3)--(163.0, 231.0)--(162.8, 230.8)--(162.8, 230.7)--(162.2, 229.9)--(162.0, 229.6)--(161.8, 229.3)--(161.7, 229.2)--(161.6, 229.0)--(161.4, 228.7)--(161.3, 228.7)--(161.4, 228.6)--(161.2, 228.3)--(161.1, 228.1)--(161.1, 228.0)--(161.1, 227.9)--(160.7, 227.2)--(160.5, 226.9)--(160.5, 226.8)--(160.6, 226.6)--(160.6, 226.5)--(160.7, 226.3)--(160.6, 226.2)--(160.7, 226.1)--(160.7, 226.0)--(160.7, 225.7)--(160.8, 225.4)--(160.8, 225.2)--(160.8, 225.1)--(160.8, 225.0)--(160.7, 225.0)--(160.6, 224.8)--(160.6, 224.7)--(160.6, 224.6)--(160.4, 224.5)--(160.4, 224.4)--(160.6, 224.5)--(160.8, 224.3)--(161.0, 224.1)--(161.1, 224.0)--(161.2, 223.8)--(161.4, 223.6)--(161.5, 223.5)--(161.5, 223.4)--(161.5, 223.3)--(161.6, 223.3)--(161.8, 223.0)--(161.7, 223.0)--(161.7, 222.9)--(161.8, 222.9)--(161.9, 222.7)--(162.1, 222.4)--(162.2, 222.1)--(162.4, 221.9)--(162.4, 221.7)--(162.4, 221.6)--(162.4, 221.5)--(162.2, 221.4)--(162.2, 221.3)--(162.5, 221.4)--(162.5, 221.3)--(162.5, 221.2)--(162.6, 221.2)--(162.6, 220.8)--(162.7, 220.6)--(162.7, 220.5)--(162.7, 220.2)--(162.8, 219.1)--(162.9, 218.7)--(162.9, 218.5)--(163.1, 218.2)--(163.5, 217.5)--(163.9, 217.1)--(164.1, 216.7)--(164.3, 216.5)--(164.5, 216.2)--(164.3, 216.1)--(164.4, 216.0)--(164.6, 216.2)--(164.9, 215.9)--(164.7, 215.6)--(164.8, 215.5)--(165.0, 215.8)--(165.5, 215.4)--(165.6, 215.4)--(165.8, 215.3)--(166.0, 215.1)--(166.2, 215.0)--(166.2, 214.9)--(166.2, 214.8)--(166.3, 214.9)--(166.4, 214.9)--(166.6, 214.8)--(166.7, 214.7)--(166.8, 214.7)--(167.0, 214.6)--(166.9, 214.4)--(167.0, 214.4)--(167.1, 214.3)--(167.3, 214.3)--(167.4, 214.2)--(167.5, 214.2)--(167.6, 214.2)--(167.8, 214.2)--(168.0, 214.1)--(168.3, 214.0)--(168.5, 213.9)--(168.9, 213.8)--(169.5, 213.7)--(169.9, 213.5)--(170.0, 213.7)--(170.1, 213.9)--(170.0, 214.0)--(169.7, 214.3)--(169.7, 214.2)--(169.6, 214.3)--(169.5, 214.4)--(169.4, 214.5)--(169.3, 214.5)--(169.2, 214.6)--(168.9, 214.7)--(168.5, 214.8)--(168.4, 214.8)--(168.3, 214.7)--(168.1, 214.7)--(168.2, 214.8)--(168.0, 214.8)--(167.9, 214.9)--(167.6, 215.0)--(167.5, 215.0)--(167.3, 215.0)--(167.0, 215.2)--(166.7, 215.3)--(166.6, 215.4)--(166.5, 215.5)--(166.4, 215.6)--(166.4, 215.7)--(166.3, 215.7)--(166.1, 215.8)--(165.9, 215.9)--(165.9, 216.0)--(166.0, 216.0)--(165.8, 216.2)--(165.4, 216.5)--(165.2, 216.7)--(164.9, 216.8)--(164.8, 216.9)--(164.7, 217.0)--(164.8, 217.1)--(164.9, 217.1)--(164.9, 217.2)--(164.8, 217.3)--(164.7, 217.4)--(164.3, 217.7)--(164.0, 218.0)--(164.0, 218.1)--(163.9, 218.3)--(163.8, 218.4)--(163.8, 218.5)--(163.7, 218.7)--(163.5, 219.1)--(163.5, 219.3)--(163.4, 219.5)--(163.4, 219.7)--(163.4, 219.8)--(163.4, 220.0)--(163.4, 220.1)--(163.4, 220.2)--(163.3, 220.4)--(163.4, 220.7)--(163.3, 220.8)--(163.3, 220.9)--(163.4, 220.9)--(163.3, 221.1)--(163.3, 221.2)--(163.2, 221.4)--(163.1, 221.7)--(163.1, 221.8)--(162.9, 222.2)--(162.7, 222.6)--(162.6, 223.0)--(162.5, 223.1)--(162.4, 223.3)--(162.8, 223.2)--(162.8, 223.3)--(162.6, 223.3)--(162.5, 223.4)--(162.4, 223.4)--(162.3, 223.5)--(162.2, 223.5)--(162.2, 223.6)--(162.2, 223.7)--(162.0, 224.1)--(161.9, 224.3)--(162.0, 224.4)--(161.9, 224.6)--(161.8, 224.6)--(161.8, 224.7)--(161.6, 225.0)--(161.5, 225.5)--(161.5, 226.3)--(161.5, 226.8)--(161.6, 227.3)--(161.8, 227.7)--(161.8, 227.8)--(161.9, 227.9)--(161.9, 228.1)--(162.0, 228.2)--(162.2, 228.4)--(162.2, 228.5)--(162.4, 228.7)--(162.6, 228.9)--(162.7, 229.0)--(163.3, 229.4)--(163.4, 229.5)--(163.5, 229.5)--(163.6, 229.6)--(164.1, 229.9)--(164.5, 230.1)--(165.1, 230.4)--(165.5, 230.6)--(165.7, 230.7)--(166.0, 230.8)--(166.3, 231.0)--(167.0, 231.4)--(167.2, 231.5)--(167.3, 231.6)--(167.5, 231.6)--(167.8, 231.7)--(167.9, 231.8)--(168.4, 231.9)--(168.6, 231.9)--(169.2, 231.9)--(169.5, 231.9)--(169.7, 231.9)--(169.9, 231.9)--(170.6, 232.0)--(172.5, 232.6)--(172.8, 232.8)--(173.9, 233.2)--(174.3, 233.3)--(175.0, 233.6)--(175.4, 233.7)--(175.9, 234.0)--(176.2, 234.1)--(176.4, 234.2)--(176.5, 234.3)--(176.8, 234.5)--(177.1, 234.7)--(177.6, 235.1)--(177.8, 235.2)--(177.9, 235.3)--(178.0, 235.3)--(178.1, 235.4)--(178.1, 235.3)--(178.2, 235.3)--(178.3, 235.4)--(178.3, 235.6)--(178.3, 235.7)--(178.7, 236.1)--(179.1, 236.6)--(179.2, 236.7)--(179.4, 237.1)--(179.6, 237.3)--(179.8, 237.6)--(179.9, 237.8)--(180.2, 238.3)--(180.2, 238.4)--(180.3, 238.5)--(180.3, 238.6)--(180.3, 238.7)--(180.3, 238.8)--(180.4, 239.2)--(180.4, 239.6)--(180.4, 239.8)--(180.4, 240.1)--(180.3, 240.2)--(180.3, 240.4)--(180.2, 240.6)--(180.3, 240.8)--(180.2, 240.8)--(180.2, 240.9)--(180.1, 241.2)--(180.0, 241.4)--(179.9, 241.5)--(179.7, 242.1)--(179.5, 242.5)--(179.4, 242.8)--(179.2, 243.2)--(179.0, 243.6)--(178.9, 243.7)--(178.6, 244.2)--(177.9, 245.1)--(177.4, 245.6)--(177.0, 246.1)--(176.6, 246.5)--(176.4, 246.6)--(176.5, 246.7)--(176.5, 246.8)--(176.4, 246.8)--(176.3, 246.7)--(176.2, 246.7)--(175.8, 246.9)--(175.5, 247.1)--(175.3, 247.2)--(175.0, 247.4)--(174.7, 247.5)--(174.6, 247.6)--(174.1, 247.8)--(173.7, 248.0)--(173.3, 248.1)--(173.2, 248.2)--(173.0, 248.3)--(172.7, 248.5)--(172.5, 248.6)--(172.4, 248.8)--(172.2, 248.9)--(172.0, 249.0)--(171.8, 249.3)--(171.4, 249.7)--(171.4, 249.8)--(171.1, 250.0)--(170.6, 250.5)--(170.5, 250.4)--(170.1, 250.7)--(169.9, 251.0)--(170.0, 251.0)--(169.7, 251.3)--(169.7, 251.4)--(169.3, 251.7)--(168.7, 252.3)--(168.6, 252.4)--(168.5, 252.5)--(168.4, 252.7)--(168.3, 252.9)--(168.1, 253.1)--(168.1, 253.3)--(168.0, 253.5)--(168.0, 253.8)--(168.0, 254.0)--(168.0, 254.3)--(168.1, 254.4)--(168.2, 254.9)--(168.3, 255.7)--(168.4, 256.6)--(168.4, 257.2)--(168.8, 259.1)--(168.9, 259.3)--(168.9, 259.4)--(169.1, 259.7)--(169.2, 259.9)--(169.3, 260.0)--(169.4, 260.1)--(169.5, 260.2)--(169.5, 260.3)--(169.6, 260.3)--(169.7, 260.4)--(169.9, 260.4)--(170.0, 260.5)--(170.3, 260.6)--(170.6, 260.7)--(170.8, 260.8)--(170.9, 260.8)--(171.0, 260.8)--(171.0, 260.9)--(171.2, 261.0)--(171.6, 261.1)--(171.8, 261.3)--(171.9, 261.3)--(171.9, 261.4)--(172.0, 261.4)--(172.1, 261.4)--(172.1, 261.5)--(172.2, 261.6)--(172.3, 261.6)--(172.4, 261.6)--(172.5, 261.6)--(172.5, 261.7)--(172.6, 261.8)--(172.7, 261.9)--(173.0, 262.0)--(173.3, 262.2)--(173.4, 262.3)--(173.5, 262.4)--(173.8, 262.5)--(173.9, 262.5)--(174.2, 262.8)--(174.3, 262.9)--(174.4, 262.9)--(174.5, 263.0)--(174.6, 263.1)--(174.7, 263.1)--(175.0, 263.3)--(176.3, 264.1)--(176.4, 264.2)--(176.6, 264.4)--(176.7, 264.5)--(176.9, 264.7)--(176.9, 264.8)--(177.0, 264.9)--(177.7, 266.0)--(178.2, 266.9)--(178.7, 267.9)--(179.3, 269.0)--(179.8, 269.7)--(180.3, 270.4)--(181.1, 271.4)--(181.9, 272.5)--(182.6, 273.4)--(183.1, 274.2)--(183.8, 275.0)--(184.6, 275.8)--(185.3, 276.4)--(185.4, 276.5)--(185.5, 276.5)--(185.6, 276.6)--(187.9, 277.5)--(188.2, 277.6)--(188.5, 277.7)--(188.8, 277.7)--(188.9, 277.8)--(190.1, 277.9)--(190.6, 277.8)--(191.5, 277.6)--(191.7, 277.5)--(191.9, 277.4)--(192.0, 277.4)--(192.1, 277.3)--(192.2, 277.2)--(192.4, 277.1)--(192.5, 276.9)--(192.6, 276.8)--(192.7, 276.7)--(192.9, 276.5)--(193.0, 276.4)--(193.1, 276.3)--(193.2, 276.2)--(193.6, 275.9)--(194.9, 275.1)--(195.2, 274.8)--(196.1, 273.8)--(196.2, 273.7)--(196.4, 273.5)--(196.6, 273.2)--(196.8, 273.0)--(196.9, 272.8)--(197.1, 272.6)--(197.2, 272.4)--(197.4, 272.1)--(197.4, 271.9)--(197.5, 271.7)--(197.7, 271.3)--(197.9, 270.8)--(198.1, 270.1)--(198.5, 268.6)--(199.1, 267.1)--(199.6, 266.0)--(200.0, 265.4)--(200.2, 264.9)--(200.5, 264.5)--(200.7, 264.3)--(200.9, 264.1)--(201.0, 263.8)--(201.3, 263.5)--(202.1, 262.7)--(202.5, 262.4)--(202.9, 262.1)--(203.3, 261.8)--(204.0, 261.4)--(204.1, 261.3)--(204.1, 261.2)--(204.3, 261.1)--(204.5, 261.1)--(205.3, 260.7)--(206.0, 260.5)--(206.7, 260.3)--(206.7, 260.2)--(206.7, 260.1)--(207.1, 260.3)--(207.8, 260.2)--(208.4, 260.2)--(208.7, 260.3)--(208.9, 260.3)--(209.4, 260.3)--(210.0, 260.4)--(210.5, 260.5)--(211.3, 260.8)--(211.7, 260.9)--(211.9, 261.0)--(212.1, 261.1)--(212.4, 261.3)--(212.6, 261.5)--(213.0, 261.7)--(213.5, 262.1)--(213.8, 262.3)--(214.2, 262.8)--(214.7, 263.2)--(215.3, 264.0)--(215.3, 264.1)--(215.5, 264.6)--(216.1, 265.9)--(216.3, 266.2)--(216.3, 266.4)--(216.4, 266.4)--(216.5, 266.9)--(216.6, 267.2)--(216.5, 267.3)--(216.6, 267.6)--(216.6, 268.4)--(216.7, 270.5)--(216.7, 270.6)--(216.7, 271.0)--(216.7, 271.2)--(216.7, 271.4)--(216.6, 271.6)--(216.6, 272.5)--(216.5, 272.7)--(216.5, 272.9)--(216.5, 273.2)--(216.5, 273.4)--(216.6, 273.6)--(216.6, 273.8)--(216.6, 274.1)--(216.7, 274.4)--(216.8, 274.6)--(216.8, 274.7)--(216.9, 274.9)--(216.9, 275.1)--(217.0, 275.2)--(217.1, 275.4)--(217.3, 275.7)--(217.4, 275.8)--(217.4, 275.9)--(217.7, 276.2)--(217.8, 276.3)--(217.9, 276.5)--(218.2, 276.8)--(220.1, 278.7)--(220.3, 278.9)--(221.5, 280.1)--(221.8, 280.3)--(221.9, 280.4)--(222.0, 280.6)--(222.2, 280.7)--(222.4, 280.8)--(222.5, 280.8)--(222.7, 280.9)--(222.9, 281.0)--(223.0, 281.1)--(223.2, 281.2)--(223.4, 281.2)--(223.6, 281.3)--(223.9, 281.3)--(224.1, 281.4)--(224.2, 281.4)--(224.4, 281.4)--(224.8, 281.4)--(225.0, 281.5)--(225.3, 281.5)--(225.6, 281.5)--(226.0, 281.4)--(226.5, 281.4)--(226.7, 281.3)--(226.9, 281.3)--(227.1, 281.3)--(227.2, 281.2)--(227.4, 281.1)--(227.6, 281.1)--(227.8, 281.0)--(228.0, 280.9)--(228.1, 280.8)--(228.4, 280.7)--(228.6, 280.5)--(228.8, 280.4)--(228.9, 280.3)--(229.1, 280.2)--(229.3, 280.0)--(229.4, 279.9)--(229.7, 279.7)--(230.0, 279.3)--(230.2, 279.0)--(230.5, 278.6)--(230.7, 278.4)--(230.9, 278.1)--(231.1, 277.6)--(231.3, 277.3)--(231.7, 276.5)--(231.6, 276.4)--(231.8, 276.0)--(231.8, 275.7)--(231.8, 275.6)--(231.9, 275.5)--(231.9, 275.3)--(232.0, 274.3)--(232.2, 272.8)--(232.4, 271.5)--(232.4, 271.1)--(232.4, 270.9)--(232.4, 270.7)--(232.4, 270.2)--(232.4, 269.3)--(232.4, 269.0)--(232.4, 268.9)--(232.4, 268.4)--(232.4, 267.9)--(232.5, 267.6)--(232.5, 267.4)--(232.6, 267.2)--(232.7, 267.0)--(232.7, 266.9)--(232.8, 266.6)--(232.9, 266.3)--(233.0, 266.1)--(233.1, 265.9)--(233.2, 265.7)--(233.5, 265.3)--(233.8, 264.7)--(234.0, 264.3)--(234.2, 264.1)--(234.3, 263.9)--(234.6, 263.4)--(235.2, 262.6)--(235.1, 262.5)--(234.7, 262.4)--(234.9, 262.2)--(234.8, 262.1)--(234.9, 262.0)--(234.9, 261.9)--(235.0, 261.8)--(235.0, 261.7)--(235.0, 261.6)--(234.9, 261.5)--(234.9, 261.4)--(234.8, 261.4)--(234.7, 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231.8)--(122.6, 231.9)--(122.4, 232.1)--(122.4, 232.2)--(122.4, 232.4)--(122.3, 232.5)--(122.3, 232.6)--(122.3, 232.7)--(122.3, 233.0)--(122.3, 233.2)--(122.3, 233.3)--(122.4, 233.6)--(122.3, 233.7)--(122.3, 233.8)--(122.1, 233.9)--(122.0, 234.1)--(121.9, 234.3)--(121.5, 234.7)--(121.4, 234.9)--(121.4, 235.0)--(121.3, 235.0)--(123.5, 235.6)--(124.1, 235.2)--(124.3, 235.5)--(124.7, 236.0)--(125.2, 236.7)--(126.2, 238.0)--(126.8, 238.8)--(127.4, 239.6)--(128.1, 240.5)--(128.3, 240.8)--(128.5, 241.0)--(128.8, 241.5)--(129.0, 241.7)--(129.2, 242.0)--(129.4, 242.2)--(129.8, 242.8)--(129.8, 242.9)--(130.2, 243.2)--(130.5, 243.5)--(131.1, 244.0)--(131.3, 244.2)--(131.5, 243.9)--(131.6, 243.7)--(131.7, 243.8)--(131.7, 244.0)--(132.1, 243.9)--(133.0, 243.8)--(133.6, 243.7)--(133.7, 243.7)--(134.5, 243.6)--(134.5, 243.7)--(134.8, 244.0)--(135.0, 244.2)--(135.1, 244.3)--(135.3, 244.3)--(135.5, 244.4)--(136.9, 244.7)--(137.3, 244.6)--(137.3, 244.5)--(137.5, 244.5)--(138.4, 244.6)--(138.7, 244.6)--(139.0, 244.8)--(139.4, 245.0)--(140.5, 245.2)--(141.3, 245.3)--(141.8, 245.4)--(142.1, 245.5)--(142.3, 245.5)--(142.6, 245.5)--(142.8, 245.5)--(143.7, 245.5)--(143.9, 245.5)--(144.9, 245.3)--(145.1, 244.9)--(145.2, 244.7)--(145.4, 244.4)--(145.4, 243.9)--(145.4, 243.8)--(145.5, 243.8)--(145.7, 243.8)--(146.2, 243.8)--(146.8, 243.8)--(147.0, 243.8)--(147.3, 243.8)--(147.9, 243.8)--(148.3, 243.8)--(148.7, 243.9)--(148.9, 243.9)--(149.0, 244.0)--(149.2, 244.0)--(149.4, 244.2)--(149.7, 244.3)--(149.8, 244.3)--(150.0, 244.4)--(150.1, 244.4)--(150.3, 244.5)--(150.4, 244.6)--(150.4, 244.9)--(150.6, 245.1)--(150.8, 245.1)--(150.9, 245.2)--(151.1, 245.2)--(151.3, 245.2)--(151.4, 245.2)--(151.6, 245.3)--(151.7, 245.3)--(151.8, 245.3)--(152.2, 245.3)--(152.4, 245.4)--(152.6, 245.7)--(152.8, 246.0)--(152.9, 246.1)--(153.0, 246.3)--(153.1, 246.4)--(153.2, 246.5)--(153.5, 246.6)--(153.8, 246.7)--(153.9, 246.7)--(154.1, 246.7)--(154.1, 246.5)--(154.0, 246.2)--(153.9, 245.9)--(153.8, 245.5)--(153.8, 245.2)--(153.8, 245.1)--(153.8, 245.0)--(153.8, 244.7)--(153.9, 244.5)--(154.0, 244.3)--(154.4, 243.8)--(154.7, 243.3)--(155.0, 243.0)--(155.3, 243.2)--(155.9, 242.5)--(155.8, 242.5)--(156.0, 242.0)--(156.0, 241.8)--(156.1, 241.7)--(156.3, 241.7)--(156.5, 241.6)--(156.7, 242.4)--(156.7, 242.6)--(156.9, 242.8)--(157.1, 242.8)--(158.0, 242.7)--(158.3, 242.7)--(158.7, 242.6)--(158.8, 242.3)--(158.9, 242.4)--(159.0, 242.4)--(159.2, 242.5)--(159.2, 242.6)--(159.3, 242.7)--(159.4, 242.9)--(159.5, 243.0)--(159.6, 243.1)--(159.6, 243.6)--(159.8, 243.8)--(160.0, 244.1)--(160.3, 244.7)--(160.5, 244.9)--(160.7, 245.2)--(161.0, 245.3)--(161.5, 245.5)--(161.9, 245.5)--(162.1, 245.5)--(162.3, 245.4)--(162.5, 245.3)--(162.9, 244.8)--(163.1, 244.4)--(163.2, 244.4)--(163.3, 244.4)--(163.4, 244.5)--(163.6, 244.7)--(163.6, 244.8)--(163.8, 245.1)--(163.9, 245.3)--(164.0, 245.3)--(164.2, 245.4)--(164.3, 245.5)--(164.3, 245.7)--(164.3, 245.8)--(164.2, 245.9)--(164.1, 246.0)--(164.0, 246.2)--(163.9, 246.2)--(163.9, 246.3)--(163.9, 246.4)--(164.0, 246.4)--(164.0, 246.5)--(163.8, 246.7)--(163.8, 246.8)--(163.7, 246.8)--(163.7, 246.9)--(163.7, 247.0)--(163.6, 247.0)--(163.6, 247.1)--(163.7, 247.1)--(163.7, 247.2)--(163.7, 247.3)--(163.8, 247.3)--(163.8, 247.4)--(163.8, 247.5)--(163.8, 247.6)--(163.7, 247.6)--(163.6, 247.6)--(163.5, 247.6)--(163.5, 247.7)--(163.4, 247.7)--(163.4, 247.8)--(163.5, 247.8)--(163.5, 247.9)--(163.5, 248.0)--(163.6, 248.2)--(163.6, 248.4)--(163.6, 248.5)--(163.7, 248.5)--(163.7, 248.6)--(163.7, 248.7)--(163.7, 248.8)--(163.6, 248.8)--(163.6, 248.9)--(163.5, 248.9)--(163.4, 248.9)--(163.4, 249.0)--(163.3, 249.0)--(163.3, 249.1)--(163.3, 249.3)--(163.3, 249.4)--(163.4, 249.4)--(163.4, 249.5)--(163.4, 249.6)--(163.4, 249.7)--(163.7, 250.0)--(163.7, 250.3)--(163.7, 250.4)--(163.8, 250.5)--(163.9, 250.5)--(163.9, 250.7)--(163.9, 250.8)--(164.0, 250.9)--(164.0, 251.0)--(164.1, 251.1)--(164.1, 251.2)--(164.1, 251.5)--(164.2, 251.5)--(164.3, 251.8)--(164.4, 252.2)--(164.6, 252.6)--(164.8, 252.8)--(165.0, 253.1)--(165.2, 253.3)--(165.3, 253.5)--(165.4, 253.6)--(165.5, 253.7)--(165.6, 253.7)--(165.6, 253.8)--(165.7, 253.8)--(165.8, 253.8)--(165.9, 253.8)--(166.0, 253.8)--(166.1, 253.8)--(166.2, 253.8)--(166.7, 253.8)--cycle; +Southwark = (313.4, 305.3)--(313.4, 305.4)--(313.7, 305.4)--(314.0, 305.4)--(314.1, 305.5)--(314.1, 305.3)--(314.2, 305.4)--(314.3, 305.5)--(314.4, 305.5)--(314.5, 305.5)--(314.7, 305.5)--(315.1, 305.5)--(315.5, 305.5)--(316.0, 305.6)--(316.3, 305.6)--(316.2, 305.5)--(316.3, 305.4)--(316.3, 305.3)--(316.7, 305.3)--(316.7, 305.4)--(316.7, 305.5)--(316.9, 305.5)--(317.2, 305.6)--(317.5, 305.6)--(318.5, 305.6)--(319.6, 305.6)--(319.7, 305.5)--(320.2, 305.5)--(320.8, 305.5)--(321.9, 305.4)--(322.3, 305.4)--(322.4, 305.4)--(322.6, 305.4)--(322.9, 305.3)--(323.0, 305.3)--(323.0, 305.2)--(323.2, 305.1)--(323.3, 305.1)--(323.4, 305.0)--(323.5, 305.0)--(324.1, 304.7)--(324.5, 304.6)--(325.1, 304.4)--(325.1, 304.3)--(325.2, 304.2)--(325.2, 304.3)--(325.4, 304.3)--(325.4, 304.2)--(325.8, 304.1)--(325.9, 304.2)--(326.3, 304.1)--(326.3, 303.9)--(326.2, 303.7)--(326.3, 303.6)--(326.3, 303.7)--(326.4, 303.9)--(326.4, 304.1)--(326.8, 304.0)--(327.8, 303.9)--(327.9, 303.9)--(328.0, 303.9)--(328.1, 303.9)--(328.3, 303.9)--(328.6, 303.9)--(328.8, 303.8)--(328.8, 303.7)--(329.0, 303.6)--(329.1, 303.6)--(329.6, 303.6)--(330.2, 303.6)--(330.4, 303.5)--(330.9, 303.4)--(331.0, 303.4)--(331.1, 303.4)--(331.3, 303.3)--(331.4, 303.3)--(331.7, 303.2)--(332.3, 303.1)--(332.5, 303.0)--(332.8, 303.0)--(332.8, 302.9)--(333.4, 302.7)--(333.6, 302.6)--(333.8, 302.5)--(334.0, 302.4)--(334.1, 302.5)--(334.9, 302.1)--(335.1, 302.0)--(335.1, 301.9)--(335.8, 301.6)--(335.9, 301.5)--(336.3, 301.3)--(336.4, 301.3)--(336.7, 301.2)--(336.8, 301.2)--(336.9, 301.2)--(337.5, 300.9)--(337.6, 300.8)--(338.0, 300.7)--(338.4, 300.5)--(338.5, 300.5)--(338.7, 300.4)--(339.1, 300.0)--(339.2, 300.1)--(339.4, 300.0)--(339.9, 299.4)--(339.4, 298.9)--(339.2, 298.7)--(339.1, 298.3)--(339.0, 298.1)--(338.9, 298.0)--(338.4, 297.5)--(338.2, 297.3)--(338.1, 297.1)--(338.0, 297.1)--(338.1, 297.1)--(338.3, 297.2)--(338.9, 297.8)--(339.1, 298.0)--(339.2, 298.1)--(339.3, 298.3)--(339.4, 298.4)--(339.7, 298.8)--(340.0, 299.1)--(340.7, 298.8)--(340.7, 298.9)--(340.9, 298.8)--(341.0, 298.8)--(341.3, 298.7)--(342.2, 298.2)--(342.2, 298.1)--(342.4, 298.1)--(342.5, 298.3)--(344.1, 297.8)--(344.1, 297.6)--(344.2, 297.6)--(344.9, 297.5)--(344.9, 297.4)--(345.2, 297.4)--(345.2, 297.5)--(345.8, 297.6)--(346.2, 297.6)--(346.4, 297.6)--(347.1, 297.5)--(347.1, 297.6)--(347.3, 297.6)--(347.8, 297.6)--(347.9, 297.6)--(348.1, 297.6)--(348.3, 297.6)--(348.6, 297.7)--(349.2, 297.8)--(349.4, 297.8)--(349.5, 297.8)--(350.0, 298.0)--(350.1, 298.0)--(350.1, 297.9)--(350.2, 298.0)--(350.2, 298.1)--(350.3, 298.2)--(350.4, 298.2)--(350.9, 298.5)--(351.0, 298.5)--(351.3, 298.7)--(351.6, 299.0)--(351.7, 299.0)--(351.8, 299.0)--(351.8, 298.9)--(351.9, 299.0)--(352.1, 299.1)--(352.2, 299.2)--(352.5, 299.3)--(352.8, 299.5)--(352.9, 299.6)--(353.6, 300.0)--(353.9, 300.2)--(354.1, 300.4)--(354.2, 300.5)--(354.6, 300.9)--(354.8, 301.1)--(355.2, 301.5)--(355.0, 301.7)--(355.3, 302.0)--(355.7, 301.9)--(355.7, 301.7)--(355.8, 301.6)--(355.9, 301.6)--(355.9, 301.7)--(355.9, 301.8)--(355.8, 302.1)--(355.8, 302.2)--(355.8, 302.3)--(356.1, 302.6)--(356.0, 302.7)--(356.4, 303.0)--(356.6, 302.7)--(356.7, 302.8)--(356.8, 302.9)--(357.0, 302.7)--(357.1, 302.8)--(356.9, 303.0)--(357.0, 303.1)--(356.8, 303.2)--(357.4, 303.8)--(357.6, 303.5)--(357.8, 303.7)--(357.7, 303.8)--(357.6, 304.2)--(358.2, 304.8)--(358.3, 304.8)--(358.5, 304.9)--(358.9, 305.0)--(359.1, 305.1)--(360.5, 305.5)--(360.9, 305.5)--(361.1, 305.5)--(361.3, 305.5)--(361.7, 305.5)--(362.1, 305.4)--(362.3, 305.4)--(363.1, 305.2)--(363.3, 305.2)--(363.3, 305.1)--(363.4, 305.1)--(363.5, 305.2)--(363.6, 305.1)--(363.8, 305.1)--(364.1, 305.0)--(364.3, 304.9)--(364.7, 304.7)--(364.9, 304.5)--(365.2, 304.4)--(365.5, 304.0)--(365.6, 303.9)--(365.7, 303.7)--(365.8, 303.6)--(365.9, 303.5)--(365.9, 303.4)--(366.0, 303.3)--(366.1, 303.0)--(366.2, 302.6)--(366.1, 302.5)--(366.2, 302.4)--(366.3, 302.4)--(366.3, 302.3)--(366.2, 302.2)--(366.2, 302.1)--(366.4, 302.1)--(366.5, 302.2)--(366.6, 302.1)--(366.6, 302.0)--(366.6, 301.9)--(366.6, 301.8)--(366.6, 301.7)--(366.6, 301.4)--(366.6, 301.3)--(366.7, 300.8)--(366.7, 300.7)--(366.6, 300.7)--(366.7, 300.4)--(366.7, 300.2)--(366.7, 300.0)--(366.8, 299.3)--(366.8, 299.2)--(366.8, 298.9)--(366.7, 298.6)--(366.7, 298.5)--(366.7, 297.4)--(366.6, 297.4)--(366.4, 297.4)--(366.6, 296.6)--(366.7, 296.5)--(366.6, 295.0)--(366.5, 295.0)--(366.4, 294.7)--(366.4, 293.9)--(366.6, 293.9)--(366.7, 292.9)--(366.7, 292.8)--(366.6, 292.8)--(366.5, 292.7)--(366.5, 292.6)--(366.6, 292.5)--(366.7, 292.5)--(366.7, 292.4)--(366.7, 292.2)--(366.7, 291.8)--(366.6, 291.4)--(366.6, 291.3)--(366.7, 291.1)--(366.7, 291.0)--(366.6, 291.0)--(366.5, 291.0)--(366.4, 290.9)--(366.3, 290.9)--(366.3, 290.8)--(366.4, 290.7)--(366.5, 290.7)--(366.5, 290.8)--(366.6, 290.8)--(366.7, 290.8)--(366.8, 290.5)--(366.7, 290.3)--(366.8, 290.2)--(366.9, 289.7)--(366.9, 289.6)--(366.8, 289.6)--(366.8, 289.5)--(366.6, 289.4)--(366.3, 289.4)--(366.3, 289.5)--(365.0, 289.1)--(363.5, 288.6)--(363.4, 288.6)--(363.3, 288.6)--(363.2, 288.6)--(363.1, 288.6)--(363.1, 288.7)--(363.0, 288.7)--(363.0, 288.8)--(362.9, 288.8)--(362.8, 288.9)--(362.7, 289.0)--(362.6, 289.1)--(362.5, 289.1)--(362.5, 289.2)--(362.4, 289.2)--(362.4, 289.3)--(362.3, 289.3)--(362.2, 289.3)--(362.1, 289.3)--(362.1, 289.4)--(362.0, 289.4)--(361.9, 289.4)--(361.8, 289.4)--(361.7, 289.3)--(361.4, 289.2)--(361.3, 289.2)--(361.3, 289.1)--(361.6, 288.5)--(360.9, 288.1)--(361.5, 287.2)--(360.0, 286.3)--(360.2, 286.0)--(360.1, 286.0)--(359.8, 286.0)--(358.6, 286.1)--(358.4, 286.4)--(358.4, 286.5)--(358.2, 286.4)--(358.2, 286.3)--(357.9, 286.2)--(357.7, 286.1)--(357.6, 286.2)--(357.4, 286.3)--(357.2, 286.3)--(357.2, 286.5)--(357.2, 286.8)--(356.4, 286.4)--(356.5, 286.3)--(355.3, 285.9)--(355.2, 285.9)--(355.0, 285.8)--(355.0, 285.7)--(354.6, 285.6)--(354.2, 285.4)--(354.0, 285.3)--(353.8, 285.2)--(353.5, 284.9)--(354.7, 284.0)--(354.8, 284.0)--(354.6, 283.8)--(354.6, 283.7)--(354.6, 283.6)--(354.5, 283.6)--(354.4, 283.5)--(354.3, 283.5)--(354.1, 283.4)--(353.9, 283.4)--(353.6, 283.4)--(353.4, 283.4)--(353.3, 283.4)--(353.2, 283.3)--(353.1, 283.3)--(353.0, 283.1)--(352.9, 283.2)--(352.1, 283.5)--(352.3, 283.0)--(352.4, 282.8)--(352.4, 282.7)--(352.5, 282.5)--(352.6, 282.5)--(352.6, 282.4)--(352.7, 282.4)--(352.8, 282.3)--(352.9, 282.2)--(352.9, 282.1)--(352.9, 282.0)--(352.9, 281.9)--(352.9, 281.8)--(352.9, 281.7)--(352.9, 281.5)--(352.9, 281.4)--(352.8, 281.3)--(352.7, 281.3)--(352.7, 281.2)--(352.4, 281.1)--(352.5, 280.8)--(352.6, 280.6)--(352.6, 280.5)--(352.7, 280.5)--(352.7, 280.4)--(352.8, 280.4)--(352.8, 280.1)--(352.9, 279.8)--(352.9, 279.6)--(352.8, 279.5)--(352.9, 278.7)--(353.0, 278.7)--(353.0, 278.5)--(352.9, 278.5)--(353.0, 277.9)--(353.0, 277.8)--(353.0, 277.7)--(353.1, 277.7)--(353.1, 277.6)--(353.0, 277.6)--(353.0, 277.4)--(352.9, 276.9)--(352.9, 276.6)--(352.9, 276.3)--(352.8, 275.9)--(352.8, 275.8)--(352.9, 275.8)--(352.8, 275.8)--(352.8, 275.7)--(352.8, 275.6)--(352.8, 275.5)--(352.7, 275.3)--(352.7, 275.2)--(352.6, 275.2)--(352.6, 275.1)--(352.7, 275.1)--(352.7, 275.0)--(352.6, 275.1)--(352.6, 275.0)--(352.6, 274.9)--(352.7, 274.9)--(352.7, 274.8)--(352.6, 274.6)--(352.6, 274.4)--(352.6, 274.5)--(352.5, 274.5)--(352.5, 274.4)--(352.3, 273.8)--(352.4, 273.8)--(352.1, 273.1)--(353.3, 272.5)--(353.3, 270.6)--(353.4, 267.8)--(353.4, 266.8)--(353.8, 264.9)--(354.0, 264.3)--(354.1, 263.7)--(354.2, 263.3)--(354.2, 263.2)--(353.9, 262.8)--(353.8, 262.7)--(354.0, 262.4)--(354.3, 262.0)--(354.4, 261.8)--(354.8, 261.1)--(354.8, 260.9)--(354.7, 260.8)--(354.5, 260.7)--(354.8, 260.4)--(355.1, 260.0)--(355.0, 260.0)--(354.9, 259.9)--(355.1, 259.8)--(355.3, 259.8)--(355.6, 259.7)--(355.7, 259.6)--(355.8, 259.6)--(355.9, 259.6)--(356.2, 259.4)--(356.5, 259.3)--(356.8, 259.1)--(356.5, 258.8)--(357.1, 258.4)--(357.2, 258.3)--(357.4, 258.1)--(357.6, 258.0)--(357.8, 257.8)--(357.9, 257.7)--(358.1, 257.5)--(358.2, 257.3)--(358.3, 257.0)--(358.5, 256.5)--(358.7, 255.9)--(358.8, 255.5)--(358.9, 255.0)--(359.0, 254.6)--(359.0, 254.1)--(359.1, 253.6)--(359.1, 253.3)--(359.1, 252.9)--(359.0, 252.4)--(359.0, 251.9)--(359.0, 251.6)--(359.0, 251.5)--(358.8, 250.5)--(358.7, 250.2)--(358.7, 250.1)--(358.7, 250.0)--(358.5, 249.4)--(358.4, 249.2)--(359.3, 248.9)--(359.5, 248.8)--(360.4, 248.1)--(360.9, 247.6)--(361.3, 247.3)--(361.4, 247.2)--(361.5, 247.1)--(361.6, 247.1)--(361.1, 245.8)--(360.8, 245.1)--(360.6, 245.0)--(360.3, 244.5)--(360.2, 244.2)--(359.7, 243.4)--(359.5, 242.9)--(359.4, 242.9)--(359.3, 242.8)--(359.3, 242.6)--(359.3, 242.5)--(359.1, 242.2)--(358.8, 241.7)--(358.7, 241.3)--(358.4, 241.4)--(358.1, 241.4)--(357.5, 241.5)--(357.2, 241.5)--(356.9, 241.5)--(356.6, 241.5)--(356.4, 241.4)--(356.0, 241.2)--(355.3, 240.8)--(353.9, 240.0)--(353.6, 239.9)--(353.5, 239.9)--(353.4, 239.9)--(353.3, 239.9)--(353.2, 239.9)--(353.2, 240.0)--(353.0, 240.2)--(352.7, 240.4)--(352.2, 240.7)--(351.4, 241.2)--(350.7, 241.6)--(350.4, 241.2)--(350.0, 240.7)--(349.7, 240.4)--(349.3, 240.1)--(348.4, 239.4)--(348.1, 239.1)--(347.8, 238.7)--(347.7, 238.4)--(347.6, 238.2)--(347.5, 238.0)--(347.3, 237.8)--(347.1, 237.2)--(347.0, 236.9)--(346.9, 236.5)--(346.8, 236.2)--(346.7, 235.9)--(346.7, 235.4)--(346.6, 235.1)--(346.6, 234.6)--(346.7, 233.7)--(346.6, 233.2)--(346.6, 232.8)--(346.4, 232.3)--(346.4, 232.2)--(346.3, 232.1)--(346.1, 231.7)--(345.8, 231.3)--(346.0, 231.1)--(346.3, 231.0)--(346.5, 230.9)--(346.7, 229.5)--(346.7, 228.9)--(346.8, 228.0)--(346.8, 227.8)--(346.8, 227.7)--(346.9, 227.5)--(347.0, 227.4)--(347.0, 227.3)--(346.9, 227.2)--(346.6, 227.0)--(346.5, 226.9)--(346.4, 226.8)--(346.3, 226.7)--(346.3, 226.6)--(346.3, 226.5)--(346.2, 226.4)--(346.2, 226.3)--(346.1, 225.9)--(345.9, 225.4)--(345.7, 224.9)--(345.6, 224.5)--(345.5, 224.1)--(345.4, 224.0)--(345.4, 223.9)--(345.3, 223.8)--(345.3, 223.7)--(345.2, 223.6)--(344.8, 223.2)--(344.5, 222.9)--(344.1, 222.6)--(343.6, 222.1)--(343.3, 222.0)--(342.9, 221.8)--(342.9, 221.7)--(342.6, 221.6)--(342.5, 221.6)--(342.3, 221.5)--(342.1, 221.5)--(341.9, 221.5)--(341.8, 221.5)--(341.7, 221.5)--(341.5, 221.5)--(341.1, 221.5)--(340.9, 221.4)--(340.8, 221.3)--(340.7, 221.2)--(340.6, 221.0)--(340.1, 220.2)--(339.3, 218.4)--(339.3, 218.3)--(339.2, 218.2)--(339.2, 217.8)--(339.2, 217.3)--(339.2, 217.1)--(339.3, 216.9)--(339.4, 216.6)--(339.6, 216.1)--(339.7, 215.9)--(339.9, 215.4)--(340.0, 214.8)--(340.0, 214.5)--(340.0, 214.4)--(339.9, 214.4)--(339.8, 214.3)--(339.7, 214.3)--(339.7, 214.2)--(339.6, 214.2)--(339.5, 214.2)--(339.5, 214.1)--(339.4, 214.0)--(339.4, 213.9)--(339.3, 213.9)--(339.3, 213.8)--(339.0, 213.1)--(338.9, 212.8)--(338.3, 211.1)--(337.6, 209.4)--(337.3, 208.6)--(337.2, 208.3)--(337.2, 208.2)--(336.6, 208.4)--(336.5, 208.4)--(336.5, 208.5)--(336.5, 208.6)--(336.6, 208.8)--(336.7, 209.2)--(336.3, 209.4)--(336.2, 209.6)--(336.1, 209.8)--(336.0, 209.9)--(335.9, 209.7)--(335.7, 209.7)--(335.4, 209.5)--(335.3, 209.5)--(335.1, 210.0)--(334.7, 211.1)--(334.7, 211.2)--(334.5, 211.1)--(333.9, 210.9)--(333.5, 210.8)--(333.5, 210.7)--(333.5, 210.9)--(333.5, 211.2)--(333.4, 211.5)--(333.4, 212.0)--(333.4, 212.4)--(333.3, 212.7)--(333.3, 213.0)--(333.2, 213.4)--(333.0, 213.8)--(332.8, 214.2)--(332.6, 214.6)--(332.4, 215.1)--(332.2, 215.5)--(332.1, 215.6)--(332.0, 215.7)--(331.8, 215.9)--(331.7, 215.9)--(331.6, 216.0)--(331.6, 216.1)--(331.5, 216.1)--(331.5, 216.2)--(331.6, 216.3)--(331.6, 216.5)--(331.7, 216.6)--(331.7, 216.7)--(331.7, 216.8)--(331.6, 216.8)--(331.6, 216.9)--(331.5, 216.9)--(331.2, 217.1)--(330.7, 217.2)--(330.4, 217.3)--(330.3, 217.4)--(330.1, 217.6)--(330.0, 217.7)--(329.9, 217.8)--(329.8, 218.1)--(329.5, 219.0)--(329.2, 220.4)--(329.0, 221.5)--(328.6, 223.3)--(328.5, 224.1)--(328.3, 225.2)--(328.2, 225.6)--(328.1, 225.8)--(327.9, 226.6)--(327.7, 227.1)--(327.5, 227.6)--(327.3, 228.0)--(327.2, 228.1)--(327.2, 228.2)--(327.1, 228.4)--(327.1, 228.5)--(326.9, 229.0)--(326.7, 229.8)--(326.5, 230.4)--(326.4, 230.7)--(326.3, 231.1)--(326.2, 231.4)--(326.1, 231.7)--(325.9, 232.0)--(325.8, 232.2)--(325.6, 232.4)--(325.5, 232.6)--(325.2, 233.2)--(324.9, 234.3)--(324.9, 234.5)--(324.8, 234.6)--(324.8, 234.7)--(324.8, 234.9)--(324.8, 235.1)--(324.8, 235.2)--(324.8, 235.3)--(324.8, 235.5)--(324.7, 235.6)--(324.7, 235.8)--(324.3, 236.4)--(324.2, 236.7)--(324.0, 236.9)--(323.1, 238.0)--(322.7, 238.4)--(322.4, 238.8)--(322.0, 239.1)--(321.8, 239.5)--(321.7, 239.5)--(321.7, 239.6)--(321.7, 239.8)--(321.6, 240.0)--(321.5, 240.3)--(321.5, 240.5)--(321.4, 240.6)--(321.4, 240.7)--(321.3, 240.7)--(321.3, 240.8)--(321.2, 240.8)--(321.1, 240.8)--(321.0, 240.8)--(320.7, 240.8)--(320.7, 240.9)--(320.7, 241.2)--(320.7, 241.6)--(320.6, 241.7)--(320.6, 241.8)--(320.5, 242.0)--(320.4, 242.2)--(320.3, 242.4)--(320.2, 242.5)--(320.2, 242.6)--(320.2, 242.7)--(320.1, 242.8)--(320.1, 243.0)--(320.1, 243.1)--(320.2, 243.2)--(320.2, 243.3)--(320.4, 243.7)--(320.4, 243.8)--(320.4, 243.9)--(320.4, 244.0)--(320.3, 244.6)--(320.3, 244.7)--(320.4, 245.0)--(320.5, 245.1)--(320.7, 245.3)--(321.1, 245.8)--(321.7, 246.5)--(322.1, 246.8)--(322.6, 247.2)--(322.7, 247.4)--(323.0, 247.7)--(323.4, 248.2)--(323.7, 248.5)--(323.9, 248.8)--(324.0, 249.0)--(324.1, 249.3)--(324.3, 249.9)--(324.4, 250.0)--(324.4, 250.2)--(324.5, 250.4)--(324.6, 250.6)--(324.7, 250.8)--(325.0, 251.2)--(325.4, 251.7)--(325.7, 252.2)--(325.8, 252.4)--(325.9, 252.5)--(326.0, 252.5)--(326.1, 252.7)--(326.2, 252.9)--(326.3, 253.0)--(326.8, 254.0)--(327.0, 254.6)--(327.2, 254.9)--(327.3, 255.4)--(327.7, 258.1)--(327.7, 258.4)--(327.7, 258.5)--(327.7, 258.6)--(327.6, 258.7)--(327.6, 258.8)--(326.9, 260.2)--(326.1, 261.8)--(325.7, 262.6)--(325.7, 262.8)--(325.6, 263.0)--(325.5, 263.7)--(325.5, 263.9)--(325.4, 265.1)--(325.4, 265.2)--(325.1, 264.9)--(325.0, 264.8)--(325.0, 264.7)--(325.0, 264.6)--(324.8, 263.8)--(324.8, 263.7)--(324.7, 263.6)--(324.6, 263.5)--(324.5, 263.4)--(324.3, 263.2)--(324.1, 263.0)--(323.4, 262.6)--(322.6, 263.5)--(321.4, 264.9)--(321.2, 265.2)--(321.1, 265.2)--(321.0, 265.6)--(320.9, 266.1)--(320.7, 266.5)--(320.3, 267.1)--(320.1, 267.4)--(320.1, 267.5)--(320.1, 267.7)--(320.1, 268.0)--(320.2, 269.7)--(320.3, 270.1)--(320.4, 270.2)--(320.5, 270.4)--(319.4, 271.1)--(317.9, 271.9)--(316.5, 272.8)--(315.4, 273.5)--(314.9, 273.8)--(314.8, 273.9)--(314.7, 274.0)--(314.6, 274.0)--(314.6, 274.1)--(314.7, 274.1)--(316.5, 274.5)--(316.9, 274.6)--(317.5, 274.8)--(317.1, 275.3)--(317.1, 275.4)--(317.1, 275.5)--(316.9, 275.8)--(316.7, 276.1)--(316.6, 276.1)--(316.5, 276.2)--(316.4, 276.4)--(316.1, 276.7)--(316.2, 276.8)--(316.4, 277.0)--(315.9, 277.4)--(315.5, 277.6)--(315.2, 277.8)--(315.4, 278.2)--(315.6, 278.7)--(315.7, 278.9)--(315.8, 279.2)--(314.8, 279.8)--(314.7, 279.9)--(314.6, 280.0)--(314.5, 280.1)--(314.3, 280.2)--(314.6, 280.5)--(316.3, 283.5)--(316.8, 284.4)--(317.2, 284.9)--(317.4, 285.4)--(317.7, 285.9)--(317.8, 286.1)--(317.9, 286.2)--(317.7, 286.4)--(317.8, 286.5)--(317.6, 286.8)--(317.7, 286.9)--(317.5, 287.0)--(317.4, 287.1)--(317.3, 287.1)--(317.3, 287.0)--(317.2, 287.4)--(317.1, 287.7)--(316.9, 287.9)--(316.9, 288.1)--(316.7, 288.3)--(316.7, 288.6)--(316.6, 288.8)--(316.3, 288.9)--(316.0, 289.0)--(315.0, 289.4)--(313.4, 290.3)--(312.6, 290.7)--(312.6, 290.8)--(312.5, 290.8)--(312.4, 290.8)--(312.3, 290.9)--(312.3, 291.0)--(312.0, 291.9)--(312.7, 292.3)--(312.7, 292.4)--(313.2, 294.1)--(313.3, 294.2)--(313.5, 294.9)--(314.0, 296.3)--(314.1, 296.5)--(314.3, 296.8)--(315.0, 297.7)--(315.1, 297.8)--(315.2, 297.7)--(315.6, 298.2)--(315.5, 298.3)--(315.5, 298.4)--(315.5, 298.5)--(315.5, 298.6)--(315.5, 298.8)--(315.0, 298.4)--(314.5, 299.3)--(315.3, 299.7)--(315.2, 300.0)--(315.1, 300.5)--(315.1, 300.7)--(315.2, 301.0)--(315.2, 301.2)--(315.2, 301.3)--(315.2, 301.5)--(315.1, 301.8)--(314.5, 303.8)--(313.9, 303.6)--(313.9, 303.7)--(313.8, 303.8)--(313.7, 304.2)--(313.7, 304.5)--(313.6, 304.5)--(313.5, 304.5)--(313.4, 304.9)--cycle; +Sutton = (285.5, 96.6)--(284.0, 99.3)--(283.8, 99.2)--(283.7, 99.3)--(282.6, 98.4)--(281.5, 99.8)--(281.0, 100.4)--(280.6, 100.9)--(280.1, 101.5)--(280.5, 101.7)--(280.4, 102.1)--(280.0, 103.2)--(279.9, 103.3)--(279.9, 103.7)--(279.9, 103.8)--(280.0, 103.8)--(280.2, 103.9)--(280.2, 104.2)--(280.2, 104.3)--(280.4, 104.5)--(280.7, 104.6)--(281.3, 104.7)--(280.7, 106.2)--(279.1, 105.8)--(277.0, 105.4)--(276.2, 105.3)--(275.9, 105.2)--(275.9, 105.7)--(276.0, 106.1)--(275.9, 106.0)--(275.8, 106.0)--(274.9, 105.9)--(274.6, 105.8)--(274.2, 105.8)--(274.1, 105.8)--(274.0, 105.8)--(273.9, 105.8)--(273.9, 105.9)--(273.8, 106.2)--(273.7, 106.5)--(273.5, 107.1)--(273.3, 107.7)--(273.1, 108.3)--(272.9, 109.3)--(272.6, 109.2)--(272.5, 109.6)--(272.4, 109.8)--(272.3, 109.9)--(271.9, 110.3)--(271.7, 110.5)--(271.5, 110.6)--(271.4, 110.7)--(269.8, 112.0)--(269.6, 112.2)--(269.2, 112.5)--(269.0, 112.8)--(268.2, 113.5)--(267.6, 114.1)--(267.5, 114.2)--(267.3, 114.3)--(267.1, 114.5)--(266.2, 115.1)--(265.5, 115.8)--(265.3, 115.9)--(264.6, 116.5)--(263.6, 117.3)--(263.5, 117.4)--(263.5, 117.5)--(263.2, 117.6)--(262.8, 117.8)--(262.6, 117.9)--(262.0, 118.2)--(261.5, 118.5)--(261.0, 118.8)--(260.8, 118.9)--(259.4, 119.4)--(258.6, 119.7)--(257.4, 120.2)--(257.4, 120.1)--(257.3, 119.8)--(257.0, 120.0)--(257.0, 120.2)--(257.1, 120.3)--(256.7, 120.4)--(256.5, 120.4)--(256.4, 120.3)--(256.3, 120.3)--(256.2, 120.2)--(256.2, 120.1)--(256.0, 119.8)--(255.8, 119.4)--(255.8, 119.1)--(255.7, 119.0)--(255.6, 118.8)--(255.3, 118.5)--(255.2, 118.5)--(255.0, 118.5)--(254.7, 117.7)--(254.3, 117.0)--(253.9, 116.2)--(253.8, 116.3)--(253.8, 116.2)--(253.7, 116.2)--(253.6, 116.2)--(253.4, 116.6)--(253.3, 116.5)--(253.1, 116.5)--(252.9, 116.3)--(252.7, 116.2)--(252.3, 115.9)--(251.8, 115.5)--(250.1, 114.3)--(250.0, 114.3)--(248.1, 112.9)--(247.6, 112.5)--(247.2, 112.3)--(246.8, 112.0)--(246.6, 111.8)--(245.8, 111.2)--(246.4, 109.6)--(246.6, 108.7)--(246.3, 107.9)--(246.1, 107.6)--(246.1, 107.5)--(245.7, 106.6)--(245.2, 105.6)--(245.2, 105.4)--(245.1, 105.3)--(245.0, 105.3)--(243.9, 104.8)--(243.8, 104.8)--(243.3, 104.8)--(243.0, 104.8)--(242.0, 104.8)--(241.5, 104.8)--(241.1, 104.7)--(240.6, 104.7)--(240.6, 104.8)--(240.4, 104.8)--(240.0, 105.4)--(239.6, 106.0)--(238.9, 106.8)--(238.1, 107.6)--(237.0, 108.7)--(236.1, 109.6)--(236.2, 109.7)--(235.8, 110.0)--(235.2, 110.6)--(234.9, 110.9)--(234.5, 111.3)--(234.1, 112.0)--(235.0, 112.7)--(235.1, 112.7)--(235.0, 112.8)--(235.4, 113.1)--(236.0, 113.5)--(236.1, 113.6)--(236.2, 113.8)--(236.6, 114.3)--(236.9, 114.8)--(237.7, 115.8)--(238.0, 116.0)--(238.2, 116.0)--(238.5, 115.9)--(239.2, 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97.0)--(286.0, 96.8)--(285.8, 96.7)--(285.6, 96.6)--cycle; +Tower_Hamlets = (333.9, 305.2)--(334.0, 305.2)--(334.1, 305.6)--(334.1, 305.7)--(334.1, 305.8)--(334.2, 306.1)--(334.3, 306.3)--(334.3, 306.4)--(334.0, 306.5)--(334.1, 306.6)--(334.2, 306.9)--(334.4, 307.0)--(334.5, 307.2)--(334.7, 307.5)--(334.8, 307.6)--(334.9, 307.7)--(335.1, 307.8)--(335.4, 307.8)--(335.5, 307.8)--(335.5, 307.9)--(335.5, 308.0)--(335.5, 308.1)--(335.5, 308.2)--(335.8, 308.2)--(335.9, 308.2)--(336.0, 308.1)--(336.1, 307.7)--(336.3, 307.8)--(336.3, 307.7)--(336.3, 307.5)--(336.4, 307.5)--(336.4, 307.4)--(336.5, 307.4)--(336.6, 307.4)--(337.0, 307.6)--(337.1, 307.6)--(337.3, 307.6)--(337.4, 307.7)--(337.5, 307.7)--(337.7, 307.8)--(337.9, 307.8)--(338.1, 307.8)--(338.2, 307.8)--(338.3, 307.8)--(338.3, 307.9)--(338.4, 308.0)--(338.4, 308.1)--(338.4, 308.4)--(338.4, 308.6)--(338.4, 308.7)--(338.3, 308.9)--(338.3, 309.1)--(338.2, 309.3)--(338.2, 309.5)--(338.1, 309.7)--(338.1, 310.3)--(338.0, 310.9)--(337.9, 311.4)--(337.8, 311.8)--(337.7, 312.3)--(337.4, 312.6)--(337.2, 312.9)--(337.1, 313.0)--(336.8, 313.3)--(336.6, 313.6)--(335.7, 314.6)--(335.4, 314.9)--(335.4, 315.0)--(335.3, 315.1)--(335.0, 315.8)--(335.0, 315.9)--(334.8, 316.3)--(334.8, 316.5)--(334.7, 316.5)--(334.7, 316.6)--(334.6, 316.8)--(334.4, 317.0)--(334.5, 317.0)--(334.5, 317.6)--(334.3, 317.5)--(333.5, 317.4)--(333.7, 318.1)--(333.8, 318.8)--(333.9, 319.4)--(333.9, 319.5)--(333.9, 319.9)--(334.1, 320.4)--(334.3, 320.8)--(334.5, 320.9)--(335.0, 321.1)--(335.3, 321.2)--(335.5, 321.0)--(335.7, 321.1)--(335.7, 321.2)--(335.7, 321.4)--(335.8, 321.6)--(335.9, 322.0)--(335.9, 322.1)--(335.9, 322.3)--(336.0, 322.6)--(336.0, 322.7)--(335.9, 323.0)--(335.8, 323.6)--(335.4, 323.5)--(335.3, 324.7)--(335.3, 325.1)--(335.3, 325.4)--(335.2, 325.6)--(335.2, 325.7)--(335.2, 325.8)--(335.3, 326.0)--(335.4, 326.4)--(335.4, 326.7)--(334.8, 326.8)--(335.0, 327.1)--(335.4, 327.5)--(335.6, 327.8)--(335.8, 328.1)--(335.9, 328.1)--(336.0, 328.3)--(336.1, 328.5)--(336.3, 329.2)--(336.5, 329.7)--(336.7, 329.9)--(336.8, 330.1)--(336.9, 330.1)--(337.1, 330.3)--(337.5, 330.5)--(337.7, 330.6)--(337.8, 330.7)--(338.1, 330.9)--(338.4, 330.9)--(338.6, 331.0)--(338.9, 331.0)--(339.3, 331.1)--(339.8, 331.1)--(340.2, 331.1)--(341.1, 331.2)--(341.6, 331.3)--(342.3, 331.4)--(342.3, 331.7)--(342.5, 332.3)--(342.7, 332.8)--(342.9, 333.3)--(343.0, 333.6)--(343.1, 333.3)--(344.3, 333.5)--(344.7, 333.7)--(345.0, 333.7)--(345.0, 333.9)--(344.9, 334.1)--(344.8, 334.3)--(344.7, 334.7)--(344.6, 334.8)--(344.6, 335.3)--(344.5, 335.7)--(344.6, 335.8)--(344.7, 336.0)--(344.8, 336.2)--(345.1, 336.2)--(345.3, 336.0)--(345.6, 335.7)--(345.9, 335.3)--(346.1, 335.1)--(346.2, 335.0)--(346.3, 334.9)--(346.4, 334.9)--(346.6, 334.8)--(346.8, 334.8)--(347.0, 334.7)--(347.2, 334.7)--(347.4, 334.8)--(347.9, 334.9)--(348.2, 335.0)--(348.4, 335.1)--(348.7, 335.3)--(349.2, 335.6)--(349.5, 335.8)--(349.7, 335.8)--(349.8, 335.9)--(350.0, 335.9)--(350.1, 335.9)--(350.2, 336.0)--(350.4, 336.0)--(350.6, 336.0)--(350.8, 336.0)--(351.1, 335.9)--(351.4, 335.8)--(351.5, 335.8)--(351.5, 336.0)--(351.6, 336.4)--(351.6, 336.7)--(351.8, 336.9)--(351.9, 337.1)--(352.1, 337.2)--(352.2, 337.3)--(352.3, 337.4)--(352.4, 337.4)--(352.8, 337.5)--(352.9, 337.5)--(353.0, 337.5)--(353.1, 337.5)--(353.2, 337.5)--(353.5, 337.5)--(353.7, 337.4)--(353.9, 337.2)--(354.2, 337.0)--(354.4, 336.8)--(354.5, 336.8)--(354.5, 336.7)--(354.6, 336.7)--(354.7, 336.6)--(354.8, 336.6)--(354.9, 336.6)--(355.1, 336.5)--(355.3, 336.5)--(355.4, 336.5)--(355.6, 336.5)--(356.2, 336.6)--(357.0, 336.9)--(357.1, 337.0)--(357.4, 337.3)--(357.6, 337.6)--(357.7, 337.7)--(357.8, 337.7)--(357.9, 337.6)--(357.9, 337.7)--(358.7, 338.3)--(359.3, 338.8)--(359.5, 339.1)--(359.8, 339.5)--(360.2, 340.2)--(360.9, 341.3)--(361.2, 341.8)--(361.6, 342.5)--(361.9, 343.0)--(362.0, 343.1)--(362.0, 343.2)--(362.1, 343.2)--(362.2, 343.3)--(361.7, 343.7)--(362.4, 344.4)--(363.0, 345.1)--(363.5, 345.5)--(363.9, 345.8)--(364.2, 346.3)--(364.4, 346.4)--(364.4, 346.6)--(364.5, 346.7)--(364.7, 346.9)--(364.8, 347.0)--(364.9, 347.0)--(365.0, 346.9)--(365.7, 346.3)--(366.0, 346.0)--(366.3, 345.8)--(366.4, 345.7)--(366.7, 345.6)--(367.1, 345.4)--(367.3, 345.3)--(367.5, 345.2)--(367.5, 345.0)--(367.7, 344.7)--(367.8, 344.4)--(368.0, 344.5)--(368.2, 344.5)--(368.6, 344.7)--(368.9, 344.8)--(369.1, 344.9)--(369.5, 345.1)--(370.0, 345.4)--(370.1, 345.5)--(370.1, 345.7)--(370.2, 345.6)--(370.3, 345.5)--(371.3, 345.6)--(371.9, 345.7)--(372.1, 345.7)--(372.2, 345.8)--(372.4, 345.8)--(372.4, 345.9)--(373.0, 345.9)--(373.8, 346.0)--(374.1, 346.0)--(374.5, 346.0)--(375.0, 346.0)--(375.5, 345.9)--(375.8, 345.9)--(376.1, 345.9)--(376.2, 345.8)--(376.3, 345.9)--(376.4, 345.8)--(376.4, 345.6)--(376.4, 345.5)--(376.5, 345.4)--(376.6, 345.2)--(376.7, 344.9)--(376.7, 344.8)--(376.7, 344.7)--(376.6, 344.5)--(376.5, 344.0)--(376.5, 343.8)--(376.4, 343.6)--(376.3, 343.5)--(376.2, 343.4)--(376.0, 343.3)--(375.8, 343.3)--(375.7, 343.2)--(375.6, 343.1)--(375.6, 343.0)--(375.6, 342.6)--(375.6, 342.4)--(375.6, 342.3)--(375.5, 342.1)--(375.4, 342.0)--(375.3, 341.8)--(375.1, 341.6)--(375.0, 341.4)--(374.9, 341.3)--(374.8, 341.1)--(374.7, 340.7)--(374.7, 340.4)--(374.6, 340.1)--(374.5, 340.1)--(374.4, 339.9)--(374.3, 339.9)--(374.2, 339.8)--(374.0, 339.7)--(373.8, 339.5)--(373.6, 339.3)--(373.4, 339.0)--(373.3, 338.8)--(373.3, 338.7)--(373.3, 338.6)--(373.4, 338.5)--(373.4, 338.3)--(373.4, 338.1)--(373.5, 337.9)--(373.6, 337.8)--(373.8, 337.7)--(373.9, 337.7)--(374.1, 337.6)--(374.3, 337.4)--(374.4, 337.2)--(374.6, 337.0)--(374.7, 336.7)--(374.8, 336.4)--(374.9, 336.2)--(374.9, 336.0)--(375.0, 335.8)--(375.0, 335.6)--(375.0, 335.5)--(375.1, 335.4)--(375.2, 335.2)--(375.4, 335.0)--(375.6, 334.9)--(375.6, 334.8)--(375.7, 334.6)--(375.7, 334.4)--(375.7, 334.3)--(375.7, 334.2)--(375.6, 334.0)--(375.6, 333.8)--(375.6, 333.7)--(375.6, 333.6)--(375.7, 333.4)--(375.9, 333.1)--(376.1, 332.7)--(376.1, 332.6)--(376.3, 332.4)--(376.6, 332.2)--(376.8, 332.1)--(377.5, 331.7)--(377.9, 331.6)--(378.2, 331.4)--(378.4, 331.3)--(378.4, 331.2)--(378.5, 331.1)--(378.5, 331.0)--(378.6, 330.9)--(378.8, 330.8)--(379.2, 330.7)--(379.5, 330.6)--(380.1, 330.6)--(380.6, 330.5)--(380.7, 330.4)--(380.8, 330.3)--(380.9, 330.2)--(381.0, 330.0)--(381.0, 329.8)--(381.1, 329.6)--(381.1, 329.5)--(381.3, 329.3)--(381.4, 329.2)--(381.8, 328.8)--(382.2, 328.6)--(382.3, 328.4)--(382.4, 328.3)--(382.5, 328.0)--(382.6, 327.8)--(382.7, 327.6)--(382.8, 327.4)--(383.0, 327.2)--(383.1, 327.1)--(383.2, 326.9)--(383.2, 326.7)--(383.2, 326.4)--(383.2, 326.3)--(383.1, 326.1)--(383.0, 326.0)--(383.0, 325.8)--(382.9, 325.5)--(382.9, 324.6)--(382.8, 324.3)--(382.8, 324.0)--(382.8, 323.8)--(382.9, 323.7)--(383.0, 323.4)--(383.0, 323.2)--(383.0, 323.1)--(382.9, 323.1)--(382.7, 323.1)--(382.7, 322.8)--(382.6, 322.4)--(382.6, 322.2)--(382.6, 322.0)--(382.1, 322.0)--(382.1, 321.7)--(382.2, 321.6)--(382.2, 321.5)--(382.3, 321.3)--(382.2, 321.0)--(382.2, 320.9)--(382.2, 320.5)--(382.2, 320.1)--(382.2, 319.9)--(382.2, 319.8)--(382.3, 319.7)--(382.3, 319.6)--(382.6, 319.4)--(383.1, 319.1)--(383.4, 318.8)--(383.5, 318.7)--(383.7, 318.4)--(383.8, 318.3)--(384.0, 318.1)--(384.1, 317.9)--(384.2, 317.8)--(384.2, 317.7)--(384.3, 317.5)--(384.3, 317.3)--(384.3, 317.2)--(384.3, 317.0)--(384.3, 316.8)--(384.3, 316.5)--(384.4, 316.4)--(384.4, 316.3)--(384.4, 316.2)--(384.5, 316.1)--(384.6, 316.0)--(384.7, 316.0)--(384.7, 315.9)--(384.8, 315.9)--(384.9, 315.9)--(385.0, 315.8)--(385.1, 315.9)--(385.3, 315.9)--(385.7, 316.1)--(385.9, 316.5)--(386.3, 317.3)--(386.3, 317.4)--(386.4, 317.5)--(386.5, 317.6)--(386.6, 317.6)--(386.7, 317.6)--(386.8, 317.6)--(386.9, 317.6)--(387.1, 317.5)--(387.1, 317.4)--(388.0, 316.8)--(388.2, 316.6)--(388.3, 316.6)--(388.6, 316.3)--(388.7, 316.2)--(388.9, 316.1)--(389.3, 315.9)--(390.0, 315.6)--(390.4, 315.3)--(390.6, 315.2)--(391.0, 314.8)--(391.1, 314.7)--(391.2, 314.5)--(391.2, 314.4)--(391.3, 314.3)--(391.3, 314.2)--(391.3, 314.0)--(391.2, 313.6)--(391.1, 313.4)--(391.0, 313.2)--(391.0, 313.1)--(390.7, 313.0)--(390.7, 312.9)--(390.6, 312.9)--(390.6, 312.8)--(390.6, 312.6)--(390.6, 312.3)--(390.1, 311.0)--(390.1, 310.6)--(390.1, 310.4)--(390.0, 310.3)--(389.8, 310.3)--(389.8, 310.2)--(390.0, 310.1)--(390.1, 310.0)--(390.2, 309.9)--(390.3, 309.7)--(390.4, 309.6)--(390.6, 309.6)--(391.0, 309.5)--(391.4, 309.5)--(391.5, 309.5)--(391.5, 309.6)--(391.7, 309.7)--(391.9, 309.7)--(391.9, 309.8)--(392.0, 309.9)--(392.0, 310.0)--(392.1, 310.0)--(392.1, 310.2)--(392.2, 310.5)--(392.2, 310.6)--(392.3, 310.7)--(392.3, 310.8)--(392.3, 311.0)--(392.4, 311.3)--(392.4, 311.5)--(392.5, 312.3)--(392.5, 312.5)--(392.5, 312.6)--(392.5, 312.8)--(392.6, 312.9)--(392.6, 313.0)--(392.7, 313.1)--(392.7, 313.2)--(392.7, 313.3)--(392.8, 313.4)--(392.9, 313.5)--(393.0, 313.5)--(393.1, 313.5)--(393.2, 313.5)--(393.4, 313.5)--(393.5, 313.5)--(393.5, 313.4)--(393.6, 313.4)--(393.6, 313.3)--(393.7, 313.2)--(393.7, 313.1)--(393.8, 312.9)--(393.9, 312.4)--(393.9, 312.3)--(393.9, 312.1)--(393.8, 311.6)--(393.6, 311.0)--(393.4, 310.5)--(393.1, 310.1)--(392.9, 309.8)--(392.8, 309.7)--(392.8, 309.6)--(392.8, 309.4)--(392.7, 309.3)--(392.8, 309.2)--(392.8, 309.0)--(392.9, 308.9)--(393.1, 308.6)--(393.3, 308.6)--(393.3, 308.4)--(393.9, 308.3)--(393.9, 308.4)--(394.6, 308.2)--(395.0, 308.0)--(395.1, 308.0)--(395.1, 307.9)--(395.2, 307.8)--(395.3, 307.7)--(395.3, 307.5)--(395.3, 307.4)--(395.3, 307.3)--(395.3, 307.2)--(395.3, 307.1)--(395.2, 307.0)--(395.2, 306.9)--(395.2, 306.8)--(395.3, 306.7)--(395.1, 306.2)--(395.0, 306.2)--(394.7, 306.4)--(393.9, 306.7)--(393.8, 306.7)--(393.7, 306.7)--(393.2, 306.9)--(393.2, 307.1)--(393.0, 307.1)--(392.9, 307.0)--(391.7, 307.2)--(391.7, 307.3)--(391.5, 307.3)--(391.4, 306.9)--(391.3, 306.9)--(391.2, 306.9)--(391.2, 307.0)--(391.2, 307.1)--(391.1, 307.1)--(391.0, 307.1)--(391.0, 307.0)--(390.9, 307.0)--(390.9, 306.9)--(389.9, 306.8)--(389.7, 306.8)--(388.0, 306.2)--(387.8, 306.1)--(387.8, 306.2)--(387.5, 306.0)--(387.5, 305.9)--(387.4, 305.9)--(387.4, 305.8)--(387.3, 305.8)--(387.2, 305.8)--(387.0, 305.7)--(387.0, 305.6)--(387.0, 305.5)--(386.8, 305.3)--(386.6, 305.2)--(386.7, 305.1)--(386.4, 304.9)--(386.2, 304.5)--(385.7, 303.9)--(385.4, 303.5)--(385.3, 303.4)--(385.2, 303.5)--(385.1, 303.6)--(385.1, 303.5)--(385.1, 303.4)--(385.1, 303.3)--(385.2, 303.2)--(385.0, 302.8)--(384.9, 302.7)--(384.8, 302.5)--(384.6, 302.3)--(384.6, 302.2)--(384.5, 302.2)--(384.4, 302.1)--(384.4, 302.0)--(384.4, 301.9)--(384.5, 301.8)--(384.3, 301.6)--(384.2, 301.5)--(384.1, 301.3)--(384.0, 300.8)--(384.0, 300.5)--(383.9, 300.2)--(383.9, 300.0)--(383.8, 299.6)--(383.8, 299.4)--(383.3, 299.2)--(383.0, 299.3)--(382.9, 299.2)--(382.9, 299.1)--(383.0, 299.0)--(383.3, 299.0)--(383.8, 298.7)--(383.8, 298.5)--(383.8, 298.3)--(383.7, 298.3)--(383.8, 297.9)--(383.9, 297.9)--(383.9, 297.8)--(383.8, 297.8)--(383.8, 297.7)--(383.8, 297.6)--(383.9, 297.7)--(383.9, 297.4)--(384.1, 296.3)--(384.1, 296.1)--(384.2, 296.0)--(384.2, 295.9)--(384.4, 295.9)--(384.5, 295.6)--(384.8, 294.4)--(384.9, 294.4)--(385.4, 293.2)--(385.0, 293.0)--(385.1, 293.0)--(385.0, 292.9)--(385.2, 292.9)--(385.3, 292.7)--(385.4, 292.5)--(385.5, 292.3)--(385.6, 291.9)--(385.8, 291.9)--(385.9, 291.7)--(386.1, 291.8)--(386.5, 290.9)--(386.8, 289.9)--(387.2, 289.1)--(387.2, 288.9)--(387.2, 288.7)--(387.3, 288.4)--(387.4, 287.8)--(387.4, 287.6)--(387.4, 287.5)--(387.3, 286.9)--(387.3, 286.7)--(387.2, 286.5)--(387.2, 286.3)--(387.0, 285.8)--(387.0, 285.6)--(386.9, 285.5)--(386.9, 285.4)--(386.7, 285.2)--(386.6, 284.9)--(386.5, 284.8)--(386.4, 284.8)--(385.9, 285.1)--(385.9, 285.0)--(386.3, 284.6)--(386.1, 284.3)--(385.9, 284.2)--(385.8, 284.1)--(385.7, 284.0)--(385.3, 283.6)--(385.2, 283.6)--(385.1, 283.5)--(384.6, 283.2)--(384.1, 283.0)--(383.9, 282.9)--(383.7, 282.8)--(383.6, 282.8)--(382.8, 282.5)--(382.7, 282.7)--(382.5, 282.7)--(382.5, 282.5)--(381.9, 282.3)--(381.5, 282.3)--(381.0, 282.2)--(380.6, 282.1)--(380.6, 282.2)--(380.5, 282.2)--(380.5, 282.1)--(380.0, 282.1)--(379.9, 282.1)--(379.9, 282.0)--(379.3, 282.1)--(379.1, 282.1)--(379.0, 282.1)--(378.2, 282.2)--(378.1, 282.2)--(378.0, 282.3)--(377.9, 282.3)--(377.2, 282.5)--(377.0, 282.5)--(376.8, 282.6)--(376.7, 282.6)--(376.5, 282.7)--(376.3, 282.8)--(375.2, 283.4)--(374.7, 283.6)--(374.5, 283.7)--(373.8, 284.3)--(373.6, 284.5)--(373.3, 284.8)--(373.2, 285.0)--(373.0, 285.2)--(372.8, 285.4)--(372.6, 285.6)--(372.2, 286.1)--(371.8, 286.7)--(371.7, 286.8)--(371.6, 287.0)--(371.5, 287.3)--(371.3, 287.7)--(371.1, 288.2)--(371.0, 288.6)--(371.0, 288.7)--(370.8, 289.0)--(370.6, 289.7)--(370.4, 290.4)--(370.4, 290.5)--(370.8, 290.7)--(371.7, 290.7)--(371.6, 290.9)--(370.8, 290.9)--(370.7, 290.9)--(370.6, 290.9)--(370.1, 291.0)--(370.1, 291.1)--(370.0, 291.2)--(370.0, 291.5)--(370.0, 291.7)--(370.0, 291.9)--(370.0, 292.5)--(370.1, 293.7)--(370.2, 294.8)--(370.2, 294.9)--(370.2, 295.2)--(370.2, 295.5)--(370.1, 295.7)--(370.1, 296.8)--(370.0, 296.8)--(370.0, 297.0)--(369.9, 297.1)--(370.1, 297.1)--(370.1, 298.5)--(370.1, 299.6)--(370.1, 299.8)--(370.2, 300.0)--(370.4, 300.9)--(370.6, 300.9)--(370.8, 300.9)--(370.9, 300.9)--(371.0, 300.9)--(371.1, 300.8)--(371.2, 300.8)--(371.2, 300.9)--(371.2, 301.0)--(370.9, 301.0)--(370.8, 301.1)--(370.5, 301.1)--(370.4, 301.2)--(370.4, 301.3)--(370.3, 301.4)--(370.2, 301.5)--(370.1, 301.8)--(369.9, 302.3)--(369.9, 302.4)--(369.9, 302.6)--(369.8, 302.8)--(369.7, 303.0)--(369.5, 303.3)--(369.8, 303.4)--(369.8, 303.7)--(369.5, 303.6)--(369.3, 303.9)--(369.1, 304.3)--(369.0, 304.3)--(369.0, 304.4)--(369.0, 304.5)--(369.6, 304.7)--(369.6, 304.8)--(369.5, 304.9)--(368.9, 304.6)--(368.8, 304.6)--(368.3, 305.2)--(368.4, 305.3)--(368.1, 305.6)--(367.9, 305.4)--(367.1, 306.0)--(367.3, 306.3)--(367.6, 306.7)--(367.9, 306.9)--(368.2, 307.1)--(368.2, 307.2)--(368.1, 307.2)--(367.8, 307.0)--(367.5, 306.9)--(367.4, 306.8)--(367.3, 306.7)--(367.2, 306.7)--(367.1, 306.7)--(367.1, 306.5)--(367.0, 306.4)--(366.8, 306.5)--(366.6, 306.5)--(366.4, 306.5)--(366.3, 306.4)--(366.0, 306.6)--(365.9, 306.7)--(365.9, 306.8)--(365.8, 306.9)--(365.8, 306.8)--(364.8, 307.2)--(364.5, 307.3)--(364.3, 307.3)--(364.0, 307.4)--(364.0, 307.6)--(363.8, 307.7)--(363.7, 307.5)--(362.9, 307.7)--(362.9, 307.8)--(362.9, 307.9)--(362.8, 307.8)--(362.7, 307.8)--(362.7, 307.7)--(362.6, 307.7)--(362.6, 307.8)--(362.5, 307.8)--(362.5, 307.9)--(362.3, 307.9)--(362.0, 308.0)--(361.8, 308.0)--(361.4, 308.0)--(361.3, 308.0)--(360.6, 308.1)--(360.3, 308.1)--(360.1, 308.1)--(360.1, 308.3)--(360.0, 308.3)--(359.9, 308.3)--(360.0, 308.0)--(359.7, 308.0)--(359.1, 307.9)--(358.5, 307.8)--(358.5, 307.7)--(358.4, 307.7)--(357.7, 307.6)--(357.7, 307.5)--(357.4, 307.5)--(357.4, 307.3)--(357.2, 307.3)--(356.5, 306.8)--(356.4, 307.0)--(355.7, 306.6)--(355.6, 306.5)--(355.4, 306.3)--(355.2, 306.1)--(355.1, 306.1)--(355.0, 306.0)--(354.9, 305.9)--(354.8, 305.9)--(354.6, 305.9)--(354.5, 305.9)--(354.5, 305.8)--(354.4, 305.8)--(354.5, 305.7)--(354.7, 305.7)--(354.7, 305.6)--(354.7, 305.4)--(354.6, 305.4)--(354.5, 305.4)--(354.4, 305.4)--(353.8, 304.8)--(353.7, 304.9)--(353.3, 304.5)--(353.4, 304.4)--(353.0, 304.1)--(352.7, 303.8)--(352.6, 303.8)--(352.5, 303.7)--(352.4, 303.8)--(352.2, 303.6)--(352.3, 303.5)--(351.7, 302.9)--(351.6, 303.0)--(351.4, 302.9)--(351.5, 302.7)--(351.1, 302.2)--(351.0, 302.3)--(350.7, 301.9)--(350.6, 301.8)--(350.0, 301.1)--(349.9, 301.0)--(349.6, 300.7)--(349.3, 300.5)--(348.9, 300.3)--(348.7, 300.2)--(348.6, 300.2)--(348.5, 300.2)--(348.5, 300.1)--(348.3, 300.1)--(347.9, 300.0)--(347.6, 300.0)--(347.3, 300.0)--(347.2, 299.9)--(347.1, 300.0)--(347.0, 300.0)--(346.9, 300.0)--(346.7, 300.0)--(346.5, 300.0)--(346.2, 300.1)--(345.9, 300.2)--(345.9, 300.1)--(345.6, 300.2)--(345.5, 300.2)--(345.5, 300.3)--(345.5, 300.4)--(345.4, 300.4)--(345.4, 300.3)--(345.3, 300.3)--(345.1, 300.4)--(345.1, 300.5)--(344.3, 300.8)--(344.1, 300.9)--(343.9, 301.0)--(343.8, 300.9)--(343.3, 301.1)--(342.7, 301.4)--(342.7, 301.3)--(342.1, 301.5)--(341.9, 301.6)--(341.9, 301.7)--(341.8, 301.7)--(341.8, 301.8)--(341.8, 301.9)--(341.9, 302.0)--(342.0, 302.1)--(341.9, 302.2)--(341.8, 302.1)--(341.8, 302.0)--(341.7, 301.9)--(341.6, 301.7)--(341.2, 301.9)--(340.5, 302.2)--(340.4, 302.3)--(339.9, 302.6)--(339.6, 302.8)--(339.5, 302.9)--(339.6, 303.0)--(339.5, 303.1)--(339.6, 303.2)--(339.5, 303.3)--(339.4, 303.1)--(339.1, 303.1)--(339.0, 303.1)--(338.8, 303.2)--(338.9, 303.2)--(338.8, 303.3)--(338.7, 303.2)--(338.6, 303.1)--(338.6, 303.2)--(338.6, 303.3)--(338.1, 303.5)--(338.0, 303.5)--(337.9, 303.5)--(337.9, 303.4)--(337.2, 303.7)--(337.3, 303.9)--(337.0, 304.0)--(336.7, 304.0)--(336.6, 304.1)--(336.5, 304.1)--(336.1, 304.3)--(335.4, 304.5)--(335.1, 304.6)--(334.9, 304.6)--(334.8, 304.7)--(334.4, 304.8)--(334.2, 305.0)--(334.1, 305.1)--(334.0, 305.1)--cycle; +Waltham_Forest = (399.2, 418.6)--(399.3, 418.5)--(399.4, 418.2)--(399.5, 418.0)--(399.4, 418.0)--(399.2, 418.1)--(399.1, 418.1)--(398.9, 417.4)--(398.6, 416.7)--(398.4, 416.2)--(398.5, 416.1)--(398.4, 416.0)--(398.4, 415.9)--(398.1, 415.4)--(398.0, 415.4)--(397.9, 415.1)--(397.8, 414.8)--(397.0, 412.8)--(396.4, 411.4)--(395.9, 410.4)--(395.7, 409.8)--(395.4, 409.3)--(395.3, 408.9)--(395.2, 408.5)--(395.1, 408.3)--(395.1, 407.6)--(395.1, 406.8)--(395.0, 405.9)--(395.0, 405.4)--(395.0, 404.8)--(395.0, 404.2)--(394.9, 403.6)--(394.9, 403.4)--(394.9, 403.3)--(394.9, 403.1)--(394.9, 403.0)--(394.9, 402.8)--(394.8, 402.1)--(394.8, 402.0)--(394.8, 401.6)--(394.8, 401.2)--(394.8, 400.6)--(394.9, 400.5)--(394.9, 400.1)--(394.9, 400.0)--(394.9, 399.8)--(395.0, 399.6)--(395.0, 399.3)--(395.0, 399.1)--(395.1, 398.6)--(395.1, 398.0)--(395.2, 397.5)--(395.2, 397.1)--(395.3, 396.7)--(395.4, 395.6)--(395.5, 394.9)--(395.6, 393.8)--(395.7, 393.1)--(395.8, 392.2)--(395.9, 391.7)--(395.9, 391.3)--(396.4, 390.9)--(397.0, 390.9)--(397.4, 390.9)--(397.4, 390.5)--(397.3, 390.2)--(397.4, 390.0)--(397.3, 390.0)--(397.4, 389.6)--(397.4, 389.5)--(397.5, 389.4)--(397.6, 389.2)--(397.4, 389.1)--(397.3, 389.1)--(397.8, 387.3)--(398.1, 386.4)--(398.3, 386.3)--(398.9, 384.5)--(398.9, 384.4)--(398.9, 383.3)--(398.9, 382.6)--(398.9, 382.1)--(398.9, 381.6)--(399.0, 381.2)--(399.0, 381.0)--(398.9, 380.8)--(398.9, 380.7)--(398.6, 380.1)--(398.4, 379.5)--(398.3, 379.3)--(398.4, 379.3)--(398.5, 379.3)--(398.8, 379.1)--(399.0, 379.1)--(399.1, 379.0)--(399.2, 378.8)--(399.3, 378.7)--(399.4, 378.5)--(399.5, 378.3)--(400.0, 378.7)--(400.1, 378.7)--(400.1, 378.6)--(400.1, 378.1)--(400.2, 377.1)--(400.3, 376.5)--(400.3, 376.0)--(400.3, 375.9)--(400.1, 376.0)--(400.0, 376.1)--(399.8, 376.2)--(399.7, 376.3)--(399.6, 376.2)--(399.5, 376.2)--(399.4, 376.2)--(399.3, 376.2)--(399.3, 376.1)--(399.3, 376.0)--(399.5, 375.5)--(399.7, 374.9)--(399.9, 374.2)--(400.0, 373.7)--(400.3, 373.0)--(400.3, 372.9)--(400.3, 372.8)--(400.3, 372.7)--(400.4, 372.6)--(400.4, 372.5)--(400.3, 372.4)--(400.3, 372.3)--(400.3, 372.2)--(400.3, 372.1)--(400.2, 372.0)--(400.2, 371.9)--(400.1, 371.8)--(400.0, 371.7)--(399.9, 371.6)--(399.8, 371.5)--(399.7, 371.4)--(399.5, 371.3)--(399.3, 371.2)--(399.1, 371.2)--(398.6, 371.2)--(398.0, 371.2)--(397.2, 371.1)--(397.2, 370.7)--(397.2, 370.1)--(397.2, 369.8)--(397.2, 369.2)--(397.2, 369.1)--(397.1, 368.7)--(397.1, 368.2)--(397.1, 367.4)--(397.1, 366.9)--(397.2, 366.4)--(397.4, 365.3)--(397.6, 364.3)--(397.8, 364.3)--(398.1, 364.3)--(398.2, 364.3)--(398.3, 364.2)--(398.5, 364.2)--(398.7, 364.1)--(398.9, 363.9)--(399.0, 363.8)--(399.3, 363.6)--(399.7, 363.2)--(399.8, 363.1)--(399.9, 363.0)--(400.0, 362.9)--(400.1, 362.8)--(400.4, 362.5)--(400.6, 362.1)--(401.2, 361.4)--(401.4, 361.1)--(401.6, 360.9)--(402.3, 359.7)--(402.2, 359.6)--(402.5, 359.4)--(401.5, 358.8)--(401.0, 358.5)--(401.0, 358.4)--(400.4, 358.2)--(400.0, 357.9)--(399.3, 357.4)--(399.2, 357.6)--(398.8, 357.4)--(398.6, 357.8)--(398.4, 358.1)--(398.4, 358.2)--(397.6, 359.3)--(397.2, 359.0)--(395.6, 358.1)--(395.2, 358.6)--(394.2, 357.5)--(395.1, 356.2)--(394.8, 355.9)--(394.6, 355.7)--(394.3, 355.5)--(394.2, 355.3)--(394.2, 355.2)--(394.2, 355.1)--(393.3, 354.9)--(393.0, 354.8)--(392.9, 354.8)--(392.4, 354.6)--(391.7, 354.3)--(391.5, 354.4)--(391.5, 354.6)--(391.6, 354.8)--(391.6, 355.0)--(391.6, 355.3)--(391.7, 355.5)--(391.6, 355.7)--(391.3, 355.7)--(391.2, 355.6)--(391.2, 356.0)--(391.0, 356.0)--(391.1, 355.7)--(390.3, 355.6)--(390.3, 355.7)--(390.3, 355.8)--(390.1, 355.8)--(389.9, 355.7)--(389.8, 355.8)--(389.8, 355.9)--(389.3, 355.8)--(389.1, 355.7)--(388.5, 355.6)--(387.9, 355.5)--(387.4, 355.4)--(387.0, 355.3)--(386.9, 355.3)--(386.5, 355.1)--(385.8, 354.8)--(385.1, 354.5)--(385.0, 354.7)--(384.1, 354.3)--(383.7, 354.0)--(383.5, 353.9)--(383.3, 353.8)--(382.8, 353.7)--(382.1, 353.4)--(381.9, 353.3)--(381.8, 353.3)--(381.7, 353.3)--(381.6, 353.3)--(381.5, 353.7)--(381.4, 354.0)--(381.3, 354.3)--(381.3, 354.6)--(381.3, 354.8)--(381.3, 355.0)--(381.3, 355.2)--(381.2, 355.3)--(381.0, 355.8)--(380.8, 355.9)--(380.7, 356.0)--(380.5, 356.1)--(380.4, 356.2)--(380.3, 356.2)--(380.2, 356.2)--(380.1, 356.2)--(379.9, 356.1)--(379.4, 355.9)--(378.6, 355.6)--(378.5, 355.6)--(378.4, 355.5)--(378.2, 355.5)--(377.6, 355.3)--(376.7, 355.1)--(375.9, 355.0)--(375.9, 354.8)--(375.8, 354.7)--(375.7, 354.8)--(375.7, 354.9)--(375.7, 355.0)--(375.7, 355.1)--(375.6, 355.8)--(375.4, 357.0)--(375.3, 357.3)--(375.2, 357.6)--(375.1, 357.9)--(375.0, 357.7)--(374.7, 358.9)--(374.5, 359.1)--(374.2, 359.3)--(373.1, 360.1)--(372.3, 360.6)--(371.8, 360.9)--(371.5, 361.2)--(371.4, 361.2)--(371.1, 361.2)--(370.9, 361.3)--(370.8, 361.3)--(370.8, 361.5)--(370.8, 361.9)--(370.6, 362.3)--(370.6, 362.5)--(370.5, 362.7)--(370.5, 362.8)--(370.4, 362.9)--(370.3, 363.0)--(370.2, 363.1)--(370.1, 363.3)--(370.0, 363.5)--(369.7, 363.8)--(369.4, 364.1)--(369.2, 364.2)--(369.1, 364.3)--(369.0, 364.4)--(368.8, 364.5)--(368.3, 364.9)--(368.0, 365.0)--(367.9, 365.1)--(367.6, 365.3)--(367.1, 365.5)--(367.0, 365.6)--(366.8, 365.6)--(366.7, 365.6)--(366.5, 365.6)--(366.4, 365.6)--(366.3, 365.6)--(366.2, 365.5)--(366.1, 365.5)--(366.0, 365.5)--(366.0, 365.4)--(365.9, 365.4)--(365.8, 365.4)--(365.8, 365.3)--(365.7, 365.3)--(365.4, 365.1)--(365.0, 364.9)--(364.8, 364.8)--(364.6, 364.7)--(364.3, 364.7)--(364.1, 364.7)--(364.0, 364.7)--(363.9, 364.8)--(363.6, 365.0)--(363.3, 365.2)--(363.1, 365.3)--(362.9, 365.4)--(362.4, 365.6)--(362.1, 365.6)--(362.0, 365.6)--(361.9, 365.6)--(361.8, 365.6)--(361.7, 365.6)--(361.6, 365.6)--(361.5, 365.5)--(361.2, 365.3)--(361.0, 365.3)--(360.9, 365.2)--(360.7, 365.2)--(360.5, 365.2)--(360.4, 365.2)--(360.3, 365.2)--(360.2, 365.3)--(360.1, 365.4)--(360.0, 365.5)--(359.8, 365.7)--(359.7, 365.8)--(359.5, 365.9)--(359.4, 365.9)--(359.3, 365.8)--(359.2, 365.8)--(359.1, 365.8)--(358.9, 365.7)--(358.4, 365.5)--(358.3, 365.4)--(358.1, 365.4)--(358.0, 365.4)--(357.8, 365.4)--(357.7, 365.4)--(357.3, 365.6)--(357.0, 365.7)--(356.7, 365.7)--(356.6, 365.7)--(356.5, 365.7)--(356.3, 365.7)--(356.1, 365.7)--(355.9, 365.9)--(355.6, 366.3)--(355.4, 366.3)--(355.0, 366.4)--(354.9, 366.4)--(354.9, 366.5)--(354.8, 366.5)--(354.7, 366.6)--(354.6, 366.6)--(354.6, 366.7)--(354.5, 366.7)--(354.5, 366.8)--(354.5, 366.9)--(354.5, 367.0)--(354.5, 367.1)--(354.5, 367.2)--(354.5, 367.3)--(354.6, 367.3)--(354.6, 367.4)--(354.7, 367.6)--(354.8, 367.8)--(354.8, 367.9)--(354.9, 367.9)--(354.9, 368.0)--(354.9, 368.1)--(354.9, 368.2)--(354.9, 368.3)--(354.9, 368.4)--(354.9, 368.5)--(354.9, 368.6)--(354.8, 368.7)--(354.8, 368.8)--(354.3, 369.5)--(354.3, 369.6)--(354.2, 369.7)--(353.8, 370.0)--(353.7, 370.1)--(353.6, 370.2)--(353.5, 370.3)--(353.3, 370.6)--(353.2, 370.7)--(353.1, 370.8)--(352.6, 371.3)--(352.2, 371.7)--(352.0, 372.0)--(351.9, 372.1)--(351.7, 372.3)--(351.1, 372.6)--(350.5, 373.0)--(350.3, 373.2)--(350.2, 373.3)--(350.1, 373.4)--(350.0, 373.7)--(349.5, 374.8)--(349.4, 375.0)--(349.3, 375.1)--(349.2, 375.2)--(349.1, 375.2)--(349.0, 375.3)--(348.7, 375.3)--(348.5, 375.3)--(348.4, 375.3)--(348.3, 375.3)--(348.2, 375.4)--(348.1, 375.5)--(348.0, 375.6)--(347.8, 375.8)--(347.7, 376.2)--(347.4, 376.5)--(347.1, 376.8)--(346.7, 377.3)--(346.5, 377.5)--(346.5, 377.7)--(346.4, 377.9)--(346.4, 378.2)--(346.3, 378.6)--(346.2, 378.8)--(346.0, 379.2)--(345.8, 379.6)--(345.6, 380.1)--(345.5, 380.2)--(345.3, 380.7)--(345.3, 380.9)--(345.1, 381.5)--(344.8, 382.2)--(344.6, 382.9)--(344.4, 383.3)--(344.4, 383.4)--(344.2, 383.8)--(343.9, 384.3)--(343.9, 384.4)--(343.8, 384.4)--(343.8, 384.5)--(343.7, 384.6)--(343.7, 384.7)--(343.7, 384.8)--(343.7, 384.9)--(343.6, 384.9)--(343.6, 385.0)--(343.6, 385.1)--(343.6, 385.2)--(343.7, 385.2)--(343.7, 385.3)--(343.7, 385.4)--(343.8, 385.5)--(343.8, 385.6)--(343.8, 385.7)--(343.9, 385.8)--(343.9, 385.9)--(344.0, 386.0)--(344.0, 386.1)--(344.1, 386.2)--(344.2, 386.3)--(344.2, 386.4)--(344.3, 386.5)--(344.4, 386.6)--(344.5, 386.7)--(344.5, 386.8)--(344.6, 386.8)--(345.1, 387.3)--(345.3, 387.4)--(345.6, 387.7)--(346.2, 388.2)--(346.3, 388.3)--(346.4, 388.4)--(346.5, 388.4)--(346.5, 388.5)--(346.6, 388.5)--(346.6, 388.6)--(346.6, 388.7)--(346.7, 388.7)--(346.7, 388.9)--(346.8, 389.0)--(346.8, 389.1)--(346.8, 389.2)--(346.9, 389.2)--(346.9, 389.3)--(347.0, 389.4)--(347.0, 389.5)--(347.1, 389.5)--(347.2, 389.6)--(347.3, 389.7)--(347.4, 389.8)--(347.5, 389.9)--(347.6, 390.0)--(347.7, 390.1)--(347.8, 390.2)--(347.8, 390.3)--(347.9, 390.3)--(347.9, 390.4)--(348.0, 390.5)--(348.0, 390.6)--(348.1, 390.7)--(348.1, 390.8)--(348.2, 390.9)--(348.2, 391.0)--(348.2, 391.1)--(348.3, 391.1)--(348.3, 391.2)--(348.3, 391.3)--(348.3, 391.4)--(348.3, 391.5)--(348.4, 391.6)--(348.4, 391.7)--(348.4, 391.8)--(348.4, 391.9)--(348.5, 392.0)--(348.5, 392.1)--(348.6, 392.2)--(348.6, 392.3)--(348.7, 392.4)--(348.7, 392.5)--(348.8, 392.6)--(348.8, 392.7)--(348.9, 392.7)--(348.9, 392.8)--(349.2, 393.1)--(349.6, 393.5)--(349.7, 393.6)--(350.0, 393.9)--(350.1, 393.9)--(350.1, 394.0)--(350.2, 394.0)--(350.2, 394.1)--(350.3, 394.1)--(350.3, 394.2)--(350.3, 394.3)--(350.4, 394.3)--(350.4, 394.4)--(350.4, 394.5)--(350.4, 394.6)--(350.4, 394.7)--(350.4, 394.8)--(350.3, 394.8)--(350.3, 394.9)--(350.2, 395.1)--(350.1, 395.2)--(350.1, 395.4)--(350.0, 395.5)--(350.0, 395.6)--(349.8, 395.9)--(349.6, 396.3)--(349.5, 396.6)--(349.4, 396.9)--(349.4, 397.2)--(349.4, 397.5)--(349.4, 397.9)--(349.5, 398.2)--(349.6, 398.5)--(349.7, 399.0)--(350.0, 399.7)--(350.0, 400.0)--(350.1, 400.6)--(350.1, 401.2)--(350.1, 401.3)--(350.1, 401.4)--(350.1, 401.5)--(350.2, 401.6)--(350.2, 401.7)--(350.2, 401.8)--(350.3, 401.8)--(350.3, 401.9)--(350.4, 402.0)--(350.4, 402.1)--(350.5, 402.1)--(350.5, 402.2)--(350.6, 402.3)--(350.7, 402.4)--(350.8, 402.5)--(350.8, 402.6)--(350.8, 402.7)--(350.9, 402.7)--(350.9, 402.8)--(350.9, 402.9)--(350.9, 403.4)--(350.9, 403.6)--(351.0, 403.7)--(351.0, 403.8)--(351.0, 403.9)--(351.1, 404.0)--(351.1, 404.1)--(351.2, 404.2)--(351.2, 404.3)--(351.2, 404.4)--(351.3, 404.4)--(351.3, 404.5)--(351.4, 404.6)--(351.4, 404.7)--(351.5, 404.8)--(351.6, 404.9)--(351.7, 405.0)--(351.8, 405.1)--(351.9, 405.3)--(352.4, 405.9)--(353.1, 406.8)--(353.7, 407.6)--(354.0, 408.0)--(354.2, 408.4)--(354.4, 408.8)--(355.1, 409.9)--(355.6, 410.8)--(355.9, 411.4)--(356.2, 412.0)--(356.5, 412.7)--(356.8, 413.4)--(356.8, 413.5)--(356.8, 413.6)--(356.9, 413.7)--(356.9, 413.9)--(357.0, 414.0)--(357.0, 414.1)--(357.1, 414.2)--(357.1, 414.3)--(357.2, 414.4)--(357.2, 414.5)--(357.3, 414.6)--(357.3, 414.7)--(357.4, 414.7)--(357.4, 414.8)--(357.5, 414.9)--(357.6, 415.0)--(357.7, 415.1)--(357.8, 415.1)--(357.8, 415.2)--(357.9, 415.2)--(359.2, 416.1)--(359.7, 416.4)--(360.7, 417.0)--(360.9, 417.1)--(361.0, 417.2)--(361.1, 417.3)--(361.1, 417.4)--(361.2, 417.4)--(361.3, 417.5)--(361.4, 417.5)--(361.4, 417.6)--(361.5, 417.6)--(361.5, 417.7)--(361.6, 417.7)--(361.7, 417.8)--(361.8, 417.9)--(361.8, 418.0)--(361.9, 418.2)--(362.0, 418.2)--(362.0, 418.3)--(362.1, 418.4)--(362.1, 418.5)--(362.2, 418.5)--(362.2, 418.6)--(362.2, 418.7)--(362.3, 418.8)--(362.3, 418.9)--(362.3, 419.0)--(362.3, 419.1)--(362.3, 419.2)--(362.3, 419.3)--(362.3, 419.4)--(362.3, 419.5)--(362.3, 419.6)--(362.3, 419.7)--(362.2, 419.7)--(362.2, 419.8)--(362.2, 419.9)--(362.2, 420.0)--(362.1, 420.0)--(362.1, 420.1)--(362.1, 420.2)--(362.0, 420.2)--(361.7, 420.8)--(361.6, 420.9)--(361.6, 421.0)--(361.5, 421.0)--(361.5, 421.1)--(361.4, 421.2)--(361.4, 421.3)--(361.3, 421.4)--(361.3, 421.5)--(361.3, 421.6)--(361.2, 421.6)--(361.2, 421.7)--(361.2, 421.8)--(361.1, 421.9)--(361.1, 422.0)--(361.1, 422.1)--(361.1, 422.2)--(361.1, 422.3)--(361.2, 422.4)--(361.2, 422.5)--(361.2, 422.6)--(361.2, 422.7)--(361.2, 422.8)--(361.3, 422.8)--(361.3, 422.9)--(361.3, 423.0)--(361.3, 423.1)--(361.3, 423.2)--(361.4, 423.3)--(361.4, 423.4)--(361.4, 423.5)--(361.5, 423.6)--(361.8, 424.1)--(362.2, 425.0)--(362.6, 425.8)--(362.9, 426.4)--(363.0, 426.5)--(363.0, 426.6)--(363.1, 426.8)--(363.1, 426.9)--(363.2, 427.0)--(363.3, 427.1)--(363.4, 427.3)--(363.6, 427.5)--(363.7, 427.7)--(363.8, 427.9)--(364.0, 428.1)--(364.6, 428.8)--(365.0, 429.2)--(365.2, 429.5)--(365.7, 430.3)--(366.2, 431.0)--(366.3, 431.1)--(366.5, 431.5)--(366.6, 431.6)--(366.8, 431.9)--(367.3, 432.6)--(367.4, 432.7)--(367.5, 432.8)--(367.8, 433.2)--(367.9, 433.5)--(368.0, 433.7)--(368.1, 434.0)--(368.2, 434.4)--(368.3, 434.8)--(368.3, 435.0)--(368.4, 435.2)--(368.5, 435.5)--(368.6, 435.8)--(368.8, 436.1)--(368.9, 436.3)--(369.1, 436.7)--(369.2, 436.8)--(369.3, 437.1)--(369.5, 437.4)--(369.8, 438.1)--(371.5, 441.8)--(372.3, 443.4)--(372.8, 444.7)--(373.3, 445.8)--(373.5, 446.1)--(373.6, 446.4)--(374.0, 447.3)--(374.4, 448.0)--(374.5, 448.3)--(374.7, 448.8)--(374.8, 449.2)--(374.8, 449.4)--(374.8, 449.7)--(374.9, 450.0)--(375.3, 451.2)--(375.6, 452.4)--(375.7, 452.9)--(375.8, 453.2)--(375.8, 454.0)--(376.0, 457.3)--(376.3, 460.3)--(376.4, 460.3)--(376.7, 460.5)--(376.8, 460.5)--(377.1, 460.5)--(377.3, 460.5)--(377.6, 460.4)--(377.7, 460.7)--(378.3, 460.7)--(378.5, 460.7)--(378.6, 460.7)--(378.8, 460.6)--(379.0, 460.6)--(379.2, 460.5)--(379.5, 460.4)--(379.7, 460.3)--(379.4, 460.0)--(379.2, 459.6)--(379.3, 459.5)--(379.3, 459.3)--(379.2, 459.2)--(379.1, 459.0)--(379.0, 458.7)--(378.9, 458.4)--(379.1, 458.4)--(379.1, 458.2)--(379.1, 458.1)--(379.1, 457.8)--(379.1, 457.6)--(379.1, 457.5)--(379.2, 457.4)--(379.3, 457.4)--(379.8, 457.3)--(380.2, 457.2)--(380.5, 457.2)--(380.7, 457.2)--(380.8, 457.2)--(380.9, 457.2)--(381.0, 457.3)--(381.7, 457.4)--(381.8, 457.2)--(381.8, 457.1)--(381.8, 457.0)--(381.9, 456.9)--(382.0, 456.7)--(382.1, 456.5)--(382.3, 456.7)--(382.8, 457.0)--(383.3, 457.3)--(383.6, 457.5)--(384.1, 457.7)--(384.2, 457.7)--(384.5, 457.7)--(384.6, 457.6)--(384.5, 457.5)--(384.6, 457.5)--(384.6, 457.4)--(384.6, 457.3)--(384.6, 457.2)--(384.9, 456.5)--(385.0, 456.3)--(385.1, 456.1)--(385.2, 456.0)--(385.3, 455.9)--(385.4, 455.7)--(385.5, 455.6)--(385.6, 455.6)--(385.7, 455.6)--(385.9, 455.6)--(386.0, 455.5)--(386.3, 455.5)--(386.6, 455.5)--(386.8, 455.5)--(386.9, 455.5)--(387.0, 455.4)--(387.1, 455.4)--(387.3, 455.5)--(387.3, 455.4)--(387.9, 455.4)--(388.4, 455.4)--(388.7, 455.5)--(388.9, 455.5)--(389.1, 455.6)--(389.3, 455.6)--(389.4, 455.6)--(389.5, 455.6)--(389.7, 455.5)--(390.0, 455.5)--(390.7, 455.4)--(391.1, 455.4)--(391.6, 455.4)--(391.7, 455.4)--(391.8, 455.4)--(392.3, 455.3)--(392.5, 455.3)--(392.6, 455.2)--(392.8, 455.1)--(392.9, 455.1)--(393.0, 455.1)--(393.3, 455.1)--(393.5, 455.1)--(393.8, 455.0)--(394.2, 454.8)--(394.6, 454.8)--(394.7, 454.8)--(394.9, 454.7)--(395.0, 454.6)--(395.2, 454.6)--(395.4, 454.5)--(395.9, 454.4)--(396.3, 454.3)--(396.5, 454.3)--(397.0, 454.2)--(397.5, 454.2)--(398.3, 454.3)--(398.8, 454.5)--(400.0, 455.0)--(400.6, 455.3)--(400.6, 455.2)--(400.7, 454.7)--(400.8, 454.1)--(400.8, 453.5)--(400.9, 452.6)--(400.9, 451.7)--(401.0, 451.5)--(401.6, 451.3)--(402.1, 451.2)--(402.2, 451.2)--(402.3, 451.1)--(402.4, 451.1)--(402.3, 450.9)--(402.3, 450.8)--(402.2, 450.6)--(402.2, 450.5)--(402.2, 450.4)--(402.2, 450.3)--(402.1, 450.0)--(402.1, 449.9)--(402.1, 449.7)--(402.2, 449.6)--(402.2, 449.5)--(402.2, 449.4)--(402.3, 449.1)--(402.4, 449.1)--(402.5, 448.8)--(402.7, 448.6)--(402.8, 448.3)--(402.9, 448.0)--(402.8, 447.8)--(402.7, 447.5)--(402.4, 446.8)--(402.2, 446.2)--(402.1, 445.9)--(402.0, 445.6)--(401.8, 445.0)--(401.7, 444.4)--(401.7, 444.2)--(401.7, 444.0)--(401.7, 443.9)--(401.8, 443.9)--(401.7, 443.8)--(401.6, 443.7)--(400.8, 442.7)--(400.6, 442.3)--(400.4, 441.6)--(400.2, 441.7)--(400.0, 441.8)--(400.0, 441.7)--(400.1, 441.7)--(400.1, 441.6)--(400.0, 441.6)--(399.9, 441.6)--(399.8, 441.7)--(399.8, 441.6)--(399.7, 441.5)--(399.7, 441.4)--(399.8, 441.3)--(399.9, 441.2)--(399.9, 441.1)--(399.9, 441.0)--(399.8, 441.0)--(399.7, 441.0)--(399.6, 441.0)--(399.6, 440.9)--(399.6, 440.8)--(399.5, 440.6)--(399.5, 440.5)--(399.5, 440.3)--(399.5, 440.2)--(399.5, 440.1)--(399.6, 440.0)--(399.6, 439.9)--(399.5, 439.9)--(399.5, 439.8)--(399.4, 439.8)--(399.3, 439.8)--(399.1, 439.5)--(399.1, 439.4)--(399.1, 439.3)--(399.3, 439.1)--(399.3, 439.0)--(399.3, 438.9)--(399.2, 438.9)--(399.2, 438.7)--(399.2, 438.6)--(399.1, 438.5)--(399.0, 438.5)--(399.1, 438.4)--(399.1, 438.3)--(398.9, 438.2)--(398.8, 438.1)--(398.8, 438.0)--(398.8, 437.9)--(398.8, 437.8)--(398.9, 437.7)--(398.8, 437.5)--(398.6, 437.3)--(398.7, 437.2)--(398.6, 437.2)--(398.6, 437.1)--(398.6, 437.0)--(398.6, 436.5)--(398.5, 436.4)--(398.3, 436.4)--(398.3, 436.3)--(398.3, 436.2)--(398.4, 436.2)--(398.4, 436.1)--(398.4, 436.0)--(398.4, 435.9)--(398.5, 435.9)--(398.5, 435.8)--(398.6, 435.9)--(398.6, 435.8)--(398.6, 435.7)--(398.5, 435.7)--(398.4, 435.7)--(398.3, 435.7)--(398.2, 435.6)--(398.3, 435.5)--(398.4, 435.5)--(398.4, 435.4)--(398.5, 435.4)--(398.5, 435.3)--(398.4, 435.2)--(398.3, 435.1)--(398.3, 435.0)--(398.3, 434.9)--(398.2, 434.8)--(398.1, 434.7)--(398.1, 434.6)--(398.0, 434.5)--(397.9, 434.5)--(397.7, 434.3)--(397.6, 434.3)--(397.5, 434.3)--(397.5, 434.4)--(397.4, 434.4)--(397.4, 434.3)--(397.3, 434.3)--(397.3, 434.2)--(397.3, 434.1)--(397.3, 434.0)--(397.3, 433.9)--(397.2, 433.9)--(397.2, 433.8)--(397.0, 433.7)--(396.8, 433.7)--(396.7, 433.6)--(396.7, 433.5)--(396.7, 433.3)--(396.8, 433.3)--(396.8, 433.2)--(396.8, 433.1)--(396.7, 433.0)--(396.6, 433.0)--(396.6, 432.9)--(396.5, 432.7)--(396.4, 432.7)--(396.3, 432.6)--(396.2, 432.6)--(396.1, 432.6)--(396.0, 432.5)--(395.9, 432.4)--(395.8, 432.3)--(395.7, 432.1)--(395.6, 432.1)--(395.5, 432.1)--(395.5, 432.0)--(395.4, 432.0)--(395.5, 431.8)--(395.6, 431.7)--(395.7, 431.6)--(395.7, 431.4)--(395.8, 431.1)--(395.7, 431.0)--(395.6, 430.4)--(395.5, 430.4)--(395.4, 430.4)--(395.3, 430.4)--(394.3, 430.6)--(394.0, 430.6)--(393.8, 430.6)--(393.5, 430.6)--(393.3, 430.5)--(393.1, 430.5)--(393.0, 430.4)--(392.8, 430.5)--(392.7, 430.6)--(392.6, 430.6)--(392.6, 430.5)--(392.5, 430.5)--(392.4, 430.5)--(392.4, 430.6)--(392.3, 430.6)--(392.1, 430.6)--(391.8, 430.6)--(391.6, 430.5)--(391.8, 430.4)--(391.9, 430.4)--(392.9, 429.9)--(395.0, 429.1)--(395.8, 428.7)--(396.2, 428.6)--(396.6, 428.4)--(396.7, 428.3)--(396.8, 428.2)--(396.9, 428.2)--(400.0, 423.8)--(400.4, 423.4)--(400.5, 423.2)--(400.5, 423.1)--(400.5, 423.0)--(400.6, 422.9)--(400.6, 422.7)--(400.6, 422.6)--(400.6, 422.4)--(400.6, 422.2)--(400.5, 422.0)--(400.6, 421.9)--(400.5, 421.5)--(400.4, 421.2)--(400.3, 421.3)--(400.1, 420.9)--(400.2, 420.8)--(400.0, 420.4)--(400.0, 420.3)--(399.8, 420.4)--(399.6, 419.6)--(399.5, 419.4)--(399.4, 419.2)--(399.5, 419.2)--(399.5, 419.1)--(399.5, 419.0)--(399.4, 418.8)--(399.4, 418.7)--(399.3, 418.7)--cycle; +Wandsworth = (234.9, 262.2)--(235.0, 262.2)--(235.1, 262.3)--(235.4, 262.5)--(236.1, 261.6)--(237.0, 260.5)--(237.5, 260.0)--(238.0, 259.4)--(238.0, 259.3)--(239.1, 258.3)--(239.6, 257.9)--(240.5, 257.2)--(240.7, 257.1)--(241.1, 256.8)--(241.0, 256.8)--(242.0, 256.1)--(242.4, 255.9)--(242.4, 255.8)--(242.5, 255.8)--(242.6, 255.8)--(243.2, 255.5)--(244.1, 255.1)--(245.1, 254.6)--(246.3, 254.1)--(246.5, 254.0)--(247.4, 253.7)--(249.0, 253.3)--(250.0, 253.1)--(251.2, 253.0)--(251.4, 252.9)--(251.6, 252.9)--(252.0, 252.9)--(252.0, 253.0)--(254.5, 253.1)--(254.8, 252.6)--(254.8, 252.3)--(254.8, 252.0)--(254.8, 251.7)--(254.9, 251.1)--(254.8, 250.7)--(254.8, 250.6)--(254.9, 250.4)--(254.8, 250.0)--(254.8, 249.9)--(255.0, 249.8)--(255.3, 249.8)--(255.6, 249.8)--(255.6, 249.9)--(255.3, 249.9)--(255.2, 250.0)--(255.1, 250.1)--(255.1, 251.4)--(255.1, 251.5)--(255.0, 251.7)--(255.0, 251.8)--(255.0, 251.9)--(255.1, 251.8)--(255.2, 251.7)--(255.2, 251.6)--(255.3, 251.5)--(255.7, 251.6)--(255.3, 252.0)--(255.3, 252.1)--(255.2, 252.2)--(255.2, 252.4)--(255.2, 252.5)--(255.2, 252.6)--(255.0, 252.9)--(255.0, 253.1)--(255.3, 253.1)--(255.4, 252.9)--(255.5, 253.0)--(255.5, 253.2)--(256.6, 253.4)--(256.7, 253.4)--(257.2, 253.5)--(257.5, 253.6)--(257.8, 253.7)--(258.1, 253.8)--(258.3, 253.8)--(259.1, 254.1)--(259.4, 254.2)--(260.2, 254.6)--(260.3, 254.7)--(262.4, 256.1)--(262.8, 256.4)--(263.4, 257.1)--(263.6, 257.2)--(263.7, 257.3)--(263.8, 257.5)--(264.2, 258.1)--(264.3, 258.2)--(264.4, 258.4)--(264.5, 258.6)--(264.6, 258.8)--(264.7, 258.9)--(265.0, 259.6)--(265.6, 261.0)--(265.7, 261.4)--(265.8, 261.4)--(265.9, 261.6)--(266.1, 261.5)--(266.2, 261.6)--(265.9, 261.7)--(266.1, 262.1)--(266.5, 262.0)--(266.5, 262.1)--(266.1, 262.2)--(266.2, 262.6)--(266.3, 262.8)--(266.5, 263.9)--(266.6, 264.2)--(266.7, 264.5)--(266.9, 264.4)--(266.9, 264.5)--(266.7, 264.6)--(266.7, 264.8)--(266.8, 264.9)--(266.8, 265.2)--(266.8, 265.4)--(266.9, 265.6)--(266.9, 265.7)--(267.0, 266.0)--(267.0, 266.1)--(267.0, 266.2)--(267.1, 266.7)--(267.2, 267.1)--(267.3, 267.3)--(267.5, 267.2)--(267.7, 268.0)--(267.8, 268.2)--(267.9, 268.2)--(267.9, 268.3)--(267.8, 268.3)--(267.7, 268.4)--(267.7, 268.5)--(267.7, 268.6)--(267.9, 269.0)--(268.0, 269.0)--(267.8, 269.1)--(268.0, 269.6)--(268.0, 269.7)--(268.2, 270.0)--(268.2, 270.2)--(268.4, 270.5)--(268.6, 270.8)--(268.7, 271.1)--(268.9, 271.3)--(269.1, 271.6)--(269.2, 271.7)--(269.4, 271.9)--(269.6, 272.1)--(270.1, 272.5)--(270.3, 272.7)--(270.4, 272.7)--(270.6, 272.8)--(270.8, 272.9)--(271.0, 273.0)--(271.1, 273.0)--(271.2, 273.0)--(271.3, 273.1)--(272.7, 273.6)--(273.2, 273.8)--(273.3, 273.8)--(273.3, 273.9)--(274.3, 274.1)--(274.6, 274.2)--(274.7, 274.3)--(275.2, 274.4)--(275.2, 274.3)--(275.3, 274.3)--(277.6, 274.9)--(280.1, 275.5)--(281.2, 275.8)--(282.8, 276.2)--(283.6, 276.3)--(283.6, 276.4)--(284.1, 276.5)--(285.4, 276.8)--(285.4, 276.7)--(285.5, 276.8)--(285.5, 276.9)--(285.9, 277.0)--(286.1, 277.0)--(286.2, 277.0)--(286.9, 277.0)--(287.5, 277.0)--(287.8, 277.0)--(288.1, 277.0)--(288.2, 277.0)--(290.1, 276.6)--(291.4, 276.4)--(292.1, 276.2)--(292.2, 276.2)--(292.3, 276.2)--(292.5, 276.2)--(293.2, 276.2)--(293.6, 276.2)--(293.8, 276.2)--(294.0, 276.2)--(294.2, 276.2)--(294.4, 276.2)--(294.4, 275.9)--(294.7, 275.9)--(294.7, 276.1)--(294.7, 276.2)--(294.7, 276.3)--(294.8, 276.3)--(295.2, 276.5)--(295.3, 276.5)--(295.4, 276.5)--(295.5, 276.5)--(296.3, 276.8)--(296.3, 276.9)--(296.6, 277.0)--(297.2, 277.2)--(297.3, 277.2)--(297.5, 277.3)--(297.6, 277.2)--(297.7, 277.3)--(297.9, 277.4)--(298.0, 277.3)--(298.1, 277.4)--(298.0, 277.5)--(298.3, 277.7)--(298.4, 277.8)--(298.5, 277.9)--(298.6, 278.0)--(298.9, 278.2)--(299.0, 278.3)--(299.1, 278.3)--(299.4, 278.4)--(299.6, 278.5)--(299.9, 278.6)--(300.0, 278.7)--(300.2, 278.8)--(300.4, 278.9)--(300.5, 278.9)--(300.5, 278.8)--(300.7, 278.6)--(300.8, 278.6)--(301.0, 278.6)--(301.5, 278.5)--(301.8, 278.4)--(301.9, 278.3)--(301.6, 277.7)--(301.5, 277.4)--(301.4, 277.1)--(301.3, 276.5)--(301.3, 276.0)--(301.2, 275.6)--(300.9, 275.4)--(300.4, 275.1)--(300.1, 274.9)--(299.9, 274.8)--(299.5, 274.6)--(299.4, 274.0)--(299.3, 274.0)--(299.3, 273.9)--(299.2, 273.8)--(299.1, 273.5)--(298.9, 273.2)--(298.9, 273.0)--(298.7, 272.8)--(298.6, 272.7)--(298.6, 272.6)--(298.5, 272.4)--(298.4, 272.3)--(298.4, 272.1)--(298.3, 271.8)--(298.3, 271.7)--(298.2, 271.6)--(298.2, 271.5)--(298.1, 271.5)--(298.0, 271.2)--(297.8, 270.6)--(297.5, 269.9)--(297.3, 269.6)--(297.1, 269.0)--(297.0, 268.7)--(297.0, 268.6)--(297.0, 268.4)--(296.9, 268.4)--(296.8, 268.3)--(296.8, 268.0)--(296.7, 267.8)--(296.5, 267.5)--(296.4, 267.6)--(296.2, 267.3)--(296.3, 267.3)--(296.1, 266.8)--(296.4, 266.6)--(296.6, 266.5)--(296.6, 266.4)--(296.6, 266.3)--(296.6, 266.1)--(296.6, 265.9)--(296.4, 265.8)--(296.2, 265.5)--(296.1, 265.4)--(296.0, 265.3)--(295.9, 265.1)--(295.8, 265.1)--(295.7, 265.1)--(295.6, 265.1)--(295.5, 265.1)--(295.4, 265.1)--(295.3, 265.1)--(295.2, 265.2)--(295.1, 265.3)--(294.9, 265.5)--(294.5, 264.9)--(294.3, 264.6)--(294.2, 264.4)--(294.0, 264.2)--(293.9, 264.1)--(293.7, 264.1)--(293.7, 264.2)--(293.5, 264.1)--(293.4, 264.1)--(293.3, 264.1)--(293.2, 264.0)--(293.1, 263.9)--(293.2, 263.8)--(293.3, 263.7)--(293.0, 263.5)--(292.9, 263.5)--(292.6, 263.8)--(292.2, 263.4)--(291.6, 263.0)--(291.8, 262.7)--(291.0, 262.0)--(290.7, 262.5)--(290.6, 262.5)--(288.6, 261.1)--(287.8, 260.6)--(286.4, 259.6)--(285.8, 259.2)--(285.6, 259.2)--(285.6, 258.9)--(285.6, 258.8)--(285.6, 257.7)--(285.7, 257.3)--(285.2, 257.2)--(285.4, 255.4)--(285.4, 255.1)--(285.4, 254.9)--(285.6, 254.1)--(285.6, 253.9)--(285.7, 253.7)--(285.8, 253.4)--(285.8, 253.3)--(285.9, 253.2)--(286.1, 252.8)--(286.2, 252.6)--(286.3, 252.4)--(286.4, 252.2)--(286.4, 252.1)--(286.7, 250.0)--(286.7, 249.7)--(286.8, 249.5)--(287.0, 249.0)--(287.2, 248.4)--(287.4, 247.8)--(287.4, 247.7)--(287.4, 247.6)--(287.6, 246.7)--(287.7, 246.2)--(287.8, 245.3)--(287.8, 245.1)--(287.7, 245.0)--(287.8, 244.9)--(287.9, 243.9)--(288.0, 242.9)--(288.1, 242.9)--(288.2, 242.9)--(288.0, 242.2)--(288.3, 242.1)--(288.8, 241.9)--(290.2, 241.3)--(290.8, 241.0)--(290.9, 241.0)--(291.2, 240.9)--(291.8, 240.8)--(291.8, 240.5)--(291.7, 240.3)--(291.6, 240.1)--(291.5, 239.8)--(291.4, 239.6)--(291.3, 239.5)--(291.1, 239.1)--(290.9, 238.9)--(290.6, 238.6)--(290.2, 238.2)--(290.2, 238.0)--(290.1, 237.7)--(289.9, 237.1)--(289.8, 236.7)--(289.8, 236.2)--(289.7, 235.6)--(289.7, 235.2)--(289.6, 235.1)--(289.6, 235.0)--(289.7, 234.5)--(289.8, 234.0)--(290.0, 233.5)--(290.2, 233.1)--(290.3, 232.9)--(290.4, 232.7)--(290.5, 232.4)--(290.6, 232.2)--(290.9, 231.5)--(291.0, 231.1)--(291.2, 230.6)--(291.4, 230.6)--(291.8, 230.6)--(292.0, 230.6)--(292.6, 230.6)--(293.1, 230.6)--(294.3, 230.5)--(295.1, 230.6)--(295.5, 230.6)--(295.9, 230.7)--(296.2, 230.8)--(296.4, 230.9)--(296.5, 230.9)--(296.5, 230.7)--(296.4, 230.4)--(296.2, 230.0)--(296.0, 229.5)--(295.9, 229.4)--(295.8, 229.0)--(295.5, 229.1)--(295.5, 228.9)--(295.5, 228.8)--(295.4, 228.8)--(295.5, 228.7)--(295.6, 228.5)--(295.6, 228.4)--(295.6, 228.3)--(295.6, 228.0)--(295.6, 227.7)--(295.5, 227.4)--(295.4, 227.1)--(295.3, 226.9)--(295.2, 226.7)--(295.0, 226.5)--(294.8, 226.4)--(294.5, 226.2)--(294.4, 226.1)--(294.3, 226.0)--(294.2, 225.9)--(294.0, 225.6)--(293.9, 225.3)--(293.8, 225.2)--(293.9, 224.5)--(294.0, 224.1)--(294.0, 223.8)--(294.1, 223.6)--(294.1, 223.4)--(294.2, 223.3)--(294.5, 222.9)--(295.1, 222.1)--(295.4, 221.8)--(295.6, 221.8)--(296.0, 221.9)--(296.2, 221.9)--(296.3, 221.9)--(296.4, 221.8)--(296.6, 220.7)--(296.8, 219.8)--(297.0, 218.9)--(297.2, 218.3)--(297.3, 218.0)--(297.2, 217.9)--(297.1, 217.9)--(296.8, 217.9)--(296.3, 218.0)--(295.9, 218.1)--(295.6, 218.1)--(295.1, 218.2)--(295.1, 218.1)--(295.1, 218.0)--(295.1, 217.9)--(295.1, 217.5)--(295.1, 217.1)--(295.1, 216.5)--(295.1, 216.2)--(295.1, 215.7)--(295.1, 215.3)--(295.0, 214.0)--(294.9, 213.2)--(294.9, 212.3)--(294.9, 211.7)--(294.9, 211.5)--(295.0, 210.7)--(295.0, 210.5)--(295.2, 209.4)--(295.2, 209.3)--(295.4, 208.7)--(295.8, 207.7)--(295.5, 207.4)--(295.3, 207.1)--(294.9, 206.6)--(294.6, 206.3)--(294.0, 205.6)--(293.8, 205.4)--(293.7, 205.3)--(293.6, 205.2)--(293.5, 205.1)--(293.4, 205.0)--(293.2, 204.9)--(293.1, 204.8)--(292.9, 204.7)--(292.5, 204.4)--(292.4, 204.3)--(292.2, 204.2)--(291.9, 204.1)--(291.7, 204.0)--(291.6, 203.9)--(291.2, 203.8)--(291.1, 203.9)--(290.9, 204.0)--(290.7, 204.1)--(290.4, 204.2)--(290.3, 204.3)--(290.3, 204.4)--(290.2, 204.4)--(290.2, 204.5)--(290.1, 204.5)--(290.0, 204.7)--(289.8, 204.9)--(289.6, 205.1)--(289.5, 205.1)--(289.4, 205.1)--(289.1, 205.1)--(289.0, 205.1)--(288.8, 205.3)--(288.6, 205.5)--(288.3, 205.8)--(287.9, 206.0)--(287.8, 206.1)--(287.7, 206.1)--(287.5, 206.2)--(287.4, 206.2)--(287.2, 206.3)--(287.1, 206.3)--(287.0, 206.3)--(286.8, 206.2)--(286.6, 206.1)--(286.5, 206.0)--(286.4, 206.0)--(286.0, 206.2)--(285.0, 206.7)--(283.6, 207.4)--(283.3, 207.5)--(283.0, 207.6)--(282.5, 207.8)--(282.0, 208.0)--(281.6, 208.3)--(281.2, 208.6)--(280.8, 208.8)--(280.6, 208.9)--(280.5, 208.7)--(280.5, 208.8)--(280.5, 208.9)--(280.4, 209.0)--(280.4, 208.6)--(280.3, 208.6)--(280.3, 208.5)--(280.4, 208.5)--(280.3, 208.2)--(280.4, 208.2)--(280.3, 208.2)--(280.3, 208.1)--(280.1, 208.2)--(280.1, 208.1)--(279.7, 207.1)--(279.5, 206.7)--(279.3, 205.9)--(279.2, 205.9)--(278.5, 205.9)--(277.9, 205.9)--(277.5, 206.0)--(277.2, 206.0)--(276.8, 206.1)--(276.9, 206.2)--(277.0, 206.1)--(277.0, 206.3)--(276.8, 206.4)--(276.1, 206.5)--(275.8, 206.5)--(275.7, 206.6)--(275.5, 206.6)--(275.4, 206.6)--(275.1, 206.7)--(274.5, 206.9)--(274.5, 207.0)--(274.0, 207.2)--(273.6, 207.4)--(272.9, 207.8)--(272.7, 207.9)--(272.6, 208.0)--(272.4, 208.2)--(272.3, 208.3)--(271.9, 208.6)--(271.8, 208.7)--(271.7, 208.8)--(271.4, 209.0)--(271.0, 209.2)--(270.8, 209.3)--(270.8, 209.2)--(270.7, 209.2)--(270.1, 209.5)--(270.1, 209.6)--(270.0, 209.6)--(269.8, 209.7)--(269.4, 209.9)--(269.3, 209.9)--(268.8, 210.1)--(268.1, 210.3)--(267.8, 210.4)--(267.2, 210.5)--(267.0, 210.6)--(265.8, 210.7)--(265.0, 210.9)--(264.1, 211.8)--(263.6, 212.5)--(263.5, 212.7)--(263.3, 213.4)--(263.1, 214.1)--(263.1, 215.2)--(263.1, 215.8)--(263.1, 216.2)--(263.2, 217.3)--(263.3, 217.8)--(263.3, 218.1)--(263.2, 218.2)--(263.1, 218.2)--(263.1, 218.3)--(263.1, 218.4)--(263.1, 218.5)--(263.1, 218.7)--(263.0, 218.9)--(262.9, 219.1)--(262.7, 219.2)--(262.5, 219.5)--(262.4, 219.7)--(262.1, 220.2)--(261.7, 220.0)--(261.6, 220.0)--(261.1, 220.0)--(261.0, 219.9)--(261.0, 219.0)--(260.4, 219.0)--(260.3, 219.0)--(259.7, 218.7)--(259.8, 218.4)--(259.4, 218.2)--(259.2, 218.4)--(259.0, 218.6)--(259.0, 218.7)--(258.9, 218.8)--(258.9, 219.0)--(258.9, 219.1)--(258.8, 219.3)--(258.7, 219.5)--(258.6, 219.7)--(258.4, 220.0)--(258.4, 220.3)--(258.3, 220.4)--(258.3, 220.5)--(258.4, 220.7)--(258.4, 220.8)--(258.5, 220.8)--(258.5, 220.9)--(258.6, 221.0)--(258.7, 221.2)--(258.9, 221.8)--(258.9, 222.0)--(259.0, 222.2)--(259.1, 222.5)--(259.2, 223.1)--(259.2, 223.5)--(259.3, 224.0)--(259.4, 224.8)--(259.5, 225.1)--(259.5, 225.7)--(259.6, 226.2)--(259.3, 226.2)--(259.2, 226.2)--(259.1, 226.4)--(258.8, 226.5)--(258.6, 226.5)--(258.5, 226.6)--(258.6, 226.6)--(258.6, 226.8)--(258.7, 227.0)--(258.8, 227.0)--(258.8, 227.1)--(259.0, 227.5)--(259.0, 227.6)--(259.1, 227.8)--(259.0, 228.2)--(258.9, 228.6)--(258.9, 228.9)--(258.9, 229.1)--(258.9, 229.2)--(258.9, 229.3)--(258.9, 229.4)--(258.7, 229.3)--(257.2, 228.9)--(257.0, 228.9)--(256.6, 228.8)--(255.9, 228.5)--(255.8, 228.5)--(255.7, 228.6)--(254.5, 228.2)--(254.3, 228.0)--(254.0, 228.0)--(253.1, 227.6)--(250.9, 227.0)--(250.1, 226.7)--(250.0, 226.6)--(249.5, 226.5)--(249.1, 226.4)--(248.9, 226.4)--(248.3, 226.2)--(248.0, 226.1)--(247.4, 226.0)--(247.0, 225.8)--(246.9, 225.8)--(246.6, 225.7)--(246.5, 225.7)--(246.3, 225.7)--(246.0, 225.6)--(245.7, 225.6)--(245.2, 225.4)--(244.9, 225.4)--(244.4, 225.3)--(244.0, 225.1)--(243.7, 225.0)--(243.6, 225.0)--(243.4, 225.0)--(243.2, 225.0)--(243.1, 225.0)--(242.9, 225.0)--(242.5, 225.0)--(242.2, 224.9)--(241.6, 224.9)--(241.3, 224.8)--(241.2, 224.8)--(241.0, 225.0)--(241.0, 225.1)--(240.9, 225.1)--(240.9, 225.2)--(240.8, 225.2)--(240.8, 225.3)--(240.6, 225.5)--(240.3, 225.6)--(240.2, 225.7)--(240.0, 225.7)--(240.0, 225.8)--(239.9, 225.8)--(239.8, 225.8)--(239.7, 225.8)--(239.6, 225.8)--(239.5, 225.8)--(239.4, 225.8)--(239.3, 225.8)--(239.2, 225.8)--(239.1, 225.8)--(239.0, 225.8)--(238.7, 225.7)--(238.4, 225.6)--(237.9, 225.5)--(237.4, 225.6)--(235.9, 225.8)--(235.2, 225.9)--(235.0, 225.6)--(234.8, 225.7)--(234.6, 225.8)--(234.3, 225.8)--(234.1, 225.7)--(233.3, 225.6)--(232.9, 225.5)--(232.0, 225.3)--(231.7, 225.3)--(231.2, 225.1)--(230.9, 225.1)--(230.6, 225.1)--(230.5, 225.0)--(230.3, 225.0)--(230.1, 224.9)--(229.7, 224.7)--(229.0, 224.3)--(227.7, 223.6)--(227.0, 223.3)--(226.5, 223.0)--(226.0, 222.7)--(225.4, 222.4)--(224.9, 222.1)--(224.5, 221.9)--(224.0, 221.7)--(223.9, 221.6)--(223.5, 221.4)--(223.2, 221.3)--(222.8, 221.1)--(222.4, 220.9)--(222.1, 220.8)--(221.8, 220.7)--(221.5, 220.6)--(220.9, 220.3)--(220.2, 220.0)--(219.9, 219.8)--(219.2, 219.5)--(218.7, 219.3)--(218.3, 219.2)--(217.7, 218.9)--(217.3, 218.7)--(217.0, 218.6)--(216.9, 218.5)--(216.8, 218.5)--(216.7, 218.5)--(216.7, 218.8)--(216.7, 219.2)--(216.7, 219.4)--(216.7, 219.5)--(216.7, 219.8)--(216.6, 220.2)--(216.5, 220.3)--(216.4, 220.4)--(216.4, 220.5)--(216.3, 220.6)--(216.2, 220.7)--(216.0, 220.8)--(215.6, 221.1)--(215.5, 221.2)--(215.4, 221.2)--(215.4, 221.3)--(215.3, 221.4)--(215.2, 221.5)--(215.1, 221.6)--(215.1, 221.8)--(215.0, 221.9)--(215.0, 222.1)--(215.0, 222.3)--(215.0, 222.6)--(215.0, 222.9)--(214.9, 223.0)--(214.9, 223.1)--(214.8, 223.3)--(214.7, 223.4)--(214.5, 223.6)--(214.5, 223.7)--(215.1, 224.1)--(215.9, 224.6)--(216.5, 225.0)--(217.9, 226.0)--(218.7, 226.5)--(220.1, 227.5)--(220.7, 227.9)--(221.1, 228.2)--(221.6, 228.5)--(222.0, 228.8)--(222.3, 229.0)--(222.5, 229.3)--(222.8, 229.9)--(223.0, 230.3)--(223.0, 230.4)--(222.7, 230.9)--(222.3, 231.5)--(222.0, 232.1)--(221.1, 233.6)--(220.9, 233.9)--(220.5, 234.5)--(220.2, 235.0)--(219.8, 235.6)--(219.4, 236.3)--(219.1, 236.7)--(218.9, 236.9)--(218.9, 237.0)--(217.5, 238.0)--(215.3, 239.7)--(213.5, 241.0)--(210.5, 243.2)--(210.6, 243.6)--(210.6, 244.0)--(210.6, 244.4)--(210.8, 244.9)--(210.9, 245.4)--(211.0, 245.9)--(211.1, 246.3)--(211.4, 247.0)--(211.7, 247.5)--(211.7, 247.7)--(211.8, 247.9)--(211.8, 248.1)--(211.9, 248.3)--(211.9, 248.4)--(212.0, 248.6)--(212.0, 248.7)--(211.2, 249.2)--(211.2, 249.4)--(211.5, 250.0)--(211.8, 250.7)--(212.1, 251.5)--(212.5, 252.3)--(212.8, 252.8)--(213.9, 254.1)--(214.1, 254.4)--(214.6, 254.7)--(215.3, 254.8)--(215.9, 254.8)--(217.0, 254.6)--(217.3, 254.6)--(217.6, 254.6)--(217.9, 254.6)--(220.0, 254.9)--(220.8, 254.9)--(221.5, 254.9)--(222.5, 254.7)--(226.6, 254.1)--(226.9, 254.1)--(227.1, 254.0)--(227.4, 254.0)--(228.2, 253.7)--(228.2, 254.0)--(228.2, 254.2)--(228.1, 254.7)--(228.0, 255.2)--(228.0, 255.6)--(227.9, 256.0)--(228.0, 256.1)--(228.1, 256.2)--(228.2, 256.3)--(228.5, 256.7)--(228.1, 257.7)--(228.1, 257.8)--(228.1, 258.0)--(228.0, 258.3)--(228.1, 258.6)--(228.2, 259.1)--(228.2, 259.5)--(228.3, 260.1)--(228.3, 260.5)--(228.3, 260.8)--(228.3, 261.3)--(228.3, 261.8)--(228.3, 262.3)--(228.2, 262.7)--(228.2, 262.8)--(228.3, 262.8)--(228.4, 263.1)--(228.4, 263.2)--(228.5, 263.3)--(228.8, 263.5)--(229.1, 263.5)--(229.3, 263.6)--(229.5, 263.6)--(229.6, 263.6)--(229.7, 263.5)--(229.8, 263.5)--(230.0, 263.3)--(230.3, 263.1)--(230.5, 262.9)--(230.6, 262.9)--(230.7, 262.8)--(230.8, 262.8)--(230.9, 262.8)--(231.0, 262.8)--(231.2, 262.8)--(231.3, 262.7)--(231.4, 262.6)--(231.5, 262.6)--(231.6, 262.6)--(231.9, 262.4)--(232.4, 262.1)--(232.6, 262.0)--(232.9, 262.0)--(233.3, 261.8)--(233.4, 261.7)--(234.3, 261.5)--(234.7, 261.4)--(234.8, 261.4)--(234.9, 261.4)--(234.9, 261.5)--(235.0, 261.6)--(235.0, 261.7)--(235.0, 261.8)--(234.9, 261.9)--(234.9, 262.0)--(234.8, 262.1)--cycle; +Westminster = (285.5, 279.0)--(285.4, 279.5)--(285.4, 279.6)--(285.3, 279.7)--(285.2, 279.9)--(284.5, 280.6)--(284.3, 280.8)--(283.0, 282.0)--(282.0, 282.9)--(281.3, 283.5)--(281.8, 284.0)--(281.6, 284.5)--(281.5, 284.7)--(281.4, 285.0)--(281.4, 285.1)--(281.3, 285.6)--(282.0, 286.2)--(281.1, 287.5)--(281.8, 288.3)--(281.1, 289.0)--(281.1, 289.1)--(281.2, 289.2)--(281.2, 289.3)--(281.3, 289.4)--(281.5, 289.6)--(281.4, 289.9)--(281.3, 290.1)--(281.3, 290.2)--(281.0, 292.3)--(280.9, 292.6)--(280.1, 293.5)--(279.7, 294.0)--(279.2, 296.2)--(279.2, 296.4)--(279.1, 296.4)--(279.1, 296.5)--(279.1, 296.6)--(279.1, 296.7)--(279.1, 297.5)--(278.7, 297.4)--(277.7, 297.0)--(277.6, 296.9)--(277.5, 296.8)--(277.4, 296.8)--(277.3, 296.6)--(277.2, 296.5)--(277.0, 296.3)--(276.5, 295.5)--(276.3, 295.3)--(276.2, 295.1)--(274.4, 293.5)--(274.1, 294.1)--(272.9, 293.0)--(272.7, 293.2)--(272.6, 293.6)--(272.5, 293.7)--(272.3, 293.8)--(271.5, 293.6)--(270.2, 293.3)--(270.0, 293.3)--(270.0, 293.2)--(269.5, 293.1)--(269.5, 293.0)--(268.8, 292.9)--(268.8, 292.8)--(264.6, 292.2)--(264.5, 293.2)--(264.0, 296.3)--(263.6, 296.2)--(263.3, 296.2)--(263.1, 296.2)--(262.8, 296.2)--(262.6, 296.3)--(262.4, 296.3)--(262.1, 296.3)--(261.3, 296.6)--(261.1, 296.9)--(260.2, 300.0)--(259.3, 303.2)--(258.5, 305.8)--(258.0, 305.8)--(257.2, 305.7)--(256.4, 305.6)--(255.9, 305.5)--(255.6, 305.4)--(254.9, 307.6)--(254.5, 307.4)--(254.2, 307.5)--(253.9, 308.5)--(253.4, 311.1)--(252.9, 311.0)--(250.5, 310.5)--(250.0, 313.0)--(249.4, 314.3)--(248.9, 314.1)--(248.2, 315.2)--(247.8, 315.9)--(247.6, 316.2)--(247.6, 316.4)--(247.5, 316.4)--(247.5, 316.6)--(247.5, 316.8)--(247.4, 316.8)--(247.3, 317.0)--(247.2, 317.1)--(247.2, 317.2)--(247.4, 317.3)--(247.7, 317.4)--(248.0, 317.5)--(248.3, 317.5)--(248.6, 317.5)--(248.7, 317.5)--(249.0, 317.5)--(249.3, 317.5)--(249.3, 317.3)--(249.4, 317.3)--(249.4, 317.4)--(249.0, 318.2)--(248.8, 318.7)--(248.7, 318.7)--(248.7, 318.8)--(248.7, 319.1)--(248.6, 319.4)--(248.6, 319.6)--(248.0, 319.9)--(247.9, 319.9)--(247.8, 319.9)--(247.7, 320.0)--(247.3, 320.2)--(247.1, 320.3)--(246.9, 320.5)--(246.7, 320.8)--(246.6, 321.0)--(246.2, 321.7)--(246.1, 321.9)--(245.9, 322.4)--(245.7, 322.6)--(245.6, 322.8)--(245.4, 323.1)--(245.2, 323.2)--(245.0, 323.3)--(244.8, 323.3)--(244.6, 323.4)--(244.4, 323.4)--(244.2, 323.4)--(244.1, 323.4)--(244.0, 323.4)--(243.5, 323.6)--(243.3, 323.6)--(243.0, 323.7)--(242.7, 323.8)--(242.5, 323.8)--(241.8, 323.9)--(241.4, 323.9)--(240.8, 324.0)--(240.4, 324.0)--(240.1, 324.0)--(239.5, 323.9)--(239.2, 323.8)--(238.9, 323.8)--(238.8, 324.0)--(238.7, 324.4)--(238.6, 324.5)--(238.6, 324.6)--(238.6, 324.7)--(238.6, 324.8)--(238.7, 325.0)--(238.5, 325.1)--(238.4, 326.2)--(238.4, 326.4)--(238.5, 326.6)--(238.5, 326.9)--(238.6, 327.1)--(238.7, 327.2)--(238.8, 327.2)--(238.9, 327.3)--(239.0, 327.4)--(239.0, 327.5)--(239.1, 327.6)--(239.1, 327.7)--(239.1, 327.8)--(239.1, 327.9)--(239.0, 328.0)--(238.9, 328.5)--(238.9, 328.6)--(238.9, 328.7)--(238.9, 328.8)--(238.9, 328.9)--(239.0, 329.0)--(239.3, 329.1)--(239.7, 329.2)--(239.8, 329.2)--(239.9, 329.3)--(240.1, 329.3)--(240.3, 329.4)--(240.5, 329.4)--(240.6, 329.4)--(240.7, 329.5)--(241.0, 329.5)--(241.4, 329.5)--(241.6, 329.6)--(241.8, 329.7)--(242.0, 329.7)--(242.4, 330.0)--(242.5, 330.0)--(242.8, 330.2)--(243.0, 330.3)--(243.3, 330.4)--(243.5, 330.5)--(244.0, 330.6)--(244.2, 330.7)--(244.4, 330.7)--(244.6, 330.7)--(244.8, 330.7)--(244.9, 330.7)--(245.1, 330.8)--(245.4, 330.8)--(245.6, 330.9)--(245.8, 330.9)--(246.3, 330.9)--(246.7, 330.8)--(247.0, 330.8)--(247.2, 330.8)--(247.4, 330.8)--(247.4, 330.9)--(247.5, 330.8)--(247.7, 330.7)--(247.9, 330.6)--(248.1, 330.5)--(248.2, 330.5)--(248.3, 330.5)--(248.4, 330.3)--(248.5, 330.2)--(248.5, 330.1)--(248.5, 330.0)--(248.6, 329.9)--(248.7, 329.8)--(248.8, 329.7)--(248.9, 329.7)--(249.1, 329.6)--(249.2, 329.5)--(249.6, 329.2)--(249.8, 329.1)--(249.9, 329.0)--(250.0, 328.9)--(250.1, 328.8)--(250.2, 328.7)--(250.3, 328.6)--(250.4, 328.6)--(250.5, 328.5)--(250.5, 328.4)--(250.7, 328.2)--(250.8, 328.1)--(251.0, 327.8)--(251.0, 327.6)--(251.0, 327.3)--(251.0, 326.8)--(251.0, 326.7)--(251.0, 326.6)--(251.0, 326.4)--(251.0, 326.3)--(250.9, 326.2)--(251.0, 325.9)--(251.1, 325.4)--(251.4, 325.3)--(251.5, 325.4)--(251.6, 325.4)--(251.7, 325.4)--(251.8, 325.4)--(251.8, 325.3)--(251.8, 325.2)--(252.0, 325.1)--(252.0, 325.2)--(252.1, 325.4)--(252.6, 326.4)--(253.5, 328.3)--(254.0, 329.3)--(254.3, 329.8)--(254.4, 330.2)--(254.6, 330.5)--(254.7, 330.7)--(254.8, 331.0)--(254.8, 331.2)--(254.8, 331.4)--(254.7, 331.5)--(254.5, 331.9)--(254.3, 332.2)--(254.3, 332.3)--(255.3, 334.8)--(255.4, 334.6)--(255.6, 334.4)--(255.8, 334.2)--(256.1, 334.0)--(256.5, 333.7)--(256.6, 333.6)--(256.7, 333.6)--(256.8, 333.5)--(256.8, 333.4)--(256.9, 333.3)--(257.0, 333.2)--(257.2, 332.9)--(257.4, 333.0)--(257.8, 333.3)--(257.8, 333.4)--(257.9, 333.4)--(257.9, 333.5)--(259.4, 335.9)--(259.5, 336.0)--(260.0, 336.7)--(260.0, 336.8)--(260.1, 336.9)--(260.3, 337.1)--(260.4, 337.1)--(260.5, 337.3)--(261.3, 337.7)--(262.1, 338.2)--(262.9, 338.6)--(263.0, 338.6)--(263.1, 338.7)--(263.3, 338.8)--(263.6, 338.8)--(263.8, 338.9)--(264.0, 338.9)--(264.2, 338.9)--(264.3, 338.9)--(266.1, 338.8)--(267.4, 338.7)--(267.3, 337.3)--(267.3, 337.2)--(267.3, 337.1)--(267.4, 337.1)--(267.7, 336.6)--(268.8, 337.5)--(269.5, 336.8)--(270.5, 337.8)--(270.5, 337.7)--(270.6, 337.7)--(270.8, 337.6)--(271.2, 337.2)--(271.4, 337.1)--(271.5, 337.0)--(272.6, 335.9)--(273.7, 334.7)--(274.5, 335.2)--(274.6, 335.3)--(274.8, 335.4)--(275.8, 336.0)--(276.8, 336.5)--(277.5, 335.5)--(278.1, 335.6)--(278.8, 335.9)--(280.0, 336.3)--(280.2, 336.5)--(280.4, 336.5)--(280.6, 336.6)--(280.7, 336.6)--(281.0, 336.7)--(281.1, 336.7)--(281.3, 336.7)--(281.7, 336.8)--(282.1, 336.9)--(283.1, 333.7)--(283.5, 332.2)--(284.3, 329.9)--(286.3, 323.1)--(287.3, 323.4)--(287.7, 321.9)--(288.0, 322.0)--(288.1, 322.0)--(288.5, 322.0)--(288.4, 322.7)--(288.6, 322.7)--(288.9, 322.7)--(289.0, 322.6)--(288.9, 322.0)--(288.9, 321.9)--(289.1, 321.9)--(289.2, 321.8)--(289.3, 321.9)--(290.4, 320.5)--(291.1, 319.7)--(291.6, 319.1)--(292.0, 318.5)--(292.3, 318.1)--(292.6, 317.8)--(292.7, 317.7)--(292.7, 317.6)--(292.8, 317.4)--(293.6, 316.4)--(294.0, 316.7)--(294.4, 316.2)--(294.8, 316.5)--(295.4, 315.5)--(295.4, 315.4)--(295.6, 315.0)--(295.7, 315.0)--(295.8, 315.0)--(296.0, 315.2)--(296.4, 314.8)--(296.7, 314.6)--(296.8, 314.1)--(297.4, 314.3)--(297.7, 314.4)--(297.7, 314.5)--(297.9, 314.5)--(298.2, 314.0)--(298.2, 313.9)--(298.2, 313.8)--(298.2, 313.5)--(298.2, 313.4)--(298.2, 313.2)--(298.7, 311.6)--(298.8, 311.2)--(299.0, 310.5)--(299.2, 310.5)--(299.3, 310.5)--(299.4, 310.4)--(299.7, 310.2)--(299.8, 310.1)--(300.0, 309.9)--(300.2, 309.7)--(300.3, 309.6)--(300.4, 309.7)--(300.5, 309.7)--(300.6, 309.7)--(300.7, 309.9)--(300.7, 310.0)--(300.8, 310.1)--(300.9, 310.2)--(301.2, 310.5)--(301.9, 310.9)--(302.4, 311.4)--(302.7, 311.7)--(303.0, 312.0)--(303.1, 312.1)--(303.5, 312.5)--(304.0, 311.9)--(304.2, 312.2)--(304.6, 312.1)--(305.3, 311.9)--(305.8, 311.8)--(306.0, 311.9)--(306.3, 312.0)--(306.4, 312.1)--(306.4, 312.2)--(306.4, 312.4)--(306.6, 312.4)--(307.1, 312.7)--(307.4, 312.9)--(309.1, 313.6)--(309.7, 312.3)--(310.8, 312.7)--(310.8, 312.8)--(311.2, 312.9)--(311.3, 312.9)--(311.3, 312.8)--(311.4, 312.3)--(311.7, 311.4)--(311.7, 311.2)--(311.5, 311.2)--(311.1, 311.1)--(311.2, 310.8)--(311.3, 310.5)--(310.9, 310.3)--(310.9, 310.2)--(310.8, 310.1)--(310.9, 310.1)--(310.8, 310.0)--(311.0, 309.8)--(311.1, 309.6)--(311.0, 309.5)--(311.2, 309.2)--(311.3, 309.2)--(311.4, 308.9)--(311.5, 308.8)--(311.4, 308.6)--(311.4, 308.4)--(311.4, 308.0)--(311.5, 307.8)--(311.4, 307.8)--(311.4, 307.9)--(311.2, 307.9)--(311.2, 307.8)--(311.0, 307.8)--(310.8, 307.7)--(310.7, 307.8)--(310.4, 307.8)--(310.2, 307.7)--(309.6, 307.6)--(309.0, 307.4)--(308.9, 307.4)--(308.8, 307.4)--(308.5, 307.3)--(308.3, 307.2)--(308.2, 307.1)--(307.8, 306.9)--(307.8, 306.8)--(307.2, 306.5)--(307.2, 306.6)--(307.1, 306.5)--(306.9, 306.4)--(306.8, 306.3)--(306.8, 306.4)--(306.7, 306.3)--(306.4, 306.1)--(306.1, 305.9)--(305.7, 305.5)--(305.5, 305.3)--(305.6, 305.2)--(305.4, 304.9)--(305.3, 305.0)--(305.1, 304.6)--(305.0, 304.4)--(304.7, 303.9)--(304.8, 303.9)--(304.8, 303.8)--(304.7, 303.7)--(304.6, 303.4)--(304.6, 303.3)--(304.6, 303.2)--(304.5, 303.0)--(304.4, 302.8)--(304.3, 302.8)--(304.2, 302.3)--(304.1, 302.4)--(304.0, 302.1)--(304.0, 301.8)--(303.9, 301.5)--(303.8, 301.0)--(303.7, 300.5)--(303.7, 300.1)--(303.7, 300.0)--(303.5, 298.4)--(303.4, 297.7)--(303.3, 297.7)--(303.3, 297.0)--(303.4, 296.9)--(303.4, 296.6)--(303.4, 296.4)--(303.3, 296.3)--(303.3, 296.2)--(303.4, 296.2)--(303.1, 293.5)--(302.9, 291.3)--(302.9, 291.2)--(302.9, 291.0)--(302.9, 290.8)--(302.8, 289.8)--(302.8, 289.7)--(302.7, 288.6)--(302.7, 287.9)--(302.8, 287.8)--(302.7, 287.5)--(302.7, 287.4)--(302.5, 286.5)--(302.4, 286.3)--(302.4, 286.2)--(302.3, 286.0)--(302.1, 285.7)--(301.7, 284.8)--(301.6, 284.6)--(301.5, 284.3)--(301.4, 284.2)--(301.4, 284.1)--(301.3, 284.1)--(301.2, 284.0)--(301.1, 283.9)--(301.0, 283.7)--(300.9, 283.4)--(300.4, 282.7)--(300.4, 282.5)--(300.3, 282.3)--(300.0, 282.0)--(298.5, 280.3)--(298.4, 280.3)--(298.3, 280.2)--(298.2, 280.1)--(298.0, 279.9)--(297.9, 279.8)--(297.8, 279.7)--(297.7, 279.7)--(297.5, 279.6)--(297.0, 279.3)--(296.7, 279.2)--(296.6, 279.2)--(296.4, 279.1)--(296.3, 279.1)--(296.0, 279.0)--(295.9, 279.1)--(295.2, 278.9)--(295.2, 278.8)--(294.8, 278.7)--(294.6, 278.6)--(294.4, 278.6)--(293.8, 278.5)--(293.5, 278.5)--(293.2, 278.5)--(293.1, 278.5)--(292.5, 278.5)--(292.2, 278.5)--(292.1, 278.5)--(291.8, 278.6)--(290.9, 278.8)--(290.5, 278.9)--(290.0, 279.1)--(289.9, 279.1)--(289.7, 279.1)--(289.5, 279.2)--(288.6, 279.3)--(288.2, 279.3)--(287.7, 279.3)--(287.0, 279.2)--(286.7, 279.2)--(286.2, 279.1)--(286.2, 279.3)--(286.2, 279.4)--(286.2, 279.5)--(286.2, 279.8)--(286.1, 279.8)--(286.1, 279.9)--(286.1, 279.8)--(286.0, 279.8)--(285.9, 279.9)--(285.8, 279.8)--(285.7, 279.7)--(285.8, 279.6)--(286.0, 279.5)--(286.0, 279.4)--(286.0, 279.3)--(286.0, 279.2)--(286.0, 279.1)--cycle; diff --git a/info/drawing-with-metapost/src/magic-square-14.mp b/info/drawing-with-metapost/src/magic-square-14.mp new file mode 100644 index 0000000000..09be4d36cd --- /dev/null +++ b/info/drawing-with-metapost/src/magic-square-14.mp @@ -0,0 +1,40 @@ +\documentclass[border=2mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +beginfig(1); +z1 = (10,10); +z4 = 144 right rotated 12; +z5 = z4 shifted (2, 78); +z7 = z4 reflectedabout(origin, (1,1)); + +z2 = 1/2 [z5, z7]; +z9 = whatever [z1, z4]; +z2-z9 = whatever * (z7-z1); +z8 = whatever [z1, z5] = whatever [z2, z4]; +z3 = whatever [z2, z9] = whatever [z4, z7]; +z6 = whatever [z1, z7] = whatever [z3, z5]; + +drawoptions(dashed withdots scaled 1/4 withcolor Blues 7 7); +draw z1 -- z4 -- z7 -- z1 -- z5 -- z6; +draw z4 -- z2 -- z9; +draw z5 -- z7; +drawoptions(); + +for i=1 upto 9: + fill fullcircle scaled 12 shifted z[i] withcolor Blues 7 2; + draw fullcircle scaled 12 shifted z[i] withcolor Blues 7 7; + label("\textsf{" & decimal i & "}", z[i]); +endfor + +for i=1 upto 2: + draw (50i, 0) -- (50i, 150) dashed withdots scaled 1/4 withcolor Reds 7 7; + draw (0, 50i) -- (150, 50i) dashed withdots scaled 1/4 withcolor Reds 7 7; +endfor + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/magic-square-14.pdf b/info/drawing-with-metapost/src/magic-square-14.pdf Binary files differnew file mode 100644 index 0000000000..a3c04cebae --- /dev/null +++ b/info/drawing-with-metapost/src/magic-square-14.pdf diff --git a/info/drawing-with-metapost/src/marked-up-photo-with-grid.pdf b/info/drawing-with-metapost/src/marked-up-photo-with-grid.pdf Binary files differnew file mode 100644 index 0000000000..ffd5a71b83 --- /dev/null +++ b/info/drawing-with-metapost/src/marked-up-photo-with-grid.pdf diff --git a/info/drawing-with-metapost/src/marked-up-photo.mp b/info/drawing-with-metapost/src/marked-up-photo.mp new file mode 100644 index 0000000000..c208e79708 --- /dev/null +++ b/info/drawing-with-metapost/src/marked-up-photo.mp @@ -0,0 +1,32 @@ +\documentclass[border=1mm]{standalone} +\usepackage{luamplib} +\usepackage{graphicx} +\usepackage{fontspec}\setmainfont[Scale=0.6]{Helvetica} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +draw btex \includegraphics[width=5in]{glenshiel.jpg} etex; +if false: +for i=1 upto 36: + draw (origin -- 180 up) shifted (10i, 0) withcolor if i mod 10 = 0: red else: 1/2 fi; +endfor +for i=1 upto 18: + draw (origin -- 360 right) shifted (0, 10i) withcolor if i mod 10 = 0: red else: 1/2 fi; +endfor +fi +vardef callout@#(expr t, p, o) = + save T; picture T; T = thelabel.@#(t, p+o); + draw T; drawarrow p+o -- p cutbefore bbox T; +enddef; +ahangle := 20; ahlength := 2; drawoptions(withpen pencircle scaled 1/4 withcolor 1/2 blue); +callout.top("Sgurr na Ciste Dubhe", (80, 96), (-10, 20)); +callout.top("Sgurr nan Spainteach", (100, 91), (6, 12)); +label.top("\tiny Cuillin Ridge, Isle of Skye", (140, 81)); +label.top("Sgurr na Carnach", (190, 90)); +label.top("Sgurr Fhuaran", (282, 94)); +label.bot("\itshape View looking west from the summit of Saileag – 19 April 2005", point 5/2 of bbox currentpicture shifted 4 down); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/marked-up-photo.pdf b/info/drawing-with-metapost/src/marked-up-photo.pdf Binary files differnew file mode 100644 index 0000000000..fc3155d855 --- /dev/null +++ b/info/drawing-with-metapost/src/marked-up-photo.pdf diff --git a/info/drawing-with-metapost/src/mediation-pitfall.mp b/info/drawing-with-metapost/src/mediation-pitfall.mp new file mode 100644 index 0000000000..84c3839f64 --- /dev/null +++ b/info/drawing-with-metapost/src/mediation-pitfall.mp @@ -0,0 +1,27 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +pair a,b; +a = 42 up; b = 72 right; +beginfig(1); + +draw (-1/4)[origin,b]--5/4[origin,b] withcolor .8 white; +draw (-1/4)[origin,a]--5/4[origin,a] withcolor .8 white; + +draw (-1)[a,b]--a dashed evenly; +draw a--b withcolor .67 red; +draw b--2[a,b] dashed evenly; + +dotlabel.urt(btex $a$ etex, a); +dotlabel.urt(btex $b$ etex, b); +dotlabel.urt(btex ${1\over2}[a,b]$ etex, 1/2[a,b]); +dotlabel.urt(btex $\left(-{1\over2}\right)[a,b]$ etex, (-1/2)[a,b]); +dotlabel.llft(btex $-{1\over2}[a,b]$ etex, -1/2[a,b]); +label(btex Probably not what was intended\dots etex, -1/2[a,b] shifted (0,-24)) withcolor .58 blue; +dotlabel.urt(btex ${3\over2}[a,b]$ etex, (3/2)[a,b]); +dotlabel.lrt(btex $(0,0)$ etex, origin); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/mediation-pitfall.pdf b/info/drawing-with-metapost/src/mediation-pitfall.pdf Binary files differnew file mode 100644 index 0000000000..d56116c5b3 --- /dev/null +++ b/info/drawing-with-metapost/src/mediation-pitfall.pdf diff --git a/info/drawing-with-metapost/src/mediation-sallows.mp b/info/drawing-with-metapost/src/mediation-sallows.mp new file mode 100644 index 0000000000..6814522e77 --- /dev/null +++ b/info/drawing-with-metapost/src/mediation-sallows.mp @@ -0,0 +1,44 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(3); +path t; t = (0,0) -- (220,42) -- (180,-90) -- cycle; +pair centroid; +centroid = 2/3[point 0 of t,point 3/2 of t]; + +path s[]; +for i=0 upto 5: + s[i] = point 1/2 i of t -- point 1/2 + 1/2 i of t -- centroid -- cycle; +endfor + +color c[]; +c[0] = (1,0.96863,0.73725); +c[1] = (0.99608,0.76863,0.3098); +c[2] = (0.85098,0.37255,0.054902); + +picture a,b; +a = image( +for i=0 upto 2: + fill s[2i] withcolor .6[c[i],white]; + fill s[2i+1] withcolor .4[c[i],white]; +endfor +for i=0 upto 2: +draw point i of t -- point 3/2+i of t; +endfor +draw t;); +b = image( +for i=0 upto 2: + fill s[2i] withcolor .6[c[i],white]; + fill s[2i+1] rotatedabout(point i+1/2 of t, 180) withcolor .4[c[i],white]; +endfor +for i=0 upto 2: + draw point i of t -- centroid -- centroid rotatedabout(point i+1/2 of t,180) -- cycle; +endfor +draw t;); +draw a; +draw b shifted 222 right; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/mediation-sallows.pdf b/info/drawing-with-metapost/src/mediation-sallows.pdf Binary files differnew file mode 100644 index 0000000000..de8d8d3a74 --- /dev/null +++ b/info/drawing-with-metapost/src/mediation-sallows.pdf diff --git a/info/drawing-with-metapost/src/mediation-shapes.mp b/info/drawing-with-metapost/src/mediation-shapes.mp new file mode 100644 index 0000000000..77e2825dfa --- /dev/null +++ b/info/drawing-with-metapost/src/mediation-shapes.mp @@ -0,0 +1,29 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{dwmpcode} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +pair a,b; +a = 42 up; b = 72 right; +path t,s,p,c; +t = origin--a--b--cycle; +s = unitsquare scaled 40 shifted 92 right; +p = (for i=0 upto 4: down scaled 21 rotated 72i -- endfor cycle) + shifted 180 right shifted 21 up; +c = fullcircle scaled 42 shifted 250 right shifted 21 up; + +beginfig(2); +forsuffixes $=t,s,p,c: + draw $; + draw point 0 of $ -- point 1/2 length $ of $ + dashed withdots scaled 1/3 + withcolor .67 red; +endfor +label.bot("\textsl{Dotted lines drawn with}: \mpl{point 0 of p -- point 1/2 length p of p}", + point 1/2 of bbox currentpicture); + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/mediation-shapes.pdf b/info/drawing-with-metapost/src/mediation-shapes.pdf Binary files differnew file mode 100644 index 0000000000..4b40c2506a --- /dev/null +++ b/info/drawing-with-metapost/src/mediation-shapes.pdf diff --git a/info/drawing-with-metapost/src/minus.mp b/info/drawing-with-metapost/src/minus.mp new file mode 100644 index 0000000000..ebf8f8cb04 --- /dev/null +++ b/info/drawing-with-metapost/src/minus.mp @@ -0,0 +1,19 @@ +\documentclass[border=2mm]{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); + + u = 8mm; + defaultfont := "texnansi-lmr10"; + string minus_sign; + minus_sign := char 12; % if you are using the 8r encoding + minus_sign := char 143; % if you are using the texnansi encoding + for x=-3 upto 3: label(decimal x, (x*u,+9)) withcolor .67 red; endfor + for x=-3 upto 3: label(if x<0: minus_sign & fi decimal abs(x), (x*u,-5)) withcolor .58 blue; endfor + label.lft("with plain decimal:" infont "texnansi-lmss10" scaled 0.9, (-3.6u,9)) withcolor .67 red; + label.lft("with this hack:" infont "texnansi-lmss10" scaled 0.9, (-3.6u,-5)) withcolor .58 blue; + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/minus.pdf b/info/drawing-with-metapost/src/minus.pdf Binary files differnew file mode 100644 index 0000000000..05d57963b4 --- /dev/null +++ b/info/drawing-with-metapost/src/minus.pdf diff --git a/info/drawing-with-metapost/src/multi-line-labels.mp b/info/drawing-with-metapost/src/multi-line-labels.mp new file mode 100644 index 0000000000..497189ed16 --- /dev/null +++ b/info/drawing-with-metapost/src/multi-line-labels.mp @@ -0,0 +1,23 @@ +\documentclass[border=0mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +label("\vbox{\hsize 4in It is a truth universally acknowledged," + & " that a single man in possession of a good fortune," + & " must be in want of a wife.\par}", origin); +label(btex \vbox{\hsize 4in + It is a truth universally acknowledged, + that a single man in possession of a good fortune, + must be in want of a wife. + \par} etex, 128 down); +label(btex + \begin{tabular}{c} + A way to get simple\\ + two line labels + \end{tabular} etex, 256 down); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/multi-line-labels.pdf b/info/drawing-with-metapost/src/multi-line-labels.pdf Binary files differnew file mode 100644 index 0000000000..ecc0cb7611 --- /dev/null +++ b/info/drawing-with-metapost/src/multi-line-labels.pdf diff --git a/info/drawing-with-metapost/src/neo-labels-tte.mp b/info/drawing-with-metapost/src/neo-labels-tte.mp new file mode 100644 index 0000000000..1838e838f5 --- /dev/null +++ b/info/drawing-with-metapost/src/neo-labels-tte.mp @@ -0,0 +1,28 @@ +\documentclass[border=0mm]{standalone} +\usepackage{fontspec} +\setmainfont{TeX Gyre Pagella} +\usepackage{luamplib} +\mplibtextextlabel{enable} % <-- added option +\begin{document} +\begin{mplibcode} + beginfig(1); + for x = 0 upto 1: + draw (80x,16) -- (80x, -68) withcolor 3/4[red, white]; + endfor + for y = 0 upto 3: + draw (0, -20y) -- (160, -20y) withcolor 3/4[red, white]; + endfor + + string s; s = "Hand gloves"; + draw s infont defaultfont shifted (0, 0); + draw s infont "phvr8r" shifted (0, -20); + draw TEX(s) shifted (0, -40); + draw btex Hand gloves etex shifted (0, -60); + + dotlabel.urt(s, (80, 0)); + dotlabel.urt(s infont "phvr8r", (80, -20)); + dotlabel.urt(TEX(s), (80, -40)); + dotlabel.urt(btex Hand gloves etex, (80, -60)); + endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/neo-labels-tte.pdf b/info/drawing-with-metapost/src/neo-labels-tte.pdf Binary files differnew file mode 100644 index 0000000000..5ebad8d274 --- /dev/null +++ b/info/drawing-with-metapost/src/neo-labels-tte.pdf diff --git a/info/drawing-with-metapost/src/neo-labels.mp b/info/drawing-with-metapost/src/neo-labels.mp new file mode 100644 index 0000000000..8b1d60b840 --- /dev/null +++ b/info/drawing-with-metapost/src/neo-labels.mp @@ -0,0 +1,27 @@ +\documentclass[border=5mm]{standalone} +\usepackage{fontspec} +\setmainfont{TeX Gyre Pagella} % <-- note chosen font +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} + beginfig(1); + for x = 0 upto 1: + draw (80x,16) -- (80x, -68) withcolor 3/4[red, white]; + endfor + for y = 0 upto 3: + draw (0, -20y) -- (160, -20y) withcolor 3/4[red, white]; + endfor + + string s; s = "Hand gloves"; + draw s infont defaultfont shifted (0, 0); + draw s infont "phvr8r" shifted (0, -20); + draw TEX(s) shifted (0, -40); + draw btex Hand gloves etex shifted (0, -60); + + dotlabel.urt(s, (80, 0)); + dotlabel.urt(s infont "phvr8r", (80, -20)); + dotlabel.urt(TEX(s), (80, -40)); + dotlabel.urt(btex Hand gloves etex, (80, -60)); + endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/neo-labels.pdf b/info/drawing-with-metapost/src/neo-labels.pdf Binary files differnew file mode 100644 index 0000000000..e5a85a74cc --- /dev/null +++ b/info/drawing-with-metapost/src/neo-labels.pdf diff --git a/info/drawing-with-metapost/src/nine-point-circle.mp b/info/drawing-with-metapost/src/nine-point-circle.mp new file mode 100644 index 0000000000..8366bb3c1b --- /dev/null +++ b/info/drawing-with-metapost/src/nine-point-circle.mp @@ -0,0 +1,54 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + pair A, B, C, D, N, M, p, q, r; + A = origin; B = 377 dir 10; C = 233 dir 70; + % pedal points + p = whatever[B, C]; A - p = whatever * (B-C) rotated 90; + q = whatever[C, A]; B - q = whatever * (C-A) rotated 90; + r = whatever[A, B]; C - r = whatever * (A-B) rotated 90; + + D = whatever[A, p] = whatever[B, q]; + N = 1/4(A + B + C + D); + M = D rotatedabout(N, 180); % M is also the circumcentre + + path circumcircle, nine_point_circle; + nine_point_circle = fullcircle scaled 2 abs(N - 1/2[A, B]) shifted N; + circumcircle = fullcircle scaled 2 abs(M - A) shifted M; + + draw nine_point_circle withcolor 3/4 red; + draw circumcircle withcolor 1/2[3/4 blue, white]; + + drawoptions(dashed evenly scaled 1/4 withcolor 1/2); + draw 1/2[A,B] -- M -- 1/2[B, C]; + draw 1/2[C,A] -- M -- D; + draw A -- p; + draw B -- q; + draw C -- r; + + drawoptions(withpen pencircle scaled 1/4); + draw fullcircle scaled 2 shifted 1/2[A, B]; + draw fullcircle scaled 2 shifted 1/2[A, C]; + draw fullcircle scaled 2 shifted 1/2[A, D]; + draw fullcircle scaled 2 shifted 1/2[B, C]; + draw fullcircle scaled 2 shifted 1/2[B, D]; + draw fullcircle scaled 2 shifted 1/2[C, D]; + draw fullcircle scaled 2 shifted p; + draw fullcircle scaled 2 shifted q; + draw fullcircle scaled 2 shifted r; + + drawoptions(); + draw A--B--C--cycle; + dotlabel.llft("$A$", A); + dotlabel.rt("$B$", B); + dotlabel.ulft("$C$", C); + dotlabel.urt("\ $D$", D); + dotlabel.llft("$M$", M); + dotlabel.llft("$N$", N); + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/nine-point-circle.pdf b/info/drawing-with-metapost/src/nine-point-circle.pdf Binary files differnew file mode 100644 index 0000000000..8436ae129b --- /dev/null +++ b/info/drawing-with-metapost/src/nine-point-circle.pdf diff --git a/info/drawing-with-metapost/src/numbered-axes.mp b/info/drawing-with-metapost/src/numbered-axes.mp new file mode 100644 index 0000000000..47ee8f4ec9 --- /dev/null +++ b/info/drawing-with-metapost/src/numbered-axes.mp @@ -0,0 +1,27 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{dwmpcode} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + numeric u, v; u = 40; v = 29; + path xx, yy; + xx = (3/2 left -- 5 right) scaled u; + yy = (3/2 down -- 4 up) scaled v; + for x=-1, 1, 2, 3, 4: + draw yy shifted (x * u, 0) withcolor 7/8; % grid + draw (down--up) shifted (x * u, 0); % ticks + label("$" & decimal x & "$", (x * u, -8)); + endfor + for y=-1, 1, 2, 3: + draw xx shifted (0, y * v) withcolor 7/8; % grid + draw (left--right) shifted (0, y * v); % ticks + label("$" & decimal y & "$", (-10, y * v)); + endfor + drawarrow xx; label.rt("$x$", point 1 of xx); + drawarrow yy; label.top("$y$", point 1 of yy); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/numbered-axes.pdf b/info/drawing-with-metapost/src/numbered-axes.pdf Binary files differnew file mode 100644 index 0000000000..d8e68dab34 --- /dev/null +++ b/info/drawing-with-metapost/src/numbered-axes.pdf diff --git a/info/drawing-with-metapost/src/open-triangle.mp b/info/drawing-with-metapost/src/open-triangle.mp new file mode 100644 index 0000000000..5535611558 --- /dev/null +++ b/info/drawing-with-metapost/src/open-triangle.mp @@ -0,0 +1,22 @@ +\documentclass{standalone} +\usepackage{dwmpcode} +\usepackage{luamplib} +\mplibtextextlabel{enable} +\begin{document} +\begin{mplibcode} +beginfig(1); +ahangle := 30; +path xx, yy; xx = left scaled 8 -- right scaled 89; yy = down scaled 8 -- up scaled 55; +path u; u = origin {right} .. (55,0) .. (55,34) .. {-2, -1} origin; +drawarrow xx withcolor .67 white; +drawarrow yy withcolor .67 white; +drawarrow u cutafter fullcircle scaled 4; +dotlabeldiam := 2; +dotlabel.urt("$\scriptstyle (55,34)$", point 2 of u); +dotlabel.lrt("$\scriptstyle (55,0)$", point 1 of u); +dotlabel.llft("$\scriptstyle (0,0)$", point 0 of u); +label.bot(TEX("\mpl{drawarrow u cutafter fullcircle scaled 4};") scaled 0.8, +point 1/2 of bbox currentpicture shifted 13 down); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/open-triangle.pdf b/info/drawing-with-metapost/src/open-triangle.pdf Binary files differnew file mode 100644 index 0000000000..16986eacca --- /dev/null +++ b/info/drawing-with-metapost/src/open-triangle.pdf diff --git a/info/drawing-with-metapost/src/overlaps-default-fillers.mp b/info/drawing-with-metapost/src/overlaps-default-fillers.mp new file mode 100644 index 0000000000..5137cdf984 --- /dev/null +++ b/info/drawing-with-metapost/src/overlaps-default-fillers.mp @@ -0,0 +1,42 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{dwmpcode} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(3); + path A, B; picture p[]; + A = fullcircle scaled 2.5cm; + B = fullcircle scaled 1.8cm shifted (1cm,0); + p1 = image(fill buildcycle(A,B) withcolor .8[blue,white]; drawarrow A; drawarrow B;); + A := A rotated 180; + p2 = image(fill buildcycle(A,B) withcolor .8[blue,white]; drawarrow A; drawarrow B;); + B := B rotatedabout(center B,180); + p3 = image(fill buildcycle(A,B) withcolor .8[blue,white]; drawarrow A; drawarrow B;); + A := A rotated 180; + p4 = image(fill buildcycle(A,B) withcolor .8[blue,white]; drawarrow A; drawarrow B;); + B := B rotatedabout(center B, 180); + A := reverse A; + p5 = image(fill buildcycle(A,B) withcolor .8[blue,white]; drawarrow A; drawarrow B;); + A := A rotated 180; + p6 = image(fill buildcycle(A,B) withcolor .8[blue,white]; drawarrow A; drawarrow B;); + B := B rotatedabout(center B,180); + p7 = image(fill buildcycle(A,B) withcolor .8[blue,white]; drawarrow A; drawarrow B;); + A := A rotated 180; + p8 = image(fill buildcycle(A,B) withcolor .8[blue,white]; drawarrow A; drawarrow B;); + + for i=1 upto 4: draw p[i] scaled .75 shifted (80i, 0); label(decimal i, (5mm+80i,0)); endfor + for i=1 upto 4: draw p[i+4] scaled .75 shifted (80i,-68); label(decimal (4+i), (5mm+80i,-68)); endfor + + picture L; + L = image( + for i=0 upto 3: fill unitsquare scaled 2 shifted (1/2,1/2) rotated (45+90i); endfor + label.rt("\small To rotate a circular path, you can use: \mpl{p rotatedaround(center p, 180)}", + 3 right); + ); + label(L, point 1/2 of bbox currentpicture shifted 21 down); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/overlaps-default-fillers.pdf b/info/drawing-with-metapost/src/overlaps-default-fillers.pdf Binary files differnew file mode 100644 index 0000000000..51a3e0ef52 --- /dev/null +++ b/info/drawing-with-metapost/src/overlaps-default-fillers.pdf diff --git a/info/drawing-with-metapost/src/overlaps-missing-filler.mp b/info/drawing-with-metapost/src/overlaps-missing-filler.mp new file mode 100644 index 0000000000..edee192c1c --- /dev/null +++ b/info/drawing-with-metapost/src/overlaps-missing-filler.mp @@ -0,0 +1,16 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(2); + path A, B; + A = fullcircle scaled 2.5cm; + B = fullcircle scaled 1.8cm shifted (1cm,0); + fill buildcycle(A,B) withcolor .8[blue,white]; drawarrow A; drawarrow B; + label.rt(btex Where has the fill colour gone? etex, point 0 of B shifted (1cm,0)); + label.ulft(btex $A$ etex, point 3 of A); + label.urt(btex $B$ etex, point 1 of B); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/overlaps-missing-filler.pdf b/info/drawing-with-metapost/src/overlaps-missing-filler.pdf Binary files differnew file mode 100644 index 0000000000..4bf91ed568 --- /dev/null +++ b/info/drawing-with-metapost/src/overlaps-missing-filler.pdf diff --git a/info/drawing-with-metapost/src/overlaps.mp b/info/drawing-with-metapost/src/overlaps.mp new file mode 100644 index 0000000000..d08b934249 --- /dev/null +++ b/info/drawing-with-metapost/src/overlaps.mp @@ -0,0 +1,97 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +% is point "p" inside cyclic path "ring" ? +vardef inside(expr p, ring) = + save t, count, test_line; + count := 0; + path test_line; + test_line = p -- (infinity, ypart p); + for i = 1 upto length ring: + t := xpart (subpath(i-1,i) of ring intersectiontimes test_line); + if ((0 <= t) and (t<1)): count := count + 1; fi + endfor + odd(count) + enddef; + +vardef front_half primary p = subpath(0, 1/2 length p) of p enddef; +vardef back_half primary p = subpath(1/2 length p, length p) of p enddef; + +% a and b should be cyclic paths... +vardef xoverlap(expr a, b) = + boolean p, q; + p = inside(point 0 of a, b); + q = inside(point 0 of b, a); + if ((not p) and (not q)): + buildcycle(a,b) + elseif (not p): + buildcycle(front_half b, a, back_half b) + elseif (not q): + buildcycle(front_half a, b, back_half a) + else: + buildcycle(front_half a, back_half b, front_half b, back_half a) + fi + enddef; + +vardef xcombine(expr a, b) = + boolean p, q; + p = inside(point 0 of a, b); + q = inside(point 0 of b, a); + if (p and q): + buildcycle(a,b) + elseif p: + buildcycle(front_half b, a, back_half b) + elseif q: + buildcycle(front_half a, b, back_half a) + else: + buildcycle(front_half a, back_half b, front_half b, back_half a) + fi + enddef; + +vardef overlap(expr a, b) = + save p, q, A, B; + boolean p, q; + p = not inside(point 0 of a, b); + q = not inside(point 0 of b, a); + path A, B; + A = counterclockwise a; + B = counterclockwise b; + if (p and q): + buildcycle(A,B) + elseif p: + buildcycle(front_half B, A, back_half B) + elseif q: + buildcycle(front_half A, B, back_half A) + else: + buildcycle(front_half A, back_half B, front_half B, back_half A) + fi + enddef; + +beginfig(4); + path A, B; picture p[]; + A = fullcircle scaled 2.5cm; + B = fullcircle scaled 1.8cm shifted (1cm,0); + p1 = image(fill overlap(A,B) withcolor .8[red,white]; drawarrow A; drawarrow B;); + A := A rotated 180; + p2 = image(fill overlap(A,B) withcolor .8[red,white]; drawarrow A; drawarrow B;); + B := B rotatedabout(center B,180); + p3 = image(fill overlap(A,B) withcolor .8[red,white]; drawarrow A; drawarrow B;); + A := A rotated 180; + p4 = image(fill overlap(A,B) withcolor .8[red,white]; drawarrow A; drawarrow B;); + B := B rotatedabout(center B, 180); + A := reverse A; + p5 = image(fill overlap(A,B) withcolor .8[red,white]; drawarrow A; drawarrow B;); + A := A rotated 180; + p6 = image(fill overlap(A,B) withcolor .8[red,white]; drawarrow A; drawarrow B;); + B := B rotatedabout(center B,180); + p7 = image(fill overlap(A,B) withcolor .8[red,white]; drawarrow A; drawarrow B;); + A := A rotated 180; + p8 = image(fill overlap(A,B) withcolor .8[red,white]; drawarrow A; drawarrow B;); + + for i=1 upto 4: draw p[i] scaled .75 shifted (80i, 0); label(decimal i, (5mm+80i,0)); endfor + for i=1 upto 4: draw p[i+4] scaled .75 shifted (80i,-68); label(decimal (4+i), (5mm+80i,-68)); endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/overlaps.pdf b/info/drawing-with-metapost/src/overlaps.pdf Binary files differnew file mode 100644 index 0000000000..f076ef2b7c --- /dev/null +++ b/info/drawing-with-metapost/src/overlaps.pdf diff --git a/info/drawing-with-metapost/src/parallel.mp b/info/drawing-with-metapost/src/parallel.mp new file mode 100644 index 0000000000..802b1b1801 --- /dev/null +++ b/info/drawing-with-metapost/src/parallel.mp @@ -0,0 +1,48 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + z0 = origin; + z1 = 333 dir -18; + z2 = 104 dir 45; + z3 = 233 dir 34; + z4 = 166 dir 12; + + z5 = whatever[z0, z1]; + z6 = whatever[z0, z1]; + z7 = whatever[z0, z1]; + + z5 - z4 = whatever * (z3 - z2); + z6 - z4 = whatever * (z3 - z2) rotated 90; + z7 - z4 = whatever * (z1 - z0) rotated 90; + + z9 = whatever [z2,z3] = whatever [z4,z6]; + + numeric t, u, o; + t = abs(z4-z5); + u = abs(z6-z7); + o = 12; + drawoptions(withpen pencircle scaled 1/4 withcolor 1/2 white); + draw unitsquare scaled 5 rotated angle (z6 - z4) shifted z9; + draw unitsquare scaled 5 rotated angle (z4 - z6) shifted z4; + draw (1+o/t)[z5,z4] -- (1+o/t)[z4,z5]; + draw (1+o/t)[z9,z6] -- (1+o/t)[z6,z9]; + drawoptions(); + + draw z0--z1; + draw z2--z3; + + interim labeloffset := 6; + forsuffixes $=0,1,2,3,4,5,6,7: + dotlabel.top("$\;\;" & char ($ + ASCII "A") & "$", z$) + if $=5: withcolor 2/3 red fi + if $=6: withcolor 2/3 blue fi + if $=7: withcolor 1/2 green fi + ; + endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/parallel.pdf b/info/drawing-with-metapost/src/parallel.pdf Binary files differnew file mode 100644 index 0000000000..0fd68348c0 --- /dev/null +++ b/info/drawing-with-metapost/src/parallel.pdf diff --git a/info/drawing-with-metapost/src/penrose-P3.mp b/info/drawing-with-metapost/src/penrose-P3.mp new file mode 100644 index 0000000000..177581a305 --- /dev/null +++ b/info/drawing-with-metapost/src/penrose-P3.mp @@ -0,0 +1,11 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input penrose +beginfig(1); + draw penrose(3); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/penrose-P3.pdf b/info/drawing-with-metapost/src/penrose-P3.pdf Binary files differnew file mode 100644 index 0000000000..71c9c5d98b --- /dev/null +++ b/info/drawing-with-metapost/src/penrose-P3.pdf diff --git a/info/drawing-with-metapost/src/penrose-stages.mp b/info/drawing-with-metapost/src/penrose-stages.mp new file mode 100644 index 0000000000..94f3800b4c --- /dev/null +++ b/info/drawing-with-metapost/src/penrose-stages.mp @@ -0,0 +1,39 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input penrose +beginfig(1); + path s, t; phi = 1.61803398875; + s = unitsquare shifted -(1/2, 1/2) rotated 45 xscaled phi yscaled 2 sind(36) scaled 20; + t = unitsquare shifted -(1/2, 1/2) rotated 45 xscaled (1/phi) yscaled 2 sind(72) scaled 20; + picture p[]; + p1 = image( + draw (left--right) scaled 30 dashed evenly; + draw s; + ); + p3 = image( + draw (left--right) scaled 13 dashed evenly; + draw t; + ) rotated 90 shifted 130 right; + p2 = image( + draw subpath (1, 3) of s -- cycle dashed withdots scaled 1/16 withpen pencircle scaled 1/8 withcolor 1/2; + inflate_wide(3, 1, point 1 of s, point 2 of s, point 3 of s); + ) shifted 64 right; + p4 = image( + draw subpath (1, 3) of t -- cycle dashed withdots scaled 1/16 withpen pencircle scaled 1/8 withcolor 1/2; + inflate_tall(3, 1, point 1 of t, point 2 of t, point 3 of t); + ) rotated 90 shifted 200 right; + ahangle := 30; + draw p1; + draw p2; + draw p3; + draw p4; + drawarrow (7 left -- 8 right) shifted point 7/4 of bbox p1 withcolor .67 red; + drawarrow (1 left -- 14 right) shifted point 7/4 of bbox p3 withcolor .67 red; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/penrose-stages.pdf b/info/drawing-with-metapost/src/penrose-stages.pdf Binary files differnew file mode 100644 index 0000000000..ae8b8bc3a6 --- /dev/null +++ b/info/drawing-with-metapost/src/penrose-stages.pdf diff --git a/info/drawing-with-metapost/src/penrose.mp b/info/drawing-with-metapost/src/penrose.mp new file mode 100644 index 0000000000..930a66680c --- /dev/null +++ b/info/drawing-with-metapost/src/penrose.mp @@ -0,0 +1,84 @@ +input colorbrewer-rgb +numeric psi, size; +psi = (sqrt 5 - 1) / 2; +size = 800; +vardef inflate_tall(expr mode, level, a, b, c) = + save d, e; pair d, e; + if level = 0: + drawoptions(withpen pencircle scaled 1/8); + filldraw a--b--c--cycle withcolor (abs(b)/size)[Reds 7 2, Blues 8 2]; + if mode = 2: + draw c--a--b; + else: + % save arc; path arc; + % arc = fullcircle rotated angle (c-a) scaled 1/2 abs(b-a) shifted a + % if turningnumber (a--b--c--cycle) < 0: cutafter else: cutbefore fi (a--b); + % cutdraw arc withpen pencircle scaled 3/2 withcolor Reds 7 4; + % cutdraw arc reflectedabout(b, 1/2[a,c]) withpen pencircle scaled 3/2 withcolor Blues 8 8; + % draw arc reflectedabout(b, 1/2[a,c]) withpen pencircle scaled 1/2 withcolor Blues 8 2; + draw a--b--c; + fi + drawoptions(); + else: + if mode = 3: + d = psi[b,a]; + inflate_tall(mode, level - 1, d, c, a); + inflate_wide(mode, level - 1, c, d, b); + elseif mode = 2: + d = psi[a, b]; + e = psi[b, c]; + inflate_tall(mode, level - 1, c, a, e); + inflate_tall(mode, level - 1, d, a, e); + inflate_wide(mode, level - 1, e, d, b); + fi + fi +enddef; +vardef inflate_wide(expr mode, level, a, b, c) = + save d, e; pair d, e; + if level = 0: + drawoptions(withpen pencircle scaled 1/8); + filldraw a--b--c--cycle withcolor (abs(b)/size)[Reds 7 3, white]; + if mode = 2: + draw c--a--b; + else: + % save arc; path arc; + % arc = fullcircle rotated angle (c-a) scaled 3/2 abs (b-a) shifted a + % if turningnumber (a--b--c--cycle) < 0: cutafter else: cutbefore fi (a--b); + % cutdraw arc withpen pencircle scaled 3/2 withcolor Reds 7 4; + % path arc; + % arc = fullcircle rotated angle (a-c) scaled 1/2 abs (b-a) shifted c + % if turningnumber (a--b--c--cycle) > 0: cutafter else: cutbefore fi (c--b); + % cutdraw arc withpen pencircle scaled 3/2 withcolor Blues 8 8; + % draw arc withpen pencircle scaled 1/2 withcolor Blues 8 2; + draw a--b--c; + + fi + drawoptions(); + else: + if mode = 3: + d = psi[a,b]; e = psi[a,c]; + inflate_tall(mode, level - 1, d, e, b); + inflate_wide(mode, level - 1, e, d, a); + inflate_wide(mode, level - 1, c, e, b); + elseif mode = 2: + d = psi[c, a]; + inflate_tall(mode, level - 1, d, c, b); + inflate_wide(mode, level - 1, b, d, a); + fi + fi +enddef; + +vardef penrose(expr mode) = + save a, b, c, P, R; + pair a, b, c; + b = origin; + c = (sind(18), sind(72)) scaled size; + a = (-xpart c, ypart c); + picture P; P = image(inflate_tall(mode, 7, a, b, c)); + picture R; R = image( + for t = 0 upto 9: + draw P if odd t: reflectedabout(b, c) fi rotatedabout(b, 72t); + endfor); + clip R to (unitsquare shifted -(1/2, 1/2) xscaled 987 yscaled 610); + R +enddef; diff --git a/info/drawing-with-metapost/src/picture_frame.mp b/info/drawing-with-metapost/src/picture_frame.mp new file mode 100644 index 0000000000..d957c54074 --- /dev/null +++ b/info/drawing-with-metapost/src/picture_frame.mp @@ -0,0 +1,71 @@ +% Toby Thurston -- 16 Apr 2021 +% Draw a kitsch picture frame round a rectangular path + +color gold, dark, grey; +gold = 1/256(243, 197, 127); +dark = 1/256(144, 87, 50); +grey = 1/256(156, 147, 138); + +picture ball; ball = image(for i=0 upto 16: + fill interpath(i/16, + fullcircle scaled 10, + fullcircle scaled 3 shifted (-2, 2) + ) withcolor (i/16)[gold, 15/16 white]; +endfor) scaled 1/4; + +newinternal pf_width; pf_width := 21; + +vardef frame expr P = + save base, side, f, t, u, xx; + picture base, side; path f; numeric t, u, xx; + + t = arclength subpath (0,1) of bbox P; + u = arclength subpath (1,2) of bbox P; + xx = max(t, u) + 2 pf_width; + f = unitsquare xscaled xx yscaled pf_width; + + % convenience / nonce function + vardef paint_strip(expr y, wd, shade) = + draw subpath (0, 1) of f + shifted (0, if y < 0: pf_width + fi y) + withpen pencircle scaled wd + withcolor shade + enddef; + + base = image( + % background colour + fill f withcolor gold; + % grey strips + paint_strip(2, 3, 5/4 grey); + paint_strip(3.5, 1/4, grey); + paint_strip(5, 1/4, 1/2[gold, dark]); + paint_strip(-6.5, 1/4, 1/2[gold, dark]); + paint_strip(-6, 1/4, 1/2[gold, dark]); + paint_strip(-2, 2, 5/4 grey); + % spatter with random spots + for i=0 upto 4 * arclength(subpath (0,1) of f): + fill fullcircle scaled uniformdeviate 3/4 + shifted (uniformdeviate xx, uniformdeviate pf_width) + withcolor dark; + endfor + % decorative balls + for x = 2 step 3 until xx: + draw ball shifted (x, 2); + endfor + ); + + % make two trapezium shapes + side = base; + clip side to (pf_width, 0) -- (pf_width + u, 0) + -- (2 pf_width + u, pf_width) -- (0, pf_width) -- cycle; + clip base to (pf_width, 0) -- (pf_width + t, 0) + -- (2 pf_width + t, pf_width) -- (0, pf_width) -- cycle; + + % arrange the pieces into a square + image( + draw base rotated 180 shifted point 1 of bbox P shifted (+pf_width, 0); + draw base rotated 0 shifted point 3 of bbox P shifted (-pf_width, 0); + draw side rotated 90 shifted point 0 of bbox P shifted (0, -pf_width); + draw side rotated 270 shifted point 2 of bbox P shifted (0, +pf_width); + ) +enddef; diff --git a/info/drawing-with-metapost/src/pinwheel.mp b/info/drawing-with-metapost/src/pinwheel.mp new file mode 100644 index 0000000000..83a957d415 --- /dev/null +++ b/info/drawing-with-metapost/src/pinwheel.mp @@ -0,0 +1,36 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +vardef pinwheel(expr level, a, b, c, s) = + if level = 0: + fill a--b--c--cycle withcolor s; + draw a--b--c--cycle withpen pencircle scaled 1/16 withcolor Blues 9 1; + % path t; t = a--b--9/10[b,c]--cycle; + % fill t withcolor s; + % draw t withpen pencircle scaled 1/16 withcolor Greens 9 1; + % draw point 5/2 of t withpen pencircle scaled 2 withcolor (1/4 + greenpart s, redpart s, bluepart s); + else: + save d, e, f, g; + pair d, e, f, g; + d = 2/5[b, c]; e = 4/5[b,c]; f = 1/2[e, a]; g = 1/2[a, b]; + pinwheel(level - 1, e, a, c, Blues 9 4); + pinwheel(level - 1, f, g, a, Blues 9 3); + pinwheel(level - 1, f, g, e, Blues 9 2); + pinwheel(level - 1, d, e, g, Blues 9 5); + pinwheel(level - 1, d, b, g, Blues 9 6); + fi +enddef; +beginfig(1); + numeric a, b; b = 6.4in; 2a = b; + pinwheel(5, (0, 0), (b, 0), (0, a), ""); + pinwheel(5, (b, a), (0, a), (b, 0), ""); + currentpicture := currentpicture rotated 90; + % for i=1 upto 1: + % draw currentpicture shifted (lrcorner currentpicture - llcorner currentpicture); + % endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/pinwheel.pdf b/info/drawing-with-metapost/src/pinwheel.pdf Binary files differnew file mode 100644 index 0000000000..d0723c21c4 --- /dev/null +++ b/info/drawing-with-metapost/src/pinwheel.pdf diff --git a/info/drawing-with-metapost/src/plain-axes.mp b/info/drawing-with-metapost/src/plain-axes.mp new file mode 100644 index 0000000000..e535290d21 --- /dev/null +++ b/info/drawing-with-metapost/src/plain-axes.mp @@ -0,0 +1,17 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + path xx, yy; + xx = (left -- right) scaled 130; + yy = (down -- up) scaled 80; + drawarrow xx; + drawarrow yy; + label.rt("$x$", point 1 of xx); + label.top("$y$", point 1 of yy); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/plain-axes.pdf b/info/drawing-with-metapost/src/plain-axes.pdf Binary files differnew file mode 100644 index 0000000000..fc1a766613 --- /dev/null +++ b/info/drawing-with-metapost/src/plain-axes.pdf diff --git a/info/drawing-with-metapost/src/pole-and-polar.mp b/info/drawing-with-metapost/src/pole-and-polar.mp new file mode 100644 index 0000000000..6673ca6341 --- /dev/null +++ b/info/drawing-with-metapost/src/pole-and-polar.mp @@ -0,0 +1,59 @@ +\documentclass[border=3mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + path C, L; + numeric r; + r = 89; + C = fullcircle scaled 2r; + L = (up--down) scaled 138 shifted 180 right rotated 34; + pair a, b, o; + a = point 0 of L; b = point 1 of L; + o = center C; + + pair P, Q, R, P', m; + P = whatever[a, b]; o - P = whatever * (a - b) rotated 90; + Q = C intersectionpoint halfcircle zscaled (P-o) shifted 1/2[P, o]; + R = Q reflectedabout(P, o); + P' = whatever[o, P] = whatever [Q, R]; + + draw fullcircle scaled abs(P' - o) shifted 1/2[o, P'] + dashed withdots scaled 1/4 + withpen pencircle scaled 1/4 + withcolor 1/3 blue; + + draw unitsquare scaled 5 rotated (270 + angle (P-Q)) shifted Q withcolor 3/4 white; + draw unitsquare scaled 5 rotated (90 + angle (P-o)) shifted P withcolor 3/4 white; + draw unitsquare scaled 5 rotated (90 + angle (P-o)) shifted P' withcolor 3/4 white; + + draw P -- Q -- o -- R -- cycle withcolor 1/2 white; + draw Q--R; draw o -- P; + draw L withcolor 2/3 blue; + + draw C dashed evenly scaled 1/2 withcolor 1/2[2/3 blue, white]; + + dotlabel.top("$Q$", Q); + dotlabel.lrt("$R$", R); + dotlabel.urt("$P$", P); + dotlabel.llft("$O$", o); fill fullcircle scaled 3/4 dotlabeldiam shifted o withcolor white; + label.lft("$r$", 1/2[o, Q]); + + drawdot P' withpen pencircle scaled dotlabeldiam; + label("$P'$", P' shifted 10 dir 68) withcolor 2/3 blue; + + drawoptions(withcolor 2/3 blue); + label.bot("\textit{circle of inversion}", point 6 of C); + label.urt("\textit{polar}", point 1/4 of L); + z0 = P' + 20 dir -20; draw z0 -- P' cutafter fullcircle scaled 8 shifted P' + withpen pencircle scaled 1/4; + label.rt("\textit{pole}", z0); + drawoptions(); + + label("$\displaystyle {r\over OP} = {OP' \over r}$", 1/2[point 0 of C, point 1 of L] + 12 down); + label("$\displaystyle r^2 = OP \times OP'$", 1/2[point 0 of C, point 1 of L] + 36 down); + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/pole-and-polar.pdf b/info/drawing-with-metapost/src/pole-and-polar.pdf Binary files differnew file mode 100644 index 0000000000..5bb43987cc --- /dev/null +++ b/info/drawing-with-metapost/src/pole-and-polar.pdf diff --git a/info/drawing-with-metapost/src/porson.mp b/info/drawing-with-metapost/src/porson.mp new file mode 100644 index 0000000000..59ac82181b --- /dev/null +++ b/info/drawing-with-metapost/src/porson.mp @@ -0,0 +1,22 @@ +%prologues := 3; +%outputtemplate := "%j%c.%{outputformat}"; +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +interim bboxmargin := 0; +vardef boxed(expr p) = save P; picture P; P = image(draw bbox p withcolor 7/8[red, white]; draw p); P enddef; +beginfig(1); + for i=2 upto 15: + numeric y; y = -14i - 4 floor(i/4); + label.lft(decimal 16i infont "phvr8r" scaled 0.6, (-6, y+2)) withcolor 0.54 red; + for j = 0 upto 15: + draw boxed(char (16i+j) infont "gporsonrg6r") shifted (12j + 2 floor(j/4), y); + endfor + endfor + for x = 0 upto 4: + draw (50x-4, -18) -- (50x-4, -224) withcolor 0.54 red; + endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/porson.pdf b/info/drawing-with-metapost/src/porson.pdf Binary files differnew file mode 100644 index 0000000000..7a26026f46 --- /dev/null +++ b/info/drawing-with-metapost/src/porson.pdf diff --git a/info/drawing-with-metapost/src/projections-code.mp b/info/drawing-with-metapost/src/projections-code.mp new file mode 100644 index 0000000000..e10619e2b6 --- /dev/null +++ b/info/drawing-with-metapost/src/projections-code.mp @@ -0,0 +1,34 @@ +% define the end points of the three rays +z1 = right scaled 200 rotated 10; +z2 = right scaled 100 rotated 135; +z3 = right scaled 225 rotated -17.5; +% define the other points, relative to Q +pair A, B, P, Q, R; +Q = 0.8125 z3; +A = whatever[origin, z1]; A-Q = whatever * z1 rotated 90; +P = whatever[origin, z2]; P-Q = whatever * z2 rotated 90; +B = whatever[origin, z1]; B-P = whatever * z1 rotated 90; +R = whatever[A,Q]; R-P = whatever * (B-P) rotated 90; +% mark the angles +drawoptions(withcolor .67 blue); +path c; c = fullcircle scaled 30; +draw c rotated angle (Q-P) shifted P cutafter (P--B); +draw c rotated angle (P-Q) shifted Q cutafter (Q--R); +draw c rotated angle P cutafter (origin--z1); +drawoptions(); +% draw the rays and A--Q +drawarrow origin -- z1; label(btex $r_1$ etex, z1 scaled 1.05); +drawarrow origin -- z2; label(btex $r_2$ etex, z2 scaled 1.08); +drawarrow origin -- z3; label(btex $r_3$ etex, z3 scaled 1.05); +draw A--Q; +% draw the dashed lines +drawoptions(dashed evenly); +draw B--P--R--Q--P; drawarrow origin -- P scaled 4/3; +drawoptions(); +% label the points +dotlabel.urt(btex $Q$ etex, Q); +dotlabel.top(btex $A$ etex, A); +dotlabel.lrt(btex $R$ etex, R) withcolor .67 blue; +label.top (btex $B$ etex, B); +label.llft(btex $P$ etex, P); +label.llft(btex $O$ etex, origin); diff --git a/info/drawing-with-metapost/src/projections.mp b/info/drawing-with-metapost/src/projections.mp new file mode 100644 index 0000000000..42d683c89d --- /dev/null +++ b/info/drawing-with-metapost/src/projections.mp @@ -0,0 +1,10 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +input projections-code +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/projections.pdf b/info/drawing-with-metapost/src/projections.pdf Binary files differnew file mode 100644 index 0000000000..327f5caf95 --- /dev/null +++ b/info/drawing-with-metapost/src/projections.pdf diff --git a/info/drawing-with-metapost/src/qed.mp b/info/drawing-with-metapost/src/qed.mp new file mode 100644 index 0000000000..e8ef24e4bd --- /dev/null +++ b/info/drawing-with-metapost/src/qed.mp @@ -0,0 +1,42 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +def perpendicular expr t of p = + direction t of p rotated 90 shifted point t of p + enddef; + +vardef block(expr r) = + save u, n, s, theta, base, ray; + numeric u, n, s, theta; + u = 5mm; n = 32; s = 8u; theta = -45; + + path base, ray[]; + base = origin for i=1 upto n-1: -- (i/n*s,r*normaldeviate) endfor + -- (s,0) -- (s,-u) -- (0,-u) -- cycle; + + image( + fill base withcolor .8[blue,white]; + draw base withcolor .67 blue; + + for i=2 upto 6: + ray[i] = (left--right) scaled 2/3 s rotated theta shifted (i*u,0); + b := ypart(ray[i] intersectiontimes base); + ray[i] := point 0 of ray[i] + -- point b of base + -- point 0 of ray[i] reflectedabout(point b of base, perpendicular b of base); + drawarrow ray[i]; + endfor + + label("r=" & decimal r, center base); + ) enddef; + +beginfig(0); + + draw block(0); draw block(0.42) shifted 180 right; + + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/qed.pdf b/info/drawing-with-metapost/src/qed.pdf Binary files differnew file mode 100644 index 0000000000..361fa0686a --- /dev/null +++ b/info/drawing-with-metapost/src/qed.pdf diff --git a/info/drawing-with-metapost/src/radical-axis.mp b/info/drawing-with-metapost/src/radical-axis.mp new file mode 100644 index 0000000000..ee1c7fd6fe --- /dev/null +++ b/info/drawing-with-metapost/src/radical-axis.mp @@ -0,0 +1,72 @@ +\documentclass[border=1mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +vardef radical_axis(expr ca, cb) = + numeric t, d, ra, rb; + ra = abs(center ca - point 0 of ca); + rb = abs(center cb - point 0 of cb); + d = abs(center cb - center ca); + 2t = 1 + (ra+rb) / d * (ra-rb) / d; + (up -- down) scaled 89 + rotated angle (center cb - center ca) + shifted t[center ca, center cb] +enddef; + +vardef tangent_times(expr C, p) = + save m, a, b, G, H; + pair m; numeric a, b; path G, H; + m = 1/2[p, center C]; + H = halfcircle scaled abs (p - center C) + rotated angle (p - center C) shifted m; + G = H rotatedabout(m, 180); + (a, whatever) = C intersectiontimes H; + (b, whatever) = C intersectiontimes G; + (a, b if b < a: + 8 fi) +enddef; + +beginfig(1); + path c[], a[]; + z1 = origin; z2 = 233 right rotated 4; z3 = 209 right rotated -42; + c1 = fullcircle scaled 202 shifted z1; + c2 = fullcircle scaled 106 shifted z2; + c3 = fullcircle scaled 62 shifted z3; + a1 = radical_axis(c1, c2); + a2 = radical_axis(c2, c3); + a3 = radical_axis(c3, c1); + + z0 = whatever [point 0 of a1, point 1 of a1] + = whatever [point 0 of a2, point 1 of a2]; + + numeric t; + (t, whatever) = tangent_times(c1, z0); + + drawoptions(withpen pencircle scaled 1 withcolor 3/4[blue, white]); + draw c1; draw c2; draw c3; + + drawoptions(withcolor 3/4[blue, white]); + draw z1 -- point t of c1 dashed evenly; + + drawoptions(withpen pencircle scaled 1/4); + draw z1 -- z2 -- z3 -- cycle; + drawoptions(); + draw a1; draw a2; draw a3; + + drawoptions(withcolor 2/3 red); + draw fullcircle scaled 2 abs (point t of c1 - z0) shifted z0; + draw z0 -- point t of c1 dashed evenly; + drawdot z0 withpen pencircle scaled dotlabeldiam; + + drawoptions(withcolor 1/4[blue, white]); + drawdot z1 withpen pencircle scaled dotlabeldiam; + drawdot z2 withpen pencircle scaled dotlabeldiam; + drawdot z3 withpen pencircle scaled dotlabeldiam; + label.urt(btex $C_1$ etex, point 1 of c1); + label.urt(btex $C_2$ etex, point 1 of c2); + label.rt (btex $C_3$ etex, point 0 of c3); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/radical-axis.pdf b/info/drawing-with-metapost/src/radical-axis.pdf Binary files differnew file mode 100644 index 0000000000..fc643c1d71 --- /dev/null +++ b/info/drawing-with-metapost/src/radical-axis.pdf diff --git a/info/drawing-with-metapost/src/random-freehand-circumcircle.mp b/info/drawing-with-metapost/src/random-freehand-circumcircle.mp new file mode 100644 index 0000000000..290ba81b7e --- /dev/null +++ b/info/drawing-with-metapost/src/random-freehand-circumcircle.mp @@ -0,0 +1,43 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +def freehand_segment(expr p) = + point 0 of p {direction 0 of p rotated (4+normaldeviate)} .. + point 1 of p {direction 1 of p rotated (4+normaldeviate)} +enddef; +def freehand_path(expr p) = + freehand_segment(subpath(0,1) of p) + for i=1 upto length(p)-1: + & freehand_segment(subpath(i,i+1) of p) + endfor + if cycle p: & cycle fi +enddef; +defaultfont := "eurm10"; +color sepia; sepia = (0.44, 0.26, 0.08); +picture marker; marker = image(for s=-1/2, 1/2: + draw (left--right) scaled 2 rotated 60 shifted (s,0); +endfor); +def moved_along expr x of p = rotated angle direction x of p + shifted point x of p enddef; + +beginfig(1); +pair A, B, C, D; +A = (0,-30); B = (180,0); C = (120,90); D = (1/2 + 1/40 normaldeviate)[A, B]; +path triangle, circumcircle, bisector; +triangle = freehand_path(A--D--B--C--cycle); +bisector = freehand_segment(C--D); +circumcircle = freehand_path(A..B..C..cycle); +draw triangle; +draw bisector; +draw circumcircle withcolor .67 red; +draw marker moved_along 1/2 of triangle withcolor .67 red; +draw marker moved_along 3/2 of triangle withcolor .67 red; +label.lft("A", A); +label.rt ("B", B); +label.top("C", C); +label.bot("D", D); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/random-freehand-circumcircle.pdf b/info/drawing-with-metapost/src/random-freehand-circumcircle.pdf Binary files differnew file mode 100644 index 0000000000..d1e3586cd0 --- /dev/null +++ b/info/drawing-with-metapost/src/random-freehand-circumcircle.pdf diff --git a/info/drawing-with-metapost/src/random-freehand-through.mp b/info/drawing-with-metapost/src/random-freehand-through.mp new file mode 100644 index 0000000000..0bbacf4436 --- /dev/null +++ b/info/drawing-with-metapost/src/random-freehand-through.mp @@ -0,0 +1,42 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +vardef freehand_through@#(expr a, b) = + save t; pair t; t = @# * unitvector(b - a) rotated (4 + normaldeviate); + a - t .. a {t} .. b {t} .. b + t +enddef; + +vardef mid_arc@#(expr p, a, b) = + save c; path c; c = fullcircle scaled (2*@#) shifted p cutbefore a cutafter b; + point arctime 1/2 arclength c of c of c +enddef; + +beginfig(1); +defaultfont := "eurm10"; color sepia; sepia = (0.44, 0.26, 0.08); +pair A, B, C; +A = (0,-30); B = (180,0); C = (120,90); + +path a, b, c; +a = freehand_through 7 (A, B); +b = freehand_through 8 (B, C); +c = freehand_through 6 (C, A); + +drawoptions(withcolor sepia); +draw a; draw b; draw c; + +drawoptions(withcolor .67 blue); +label.bot ("a", point 3/2 of a); +label.rt ("b", point 3/2 of b); +label.ulft("c", point 3/2 of c); + +drawoptions(withcolor .5 red); +label(char 11, mid_arc 16 (A, a, c)); +label(char 12, mid_arc 14 (B, b, a)); +label(char 13, mid_arc 14 (C, c, b)); + +drawoptions(); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/random-freehand-through.pdf b/info/drawing-with-metapost/src/random-freehand-through.pdf Binary files differnew file mode 100644 index 0000000000..55e5548095 --- /dev/null +++ b/info/drawing-with-metapost/src/random-freehand-through.pdf diff --git a/info/drawing-with-metapost/src/random-function.mp b/info/drawing-with-metapost/src/random-function.mp new file mode 100644 index 0000000000..d2c13dd10b --- /dev/null +++ b/info/drawing-with-metapost/src/random-function.mp @@ -0,0 +1,33 @@ +\documentclass[border=2mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +beginfig(1); +path xx, yy, ff; +xx = origin -- 233 right; +yy = origin -- 144 up; + +z1 = (72, 62); +z2 = (180, 110); + +ff = (xpart point 0 of xx, ypart point 0 of yy) {dir 76} + .. z1 {dir 64} .. z2 {dir 64} .. (xpart point 1 of xx, ypart point 1 of yy) {dir 30}; + +draw ff withcolor 0.54 red; +drawarrow xx; +drawarrow yy; +forsuffixes $=1,2: + draw (0, y$) -- z$ -- (x$,0) dashed evenly withcolor 1/2; + dotlabel.lft("\texttt{(0,y" & decimal $ & ")}", (0, y$)); + dotlabel.bot("\texttt{(x" & decimal $ & ",0)}", (x$, 0)); + dotlabel.rt("$z_" & decimal $ & "$", z$); +endfor + + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/random-function.pdf b/info/drawing-with-metapost/src/random-function.pdf Binary files differnew file mode 100644 index 0000000000..8ec56fba8e --- /dev/null +++ b/info/drawing-with-metapost/src/random-function.pdf diff --git a/info/drawing-with-metapost/src/random-selection.mp b/info/drawing-with-metapost/src/random-selection.mp new file mode 100644 index 0000000000..d77f7fc141 --- /dev/null +++ b/info/drawing-with-metapost/src/random-selection.mp @@ -0,0 +1,25 @@ +\documentclass[border=0mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + numeric u; u = 72; + for x = -2u step 1/8 until 2u: + drawdot (x, normaldeviate * 1/2u) withpen pencircle scaled 2; + endfor + path c; + c = fullcircle scaled u shifted (u, 1/2u); + picture P; P = currentpicture; clip P to c; + draw c withcolor .54 red; + draw P withcolor .84 red; + + picture B; B = currentpicture; + path S; + S = unitsquare shifted (-1/2, -1/2) scaled 4u; + fill S withcolor (1,1,31/32); draw S; clip B to S; draw B; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/random-selection.pdf b/info/drawing-with-metapost/src/random-selection.pdf Binary files differnew file mode 100644 index 0000000000..54b7a91dc6 --- /dev/null +++ b/info/drawing-with-metapost/src/random-selection.pdf diff --git a/info/drawing-with-metapost/src/random-shapes.mp b/info/drawing-with-metapost/src/random-shapes.mp new file mode 100644 index 0000000000..de581a58fe --- /dev/null +++ b/info/drawing-with-metapost/src/random-shapes.mp @@ -0,0 +1,24 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +numeric desired_length, n, s; +desired_length = 180; n = 30; s = 80; +for r=0 upto 8: + path shape; + shape = for i=1 upto n: + (s + r * normaldeviate, 0) rotated (360/n*i) .. + endfor cycle; + + shape := shape scaled (desired_length/arclength shape); + + draw shape shifted (r*s, 0) withcolor (r/8)[black,red]; + label(decimal r, (r*s, 0)); + +endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/random-shapes.pdf b/info/drawing-with-metapost/src/random-shapes.pdf Binary files differnew file mode 100644 index 0000000000..540fdb680c --- /dev/null +++ b/info/drawing-with-metapost/src/random-shapes.pdf diff --git a/info/drawing-with-metapost/src/random-walks-red-blue.mp b/info/drawing-with-metapost/src/random-walks-red-blue.mp new file mode 100644 index 0000000000..a11a5f22f3 --- /dev/null +++ b/info/drawing-with-metapost/src/random-walks-red-blue.mp @@ -0,0 +1,45 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + randomseed := 3.1415; + color tr, tb; tr = 0.84 red; tb = 1.2(.2,.2,.7); + numeric w, h, n; w = 377; h = 80; n = 500; + pair zenith, nadir; zenith = nadir = origin; + path walk[]; + + draw (origin--right) scaled w; + draw (origin--right) scaled w shifted (0,+h/2) withcolor 3/4; + draw (origin--right) scaled w shifted (0,-h/2) withcolor 3/4; + + for i=1 upto 8: + numeric y; y = 0; + walk[i] = origin for x=w/n step w/n until w: + hide( + y := y if uniformdeviate 1 < 1/2: + else: - fi 1; + if y > ypart zenith: zenith := (x,y) ; fi + if y < ypart nadir: nadir := (x,y) ; fi + ) + -- (x,y) + endfor; + undraw walk[i] withpen pencircle scaled 3/4; + draw walk[i] withcolor (1/2+y/h)[tr, tb]; + label.rt("$\scriptstyle" & decimal i & "$", + point infinity of walk[i] + if i=1: shifted (-3, 3) + elseif i=3: shifted (1,3) + elseif i=5: shifted (-3, 3) + elseif i=7: shifted (-1, -4) fi); + endfor + + drawarrow (12 up -- 2 up ) shifted zenith withcolor tb; + drawarrow (12 down -- 2 down) shifted nadir withcolor tr; + + fill fullcircle scaled 2; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/random-walks-red-blue.pdf b/info/drawing-with-metapost/src/random-walks-red-blue.pdf Binary files differnew file mode 100644 index 0000000000..c76047183c --- /dev/null +++ b/info/drawing-with-metapost/src/random-walks-red-blue.pdf diff --git a/info/drawing-with-metapost/src/rec-bush.mp b/info/drawing-with-metapost/src/rec-bush.mp new file mode 100644 index 0000000000..594a2eebf5 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-bush.mp @@ -0,0 +1,26 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +vardef bush(expr start, aim, level, limit) = + save s, target; + numeric s; s = level / limit; + for a = -32, -8, 8, 16: + pair target; + target = aim scaled ((32 + 16 normaldeviate) * s) rotated a shifted start; + draw start -- target + withpen pencircle scaled s + withcolor BrBG[limit][limit-level]; + if level > 1: + bush(target, aim rotated a, level - 1, limit); + fi + endfor +enddef; +beginfig(1); +bush(origin, dir 80, 6, 8); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-bush.pdf b/info/drawing-with-metapost/src/rec-bush.pdf Binary files differnew file mode 100644 index 0000000000..415b01c391 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-bush.pdf diff --git a/info/drawing-with-metapost/src/rec-dragon-golden.mp b/info/drawing-with-metapost/src/rec-dragon-golden.mp new file mode 100644 index 0000000000..df4dc37c48 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-dragon-golden.mp @@ -0,0 +1,40 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{dwmpcode} +\mplibtextextlabel{enable} +\begin{document} +\begin{mplibcode} +vardef acosd(expr a) = angle (a, 1 +-+ a) enddef; +numeric r, phi, theta; +phi = 0.61803398875; +r = phi**phi; +theta = acosd(1/2(1/r + r - r * r * r)); + +vardef dragon(expr a, b) = + if abs(a-b) > 1: + save p; pair p; + p = r[a, b] rotatedabout(a, theta); + dragon(a, p) & reverse dragon(b, p) + else: + a -- b + fi +enddef; + +beginfig(1); + path p; p = dragon(origin, 518 right); + draw p withpen pencircle scaled 1/8 withcolor \mpcolor{carrot}; + picture base; base = image( + draw 10 left -- 90 right withcolor 3/4; + path a; a = (origin -- r * dir theta -- right) scaled 80; + draw a; + for t=0 upto 2: draw point t of a withpen pencircle scaled 2 withcolor .54 red; endfor + label.ulft("$r$", point 1/2 of a); + label.urt ("$r^2$", point 3/2 of a); + label.bot ("$1$", 1/2[point 0 of a, point 2 of a]); + label("$\theta$", 18 dir 1/2 theta); + ); + label.lrt(base, ulcorner currentpicture); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-dragon-golden.pdf b/info/drawing-with-metapost/src/rec-dragon-golden.pdf Binary files differnew file mode 100644 index 0000000000..810ad03560 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-dragon-golden.pdf diff --git a/info/drawing-with-metapost/src/rec-dragon.mp b/info/drawing-with-metapost/src/rec-dragon.mp new file mode 100644 index 0000000000..60b8549cf1 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-dragon.mp @@ -0,0 +1,40 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +numeric r, theta; r = sqrt 1/2; theta = 45; +vardef dragon(expr level, a, b) = + if level > 0: + save p; pair p; + p = r[a, b] rotatedabout(a, theta); + dragon(level - 1, a, p) & reverse dragon(level - 1, b, p) + else: + a .. b + fi +enddef; + +vardef rounded_corners expr p = + save r, n; numeric r, n; r = 1/3; n = length p; + subpath (0, 1-r) of p + for t=1 upto n-1: + .. subpath (t+r, t+1-r) of p + endfor .. subpath (n-r, n) of p +enddef; + +beginfig(1); + + path d; d = dragon(15, origin, 240 right); + draw d withpen pencircle scaled 1/4 withcolor (.2, .2, .7); + label.top("The dragon curve at level 15", point 1/3 of bbox currentpicture); + + draw rounded_corners dragon(10, origin, 240 right) + shifted 280 down + withcolor .54 red; + + label.top("\dots\ and at level 10 with rounded corners", point 1/3 of bbox currentpicture); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-dragon.pdf b/info/drawing-with-metapost/src/rec-dragon.pdf Binary files differnew file mode 100644 index 0000000000..3338a7adde --- /dev/null +++ b/info/drawing-with-metapost/src/rec-dragon.pdf diff --git a/info/drawing-with-metapost/src/rec-flowsnake-construction.mp b/info/drawing-with-metapost/src/rec-flowsnake-construction.mp new file mode 100644 index 0000000000..ec543a7512 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-flowsnake-construction.mp @@ -0,0 +1,53 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{dwmpcode} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path h[], snake; +h0 = (for i = 1 upto 6: 40 dir 60i -- endfor cycle) rotated angle (9, -sqrt(3)); +for i = 1 upto 6: + h[i] = h0 shifted (point 0 of h0 - point 2 of h0) rotated 60i; +endfor +snake = point 0 of h3 -- point 1 of h2 -- point 2 of h1 + -- point 1 of h6 -- point 2 of h5 -- point 3 of h0 + -- point 1 of h4 -- point 0 of h5; + +for i=1 upto 6: + draw subpath (4, 9) of h[i] withcolor 3/4; +endfor + +% path usnake; +% usnake = snake +% shifted - point 0 of h3 +% scaled (1/abs(point 0 of h3 - point 0 of h5)); + +% def zx(expr a, b) = zscaled (point b of snake - point a of snake) +% shifted point a of snake enddef; + +% draw usnake zx(1, 0) dashed withdots scaled 1/2 withcolor 2/3 red; +% draw usnake zx(1, 2) dashed withdots scaled 1/2 withcolor 2/3 red; +% draw usnake zx(2, 3) dashed withdots scaled 1/2 withcolor 2/3 red; +% draw usnake zx(3, 4) dashed withdots scaled 1/2 withcolor 2/3 red; +% draw usnake zx(5, 4) dashed withdots scaled 1/2 withcolor 2/3 red; +% draw usnake zx(6, 5) dashed withdots scaled 1/2 withcolor 2/3 red; +% draw usnake zx(6, 7) dashed withdots scaled 1/2 withcolor 2/3 red; + + +draw 2/3[point 0 of h3, point 0 of h5] -- point 0 of h5 withcolor 1/2; +drawarrow point 0 of h3 -- 2/3[point 0 of h3, point 0 of h5] withcolor 1/2; +draw snake withcolor 2/3 red; +for t=1, 5, 6: + drawarrow subpath (t, t-2/3) of snake withcolor \mpcolor{textred}; +endfor +for t=1, 2, 3, 6: + drawarrow subpath (t, t+2/3) of snake withcolor \mpcolor{textred}; +endfor + +dotlabel.llft("$(0, 0)$", point 0 of h3); +dotlabel.lrt ("$(1, 0)$", point 0 of h5); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-flowsnake-construction.pdf b/info/drawing-with-metapost/src/rec-flowsnake-construction.pdf Binary files differnew file mode 100644 index 0000000000..06f111a067 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-flowsnake-construction.pdf diff --git a/info/drawing-with-metapost/src/rec-flowsnake.mp b/info/drawing-with-metapost/src/rec-flowsnake.mp new file mode 100644 index 0000000000..be94b53065 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-flowsnake.mp @@ -0,0 +1,70 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} + +path h[], snake; +h0 = for i = 1 upto 6: dir 60i -- endfor cycle; +for i = 1 upto 6: + h[i] = h0 shifted (point 0 of h0 - point 2 of h0) rotated 60i; +endfor +snake = point 0 of h3 -- point 1 of h2 -- point 2 of h1 + -- point 1 of h6 -- point 2 of h5 -- point 3 of h0 + -- point 1 of h4 -- point 0 of h5; +snake := snake shifted - point 0 of h3; +snake := snake rotated - angle (point 0 of h5 - point 0 of h3); +snake := snake scaled (1 / length (point 0 of h5 - point 0 of h3)); + +vardef hexon(expr a, b) = + pair c, m; + c = b rotatedaround(a, 60); + m = 2/3[c, 1/2[a, b]]; + a -- a rotatedaround(m, 60) -- + b -- b rotatedaround(m, 60) -- + c -- c rotatedaround(m, 60) -- cycle +enddef; + +path w; +w = origin -- right; +w := w -- point 0 of w rotatedabout(point 1 of w, 120); +w := w -- point 1 of w rotatedabout(point 2 of w, -120); +w := w rotated - angle (point 3 of w - point 0 of w); +w := w scaled (1/abs (point 3 of w - point 0 of w)); + +pair a, b; +a = 144 down rotated -60; +b = 144 down rotated +60; +numeric n; n = 4; + +path boundary; +boundary = hexon(a, b); +for i = 1 upto n: + boundary := for t = 1 upto length boundary: + subpath (0, 2) of w + zscaled (point t of boundary - point t-1 of boundary) + shifted point t-1 of boundary -- endfor cycle; +endfor + + +vardef rattle(expr level, a, b) = + if level > 0: + save s; path s; s = snake zscaled (b-a) shifted a; + reverse rattle(level - 1, point 1 of s, a) & + rattle(level - 1, point 1 of s, point 2 of s) & + rattle(level - 1, point 2 of s, point 3 of s) & + rattle(level - 1, point 3 of s, point 4 of s) & + reverse rattle(level - 1, point 5 of s, point 4 of s) & + reverse rattle(level - 1, point 6 of s, point 5 of s) & + rattle(level - 1, point 6 of s, b) + else: + a -- b + fi +enddef; +beginfig(1); + path s; s = rattle(n, a, b); + fill a + (-40, 0) -- s -- b + (40, 0) -- + b + (40, 224) -- a + (-40, 224) -- cycle withcolor 3/4[1/4 blue, white]; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-flowsnake.pdf b/info/drawing-with-metapost/src/rec-flowsnake.pdf Binary files differnew file mode 100644 index 0000000000..3bcc884ad1 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-flowsnake.pdf diff --git a/info/drawing-with-metapost/src/rec-general-tree-deviate.mp b/info/drawing-with-metapost/src/rec-general-tree-deviate.mp new file mode 100644 index 0000000000..f6e1ac1e7b --- /dev/null +++ b/info/drawing-with-metapost/src/rec-general-tree-deviate.mp @@ -0,0 +1,34 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +r = 0.75; theta = 17; +vardef make_tree(expr bar) = + save a; numeric a; a = abs(point 1 of bar - point 0 of bar); + cutdraw bar withpen pencircle scaled (a/8); + if a > 5: + save s; pair s; s = 1/32 a * r * unitvector(direction 1 of bar) rotated 90; + make_tree( + bar shifted - point 0 of bar shifted s + scaled (r + 1/16 normaldeviate) + rotated (theta + 8 normaldeviate) + shifted point 1 of bar + ); + make_tree( + bar shifted - point 0 of bar shifted -s + scaled (r + 1/16 normaldeviate) + rotated -(theta + 8 normaldeviate) + shifted point 1 of bar + ); + fi +enddef; +beginfig(1); +for i=1 upto 4: + picture T; T = image(make_tree(origin -- 40 up)); draw T shifted (170i, 0); +endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-general-tree-deviate.pdf b/info/drawing-with-metapost/src/rec-general-tree-deviate.pdf Binary files differnew file mode 100644 index 0000000000..3803c073ca --- /dev/null +++ b/info/drawing-with-metapost/src/rec-general-tree-deviate.pdf diff --git a/info/drawing-with-metapost/src/rec-general-tree.mp b/info/drawing-with-metapost/src/rec-general-tree.mp new file mode 100644 index 0000000000..c4e31178af --- /dev/null +++ b/info/drawing-with-metapost/src/rec-general-tree.mp @@ -0,0 +1,34 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef make_tree(expr bar) = + save a; numeric a; a = abs(point 1 of bar - point 0 of bar); + cutdraw bar withpen pencircle scaled 1.2(1/8 a) withcolor background; + cutdraw bar withpen pencircle scaled (1/8 a) withcolor 1/256(57, 35, 32); + if a > leaf: + save s; pair s; s = 1/32 a * r * unitvector(direction 1 of bar) rotated 90; + make_tree( + bar shifted - point 0 of bar + shifted s scaled r rotated theta + shifted point 1 of bar + ); + make_tree( + bar shifted - point 0 of bar + shifted -s scaled r rotated -theta + shifted point 1 of bar + ); + else: + draw point 1 of bar withpen pencircle scaled 1 withcolor 2/3 green; + fi +enddef; +beginfig(1); +numeric stem, leaf, r, theta; +r = 0.75; theta = 14; +stem = 100; leaf = 3; % max(1, round(stem * (r ** 12))); +make_tree(origin -- stem * up); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-general-tree.pdf b/info/drawing-with-metapost/src/rec-general-tree.pdf Binary files differnew file mode 100644 index 0000000000..8db8f8a29e --- /dev/null +++ b/info/drawing-with-metapost/src/rec-general-tree.pdf diff --git a/info/drawing-with-metapost/src/rec-h-tree.mp b/info/drawing-with-metapost/src/rec-h-tree.mp new file mode 100644 index 0000000000..a131038491 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-h-tree.mp @@ -0,0 +1,30 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +r = 0; +vardef make_H(expr level, bar, thickness) = + cutdraw bar withpen pencircle scaled thickness + withcolor Blues[9][9 - floor(level/2)]; + + % save spike, m; path spike; pair m; m = unitvector(direction 1/2 of bar rotated 90); + % spike = point 0 of bar .. m scaled +thickness shifted point 1/2 of bar .. point 1 of bar & + % point 1 of bar .. m scaled -thickness shifted point 1/2 of bar .. point 0 of bar & cycle; + % fill spike withcolor Reds 9 7; + + if level > 0: + save foo, fo; + path foo, fo; + foo = bar shifted - point 1/2 of bar scaled 0.7071067811865475; + make_H(level - 1, foo rotated 90 shifted point r of bar, thickness); + make_H(level - 1, foo rotated 90 shifted point 1-r of bar, thickness); + fi +enddef; +beginfig(1); +make_H(8, (left -- right) scaled 200, 1/2); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-h-tree.pdf b/info/drawing-with-metapost/src/rec-h-tree.pdf Binary files differnew file mode 100644 index 0000000000..5efa7ed845 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-h-tree.pdf diff --git a/info/drawing-with-metapost/src/rec-heighway-stages.mp b/info/drawing-with-metapost/src/rec-heighway-stages.mp new file mode 100644 index 0000000000..c5d070a23d --- /dev/null +++ b/info/drawing-with-metapost/src/rec-heighway-stages.mp @@ -0,0 +1,30 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +picture s[]; +path p; p = origin -- 42 right; +s1 = image(undraw (origin -- up) scaled 21; draw p;); +for i=2 upto 5: + s[i] = image( + draw p withcolor 7/8; + p := p scaled sqrt 1/2; + p := p rotated 45 .. reverse p rotated 135 shifted 42 right; + draw p; + ) shifted (80 * i - 80, 0); +endfor + +beginfig(1); +draw s1; draw s2; draw s3; draw s4; draw s5; +ahangle := 30; y = 10; +drawoptions(withcolor (.2,.2,.7)); +drawarrow (50,y) -- (70, y); +drawarrow (130,y) -- (150, y); +drawarrow (205,y) -- (220, y); +drawarrow (285,y) -- (300, y); +drawoptions(); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-heighway-stages.pdf b/info/drawing-with-metapost/src/rec-heighway-stages.pdf Binary files differnew file mode 100644 index 0000000000..6ef4716a86 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-heighway-stages.pdf diff --git a/info/drawing-with-metapost/src/rec-koch-steps.mp b/info/drawing-with-metapost/src/rec-koch-steps.mp new file mode 100644 index 0000000000..e696eac4f8 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-koch-steps.mp @@ -0,0 +1,32 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} + +vardef koch(expr level, a, b) = + if level = 0: + a -- b + else: + save p, q, r; pair p, q, r; + p = 1/3[a,b]; r = 2/3[a,b]; q = r rotatedabout(p, 60); + koch(level-1, a, p) & + koch(level-1, p, q) & + koch(level-1, q, r) & + koch(level-1, r, b) + fi +enddef; +beginfig(1); + wd = 300; + for n=0 upto 4: + numeric y; y = -1/3 wd * n; + path k; k = koch(n, origin, (wd, 0)) shifted (0, y); + draw k withcolor 2/3 blue; + label.urt("\small\textsf{Level " & decimal n & "}", point infinity of k); + label.top("$\scriptstyle " & decimal length k & "/" & decimal arclength k & "$", (wd/2, y)); + endfor + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-koch-steps.pdf b/info/drawing-with-metapost/src/rec-koch-steps.pdf Binary files differnew file mode 100644 index 0000000000..d0be311ae6 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-koch-steps.pdf diff --git a/info/drawing-with-metapost/src/rec-mink-sausage.mp b/info/drawing-with-metapost/src/rec-mink-sausage.mp new file mode 100644 index 0000000000..fbde07eafa --- /dev/null +++ b/info/drawing-with-metapost/src/rec-mink-sausage.mp @@ -0,0 +1,30 @@ +% ref Mandlebrot, FGoN, p.32 and p.48 +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +path gene; gene = (0,0) -- (1,0) -- (1, -1) -- (2, -1); +gene := gene rotated - angle point length gene of gene; +gene := gene scaled (1 / abs point length gene of gene); +beginfig(1); +for n=5, 6: + path p; p = gene scaled 300; + for i = 1 upto n: + p := for t = 1 upto length p: + subpath (0, 2) of gene + zscaled (point t of p - point t-1 of p) + shifted point t-1 of p -- endfor point length p of p; + endfor + p := p shifted (0, -144n); + for i = 0 upto length p: + fill fullcircle scaled 4 + shifted point i of p + withcolor 7/8; + endfor + draw p withcolor 1/2; +endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-mink-sausage.pdf b/info/drawing-with-metapost/src/rec-mink-sausage.pdf Binary files differnew file mode 100644 index 0000000000..bc6fe1f25d --- /dev/null +++ b/info/drawing-with-metapost/src/rec-mink-sausage.pdf diff --git a/info/drawing-with-metapost/src/rec-sierpinski-pentagon.mp b/info/drawing-with-metapost/src/rec-sierpinski-pentagon.mp new file mode 100644 index 0000000000..1829c8003c --- /dev/null +++ b/info/drawing-with-metapost/src/rec-sierpinski-pentagon.mp @@ -0,0 +1,33 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +vardef make_gasket(expr t, s, level, c) = + if level = 0: + fill t withcolor Reds[7][7-c]; + draw t; + else: + save little_t; path little_t; little_t = t scaled s; + for i=1 upto length t: + make_gasket(little_t shifted (point i of t - point i of little_t), s, level-1, c); + endfor + fi +enddef; +vardef sf(expr n) = + numeric s; s = 0; + for k=1 upto floor (n/4): + s := s + cosd(360 / n * k); + endfor + 1 / (2 * (1 + s)) +enddef; +beginfig(1); + for n=5 upto 9: + draw image(make_gasket(for i=1 upto n: 64 up rotated (360/n*i) -- endfor cycle, sf(n), 3, n-4)) + shifted (144n, 0); + endfor + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/rec-sierpinski-pentagon.pdf b/info/drawing-with-metapost/src/rec-sierpinski-pentagon.pdf Binary files differnew file mode 100644 index 0000000000..718d201949 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-sierpinski-pentagon.pdf diff --git a/info/drawing-with-metapost/src/rec-sierpinski-triangle.mp b/info/drawing-with-metapost/src/rec-sierpinski-triangle.mp new file mode 100644 index 0000000000..c0bbc2ec1a --- /dev/null +++ b/info/drawing-with-metapost/src/rec-sierpinski-triangle.mp @@ -0,0 +1,21 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef make_gasket(expr t, s, limit) = + if length (point 1 of t - point 0 of t) < limit: + fill t; + else: + save little_t; path little_t; little_t = t scaled s; + for i=1 upto length t: + make_gasket(little_t shifted (point i of t - point i of little_t), s, limit); + endfor + fi +enddef; +beginfig(1); + path T; T = for i = 1 upto 3: 220 up rotated (120i) -- endfor cycle; + make_gasket(T, 1/2, 20); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/rec-sierpinski-triangle.pdf b/info/drawing-with-metapost/src/rec-sierpinski-triangle.pdf Binary files differnew file mode 100644 index 0000000000..7a2c122f2c --- /dev/null +++ b/info/drawing-with-metapost/src/rec-sierpinski-triangle.pdf diff --git a/info/drawing-with-metapost/src/rec-simple-tree.mp b/info/drawing-with-metapost/src/rec-simple-tree.mp new file mode 100644 index 0000000000..59657cb1b8 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-simple-tree.mp @@ -0,0 +1,33 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\usepackage{dwmpcode} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef make_tree(expr level, bar) = + draw bar; + if level > 0: + for t=-theta, theta: + make_tree(level - 1, bar shifted - point 0 of bar + scaled r rotated t shifted point 1 of bar + ); + endfor + fi +enddef; +beginfig(1); +picture T[]; +numeric r, theta; +r = 0.58; theta = 60; T1 = image( + make_tree(3, origin -- 100 up); + label.urt("\ $r=" & decimal r & "$, $\theta=" & decimal theta & "$", origin) withcolor \mpcolor{textred}; +); +r := 0.75; theta := 14; T2 = image( + make_tree(10, origin -- 100 up); + label.urt("\ $r=" & decimal r & "$, $\theta=" & decimal theta & "$", origin) withcolor \mpcolor{textred}; +); +draw T1 shifted 32 up; +draw T2 shifted 128 right; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/rec-simple-tree.pdf b/info/drawing-with-metapost/src/rec-simple-tree.pdf Binary files differnew file mode 100644 index 0000000000..1c3c848604 --- /dev/null +++ b/info/drawing-with-metapost/src/rec-simple-tree.pdf diff --git a/info/drawing-with-metapost/src/rope.mp b/info/drawing-with-metapost/src/rope.mp new file mode 100644 index 0000000000..09bf92a56a --- /dev/null +++ b/info/drawing-with-metapost/src/rope.mp @@ -0,0 +1,21 @@ +vardef rope expr c = + save s, w, hemp, n, a, b, A; + color hemp; hemp = 1/256 (192, 149, 82); + numeric s, w, n, A; + w = -1; n = -1; A = arclength c; s = A/floor(A/2); + path a[]; + for t=0 step s until A + 1: + a[incr n] = (0,+w) rotated angle direction arctime t-3/2s of c of c shifted point arctime t-3/2s of c of c + .. (0,+w) rotated angle direction arctime t-1/2s of c of c shifted point arctime t-1/2s of c of c + .. (0,-w) rotated angle direction arctime t+1/2s of c of c shifted point arctime t+1/2s of c of c + .. (0,-w) rotated angle direction arctime t+3/2s of c of c shifted point arctime t+3/2s of c of c; + endfor + image( + for i=1 upto n: + path b; b = buildcycle(a[i-1], reverse a[i]); + fill b withcolor 1/2[white, hemp]; + draw b withpen pencircle scaled 1/8; + endfor + ) + enddef; + diff --git a/info/drawing-with-metapost/src/ruler-cm.mp b/info/drawing-with-metapost/src/ruler-cm.mp new file mode 100644 index 0000000000..bdbaad10b5 --- /dev/null +++ b/info/drawing-with-metapost/src/ruler-cm.mp @@ -0,0 +1,6 @@ +numeric u[]; +u0 = 1 cm; +u1 = 1 mm; +drawoptions(withcolor 0.54 red); +input ruler +drawoptions(); diff --git a/info/drawing-with-metapost/src/ruler-inch.mp b/info/drawing-with-metapost/src/ruler-inch.mp new file mode 100644 index 0000000000..54fb5a654d --- /dev/null +++ b/info/drawing-with-metapost/src/ruler-inch.mp @@ -0,0 +1,6 @@ +numeric u[]; +u0 = 1 in; +u1 = 6 bp; +drawoptions(withcolor 0.54 red); +input ruler +drawoptions(); diff --git a/info/drawing-with-metapost/src/ruler-pt.mp b/info/drawing-with-metapost/src/ruler-pt.mp new file mode 100644 index 0000000000..0b0992bae4 --- /dev/null +++ b/info/drawing-with-metapost/src/ruler-pt.mp @@ -0,0 +1,6 @@ +numeric u[]; +u0 = 12 bp; +u1 = 1 bp; +drawoptions(withcolor 0.54 red); +input ruler +drawoptions(); diff --git a/info/drawing-with-metapost/src/ruler.mp b/info/drawing-with-metapost/src/ruler.mp new file mode 100644 index 0000000000..9d11af25d3 --- /dev/null +++ b/info/drawing-with-metapost/src/ruler.mp @@ -0,0 +1,19 @@ +% add a ruler along the left hand and lower edges +% of the bounding box of the currentpicture +% ** meant to be embedded by something that sets u[] appropriately ** +path B; B = bbox currentpicture; +for s=0, 1: + path p; numeric a; pair o; + p = subpath (0, 1) of if s=0: reverse fi B; + a = arclength p; + o = if s=0: left else: down fi; + for i=0 upto 3: + exitif not known u[i]; + for j=0 upto floor(a/u[i]): + pair t; t = point arctime j*u[i] of p of p; + draw (origin -- (6 - 2i) * o) shifted t; + if i=0: label(decimal j, t shifted 12 o); fi + endfor + endfor + draw p; +endfor diff --git a/info/drawing-with-metapost/src/saturn-code.mp b/info/drawing-with-metapost/src/saturn-code.mp new file mode 100644 index 0000000000..9e80e0a492 --- /dev/null +++ b/info/drawing-with-metapost/src/saturn-code.mp @@ -0,0 +1,17 @@ +path globe, gap, ring[], limb[]; +globe = fullcircle scaled 2cm; +gap = fullcircle xscaled 3cm yscaled .8cm; +ring1 = fullcircle xscaled 4cm yscaled 1.2cm; +ring2 = ring1 scaled 0.93; +ring3 = ring1 scaled 0.89; +limb1 = buildcycle(subpath(5,7) of ring1, subpath(8,4) of globe); +limb2 = buildcycle(subpath(5,7) of gap, subpath(-2,6) of globe); +picture saturn; saturn = image( + fill ring1 withcolor .1 red + .1 green + .4 white; + fill ring2 withcolor .2 white; + fill ring3 withcolor .1 red + .1 green + .6 white; + unfill gap; + fill limb1 withcolor .2 red + .1 green + .7 white; + fill limb2 withcolor .2 red + .1 green + .7 white; +); +draw saturn rotated 30; diff --git a/info/drawing-with-metapost/src/saturn.mp b/info/drawing-with-metapost/src/saturn.mp new file mode 100644 index 0000000000..08825c4f94 --- /dev/null +++ b/info/drawing-with-metapost/src/saturn.mp @@ -0,0 +1,10 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +input saturn-code +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/saturn.pdf b/info/drawing-with-metapost/src/saturn.pdf Binary files differnew file mode 100644 index 0000000000..3ba35ab965 --- /dev/null +++ b/info/drawing-with-metapost/src/saturn.pdf diff --git a/info/drawing-with-metapost/src/shadows.mp b/info/drawing-with-metapost/src/shadows.mp new file mode 100644 index 0000000000..9134946b15 --- /dev/null +++ b/info/drawing-with-metapost/src/shadows.mp @@ -0,0 +1,37 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + +path b, w; +b = ((-3,-4)--(3,-2)--(3,+2)--(-3,4)--cycle) scaled 5; +w = b reflectedabout(up, down); + +numeric n; +n = 128; + +picture B, W; +B = image(for i=0 step 1/n until 1: + draw point 4-i of b -- point 1+i**2 of b + withcolor 1-i**8; + endfor); + + W = image(for i=0 step 1/n until 1: + draw point 4-i of w -- point 1+i**2 of w + withcolor 3/4-i**8; + endfor); + +for i=-9 upto 9: + for j=-4 upto 4: + draw if odd (i+j): W else: B fi shifted (i*30,j*30); + endfor +endfor + +clip currentpicture to bbox currentpicture yscaled 7/8; + + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/shadows.pdf b/info/drawing-with-metapost/src/shadows.pdf Binary files differnew file mode 100644 index 0000000000..ea0b67ef2b --- /dev/null +++ b/info/drawing-with-metapost/src/shadows.pdf diff --git a/info/drawing-with-metapost/src/shady-circles.mp b/info/drawing-with-metapost/src/shady-circles.mp new file mode 100644 index 0000000000..62737f38fe --- /dev/null +++ b/info/drawing-with-metapost/src/shady-circles.mp @@ -0,0 +1,24 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path c, C; numeric r; r = 60; +c = fullcircle scaled 2r shifted (-r, 0); +C = fullcircle scaled 4r shifted (2r, 0); +numeric t, u; +(t, whatever) = C intersectiontimes C shifted (center c - center C); +(u, whatever) = c intersectiontimes (point t of C -- center c); +path s; +s = subpath (0, u) of c -- subpath (t, 4) of C -- cycle; +for i=0 upto r: + draw (left--right) scaled 2r rotated -42 shifted (3i, 0) + withpen pencircle scaled 1/4 withcolor 2/3 blue; +endfor +clip currentpicture to s; +draw c; draw C; +draw center c -- center C -- point t of C -- cycle; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/shady-circles.pdf b/info/drawing-with-metapost/src/shady-circles.pdf Binary files differnew file mode 100644 index 0000000000..df8eef9b56 --- /dev/null +++ b/info/drawing-with-metapost/src/shady-circles.pdf diff --git a/info/drawing-with-metapost/src/simple-circle-labels.mp b/info/drawing-with-metapost/src/simple-circle-labels.mp new file mode 100644 index 0000000000..77aa5e450b --- /dev/null +++ b/info/drawing-with-metapost/src/simple-circle-labels.mp @@ -0,0 +1,15 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path C; C = fullcircle scaled 100; draw C withcolor 2/3 red; +for i=0 upto 11: + drawdot point 2/3 i of C withpen pencircle scaled dotlabeldiam; + label("$p_{" & decimal i & "}$", point 2/3 i of C scaled 1.2); +endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/simple-circle-labels.pdf b/info/drawing-with-metapost/src/simple-circle-labels.pdf Binary files differnew file mode 100644 index 0000000000..209463a97a --- /dev/null +++ b/info/drawing-with-metapost/src/simple-circle-labels.pdf diff --git a/info/drawing-with-metapost/src/simple-number-line.mp b/info/drawing-with-metapost/src/simple-number-line.mp new file mode 100644 index 0000000000..83ef3d2833 --- /dev/null +++ b/info/drawing-with-metapost/src/simple-number-line.mp @@ -0,0 +1,14 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + draw (left--right) scaled 2in withcolor 2/3 red; + for i=-4 upto 4: + dotlabel.top("$" & decimal i & "$", (32i, 0)); + endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/simple-number-line.pdf b/info/drawing-with-metapost/src/simple-number-line.pdf Binary files differnew file mode 100644 index 0000000000..bb798247c5 --- /dev/null +++ b/info/drawing-with-metapost/src/simple-number-line.pdf diff --git a/info/drawing-with-metapost/src/split-labels.mp b/info/drawing-with-metapost/src/split-labels.mp new file mode 100644 index 0000000000..ca298e5e99 --- /dev/null +++ b/info/drawing-with-metapost/src/split-labels.mp @@ -0,0 +1,25 @@ +\documentclass[border=0mm]{standalone} +\usepackage{luamplib} +\def\s#1{\let\\\cr\vbox{\halign{\hfil\strut ##\hfil\cr#1\crcr}}} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +beginfig(1); +z0 = origin; z1 = 100 down; z2 = 144 down; +picture L[]; +L0 = thelabel(btex \vbox{\hsize 2in\parindent 0pt\raggedright + An extended caption or label that will be set as a + small paragraph with automatic hyphenation and + line-wrapping. +} etex, z0); +L1 = thelabel(btex \s{Single line} etex, z1); +L2 = thelabel(btex \s{Longer text split\\onto a new line} etex, z2); +forsuffixes $=0,1,2: + draw L$; draw bbox L$ withcolor Blues 8 2; + draw fullcircle scaled 2 shifted z$ withcolor Reds 8 7; +endfor +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/split-labels.pdf b/info/drawing-with-metapost/src/split-labels.pdf Binary files differnew file mode 100644 index 0000000000..d1a1ba6fd5 --- /dev/null +++ b/info/drawing-with-metapost/src/split-labels.pdf diff --git a/info/drawing-with-metapost/src/tangent-times-on-circle.mp b/info/drawing-with-metapost/src/tangent-times-on-circle.mp new file mode 100644 index 0000000000..b59986110c --- /dev/null +++ b/info/drawing-with-metapost/src/tangent-times-on-circle.mp @@ -0,0 +1,24 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input tangent-times + +beginfig(1); +path C; pair p; numeric a, b; +C = fullcircle scaled 100; +p = 189 right rotated 30; +(a, b) = tangent_times(C, p); + +draw C withpen pencircle scaled 2 withcolor 3/4[(.2, .2, .7), white]; +draw p -- subpath (a, b) of C -- cycle dashed evenly; +label.llft("$C$", point 5 of C); +dotlabel.urt("$p$", p); +dotlabel.ulft("point $a$ of $C$", point a of C); +dotlabel.lrt("point $b$ of $C$", point b of C); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tangent-times-on-circle.pdf b/info/drawing-with-metapost/src/tangent-times-on-circle.pdf Binary files differnew file mode 100644 index 0000000000..074d756042 --- /dev/null +++ b/info/drawing-with-metapost/src/tangent-times-on-circle.pdf diff --git a/info/drawing-with-metapost/src/tangent-times.mp b/info/drawing-with-metapost/src/tangent-times.mp new file mode 100644 index 0000000000..ecbb62bb97 --- /dev/null +++ b/info/drawing-with-metapost/src/tangent-times.mp @@ -0,0 +1,11 @@ +vardef tangent_times(expr C, p) = + save m, a, b, G, H; + pair m; numeric a, b; path G, H; + m = 1/2[p, center C]; + H = halfcircle scaled abs (p - center C) + rotated angle (p - center C) shifted m; + G = H rotatedabout(m, 180); + (a, whatever) = C intersectiontimes H; + (b, whatever) = C intersectiontimes G; + (a, b if b < a: + 8 fi) +enddef; diff --git a/info/drawing-with-metapost/src/tangents-on-path.mp b/info/drawing-with-metapost/src/tangents-on-path.mp new file mode 100644 index 0000000000..c93dc1d1d6 --- /dev/null +++ b/info/drawing-with-metapost/src/tangents-on-path.mp @@ -0,0 +1,60 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{dwmpcode} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path p; numeric s; s = 144; +p := (origin {1, 3} .. (1, 0) .. {1, 2} (2, 0)) scaled s; + +ahangle := 20; +ahlength := 6; +picture P[]; +P1 = image( + draw p withcolor 0.54 red; + for t=0 upto length p: draw fullcircle scaled dotlabeldiam shifted point t of p; endfor + label.bot(TEX("\mpl{point 0 of p}") scaled 0.8, point 0 of p shifted 16 right); + label.urt(TEX("\mpl{point 1 of p}") scaled 0.8, point 1 of p); + label.top(TEX("\mpl{point 2 of p}") scaled 0.8, point 2 of p shifted 16 left); +); +P2 = image( + draw p withcolor 0.54 red; + drawoptions(withcolor 1/2); + for t=0 upto length p: + drawarrow precontrol t of p -- postcontrol t of p; + fill fullcircle scaled 2 shifted precontrol t of p; + draw fullcircle scaled 4 shifted point t of p; + endfor + label.bot(TEX("\mpl{precontrol 0 of p}") scaled 0.8, precontrol 0 of p shifted (24, -2)); + label.top(TEX("\mpl{postcontrol 0 of p}") scaled 0.8, postcontrol 0 of p shifted (24, 4)); + label.top(TEX("\mpl{precontrol 1 of p}") scaled 0.8, precontrol 1 of p); + label.bot(TEX("\mpl{postcontrol 1 of p}") scaled 0.8, postcontrol 1 of p); + label.bot(TEX("\mpl{precontrol 2 of p}") scaled 0.8, precontrol 2 of p); + label.top(TEX("\mpl{postcontrol 2 of p}") scaled 0.8, postcontrol 2 of p shifted (-24, 2)); + drawoptions(); +); +P3 = image( + draw p withcolor 0.54 red; + drawoptions(withcolor 1/2); + for t=0 step 1/4 until length p: + pair s; s = 16 unitvector(direction t of p); + drawarrow (-s -- s) shifted point t of p; + draw point t of p withpen pencircle scaled 2; + endfor + drawoptions(); + numeric x, y; x = 8; y = -12; + label.rt(TEX("\mpl{pair s;}") scaled 0.7, (x, y+10)); + label.rt(TEX("\mpl{for t=0 step 1/4 until length p:}") scaled 0.7, (x, y)); + label.rt(TEX("\quad\mpl{s := 16 unitvector(direction t of p);}") scaled 0.7, (x, y-10)); + label.rt(TEX("\quad\mpl{drawarrow (-s -- s) shifted point t of p);}") scaled 0.7, (x, y-20)); + label.rt(TEX("\quad\mpl{draw point t of p withpen pencircle scaled 2;}") scaled 0.7, (x, y-30)); + label.rt(TEX("\mpl{endfor}") scaled 0.7, (x, y-40)); +); +draw P1; +draw P2 shifted 144 down; +draw P3 shifted 248 down; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tangents-on-path.pdf b/info/drawing-with-metapost/src/tangents-on-path.pdf Binary files differnew file mode 100644 index 0000000000..6fdd9064a2 --- /dev/null +++ b/info/drawing-with-metapost/src/tangents-on-path.pdf diff --git a/info/drawing-with-metapost/src/tangents-point-to-circle.mp b/info/drawing-with-metapost/src/tangents-point-to-circle.mp new file mode 100644 index 0000000000..500e009160 --- /dev/null +++ b/info/drawing-with-metapost/src/tangents-point-to-circle.mp @@ -0,0 +1,38 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path C, C'; pair p, t, t'; +C = fullcircle scaled 100; +p = 189 right rotated 30; + +C' = fullcircle zscaled p shifted 1/2 p; +% t = C intersectionpoint C'; +% t' = reverse C intersectionpoint C'; +t = C intersectionpoint halfcircle zscaled p shifted 1/2 p; +t' = C intersectionpoint halfcircle zscaled -p shifted 1/2 p; + +drawoptions(dashed withdots scaled 1/4 withcolor (.2, .2, .7)); +draw C'; draw t -- center C -- p; + +drawoptions(withcolor (.2, .2, .7)); +dotlabel.lrt("$m$", 1/2 p); +dotlabel.llft("$o$", center C); + +drawoptions(dashed evenly); +draw t -- p -- t'; + +drawoptions(); +draw C; +label.llft("$C$", point 5 of C); + +dotlabel.urt("$p$", p); +dotlabel.ulft("$t$", t); +dotlabel.bot("\strut $t'$", t'); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tangents-point-to-circle.pdf b/info/drawing-with-metapost/src/tangents-point-to-circle.pdf Binary files differnew file mode 100644 index 0000000000..6991303955 --- /dev/null +++ b/info/drawing-with-metapost/src/tangents-point-to-circle.pdf diff --git a/info/drawing-with-metapost/src/tangents-two-circles-exterior.mp b/info/drawing-with-metapost/src/tangents-two-circles-exterior.mp new file mode 100644 index 0000000000..504fdf19e0 --- /dev/null +++ b/info/drawing-with-metapost/src/tangents-two-circles-exterior.mp @@ -0,0 +1,55 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input tangent-times +color tb; tb = (0.2, 0.2, 0.7); +beginfig(1); +path A, B; +A = fullcircle scaled 144; +B = fullcircle scaled 60 shifted (200, 140); + +numeric R, r; +R = abs (point 0 of A - center A); +r = abs (point 0 of B - center B); + +path C; numeric t, u; +C = fullcircle scaled (2R-2r) shifted center A; +(t, u) = tangent_times(C, center B); + +draw A withpen pencircle scaled 2 withcolor 3/4[blue, white]; +draw B withpen pencircle scaled 2 withcolor 3/4[blue, white]; +draw C withpen pencircle scaled 1 withcolor 7/8[blue, white]; + +draw subpath (t, u) of C -- center B -- cycle dashed evenly withcolor 1/2; +draw subpath (t, u) of A -- subpath (u-8, t) of B -- cycle; + +drawoptions(dashed withdots scaled 1/2 withcolor 1/4); +forsuffixes $=t, u: + draw point $ of C -- point $ of A; + draw center B -- point $ of B; +endfor + +drawoptions(withcolor tb); +label.urt("$A$", point 1 of A); +label.ulft("$B$", point 4 of B); +label.urt("$C$", point 1/2 of C); +drawdot center B withpen pencircle scaled dotlabeldiam; + +drawoptions(); +dotlabel.lrt("$t$", point t of C); +dotlabel.ulft("$t'$", point t of A); +dotlabel.ulft("$t''$", point t of B); +dotlabel.ulft("$u$", point u of C); +dotlabel.lrt("$u'$", point u of A); +dotlabel.lrt("$u''$", point u of B); + +label.rt("$r$", 1/2[center B, point t of B]); +label.rt("$r$", 1/2[point t of C, point t of A]); + + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tangents-two-circles-exterior.pdf b/info/drawing-with-metapost/src/tangents-two-circles-exterior.pdf Binary files differnew file mode 100644 index 0000000000..db6464e4a4 --- /dev/null +++ b/info/drawing-with-metapost/src/tangents-two-circles-exterior.pdf diff --git a/info/drawing-with-metapost/src/tangents-two-circles-interior.mp b/info/drawing-with-metapost/src/tangents-two-circles-interior.mp new file mode 100644 index 0000000000..afa46790e6 --- /dev/null +++ b/info/drawing-with-metapost/src/tangents-two-circles-interior.mp @@ -0,0 +1,59 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input tangent-times +vardef adjust_time(expr tt, AA, BB) = + tt + 1/45 angle (point 0 of AA - center AA) + - 1/45 angle (point 0 of BB - center BB) +enddef; +beginfig(1); +path A, B; +A = fullcircle scaled 144 rotated uniformdeviate 360; +B = fullcircle scaled 60 shifted 240 right rotated 36; + +numeric R, r; +R = abs (point 0 of A - center A); +r = abs (point 0 of B - center B); + +path C; +C = fullcircle scaled (2R+2r) shifted center A; % NB +ve + +numeric t, t', t'', u, u', u''; +(t, u) = tangent_times(C, center B); +t' = adjust_time(t, C, A); +u' = adjust_time(u, C, A); +t'' = adjust_time(t + 4, C, B); % Note the plus fours +u'' = adjust_time(u + 4, C, B); + +draw A withpen pencircle scaled 2 withcolor 3/4[blue, white]; +draw B withpen pencircle scaled 2 withcolor 3/4[blue, white]; +draw C withpen pencircle scaled 1 withcolor 3/4[blue, white]; + +draw subpath (t', u') of A -- subpath (u'', t'') of B -- cycle; +draw center B -- subpath (t, u) of C -- cycle dashed evenly; + +draw center B -- point t'' of B dashed withdots scaled 1/2; +draw center B -- point u'' of B dashed withdots scaled 1/2; +draw point t of C -- point t' of A dashed withdots scaled 1/2; +draw point u of C -- point u' of A dashed withdots scaled 1/2; + +dotlabel.ulft(btex $t$ etex, point t of C); +dotlabel.lrt (btex $t'$ etex, point t' of A); +dotlabel.lrt (btex $t''$ etex, point t'' of B); +dotlabel.lrt (btex $u$ etex, point u of C); +dotlabel.ulft(btex $u'$ etex, point u' of A); +dotlabel.ulft(btex $u''$ etex, point u'' of B); +drawdot center B withpen pencircle scaled dotlabeldiam; + +drawoptions(withcolor 1/2[blue, white]); + label.urt(btex $A$ etex, point 1/2(t'+u'- 7.6) of A); + label.rt (btex $B$ etex, point 1/2(t''+u''- 2) of B); + label.urt(btex $C$ etex, point 1/2(t+u-8) of C); +drawoptions(); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tangents-two-circles-interior.pdf b/info/drawing-with-metapost/src/tangents-two-circles-interior.pdf Binary files differnew file mode 100644 index 0000000000..f9faedbd97 --- /dev/null +++ b/info/drawing-with-metapost/src/tangents-two-circles-interior.pdf diff --git a/info/drawing-with-metapost/src/tikzcd-example.mp b/info/drawing-with-metapost/src/tikzcd-example.mp new file mode 100644 index 0000000000..3796618baf --- /dev/null +++ b/info/drawing-with-metapost/src/tikzcd-example.mp @@ -0,0 +1,58 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +vardef sbox primary P = + superellipse(1/2[point 1 of bbox P, point 2 of bbox P], + 1/2[point 2 of bbox P, point 3 of bbox P], + 1/2[point 3 of bbox P, point 0 of bbox P], + 1/2[point 0 of bbox P, point 1 of bbox P], 0.75) + enddef; + +beginfig(1); + r = 0.9; + z0 = (0, 0); + z1 = r*(89, -55); + z2 = r*(42, 30); + z3 = z1 + z2; + + picture FE, FEV, E, EV, M, N, U, V; + FE = thelabel("$f^*E$", (x0, y0)); + M = thelabel("$M$", (x0, y1)); + E = thelabel("$E$", (x1, y0)); + N = thelabel("$N$", (x1, y1)); + FEV = thelabel("$f^*E_V$", (x2, y2)); + U = thelabel("$U$", (x2, y3)); + EV = thelabel("$E_V$", (x3, y2)); + V = thelabel("$V$", (x3, y3)); + + forsuffixes @=E, M, N, U, V, EV, FE, FEV: draw @; endfor + + ahangle := 20; + vardef connect(expr a, b) = + save line; path line; + interim bboxmargin := 3; + line = center a .. center b + cutbefore sbox a cutafter sbox b; + cutdraw line withpen pencircle scaled 4 withcolor background; + drawarrow line + enddef; + + connect(FEV, EV); + connect(FEV, U); + connect(EV, V); + connect(U, V); + + connect(FEV, FE); + connect(EV, E); + connect(U, M); + connect(V, N); + + connect(FE, E); + connect(FE, M); + connect(E, N); + connect(M, N); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/tikzcd-example.pdf b/info/drawing-with-metapost/src/tikzcd-example.pdf Binary files differnew file mode 100644 index 0000000000..c2b95990db --- /dev/null +++ b/info/drawing-with-metapost/src/tikzcd-example.pdf diff --git a/info/drawing-with-metapost/src/tiling-hex-trig.mp b/info/drawing-with-metapost/src/tiling-hex-trig.mp new file mode 100644 index 0000000000..ce9d1565e6 --- /dev/null +++ b/info/drawing-with-metapost/src/tiling-hex-trig.mp @@ -0,0 +1,83 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +vardef make_image(expr P, n, u, v, s, arrows) = + save currentpicture; + picture currentpicture; + currentpicture := nullpicture; + for i = -n upto n: + for j = -n upto n: + draw P shifted (i * u + j * v); + endfor + endfor; + if arrows: + interim ahangle := 20; + drawarrow origin--u dashed evenly scaled 1/2 withpen pencircle scaled 1/4 withcolor Reds 8 8; + drawarrow origin--v dashed evenly scaled 1/2 withpen pencircle scaled 1/4 withcolor Blues 8 8; + draw origin withpen pencircle scaled 1; + fi + clip currentpicture to s; + draw s dashed withdots scaled 1/2; + currentpicture +enddef; +beginfig(1); + path hexagon, triangle; pair u, v; numeric n; + hexagon = for i=0 upto 5: (0, 16) rotated 60i -- endfor cycle; + triangle = for i=0 upto 2: (0, 16) rotated 120i -- endfor cycle; + + u = point 0 of triangle - point 1 of triangle; + v = u rotated -60; + + path s; s = superellipse(89 right, 89 up, 89 left, 89 down, 0.78); + n = 8; + picture P[]; + P3 = make_image(triangle, n, u, v, s, true); + P6 = make_image(hexagon, n, u, v, s, true); + + picture T; T = image( + path t'; t' = triangle reflectedabout(point 0 of triangle, point 1 of triangle); + fill triangle withcolor Reds 8 2; + fill t' withcolor Blues 8 2; + draw triangle; + draw t'; + ); + P4 = make_image(T, n, u, v, s, true); + picture H; H = image( + path ha, hb; + ha = hexagon reflectedabout(point 0 of hexagon, point 1 of hexagon); + hb = hexagon reflectedabout(point 0 of hexagon, point 5 of hexagon); + fill hexagon withcolor Oranges 8 2; + fill ha withcolor Blues 8 2; + fill hb withcolor Greens 8 2; + draw hexagon; draw ha; draw hb; + ); + P7 = make_image(H, n, u zscaled 1.732 dir 30, v zscaled 1.732 dir 30, s, true); + + picture dualt, dualh; + dualt = image( + draw triangle withcolor 1/2; + for i=0 upto 2: draw origin -- point i+1/2 of triangle withcolor Reds 7 6; endfor + draw currentpicture reflectedabout(point 0 of triangle, point 1 of triangle); + ); + dualh = image( + draw hexagon withcolor 1/2; + for i=0 upto 5: + draw origin -- point i+1/2 of hexagon withcolor Reds 7 6; + endfor + ); + P5 = make_image(dualt, n, u, v, s, false); + P8 = make_image(dualh, n, u, v, s, false); + + draw P3 shifted (-100, +100); + draw P4 shifted (-100, -100); + draw P5 shifted (-100, -300); + draw P6 shifted (+100, +100); + draw P7 shifted (+100, -100); + draw P8 shifted (+100, -300); +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tiling-hex-trig.pdf b/info/drawing-with-metapost/src/tiling-hex-trig.pdf Binary files differnew file mode 100644 index 0000000000..a4a9313e42 --- /dev/null +++ b/info/drawing-with-metapost/src/tiling-hex-trig.pdf diff --git a/info/drawing-with-metapost/src/tiling-kepler-k.mp b/info/drawing-with-metapost/src/tiling-kepler-k.mp new file mode 100644 index 0000000000..81ea654def --- /dev/null +++ b/info/drawing-with-metapost/src/tiling-kepler-k.mp @@ -0,0 +1,31 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +beginfig(1); + numeric a; a = 13; + path s; s = unitsquare shifted -(1/2, 1/2) rotated 45 shifted (sqrt(3/2), 0) scaled a; + picture unit; + ahangle := 10; + unit = image( + for i=-1 upto 1: + fill s rotated 60i withcolor Spectral[9][3+2i]; + draw s rotated 60i; + endfor + ); + pair u, v; + u = point 2 of s - point 0 of s rotated -60; + v = u rotated 60; + numeric n; n = 8; + for i = -n upto n: + for j = -n upto n: + draw unit shifted (i * u + j * v); + endfor + endfor + clip currentpicture to fullcircle scaled 400; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tiling-kepler-k.pdf b/info/drawing-with-metapost/src/tiling-kepler-k.pdf Binary files differnew file mode 100644 index 0000000000..c2b4272c60 --- /dev/null +++ b/info/drawing-with-metapost/src/tiling-kepler-k.pdf diff --git a/info/drawing-with-metapost/src/tiling-kepler-x.mp b/info/drawing-with-metapost/src/tiling-kepler-x.mp new file mode 100644 index 0000000000..4eee3b32be --- /dev/null +++ b/info/drawing-with-metapost/src/tiling-kepler-x.mp @@ -0,0 +1,33 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input colorbrewer-rgb +beginfig(1); + path s; s = unitsquare shifted -(1/2, 1/2) scaled 21; + path t; t = s rotated 45; + + vardef show_corners(expr p) = + save s; numeric s; s = abs(point 1 of p - point 0 of p) / (2 + sqrt(2)); + image( for i=0 upto 3: + fill unitsquare scaled s shifted point 0 of p rotated 90i withcolor Oranges[8][1+2i]; + draw unitsquare scaled s shifted point 0 of p rotated 90i; + endfor) + enddef; + + pair u, v; + u = point 1 of s - point 0 of s; + v = u rotated 90; + numeric n; n = 8; + for i = -n upto n: + for j = -n upto n: + draw show_corners(s) shifted (i * u + j * v); + draw show_corners(s) rotated -45 shifted (i * u + j * v); + endfor + endfor + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tiling-kepler-x.pdf b/info/drawing-with-metapost/src/tiling-kepler-x.pdf Binary files differnew file mode 100644 index 0000000000..79cd5b6ff4 --- /dev/null +++ b/info/drawing-with-metapost/src/tiling-kepler-x.pdf diff --git a/info/drawing-with-metapost/src/tiling-simple.mp b/info/drawing-with-metapost/src/tiling-simple.mp new file mode 100644 index 0000000000..6b0fdf540a --- /dev/null +++ b/info/drawing-with-metapost/src/tiling-simple.mp @@ -0,0 +1,41 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); +picture P[]; +P1 = image( +for i = -10 upto 10: + draw (left--right) scaled 200 shifted (0, 20i); + draw (down--up) scaled 200 shifted (20i, 0); +endfor +clip currentpicture to fullcircle scaled 180; +draw fullcircle scaled 180; +); + +P2 = image( +path unit; pair u, v; color a, b; +unit = unitsquare scaled 24; +u = point 1 of unit - point 0 of unit; +v = point 3 of unit - point 0 of unit; +a = 3/4[red, white]; b = 3/4[blue, white]; +for i=-5 upto 5: + for j=-5 upto 5: + fill unit shifted (i*u + j*v) + withcolor if odd (i+j): a else: b fi; + endfor +endfor +for i=-5 upto 5: + for j=-5 upto 5: + draw subpath (0,2) of unit shifted (i*u + j*v); + endfor +endfor +clip currentpicture to fullcircle scaled 180; +draw fullcircle scaled 180; +); +draw P1; draw P2 shifted 200 down; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tiling-simple.pdf b/info/drawing-with-metapost/src/tiling-simple.pdf Binary files differnew file mode 100644 index 0000000000..ef086a2e0a --- /dev/null +++ b/info/drawing-with-metapost/src/tiling-simple.pdf diff --git a/info/drawing-with-metapost/src/tiling.mp b/info/drawing-with-metapost/src/tiling.mp new file mode 100644 index 0000000000..9a82b1fd39 --- /dev/null +++ b/info/drawing-with-metapost/src/tiling.mp @@ -0,0 +1,56 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + +prologues := 3; +outputtemplate := "%j%c.eps"; +vardef poly_on(expr a,b,n) = + save s,t,c; pair c; + s = 360/n; + t = 90-180/n; + c = whatever[a,b rotatedabout(a,+t)] + = whatever[b,a rotatedabout(b,-t)]; + for i=0 upto n-1: a rotatedabout(c,i*s) -- endfor cycle +enddef; + +vardef make_uniform_tiling_shd(expr u, reps, scolor, hcolor, dcolor) = + save x,y,s, h, d, t; path s[], h[], d; picture t; + (x,y) = dir 15; + d = for i=0 upto 11: (x/2y, 1/2) scaled u rotated 30i -- endfor cycle; + for i=0 upto 2: s[i] = unitsquare scaled u shifted point -1 of d rotated (60+60i); endfor + for i=0 upto 1: h[i] = poly_on(point 9+2i of d, point 8+2i of d, 6); endfor + t = image( + fill d withcolor dcolor; + for i=0 upto 2: fill s[i] withcolor scolor; endfor + for i=0 upto 1: fill h[i] withcolor hcolor; endfor + + draw d; + for i=0 upto 2: draw s[i]; endfor + for i=0 upto 1: draw h[i]; endfor + ); + save dx, dy; pair dx, dy; + dx = (u,0) + point -1/2 of d - point 11/2 of d; + dy = dx rotated 60; + for i=-reps upto reps: + for j=-reps upto reps: + draw t shifted (i*dx) shifted (j*dy); + endfor + endfor +enddef; + +def rgb(expr r,g,b) = (r/255, g/255, b/255) enddef; +color b[]; +b1 = rgb(255,255,217); +b2 = rgb(237,248,177); +b3 = rgb(199,233,180); + +beginfig(2); +picture tt; tt = image(make_uniform_tiling_shd(4mm, 7, b1, b2, b3); fill fullcircle scaled 3 withcolor red;); +clip tt to fullcircle scaled 12cm; draw tt; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tiling.pdf b/info/drawing-with-metapost/src/tiling.pdf Binary files differnew file mode 100644 index 0000000000..b37895e16c --- /dev/null +++ b/info/drawing-with-metapost/src/tiling.pdf diff --git a/info/drawing-with-metapost/src/torn-edge-circle.mp b/info/drawing-with-metapost/src/torn-edge-circle.mp new file mode 100644 index 0000000000..9a461eb737 --- /dev/null +++ b/info/drawing-with-metapost/src/torn-edge-circle.mp @@ -0,0 +1,18 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +input torn-edge +beginfig(1); +path c; c = fullcircle scaled 200; +draw c withcolor .8 white; +y=0; n = 600; +path t; t = for i=0 upto n-1: + point i/n*length(c) of c + + (0, walkr y) rotated angle direction i/n*length(c) of c + -- +endfor cycle; +draw t withcolor .67 red; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/torn-edge-circle.pdf b/info/drawing-with-metapost/src/torn-edge-circle.pdf Binary files differnew file mode 100644 index 0000000000..2142838a3d --- /dev/null +++ b/info/drawing-with-metapost/src/torn-edge-circle.pdf diff --git a/info/drawing-with-metapost/src/torn-edge-straight.mp b/info/drawing-with-metapost/src/torn-edge-straight.mp new file mode 100644 index 0000000000..18a365c4c2 --- /dev/null +++ b/info/drawing-with-metapost/src/torn-edge-straight.mp @@ -0,0 +1,14 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +input torn-edge +beginfig(2); +path t; numeric x, y; +x = 0; y=0; +t = (x, -20) -- (x, y) for i=1 upto 288: -- (incr x, walkr y) endfor -- (x, -20) -- cycle; +fill t withcolor 1/32(32,32,31); +draw t withcolor .67 blue; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/torn-edge-straight.pdf b/info/drawing-with-metapost/src/torn-edge-straight.pdf Binary files differnew file mode 100644 index 0000000000..c0afc7349f --- /dev/null +++ b/info/drawing-with-metapost/src/torn-edge-straight.pdf diff --git a/info/drawing-with-metapost/src/torn-edge.mp b/info/drawing-with-metapost/src/torn-edge.mp new file mode 100644 index 0000000000..5d0afface0 --- /dev/null +++ b/info/drawing-with-metapost/src/torn-edge.mp @@ -0,0 +1,7 @@ +vardef signr suffix $ = + if $<0: - else: + fi uniformdeviate 1 + enddef; +vardef walkr suffix $ = + $ := $ if uniformdeviate 1 < (2**-abs($)): + else: - fi signr $; + $ + enddef; diff --git a/info/drawing-with-metapost/src/trad-font-changes.mp b/info/drawing-with-metapost/src/trad-font-changes.mp new file mode 100644 index 0000000000..7b061c8da7 --- /dev/null +++ b/info/drawing-with-metapost/src/trad-font-changes.mp @@ -0,0 +1,23 @@ +prologues := 3; +outputtemplate := "trad-font-changes.eps"; +beginfig(1); +yy := 120; +forsuffixes $ = + font_times, + font_palatino, + font_charter, + font_utopia, + font_cmbright, + font_century, + font_concrete, + font_bookman, + font_arev: + draw str $ infont "texnansi-lmtt10" shifted (-22,yy) withcolor .43 green; + write "draw btex \input " & str $ & " {\bf NB}. Learn $v=u+at$ right {\it now\/}! etex shifted (80," & decimal yy & ");" to ".mplabels"; + yy := yy - 16; +endfor + write EOF to ".mplabels"; + input .mplabels; +endfig; +end. + diff --git a/info/drawing-with-metapost/src/trad-font-changes.pdf b/info/drawing-with-metapost/src/trad-font-changes.pdf Binary files differnew file mode 100644 index 0000000000..6046a3507f --- /dev/null +++ b/info/drawing-with-metapost/src/trad-font-changes.pdf diff --git a/info/drawing-with-metapost/src/trad-font-samples.mp b/info/drawing-with-metapost/src/trad-font-samples.mp new file mode 100644 index 0000000000..44674cf604 --- /dev/null +++ b/info/drawing-with-metapost/src/trad-font-samples.mp @@ -0,0 +1,50 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); + +vardef set_line(expr f, n) = + save code, name, sample; + picture code, name, sample; + code = f infont "cs-lmtt10"; + name = n infont "cs-lmss10"; + sample = "Hand in glove 42" infont f; + yy := yy - 14; + draw code shifted (-48,yy); + draw name shifted (0,yy); + draw sample shifted (124,yy); + enddef; + +vardef set_symbol_line(expr f, n) = + save code, name, sample; + picture code, name, sample; + code = f infont "cs-lmtt10"; + name = n infont "cs-lmss10"; + sample = (char 97 & char 98 & char 99 & char 100 & char 101 & char 102) infont f; + yy := yy - 14; + draw code shifted (-48,yy); + draw name shifted (0,yy); + draw sample shifted (124,yy); + enddef; + + +beginfig(2); +yy := 150; +set_line("pagk8r","Avant Garde"); +set_line("pbkl8r","Bookman"); +set_line("pcrr8r","Courier"); +set_line("phvr8r","Helvetica"); +set_line("pncr8r","New Century Schoolbook"); +set_line("pplr8r","Palatino"); +set_line("ptmr8r","Times"); +set_line("pzcmi8r","Zapf Chancery"); +yy := yy - 6; +set_symbol_line("pzdr","Zapf Dingbats"); +set_symbol_line("psyr","Symbol"); +set_symbol_line("eurm10","Euler"); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/trad-font-samples.pdf b/info/drawing-with-metapost/src/trad-font-samples.pdf Binary files differnew file mode 100644 index 0000000000..2f5c40ddda --- /dev/null +++ b/info/drawing-with-metapost/src/trad-font-samples.pdf diff --git a/info/drawing-with-metapost/src/trad-font-tables.mp b/info/drawing-with-metapost/src/trad-font-tables.mp new file mode 100644 index 0000000000..8a9a4c6c05 --- /dev/null +++ b/info/drawing-with-metapost/src/trad-font-tables.mp @@ -0,0 +1,38 @@ +% why have I set this one using `lualatex`? +% (a) to keep my build process consistent, so that I am using luamplib for *all* +% the external pictures +% (b) to show you that you can use the full traditional MP typesetting with "infont" +% even though you are compiling with lualatex... +% Toby Thurston -- 04 Apr 2023 +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); +ypos = 0; +picture matrix; +for $="texnansi-lmr10", "pplr8r": + draw ("Font: " & $) infont "texnansi-lmss10" scaled .7 shifted (-20,ypos) withcolor .67 red; + for m=0 upto 15: + ypos := ypos - 12; + if (m mod 4)=0: + ypos := ypos - 4; + fi + label.lft(decimal 16m infont "texnansi-lmss10" scaled .7, (-7,ypos+3)) withcolor .67 red; + for c=0 upto 15: + xx := 16m+c; + if (xx<>0) and (xx<>10) and (xx<>13) and (xx<>160): + matrix := char (16m+c) infont $; + draw matrix shifted (c*14,ypos) shifted 1/2(llcorner matrix - lrcorner matrix) + withcolor if ((32<xx) and (xx<127)) or (160<xx): black else: .67 blue fi; + else: + draw unitsquare shifted 1/2 left scaled 3 shifted (c*14,ypos) withcolor .67 red; + fi + endfor + endfor + for i=0 upto 4: draw (-7+56i,ypos-4) -- (-7+56i,ypos+17*12-4) withcolor .67 red; endfor + ypos := ypos - 24; +endfor +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/trad-font-tables.pdf b/info/drawing-with-metapost/src/trad-font-tables.pdf Binary files differnew file mode 100644 index 0000000000..13fc4801b7 --- /dev/null +++ b/info/drawing-with-metapost/src/trad-font-tables.pdf diff --git a/info/drawing-with-metapost/src/trisection-classical.mp b/info/drawing-with-metapost/src/trisection-classical.mp new file mode 100644 index 0000000000..980ceae25e --- /dev/null +++ b/info/drawing-with-metapost/src/trisection-classical.mp @@ -0,0 +1,41 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + path C, base, section, aa; + numeric theta; + C = fullcircle scaled 144; + base = origin -- right scaled 172.8; + theta = 60; + + z1 = point 0 of C scaled 1.2 rotated theta; + z2 = point 4 of C rotated theta; + + draw C; + draw z1 -- origin -- base; + draw origin -- z2 -- base shifted z2 dashed evenly; + + section = base scaled 1.2 rotated 1/3 theta shifted z2 cutafter base; + draw section; + + label.ulft("$a$", 1/2 z2); + aa = subpath (eps, infinity) of section cutbefore C; + label.lrt("$a$", point 1/2 of aa); + + draw origin -- point 0 of aa dashed withdots scaled 1/2; + + drawoptions(withcolor 2/3 blue); % distinguish angle labels + label("$\theta$", 16 dir 1/2 theta); + label("$\theta/3$", 52 dir 1/6 theta shifted z2); + drawoptions(); + + drawoptions(withpen pencircle scaled 2/3 dotlabeldiam); + draw origin; draw z2; draw point 0 of aa; draw point 1 of aa; + drawoptions(); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/trisection-classical.pdf b/info/drawing-with-metapost/src/trisection-classical.pdf Binary files differnew file mode 100644 index 0000000000..7dc1d031e2 --- /dev/null +++ b/info/drawing-with-metapost/src/trisection-classical.pdf diff --git a/info/drawing-with-metapost/src/trisection-simple.mp b/info/drawing-with-metapost/src/trisection-simple.mp new file mode 100644 index 0000000000..8f55281efb --- /dev/null +++ b/info/drawing-with-metapost/src/trisection-simple.mp @@ -0,0 +1,22 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + path ray; + numeric theta; + ray = origin -- 200 right; + theta = 42; + draw ray; + draw ray rotated 1/3 theta withcolor 2/3 red; + draw ray rotated 2/3 theta withcolor 2/3 red; + draw ray rotated theta; + dotlabel.llft("$0$", origin); + label("$\theta/3$", 72 right rotated 1/6 theta) withcolor 2/3 blue; + label("$\theta/3$", 72 right rotated 3/6 theta) withcolor 2/3 blue; + label("$\theta/3$", 72 right rotated 5/6 theta) withcolor 2/3 blue; +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/trisection-simple.pdf b/info/drawing-with-metapost/src/trisection-simple.pdf Binary files differnew file mode 100644 index 0000000000..7a63ce3cf3 --- /dev/null +++ b/info/drawing-with-metapost/src/trisection-simple.pdf diff --git a/info/drawing-with-metapost/src/trisection-triangles.mp b/info/drawing-with-metapost/src/trisection-triangles.mp new file mode 100644 index 0000000000..400934e647 --- /dev/null +++ b/info/drawing-with-metapost/src/trisection-triangles.mp @@ -0,0 +1,42 @@ +\documentclass[border=0mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +randomseed := 2485.81543; +vardef measured_angle(expr p, o, q) = + (angle (p-o) - angle (q-o)) mod 360 +enddef; +beginfig(1); +picture T; +for i=0 upto 1: + for j=0 upto 1: + clearxy; + T := image( + z1 = (120 + uniformdeviate 21, 0); + z2 = (120 + uniformdeviate 21, 0) rotated 120 rotated 21 normaldeviate; + z3 = (120 + uniformdeviate 21, 0) rotated 240 rotated 21 normaldeviate; + numeric a, b, c; + a = measured_angle(z3, z1, z2); + b = measured_angle(z1, z2, z3); + c = measured_angle(z2, z3, z1); + z4 = whatever [z1, z2 rotatedabout(z1, 1/3 a)] + = whatever [z2, z3 rotatedabout(z2, 2/3 b)]; + z5 = whatever [z2, z3 rotatedabout(z2, 1/3 b)] + = whatever [z3, z1 rotatedabout(z3, 2/3 c)]; + z6 = whatever [z3, z1 rotatedabout(z3, 1/3 c)] + = whatever [z1, z2 rotatedabout(z1, 2/3 a)]; + fill z4--z5--z6--cycle withcolor 3/4[red + 1/2 green, white]; + draw z4--z5--z6--cycle; + draw z1 -- z4 -- z2 -- z5 -- z3 -- z6 -- cycle + dashed withdots scaled 1/4; + draw z1 -- z2 -- z3 -- cycle; + ); + draw T shifted (200i, 240j); + endfor +endfor +label.rt(btex \vbox{\halign{#\hfil\cr The trisectors of each angle\cr +in any given triangle form a\cr central equilateral triangle.\cr}} etex, (24, 128)); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/trisection-triangles.pdf b/info/drawing-with-metapost/src/trisection-triangles.pdf Binary files differnew file mode 100644 index 0000000000..2bf044ad78 --- /dev/null +++ b/info/drawing-with-metapost/src/trisection-triangles.pdf diff --git a/info/drawing-with-metapost/src/tufte-budget.mp b/info/drawing-with-metapost/src/tufte-budget.mp new file mode 100644 index 0000000000..6fd4c3ead1 --- /dev/null +++ b/info/drawing-with-metapost/src/tufte-budget.mp @@ -0,0 +1,65 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{unicode-math} +\setmainfont[Numbers=OldStyle]{TeX Gyre Pagella} +\setmathfont{TeX Gyre Pagella Math} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path data, p; +data = (1967, 311) + -- (1968, 332) + -- (1969, 372) + -- (1970, 385) + -- (1971, 385) + -- (1972, 393) + -- (1973, 387) + -- (1974, 381) + -- (1975, 387) + -- (1976, 400) + -- (1977, 380); + +numeric u, v; +u = xpart urcorner textext("1980\kern 0.75em"); +v = 1.414; +p = data shifted -(xpart point 0 of data, 300) xscaled u yscaled v; +draw p; + +for d = 300 step 20 until 400: + numeric y; y = (d - 300) * v; + label.lft("\strut\scriptsize\$\,\small" & decimal d, (-12, y)); + draw (-8, y) -- (-12, y); + if d > 370: + draw (xpart point 3 of p, y) -- (xpart point infinity of p + 8, y) + dashed evenly scaled 1/4 withpen pencircle scaled 1/4; + fi +endfor +path a; a = (xpart point infinity of p + 21, (380 - 300) * v) + -- (xpart point infinity of p + 21, (400 - 300) * v); +interim ahangle := 180; +interim ahlength := 2; +drawdblarrow a withpen pencircle scaled 1/4; +label("\small 5\%", point 1/2 of a shifted 12 right); + +for t=0 upto length p: + numeric x, y; (x, y) = point t of p; + undraw (x, y) withpen pencircle scaled 2 dotlabeldiam; + draw (x, y) withpen pencircle scaled dotlabeldiam; + draw (x, -8) -- (x, -12); + label("\strut\small" & decimal xpart point t of data, (x, -20)); +endfor + +label.urt(btex \vbox{\halign{\small #\hfill\cr +Per capita\cr budget expenditure\cr in constant dollars\cr}} etex, +(0, (410 - 300) * v)); + +picture p; p = currentpicture; currentpicture := nullpicture; +bboxmargin := 12; +fill bbox p withcolor 1/32(32, 32, 31); +draw p; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tufte-budget.pdf b/info/drawing-with-metapost/src/tufte-budget.pdf Binary files differnew file mode 100644 index 0000000000..9a369d27d1 --- /dev/null +++ b/info/drawing-with-metapost/src/tufte-budget.pdf diff --git a/info/drawing-with-metapost/src/tufte-currency.mp b/info/drawing-with-metapost/src/tufte-currency.mp new file mode 100644 index 0000000000..708f229696 --- /dev/null +++ b/info/drawing-with-metapost/src/tufte-currency.mp @@ -0,0 +1,363 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{unicode-math} +\setmainfont[Numbers=OldStyle]{TeX Gyre Pagella} +\setmathfont{TeX Gyre Pagella Math} +\def\tl#1{\vbox{\tiny\let\\\cr\halign{\hss ##\hss\cr#1\crcr}}} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input basedate +path euro, years, events; +euro = + (base(1999, 01, 31), 70.256200) + -- (base(1999, 02, 28), 68.85200) + -- (base(1999, 03, 31), 67.106100) + -- (base(1999, 04, 30), 66.456200) + -- (base(1999, 05, 31), 65.751900) + -- (base(1999, 06, 30), 65.054700) + -- (base(1999, 07, 31), 65.818600) + -- (base(1999, 08, 31), 66.039600) + -- (base(1999, 09, 30), 64.618500) + -- (base(1999, 10, 31), 64.604600) + -- (base(1999, 11, 30), 63.715400) + -- (base(1999, 12, 31), 62.694500) + -- (base(2000, 01, 31), 61.723600) + -- (base(2000, 02, 29), 61.416200) + -- (base(2000, 03, 31), 61.095600) + -- (base(2000, 04, 30), 59.795100) + -- (base(2000, 05, 31), 60.165300) + -- (base(2000, 06, 30), 62.91600) + -- (base(2000, 07, 31), 62.337900) + -- (base(2000, 08, 31), 60.789500) + -- (base(2000, 09, 30), 60.777700) + -- (base(2000, 10, 31), 58.965700) + -- (base(2000, 11, 30), 60.020900) + -- (base(2000, 12, 31), 61.469800) + -- (base(2001, 01, 31), 63.488500) + -- (base(2001, 02, 28), 63.3800) + -- (base(2001, 03, 31), 62.956100) + -- (base(2001, 04, 30), 62.167200) + -- (base(2001, 05, 31), 61.405900) + -- (base(2001, 06, 30), 60.874200) + -- (base(2001, 07, 31), 60.811600) + -- (base(2001, 08, 31), 62.70600) + -- (base(2001, 09, 30), 62.322400) + -- (base(2001, 10, 31), 62.422700) + -- (base(2001, 11, 30), 61.851800) + -- (base(2001, 12, 31), 61.94300) + -- (base(2002, 01, 31), 61.676200) + -- (base(2002, 02, 28), 61.136700) + -- (base(2002, 03, 31), 61.565600) + -- (base(2002, 04, 30), 61.413900) + -- (base(2002, 05, 31), 62.738200) + -- (base(2002, 06, 30), 64.39600) + -- (base(2002, 07, 31), 63.882400) + -- (base(2002, 08, 31), 63.575100) + -- (base(2002, 09, 30), 63.061600) + -- (base(2002, 10, 31), 63.008300) + -- (base(2002, 11, 30), 63.737600) + -- (base(2002, 12, 31), 64.218200) + -- (base(2003, 01, 31), 65.650100) + -- (base(2003, 02, 28), 66.890200) + -- (base(2003, 03, 31), 68.137100) + -- (base(2003, 04, 30), 68.917100) + -- (base(2003, 05, 31), 71.205200) + -- (base(2003, 06, 30), 70.409100) + -- (base(2003, 07, 31), 69.997700) + -- (base(2003, 08, 31), 69.876900) + -- (base(2003, 09, 30), 69.764400) + -- (base(2003, 10, 31), 69.859800) + -- (base(2003, 11, 30), 69.252500) + -- (base(2003, 12, 31), 70.127900) + -- (base(2004, 01, 31), 69.311900) + -- (base(2004, 02, 29), 67.658400) + -- (base(2004, 03, 31), 67.186300) + -- (base(2004, 04, 30), 66.456100) + -- (base(2004, 05, 31), 67.167300) + -- (base(2004, 06, 30), 66.304900) + -- (base(2004, 07, 31), 66.615300) + -- (base(2004, 08, 31), 66.917400) + -- (base(2004, 09, 30), 68.148700) + -- (base(2004, 10, 31), 69.067300) + -- (base(2004, 11, 30), 69.916800) + -- (base(2004, 12, 31), 69.655900) + -- (base(2005, 01, 31), 70.672300) + -- (base(2005, 02, 28), 68.954800) + -- (base(2005, 03, 31), 69.266800) + -- (base(2005, 04, 30), 68.301400) + -- (base(2005, 05, 31), 68.404700) + -- (base(2005, 06, 30), 66.887400) + -- (base(2005, 07, 31), 68.613100) + -- (base(2005, 08, 31), 68.5500) + -- (base(2005, 09, 30), 67.759400) + -- (base(2005, 10, 31), 68.186800) + -- (base(2005, 11, 30), 67.934800) + -- (base(2005, 12, 31), 67.914300) + -- (base(2006, 01, 31), 68.629700) + -- (base(2006, 02, 28), 68.299400) + -- (base(2006, 03, 31), 68.879200) + -- (base(2006, 04, 30), 69.411300) + -- (base(2006, 05, 31), 68.343200) + -- (base(2006, 06, 30), 68.7700) + -- (base(2006, 07, 31), 68.851400) + -- (base(2006, 08, 31), 67.681500) + -- (base(2006, 09, 30), 67.498800) + -- (base(2006, 10, 31), 67.294500) + -- (base(2006, 11, 30), 67.386300) + -- (base(2006, 12, 31), 67.297100) + -- (base(2007, 01, 31), 66.365300) + -- (base(2007, 02, 28), 66.73900) + -- (base(2007, 03, 31), 67.988900) + -- (base(2007, 04, 30), 67.95900) + -- (base(2007, 05, 31), 68.144800) + -- (base(2007, 06, 30), 67.589800) + -- (base(2007, 07, 31), 67.449200) + -- (base(2007, 08, 31), 67.714700) + -- (base(2007, 09, 30), 68.791700) + -- (base(2007, 10, 31), 69.624700) + -- (base(2007, 11, 30), 70.779800) + -- (base(2007, 12, 31), 72.135100) + -- (base(2008, 01, 31), 74.638700) + -- (base(2008, 02, 29), 74.922400) + -- (base(2008, 03, 31), 77.322800) + -- (base(2008, 04, 30), 79.552400) + -- (base(2008, 05, 31), 79.079900) + -- (base(2008, 06, 30), 79.094700) + -- (base(2008, 07, 31), 79.236500) + -- (base(2008, 08, 31), 79.221500) + -- (base(2008, 09, 30), 79.928300) + -- (base(2008, 10, 31), 78.440300) + -- (base(2008, 11, 30), 82.852800) + -- (base(2008, 12, 31), 90.767900) + -- (base(2009, 01, 31), 92.114300) + -- (base(2009, 02, 28), 88.967100) + -- (base(2009, 03, 31), 91.925500) + -- (base(2009, 04, 30), 89.774800) + -- (base(2009, 05, 31), 88.591200) + -- (base(2009, 06, 30), 85.739200) + -- (base(2009, 07, 31), 86.02100) + -- (base(2009, 08, 31), 86.196800) + -- (base(2009, 09, 30), 89.16800) + -- (base(2009, 10, 31), 91.597400) + -- (base(2009, 11, 30), 89.818800) + -- (base(2009, 12, 31), 89.911700) + -- (base(2010, 01, 31), 88.424400) + -- (base(2010, 02, 28), 87.548100) + -- (base(2010, 03, 31), 90.177300) + -- (base(2010, 04, 30), 87.677600) + -- (base(2010, 05, 31), 85.83100) + -- (base(2010, 06, 30), 82.888900) + -- (base(2010, 07, 31), 83.636200) + -- (base(2010, 08, 31), 82.39800) + -- (base(2010, 09, 30), 83.807500) + -- (base(2010, 10, 31), 87.57300) + -- (base(2010, 11, 30), 85.595400) + -- (base(2010, 12, 31), 84.741200) + -- (base(2011, 01, 31), 84.729300) + -- (base(2011, 02, 28), 84.686200) + -- (base(2011, 03, 31), 86.708800) + -- (base(2011, 04, 30), 88.313100) + -- (base(2011, 05, 31), 87.653300) + -- (base(2011, 06, 30), 88.668300) + -- (base(2011, 07, 31), 88.569400) + -- (base(2011, 08, 31), 87.548400) + -- (base(2011, 09, 30), 87.21700) + -- (base(2011, 10, 31), 87.015700) + -- (base(2011, 11, 30), 85.784300) + -- (base(2011, 12, 31), 84.494100) + -- (base(2012, 01, 31), 83.159600) + -- (base(2012, 02, 29), 83.664600) + -- (base(2012, 03, 31), 83.485600) + -- (base(2012, 04, 30), 82.378400) + -- (base(2012, 05, 31), 80.472400) + -- (base(2012, 06, 30), 80.644300) + -- (base(2012, 07, 31), 78.858800) + -- (base(2012, 08, 31), 78.862900) + -- (base(2012, 09, 30), 79.89500) + -- (base(2012, 10, 31), 80.69300) + -- (base(2012, 11, 30), 80.362500) + -- (base(2012, 12, 31), 81.282400) + -- (base(2013, 01, 31), 83.14100) + -- (base(2013, 02, 28), 86.2300) + -- (base(2013, 03, 31), 85.989300) + -- (base(2013, 04, 30), 85.050300) + -- (base(2013, 05, 31), 84.83100) + -- (base(2013, 06, 30), 85.192200) + -- (base(2013, 07, 31), 86.163400) + -- (base(2013, 08, 31), 85.98600) + -- (base(2013, 09, 30), 84.243200) + -- (base(2013, 10, 31), 84.770300) + -- (base(2013, 11, 30), 83.829200) + -- (base(2013, 12, 31), 83.68200) + -- (base(2014, 01, 31), 82.752100) + -- (base(2014, 02, 28), 82.464100) + -- (base(2014, 03, 31), 83.143900) + -- (base(2014, 04, 30), 82.506500) + -- (base(2014, 05, 31), 81.574400) + -- (base(2014, 06, 30), 80.463800) + -- (base(2014, 07, 31), 79.300800) + -- (base(2014, 08, 31), 79.758700) + -- (base(2014, 09, 30), 79.150400) + -- (base(2014, 10, 31), 78.837400) + -- (base(2014, 11, 30), 79.037400) + -- (base(2014, 12, 31), 78.805100) + -- (base(2015, 01, 31), 76.705900) + -- (base(2015, 02, 28), 74.11400) + -- (base(2015, 03, 31), 72.317800) + -- (base(2015, 04, 30), 72.364700) + -- (base(2015, 05, 31), 72.255300) + -- (base(2015, 06, 30), 72.076600) + -- (base(2015, 07, 31), 70.762500) + -- (base(2015, 08, 31), 71.401100) + -- (base(2015, 09, 30), 73.266100) + -- (base(2015, 10, 31), 73.267700) + -- (base(2015, 11, 30), 70.662300) + -- (base(2015, 12, 31), 72.61500) + -- (base(2016, 01, 31), 75.332500) + -- (base(2016, 02, 29), 77.556300) + -- (base(2016, 03, 31), 78.041900) + -- (base(2016, 04, 30), 79.311700) + -- (base(2016, 05, 31), 77.868700) + -- (base(2016, 06, 30), 79.00800) + -- (base(2016, 07, 31), 84.173500) + -- (base(2016, 08, 31), 85.603400) + -- (base(2016, 09, 30), 85.237500) + -- (base(2016, 10, 31), 89.187500) + -- (base(2016, 11, 30), 86.6400) + -- (base(2016, 12, 31), 84.447800) + -- (base(2017, 01, 31), 86.087300) + -- (base(2017, 02, 28), 85.299700) + -- (base(2017, 03, 31), 86.653800) + -- (base(2017, 04, 30), 84.7200) + -- (base(2017, 05, 31), 85.610300) + -- (base(2017, 06, 30), 87.732600) + -- (base(2017, 07, 31), 88.646200) + -- (base(2017, 08, 31), 91.20400) + -- (base(2017, 09, 30), 89.478400) + -- (base(2017, 10, 31), 89.03800) + -- (base(2017, 11, 30), 88.873100) + -- (base(2017, 12, 31), 88.37100) + -- (base(2018, 01, 31), 88.307800) + -- (base(2018, 02, 28), 88.380700) + -- (base(2018, 03, 31), 88.297700) + -- (base(2018, 04, 30), 87.298900) + -- (base(2018, 05, 31), 87.835600) + -- (base(2018, 06, 30), 87.852700) + -- (base(2018, 07, 31), 88.778300) + -- (base(2018, 08, 31), 89.686900) + -- (base(2018, 09, 30), 89.369500) + -- (base(2018, 10, 31), 88.273600) + -- (base(2018, 11, 30), 88.099300) + -- (base(2018, 12, 31), 89.733600) + -- (base(2019, 01, 31), 88.56500) + -- (base(2019, 02, 28), 87.257800) + -- (base(2019, 03, 31), 85.830800) + -- (base(2019, 04, 30), 86.259500) + -- (base(2019, 05, 31), 87.048600) + -- (base(2019, 06, 30), 89.14400) + -- (base(2019, 07, 31), 89.935100) + -- (base(2019, 08, 31), 91.649700) + -- (base(2019, 09, 30), 89.223500) + -- (base(2019, 10, 31), 87.21200) + -- (base(2019, 11, 30), 85.81800) + -- (base(2019, 12, 31), 84.903300) + -- (base(2020, 01, 31), 84.880700) + -- (base(2020, 02, 29), 84.179300) + -- (base(2020, 03, 31), 89.452900) + -- (base(2020, 04, 30), 87.50400) + -- (base(2020, 05, 31), 88.826200) + -- (base(2020, 06, 30), 89.960700) + -- (base(2020, 07, 31), 90.43400) + -- (base(2020, 08, 31), 90.034100) + -- (base(2020, 09, 30), 91.040100) + -- (base(2020, 10, 31), 90.656300) + -- (base(2020, 11, 30), 89.649300) + -- (base(2020, 12, 31), 90.677900) + -- (base(2021, 01, 31), 89.174800) + -- (base(2021, 02, 28), 87.189400) + -- (base(2021, 03, 31), 85.864500) + -- (base(2021, 04, 30), 86.550500) + -- (base(2021, 05, 31), 86.335700) + -- (base(2021, 06, 30), 85.900900) + -- (base(2021, 07, 31), 85.619400) + -- (base(2021, 08, 31), 85.315600) + -- (base(2021, 09, 30), 85.644300) + -- (base(2021, 10, 31), 84.744100) + -- (base(2021, 11, 30), 84.789100) + -- (base(2021, 12, 31), 84.94200) + -- (base(2022, 01, 31), 83.52100) + -- (base(2022, 02, 28), 83.827900) + -- (base(2022, 03, 31), 83.615400) + -- (base(2022, 04, 30), 83.557800) + -- (base(2022, 05, 22), 84.892600) + ; + +years = (base(1999, 01, 01), 1999) for y=2000 upto 2022: -- (base(y, 1, 1), y) endfor; + +numeric offset; +offset = base(1999, 1, 1); +euro := euro shifted -(offset, 0) xscaled 42 yscaled 3; +years := years shifted -(offset, 0) xscaled 42; + +beginfig(1); +draw euro withcolor 3/4 blue; +numeric baseline; baseline = 54*3; +draw (xpart point 0 of euro, baseline) -- (xpart point infinity of euro, baseline); +for t = 1 upto length years: + numeric x, y, q; (x, y) = point t of years; 4q = xpart (point t-1 of years - point t of years); + draw (x, baseline) -- (x, baseline + 4); + for i=1 upto 3: + draw (x + i*q, baseline) -- (x + i*q, baseline + 2); + endfor + if not odd y: + label.bot("\small\strut " & decimal y, (x, baseline)); + fi +endfor +numeric minx, miny, maxx, maxy; +miny = 1000; maxy = 0; +minx = maxx = 0; +for i=0 upto length euro: + numeric x, y; (x, y) = point i of euro; + if y > maxy: maxy := y; maxx := x; fi + if y < miny: miny := y; minx := x; fi +endfor +draw (2 up -- up) scaled 2 shifted (maxx, maxy) withcolor red; +draw (2 down -- down) scaled 2 shifted (minx, miny) withcolor red; + +numeric left_axis, right_axis; left_axis = -16; right_axis = 368; + +draw (left_axis, miny) -- (left_axis, maxy); +for y = 60 step 10 until 100: + draw (left_axis, 3y) -- (left_axis + 2, 3y); + label.lft("\small\strut " & if y < 100: "0." & decimal y else: "1.00" fi, (left_axis, 3y)); +endfor + +draw (right_axis, miny) -- (right_axis, maxy); +for eur = 100 step 10 until 170: + numeric y; y = 100/eur * 100; + draw (right_axis, 3y) -- (right_axis - 2, 3y); + label.rt("\small\strut " & if eur=100: "1.00" else: "1." & decimal (eur mod 100) fi, (right_axis, 3y)); +endfor + +label.top("\small £/€", (left_axis - 10, 310)); +label.top("\small €/£", (right_axis + 10, 310)); + +interim dotlabeldiam := 3/2; +dotlabel.top("\tiny Creation", (base(1999, 1, 1) - offset, 71) xscaled 42 yscaled 3); +dotlabel.top("\tiny 9/11", (base(2001, 9, 11) - offset, 64) xscaled 42 yscaled 3); +dotlabel.lrt("\tl{Lehmans\\collapse}", (base(2008, 9, 15) - offset, 77) xscaled 42 yscaled 3); +dotlabel.lrt("\tl{Brexit\\vote}", (base(2016, 6, 24) - offset, 77) xscaled 42 yscaled 3); +dotlabel.bot("\tl{UK leaves\\EU}", (base(2020, 1, 31) - offset, 83) xscaled 42 yscaled 3); +dotlabel.top("\tl{WHO declares\\pandemic}", (base(2020, 3, 11) - offset, 91) xscaled 42 yscaled 3); +dotlabel.bot("\tl{Ukraine\\invaded}", (base(2022, 2, 24) - offset, 82.5) xscaled 42 yscaled 3); + + +picture p; p = currentpicture; currentpicture := nullpicture; +bboxmargin := 12; +fill bbox p withcolor 1/32(32, 32, 31); +draw p; + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/tufte-currency.pdf b/info/drawing-with-metapost/src/tufte-currency.pdf Binary files differnew file mode 100644 index 0000000000..18bb4f461b --- /dev/null +++ b/info/drawing-with-metapost/src/tufte-currency.pdf diff --git a/info/drawing-with-metapost/src/tufte-mpg.mp b/info/drawing-with-metapost/src/tufte-mpg.mp new file mode 100644 index 0000000000..ab52f8336f --- /dev/null +++ b/info/drawing-with-metapost/src/tufte-mpg.mp @@ -0,0 +1,55 @@ +\documentclass{standalone} +\usepackage{luamplib} +\usepackage{unicode-math} +\setmainfont[Numbers=OldStyle]{TeX Gyre Pagella} +\setmathfont{TeX Gyre Pagella Math} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); +path data, p; +data = (1978, 18) + -- (1979, 19) + -- (1980, 20) + -- (1981, 22) + -- (1982, 24) + -- (1983, 26) + -- (1984, 27) + -- (1985, 27.5); +numeric u, v; +u = xpart urcorner textext("1980\kern 0.75em"); +v = 8; +p = data shifted -(xpart point 0 of data, 0) xscaled u yscaled v; +draw (xpart point 0 of p, 20) -- p -- (xpart point infinity of p, 20); +draw (xpart point 0 of p, 0) -- (xpart point infinity of p, 0); + +for t=0 upto length p: + numeric x, y; (x, y) = point t of p; + undraw (x, y) withpen pencircle scaled 2 dotlabeldiam; + draw (x, y) withpen pencircle scaled dotlabeldiam; + label("\strut" & decimal ypart point t of data, (x, y + 8)); + draw (x, 0) -- (x, 4); + label("\strut" & decimal xpart point t of data, (x, 12)); +endfor + +dotlabel.rt(btex \vbox to 6pt{\halign{\small #\hss\cr +13.7 mpg, average\cr for all cars on\cr road, 1978\cr}\vss} etex, +(xpart point 0 of p, 13.7v)); +dotlabel.lft(btex \vbox to 6pt{\halign{\small #\hss\cr +19.5 mpg, expected\cr average for all cars\cr on road, 1985\cr}\vss} etex, +(xpart point infinity of p, 19.5v)); + +label.top(btex \vbox{\halign{\hss\textsc{#}\hss\cr +required fuel economy standards:\cr +new cars built from 1978 to 1985\cr}} etex, +point 5/2 of bbox currentpicture shifted 21 up); + +picture p; p = currentpicture; currentpicture := nullpicture; +bboxmargin := 12; +fill bbox p withcolor 1/32(32, 32, 31); +draw p; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/tufte-mpg.pdf b/info/drawing-with-metapost/src/tufte-mpg.pdf Binary files differnew file mode 100644 index 0000000000..7eac49d2b4 --- /dev/null +++ b/info/drawing-with-metapost/src/tufte-mpg.pdf diff --git a/info/drawing-with-metapost/src/twister.mp b/info/drawing-with-metapost/src/twister.mp new file mode 100644 index 0000000000..802d1a1ab8 --- /dev/null +++ b/info/drawing-with-metapost/src/twister.mp @@ -0,0 +1,31 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +input compass-rose +input colorbrewer-rgb +beginfig(1); +picture O; +O = compass_rose(Blues 8 8, Reds 8 7) scaled 1/2; +%O = thelabel(TEX("\TeX"), origin) scaled 3; + +numeric s; pair p; +s = 108; +p = origin; +draw O shifted p; draw bbox O withcolor 3/4[red, white]; label.urt("\small\texttt{bbox P}", point -1 of bbox O) withcolor 3/4[red, white]; +p := (1s, 0); draw O rotated 30 shifted p; label("\small\texttt{rotated 30}", p shifted 54 down); +p := (2s,+15); draw O shifted p; label("\small\texttt{shifted 15 up}", p shifted 54 down); + +p := ( 0, -1s); draw O scaled 3/4 shifted p; label("\small\texttt{scaled 3/4}", p shifted 54 down); +p := (1s, -1s); draw O xscaled 3/4 shifted p; label("\small\texttt{xscaled 3/4}", p shifted 54 down); +p := (2s, -1s); draw O yscaled 3/4 shifted p; label("\small\texttt{yscaled 3/4}", p shifted 54 down); + +p := ( 0, -2s); draw O slanted 3/4 shifted p; label("\small\texttt{slanted 3/4}", p shifted 54 down); +p := (1s, -2s); draw O zscaled 3/4 dir 21 shifted p; label("\small\texttt{zscaled 3/4 dir 21}", p shifted 54 down); +p := (2s, -2s); draw O reflectedabout(up, down) shifted p; label("\small\texttt{reflectedabout(up, down)}", p shifted 54 down); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/twister.pdf b/info/drawing-with-metapost/src/twister.pdf Binary files differnew file mode 100644 index 0000000000..24b6fd8341 --- /dev/null +++ b/info/drawing-with-metapost/src/twister.pdf diff --git a/info/drawing-with-metapost/src/unicode.mp b/info/drawing-with-metapost/src/unicode.mp new file mode 100644 index 0000000000..629ebe18e2 --- /dev/null +++ b/info/drawing-with-metapost/src/unicode.mp @@ -0,0 +1,26 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\usepackage{fontspec} \newfontface\polytonic{GFS Porson} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + +label("café noir £2.50", origin); +label(btex \vbox{\hsize 4in + Nous étions à l'Étude, quand le Proviseur entra, suivi d'un + \textit{nouveau} habillé en bourgeois et d'un garçon de classe + qui portait un grand pupitre. Ceux qui dormaient se réveillèrent, + et chacun se leva comme surpris dans son travail. +\par} etex, 60 down); +label(btex \vbox{\polytonic\halign{#\hfil\cr +μῆνιν ἄειδε θεὰ Πηληϊάδεω Ἀχιλῆος\cr +οὐλομένην, ἣ μυρί᾽ Ἀχαιοῖς ἄλγε᾽ ἔθηκε,\cr +πολλὰς δ᾽ ἰφθίμους ψυχὰς Ἄϊδι προΐαψεν\cr +ἡρώων, αὐτοὺς δὲ ἑλώρια τεῦχε κύνεσσιν\cr +οἰωνοῖσί τε πᾶσι, Διὸς δ᾽ ἐτελείετο βουλή,\cr +ἐξ οὗ δὴ τὰ πρῶτα διαστήτην ἐρίσαντε\cr +Ἀτρεΐδης τε ἄναξ ἀνδρῶν καὶ δῖος Ἀχιλλεύς.\cr}} etex, 220 down); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/unicode.pdf b/info/drawing-with-metapost/src/unicode.pdf Binary files differnew file mode 100644 index 0000000000..53ca0238c1 --- /dev/null +++ b/info/drawing-with-metapost/src/unicode.pdf diff --git a/info/drawing-with-metapost/src/utf8.mp b/info/drawing-with-metapost/src/utf8.mp new file mode 100644 index 0000000000..5606860388 --- /dev/null +++ b/info/drawing-with-metapost/src/utf8.mp @@ -0,0 +1,35 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +beginfig(1); + +def U primary s = if string s: decode(s) fi enddef; + +vardef decode(expr given) = + save a,b,i,s,out; string s, out; numeric a, b, i; + out = ""; i=0; + forever: + i := i+1; s := substring(i-1,i) of given; a := ASCII s; + if a < 128: + elseif a = 194: + i := i+1; s := substring (i-1,i) of given; + elseif a = 195: + i := i+1; s := char (64 + ASCII substring (i-1,i) of given); + else: + s := "?"; + fi + out := out & s; + exitif i >= length given; + endfor + out +enddef; +beginfig(2); + draw "café noir £2.50" infont "pncr8r"; + draw U"café noir £2.50" infont "pncr8r" shifted 12 down; + defaultfont := "pncr8r"; + label.rt("café noir £2.50", 24 down); + label.rt(U"café noir £2.50", 36 down); +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/utf8.pdf b/info/drawing-with-metapost/src/utf8.pdf Binary files differnew file mode 100644 index 0000000000..521f8dfc66 --- /dev/null +++ b/info/drawing-with-metapost/src/utf8.pdf diff --git a/info/drawing-with-metapost/src/verbatim-listing.mp b/info/drawing-with-metapost/src/verbatim-listing.mp new file mode 100644 index 0000000000..448246aac3 --- /dev/null +++ b/info/drawing-with-metapost/src/verbatim-listing.mp @@ -0,0 +1,30 @@ +prologues := 3; +outputtemplate := "%j.eps"; +verbatimtex +\documentclass{article} +\usepackage{listings} +\newcommand\mpstyle{\lstset{ +language=Metapost, basicstyle=\ttfamily, +frame=none, columns=fullflexible, +keepspaces=true, showstringspaces=false}} +\lstnewenvironment{code}[1][]{\mpstyle\lstset{#1}}{} +\begin{document} +etex +beginfig(1); +picture P; +P = thelabel(btex \vbox{\begin{code} + % special operators + vardef incr suffix $ = $:=$+1; $ enddef; + vardef decr suffix $ = $:=$-1; $ enddef; + + def reflectedabout(expr w,z) = % reflects about the line w..z + transformed begingroup transform T_; + w transformed T_ = w; + z transformed T_ = z; + xxpart T_ = -yypart T_; + xypart T_ = yxpart T_; % T_ is a reflection + T_ endgroup enddef; +\end{code}} etex, origin); +fill bbox P withcolor (1,1,31/32); draw P; draw bbox P; +endfig; +end. diff --git a/info/drawing-with-metapost/src/verbatim-listing.pdf b/info/drawing-with-metapost/src/verbatim-listing.pdf Binary files differnew file mode 100644 index 0000000000..438c727086 --- /dev/null +++ b/info/drawing-with-metapost/src/verbatim-listing.pdf diff --git a/info/drawing-with-metapost/src/verbatim-with-infont.mp b/info/drawing-with-metapost/src/verbatim-with-infont.mp new file mode 100644 index 0000000000..7e4081a8e8 --- /dev/null +++ b/info/drawing-with-metapost/src/verbatim-with-infont.mp @@ -0,0 +1,13 @@ +prologues := 3; +outputtemplate := "%j.eps"; +beginfig(1); + string s; s = "\TeX\ sets maths like this $e=mc^2$"; + draw ("1. " & s) infont defaultfont; + draw ("2. " & s) infont "texnansi-lmr10" shifted 20 down; + draw ("3. " & s) infont "cmtt10" shifted 40 down; + draw ("4. " & s) infont "texnansi-lmtt10" shifted 60 down; + bboxmargin := 14; + picture p; p = currentpicture; fill bbox currentpicture withcolor (1,1,31/32); draw p; + draw bbox p; +endfig; +end. diff --git a/info/drawing-with-metapost/src/verbatim-with-infont.pdf b/info/drawing-with-metapost/src/verbatim-with-infont.pdf Binary files differnew file mode 100644 index 0000000000..b2b3408c13 --- /dev/null +++ b/info/drawing-with-metapost/src/verbatim-with-infont.pdf diff --git a/info/drawing-with-metapost/src/whatever.mp b/info/drawing-with-metapost/src/whatever.mp new file mode 100644 index 0000000000..70b4678182 --- /dev/null +++ b/info/drawing-with-metapost/src/whatever.mp @@ -0,0 +1,48 @@ +\documentclass[border=1mm]{standalone} +\usepackage{luatex85} +\usepackage{luamplib} +\usepackage{dwmpcode} +\mplibtextextlabel{enable} +\begin{document} +\hbox to 5.3in{$\vcenter{% +\begin{mplibcode} +beginfig(1); + z1 = (10, 50); + z2 = (80, 190); + z3 = (0, 170); + z4 = (60, 10); + + draw z1--z2; + draw z3--z4; + + z0 = alpha [z1, z2] = beta [z3, z4]; + + forsuffixes @=0,1,2,3,4: + dotlabel.rt("$z_" & decimal @ & "$", z@) + withcolor 2/3 if @=0: red else: blue fi; + endfor + + %label.urt(decimal alpha, (x2, y0)); + %label.lrt(decimal beta, (x2, y0)); + +endfig; +\end{mplibcode}} +\qquad\vcenter{ +\begin{code} +beginfig(1); + z1 = (10, 50); + z2 = (80, 190); + z3 = (0, 170); + z4 = (60, 10); + + z0 = whatever [z1, z2] = whatever [z3, z4]; + + draw z1--z2; + draw z3--z4; + forsuffixes @=0,1,2,3,4: + dotlabel.rt("$z_" & decimal @ & "$", z@) + withcolor 2/3 if @=0: red else: blue fi; + endfor +endfig; +\end{code}}$\hss} +\end{document} diff --git a/info/drawing-with-metapost/src/whatever.pdf b/info/drawing-with-metapost/src/whatever.pdf Binary files differnew file mode 100644 index 0000000000..e02bd9c354 --- /dev/null +++ b/info/drawing-with-metapost/src/whatever.pdf diff --git a/info/drawing-with-metapost/src/workflow-plain.mp b/info/drawing-with-metapost/src/workflow-plain.mp new file mode 100644 index 0000000000..3d29b50e80 --- /dev/null +++ b/info/drawing-with-metapost/src/workflow-plain.mp @@ -0,0 +1,46 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\def\tll#1{$\vcenter{\let\\\cr\halign{\hss\textsf{##}\hss\cr#1\cr}}$} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +path paper; paper = origin {dir -16} .. (55, 0) {dir -13} -- (55,34) -- (0, 34) -- cycle; +path source, EPS, pdf; +source = paper; +EPS = paper shifted 105 right; +pdf = paper shifted 210 right; +beginfig(1); + fill source withcolor (1,1,7/8); + draw source; + label("\tll{MP source}", center source); + label.bot("\tll{Edit with\\MacVim}", point 1/2 of source shifted 8 down); + + fill EPS withcolor (15/16, 15/16, 1); + draw EPS dashed withdots scaled 1/2; label("\textsf{EPS}", center EPS); + + fill pdf withcolor (7/8, 7/8, 1); + draw pdf; label("\textsf{PDF}", center pdf); + picture p, u; + p = thelabel.bot("\tll{Preview with\\Skim.app}", point 1/2 of pdf shifted 8 down); + u = thelabel.top("\tll{Use in \LaTeX with\\\texttt{\textbackslash includegraphics}}", point 5/2 of pdf shifted 34 up); + draw p; draw u; + + interim ahangle := 30; + drawarrow (origin -- 34 up) shifted point 5/2 of pdf dashed evenly scaled 1/2; + drawarrow point 3/2 of bbox source -- point 7/2 of bbox EPS; + label.top("\texttt{mpost}", 1/2[center source, center EPS]); + drawarrow point 3/2 of bbox EPS -- point 7/2 of bbox pdf; + label.top("\texttt{epstopdf}", 1/2[center EPS, center pdf]); + + drawarrow center EPS -- center u + cutbefore bbox EPS cutafter bbox u + dashed withdots scaled 1/2; + + draw center EPS -- center p + cutbefore bbox EPS cutafter bbox p + dashed withdots scaled 1/2; + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/workflow-plain.pdf b/info/drawing-with-metapost/src/workflow-plain.pdf Binary files differnew file mode 100644 index 0000000000..fc0eac7888 --- /dev/null +++ b/info/drawing-with-metapost/src/workflow-plain.pdf diff --git a/info/drawing-with-metapost/src/workflow.mp b/info/drawing-with-metapost/src/workflow.mp new file mode 100644 index 0000000000..0afa3396c7 --- /dev/null +++ b/info/drawing-with-metapost/src/workflow.mp @@ -0,0 +1,30 @@ +\documentclass[border=5mm]{standalone} +\usepackage{luamplib} +\def\tll#1{$\vcenter{\let\\\cr\halign{\hss\textsf{##}\hss\cr#1\cr}}$} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +path paper; paper = origin {dir -16} .. (55, 0) {dir -13} -- (55,34) -- (0, 34) -- cycle; +path source, pdf; +source = paper; +pdf = paper shifted 210 right; +beginfig(1); + fill source withcolor (1,1,7/8); + draw source; + label("\tll{MP source\\in wrapper}", center source); + label.bot("\tll{Edit with\\MacVim}", point 1/2 of source shifted 8 down); + + fill pdf withcolor (7/8, 7/8, 1); + draw pdf; label("\textsf{PDF}", center pdf); + label.bot("\tll{Preview with\\Skim.app}", point 1/2 of pdf shifted 8 down); + label.top("\tll{Use in \LaTeX with\\\texttt{\textbackslash includegraphics}}", point 5/2 of pdf shifted 34 up); + + interim ahangle := 30; + drawarrow (origin -- 34 up) shifted point 5/2 of pdf dashed evenly scaled 1/2; + drawarrow point 3/2 of bbox source -- point 7/2 of bbox pdf; + label.top("\texttt{lualatex}", 1/2[center source, center pdf]); + +endfig; +\end{mplibcode} +\end{document} + diff --git a/info/drawing-with-metapost/src/workflow.pdf b/info/drawing-with-metapost/src/workflow.pdf Binary files differnew file mode 100644 index 0000000000..705656fafb --- /dev/null +++ b/info/drawing-with-metapost/src/workflow.pdf diff --git a/info/drawing-with-metapost/src/xypic-example.mp b/info/drawing-with-metapost/src/xypic-example.mp new file mode 100644 index 0000000000..074014c82c --- /dev/null +++ b/info/drawing-with-metapost/src/xypic-example.mp @@ -0,0 +1,53 @@ +\documentclass{standalone} +\usepackage{luamplib} +\begin{document} +\mplibtextextlabel{enable} +\begin{mplibcode} +beginfig(1); + + picture U, XY, X, Y, Z; + z1 = -z2 = (-61, 42); + + U = thelabel("$U$", z1); + XY = thelabel("$X\times_ZY$", origin); + X = thelabel("$X$", (x2, 0)); + Y = thelabel("$Y$", (0, y2)); + Z = thelabel("$Z$", z2); + + forsuffixes @=U, XY, X, Y, Z: draw @; endfor + + ahangle := 20; + vardef connect@#(expr s, a, b) = + curved_connect@#(s, a, b, center b - center a) + enddef; + + vardef curved_connect@#(expr s, a, b, d) = + save line, mark; + + path line; + line = center a {d} .. center b; + interim bboxmargin := 4; + drawarrow line cutbefore bbox a cutafter bbox b; + + picture mark; + mark = thelabel@#("$\scriptstyle " & s & "$", point 1/2 of line); + interim bboxmargin := 1; + unfill bbox mark; draw mark; + enddef; + + connect.bot("p", XY, X); + connect.rt ("q", XY, Y); + connect.top("g", Y, Z); + connect.lft("f", X, Z); + + curved_connect.urt("x", U, X, right); + curved_connect.llft("y", U, Y, dir -80); + + drawoptions(dashed withdots scaled 1/2); + connect("(x,y)", U, XY); + drawoptions(); + + +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/xypic-example.pdf b/info/drawing-with-metapost/src/xypic-example.pdf Binary files differnew file mode 100644 index 0000000000..fb5cb9d295 --- /dev/null +++ b/info/drawing-with-metapost/src/xypic-example.pdf diff --git a/info/drawing-with-metapost/src/youth.jpg b/info/drawing-with-metapost/src/youth.jpg Binary files differnew file mode 100644 index 0000000000..16f0438eea --- /dev/null +++ b/info/drawing-with-metapost/src/youth.jpg diff --git a/info/drawing-with-metapost/src/youth.mp b/info/drawing-with-metapost/src/youth.mp new file mode 100644 index 0000000000..77830d6629 --- /dev/null +++ b/info/drawing-with-metapost/src/youth.mp @@ -0,0 +1,12 @@ +\documentclass[border=5mm]{standalone} +\usepackage{graphicx} +\usepackage{luamplib} +\begin{document} +\begin{mplibcode} +input picture_frame +beginfig(1); + picture F; F = thelabel(TEX("\includegraphics[width=200pt]{youth.jpg}"), origin); + draw F; draw frame F; +endfig; +\end{mplibcode} +\end{document} diff --git a/info/drawing-with-metapost/src/youth.pdf b/info/drawing-with-metapost/src/youth.pdf Binary files differnew file mode 100644 index 0000000000..ad0aee97ca --- /dev/null +++ b/info/drawing-with-metapost/src/youth.pdf |