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+\documentclass{article} % article is NOT the original style
+\usepackage{makeidx}
+\usepackage{fancyvrb}
+\usepackage{ctabbing}
+\RecustomVerbatimEnvironment
+ {verbatim}{BVerbatim}{baseline=c}
+\usepackage{epsf}
+\usepackage[textwidth=6in,textheight=8.75in]{geometry}
+\usepackage{tocloft}
+\setlength\cftbeforesecskip{1.5ex plus 0.2ex minus 0.1ex}
+
+ \makeatletter
+ \def\logo{\global\font\logo=logo10 at1\@ptsize\p@ \logo}
+ \def\logosl{\global\font\logosl=logosl10 at1\@ptsize\p@ \logosl}
+ \def\MF{{\ifdim \fontdimen\@ne\font >\z@ \def\logo{\logosl}\fi
+ {\logo META}\-{\logo FONT}}}
+ \makeatother
+
+% \def\MF{{META\-FONT}} % Replacement for the above when using times.sty
+
+
+\newfont\psyvii{rpsyr at 7pt}
+\newcommand\reg{$^{\hbox{\psyvii\char'322}}$} % Registered trademark
+
+\newcommand\descr[1]{{\langle\hbox{#1}\rangle}}
+\newcommand\invisgap{\nobreak\hskip0pt\relax}
+\newcommand\tdescr[1]{$\langle$\invisgap#1\invisgap$\rangle$}
+
+\newcommand\pl{\dag}
+\newcommand\bx{\ddag}
+
+\newcommand\mathcenter[1]{\vcenter{\hbox{#1}}}
+
+
+\renewcommand{\topfraction}{.85}
+\renewcommand{\bottomfraction}{.7}
+\renewcommand{\textfraction}{.15}
+\renewcommand{\floatpagefraction}{.66}
+\renewcommand{\dbltopfraction}{.66}
+\renewcommand{\dblfloatpagefraction}{.66}
+\setcounter{topnumber}{9}
+\setcounter{bottomnumber}{9}
+\setcounter{totalnumber}{20}
+\setcounter{dbltopnumber}{9}
+
+\makeindex
+
+\begin{document}
+\VerbatimFootnotes
+\author{John D. Hobby}
+\title{A User's Manual for MetaPost}
+\date{}
+\maketitle
+
+\begin{abstract}
+The MetaPost system implements a picture-drawing language very much like Knuth's
+\MF\ except that it outputs PostScript commands instead of run-length-encoded
+bitmaps. MetaPost is a powerful language for producing figures for documents
+to be printed on PostScript printers. It provides easy access to all the
+features of PostScript and it includes facilities for integrating text and
+graphics.
+
+This document serves as an introductory user's manual. It does not require
+knowledge of \MF\ or access to {\it The \MF book}, but both are beneficial.
+An appendix explains the differences between MetaPost and \MF.
+\end{abstract}
+\thispagestyle{empty}
+\newpage
+\setcounter{page}{1}
+\pagestyle{plain}
+\pagenumbering{roman}
+\tableofcontents
+\newpage
+\setcounter{page}{1}
+\pagestyle{headings}
+\pagenumbering{arabic}
+\setlength{\parskip}{1ex plus 0.5ex minus 0.2ex}
+
+\section{Introduction}
+\label{intro}
+
+MetaPost is a programming language much like Knuth's \MF\footnote{\MF\ is a
+trademark of Addison Wesley Publishing company.}\index{metafont?\MF}~\cite{kn:c}
+except that it outputs PostScript programs instead of bitmaps. Borrowed from \MF\
+are the basic tools for creating and manipulating pictures. These include numbers,
+coordinate pairs, cubic splines, affine transformations, text strings, and boolean
+quantities. Additional features facilitate integrating text and graphics and
+accessing special features of PostScript\footnote{PostScript is a
+trademark of Adobe Systems Inc.}\index{PostScript} such as clipping, shading, and
+dashed lines.
+Another feature borrowed from \MF\ is the ability to solve linear equations
+that are given implicitly, thus allowing many programs to be written in a
+largely declarative style. By building complex operations
+from simpler ones, MetaPost achieves both power and flexibility.
+
+MetaPost is particularly
+well-suited to generating figures for technical documents where some aspects of a
+picture may be controlled by mathematical or geometrical constraints that are
+best expressed symbolically. In other words, MetaPost is not meant to take the
+place of a freehand drawing tool or even an interactive graphics editor.
+It is really a programming language for generating graphics, especially figures
+for \TeX\footnote{\TeX\ is a trademark of the American Mathematical
+Society.}\index{TeX?\TeX} and troff\index{troff} documents.
+The figures can be integrated into a \TeX\ document via a freely available
+program called {\tt dvips}\index{dvips} as shown in
+Figure~\ref{fig0}.\footnote{The C
+source for {\tt dvips} comes with the web2c \TeX\ distribution. Similar programs
+are available from other sources.} A similar procedure works with troff: the
+{\tt dpost} output processor includes PostScript figures when they are
+requested via troff's {\tt \char`\\X} command.
+
+\begin{figure}[htp]
+$$ \def\fbox#1{\hbox{\vrule
+ \vbox{\hrule\kern5pt\hbox{\kern5pt\hbox{#1}\kern5pt}\kern5pt\hrule}%
+ \vrule}}
+ \vbox{
+ \halign{$\hfil#\hfil$&\hskip1in$\hfil#\hfil$\cr
+ \hbox{Figures in MetaPost}&
+ \hbox{\TeX\ Document}
+ \cr
+ \bigg\downarrow&
+ \bigg\downarrow
+ \cr
+ \fbox{\vrule height.2in depth.133in width0pt
+ \kern .1in MetaPost\kern.1in}
+ &
+ \fbox{\vrule height.2in depth.133in width0pt
+ \kern .167in \TeX\kern.167in}
+ \cr
+ \bigg\downarrow&
+ \bigg\downarrow
+ \cr
+ \hbox{Figures in PostScript}&
+ \hbox{{\tt dvi} file}
+ \cr
+ \bigg\downarrow&
+ \bigg\downarrow
+ \cr
+ \fbox{\vrule height.2in depth.133in width0pt
+ \kern 1in {\tt dvips} \kern1in}
+ \span\omit\cr
+ \bigg\downarrow\span\omit\cr
+ \hbox{PostScript}\span\omit\cr}}
+$$
+\caption[A diagram of the processing for a document with MetaPost figures]
+ {A diagram of the processing for a \TeX\ document with figures
+ in MetaPost}
+\label{fig0}
+\end{figure}
+
+To use MetaPost, you prepare an input file containing MetaPost code and then
+invoke MetaPost, usually by giving a command of the form\index{mp?\texttt{mp}}
+$$ {\tt mp}\, \descr{file name} $$
+(This syntax could be system dependent).
+MetaPost input files\index{files!input} normally have names ending ``{\tt .mp}''
+but this part of the name can be omitted when invoking MetaPost. For an input
+file {\tt foo.mp}
+$$ \hbox{\tt mp foo} $$
+invokes MetaPost and produces output files with names like {\tt foo.1} and
+{\tt foo.2}. Any terminal I/O is summarized in a
+transcript\index{files!transcript}\index{transcript file}
+file called {\tt foo.log}. This includes
+error messages and any MetaPost commands entered interactively.%
+\footnote{A {\tt *}\index{*?\texttt{*}} prompt is used for interactive input and a
+{\tt **}\index{**?\texttt{**}} prompt
+indicates that an input file name is expected. This can be avoided by invoking
+MetaPost on a file that ends with an {\tt end}\index{end?\texttt{end}} command.}
+The transcript file starts with a banner line that tells what version of MetaPost
+you are using.
+
+This document introduces the MetaPost language, beginning with the features that
+are easiest to use and most important for simple applications. The first few
+sections describe the language as it appears to the novice user with key parameters
+at their default values. Some features described in these sections are part of a
+predefined macro package called Plain. Later sections summarize the
+complete language and distinguish between primitives and preloaded macros
+from the Plain macro package\index{Plain macros}.
+Since much of the language is identical to Knuth's \MF, the appendix gives a
+detailed comparison so that advanced users can learn more about MetaPost by
+reading {\sl The \MF book\/}.~\cite{kn:c}
+
+
+\section{Basic Drawing Statements}
+\label{basic}
+
+The simplest drawing statements are the ones that generate straight lines.
+Thus\index{draw?\texttt{draw}}\index{--?\texttt{--}}
+$$ \hbox{\verb|draw (20,20)--(0,0)|} $$
+draws\index{draw?\texttt{draw}} a diagonal line and
+$$ \hbox{\verb|draw (20,20)--(0,0)--(0,30)--(30,0)--(0,0)|} $$
+draws a polygonal line like this:
+$$ \epsfbox{manfig.0} $$
+
+What is meant by coordinates like \verb|(30,0)|? MetaPost uses the same default
+coordinate system that PostScript\index{PostScript} does. This means that
+\verb|(30,0)| is 30 units
+to the right of the origin, where a unit is $1\over72$ of an inch. We shall refer
+to this default unit as a
+{\sl PostScript point\/}\index{PostScript!point}\index{point!PostScript}
+to distinguish it from the standard printer's point\index{point!printer's}
+which is $1\over72.27$ inches.
+
+MetaPost uses the same names for units of measure that \TeX\ and \MF\ do. Thus
+\verb|bp|\index{bp?\texttt{bp}}\label{Dbp} refers to PostScript points (``big points'')
+and \verb|pt|\index{pt?\texttt{pt}}\label{Dpt} refers to printer's points.
+Other units of measure
+include \verb|in|\index{in?\texttt{in}}\label{Din} for inches,
+\verb|cm|\index{cm?\texttt{cm}}\label{Dcm} for centimeters,
+and \verb|mm|\index{mm?\texttt{mm}}\label{Dmm} for
+millimeters. For example,
+$$ \hbox{\verb|(2cm,2cm)--(0,0)--(0,3cm)--(3cm,0)--(0,0)|} $$
+generates a larger version of the above diagram. It is OK to say \verb|0| instead
+\verb|0cm| because {\tt cm} is really just a conversion factor and {\tt 0cm} just
+multiplies the conversion factor by zero. (MetaPost understands constructions
+like {\tt 2cm}\index{multiplication!implicit} as shorthand for \verb|2*cm|).
+
+It is often convenient to introduce your own scale factor, say $u$.
+Then you can define coordinates in terms of $u$ and decide later whether you want
+to begin with \verb|u=1cm| or \verb|u=0.5cm|. This gives you control over what
+gets scaled and what does not so that changing $u$ will not affect features such
+as line widths.
+
+There are many ways to affect the appearance of a line besides just changing its
+width, so the width-control mechanisms allow a lot of generality that we do not need
+yet.
+This leads to the strange looking statement\index{pickup?\texttt{pickup}}\index{pencircle?\texttt{pencircle}}%
+\index{scaled?\texttt{scaled}}
+$$ \hbox{\verb|pickup pencircle scaled 4pt|} $$
+for setting the line width for subsequent \verb|draw| statements to 4 points.
+(This is about eight times the default line width).
+
+With such a wide line width, even a line of zero length comes out as a big bold
+dot\index{dots}. We can use this to make a grid of bold dots by having one
+\verb|draw| statement
+for each grid point. Such a repetitive sequence of \verb|draw| statements is
+best written as a pair of nested loops:\index{loops}%
+\index{for?\texttt{for}}\index{endfor?\texttt{endfor}}
+$$\begin{verbatim}
+for i=0 upto 2:
+ for j=0 upto 2: draw (i*u,j*u); endfor
+endfor
+\end{verbatim}
+$$
+The outer loop runs for $i=0,1,2$ and the inner loop runs for $j=0,1,2$.
+The result is a three-by-three grid of bold dots as shown in Figure~\ref{fig1}.
+The figure also includes a larger version of the polygonal line diagram that we
+saw before.
+
+\begin{figure}[htp]
+$$ \begin{verbatim}
+beginfig(2);
+u=1cm;
+draw (2u,2u)--(0,0)--(0,3u)--(3u,0)--(0,0);
+pickup pencircle scaled 4pt;
+for i=0 upto 2:
+ for j=0 upto 2: draw (i*u,j*u); endfor
+endfor
+endfig;
+\end{verbatim}
+\quad \mathcenter{\epsfbox{manfig.2}}
+$$
+\caption{MetaPost commands and the resulting output}
+\label{fig1}
+\end{figure}
+
+Note that the program in Figure~\ref{fig1} starts with
+\verb|beginfig(2)|\index{beginfig?\texttt{beginfig}} and
+ends with \verb|endfig|\index{endfig?\texttt{endfig}}.
+These are macros that perform various administrative
+functions and ensure that the results of all the \verb|draw| statements get
+packaged up and translated into PostScript. A MetaPost input file normally
+contains a sequence of \verb|beginfig|, \verb|endfig| pairs with an
+{\tt end}\index{end?\texttt{end}}
+statement after the last one. If this file is named {\tt fig.mp}, the output
+from \verb|draw| statements between \verb|beginfig(1)| and the next \verb|endfig|
+is written in a file {\tt fig.1}\index{files!output}.
+In other words, the numeric argument to the \verb|beginfig| macro determines the
+name of the corresponding output file.
+
+What does one do with all the PostScript files? They can be included as figures
+in a \TeX\index{TeX?\TeX} or troff\index{troff} document if you have an
+output driver that can handle
+encapsulated PostScript figures. If your standard \TeX\ macro directory contains
+a file {\tt epsf.tex}\index{epsf.tex?\texttt{epsf.tex}}, you can probably include {\tt fig.1}
+in a \TeX\ document as follows:
+$$ \begin{array}{c}
+ \hbox{\verb|\input epsf |}\\
+ \vdots\\
+ \hbox{\verb|$$\epsfbox{fig.1}$$|}
+ \end{array}
+$$
+The \verb|\epsfbox| macro figures out how much room to leave for the figure and
+uses \TeX's \verb|\special| command to insert a request for {\tt fig.1}.
+
+It is also possible to include MetaPost output in a {\em troff\/} document.
+The {\tt -mpictures\/} macro package defines a command \verb|.BP| that includes
+an encapsulated PostScript file. For instance, the {\em troff\/} command
+$$ \hbox{\verb|.BP fig.1 3c 3c|} $$
+includes {\tt fig.1} and specifies that its height and width are both three
+centimeters.
+
+
+\section{Curves}
+\label{curves}
+
+MetaPost is perfectly happy to draw curved lines as well as straight ones.
+A \verb|draw| statement with the points separated by \verb|..| draws
+a smooth curve through the points. For example consider the result of
+$$ \hbox{\verb|draw z0..z1..z2..z3..z4|} $$
+after defining five points as follows:
+$$\begin{verbatim}
+z0 = (0,0); z1 = (60,40);
+z2 = (40,90); z3 = (10,70);
+z4 = (30,50);
+\end{verbatim}
+$$
+Figure~\ref{fig2} shows the curve with points \verb|z0| through \verb|z4|
+labeled.
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.3}
+$$
+\caption[A curve through points 0, 1, 2, 3, and 4]
+ {The result of {\tt draw z0..z1..z2..z3..z4}}
+\label{fig2}
+\end{figure}
+
+There are many other ways to draw a curved path through the same five points.
+To make a smooth closed curve, connect \verb|z4| back to the beginning by
+appending \verb|..cycle|\index{cycle?\texttt{cycle}} to the \verb|draw| statement as shown
+in Figure~\ref{fig3}a. It is also possible in a single \verb|draw| statement
+to mix curves and straight lines as shown in Figure~\ref{fig3}b. Just use
+\verb|--| where you want straight lines and \verb|..| where you want curves.
+Thus
+$$ \hbox{\verb|draw z0..z1..z2..z3--z4--cycle|} $$
+produces a curve through points 0,~1, 2, and~3, then a polygonal line from
+point~3 to point~4 and back to point~0. The result is essentially the same
+as having two draw statements
+\begin{eqnarray*}
+ \hbox{\verb|draw z0..z1..z2..z3|}\\
+\noalign{\hbox{and}}
+ \hbox{\verb|draw z3--z4--z0|}
+\end{eqnarray*}
+
+\begin{figure}[htp]
+$$ {\epsfbox{manfig.104} \atop (a)}
+ \qquad {\epsfbox{manfig.204} \atop (b)}
+$$
+\caption[Closed curves through five points]
+ {(a)~The result of {\tt draw z0..\linebreak[0]z1..\linebreak[0]%
+ z2..\linebreak[0]z3..\linebreak[0]z4..\linebreak[0]cycle};
+ (b)~the result of {\tt draw z0..\linebreak[0]z1..\linebreak[0]%
+ z2..\linebreak[0]z3--\linebreak[0]z4--\linebreak[0]cycle}.}
+\label{fig3}
+\end{figure}
+
+\subsection{B\'ezier Cubic Curves}
+
+When MetaPost is asked to draw a smooth curve through a sequence of points,
+it constructs a piecewise cubic curve with continuous slope and approximately
+continuous curvature\index{curvature}. This means that a path specification such
+as
+$$ \hbox{\verb|z0..z1..z2..z3..z4..z5|} $$
+results in a curve that can be defined parametrically\index{parameterization}
+as $(X(t),Y(t))$ for
+$0\le t\le5$, where $X(t)$ and $Y(t)$ are piecewise cubic functions. That is,
+there is a different pair of cubic functions for each integer-bounded
+$t$-interval. If ${\tt z0}=(x_0,y_0)$, ${\tt z1}=(x_1,y_1)$,
+${\tt z2}=(x_2,y_2)$, \ldots, MetaPost selects
+B\'ezier control\index{control points} points
+$(x_0^+,y_0^+)$, $(x_1^-,y_1^-)$, $(x_1^+,y_1^+)$, \ldots, where
+\begin{eqnarray*}
+ X(t+i) &=& (1-t)^3x_i + 3t(1-t)^2x_i^+ + 3t^2(1-t)x_{i+1}^- + t^3x_{i+1},\\
+ Y(t+i) &=& (1-t)^3y_i + 3t(1-t)^2y_i^+ + 3t^2(1-t)y_{i+1}^- + t^3y_{i+1}
+\end{eqnarray*}
+for $0\le t\le1$. The precise rules for choosing the B\'ezier control points
+are described in \cite{ho:splin} and in {\sl The \MF book\/}~\cite{kn:c}.
+
+In order for the path to have a continuous slope at $(x_i,y_i)$, the incoming
+and outgoing directions at $(X(i),Y(i))$ must match. Thus the vectors
+$$ (x_i-x_i^-,\,y_i-y_i^-) \qquad \hbox{and}
+ \qquad (x_i^+-x_i,\,y_i^+-y_i)
+$$
+must have the same direction; i.e., $(x_i,y_i)$ must be on the line segment
+between $(x_i^-,y_i^-)$ and $(x_i^+,y_i^+)$. This situation is illustrated
+in Figure~\ref{fig4} where the B\'ezier control points selected by MetaPost
+are connected by dashed lines. For those who are familiar with the interesting
+properties of this construction, MetaPost allows the control points to be
+specified directly in the following format:\index{controls?\texttt{controls}}
+$$ \begin{verbatim}
+draw (0,0)..controls (26.8,-1.8) and (51.4,14.6)
+ ..(60,40)..controls (67.1,61.0) and (59.8,84.6)
+ ..(40,90)..controls (25.4,94.0) and (10.5,84.5)
+ ..(10,70)..controls ( 9.6,58.8) and (18.8,49.6)
+ ..(30,50);
+\end{verbatim}
+$$
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.5}
+$$
+\caption[A curve and the control polygon]
+ {The result of {\tt draw z0..z1..z2..z3..z4} with the
+ automatically-selected B\'ezier control polygon illustrated by dashed
+ lines.}
+\label{fig4}
+\end{figure}
+
+\subsection{Specifying Direction, Tension, and Curl}
+\label{tenscurl}
+
+MetaPost provides many ways of controlling the behavior of a curved path without
+actually specifying the control points. For instance, some points on the path
+may be selected as vertical or horizontal extrema. If \verb|z1| is to be a
+horizontal extreme and \verb|z2| is to be a vertical extreme, you can specify
+that $(X(t),Y(t))$ should go upward at \verb|z1| and to the left at \verb|z2|:
+$$ \hbox{\verb|draw z0..z1{up}..z2{left}..z3..z4;|} $$
+The resulting shown in Figure~\ref{fig5} has the desired vertical and horizontal
+directions at \verb|z1| and \verb|z2|, but it does not look as smooth as the
+curve in Figure~\ref{fig2}. The reason is the large discontinuity in
+curvature\index{curvature}
+at \verb|z1|. If it were not for the specified direction at \verb|z1|, the
+MetaPost interpreter would have chosen a direction designed to make the curvature
+above \verb|z1| almost the same as the curvature below that point.
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.6}
+$$
+\caption[A curve and the control polygon]
+ {The result of {\tt draw z0..z1\char`\{up\char`\}..z2\char`\{left\char`\}%
+ ..z3..z4}.}
+\label{fig5}
+\end{figure}
+
+How can the choice of directions at given points on a curve determine whether
+the curvature will be continuous? The reason is that curves used in MetaPost
+come from a family where a path is determined by its endpoints and the
+directions there. Figures \ref{fig6} and~\ref{fig7} give a good idea of what
+this family of curves is like.
+
+\begin{figure}[htp]
+$$ \mathcenter{\epsfbox{manfig.7}} \quad
+\begin{verbatim}
+beginfig(7)
+for a=0 upto 9:
+ draw (0,0){dir 45}..{dir -10a}(6cm,0);
+endfor
+endfig;
+\end{verbatim}
+$$
+\caption{A curve family and the MetaPost instructions for generating it}
+\label{fig6}
+\end{figure}
+
+\begin{figure}[htp]
+$$ \mathcenter{\epsfbox{manfig.8}} \quad
+\begin{verbatim}
+beginfig(8)
+for a=0 upto 7:
+ draw (0,0){dir 45}..{dir 10a}(6cm,0);
+endfor
+endfig;
+\end{verbatim}
+$$
+\caption{Another curve family with the corresponding MetaPost instructions}
+\label{fig7}
+\end{figure}
+
+Figures \ref{fig6} and~\ref{fig7} illustrate a few new MetaPost features.
+The first is the {\tt dir}\index{dir?\texttt{dir}}\label{Ddirop} operator that takes an
+angle in degrees
+and generates a unit vector in that direction. Thus \verb|dir 0| is equivalent
+to {\tt right}\index{right?\texttt{right}}\label{Dright} and \verb|dir 90| is equivalent to
+{\tt up}\index{up?\texttt{up}}\label{Dup}. There are also predefined direction vectors
+{\tt left}\index{left?\texttt{left}}\label{Dleft}
+and {\tt down}\index{down?\texttt{down}}\label{Ddown} for {\tt dir 180}
+and {\tt dir 270}.
+
+The direction
+vectors given in \verb|{}| can be of any length, and they can come before a
+point as well as after one. It is even possible for a path specification
+to have directions given before and after a point. For example a path
+specification containing
+$$ \hbox{\verb|..{dir 60}(10,0){up}..|} $$
+produces a curve with a corner at $(10,0)$.
+
+Note that some of the curves in Figure~\ref{fig6} have points of
+inflection\index{inflections}.
+This is necessary in order to produce smooth curves in situations like
+Figure~\ref{fig3}a, but it is probably not desirable when dealing with vertical
+and horizontal extreme points as in Figure~\ref{fig8}a. If \verb|z1| is supposed
+to be the topmost point on the curve, this can be achieved by using
+\verb|...|\index{...?\texttt{...}}
+instead of \verb|..| in the path specification as shown in Figure~\ref{fig8}b.
+The meaning of \verb|...| is ``choose an inflection-free path between these
+points unless the endpoint directions make this impossible.'' (It would be
+possible to avoid inflections in Figure~\ref{fig6}, but not in Figure~\ref{fig7}).
+
+\begin{figure}[htp]
+$$ {\mathcenter{\epsfbox{manfig.109}} \atop
+ \hbox{\verb|draw z0{up}..z1{right}..z2{down}|}}
+ \quad
+ {\mathcenter{\epsfbox{manfig.209}} \atop
+ \hbox{\verb|draw z0{up}...z1{right}...z2{down}|}}
+$$
+\caption{Two {\tt draw} statements and the resulting curves.}
+\label{fig8}
+\end{figure}
+
+Another way to control a misbehaving path is to increase the
+``tension''\index{tension} parameter.
+Using \verb|..| in a path specification sets the tension parameter to the default
+value~1. If this makes some part of a path a little too wild, we can selectively
+increase the tension. If Figure~\ref{fig9}a is considered ``too wild,'' a
+{\tt draw} statement of the following form increases the tension between
+{\tt z1} and {\tt z2}:
+$$ \hbox{\verb|draw z0..z1..tension 1.3..z2..z3|} $$
+This produces Figure~\ref{fig9}b. For an asymmetrical effect like
+Figure~\ref{fig9}c, the \verb|draw| statement becomes
+$$ \hbox{\verb|draw z0..z1..tension 1.6 and 1..z2..z3|} $$
+The tension parameter can be less than one, but it must be at least $3\over4$.
+
+\begin{figure}[htp]
+$$ {\mathcenter{\epsfbox{manfig.110}} \atop (a)}
+ \quad
+ {\mathcenter{\epsfbox{manfig.210}} \atop (b)}
+ \quad
+ {\mathcenter{\epsfbox{manfig.310}} \atop (c)}
+$$
+\caption[Effects of changing the tension parameter]
+ {Results of {\tt draw z0..z1..tension} $\alpha$ {\tt and} $\beta$
+ {\tt ..z2..z3} for various $\alpha$ and $\beta$:
+ (a)~$\alpha=\beta=1$; (b)~$\alpha=\beta=1.3$;
+ (c)~$\alpha=1.5$, $\beta=1$.}
+\label{fig9}
+\end{figure}
+
+MetaPost paths also have a parameter called ``curl''\index{curl?\texttt{curl}} that affects
+the ends of a
+path. In the absence of any direction specifications, the first and last segments
+of a non-cyclic path are approximately circular arcs as in the $c=1$ case of
+Figure~\ref{fig10}. To use a different value for the curl parameter, specify
+\verb|{curl c}| for some other value of $c$. Thus
+$$ \hbox{\verb|draw z0{curl c}..z1..{curl c}z2|} $$
+sets the curl parameter for \verb|z0| and \verb|z2|. Small values of the curl
+parameter reduce the curvature\index{curvature} at the indicated path endpoints,
+while large values
+increase the curvature as shown in Figure~\ref{fig10}. In particular, a curl value
+of zero makes the curvature approach zero.
+
+\begin{figure}[htp]
+$$ {\mathcenter{\epsfbox{manfig.111}} \atop c=0}
+ \qquad
+ {\mathcenter{\epsfbox{manfig.211}} \atop c=1}
+ \qquad
+ {\mathcenter{\epsfbox{manfig.311}} \atop c=2}
+ \qquad
+ {\mathcenter{\epsfbox{manfig.411}} \atop c=\infty}
+$$
+\caption[Effects of changing the curl parameter]
+ {Results of {\tt draw z0\char`\{curl c\char`\}..z1..%
+ \char`\{curl c\char`\}z2} for various values
+ of the curl parameter~$c$.}
+\label{fig10}
+\end{figure}
+
+\subsection{Summary of Path Syntax}
+
+There are a few other features of MetaPost path syntax, but they are relatively
+unimportant. Since \MF\ uses the same path syntax, interested readers can refer
+to \cite[chapter 14]{kn:c}. The summary of path syntax in Figure~\ref{sypath}
+includes everything discussed so far including the \verb|--| and \verb|...|
+constructions which \cite{kn:c} shows to be macros rather than primitives.
+A few comments on the semantics are in order here: If there is a non-empty
+$\descr{direction specifier}$ before a $\descr{path knot}$ but not after it,
+or vice versa, the specified direction (or curl amount) applies to both the
+incoming and outgoing path segments. A similar arrangement applies when a
+$\descr{controls}$ specification gives only one $\descr{pair primary}$.
+Thus
+$$ \hbox{\verb|..controls (30,20)..|} $$
+is equivalent to
+$$ \hbox{\verb|...controls (30,20) and (30,20)..|} $$
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\descr{path expression} \rightarrow
+ \descr{path subexpression}$\\
+\qquad \= ${}\mid \descr{path subexpression} \descr{direction specifier}$\\
+\> ${}\mid \descr{path subexpression} \descr{path join}$ \verb|cycle|\\
+$\descr{path subexpression} \rightarrow
+ \descr{path knot}$\\
+\> ${}\mid \descr{path expression} \descr{path join} \descr{path knot}$\\
+$\descr{path join} \rightarrow
+ \hbox{\verb|--|}$\\
+\> ${}\mid \descr{direction specifier} \descr{basic path join}
+ \descr{direction specifier}$\\
+$\descr{direction specifier} \rightarrow
+ \descr{empty}$\\
+\> ${}\mid {}$\verb|{curl| $\descr{numeric expression}$\verb|}|\\
+\> ${}\mid {}$\verb|{|$\descr{pair expression}$\verb|}|\\
+\> ${}\mid {}$\verb|{|$\descr{numeric expression}$\verb|,|%
+ $\descr{numeric expression}$\verb|}|\\
+$\descr{basic path join} \rightarrow
+ \hbox{\verb|..|}
+ \mid \hbox{\verb|...|}
+ \mid \hbox{\verb|..|}\descr{tension}\hbox{\verb|..|}
+ \mid \hbox{\verb|..|}\descr{controls}\hbox{\verb|..|}$\\
+$\descr{tension} \rightarrow
+ \hbox{\verb|tension|}\descr{numeric primary}$\\
+\> ${}\mid \hbox{\verb|tension|}\descr{numeric primary}
+ \hbox{\verb|and|}\descr{numeric primary}$\\
+$\descr{controls} \rightarrow
+ \hbox{\verb|controls|}\descr{pair primary}$\\
+\> ${}\mid \hbox{\verb|controls|}\descr{pair primary}
+ \hbox{\verb|and|}\descr{pair primary}$
+\end{ctabbing}
+\caption{The syntax for path construction}
+\label{sypath}
+\end{figure}
+
+A pair of coordinates like \verb|(30,20)| or a \verb|z| variable that represents a
+coordinate pair is what Figure~\ref{sypath} calls a $\descr{pair primary}$.
+A $\descr{path knot}$ is similar except that it can take on other forms such as
+a path expression in parentheses. Primaries and expressions of various types will
+be discussed in full generality in Section~\ref{exprs}.
+
+
+\section{Linear Equations}
+\label{lin.eq}
+
+An important feature taken from \MF\ is the ability to solve linear
+equations so that programs can be written in a partially declarative fashion.
+For example, the MetaPost interpreter can read
+$$ \hbox{\verb|a+b=3; 2*a=b+3;|} $$
+and deduce that $a=2$ and $b=1$. The same equations can be written slightly more
+compactly by stringing them together with multiple equal signs:
+$$ \hbox{\verb|a+b = 2*a-b = 3;|} $$
+Whichever way you give the equations, you can then give the command\index{show?\texttt{show}}
+$$ \hbox{\tt show a,b;} $$
+to see the values of {\tt a} and {\tt b}. MetaPost responds by typing
+$$\begin{verbatim}
+>> 2
+>> 1
+\end{verbatim}
+$$
+
+Note that {\tt =}\index{=?\texttt{=}} is not an assignment operator; it simply declares
+that the left-hand side equals the right-hand side. Thus {\tt a=a+1} produces an
+error message complaining about an
+``inconsistent equation\index{Inconsistent equation?\texttt{Inconsistent equation}}.'' The way to increase
+the value of {\tt a} is to use the assignment\index{assignment} operator
+{\tt :=}\index{:=?\texttt{:=}} as follows:
+$$ \hbox{\tt a:=a+1;} $$
+In other words, {\tt :=} is for changing existing values while {\tt =} is for
+giving linear equations to solve.
+
+There is no restriction against mixing equations and assignment operations as in
+the following example:
+$$ \hbox{\tt a = 2; b = a; a := 3; c = a;} $$
+After the first two equations set {\tt a} and~{\tt b} equal to 2, the assignment
+operation changes {\tt a} to~3 without affecting {\tt b}. The final value of
+{\tt c} is 3 since it is equated to the new value of {\tt a}. In general, an
+assignment operation is interpreted by first computing the new value, then
+eliminating the old value from all existing equations before actually assigning
+the new value.
+
+\subsection{Equations and Coordinate Pairs}
+
+MetaPost can also solve linear equations involving coordinate pairs. We have
+already seen many trivial examples of this in the form of equations like
+$$ \hbox{\verb|z1=(0,.2in)|} $$
+Each side of the equation must be formed by adding or subtracting coordinate pairs
+and multiplying or dividing them by known numeric quantities. Other ways of
+naming pair-valued variables will be discussed later, but the
+${\tt z}\descr{number}$\index{z convention?{\tt z} convention} is convenient because it is
+an abbreviation for
+$$ \hbox{\tt (x}\descr{number} \hbox{\tt, y}\descr{number}\hbox{\tt)} $$
+This makes it possible to give values to \verb|z| variables by giving equations
+involving their coordinates. For instance, points {\tt z1}, {\tt z2}, {\tt z3},
+and~{\tt z6} in Figure~\ref{fig12} were initialized via the following equations:
+\begin{eqnarray*}
+ &&\hbox{\verb|z1=-z2=(.2in,0);|} \\
+ &&\hbox{\verb|x3=-x6=.3in;|} \\
+ &&\hbox{\verb|x3+y3=x6+y6=1.1in;|}
+\end{eqnarray*}
+Exactly the same points could be obtained by setting their values directly:
+$$ \begin{verbatim}
+z1=(.2in,0); z2=(-.2in,0);
+z3=(.3in,.6in); z6=(-.3in,1.2in);
+\end{verbatim}
+$$
+
+After reading the equations, the MetaPost interpreter knows the values of
+{\tt z1}, {\tt z2},
+{\tt z3}, and~{\tt z6}. The next step in the construction of Figure~\ref{fig12}
+is to define points {\tt z4} and {\tt z5} equally spaced along the line from
+{\tt z3} to {\tt z6}. Since this operation comes up often, MetaPost has a special
+syntax for it. This mediation construction\index{mediation}
+$$ \hbox{\verb|z4=1/3[z3,z6]|} $$
+means that {\tt z4} is $1\over3$ of the way from $z3$ to $z6$; i.e.,
+$$ {\tt z4}={\tt z3}+{1\over3}({\tt z6}-{\tt z3}). $$
+Similarly
+$$ \hbox{\verb|z5=2/3[z3,z6]|} $$
+makes {\tt z5} $2\over3$ of the way from $z3$ to $z6$.
+
+\begin{figure}[htp]
+$$ \begin{verbatim}
+beginfig(13);
+z1=-z2=(.2in,0);
+x3=-x6=.3in;
+x3+y3=x6+y6=1.1in;
+z4=1/3[z3,z6];
+z5=2/3[z3,z6];
+z20=whatever[z1,z3]=whatever[z2,z4];
+z30=whatever[z1,z4]=whatever[z2,z5];
+z40=whatever[z1,z5]=whatever[z2,z6];
+draw z1--z20--z2--z30--z1--z40--z2;
+pickup pencircle scaled 1pt;
+draw z1--z2;
+draw z3--z6;
+endfig;
+\end{verbatim}
+\quad \mathcenter{\epsfbox{manfig.13}}
+$$
+\caption[MetaPost code and figure using linear equations]
+ {MetaPost commands and the resulting figure. Point labels have been
+ added to the figure for clarity.}
+\label{fig12}
+\end{figure}
+
+Mediation can also be used to say that some point is at an unknown position along
+the line between two known points. For instance, we could a introduce new
+variable {\tt aa} and write something like
+$$ \hbox{\verb|z20=aa[z1,z3];|} $$
+This says that {\tt z20} is some unknown fraction {\tt aa} of the way along the
+line between {\tt z1} and {\tt z3}. Another such equation involving a different
+line is sufficient to fix the value of {\tt z20}. To say that {\tt z20} is at
+the intersection of the {\tt z1}-{\tt z3} line and the {\tt z2}-{\tt z4} line,
+introduce another variable {\tt ab} and set
+$$ \hbox{\verb|z20=ab[z2,z4];|} $$
+This allows MetaPost to solve for {\tt x20}, {\tt y20}, {\tt aa}, and {\tt ab}.
+
+It is a little painful to keep
+thinking up new names like {\tt aa} and {\tt ab}. This can be avoided by using
+a special feature called {\tt whatever}\index{whatever?\texttt{whatever}}\label{Dwhatev}.
+This macro generates a new anonymous
+variable each time it appears. Thus the statement
+$$ \hbox{\verb|z20=whatever[z1,z3]=whatever[z2,z4]|} $$
+sets {\tt z20} as before, except it uses {\tt whatever} to generate two
+{\em different\/} anonymous variables instead of {\tt aa} and {\tt ab}.
+This is how Figure~\ref{fig12} sets {\tt z20}, {\tt z30}, and
+{\tt z40}.
+
+\subsection{Dealing with Unknowns}
+
+A system of equations such as those used in Figure~\ref{fig12} can be given in
+any order as long as all the equations are linear and all the variables can
+be determined before they are needed. This means that the equations
+\begin{eqnarray*}
+ && \hbox{\verb|z1=-z2=(.2in,0);|}\\
+ && \hbox{\verb|x3=-x6=.3in;|}\\
+ && \hbox{\verb|x3+y3=x6+y6=1.1in;|}\\
+ && \hbox{\verb|z4=1/3[z3,z6];|}\\
+ && \hbox{\verb|z5=2/3[z3,z6];|}
+\end{eqnarray*}
+suffice to determine {\tt z1} through {\tt z6}, no matter what order the equations
+are given in. On the other hand
+$$ \hbox{\verb|z20=whatever[z1,z3]|} $$
+is legal only when a known value has previously been specified for the difference
+${\tt z3}-{\tt z1}$, because the equation is equivalent
+to\index{mediation}
+$$ \hbox{\verb|z20 = z1 + whatever*(z3-z1)|} $$
+and the linearity requirement disallows multiplying unknown components of
+${\tt z3}-{\tt z1}$ by the anonymous unknown result of {\tt whatever}. The general
+rule is that you cannot multiply two unknown quantities or divide by an unknown
+quantity, nor can an unknown quantity be used in a {\tt draw} statement.
+Since only linear equations are allowed, the MetaPost interpreter can easily solve
+the equations and keep track of what values are known.
+
+The most natural way to ensure that MetaPost can handle an expression like
+$$ \hbox{\verb|whatever[z1,z3]|} $$
+is to ensure that {\tt z1} and {\tt z3} are both known. However this is not
+actually required since MetaPost may be able to deduce a known value for
+${\tt z3}-{\tt z1}$ before either of {\tt z1} and {\tt z3} are known.
+For instance, MetaPost will accept the equations
+$$ \hbox{\verb|z3=z1+(.1in,.6in); z20=whatever[z1,z3];|} $$
+but it will not be able to determine any of the components of {\tt z1}, {\tt z3},
+or {\tt z20}.
+
+These equations do give partial information about {\tt z1}, {\tt z3},
+and {\tt z20}. A good way to see this is to give another equation such as
+$$ \hbox{\verb|x20-x1=(y20-y1)/6;|} $$
+This produces the error message
+``{\tt ! Redundant equation}\index{Redundant equation?\texttt{Redundant equation}}.''
+MetaPost assumes that you are trying to tell it something new, so it will usually
+warn you when you give a redundant equation. If the new equation had been
+$$ \hbox{\verb|(x20-x1)-(y20-y1)/6=1in;|} $$
+the error message would have been\index{Inconsistent equation?\texttt{Inconsistent equation}}
+$$ \hbox{\verb|! Inconsistent equation (off by 71.99979).|} $$
+This error message illustrates
+roundoff\index{roundoff error} error in MetaPost's linear equation solving
+mechanism. Roundoff error
+is normally not a serious problem. but it is likely to cause trouble if you are
+trying to do something like find the intersection of two lines that are almost
+parallel.
+
+
+\section{Expressions}
+\label{exprs}
+
+It is now time for a more systematic view of the MetaPost language. We have seen
+that there are numeric quantities and coordinate pairs, and that these can be
+combined to specify paths for {\tt draw} statements.
+We have also seen how variables can be used in linear equations, but we have not
+discussed all the operations and data types that can be used in equations.
+
+It is possible to experiment with expressions involving any of the data types
+mentioned below by using the statement\index{show?\texttt{show}}\label{Dshow}
+$$ {\tt show}\, \descr{expression} $$
+to ask MetaPost to print a symbolic representation of the value of each expression.
+For known numeric values, each is printed on a new line preceded by ``{\tt >>} ''.
+Other types of results are printed similarly, except that complicated values are
+sometimes not printed on standard output. This produces a reference to the
+transcript file\index{files!transcript} that looks like this:
+$$ \hbox{\tt >> picture (see the transcript file)} $$
+If you want to the full results of {\tt show} statements to be printed on your
+terminal, assign a positive value to the
+internal\index{internal variables} variable\index{variables!internal}
+{\tt tracingonline}\index{tracingonline?\texttt{tracingonline}}\label{Dtonline}.
+
+\subsection{Data Types}
+
+MetaPost actually has nine basic data types\index{types}: numeric,
+pair, path, transform,
+color, string, boolean, picture, and pen. Let us consider these one at a time
+beginning with the numeric type.
+
+Numeric\index{numeric type} quantities in MetaPost are represented in fixed
+point arithmetic\index{arithmetic} as
+integer multiples of $1\over65536$. They must normally have absolute values
+less than 4096 but intermediate results can be eight times larger.
+This should not be a problem for distances or coordinate values since 4096
+PostScript points is more than 1.4~meters. If you need to work with numbers
+of magnitude 4096 or more, setting the internal variable
+{\tt warningcheck}\index{warningcheck}\label{Dwarncheck} to zero
+suppresses the warning messages about large numeric quantities.
+
+The pair\index{pair type} type is represented as a pair of numeric quantities.
+We have seen that pairs
+are used to give coordinates in {\tt draw} statements. Pairs can be added,
+subtracted, used in mediation expressions, or multiplied or divided by numerics.
+
+Paths\index{path type} have already been discussed in the context of {\tt draw}
+statements, but
+that discussion did not mention that paths are first-class objects that can be
+stored and manipulated. A path represents a straight or curved line that is
+defined parametrically.
+
+Another data type represents an arbitrary affine
+transformation\index{transform type}. A {\em transform\/} can be any combination
+of rotating, scaling, slanting,
+and shifting. If ${\tt p}=(p_x,p_y)$ is a pair and {\tt T} is a
+transform,\index{transformed?\texttt{transformed}}
+$$ \hbox{\tt p transformed T} $$
+is a pair of the form
+$$ (t_x+t_{xx}p_x+t_{xy}p_y, t_y+t_{yx}p_x+t_{yy}p_y), $$
+where the six numeric quantities $(t_x,t_y,t_{xx},t_{xy},t_{yx},t_{yy})$
+determine {\tt T}. Transforms can also be applied to paths, pictures, pens,
+and transforms.
+
+The color\index{color type} type is a lot like the pair type, except that it
+has three components
+instead of two. Like pairs, colors can be added, subtracted, used in mediation
+expressions, or multiplied or divided by numerics. Colors can be specified
+in terms of the predefined constants {\tt black}\index{black?\texttt{black}}\label{Dblack},
+{\tt white}\index{white?\texttt{white}}\label{Dwhite}, {\tt red}\index{red?\texttt{red}}\label{Dred},
+{\tt green}\index{green?\texttt{green}}\label{Dgreen},
+{\tt blue}\index{blue?\texttt{blue}}\label{Dblue}, or the red, green,
+and blue components can be given explicitly. Black is {\tt (0,0,0)} and white
+is {\tt (1,1,1)}. A level of gray such as {\tt (.4,.4,.4)} can be specified
+as {\tt 0.4white}. There is no restriction against colors ``blacker
+than black'' or ``whiter than white'' except all components are snapped
+back to the $[0,1]$ range when a color is given in a PostScript\index{PostScript}
+output file. MetaPost solves linear equations involving colors the same way it
+does for pairs.
+
+A string\index{string type} represents a sequence of characters.
+String constants\index{string constants} are given
+in double quotes \hbox{\verb|"like this"|}. String constants cannot contain
+double quotes or newlines, but there is a way to construct a string containing
+any sequence of eight-bit characters.
+
+The boolean\index{boolean type} type has the constants
+{\tt true}\index{true?\texttt{true}}\label{Dtrue} and
+{\tt false}\index{false}\label{Dfalse} and the
+operators {\tt and}\index{and?\texttt{and}}\label{Dand}, {\tt or}\index{or?\texttt{or}}\label{Dor},
+{\tt not}\index{not?\texttt{not}}\label{Dnot}. The relations \verb|=| and
+\verb|<>|\index{<>?\texttt{<>}}\label{Dcmpar}
+test objects of any type for equality and inequality\index{inequality}.
+Comparison\index{comparison} relations \verb|<|\index{<?\texttt{<}},
+\verb|<=|\index{<=?\texttt{<=}}, \verb|>|\index{>?\texttt{>}}, and \verb|>=|\index{>=?\texttt{>=}}
+are defined lexicographically for
+strings and in the obvious way for numerics. Ordering relations are also
+defined for booleans, pairs, colors, and transforms, but the comparison rules
+are not worth discussing here.
+
+The picture\index{picture type} data type is just what the name implies.
+Anything that can be drawn in MetaPost can be stored in a picture variable.
+In fact, the {\tt draw}\index{draw?\texttt{draw}}
+statement actually stores its results in a special picture variable called
+{\tt currentpicture}\index{currentpicture?\texttt{currentpicture}}. Pictures can be added to other
+pictures and operated on by transforms.
+
+Finally, there is a data type called a pen\index{pen type}. The main function
+of pens in
+MetaPost is to determine line thickness, but they can also be used to achieve
+calligraphic effects. The statement\index{pickup?\texttt{pickup}}\label{Dpickup}
+$$ {\tt pickup\ }\descr{pen expression} $$
+causes the given pen to be used in subsequent {\tt draw} statements.
+Normally, the pen expression is of the form
+$$ {\tt pencircle\ scaled\ }\descr{numeric primary}. $$
+This defines a circular pen that produces lines of constant thickness.
+If calligraphic effects are desired, the pen expression can be adjusted to give
+an elliptical pen or a polygonal pen.
+
+\subsection{Operators}
+
+There are many different ways to make expressions of the nine basic types, but
+most of the operations fit into a fairly simple syntax with four levels of
+precedence as shown in Figure~\ref{syexpr}. There are
+primaries\index{primary?\tdescr{primary}}, secondaries\index{secondary?\tdescr{secondary}},
+tertiaries\index{tertiary?\tdescr{tertiary}}, and expressions\index{expression?\tdescr{expression}}
+of each of the basic types, so the syntax rules could
+be specialized to deal with items such as \tdescr{numeric primary},
+\tdescr{boolean tertiary}, etc. This allows the result type for an operation
+to depend on the choice of operator and the types of its operands. For example,
+the {\tt <} relation is a \tdescr{tertiary binary} that can be applied
+to a \tdescr{numeric expression} and a \tdescr{numeric tertiary} to give a
+\tdescr{boolean expression}. The same operator can accept other operand types
+such as \tdescr{string expression} and \tdescr{string tertiary}, but an error
+message results if the operand types do not match.
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{primary} \rightarrow \descr{variable}$\\
+$\tt \qquad \;|\; \hbox{\tt (}\descr{expression}\hbox{\tt )}$\\
+$\tt \qquad \;|\; \descr{nullary op}$\\
+$\tt \qquad \;|\; \descr{of operator} \descr{expression}
+ of \descr{primary}$\\
+$\tt \qquad \;|\; \descr{unary op} \descr{primary}$\\
+$\tt \descr{secondary} \rightarrow \descr{primary}$\\
+$\tt \qquad \;|\; \descr{secondary} \descr{primary binop} \descr{primary}$\\
+$\tt \descr{tertiary} \rightarrow \descr{secondary}$\\
+$\tt \qquad \;|\; \descr{tertiary} \descr{secondary binop}
+ \descr{secondary}$\\
+$\tt \descr{expression} \rightarrow \descr{tertiary}$\\
+$\tt \qquad \;|\; \descr{expression} \descr{tertiary binop}
+ \descr{tertiary}$
+\end{ctabbing}
+\caption{The overall syntax rules for expressions}
+\index{unary op?\tdescr{unary op}} \index{nullary op?\tdescr{nullary op}}
+\index{primary binop?\tdescr{primary binop}} \index{secondary binop?\tdescr{secondary binop}}
+\index{tertiary binop?\tdescr{tertiary binop}}
+\label{syexpr}
+\end{figure}
+
+The multiplication and division operators {\tt *}\label{Dmldiv}
+and~{\tt /} are examples of what
+Figure~\ref{syexpr} calls a \tdescr{primary binop}. Each can accept two numeric
+operands or one numeric operand and one operand of type pair or color.
+The exponentiation operator \verb|**|\index{**?\texttt{**}}\index{exponentiation}\label{Dpow}
+is a \tdescr{primary binop} that requires two numeric operands.
+Placing this at the
+same level of precedence as multiplication
+and division has the unfortunate consequence that \verb|3*a**2| means $(3a)^2$,
+not $3(a^2)$\index{parsing irregularities}. Since unary negation\label{Dneg}
+applies at the primary level, it also turns
+out that \verb|-a**2| means $(-a)^2$. Fortunately, subtraction has lower
+precedence so that \verb|a-b**2| does mean $a-(b^2)$ instead of $(a-b)^2$.
+
+Another \tdescr{primary binop} is the
+{\tt dotprod}\index{dotprod?\texttt{dotprod}}\label{Ddprod} operator that computes the
+vector dot product of two pairs. For example, {\tt z1 dotprod z2} is equivalent
+to {\tt x1*y1 + x2*y2}.
+
+The additive operators {\tt +} and {\tt -}\label{Dadd} are
+\tdescr{secondary binops} that
+operate on numerics, pairs, or colors and produce results of the same type.
+Other operators that fall in this category are ``Pythagorean addition''
+\verb|++|\index{++?\texttt{++}}\label{Dpyadd} and
+``Pythagorean subtraction'' \verb|+-+|\index{+-+?\texttt{+-+}}\label{Dpysub}:
+\verb|a++b| means $\sqrt{a^2+b^2}$ and \verb|a+-+b| means $\sqrt{a^2-b^2}$.
+There are too many other operators to list here, but some of the most important
+are the boolean operators {\tt and}\index{and?\texttt{and}} and {\tt or}\index{or?\texttt{or}}.
+The {\tt and} operator is a
+\tdescr{primary binop} and the {\tt or} operator is a \tdescr{secondary binop}.
+
+The basic operations on strings are concatenation\index{concatenation} and
+substring construction.
+The \tdescr{tertiary binop} \verb|&|\index{&?\texttt{\&}}\label{Damp}
+implements concatenation; e.g.,
+$$ \hbox{\verb|"abc" & "de"|} $$
+produces the string \verb|"abcde"|.
+For substring construction, the
+\tdescr{of operator} {\tt substring}\index{substring of?\texttt{substring of}}\label{Dsubstr}
+is used like this:
+$$ {\tt substring}\, \descr{pair expression} \,{\tt of}\, \descr{string primary} $$
+The \tdescr{pair expression} determines what part of the string to select. For
+this purpose, the string is indexed\index{indexing} so that integer positions
+fall {\em between\/} characters. Pretend the string is written on a piece of
+graph paper
+so that the first character occupies $x$~coordinates between zero and one and the
+next character covers the range $1\le x\le2$, etc. Thus the string \verb|"abcde"|
+should be thought of like this
+$$ \epsfbox{manfig.14} $$
+and {\tt substring (2,4) of "abcde"} is {\tt "cd"}. This takes a little getting
+used to but it tends to avoid annoying ``off by one'' errors.
+
+Some operators take no arguments at all. An example of what Figure~\ref{syexpr}
+calls a \tdescr{nullary op} is
+{\tt nullpicture}\index{nullpicture?\texttt{nullpicture}}\label{Dnlpic} which
+returns a completely blank picture.
+
+The basic syntax in Figure~\ref{syexpr} only covers aspects of the expression
+syntax that are relatively type-independent. For instance, the complicated path
+syntax given in Figure~\ref{sypath} gives alternative rules for constructing a
+\tdescr{path expression}. An additional rule\index{path knot?\tdescr{path knot}}
+$$ \descr{path knot} \rightarrow \descr{pair tertiary} \;|\; \descr{path tertiary}
+$$
+explains the meaning of \tdescr{path knot} in Figure~\ref{sypath}. This means
+that the path expression
+$$ \hbox{\verb|z1+(1,1){right}..z2|} $$
+does not need parentheses around {\tt z1+(1,1)}.
+
+\subsection{Fractions, Mediation, and Unary Operators}
+
+Mediation\index{mediation} expressions do not appear in the basic expression
+syntax of Figure~\ref{syexpr}. Mediation expressions are parsed at the
+\tdescr{primary} level, so the general rule for constructing them is
+$$ \descr{primary} \rightarrow
+ \descr{numeric atom} \hbox{\tt [} \descr{expression}
+ \hbox{\tt ,} \descr{expression} \hbox{\tt ]}
+$$
+where each \tdescr{expression} can be of type numeric, pair, or color.
+The \tdescr{numeric atom}\index{numeric atom?\tdescr{numeric atom}} in a mediation
+expression is an extra simple type of \tdescr{numeric primary} as
+shown in Figure~\ref{synprim}. The meaning of all this is that the initial
+parameter in a mediation expression needs to be parenthesized when it is not
+just a variable, a positive number, or a positive fraction.
+For example,\index{parsing irregularities}
+$$ \hbox{\tt -1[a,b]} \quad {\rm and}\quad \hbox{\tt (-1)[a,b]} $$
+are very different: the former is $-b$ since it is equivalent to
+{\tt -(1[a,b])}; the latter is $a-(b-a)$ or $2a-b$.
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{numeric primary} \rightarrow \descr{numeric atom}$\\
+$\tt \qquad \;|\; \descr{numeric atom}\hbox{\tt [}
+ \descr{numeric expression}\hbox{\tt ,}\descr{numeric expression}\hbox{\tt ]}$\\
+$\tt \qquad \;|\; \descr{of operator} \descr{expression} of \descr{primary}$\\
+$\tt \qquad \;|\; \descr{unary op} \descr{primary}$\\
+$\tt \descr{numeric atom} \rightarrow \descr{numeric variable}$\\
+$\tt \qquad \;|\; \descr{number or fraction}$\\
+$\tt \qquad \;|\; \hbox{\tt (}\descr{numeric expression}\hbox{\tt )}$\\
+$\tt \qquad \;|\; \descr{numeric nullary op}$\\
+$\tt \descr{number or fraction} \rightarrow \descr{number}
+ \hbox{\tt /}\descr{number}$\\
+$\tt \qquad \;|\; \descr{number not followed by
+ `$\hbox{\tt /}\descr{number}$'}$\\
+\end{ctabbing}
+\caption{Syntax rules for numeric primaries}
+\label{synprim}
+\end{figure}
+
+A noteworthy feature of the syntax rules in Figure~\ref{synprim} is that the
+{\tt /}\index{fractions} operator binds most tightly when its operands are
+numbers. Thus {\tt 2/3} is a
+\tdescr{numeric atom}\index{numeric atom?\tdescr{numeric atom}}\index{parsing irregularities}
+while {\tt (1+1)/3} is only a \tdescr{numeric secondary}. Applying a
+\tdescr{primary binop} such as {\tt sqrt}\index{sqrt?\texttt{sqrt}}\label{Dsqrt}
+makes the difference clear:
+$$ \hbox{\tt sqrt 2/3} $$
+means $\sqrt{2\over3}$ while
+$$ \hbox{\tt sqrt(1+1)/3} $$
+means $\sqrt 2/3$.
+Operators such as {\tt sqrt} can be written in standard functional notation,
+but it is often unnecessary to parenthesize the argument. This applies to any
+function that is parsed as a \tdescr{primary binop}. For instance
+{\tt abs(x)}\index{abs?\texttt{abs}}\label{Dabs} and {\tt abs x} both compute the
+absolute value of {\tt x}. The same holds for the
+{\tt round}\index{round?\texttt{round}}\label{Dround},
+{\tt floor}\index{floor?\texttt{floor}}\label{Dfloor},
+{\tt ceiling}\index{ceiling?\texttt{ceiling}}\label{Dceil},
+{\tt sind}\index{sind?\texttt{sind}}\label{Dsind},
+and {\tt cosd}\index{cosd?\texttt{cosd}}\label{Dcosd}
+functions. The last two of these compute trigonometric functions of angles in
+degrees.
+
+Not all unary operators take numeric arguments and return numeric results.
+For instance, the {\tt abs}\index{abs?\texttt{abs}} operator can be applied to a pair
+to compute the Euclidean length of a vector. Applying the
+{\tt unitvector}\index{unitvector?\texttt{unitvector}}\label{Duvec} operator to a pair produces
+the same pair rescaled so that its Euclidean length is~1.
+The {\tt decimal}\index{decimal?\texttt{decimal}}\label{Ddecop}
+operator takes a number and returns the string representation.
+The {\tt angle}\index{angle?\texttt{angle}}\label{Dangle}
+operator takes a pair and computes the two-argument arctangent; i.e., {\tt angle}
+is the inverse of the {\tt dir} operator that was discussed in
+Section~\ref{tenscurl}. There is also an operator
+{\tt cycle}\index{cycle?\texttt{cycle}}\label{Dcycop}
+that takes a \tdescr{path primary} and returns a boolean result indicating whether
+the path is a closed curve.
+
+There is a whole class of other operators that classify expressions and return
+boolean results. A type name such as {\tt pair}\index{pair?\texttt{pair}} can operate on
+any type of \tdescr{primary} and return a boolean result indicating whether the
+argument is a {\tt pair}\label{Dpairop}. Similarly, each of the following can
+be used as a unary operator:
+{\tt numeric}\index{numeric?\texttt{numeric}}\label{Dnumop},
+{\tt boolean}\index{boolean?\texttt{boolean}}\label{Dboolop},
+{\tt color}\index{color?\texttt{color}}\label{Dcolrop},
+{\tt string}\index{string?\texttt{string}}\label{Dstrgop},
+{\tt transform}\index{transform?\texttt{transform}}\label{Dtrnfop},
+{\tt path}\index{path?\texttt{path}}\label{Dpathop},
+{\tt pen}\index{pen?\texttt{pen}}\label{Dpenop},
+and {\tt picture}\index{picture?\texttt{picture}}\label{Dpictop}.
+Besides just testing the type of a \tdescr{primary}, you can use the
+{\tt known}\index{known?\texttt{known}}\label{Dknown} and
+{\tt unknown}\index{unknown?\texttt{unknown}}\label{Dunknwn} operators to
+test if it has a completely known value.
+
+Even a number can behave like an operator in some contexts.
+This refers to the trick that allows {\tt 3x}\index{multiplication, implicit} and
+{\tt 3cm} as alternatives to {\tt 3*x} and {\tt 3*cm}. The rule is that a
+\tdescr{number or fraction} that is not followed by {\tt +}, {\tt -}, or another
+\tdescr{number or fraction} can serve as a \tdescr{primary binop}.
+Thus {\tt 2/3x}\index{parsing irregularities}
+is two thirds of {\tt x} but {\tt (2)/3x} is $2\over3x$ and {\tt 3 3} is illegal.
+
+There are also operators for extracting numeric subfields from pairs, colors,
+and even transforms. If {\tt p} is a \tdescr{pair primary},
+{\tt xpart p}\index{xpart?\texttt{xpart}}\label{Dxprt} and
+{\tt ypart p}\index{ypart}\label{Dyprt} extract its
+components so that
+$$ \hbox{\tt (xpart p, ypart p)} $$
+is equivalent to~{\tt p} even if {\tt p} is an unknown pair that is being used
+in a linear equation. Similarly, a color {\tt c} is equivalent
+to\index{redpart?\texttt{redpart}}\index{greenpart?\texttt{greenpart}}\index{bluepart?\texttt{bluepart}}\label{Drgbprt}
+$$ \hbox{\tt (redpart c, greenpart c, bluepart c)} $$
+The part specifiers for transforms will be discussed later.
+
+
+\section{Variables}
+\label{vars}
+
+MetaPost allows compound variable names such as {x.a}, {\tt x2r}, {\tt y2r},
+and {\tt z2r}, where {\tt z2r} means {\tt (x2r,y2r)} and {\tt z.a} means
+{\tt (x.a,y.a)}. In fact there is a broad class of suffixes such that
+{\tt z}\tdescr{suffix}\index{suffix?\tdescr{suffix}} means
+$$ (x\descr{suffix},\, y\descr{suffix}). $$
+Since a \tdescr{suffix} is composed of tokens, it is best to begin with a few
+comments about tokens.
+
+\subsection{Tokens}
+
+A MetaPost input file is treated as a sequence of numbers, string constants, and
+symbolic tokens\index{tokens}\index{tokens!symbolic}. A number consists of a
+sequence of digits possibly containing
+a decimal point. Technically, the minus sign in front of a negative number is
+a separate token. Since MetaPost uses fixed point arithmetic\index{arithmetic},
+it does not understand exponential notation such as {\tt 6.02E23}. MetaPost
+would interpret this as the number 6.02, followed by the symbolic token {\tt E},
+followed by the number~23.
+
+Anything between a pair of double quotes {\tt "} is a
+string constant\index{string constants}. It is
+illegal for a string constant to start on one line and end on a later line.
+Nor can a string constant contain double quotes {\tt "} or anything other than
+printable ASCII characters.
+
+Everything in a line of input other than numbers and string constants is broken
+into symbolic tokens\index{tokens!symbolic}. A symbolic token is a sequence of
+one or more similar characters, where characters are ``similar'' if they occur
+on the same row of Table~\ref{classes}.
+
+\begin{table}
+$$\begin{tabular}{c}
+\verb|ABCDEFGHIJKLMNOPQRSTUVWXYZ_abcdefghijklmnopqrstuvwxyz|\\
+{\tt :<=>|}\\
+\verb|#&@$|\\
+\verb|/*\|\\
+{\tt +-}\\
+{\tt !?}\\
+{\tt '`}\\
+\verb|^~|\\
+\verb|{}|\\
+{\tt [}\\
+{\tt ]}\\
+\end{tabular}
+$$
+\caption{Character classes for tokenization}
+\label{classes}
+\end{table}
+
+Thus \verb|A_alpha| and {\tt +-+} are symbolic tokens but {\tt !=} is interpreted
+as two tokens and {\tt x34} is a symbolic token followed by a number. Since the
+brackets {\tt [} and {\tt ]} are listed on lines by themselves, the only symbolic
+tokens involving them are {\tt [}, {\tt [[}, {\tt [[[}, etc.\ and
+{\tt ]}, {\tt ]]}, etc.
+
+Some characters are not listed in Table~\ref{classes} because they need special
+treatment. The four characters {\tt ,;()} are ``loners'': each comma, semicolon,
+or parenthesis is a separate token even when they occur consecutively. Thus
+{\tt (())} is four tokens, not one or two. The percent sign is very special
+because it introduces comments\index{comments}. The percent sign and everything
+after it up to the end of the line are ignored.
+
+Another special character is the period. Two or more periods
+together form a symbolic token, but a single period is ignored, and a period
+preceded or followed by digits is part of a number Thus {\tt ..}
+and {\tt ...} are symbolic tokens while {\tt a.b} is just two tokens {\tt a}
+and {\tt b}. It conventional to use periods to separate tokens in this fashion
+when naming a variable that is more than one token long.
+
+\subsection{Variable Declarations}
+\label{vardecl}
+
+A variable name is a symbolic token or a sequence of symbolic tokens.
+Most symbolic
+tokens are legitimate variable names, but anything with a predefined meaning like
+{\tt draw}, {\tt +}, or {\tt ..} is disallowed; i.e., variable names cannot be
+macros or MetaPost primitives. This minor restriction allows an amazingly broad
+class of variable names: {\tt alpha}, \verb|==>|, \verb|@&#$&|, and \verb|~~| are
+all legitimate variable names. Such symbolic tokens without special meanings
+are called {\em tags}\index{tags}.
+
+A variable name can be a sequence of tags like {\tt f.bot} or {\tt f.top}.
+The idea is to provide some of the functionality of Pascal records or C structures.
+It is also possible to simulate arrays by using variable names that contain
+numbers as well as symbolic tokens. For example, the variable name {\tt x2r}
+consists of the tag {\tt x}, the number 2, and the tag~{\tt r}. There can also
+be variables named {\tt x3r} and even {\tt x3.14r}. These variables can be
+treated as an array\index{arrays} via constructions like {\tt x[i]r},
+where {\tt i} has an appropriate numeric value. The overall syntax for
+variable names is shown in Figure~\ref{syvar}.
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{variable} \rightarrow \descr{tag}\descr{suffix}$\\
+$\tt \descr{suffix} \rightarrow \descr{empty} \;|\;
+ \descr{suffix}\descr{subscript} \;|\; \descr{suffix}\descr{tag}$\\
+$\tt \descr{subscript} \rightarrow \descr{number} \;|\;
+ \hbox{\tt [}\descr{numeric expression}\hbox{\tt ]}$
+\end{ctabbing}
+\caption{The syntax for variable names.}
+\index{suffix?\tdescr{suffix}}\index{subscript?\tdescr{subscript}}
+\label{syvar}
+\end{figure}
+
+Variables like {\tt x2} and {\tt y2} take on numeric values by default, so we
+can use the fact that {\tt z}\tdescr{suffix} is an abbreviation for\index{z convention?{\tt z} convention}\label{Dzconv}
+$$ (x\descr{suffix},\, y\descr{suffix}) $$
+to generate pair-valued variables when needed. It turns out that the
+{\tt beginfig}\index{beginfig?\texttt{beginfig}} macro wipes out pre-existing values variables
+that begin with the tags {\tt x} or {\tt y} so that
+{\tt beginfig} \ldots\ {\tt endfig}
+blocks do not interfere with each other when this naming scheme is used.
+In other words, variables that start with {\tt x}, {\tt y}, {\tt z} are
+local\index{variables!local}\index{locality}
+to the figure they are used in. General mechanisms for making variables local
+will be discussed in Section~\ref{grsec}.
+
+Type declarations\index{declarations}\index{type declarations}
+make it possible to use almost any naming scheme while still
+wiping out any previous value that might cause interference. For example, the
+declaration
+$$ \hbox{\tt pair pp, a.b;} $$
+makes {\tt pp} and {\tt a.b} unknown pairs. Such a declaration is not strictly
+local since {\tt pp} and {\tt a.b} are not automatically restored to their
+previous values at the end of the current figure. Of course, they are restored
+to unknown pairs if the declaration is repeated.
+
+Declarations work the same way for any of
+the other eight types: numeric, path, transform, color, string, boolean, picture,
+and pen. The only restriction is that you cannot give explicit numeric subscripts
+in a variable declaration. Do not give the illegal declaration
+$$ \hbox{\tt numeric q1, q2, q3;} $$
+use the generic subscript\index{subscript!generic} symbol {\tt []}\index{[]?\texttt{[]}}
+instead, to declare the whole array:
+$$ \hbox{\tt numeric q[];} $$
+You can also declare ``multidimensional'' arrays\index{arrays!multidimensional}.
+After the declaration
+$$ \hbox{\tt path p[]q[], pq[][];} $$
+{\tt p2q3} and {\tt pq1.4 5} are both paths.
+
+Internal\index{internal variables}\index{variables!internal}
+variables like {\tt tracingonline} cannot be declared in
+the normal fashion. All the internal variables discussed in this manual are
+predefined and do not have to be declared at all, but there is a way to declare
+that a variable should behave like a newly-created internal variable.
+The declaration is {\tt newinternal}\index{newinternal?\texttt{newinternal}}\label{Dnewint}
+followed by a list of symbolic tokens. For example,
+$$ \hbox{\tt newinternal a, b, c;} $$
+causes {\tt a}, {\tt b}, and {\tt c} to behave like internal variables. Such
+variables always have known numeric values, and these values can only be changed
+by using the assignment\index{assignment} operator {\tt:=}\index{:=?\texttt{:=}}.
+Internal variables are initially zero
+except that the Plain\index{Plain macros} macro package gives some of them nonzero
+initial values. (The Plain macros are normally preloaded automatically as
+explained in Section~\ref{intro}.)
+
+
+\section{Integrating Text and Graphics}
+\label{text}
+
+MetaPost has a number of features for including labels and other
+text\index{text and graphics}
+in the figures it generates. The simplest way to do this is to use the
+{\tt label}\index{label?\texttt{label}}\label{Dlabel} statement\index{label suffix?\tdescr{label suffix}}
+$$ {\tt label}\descr{label suffix} \hbox{\tt (}
+ \descr{string or picture expression} \hbox{\tt,}\, \descr{pair expression}
+ \hbox{\tt );}
+$$
+The \tdescr{string or picture expression} gives the label and the
+\tdescr{pair expression} says where to put it. The \tdescr{label suffix} can be
+\tdescr{empty} in which case the label is just centered on the given coordinates.
+If you are labeling some feature of a diagram you probably want to offset the
+label slightly to avoid overlapping. This is illustrated in Figure~\ref{fig16}
+where the {\tt "a"} label is placed above the midpoint of the line it refers to
+and the {\tt "b"} label is to the left of the midpoint of its line. This is
+achieved by using {\tt label.top}\index{top?\texttt{top}} for the {\tt "a"} label and
+{\tt label.lft}\index{lft?\texttt{lft}}
+for the {\tt "b"} label as shown in the figure. The \tdescr{label suffix}
+specifies the position of the label relative to the specified coordinates.
+The complete set of possibilities is\index{rt?\texttt{rt}}\index{bot?\texttt{bot}}%
+\index{ulft?\texttt{ulft}}\index{urt?\texttt{urt}}\index{llft?\texttt{llft}}\index{lrt?\texttt{lrt}}
+$$ \tt \descr{label suffix} \rightarrow
+ \descr{empty} \;|\; lft \;|\; rt \;|\; top \;|\; bot \;|\;
+ ulft \;|\;urt \;|\; llft \;|\; lrt
+$$
+where {\tt lft} and {\tt rt} mean left and right and {\tt llft}, {\tt ulft}, etc.\
+mean lower left, upper left, etc. The actual amount by which the label is offset
+in whatever direction is determined by the
+internal variable\index{internal variables}\index{variables!internal}
+{\tt labeloffset}\index{labeloffset?\texttt{labeloffset}}\label{Dlaboff}.
+
+\begin{figure}[htp]
+$$
+\begin{verbatim}
+beginfig(17);
+a=.7in; b=.5in;
+z0=(0,0);
+z1=-z3=(a,0);
+z2=-z4=(0,b);
+draw z1..z2..z3..z4..cycle;
+draw z1--z0--z2;
+label.top("a", .5[z0,z1]);
+label.lft("b", .5[z0,z2]);
+dotlabel.bot("(0,0)", z0);
+endfig;
+\end{verbatim}
+\qquad \mathcenter{\epsfbox{manfig.17}}
+$$
+\caption{MetaPost code and the resulting output}
+\label{fig16}
+\end{figure}
+
+Figure~\ref{fig16} also illustrates the
+{\tt dotlabel}\index{dotlabel?\texttt{dotlabel}}\label{Ddotlab}
+statement. This is exactly
+like the {\tt label} statement except that it adds a dot at the indicated
+coordinates. For example
+$$ \hbox{\tt dotlabel.bot("(0,0)", z0)} $$
+places a dot at {\tt z0} and then puts the label ``(0,0)'' just below the dot.
+Another alternative is the macro
+{\tt thelabel}\index{thelabel?\texttt{thelabel}}\label{Dthelab}. This has
+the same syntax as the {\tt label} and {\tt dotlabel} statements except that it
+returns the label as a \tdescr{picture primary} instead of actually drawing it.
+Thus
+$$ \hbox{\tt label.bot("(0,0)", z0)} $$
+is equivalent to
+$$ \hbox{\tt draw thelabel.bot("(0,0)", z0)} $$
+
+For simple applications of labeled figures, you can normally get by with just
+{\tt label} and {\tt dotlabel}. In fact, you may be able to use a short form of
+the {\tt dotlabel} statement that saves a lot of typing
+when you have many points {\tt z0}, {\tt z1}, {\tt z.a}, {\tt z.b}, etc.\
+and you want to use the {\tt z} suffixes as labels.
+The statement\index{dotlabels?\texttt{dotlabels}}\label{Ddotlbs}
+$$ \hbox{\tt dotlabels.rt(0, 1, a);} $$
+is equivalent to
+$$ \hbox{\tt dotlabel.rt("0",z0); dotlabel.rt("1",z1); dotlabel.rt("a",z.a);} $$
+Thus the argument to {\tt dotlabels} is a list of suffixes for which {\tt z}
+variables are known, and the \tdescr{label suffix} given with {\tt dotlabels}
+is used to position all the labels.
+
+There is also a {\tt labels}\index{labels?\texttt{labels}}\label{Dlabels} statement that is
+analogous to
+{\tt dotlabels} but its use is discouraged because it presents compatibility
+problems with \MF\index{metafont?\MF}. Some versions of the preloaded
+Plain\index{Plain macros} macro package define {\tt labels} to be synonymous
+with {\tt dotlabels}.
+
+For labeling statements such as {\tt label} and {\tt dotlabel} that use a
+string expression for the label text,
+the string gets typeset in a default font as determined by
+the string variable {\tt defaultfont}\index{defaultfont?\texttt{defaultfont}}\label{Ddffont}.
+The initial value of {\tt defaultfont}
+is likely to be {\tt "cmr10"}, but it can be changed to a different font name
+by giving an assignment such as
+$$ \hbox{\tt defaultfont:="Times-Roman"} $$
+There is also a numeric quantity called
+{\tt defaultscale}\index{defaultscale?\texttt{defaultscale}}\label{Ddfscale}
+that determines the type size.
+When {\tt default\-scale} is 1, you get the ``normal size'' which is
+usually 10 point, but this can also be changed. For instance
+$$ \hbox{\tt defaultscale := 1.2} $$
+makes labels come out twenty percent larger. If you do not know the normal size
+and you want to be sure the text comes out at some specific size, say 12 points,
+you can use the {\tt fontsize}\index{fontsize?\texttt{fontsize}}\label{Dfntsiz}
+operator to determine the normal size: e.g.,
+$$ \hbox{\tt defaultscale := 12pt/fontsize defaultfont;} $$
+
+When you change {\tt defaultfont}, the new font name should be something that
+\TeX\ would understand since MetaPost gets height and width information by reading
+the {\tt tfm}\index{tfm file?{\tt tfm} file}\index{files!tfm?{\tt tfm}} file.
+(This is explained in {\it The \TeX book\/}.~\cite{kn:a})
+It should be possible to use built-in PostScript fonts, but the names for them
+are system-dependent. Some systems may use {\tt rptmr} or {\tt ps-times-roman}
+instead of {\tt Times-Roman}.
+A \TeX\index{TeX?\TeX} font such as {\tt cmr10} is a little dangerous because it does
+not have a space character or certain ASCII symbols. In addition, MetaPost does
+not use the ligatures\index{ligatures} and kerning\index{kerning} information
+that comes with a \TeX\ font.
+
+
+\subsection{Typesetting Your Labels}
+
+\TeX\index{TeX?\TeX} may be used to format complex labels.
+If you say\index{btex?\texttt{btex}}\index{etex?\texttt{etex}}
+$$ {\tt btex}\, \descr{typesetting commands}\, {\tt etex} $$
+in a MetaPost input file, the \tdescr{typesetting commands} get processed by
+\TeX\ and translated into a picture expression
+(actually a \tdescr{picture primary}) that can be used in a {\tt label}
+or {\tt dotlabel} statement. Any spaces after {\tt btex} or before {\tt etex}
+are ignored. For instance, the statement
+$$ \hbox{\verb|label.lrt(btex $\sqrt x$ etex, (3,sqrt 3)*u)|} $$
+in Figure~\ref{fig17} places the label $\sqrt x$ at the lower right of the
+point {\tt (3,sqrt 3)*u}.
+
+\begin{figure}[htp]
+$$
+\begin{verbatim}
+beginfig(18);
+numeric u;
+u = 1cm;
+draw (0,2u)--(0,0)--(4u,0);
+pickup pencircle scaled 1pt;
+draw (0,0){up}
+ for i=1 upto 8: ..(i/2,sqrt(i/2))*u endfor;
+label.lrt(btex $\sqrt x$ etex, (3,sqrt 3)*u);
+label.bot(btex $x$ etex, (2u,0));
+label.lft(btex $y$ etex, (0,u));
+endfig;
+\end{verbatim}
+\qquad \mathcenter{\epsfbox{manfig.18}}
+$$
+\caption{MetaPost code and the resulting output}
+\label{fig17}
+\end{figure}
+
+Figure~\ref{fig18} illustrates some of the more complicated things that can
+be done with labels. Since the result of {\tt btex} \ldots {\tt etex} is
+a picture, it can be operated on like a picture. In particular, it is possible
+to apply transformations to pictures. We have not discussed the syntax for
+this yet, but a \tdescr{picture secondary}
+can be\index{rotated text}\index{rotated?\texttt{rotated}}
+$$ \descr{picture secondary}\, {\tt rotated}\, \descr{numeric primary} $$
+This is used in Figure~\ref{fig18} to rotate the label ``$y$ axis'' so that
+it runs vertically.
+
+\begin{figure}[htp]
+$$
+\begin{verbatim}
+beginfig(19);
+numeric ux, uy;
+120ux=1.2in; 4uy=2.4in;
+draw (0,4uy)--(0,0)--(120ux,0);
+pickup pencircle scaled 1pt;
+draw (0,uy){right}
+ for ix=1 upto 8:
+ ..(15ix*ux, uy*2/(1+cosd 15ix))
+ endfor;
+label.bot(btex $x$ axis etex, (60ux,0));
+label.lft(btex $y$ axis etex rotated 90,
+ (0,2uy));
+label.lft(
+ btex $\displaystyle y={2\over1+\cos x}$ etex,
+ (120ux, 4uy));
+endfig;
+\end{verbatim}
+\qquad \mathcenter{\epsfbox{manfig.19}}
+$$
+\caption{MetaPost code and the resulting output}
+\label{fig18}
+\end{figure}
+
+Another complication in Figure~\ref{fig18} is the use of the displayed equation
+$$y={2\over 1+\cos x}$$
+as a label. It would be more natural to code this as
+$$ \hbox{\verb|$$y={2\over 1+\cos x}$$|} $$
+but this would not work because
+\TeX\ typesets the labels in ``horizontal mode.''
+
+Here is how \TeX\ material gets translated into a form MetaPost understands:
+The MetaPost processor skips over
+{\tt btex}\index{btex?\texttt{btex}} \ldots\ {\tt etex}\index{etex?\texttt{etex}} blocks
+and depends on a preprocessor to translate them into low level MetaPost
+commands. If the main file is {\tt fig.mp}, the translated \TeX\
+material is placed in a file named {\tt fig.mpx}\index{files!mpx?{\tt mpx}}.
+This is normally
+done silently without any user intervention but it could fail if one of
+the {\tt btex} $\ldots$ {\tt etex} blocks contains an erroneous
+\TeX\index{TeX?\TeX!errors} command. Then the erroneous \TeX\ input
+is saved in the file {\tt mpxerr.tex}\index{mpxerr.tex?\texttt{mpxerr.tex}} and the error
+messages appear in {\tt mpxerr.log}\index{mpxerr.log?\texttt{mpxerr.log}}.
+
+\TeX\ macro definitions or any other auxiliary
+\TeX\ commands can be enclosed in a
+{\tt verbatimtex}\index{verbatimtex?\texttt{verbatimtex}} \ldots\ {\tt etex}\index{etex?\texttt{etex}} block.
+The difference between
+{\tt btex} and {\tt verbatimtex} is that the former generates a picture
+expression while the latter only adds material for \TeX\ to process.
+For instance, if you want \TeX\ to typeset labels using macros defined in
+{\tt mymac.tex}, your MetaPost input file would look something like this:
+\begin{eqnarray*}
+&& \verb|verbatimtex \input mymac etex|\\
+&& \verb|beginfig(1);|\\
+&& \qquad \vdots\\
+&& \verb|label(btex|\, \descr{\TeX\ material using \hbox{\tt mymac.tex}}\,
+ \verb|etex, | \descr{some coordinates} \hbox{\tt );}\\
+&& \qquad \vdots
+\end{eqnarray*}
+
+On Unix\footnote{Unix is a registered trademark of Unix Systems
+Laboratories.}\index{Unix\reg}
+systems, an environment variable can be used to specify that
+{\tt btex} $\ldots$ {\tt etex} and {\tt verbatimtex} $\ldots$ {\tt etex}
+blocks are in troff\index{troff} instead of \TeX. When using this option,
+it is a good idea to start your MetaPost input file with the assignment
+{\tt prologues:=1}\index{prologues?\texttt{prologues}}\label{Dprologs}. Giving this
+internal variable\index{internal variables}\index{variables!internal}
+a positive value causes causes output to be formatted as
+``structured PostScript''\index{PostScript!structured} generated on the
+assumption that text comes from built-in PostScript fonts. This makes MetaPost
+output much more portable, but it has an important drawback: It generally
+does not work when you use \TeX\ fonts, since programs that translate \TeX\
+output into PostScript\index{PostScript} need to make special provisions for
+\TeX\index{TeX?\TeX!fonts} fonts in
+included figures and the standard PostScript structuring rules do not allow
+for this. The details on how to include PostScript figures in a paper done
+in \TeX\ or troff are system-dependent. They can generally be found in
+manual pages and other on-line documentation. A file called {\tt dvips.tex}
+is distributed electronically along with the dvips\index{dvips} \TeX\ output
+processor.
+
+
+\subsection{The {\tt infont} operator}
+\label{Sinfont}
+
+Regardless of whether you use \TeX\ or troff, all the real work of adding
+text to pictures is done by a MetaPost primitive operator called
+{\tt infont}\index{infont?\texttt{infont}}. It is a
+\tdescr{primary binop}\index{primary binop?\tdescr{primary binop}} that takes a
+\tdescr{string secondary} as its
+left argument and a \tdescr{string primary} as its right argument. The left
+argument is text, and the right argument is a font name.
+The result of the operation is a \tdescr{picture secondary} that can then be
+transformed in various ways. One possibility is enlargement by a given factor
+via the syntax\index{scaled?\texttt{scaled}}
+$$ \descr{picture secondary}\, \hbox{\tt scaled}\, \descr{numeric primary} $$
+Thus {\tt label("text",z0)} is equivalent to
+$$ \hbox{\tt label("text" infont defaultfont scaled defaultscale, z0)} $$
+
+If it is not convenient to use a string constant for the left argument of
+{\tt infont}, you can use\index{char?\texttt{char}}\label{Dchar}
+$$ {\tt char}\, \descr{numeric primary} $$
+to select a character based on its numeric position in the font.
+Thus
+$$ \hbox{\tt char(n+64) infont "Times-Roman"} $$
+is a picture containing character {\tt n+64} of the Times-Roman font.
+
+\subsection{Measuring Text}
+\label{meas}
+
+MetaPost makes readily available the physical dimensions\index{size}
+of pictures generated by the {\tt infont} operator. There are
+unary operators {\tt llcorner}\index{llcorner?\texttt{llcorner}}\label{Dcornop},
+{\tt lrcorner}\index{lrcorner?\texttt{lrcorner}}, {\tt urcorner}\index{urcorner?\texttt{urcorner}},
+{\tt ulcorner}\index{ulcorner?\texttt{ulcorner}}, and {\tt center}\index{center}\label{Dcenter}
+that take a \tdescr{picture primary} and return the corners of its ``bounding
+box'' as illustrated in Figure~\ref{bbox}. The {\tt center} operator also
+accepts \tdescr{path primary} and \tdescr{pen primary} operands.
+In MetaPost Version 0.30 and higher, {\tt llcorner}, {\tt lrcorner}, etc.
+accept all three argument types as well.
+
+The argument type restrictions on the corner operators are not very important
+because their main purpose is to allow {\tt label} and {\tt dotlabel} statements
+to center their text properly.
+The predefined macro\index{bbox?\texttt{bbox}}\label{Dbbox}
+$$ {\tt bbox}\, \descr{picture primary} $$
+finds a rectangular path that represents the bounding box of a given picture.
+If {\tt p} is a picture, {\tt bbox p} equivalent to
+$$ \hbox{\tt (llcorner p--lrcorner p--urcorner p--ulcorner p--cycle)} $$
+except that it allows for a small amount of extra space around {\tt p} as specified
+by the internal variable\index{internal variables}\index{variables!internal}
+{\tt bboxmargin}\index{bboxmargin?\texttt{bboxmargin}}\label{Dbbmargin}.
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.20} $$
+\caption{A bounding box and its corner points.}
+\label{bbox}
+\end{figure}
+
+Note that MetaPost computes the bounding box of a {\tt btex}\index{btex?\texttt{btex}}
+\ldots\ {\tt etex}\index{etex?\texttt{etex}} picture just the way \TeX\index{TeX?\TeX} does.
+This is quite natural, but it has certain implications in view of the fact that
+\TeX\ has features like {\tt\string\strut}\index{strut?{\tt\string\strut}} and
+{\tt\string\rlap}\index{rlap?{\tt\string\rlap}} that allow \TeX\ users to lie about the
+dimensions of a box.
+
+When \TeX\ commands that lie about the dimensions of a box are translated in to
+low-level MetaPost code, a {\tt setbounds}\index{setbounds?\texttt{setbounds}}\label{Dsetbnd}
+statement does the lying:\index{picture variable?\tdescr{picture variable}}
+$$ {\tt setbounds}\, \descr{picture variable}\, {\tt to}\, \descr{path expression}
+$$
+makes the \tdescr{picture variable} behave as if its bounding box were the same
+as the given path. To get the true bounding box of such a picture, assign a
+positive value to the
+internal variable\index{internal variables}\index{variables!internal}
+{\tt truecorners}\index{truecorners?\texttt{truecorners}}\label{Dtruecorn}:\footnote{The
+{\tt setbounds} and
+{\tt truecorners} features are only found in MetaPost version 0.30 and higher.}
+i.e.,
+$$ \hbox{\verb|show urcorner btex $\bullet$\rlap{ A} etex|} $$
+produces ``\verb|>> (4.9813,6.8078)|'' while
+$$ \hbox{\verb|truecorners:=1; show urcorner btex $\bullet$\rlap{ A} etex|} $$
+produces ``\verb|>> (15.7742,6.8078)|.''
+
+
+\section{Advanced Graphics}
+\label{adv.gr}
+
+All the examples in the previous sections have been simple line drawings with
+labels added. This section describes shading and tools for generating
+not-so-simple line drawings.
+Shading is done with the {\tt fill}\index{fill?\texttt{fill}}\label{Dfill} statement.
+In its simplest
+form, the {\tt fill} statement requires a \tdescr{path expression} that gives
+the boundary of the region to be filled. In the syntax
+$$ {\tt fill}\, \descr{path expression} $$
+the argument should be a cyclic path, i.e., a path that describes a closed curve
+via the {\tt ..cycle} or {\tt --cycle} notation. For example, the {\tt fill}
+statement in Figure~\ref{fig20} builds a closed path by extending the roughly
+semicircular path~{\tt p}.
+This path has a counter-clockwise orientation, but that does not matter because
+the {\tt fill} statement uses PostScript's\index{PostScript} non-zero
+winding\index{winding number} number rule~\cite{ad:red}.
+
+\begin{figure}[htp]
+$$ \begin{verbatim}
+beginfig(21);
+path p;
+p = (-1cm,0)..(0,-1cm)..(1cm,0);
+fill p{up}..(0,0){-1,-2}..{up}cycle;
+draw p..(0,1cm)..cycle;
+endfig;
+\end{verbatim}
+\qquad \mathcenter{\epsfbox{manfig.21}}
+$$
+\caption{MetaPost code and the corresponding output.}
+\label{fig20}
+\end{figure}
+
+The general {\tt fill} statement\index{withcolor?\texttt{withcolor}}
+$$ {\tt fill}\, \descr{path expression}\,
+ {\tt withcolor}\, \descr{color expression}
+$$
+specifies a shade of gray or (if you have a color printer) some
+rainbow color.
+
+Figure~\ref{fig21} illustrates several applications of the fill command to fill
+areas with shades of gray. The paths involved are intersecting circles {\tt a}
+and {\tt b} and a path {\tt ab} that bounds the region inside both circles.
+Circles {\tt a} and {\tt b} are derived from a predefined path
+{\tt fullcircle}\index{fullcircle?\texttt{fullcircle}}\label{Dfcirc}
+that approximates a circle of unit diameter centered on the origin. There is
+also a predefined path {\tt halfcircle}\index{halfcircle?\texttt{halfcircle}}\label{Dhcirc}
+that is the part
+of {\tt fullcircle} above the $x$ axis. Path~{\tt ab} is the initialized
+using a predefined macro {\tt buildcycle} that will be discussed shortly.
+
+\begin{figure}[htp]
+$$ \begin{verbatim}
+beginfig(22);
+path a, b, aa, ab;
+a = fullcircle scaled 2cm;
+b = a shifted (0,1cm);
+aa = halfcircle scaled 2cm;
+ab = buildcycle(aa, b);
+picture pa, pb;
+pa = thelabel(btex $A$ etex, (0,-.5cm));
+pb = thelabel(btex $B$ etex, (0,1.5cm));
+fill a withcolor .7white;
+fill b withcolor .7white;
+fill ab withcolor .4white;
+unfill bbox pa;
+draw pa;
+unfill bbox pb;
+draw pb;
+label.lft(btex $U$ etex, (-1cm,.5cm));
+draw bbox currentpicture;
+endfig;
+\end{verbatim}
+\qquad \mathcenter{\epsfbox{manfig.22}}
+$$
+\caption{MetaPost code and the corresponding output.}
+\index{fullcircle?\texttt{fullcircle}}\index{halfcircle?\texttt{halfcircle}}\index{buildcycle?\texttt{buildcycle}}
+\label{fig21}
+\end{figure}
+
+Filling circle {\tt a} with the light gray color {\tt .7white} and then doing the
+same with circle {\tt b} doubly fills the region where the disks overlap. The
+rule is that each {\tt fill} statement assigns the given color to all points in
+the region covered, wiping out whatever was there previously including lines and
+text as well as filled regions. Thus it is important to give {\tt fill} commands
+in the right order.
+In the above example, the overlap region gets the same color twice,
+leaving it light gray after the first two {\tt fill} statements. The third fill
+statement assigns the darker color {\tt .4white} to the overlap region.
+
+At this point the circles and the overlap region have their final colors but
+there are no cutouts for the labels. The cutouts are achieved by the
+{\tt unfill}\index{unfill?\texttt{unfill}}\label{Dunfill}
+statements that effectively erase\index{erasing}
+the regions bounded by {\tt bbox pa}\index{bbox?\texttt{bbox}} and
+{\tt bbox pb}. More precisely, {\tt unfill} is shorthand for filling
+{\tt withcolor background}, where {\tt background} is normally equal to {\tt white}
+as is appropriate for printing on white paper. If necessary, you can assign a new
+color value to {\tt background}\index{background?\texttt{background}}\label{Dbground}.
+
+The labels need to be stored in pictures {\tt pa} and {\tt pb} to allow
+for measuring their bounding box before actually drawing them. The macro
+{\tt thelabel}\index{thelabel?\texttt{thelabel}} creates such
+pictures and shifts them into position so that they are ready to draw. Using the
+resulting pictures in {\tt draw} statements of the form\index{draw?\texttt{draw}}
+$$ {\tt draw}\, \descr{picture expression} $$
+adds them to {\tt currentpicture}\index{currentpicture?\texttt{currentpicture}}
+so that they overwrite a portion of what has
+already been drawn. In Figure~\ref{fig21} just the white rectangles produced by
+{\tt unfill} get overwritten.
+
+\subsection{Building Cycles}
+\label{buildcy}
+
+The {\tt buildcycle}\index{buildcycle?\texttt{buildcycle}} command constructs paths for use with
+the {\tt fill} or {\tt unfill} macros. When given two or more paths such as
+{\tt aa} and {\tt b},
+the {\tt buildcycle} macro tries to piece them together so as to form a cyclic
+path. In this case path {\tt aa} is a semicircle that starts just to the right
+of the intersection with path {\tt b}, then passes through {\tt b} and ends just
+outside the circle on the left as shown in Figure~\ref{fig22}a.
+
+Figure~\ref{fig22}b shows how {\tt buildcycle} forms a closed
+cycle from pieces of paths {\tt aa} and {\tt b}.
+The {\tt buildcycle} macro detects the two intersections\index{intersections}
+labeled 1 and 2 in
+Figure~\ref{fig22}b. Then it constructs the cyclic path shown in bold in the
+figure by going forward along path {\tt aa} from intersection~1 to
+intersection~2 and then forward around the counter-clockwise path {\tt b} back to
+intersection~1. It turns out that {\tt buildcycle(a,b)} would have produced the
+same result, but the reasoning behind this is a little confusing.
+
+
+\begin{figure}[htp]
+$$ {\epsfbox{manfig.123} \atop (a)}
+ \qquad {\epsfbox{manfig.223} \atop (b)}
+$$
+\caption[A demonstration of cycle building]
+ {(a)~The semicircular path~{\tt aa}
+ with a dashed line marking path {\tt b}; (b)~paths~{\tt aa} and {\tt b}
+ with the portions selected by {\tt buildcycle} shown by heavy lines.}
+\label{fig22}
+\end{figure}
+
+It is a easier to use the {\tt buildcycle} macro in situations like
+Figure~\ref{fig23} where there are more than two path arguments and each pair
+of consecutive paths has a unique intersection. For instance, the line~{\tt q0.5}
+and the curve~{\tt p2} intersect only at point~$P$; and the curve {\tt p2} and the
+line~{\tt q1.5} intersect only at point~$Q$. In fact, each of the points $P$,
+$Q$, $R$, $S$ is a unique intersection, and the result of\index{buildcycle?\texttt{buildcycle}}
+$$ \hbox{\tt buildcycle(q0.5, p2, q1.5, p4)} $$
+takes {\tt q0.5} from $S$ to~$P$, then {\tt p2} from $P$ to~$Q$, then {\tt q1.5}
+from $Q$ to~$R$, and finally {\tt p4} from $R$ back to~$S$. An examination of the
+MetaPost code for Figure~\ref{fig23} reveals that you have to go backwards along
+{\tt p2} in order to get from $P$ to~$Q$. This works perfectly well as long as
+the intersection\index{intersection} points are uniquely defined but it can cause
+unexpected results when pairs of paths intersect more than once.
+
+\begin{figure}[htp]
+$$ \begin{verbatim}
+beginfig(24);
+h=2in; w=2.7in;
+path p[], q[], pp;
+for i=2 upto 4: ii:=i**2;
+ p[i] = (w/ii,h){1,-ii}...(w/i,h/i)...(w,h/ii){ii,-1};
+endfor
+q0.5 = (0,0)--(w,0.5h);
+q1.5 = (0,0)--(w/1.5,h);
+pp = buildcycle(q0.5, p2, q1.5, p4);
+fill pp withcolor .7white;
+z0=center pp;
+picture lab; lab=thelabel(btex $f>0$ etex, z0);
+unfill bbox lab; draw lab;
+draw q0.5; draw p2; draw q1.5; draw p4;
+dotlabel.top(btex $P$ etex, p2 intersectionpoint q0.5);
+dotlabel.rt(btex $Q$ etex, p2 intersectionpoint q1.5);
+dotlabel.lft(btex $R$ etex, p4 intersectionpoint q1.5);
+dotlabel.bot(btex $S$ etex, p4 intersectionpoint q0.5);
+endfig;
+\end{verbatim}
+\atop \mathcenter{\epsfbox{manfig.24}}
+$$
+\caption{MetaPost code and the corresponding output.}
+\label{fig23}
+\end{figure}
+
+The general rule for the {\tt buildcycle} macro is that
+$$ \hbox{\tt buildcycle(}p_1\hbox{\tt,}\, p_2\hbox{\tt,}\,
+ p_3\hbox{\tt,}\, \ldots \hbox{\tt,} p_k \hbox{\tt )}
+$$
+chooses the intersection between each $p_i$ and $p_{i+1}$ to be as late as possible
+on $p_i$ and as early as possible on $p_{i+1}$. There is no
+simple rule for resolving conflicts between these two goals, so you should avoid
+cases where one intersection point occurs later on $p_i$ and another
+intersection\index{intersection} point occurs earlier on $p_{i+1}$.
+
+The preference for intersections as late as possible
+on $p_i$ and as early as possible on $p_{i+1}$ leads to ambiguity resolution in
+favor of forward-going subpaths. For cyclic paths such as path~{\tt b} in
+Figure~\ref{fig22} ``early'' and ``late'' are relative to a start/finish point
+which is where you get back to when you say ``{\tt ..cycle}''.
+For the path~{\tt b}, this turns out to be the rightmost point on the circle.
+
+A more direct way to deal with path intersections is via the
+\tdescr{secondary binop}\index{secondary binop?\tdescr{secondary binop}}
+{\tt intersection\-point}\index{intersectionpoint?\texttt{intersectionpoint}}\label{Disecpt}
+that finds the points $P$, $Q$, $R$, and~$S$ in Figure~\ref{fig23}.
+This macro finds a point where two given
+paths intersect. If there is more than one intersection point, it just chooses
+one; if there is no intersection, the macro generates an error message.
+
+\subsection{Dealing with Paths Parametrically}
+
+The {\tt intersectionpoint}\index{intersectionpoint?\texttt{intersectionpoint}} macro is based on a
+primitive operation called
+{\tt intersectiontimes}\index{intersectiontimes?\texttt{intersectiontimes}}\label{Disectt}.
+This \tdescr{secondary binop} is one of several
+operations that deal with paths parametrically. It locates an intersection
+between two paths by giving the ``time'' parameter on each path. This refers to
+the parameterization scheme from Section~\ref{curves} that described paths as
+piecewise cubic curves $\bigl(X(t),Y(t)\bigr)$ where $t$ ranges from zero to the
+number of curve segments. In other words, when a path is specified as passing
+through a sequence of points, where $t=0$ at the first point,
+then $t=1$ at the next, and $t=2$ at the next, etc. The result of
+$$ \hbox{\tt a intersectiontimes b} $$
+is $(-1,-1)$ if there is no intersection; otherwise you get
+a pair $(t_a,t_b)$, where $t_a$ is a time on path {\tt a} when it intersects
+path~{\tt b}, and $t_b$ is the corresponding time on path~{\tt b}.
+
+For example, suppose path~{\tt a} is denoted by the thin line in Figure~\ref{fig24}
+and path~{\tt b} is denoted by the thicker line. If the labels indicate time
+values on the paths, the pair of time values computed by
+$$ \hbox{\tt a intersectiontimes b} $$
+must be one of
+$$ (0.25,1.77),\ (0.75,1.40), {\rm or}\ (2.58,0.24), $$
+depending on which of the three intersection points is chosen by the MetaPost
+interpreter. The exact rules for choosing among multiple intersection points
+are a little complicated, but it turns out that you get the time values
+$(0.25,1.77)$ in this example. Smaller time values are preferred over larger
+ones so that $(t_a,t_b)$ is preferred to $(t'_a,t'_b)$ whenever $t'_a<t_a$ and
+$t_b<t'_b$. When no single alternative minimizes both the $t_a$ and $t_b$
+components the $t_a$ component tends to get priority, but the rules get more
+complicated when there are no integers between $t_a$
+and $t'_a$\index{intersection}.
+(For more details, see {\it The \MF book}.\cite[Chapter 14]{kn:c})
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.25} $$
+\caption{Two intersecting paths with time values marked on each path.}
+\label{fig24}
+\end{figure}
+
+The {\tt intersectiontimes} operator is more flexible than {\tt intersectionpoint}
+because there are a number of things that can be done with time values on a path.
+One of the most important is just to ask ``where is path {\tt p} at
+time {\tt t}?'' The construction\index{point of?\texttt{point of}}\label{Dpntof}
+$$ {\tt point}\, \descr{numeric expression}\, {\tt of}\, \descr{path primary} $$
+answers this question. If the \tdescr{numeric expression} is less than zero or
+greater than the time value assigned to the last point on the path, the
+{\tt point of} construction normally yields an endpoint of the path. Hence, it
+is common to use the predefined constant
+{\tt infinity}\index{infinity?\texttt{infinity}}\label{Dinf}
+(equal to 4095.99998) as the
+\tdescr{numeric expression} in a {\tt point of} construction when dealing with
+the end of a path.
+
+Such ``infinite'' time values do not work for a cyclic path, since
+time values outside of the normal range can be handled by modular arithmetic in
+that case; i.e., a cyclic path~{\tt p} through points $z_0$, $z_1$, $z_2$,
+\ldots, $z_{n-1}$ has the normal parameter range $0\le t<n$, but
+$$ \hbox{\tt point t of p} $$
+can be computed for any~$t$ by first reducing $t$ modulo~$n$. If the modulus~$n$
+is not readily available,\index{length?\texttt{length}}\label{Dlength}
+$$ {\tt length}\, \descr{path primary} $$
+gives the integer value of the upper limit of the normal time parameter range
+for the specified path.
+
+MetaPost uses the same correspondence between time values and points on a path to
+evaluate the {\tt subpath}\index{subpath?\texttt{subpath}}\label{Dsubpth} operator.
+The syntax for this operator is
+$$ {\tt subpath}\, \descr{pair expression}\, {\tt of}\, \descr{path primary} $$
+If the value of the \tdescr{pair expression} is $(t_1,t_2)$ and the
+\tdescr{path primary} is $p$, the result is a path that follows $p$ from
+{\tt point $t_1$ of $p$} to {\tt point $t_2$ of $p$}. If $t_2<t_1$, the subpath
+runs backwards along~$p$.
+
+An important operation based on the {\tt subpath} operator is the
+\tdescr{tertiary binop}\index{tertiary binop?\tdescr{tertiary binop}}
+{\tt cutbefore}\index{cutbefore?\texttt{cutbefore}}\label{Dcutb}. For intersecting
+paths $p_1$ and $p_2$,
+$$ p_1\ {\tt cutbefore}\ p_2 $$
+is equivalent to
+$$ \hbox{\tt subpath (xpart($p_1$ intersectiontimes $p_2$), length $p_1$) of $p_1$}
+$$
+except that it also sets the path variable
+{\tt cuttings}\index{cuttings?\texttt{cuttings}}\label{Dcuttings} to
+the portion of $p_1$ that gets cut off. In other words, {\tt cutbefore} returns
+its first argument with the part before the intersection cut off. With multiple
+intersections, it tries to cut off as little as possible. If the paths do not
+intersect, {\tt cutbefore} returns its first argument.
+
+There is also an analogous \tdescr{tertiary binop}\index{tertiary binop?\tdescr{tertiary binop}}
+called {\tt cutafter}\index{cutafter?\texttt{cutafter}}\label{Dcuta} that works by applying
+{\tt cutbefore} with
+time reversed along its first argument. Thus
+$$ p_1\ {\tt cutafter}\ p_2 $$
+tries to cut off the part of $p_1$ after its last intersection with $p_2$.
+
+Another operator\index{direction of?\texttt{direction of}}\label{Ddirof}
+$$ {\tt direction}\, \descr{numeric expression}\, {\tt of}\, \descr{path primary}
+$$
+finds a vector in the direction of the \tdescr{path primary}. This is defined
+for any time value analogously to the {\tt point of} construction. The resulting
+direction vector has the correct orientation and a somewhat arbitrary magnitude.
+Combining {\tt point of} and {\tt direction of} constructions yields the equation
+for a tangent line as illustrated in Figure~\ref{fig25}.
+
+\begin{figure}[htp]
+$$ \begin{verbatim}
+beginfig(26);
+numeric scf, #, t[];
+3.2scf = 2.4in;
+path fun;
+# = .1; % Keep the function single-valued
+fun = ((0,-1#)..(1,.5#){right}..(1.9,.2#){right}..{curl .1}(3.2,2#))
+ yscaled(1/#) scaled scf;
+x1 = 2.5scf;
+for i=1 upto 2:
+ (t[i],whatever) =
+ fun intersectiontimes ((x[i],-infinity)--(x[i],infinity));
+ z[i] = point t[i] of fun;
+ z[i]-(x[i+1],0) = whatever*direction t[i] of fun;
+ draw (x[i],0)--z[i]--(x[i+1],0);
+ fill fullcircle scaled 3bp shifted z[i];
+endfor
+label.bot(btex $x_1$ etex, (x1,0));
+label.bot(btex $x_2$ etex, (x2,0));
+label.bot(btex $x_3$ etex, (x3,0));
+draw (0,0)--(3.2scf,0);
+pickup pencircle scaled 1pt;
+draw fun;
+endfig;
+\end{verbatim}
+\atop \epsfbox{manfig.26}
+$$
+\caption{MetaPost code and the resulting figure}
+\label{fig25}
+\end{figure}
+
+If you know a slope and you want to find a point on a curve where the tangent
+line has that slope,
+the {\tt directiontime}\index{directiontime of?\texttt{directiontime of}}\label{Ddtimof}
+operator inverts the {\tt direction
+of} operation. Given a direction vector and a path,
+$$ {\tt directiontime}\, \descr{pair expression}\, {\tt of}\,
+ \descr{path primary}
+$$
+returns a numeric value that gives the first time~$t$ when the path has the
+indicated direction. (If there is no such time, the result is $-1$).
+For example, if {\tt a} is the path drawn as a thin curve in Figure~\ref{fig24},
+{\tt directiontime (1,1) of a} returns 0.2084.
+
+There is also an predefined macro \index{directionpoint of?\texttt{directionpoint of}}\label{Ddpntof}
+$$ {\tt directionpoint}\, \descr{pair expression}\, {\tt of}\,
+ \descr{path primary}
+$$
+that finds the first point on a path where a given direction is achieved. The
+{\tt directionpoint} macro produces an error message if the direction does not
+occur on the path.
+
+Operators {\tt arclength}\index{arclength?\texttt{arclength}}\label{Darclng} and
+{\tt arctime of}\index{arctime of?\texttt{arctime of}}\label{Darctim} relate the ``time''
+on a path is related to the more familiar concept of
+arc length.\index{arc length}\footnote{The
+{\tt arclength} and {\tt arctime} operators are only found in MetaPost
+version 0.50 and higher.}
+The expression
+$$ \hbox{{\tt arclength} \tdescr{path primary}} $$
+gives the arc length of a path. If {\tt p} is a path and {\tt a} is a number
+between 0 and {\tt arclength p},
+$$ \hbox{\tt arctime a of p} $$
+gives the time~{\tt t} such that
+$$ \hbox{\tt arclength subpath (0,t) of p} = {\tt a}. $$
+
+\subsection{Affine Transformations}
+\label{transsec}
+\index{transform type}
+
+Note how path {\tt fun} in Figure~\ref{fig25} is first constructed as
+$$ \hbox{\verb|(0,-.1)..(1,.05){right}..(1.9,.02){right}..{curl .1}(3.2,.2)|} $$
+and then the {\tt yscaled}\index{yscaled?\texttt{yscaled}} and {\tt scaled}\index{scaled?\texttt{scaled}}
+operators are used to adjust the
+shape and size of the path. As the name suggests, an expression involving
+``{\tt yscaled 10}'' multiplies $y$ coordinates by ten so that every point $(x,y)$
+on the original path corresponds to a point $(x,10y)$ on the transformed path.
+
+Including {\tt scaled} and {\tt yscaled}, there are seven transformation
+operators that take a numeric or pair argument:\index{shifted?\texttt{shifted}}%
+\index{rotated?\texttt{rotated}}\index{slanted?\texttt{slanted}}\index{scaled?\texttt{scaled}}\index{xscaled?\texttt{xscaled}}%
+\index{yscaled?\texttt{yscaled}}\index{zscaled?\texttt{zscaled}}\label{Dtranop}
+\begin{eqnarray*}
+ (x,y){\tt\ shifted\ }(a,b) &=& (x+a,\, y+b); \\
+ (x,y){\tt\ rotated\ }\theta &=& (x\cos\theta-y\sin\theta,\,
+ x\sin\theta+y\cos\theta); \\
+ (x,y){\tt\ slanted\ }a &=& (x+ay,\, y); \\
+ (x,y){\tt\ scaled\ }a &=& (ax,\, ay); \\
+ (x,y){\tt\ xscaled\ }a &=& (ax,\, y); \\
+ (x,y){\tt\ yscaled\ }a &=& (x,\, ay); \\
+ (x,y){\tt\ zscaled\ }(a,b) &=& (ax-by,\, bx+ay).
+\end{eqnarray*}
+Most of these operations are self-explanatory except for {\tt zscaled} which can
+be thought of as multiplication of complex numbers. The effect of {\tt zscaled}
+$(a,b)$ is to rotate and scale so as to map $(1,0)$ into $(a,b)$. The effect of
+{\tt rotated}~$\theta$ is rotate $\theta$ degrees counter-clockwise.
+
+Any combination of shifting, rotating, slanting, etc.\ is an affine transformation,
+the net effect of which is to transform any pair $(x,y)$ into
+$$ (t_x+t_{xx}x+t_{xy}y,\, t_y+t_{yx}x+t_{yy}y), $$
+for some sextuple $(t_x,t_y,t_{xx},t_{xy},t_{yx},t_{yy})$. This information can
+be stored in a variable of type transform so that
+{\tt transformed T}\index{transformed?\texttt{transformed}}\label{Dtrfrmd} might be equivalent to
+$$ \hbox{\tt xscaled -1 rotated 90 shifted (1,1)} $$
+if {\tt T} is an appropriate transform variable. The
+transform~{\tt T} could then be initialized with an
+expression of type transform as follows:
+$$ \begin{verbatim}
+transform T;
+T = identity xscaled -1 rotated 90 shifted (1,1);
+\end{verbatim}
+$$
+As this example indicates, transform expressions can be built up by applying
+transformation operators to other transforms. The predefined transformation
+{\tt identity}\index{identity?\texttt{identity}}\label{Dident} is a useful starting point
+for this process.
+This can be illustrated by paraphrasing the above equation for {\tt T} into
+English: ``{\tt T} should be the transform obtained by doing whatever
+{\tt identity} does, then scaling $x$~coordinates by $-1$, rotating $45^\circ$,
+and shifting by $(1,1)$.'' This works because {\tt identity} is the identity
+transformation which does nothing; i.e., {\tt transformed identity} is a no-op.
+
+The syntax for transform expressions and transformation operators is given in
+Figure~\ref{sytrans}. It includes two more options for
+\tdescr{transformer}:\index{reflectedabout?\texttt{reflectedabout}}
+$$ \hbox{\tt reflectededabout(}p, q\hbox{\tt )} $$
+reflects about the line defined by points $p$ and $q$; and\index{rotatedaround?\texttt{rotatedaround}}
+$$ \hbox{\tt rotatedaround(}p,\theta\hbox{\tt )} $$
+rotates $\theta$ degrees counter-clockwise around point $p$. For example,
+the equation for initializing transform~{\tt T} could have been
+$$ \hbox{\tt T = identity reflectedabout((2,0), (0,2))}. $$
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{pair secondary} \rightarrow
+ \descr{pair secondary} \descr{transformer}$\\
+$\tt \descr{path secondary} \rightarrow
+ \descr{path secondary} \descr{transformer}$\\
+$\tt \descr{picture secondary} \rightarrow
+ \descr{picture secondary} \descr{transformer}$\\
+$\tt \descr{pen secondary} \rightarrow
+ \descr{pen secondary} \descr{transformer}$\\
+$\tt \descr{transform secondary} \rightarrow
+ \descr{transform secondary} \descr{transformer}$\\[6pt]
+$\tt \descr{transformer} \rightarrow rotated \descr{numeric primary}$\\
+$\tt \qquad \;|\; scaled \descr{numeric primary}$\\
+$\tt \qquad \;|\; shifted \descr{pair primary}$\\
+$\tt \qquad \;|\; slanted \descr{numeric primary}$\\
+$\tt \qquad \;|\; transformed \descr{transform primary}$\\
+$\tt \qquad \;|\; xscaled \descr{numeric primary}$\\
+$\tt \qquad \;|\; yscaled \descr{numeric primary}$\\
+$\tt \qquad \;|\; zscaled \descr{pair primary}$\\
+$\tt \qquad \;|\; reflectedabout\hbox{\tt (}\descr{pair expression}
+ \hbox{\tt ,}\descr{pair expression}\hbox{\tt )}$\\
+$\tt \qquad \;|\; rotatedaround\hbox{\tt (}\descr{pair expression}
+ \hbox{\tt ,}\descr{numeric expression}\hbox{\tt )}$\\
+\end{ctabbing}
+\caption{The syntax for transforms and related operators}
+\label{sytrans}
+\end{figure}
+
+There is also a unary operator {\tt inverse}\index{inverse?\texttt{inverse}}\label{Dinv}
+that takes a
+transform and finds another transform that undoes the effect of the first
+transform. Thus if
+$$ p = q{\tt\ transformed\ }T $$
+then
+$$ q = p{\tt\ transformed\ inverse\ }T. $$
+
+It is not legal to take the {\tt inverse} of an
+unknown transform\index{transformation!unknown} but we
+have already seen that you can say
+$$ \hbox{\tt T = } \descr{transform expression} $$
+when {\tt T} has not been given a value yet. It is also possible to apply
+an unknown transform to a known pair or transform and use the result in a linear
+equation. Three such equations are sufficient to determine a transform. Thus
+the equations
+$$ \begin{verbatim}
+(0,1) transformed T' = (3,4);
+(1,1) transformed T' = (7,1);
+(1,0) transformed T' = (4,-3);
+\end{verbatim}
+$$
+allow MetaPost to determine that the transform {\tt T'} is a combination of
+rotation and scaling with
+$$\openup\jot
+ \tabskip=0pt plus 1fil
+ \halign to\displaywidth{\tabskip=0pt
+ \hfil$\displaystyle{#}$& $\displaystyle{{}#}$\hfil \qquad&
+ \hfil$\displaystyle{#}$& $\displaystyle{{}#}$\hfil
+ \tabskip=0pt plus 1fil\cr
+\noalign{\vskip-\jot}
+ t_{xx}&=4,& t_{yx}&=-3,\cr
+ t_{yx}&=3,& t_{yy}&=4,\cr
+ t_x&=0,& t_y&=0.\cr}
+$$
+
+Equations involving an unknown transform are treated as linear equations in the
+six parameters that define the transform. These six parameters can also be
+referred to directly as\index{xpart?\texttt{xpart}}\index{ypart?\texttt{ypart}}\index{xxpart?\texttt{xxpart}}%
+\index{xypart?\texttt{xypart}}\index{yxpart?\texttt{yxpart}}\index{yypart?\texttt{yypart}}\label{Dtrprt}
+$$ {\tt xpart\ T},\ {\tt ypart\ T},\ {\tt xxpart\ T},\ {\tt xypart\ T},\
+ {\tt yxpart\ T},\ {\tt yypart\ T},
+$$
+where {\tt T} is a transform. For instance, Figure~\ref{fig27} uses the
+equations
+$$ \hbox{\tt xxpart T=yypart T; yxpart T=-xypart T} $$
+to specify that {\tt T} is shape preserving; i.e., it is a combination of
+rotating, shifting, and uniform scaling.
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+beginfig(28);
+path p[];
+p1 = fullcircle scaled .6in;
+z1=(.75in,0)=-z3;
+z2=directionpoint left of p1=-z4;
+p2 = z1..z2..{curl1}z3..z4..{curl 1}cycle;
+fill p2 withcolor .4[white,black];
+unfill p1;
+draw p1;
+transform T;
+z1 transformed T = z2;
+z3 transformed T = z4;
+xxpart T=yypart T; yxpart T=-xypart T;
+picture pic;
+pic = currentpicture;
+for i=1 upto 2:
+ pic:=pic transformed T;
+ draw pic;
+endfor
+dotlabels.top(1,2,3); dotlabels.bot(4);
+endfig;
+\end{verbatim}
+\quad \mathcenter{\epsfbox{manfig.28}}
+$$
+\caption{MetaPost code and the resulting ``fractal'' figure}
+\label{fig27}
+\end{figure}
+
+
+\subsection{Dashed Lines}
+
+The MetaPost language provides many ways of changing the appearance of a line
+besides just changing its width. One way is to use dashed lines as was done in
+Figures \ref{fig4} and~\ref{fig22}. The syntax for this is\index{dashed?\texttt{dashed}}
+$$ {\tt draw}\, \descr{path expression}\, {\tt dashed}\, \descr{dash pattern} $$
+where a \tdescr{dash pattern}\index{dash pattern?\tdescr{dash pattern}} is really a special
+type of \tdescr{picture expression}. There is a predefined \tdescr{dash pattern}
+called {\tt evenly}\index{evenly?\texttt{evenly}}\label{Devenly} that makes dashes 3 PostScript
+points long separated by gaps of the same size.
+Another predefined dash pattern {\tt withdots}\index{withdots?\texttt{withdots}}\label{Dwdots}
+produces dotted lines with dots 5 PostScript points apart.\footnote{{\tt withdots}
+is only found in MetaPost version 0.50 and higher.}
+For dots further apart or longer dashes further apart, the
+\tdescr{dash pattern} can be
+scaled\index{scaled?\texttt{scaled}} as shown in Figure~\ref{fig28}
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.29} $$
+\caption[Dashed lines and the corresponding dash patters]
+ {Dashed lines each labeled with the \tdescr{dash pattern} used to create
+ it.}
+\label{fig28}
+\end{figure}
+
+Another way to change a dash pattern is to alter its phase by shifting it
+horizontally. Shifting to the right makes the dashes move forward along the
+path and shifting to the left moves them backward. Figure~\ref{fig29} illustrates
+this effect. The dash pattern can be thought of as an infinitely repeating pattern
+strung out along a horizontal line where the portion of the line to the right of
+the $y$~axis is laid out along the path to be dashed\index{dash pattern?\tdescr{dash pattern}}.
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.30} $$
+\caption[Dashed lines and the corresponding dash patters]
+ {Dashed lines and the MetaPost statements for drawing them where {\tt e4}
+ refers to the dash pattern {\tt evenly scaled 4}.}
+\label{fig29}
+\end{figure}
+
+When you shift a dash pattern so that the $y$~axis crosses the middle of a dash,
+the first dash gets truncated. Thus the line with dash pattern {\tt e4} starts
+with a dash of length 12bp followed by a 12bp gap and another 12bp dash, etc.,
+while {\tt e4 shifted (-6bp,0)} produces a 6bp dash, a 12 bp gap, then a
+12bp dash, etc. This dash pattern could be specified more directly via the
+{\tt dashpattern}\index{dash pattern?\texttt{dash pattern}}\label{Ddshpat} function:
+$$ \hbox{\tt dashpattern(on 6bp off 12bp on 6bp)} $$
+This means ``draw the first 6bp of the line, then skip the next 12bp, then draw
+another 6bp and repeat.'' If the line to be dashed is more than 30bp long, the
+last 6bp of the first copy of the dash pattern will merge with the first 6bp of
+the next copy to form a dash 12bp long. The general syntax for the
+{\tt dashpattern} function is shown in Figure~\ref{sydash}.
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{dash pattern} \rightarrow dashpattern
+ \hbox{\tt (}\descr{on/off list}\hbox{\tt )}$\\
+$\tt \descr{on/off list} \rightarrow
+ \descr{on/off list}\descr{on/off clause} \;|\; \descr{on/off clause}$\\
+$\tt \descr{on/off clause} \rightarrow on \descr{numeric tertiary}
+ \;|\; off \descr{numeric tertiary}$
+\end{ctabbing}
+\caption{The syntax for the {\tt dashpattern} function}
+\label{sydash}
+\end{figure}
+
+Since a dash pattern is really just a special kind of picture, the
+{\tt dashpattern} function returns a picture. It is not really necessary to know
+the structure of such a picture, so the casual reader will probably want to skip
+on to Section~\ref{oopt}. For those who want to know, a little experimentation
+shows that if {\tt d} is
+$$ \hbox{\tt dashpattern(on 6bp off 12bp on 6bp)}, $$
+then {\tt llcorner d} is $(0,24)$ and {\tt urcorner d} is $(24,24)$. Drawing
+{\tt d} directly without using it as a dash pattern produces two thin horizontal
+line segments like this:
+$$ \epsfbox{manfig.31} $$
+The lines in this example are specified as having width zero, but this does not
+matter because the line width is ignored when a picture is used as a dash pattern.
+
+The general rule for interpreting a picture {\tt d} as a dash pattern is that
+the line segments in {\tt d} are projected onto the $x$-axis and the resulting
+pattern is replicated to infinity in both directions by placing copies of the
+pattern end-to-end. The actual dash lengths are obtained by starting at $x=0$
+and scanning in the positive $x$ direction.
+
+To make the idea of ``replicating to infinity'' more precise, let $P({\tt d})$
+be the projection of {\tt d} onto the $x$~axis, and let
+${\rm shift}(P({\tt d}),x)$ be the result of shifting {\tt d} by~$x$.
+The pattern resulting from infinite replication is
+$$ \bigcup_{{\rm integers}\ n} {\rm shift}(P(d),\, n\cdot\ell(d)), $$
+where $\ell(d)$ measures the length of $P(d)$. The most restrictive possible
+definition of this length is $d_{\rm max}-d_{\rm min}$,
+where $[d_{\rm min},d_{\rm max}]$
+is the range of $x$~coordinates in $P(d)$. In fact, MetaPost uses
+$$ \max(\left|y_0({\tt d})\right|,\, d_{\rm max}-d_{\rm min}), $$
+where $y_0({\tt d})$ is the $y$ coordinate of the contents of {\tt d}.
+The contents of {\tt d} should lie on a horizontal line, but if they do not,
+the MetaPost interpreter just picks
+a $y$~coordinate that occurs in {\tt d}\index{dash pattern?\tdescr{dash pattern}}.
+
+A picture used as a dashed pattern must contain no text or filled regions,
+but it can contain lines that are themselves dashed. This can give small dashes
+inside of larger dashes as shown in
+Figure~\ref{fig32}\index{dash pattern?\tdescr{dash pattern}!recursive}
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+beginfig(32);
+draw dashpattern(on 15bp off 15bp) dashed evenly;
+picture p;
+p=currentpicture;
+currentpicture:=nullpicture;
+draw fullcircle scaled 1cm xscaled 3 dashed p;
+endfig;
+\end{verbatim}
+\quad \mathcenter{\epsfbox{manfig.32}}
+$$
+\caption{MetaPost code and the corresponding output}
+\label{fig32}
+\end{figure}
+
+
+\subsection{Other Options}
+\label{oopt}
+
+You might have noticed that the dashed lines produced by
+{\tt dashed evenly}\index{evenly?\texttt{evenly}} appear
+to have more black than white. This is an effect of the
+{\tt linecap}\index{linecap?\texttt{linecap}}\label{Dlinecap} parameter
+that controls the appearance of the ends of lines as well as the ends of dashes.
+There are also a number of other ways to affect the appearance of things drawn
+with MetaPost.
+
+The {\tt linecap} parameter has three different settings just as in PostScript.
+Plain MetaPost gives this
+internal variable\index{internal variables}\index{variables!internal} the
+default value {\tt rounded}\index{rounded?\texttt{rounded}}
+which causes line segments to be drawn with rounded ends like the segment from
+{\tt z0} to {\tt z3} in Figure~\ref{fig33}. Setting
+${\tt linecap}\mathrel{\hbox{\tt:=}}{\tt butt}$\index{butt?\texttt{butt}}\label{Dbutt}
+cuts the ends off
+flush so that dashes produced by {\tt dashed evenly}\index{evenly?\texttt{evenly}} have
+length 3bp, not 3bp plus the line width. You can also get squared-off ends
+that extend past the specified endpoints by setting
+${\tt linecap}\mathrel{\hbox{\tt:=}}{\tt squared}$\index{squared?\texttt{squared}}\label{Dsqred}
+as was done in the line from {\tt z2} to {\tt z5} in Figure~\ref{fig33}.
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+beginfig(33);
+for i=0 upto 2:
+ z[i]=(0,40i); z[i+3]-z[i]=(100,30);
+endfor
+pickup pencircle scaled 18;
+draw z0..z3 withcolor .8white;
+linecap:=butt;
+draw z1..z4 withcolor .8white;
+linecap:=squared;
+draw z2..z5 withcolor .8white;
+dotlabels.top(0,1,2,3,4,5);
+endfig; linecap:=rounded;
+\end{verbatim}
+\qquad
+\mathcenter{\epsfbox{manfig.33}}
+$$
+\caption{MetaPost code and the corresponding output}
+\label{fig33}
+\end{figure}
+
+Another parameter borrowed from PostScript affects the way a {\tt draw} statement
+treats sharp corners\index{corners} in the path to be drawn.
+The {\tt linejoin}\index{linejoin?\texttt{linejoin}}\label{Dlinejoin} parameter can
+be {\tt rounded}\index{rounded?\texttt{rounded}}\label{Drnded},
+{\tt beveled}\index{beveled?\texttt{beveled}}\label{Dbvled},
+or {\tt mitered}\index{mitered?\texttt{mitered}}\label{Dmitred} as shown in Figure~\ref{fig34}.
+The default value for plain MetaPost is {\tt rounded} which gives the effect of
+drawing with a circular brush.
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+beginfig(34);
+for i=0 upto 2:
+ z[i]=(0,50i); z[i+3]-z[i]=(60,40);
+ z[i+6]-z[i]=(120,0);
+endfor
+pickup pencircle scaled 24;
+draw z0--z3--z6 withcolor .8white;
+linejoin:=mitered;
+draw z1..z4--z7 withcolor .8white;
+linejoin:=beveled;
+draw z2..z5--z8 withcolor .8white;
+dotlabels.bot(0,1,2,3,4,5,6,7,8);
+endfig; linejoin:=rounded;
+\end{verbatim}
+\qquad
+\mathcenter{\epsfbox{manfig.34}}
+$$
+\caption{MetaPost code and the corresponding output}
+\label{fig34}
+\end{figure}
+
+When {\tt linejoin} is {\tt mitered}, sharp corners generate long pointed features
+as shown in Figure~\ref{fig35}. Since this might be undesirable, there is an
+internal variable\index{internal variables}\index{variables!internal}
+called {\tt miterlimit}\index{miterlimit?\texttt{miterlimit}}\label{Dmiterlim} that controls how
+extreme the situation can get before the mitered join is replaced by a beveled
+join. For Plain MetaPost, {\tt miterlimit} has a default value of 10.0 and line
+joins revert to beveled when the ratio of miter length to line width
+reaches this value.
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.35} $$
+\caption{The miter length and line width whose ratio is limited by
+ {\tt miterlimit}.}
+\label{fig35}
+\end{figure}
+
+The {\tt linecap}, {\tt linejoin}, and {\tt miterlimit} parameters are especially
+important because they also affect things that get drawn behind the scenes.
+For instance, Plain MetaPost has statements for drawing
+arrows\index{arrows}, and the arrowheads are slightly rounded when {\tt linejoin}
+is {\tt rounded}. The effect depends on the line width and is quite subtle at the
+default line width of 0.5bp as shown in Figure~\ref{fig36}.
+
+\begin{figure}[htp]
+$$\epsfbox{manfig.36}$$
+\caption{Three ways of drawing arrows.}
+\label{fig36}
+\end{figure}
+
+Drawing arrows like the ones in Figure~\ref{fig36} is simply a matter of
+saying\index{drawarrow?\texttt{drawarrow}}\label{Ddrwarr}
+$$ {\tt drawarrow}\, \descr{path expression} $$
+instead of {\tt draw} \tdescr{path expression}. This draws the given path with
+an arrowhead at the last point on the path. If you want the arrowhead at the
+beginning of the path, just use the unary operator
+{\tt reverse}\index{reverse?\texttt{reverse}}\label{Drevrse} to take the
+original path and make a new one with its time sense reversed; i.e., for a
+path~{\tt p} with {\tt length p}${}=n$,
+$$ {\tt point\ } t {\tt\ of\ reverse\ p}
+ \quad {\rm and} \quad
+ {\tt point\ } n-t {\tt\ of\ p}
+$$ are synonymous.
+
+As shown in Figure~\ref{fig36}, a statement beginning\index{drawdblarrow?\texttt{drawdblarrow}}%
+\index{arrows!double-headed}\label{Ddrwdar}
+$$ {\tt drawdblarrow}\, \descr{path expression} $$
+draws a double-headed arrow. The size of the arrowhead is guaranteed to be
+larger than the line width, but it might need adjusting if the line width is
+very great. This is done by assigning a new value to the
+internal variable\index{internal variables}\index{variables!internal}
+{\tt ahlength}\index{ahlength?\texttt{ahlength}}\label{Dahlength}
+that determines arrowhead length as shown in Figure~\ref{fig37}.
+Increasing {\tt ahlength} from the default value of 4 PostScript points to
+1.5 centimeters produces the large arrowhead in Figure~\ref{fig37}. There
+is also an {\tt ahangle}\index{ahangle?\texttt{ahangle}}\label{Dahangle}
+parameter that controls the angle
+at the tip of the arrowhead. The default value of this angle is 45 degrees
+as shown in the figure.
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.37} $$
+\caption[A large arrowhead with key parameters labeled.]
+ {A large arrowhead with key parameters labeled and paths used to
+ draw it marked with white lines.}
+\label{fig37}
+\end{figure}
+
+The arrowhead is created by filling the triangular region that is outlined
+in white in Figure~\ref{fig37} and then drawing around it with the currently
+picked up pen. This combination of filling and drawing can be combined into
+a single {\tt filldraw} statement\index{filldraw?\texttt{filldraw}}\label{Dfildrw}:
+$$ {\tt filldraw}\, \descr{path expression}\,
+ \descr{optional {\tt dashed} and {\tt withcolor} and {\tt withpen} clauses};
+$$
+The \tdescr{path expression} should be a closed cycle like the triangular path
+in Figure~\ref{fig37}. This path should not be confused with the path argument
+to {\tt drawarrow} which is indicated by a white line in the figure.
+
+White lines like the ones in the figure can be created by an
+{\tt undraw}\index{undraw?\texttt{undraw}}\label{Dundraw} statement.
+This is an erasing\index{erasing}
+version of {\tt draw} that draws {\tt withcolor background}\index{background?\texttt{background}}
+just as the {\tt unfill} statement does. There is also an
+{\tt unfilldraw}\index{unfilldraw?\texttt{unfilldraw}}\label{Dunfdrw}
+statement just in case someone finds a
+use for it.
+
+The {\tt filldraw}, {\tt undraw} and {\tt unfilldraw} statements and all the
+arrow drawing statements are like the {\tt fill} and {\tt draw} statements in that
+they take {\tt dashed}\index{dashed?\texttt{dashed}}, {\tt withpen}\index{withpen?\texttt{withpen}},
+and {\tt withcolor}\index{withcolor?\texttt{withcolor}} options.
+When you have a lot of drawing statements it is
+nice to be able to apply an option such as {\tt withcolor 0.8white} to all of
+them without having to type this repeatedly as was done in Figures \ref{fig33}
+and~\ref{fig34}. The statement for this purpose is\index{drawoptions?\texttt{drawoptions}}\label{Ddropts}
+$$ \hbox{\tt drawoptions(} \descr{text} \hbox{\tt )} $$
+where the \tdescr{text} argument gives a sequence of {\tt dashed}, {\tt withcolor},
+and {\tt withpen} options to be applied automatically to all drawing statements.
+If you specify
+$$ \hbox{\tt drawoptions(withcolor .5[black,white])} $$
+and then want to draw a black line, you can override the {\tt drawoptions}
+by specifying
+$$ {\tt draw}\, \descr{path expression}\, {\tt withcolor\ black} $$
+To turn off {\tt drawoptions} all together, just give an empty list:
+$$ \hbox{\tt drawoptions()} $$
+(This is done automatically by the {\tt beginfig}\index{beginfig?\texttt{beginfig}} macro).
+
+Since irrelevant options are ignored, there is no harm in giving a statement
+like
+$$ \hbox{\tt drawoptions(dashed evenly)} $$
+followed by a sequence of {\tt draw} and {\tt fill} commands. It does not make
+sense to use a dash pattern when filling so the {\tt dashed evenly} gets ignored
+for {\tt fill} statements. It turns out that
+$$ \hbox{\tt drawoptions(withpen } \descr{pen expression} \hbox{\tt )} $$
+does affect {\tt fill} statements as well as {\tt draw} statements.
+In fact there is a special pen variable called
+{\tt currentpen}\index{currentpen?\texttt{currentpen}} such that
+{\tt fill} \ldots\ {\tt withpen currentpen} is equivalent to a {\tt filldraw}
+statement.
+
+Precisely what does it mean to say that drawing options affect those statements
+where they make sense? The {\tt dashed} \tdescr{dash pattern} option only affects
+$$ {\tt draw}\, \descr{path expression} $$
+statements, and text appearing in the \tdescr{picture expression} argument to
+$$ {\tt draw}\, \descr{picture expression} $$
+statement is only affected by the {\tt withcolor} \tdescr{color expression} option.
+For all other combinations of drawing statements and options, there is some effect.
+An option applied to a {\tt draw} \tdescr{picture expression} statement will in
+general affect some parts of the picture but not others. For instance,
+a {\tt dashed} or {\tt withpen} option will affect all the lines in the picture
+but none of the labels.
+
+
+\subsection{Pens}
+
+Previous sections have given numerous examples of {\tt pickup}
+\tdescr{pen expression} and {\tt withpen} \tdescr{pen expression}, but there have
+not been any examples of pen expressions other than
+$$ {\tt pencircle\ scaled}\, \descr{numeric primary} $$
+which produces lines of a specified width. For calligraphic effects such in
+Figure~\ref{fig38}, you can apply any of the transformation operators discussed
+in Section~\ref{transsec}. The starting point for such transformations is
+{\tt pencircle}\index{pencircle?\texttt{pencircle}}\label{Dpncirc},
+a circle one PostScript point in diameter. Thus affine
+transformations produce a circular or elliptical\index{pens!elliptical} pen shape.
+The width of lines drawn with the pen depends on how nearly perpendicular the line
+is to the long axis of the ellipse.
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+beginfig(38);
+pickup pencircle scaled .2in yscaled .08 rotated 30;
+x0=x3=x4;
+z1-z0 = .45in*dir 30;
+z2-z3 = whatever*(z1-z0);
+z6-z5 = whatever*(z1-z0);
+z1-z6 = 1.2*(z3-z0);
+rt x3 = lft x2;
+x5 = .55[x4,x6];
+y4 = y6;
+lft x3 = bot y5 = 0;
+top y2 = .9in;
+draw z0--z1--z2--z3--z4--z5--z6 withcolor .7white;
+dotlabels.top(0,1,2,3,4,5,6);
+endfig;
+\end{verbatim}
+\quad \mathcenter{\epsfbox{manfig.38}}
+$$
+\caption{MetaPost code and the resulting ``calligraphic'' figure.}
+\label{fig38}\index{lft?\texttt{lft}}\index{bot?\texttt{bot}}\index{top?\texttt{top}}
+\end{figure}
+
+Figure~\ref{fig38} demonstrates operators {\tt lft}\index{lft?\texttt{lft}}\label{Dlft},
+{\tt rt}\index{rt?\texttt{rt}}\label{Drt}, {\tt top}\index{top?\texttt{top}}\label{Dtop},
+and {\tt bot}\index{bot?\texttt{bot}}\label{Dbot}
+that answer the question, ``If the current pen is placed at the position
+given by the argument, where will its left, right, top, or bottom edge be?''
+In this case the current pen is the ellipse given in the {\tt pickup} statement
+and its bounding box is 0.1734 inches wide and 0.1010 inches high, so
+{\tt rt x3} is ${\tt x3}+{\tt 0.0867in}$ and {\tt bot y5} is
+${\tt y5}-{\tt 0.0505in}$.
+The {\tt lft}, {\tt rt}, {\tt top}, and {\tt bot} operators also accept arguments
+of type pair in which case they compute the $x$ and~$y$ coordinates of the
+leftmost, rightmost, topmost, or bottommost point on the pen shape. For example,
+$$ {\tt rt}(x,y) = (x,y)+({\tt 0.0867in}, {\tt 0.0496in}) $$
+for the pen in Figure~\ref{fig38}. Note that {\tt beginfig}\index{beginfig?\texttt{beginfig}}
+resets the current pen to a default value of
+$$ \hbox{\tt pencircle scaled 0.5bp} $$
+at the beginning of each figure. This value can be reselected at any time
+by giving the command
+{\tt pickup defaultpen}\index{defaultpen?\texttt{defaultpen}}\label{Ddefaultpen}.
+
+This would be the end of the story on pens, except that
+for compatibility with \MF\index{metafont?\MF}, MetaPost also allows pen shapes to be
+polygonal\index{pens!polygonal}.
+There is a predefined pen called
+{\tt pensquare}\index{pensquare?\texttt{pensquare}}\label{Dpnsqr} that
+can be transformed to yield pens shaped like parallelograms. In fact, there is
+even an operator called {\tt makepen}\index{makepen?\texttt{makepen}}\label{Dmkpen} that takes
+a convex-polygon-shaped path and makes a pen that shape and size. If the path is
+not exactly convex or polygonal, the {\tt makepen} operator will straighten the
+edges and/or drop some of the vertices.
+In particular, {\tt pensquare} is equivalent to
+$$ \hbox{\tt makepen((-.5,-.5)--(.5,-.5)--(.5,.5)--(-.5,.5)--cycle)} $$
+
+The inverse of {\tt makepen} is the
+{\tt makepath}\index{makepath?\texttt{makepath}}\label{Dmkpath} operator
+that takes a \tdescr{pen primary} and returns the corresponding path. Thus
+{\tt makepath pencircle} produces a circular path identical to
+{\tt fullcircle}\index{fullcircle?\texttt{fullcircle}}. This also works for a polygonal pen
+so that
+$$ {\tt makepath\ makepen}\, \descr{path expression} $$
+will take any cyclic path and turn it into a convex polygon\index{convex polygons}.
+
+
+\subsection{Clipping and Low-Level Drawing Commands}
+
+Drawing statements such as {\tt draw}, {\tt fill}, {\tt filldraw}, and {\tt unfill}
+are part of the Plain macro\index{Plain macros} package and are defined in terms
+of more primitive statements. The main difference between the drawing
+statements discussed in previous sections and the more primitive versions is that
+the primitive drawing statements all require you to specify a picture variable to
+hold the results. For {\tt fill}, {\tt draw}, and related statements, the results
+always go to a picture variable called
+{\tt currentpicture}\index{currentpicture?\texttt{currentpicture}}\label{Dcurpic}.
+The syntax for the primitive
+drawing statements that allow you to specify a picture variable is shown in
+Figure~\ref{sydraw}.
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{addto command} \rightarrow$\\
+$\tt \qquad addto \descr{picture variable} also
+ \descr{picture expression} \descr{option list}$\\
+$\tt \qquad \;|\; addto \descr{picture variable}
+ contour \descr{path expression} \descr{option list}$\\
+$\tt \qquad \;|\; addto \descr{picture variable}
+ doublepath \descr{path expression} \descr{option list}$\\
+$\tt \descr{option list} \rightarrow \descr{empty} \;|\;
+ \descr{drawing option} \descr{option list}$\\
+$\tt \descr{drawing option} \rightarrow withcolor \descr{color expression}$\\
+$\tt \qquad \;|\; withpen \descr{pen expression} \;|\;
+ dashed \descr{picture expression}$
+\end{ctabbing}
+\caption{The syntax for primitive drawing statements}
+\label{sydraw}
+\index{option list?\tdescr{option list}}\index{addto also?\texttt{addto also}}\index{addto contour?\texttt{addto contour}}%
+\index{addto doublepath?\texttt{addto doublepath}}\index{withcolor?\texttt{withcolor}}\index{withpen?\texttt{withpen}}%
+\index{dashed?\texttt{dashed}}\index{drawing option?\tdescr{drawing option}}
+\end{figure}
+
+The syntax for primitive drawing commands is compatible with
+\MF\index{metafont?\MF}. Table~\ref{draweqv} shows how the primitive drawing statements
+relate to the familiar {\tt draw} and {\tt fill} statements. Each of the
+statements in the first column of the table could be ended with an
+\tdescr{option list} of its own, which is equivalent to appending the
+\tdescr{option list} to the corresponding entry in the second column of the table.
+For example,
+$$ {\tt draw}\ p\ {\tt withpen\ pencircle} $$
+is equivalent to
+$$ {\tt addto\ currentpicture\ doublepath}\ p\
+ {\tt withpen\ currentpen\ withpen\ pencircle}
+$$
+where {\tt currentpen}\index{currentpen?\texttt{currentpen}}\label{Dcurpen} is a special
+pen variable that always holds the last pen picked up.
+The second {\tt withpen} option silently overrides the {\tt withpen currentpen}
+from the expansion of {\tt draw}.
+
+\begin{table}[htp]
+$$\begin{tabular}{|l|l|} \hline
+\multicolumn1{|c|}{statement}& \multicolumn1{c|}{equivalent primitives}\\ \hline
+{\tt draw} {\it pic}& {\tt addto currentpicture also} {\it pic}\\
+{\tt draw} $p$& {\tt addto currentpicture doublepath} $p$
+ {\tt withpen} $q$\\
+{\tt fill} $c$& {\tt addto currentpicture contour} $c$\\
+{\tt filldraw} $c$& {\tt addto currentpicture contour} $c$ {\tt withpen} $q$\\
+{\tt undraw} {\it pic}& {\tt addto currentpicture also} {\it pic}
+ {\tt withcolor} $b$\\
+{\tt undraw} $p$& {\tt addto currentpicture doublepath} $p$
+ {\tt withpen} $q$
+ {\tt withcolor} $b$\\
+{\tt unfill} $c$& {\tt addto currentpicture contour} $c$
+ {\tt withcolor} $b$\\
+{\tt unfilldraw} $c$& {\tt addto currentpicture contour} $c$ {\tt withpen} $q$
+ {\tt withcolor} $b$\\ \hline
+\end{tabular}
+$$
+\caption[Drawing statements and equivalent primitive commands]
+ {Common drawing statements and equivalent primitive versions, where
+ $q$ stands for {\tt currentpen}, $b$ stands for {\tt background},
+ $p$ stands for any path, $c$ stands for a cyclic path, and {\it pic} stands
+ for a \tdescr{picture expression}. Note that nonempty {\tt drawoptions}
+ would complicate the entries in the second column.}
+\label{draweqv}
+\index{drawoptions?\texttt{drawoptions}}
+\end{table}
+
+There are two more primitive drawing commands that do not accept any drawing
+options. One is the {\tt setbounds} command that was discussed in
+Section~\ref{meas}; the other is the
+{\tt clip} command\index{clip?\texttt{clip}}\label{Dclip}:
+$$ {\tt clip}\, \descr{picture variable}\, {\tt to}\, \descr{path expression} $$
+Given a cyclic path, this statement trims the contents of the
+\tdescr{picture variable} to eliminate everything outside of the cyclic path.
+There is no ``high level'' version of this statement, so you have to use
+$$ {\tt clip\ currentpicture\ to}\, \descr{path expression} $$
+if you want to clip {\tt currentpicture}\index{currentpicture?\texttt{currentpicture}}.
+Figure~\ref{fig40} illustrates clipping.
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+beginfig(40);
+path p[];
+p1 = (0,0){curl 0}..(5pt,-3pt)..{curl 0}(10pt,0);
+p2 = p1..(p1 yscaled-1 shifted(10pt,0));
+p0 = p2;
+for i=1 upto 3: p0:=p0.. p2 shifted (i*20pt,0);
+ endfor
+for j=0 upto 8: draw p0 shifted (0,j*10pt);
+ endfor
+p3 = fullcircle shifted (.5,.5) scaled 72pt;
+clip currentpicture to p3;
+draw p3;
+endfig;
+\end{verbatim}
+\qquad
+\mathcenter{\epsfbox{manfig.40}}
+$$
+\caption{MetaPost code and the resulting ``clipped'' figure.}
+\label{fig40}
+\end{figure}
+
+All the primitive drawing operations would be useless without one last operation
+called {\tt shipout}. The statement\index{shipout?\texttt{shipout}}\label{Dship}
+$$ {\tt shipout}\, \descr{picture expression} $$
+This writes out a picture as a PostScript\index{PostScript} file whose name ends
+{\tt.}{\it nnn}, where {\tt nnn} is the decimal representation of the value of
+the internal variable\index{internal variables}\index{variables!internal}
+{\tt charcode}\index{charcode?\texttt{charcode}}\label{Dcharcode}.
+(The name ``{\tt charcode}'' is for compatibility with \MF\index{metafont?\MF}.)
+Normally, {\tt beginfig}\index{beginfig?\texttt{beginfig}} sets {\tt charcode}, and
+{\tt endfig}\index{endfig?\texttt{endfig}} invokes {\tt shipout}.
+
+\section{Macros}
+\label{macros}
+
+As alluded to earlier, MetaPost has a set of automatically included macros called
+the Plain macro package\index{Plain macros}, and some of the commands discussed in
+previous sections are defined as macros instead of being built into MetaPost.
+The purpose of this section is to explain how to write such macros.
+
+Macros with no arguments are very simple.
+A macro definition\index{replacement text?\tdescr{replacement text}}%
+\index{def?\texttt{def}}\index{enddef?\texttt{enddef}}
+$$ {\tt def}\, \descr{symbolic token}\, \hbox{\tt =}\,
+ \descr{replacement text}\, {\tt enddef}
+$$
+makes the \tdescr{symbolic token} an abbreviation for the \tdescr{replacement text},
+where the \tdescr{replacement text} can be virtually any sequence of tokens. For
+example, the Plain macro package could almost define the {\tt fill} statement like
+this\index{fill?\texttt{fill}}:
+$$ \hbox{\tt def fill = addto currentpicture contour enddef} $$
+
+Macros with arguments are similar, except they have formal parameters that tell
+how to use the arguments in the \tdescr{replacement text}. For example, the
+{\tt rotatedaround}\index{rotatedaround?\texttt{rotatedaround}} macro is defined like this:
+$$\begin{verbatim}
+def rotatedaround(expr z, d) =
+ shifted -z rotated d shifted z enddef;
+\end{verbatim}
+$$
+The {\tt expr}\index{expr?\texttt{expr}} in this definition means that formal parameters
+{\tt z} and {\tt d} can be arbitrary expressions. (They should be pair expressions
+but the MetaPost interpreter does not immediately check for that.)
+
+Since MetaPost is an interpreted language, macros with arguments are a lot like
+subroutines\index{subroutines}. MetaPost macros are often used like subroutines,
+so the language includes programming concepts to support this.
+These concepts include local variables, loops, and conditional statements.
+
+\subsection{Grouping}
+\label{grsec}
+
+Grouping in MetaPost is essential for functions\index{functions} and
+local\index{variables!local}\index{locality} variables.
+The basic idea is that a group is
+a sequence of statements possibly followed by an expression with the provision
+that certain symbolic tokens\index{tokens!symbolic} can have their old meanings
+restored at the end of the group. If the group ends with an expression, the
+group behaves like a function call that returns that expression. Otherwise,
+the group is just a compound statement\index{compound statement}.
+The syntax for a group is\index{begingroup?\texttt{begingroup}}\index{endgroup?\texttt{endgroup}}
+$$ {\tt begingroup}\, \descr{statement list}\, {\tt endgroup} $$
+or
+$$ {\tt begingroup}\, \descr{statement list}\, \descr{expression}\, {\tt endgroup}
+$$
+where a \tdescr{statement list} is a sequence of statements each followed by a
+semicolon. A group with an \tdescr{expression} after the \tdescr{statement list}
+behaves like a \tdescr{primary} in Figure~\ref{syexpr} or like a
+\tdescr{numeric atom} in Figure~\ref{synprim}.
+
+Since the \tdescr{replacement text} for the {\tt beginfig}\index{beginfig?\texttt{beginfig}}
+macro starts with {\tt begingroup} and the \tdescr{replacement text} for
+{\tt endfig}\index{endfig?\texttt{endfig}} ends with {\tt endgroup},
+each figure in a MetaPost input file behaves like a
+group. This is what allows figures can have local variables.
+We have already seen in Section~\ref{vardecl} that
+variable names beginning with {\tt x} or {\tt y} are local in the sense that they
+have unknown values at the beginning of each figure and these values are forgotten
+at the end of each figure. The following example illustrates how locality works:
+\begin{eqnarray*}
+&& \hbox{\tt x23 = 3.1;}\\
+&& \hbox{\tt beginfig(17);}\\
+&& \qquad \vdots\\
+&& \hbox{\tt y3a=1; x23=2;}\\
+&& \qquad \vdots\\
+&& \hbox{\tt endfig;}\\
+&& \hbox{\tt show x23, y3a;}
+\end{eqnarray*}
+The result of the {\tt show}\index{show} command is
+$$\begin{verbatim}
+>> 3.1
+>> y3a
+\end{verbatim}
+$$
+indicating that {\tt x23} has returned to its former value of {\tt 3.1} and
+{\tt y3a} is completely unknown as it was at {\tt beginfig(17)}.
+
+The locality of {\tt x} and {\tt y} variables is achieved by the
+statement\index{save?\texttt{save}}\label{Dsave}
+$$ \hbox{\tt save x,y} $$
+in the \tdescr{replacement text} for {\tt beginfig}\index{beginfig?\texttt{beginfig}}.
+In general, variables are made local by the statement
+$$ {\tt save}\, \descr{symbolic token list} $$
+where \tdescr{symbolic token list} is a comma-separated list of
+tokens:\index{tokens!symbolic}
+\begin{ctabbing}
+$\tt \descr{symbolic token list} \rightarrow \descr{symbolic token}$\\
+ $\tt \qquad \;|\; \descr{symbolic token}\hbox{\tt ,}
+ \descr{symbolic token list}$
+\end{ctabbing}
+All variables whose names begin with one of the specified symbolic tokens become
+unknown numerics and their present values are saved for restoration at the end
+of the current group. If the {\tt save} statement is used outside of a group, the
+original values are simply discarded.
+
+The main purpose of the {\tt save} statement is to allow macros to use variables
+without interfering with existing variables or variables in other calls to the
+same macro. For example, the predefined macro {\tt whatever}\index{whatever}
+has the \tdescr{replacement text}
+$$ \hbox{\tt begingroup save ?; ? endgroup} $$
+This returns an unknown numeric quantity, but it is no longer called question
+mark since that name was local to the group. Asking the name via
+{\tt show\index{show?\texttt{show}} whatever} yields\index{CAPSULE?\texttt{CAPSULE}}
+$$ \hbox{\tt >> \%CAPSULE}{\it nnnn} $$
+where {\it nnnn} is an identification number that is chosen when {\tt save}
+makes the name question mark disappear.
+
+In spite of the versatility of {\tt save}, it cannot be used to make local changes
+to any of MetaPost's
+internal variables\index{internal variables}\index{variables!internal}.
+A statement such as\index{linecap?\texttt{linecap}}
+$$ \hbox{\tt save linecap} $$
+would cause MetaPost to temporarily forget the special meaning of this variable
+and just make it an unknown numeric. If you want to draw one dashed line with
+{\tt linecap:=butt} and then go back to the previous value, you can use the
+{\tt interim}\index{interim?\texttt{interim}}\label{Dinterm} statement as follows:
+\begin{eqnarray*}
+&& \hbox{\tt begingroup interim linecap:=butt;}\\
+&& {\tt draw}\, \descr{path expression}\, \hbox{\tt dashed evenly; endgroup}
+\end{eqnarray*}
+This saves the value of the
+internal variable\index{internal variables}\index{variables!internal}
+{\tt linecap} and temporarily
+gives it a new value without forgetting that {\tt linecap} is an internal
+variable. The general syntax is
+$$ {\tt interim}\, \descr{internal variable} \mathrel{\hbox{\tt:=}}
+ \descr{numeric expression}
+$$
+
+
+\subsection{Parameterized Macros}
+
+The basic idea behind parameterized macros is to achieve greater flexibility by
+allowing auxiliary information to be passed to a macro. We have already seen
+that macro definitions can have formal parameters that represent expressions
+to be given when the macro is called. For instance a definition such as
+$$ \hbox{\tt def rotatedaround(expr z, d) = } \descr{replacement text}\,
+ {\tt enddef}
+$$
+allows the MetaPost interpreter to understand macro calls of the form
+$$\tt rotatedaround\hbox{\tt (}
+ \descr{expression}\hbox{\tt ,} \descr{expression}\hbox{\tt )}
+$$
+
+The keyword {\tt expr}\index{expr?\texttt{expr}}\index{parameter!expr} in the macro
+definition means that the
+parameters can be expressions of any type. When the definition specifies
+{\tt (expr z, d)}, the formal parameters {\tt z} and {\tt d} behave like
+variables of the appropriate
+types. Within the \tdescr{replacement text}, they can be used in expressions
+just like variables, but they cannot be redeclared or assigned to. There is no
+restriction against unknown or partially known arguments. Thus the
+definition\index{midpoint?\texttt{midpoint}}
+$$ \hbox{\tt def midpoint(expr a, b) = (.5[a,b]) enddef} $$
+works perfectly well when {\tt a} and {\tt b} are unknown. An
+equation such as
+$$ \hbox{\tt midpoint(z1,z2) = (1,1)} $$
+could be used to help determine {\tt z1} and {\tt z2}.
+
+Notice that the above definition for {\tt midpoint} works for numerics, pairs,
+or colors as long as both parameters have the same type. If for some reason we
+want a {\tt middlepoint}\index{middlepoint?\texttt{middlepoint}} macro that works for
+a single path or picture, it would be
+necessary to do an {\tt if}\index{if?\texttt{if}} test on the argument type. This uses
+the fact there is a unary operator\index{path?\texttt{path}}
+$$ {\tt path}\, \descr{primary} $$
+that returns a boolean result indicating whether its argument is a path. Since
+the basic {\tt if} test has the syntax\index{else?\texttt{else}}\index{fi?\texttt{fi}}
+$$ {\tt if}\, \descr{boolean expression}\hbox{\tt:}\, \descr{balanced tokens}\,
+ \hbox{\tt else:}\, \descr{balanced tokens}\, {\tt fi}
+$$
+where the \tdescr{balanced tokens}\index{balanced tokens?\tdescr{balanced tokens}} can be anything
+that is balanced with respect to {\tt if} and {\tt fi}, the complete
+{\tt middlepoint}\index{midpoint?\texttt{midpoint}} macro with type test looks like this:
+$$\begin{verbatim}
+def middlepoint(expr a) = if path a: (point .5*length a of a)
+ else: .5(llcorner a + urcorner a) fi enddef;
+\end{verbatim}
+$$
+The complete syntax for {\tt if} tests is shown in Figure~\ref{syif}.
+It allows multiple {\tt if} tests like
+$$ \hbox{\tt if $e_1$: \ldots\ else: if $e_2$: \ldots\ else: \ldots\ fi fi} $$
+to be shortened to\index{elseif?\texttt{elseif}}
+$$ \hbox{\tt if $e_1$: \ldots\ elseif $e_2$: \ldots\ else: \ldots\ fi} $$
+where $e_1$ and $e_2$ represent boolean expressions.
+
+Note that {\tt if} tests are not statements and the \tdescr{balanced tokens} in
+the syntax rules can be any sequence of balanced tokens even if they do not form
+a complete expression or statement. Thus we could have saved two tokens at the
+expense of clarity by defining {\tt midpoint} like this:
+$$\begin{verbatim}
+def midpoint(expr a) = if path a: (point .5*length a of
+ else: .5(llcorner a + urcorner fi a) enddef;
+\end{verbatim}
+$$
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{if test} \rightarrow if \descr{boolean expression} \hbox{\tt :}
+ \descr{balanced tokens} \descr{alternatives} fi$\\
+$\tt \descr{alternatives} \rightarrow \descr{empty}$\\
+$\tt \qquad \;|\; else\hbox{\tt :} \descr{balanced tokens}$\\
+$\tt \qquad \;|\; elseif \descr{boolean expression} \hbox{\tt :}
+ \descr{balanced tokens} \descr{alternatives}$
+\end{ctabbing}
+\caption{The syntax for {\tt if} tests.}
+\label{syif}
+\end{figure}
+
+The real purpose of macros and {\tt if} tests is to automate repetitive tasks and
+allow important subtasks to be solved separately. For example, Figure~\ref{fig42}
+uses macros \verb|draw_marked|, \verb|mark_angle|, and \verb|mark_rt_angle| to
+mark lines and angles that appear in the figure.
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+beginfig(42);
+pair a,b,c,d;
+b=(0,0); c=(1.5in,0); a=(0,.6in);
+d-c = (a-b) rotated 25;
+dotlabel.lft("a",a);
+dotlabel.lft("b",b);
+dotlabel.bot("c",c);
+dotlabel.llft("d",d);
+z0=.5[a,d];
+z1=.5[b,c];
+(z.p-z0) dotprod (d-a) = 0;
+(z.p-z1) dotprod (c-b) = 0;
+draw a--d;
+draw b--c;
+draw z0--z.p--z1;
+draw_marked(a--b, 1);
+draw_marked(c--d, 1);
+draw_marked(a--z.p, 2);
+draw_marked(d--z.p, 2);
+draw_marked(b--z.p, 3);
+draw_marked(c--z.p, 3);
+mark_angle(z.p, b, a, 1);
+mark_angle(z.p, c, d, 1);
+mark_angle(z.p, c, b, 2);
+mark_angle(c, b, z.p, 2);
+mark_rt_angle(z.p, z0, a);
+mark_rt_angle(z.p, z1, b);
+endfig;
+\end{verbatim}
+\quad \mathcenter{\epsfbox{manfig.42}}
+$$
+\caption{MetaPost code and the corresponding figure}
+\label{fig42}
+\end{figure}
+
+The task of the \verb|draw_marked|\index{draw_marked?\texttt{draw\_marked}} macro is to draw a path
+with a given number of cross marks near its midpoint. A convenient starting place
+is the subproblem of drawing a single cross mark perpendicular to a path {\tt p}
+at some time {\tt t}. The \verb|draw_mark|\index{draw_mark?\texttt{draw\_mark}} macro in
+Figure~\ref{drawmarked} does this by first finding a vector {\tt dm} perpendicular
+to~{\tt p} at~{\tt t}. To simplify positioning the cross mark,
+the \verb|draw_marked| macro is defined to take an arc length\index{arc length}
+{\tt a} along {\tt p} and use the {\tt arctime}\index{arctime} operator to
+compute~{\tt t}
+
+With the subproblem of drawing a single mark out of the way, the \verb|draw_marked|
+macro only needs to draw the path and call \verb|draw_mark| with the appropriate
+arc length values. The \verb|draw_marked| macro in Figure~\ref{drawmarked}
+uses {\tt n} equally-spaced {\tt a} values centered on
+{\tt .5*arclength~p}\index{arclength?\texttt{arclength}}.
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+marksize=4pt;
+
+def draw_mark(expr p, a) =
+ begingroup
+ save t, dm; pair dm;
+ t = arctime a of p;
+ dm = marksize*unitvector direction t of p
+ rotated 90;
+ draw (-.5dm.. .5dm) shifted point t of p;
+ endgroup
+enddef;
+
+def draw_marked(expr p, n) =
+ begingroup
+ save amid;
+ amid = .5*arclength p;
+ for i=-(n-1)/2 upto (n-1)/2:
+ draw_mark(p, amid+.6marksize*i);
+ endfor
+ draw p;
+ endgroup
+enddef;
+\end{verbatim}
+$$
+\caption{Macros for drawing a path {\tt p} with {\tt n} cross marks.}
+\label{drawmarked}
+\end{figure}
+
+Since \verb|draw_marked| works for curved lines, it can be used to draw the arcs
+that the \verb|mark_angle|\index{mark_angle?\texttt{mark\_angle}} macro generates. Given points
+{\tt a}, {\tt b}, and {\tt c} that define a counter-clockwise angle at {\tt b},
+the \verb|mark_angle| needs to generate a small arc from segment {\tt ba} to
+segment {\tt bc}. The macro definition in Figure~\ref{markangle} does this by
+creating an arc {\tt p} of radius one and then computing a scale factor {\tt s}
+that makes it big enough to see clearly.
+
+The \verb|mark_rt_angle|\index{mark_rt_angle?\texttt{mark\_rt\_angle}} macro is much simpler.
+It takes a generic right-angle corner and uses the {\tt zscaled}\index{zscaled?\texttt{zscaled}}
+operator to rotate it and scale it as necessary.
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+angle_radius=8pt;
+
+def mark_angle(expr a, b, c, n) =
+ begingroup
+ save s, p; path p;
+ p = unitvector(a-b){(a-b)rotated 90}..unitvector(c-b);
+ s = .9marksize/length(point 1 of p - point 0 of p);
+ if s<angle_radius: s:=angle_radius; fi
+ draw_marked(p scaled s shifted b, n);
+ endgroup
+enddef;
+
+def mark_rt_angle(expr a, b, c) =
+ draw ((1,0)--(1,1)--(0,1))
+ zscaled (angle_radius*unitvector(a-b)) shifted b
+enddef;
+\end{verbatim}
+$$
+\caption{Macros for marking angles.}
+\label{markangle}
+\end{figure}
+
+
+\subsection{Suffix and Text Parameters}
+
+Macro parameters need not always be expressions as in the previous examples.
+Replacing the keyword {\tt expr} with {\tt suffix}\index{suffix?\texttt{suffix}} or
+{\tt text}\index{text?\texttt{text}} in a macro definition declares the parameters to be
+variable names or arbitrary sequences of tokens. For example, there is a
+predefined macro called {\tt hide}\index{hide?\texttt{hide}} that takes a
+text parameter\index{parameter!text} and
+interprets it as a sequence of statements while ultimately producing an empty
+\tdescr{replacement text}. In other words, {\tt hide} executes its argument and
+then gets the next token as if nothing happened. Thus
+$$ \hbox{\tt show hide(numeric a,b; a+b=3; a-b=1) a;} $$
+prints ``\verb|>> 2|.''
+
+If the {\tt hide} macro were not predefined, it could be defined like this:
+$$\begin{verbatim}
+def ignore(expr a) = enddef;
+def hide(text t) = ignore(begingroup t; 0 endgroup) enddef;
+\end{verbatim}
+$$
+The statements represented by the text parameter {\tt t} would be evaluated as part
+of the group that forms the argument to {\tt ignore}. Since {\tt ignore} has an
+empty \tdescr{replacement text}, expansion of the {\tt hide} macro ultimately
+produces nothing.
+
+Another example of a predefined macro with a text parameter is
+{\tt dashpattern}\index{dashpattern?\texttt{dashpattern}}. The definition of {\tt dashpattern}
+starts
+$$\begin{verbatim}
+def dashpattern(text t) =
+ begingroup save on, off;
+\end{verbatim}
+$$
+then it defines {\tt on} and {\tt off} to be macros that create the desired
+picture when the text parameter~{\tt t} appears in the replacement text.
+
+Text parameters are very general, but their generality sometimes gets in the way.
+If you just want to pass a variable name to a macro, it is better to declare it
+as a suffix parameter\index{parameter!suffix}. For example,\index{incr?\texttt{incr}}
+$$ \hbox{\verb|def incr(suffix $) = begingroup $:=$+1; $ endgroup enddef;|} $$
+defines a macro that will take any numeric variable, add one to it, and return
+the new value. Since variable names can be more than one token long,
+$$ \hbox{\tt incr(a3b)} $$
+is perfectly acceptable if {\tt a3b} is a numeric variable.
+Suffix parameters are slightly more general than variable names because the
+definition in Figure~\ref{syvar} allows a \tdescr{suffix}\index{suffix?\tdescr{suffix}}
+to start with a \tdescr{subscript}\index{subscript?\tdescr{subscript}}.
+
+Figure~\ref{fig45} shows how suffix and expr parameters can be used together.
+The {\tt getmid}\index{getmid?\texttt{getmid}} macro takes a path variable and creates arrays
+of points and directions whose names are obtained by appending {\tt mid},
+{\tt off}, and {\tt dir} to the path variable. The {\tt joinup}\index{joinup?\texttt{joinup}}
+macro takes arrays of points and directions and creates a path of length {\tt n}
+that passes through each {\tt pt[i]} with direction {\tt d[i]} or
+$-\hbox{\tt d[i]}$.
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+def getmid(suffix p) =
+ pair p.mid[], p.off[], p.dir[];
+ for i=0 upto 36:
+ p.dir[i] = dir(5*i);
+ p.mid[i]+p.off[i] = directionpoint p.dir[i] of p;
+ p.mid[i]-p.off[i] = directionpoint -p.dir[i] of p;
+ endfor
+enddef;
+
+def joinup(suffix pt, d)(expr n) =
+ begingroup
+ save res, g; path res;
+ res = pt[0]{d[0]};
+ for i=1 upto n:
+ g:= if (pt[i]-pt[i-1]) dotprod d[i] <0: - fi 1;
+ res := res{g*d[i-1]}...{g*d[i]}pt[i];
+ endfor
+ res
+ endgroup
+enddef;
+
+beginfig(45)
+path p, q;
+p = ((5,2)...(3,4)...(1,3)...(-2,-3)...(0,-5)...(3,-4)
+ ...(5,-3)...cycle) scaled .3cm shifted (0,5cm);
+getmid(p);
+draw p;
+draw joinup(p.mid, p.dir, 36)..cycle;
+q = joinup(p.off, p.dir, 36);
+draw q..(q rotated 180)..cycle;
+drawoptions(dashed evenly);
+for i=0 upto 3:
+ draw p.mid[9i]-p.off[9i]..p.mid[9i]+p.off[9i];
+ draw -p.off[9i]..p.off[9i];
+endfor
+endfig;
+\end{verbatim}
+\quad \mathcenter{\epsfbox{manfig.45}}
+$$
+\caption{MetaPost code and the corresponding figure}
+\label{fig45}
+\end{figure}
+
+A definition that starts
+$$ \hbox{\tt def joinup(suffix pt, d)(expr n) =} $$
+might suggest that calls to the {\tt joinup} macro should have two sets of
+parentheses as in
+$$ \hbox{\tt joinup(p.mid, p.dir)(36)} $$
+instead of
+$$ \hbox{\tt joinup(p.mid, p.dir, 36)} $$
+In fact, both forms are acceptable. Parameters in a macro call can be separated
+by commas or by {\tt )(} pairs. The only restriction is that a
+text parameter\index{parameter!text}
+must be followed by a right parenthesis. For instance, a macro {\tt foo} with one
+text parameter and one expr parameter can be called
+$$ \hbox{\tt foo(a,b)(c)} $$
+in which case the text parameter is ``{\tt a,b}'' and the expr parameter is
+{\tt c}, but
+$$ \hbox{\tt foo(a,b,c)} $$
+sets the text parameter to ``{\tt a,b,c}'' and leaves the MetaPost interpreter
+still looking for the expr parameter.
+
+
+\subsection{Vardef Macros}
+
+A macro definition can begin with {\tt vardef}\index{vardef?\texttt{vardef}} instead of
+{\tt def}. Macros defined in this way are called vardef macros. They are
+particularly well-suited to applications where macros are being used like functions
+or subroutines. The main idea is that a vardef macro is like a variable of type
+``macro.''
+
+Instead of {\tt def} \tdescr{symbolic token}, a vardef macro begins
+$$ {\tt vardef}\, \descr{generic variable} $$
+where a \tdescr{generic variable}\index{generic variable?\tdescr{generic variable}} is a variable
+name with numeric subscripts replaced by the
+generic subscript\index{subscript!generic} symbol {\tt []}\index{[]?\texttt{[]}}.
+In other words, the name following {\tt vardef} obeys exactly the same syntax as
+the name given in a variable declaration. It is a sequence of tags and generic
+subscript symbols starting with a tag, where a tag\index{tags} is a symbolic token
+that is not a macro or a primitive operator as explained in Section~\ref{vardecl}.
+
+The simplest case is when the name of a vardef macro consists of a single tag.
+Under such circumstances, {\tt def} and {\tt vardef} provide roughly the same
+functionality. The most obvious difference is that
+{\tt begingroup}\index{begingroup?\texttt{begingroup}} and {\tt endgroup}\index{endgroup?\texttt{endgroup}}
+are automatically inserted at the beginning and end of the
+\tdescr{replacement text} of every vardef macro. This makes the
+\tdescr{replacement text} a group so that a vardef
+macro behaves like a subroutine or a function call.
+
+Another property of vardef macros is that they allow multi-token macro
+names and macro names involving generic subscripts.
+When a vardef macro name has generic subscripts, numeric values have to be given
+when the macro is called. After a macro definition
+$$ \hbox{\tt vardef a[]b(expr p) =}\, \descr{replacement text}\,
+ \hbox{\tt enddef;}
+$$
+{\tt a2b((1,2))} and {\tt a3b((1,2)..(3,4))} are macro calls. But how can the
+\tdescr{replacement text} tell the difference between {\tt a2b} and {\tt a3b}?
+Two implicit suffix parameters\index{parameter!suffix} are automatically
+provided for this purpose.
+Every vardef macro has suffix parameters \verb|#@|\index{#@?\texttt{\#@}}
+and \verb|@|\index{@?\texttt{@}}, where \verb|@| is the last token in the name from the
+macro call and \verb|#@| is everything preceding the last token. Thus \verb|#@|
+is {\tt a2} when the name is given as {\tt a2b} and {\tt a3} when the name is
+given as {\tt a3b}.
+
+Suppose, for example, that the {\tt a[]b} macro is to take its argument and
+shift it by an amount that depends on the macro name. The macro could be defined
+like this:
+$$ \hbox{\verb|vardef a[]b(expr p) = p shifted (#@,b) enddef;|} $$
+Then {\tt a2b((1,2))} means {\tt (1,2) shifted (a2,b)}
+and {\tt a3b((1,2)..(3,4))} means
+$$ \hbox{\tt ((1,2)..(3,4)) shifted (a3,b)}. $$
+
+If the macro had been {\tt a.b[]}, \verb|#@| would always be {\tt a.b} and the
+\verb|@| parameter would give the numeric subscript. Then {\tt a@} would refer to
+an element of the array {\tt a[]}. Note that \verb|@| is a suffix parameter, not
+an expr parameter, so an expression like {\tt @+1} would be illegal. The only way
+to get at the numeric values of subscripts in a
+suffix parameter\index{parameter!suffix} is by extracting
+them from the string returned by the {\tt str}\index{str?\texttt{str}}\label{Dstr}
+operator. This operator takes a suffix and returns a string
+representation of a suffix. Thus {\tt str @} would be \verb|"3"| in {\tt a.b3}
+and \verb|"3.14"| in {\tt a.b3.14} or {\tt a.b[3.14]}. Since the syntax for a
+\tdescr{suffix}\index{suffix?\tdescr{suffix}} in Figure~\ref{syvar} requires negative
+subscripts to be in brackets, {\tt str @} returns {\tt "[-3]"} in {\tt a.b[-3]}.
+
+The {\tt str} operator is generally for emergency use only. It is better to
+use suffix parameters only as variable names or suffixes. The best example of a
+vardef macro involving suffixes is the {\tt z} macro that defines the
+{\tt z} convention\index{z convention?{\tt z} convention}. The definition involves a special
+token \verb|@#|\index{@#?\texttt{@\#}} that refers to the suffix following the macro name:
+$$ \hbox{\verb|vardef z@#=(x@#,y@#) enddef;|} $$
+This means that any variable name whose first token is {\tt z} is equivalent to
+a pair of variables whose names are obtained by replacing {\tt z} with {\tt x}
+and~{\tt y}. For instance, {\tt z.a1} calls the {\tt z} macro with the suffix
+parameter \verb|@#| set to {\tt a1}.
+
+In general,
+$$ {\tt vardef}\, \descr{generic variable} \hbox{\verb|@#|} $$
+is an alternative to {\tt vardef} \tdescr{generic variable} that causes the
+MetaPost interpreter
+to look for a suffix following the name given in the macro call and makes this
+available as the \verb|@#| suffix parameter.
+
+To summarize the special features of vardef macros, they allow a broad class of
+macro names as well as macro names followed by a special suffix parameter.
+Furthermore, {\tt begingroup} and {\tt endgroup} are automatically added to the
+\tdescr{replacement text} of a vardef macro. Thus using {\tt vardef}
+instead of {\tt def} to define the {\tt joinup}\index{joinup?\texttt{joinup}} macro in
+Figure~\ref{fig45} would have avoided the need to include {\tt begingroup} and
+{\tt endgroup} explicitly in the macro definition.
+
+In fact, most of the macro definitions given in previous examples could equally
+well use {\tt vardef} instead of {\tt def}. It usually does not matter very much
+which you use, but a good general rule is to use {\tt vardef} if you intend the
+macro to be used like a function or a subroutine. The following comparison
+should help in deciding when to use {\tt vardef}.
+
+\begin{itemize}
+\item Vardef macros are automatically surrounded by {\tt begingroup}
+and {\tt endgroup}.
+\item The name of a vardef macro can be more than one token long and it can
+contain subscripts.
+\item A vardef macro can have access to the suffix that follows the macro name
+when the macro is called.
+\item When a symbolic token is used in the name of a vardef macro it remains
+a tag\index{tags} and can still be used in other variable names. Thus {\tt p5dir}
+is a legal variable name even though {\tt dir} is a vardef macro, but an ordinary
+macro such as {\tt ...}\index{...?\texttt{...}} cannot be used in a variable name.
+(This is fortunate since {\tt z5...z6} is supposed to be a path expression, not
+an elaborate variable name).
+\end{itemize}
+
+
+\subsection{Defining Unary and Binary Macros}
+
+It has been mentioned several times that some of the operators and commands
+discussed so far are actually predefined macros. These include unary operators
+such as {\tt round}\index{round?\texttt{round}} and {\tt unitvector}\index{unitvector?\texttt{unitvector}},
+statements such as {\tt fill}\index{fill?\texttt{fill}} and {\tt draw}\index{draw?\texttt{draw}},
+and binary operators such as {\tt dotprod}\index{dotprod?\texttt{dotprod}} and
+{\tt intersectionpoint}\index{intersectionpoint?\texttt{intersectionpoint}}. The main difference
+between these macros and the ones we already know how to define is their argument
+syntax.
+
+The {\tt round} and {\tt unitvector} macros are examples of what
+Figure~\ref{syexpr} calls \tdescr{unary op}. That is, they are followed by a
+primary expression. To specify a macro argument of this type, the macro definition
+should look like this:
+$$ \hbox{\tt vardef round primary u =}\, \descr{replacement text}\,
+ \hbox{\tt enddef;}
+$$
+The {\tt u} parameter is an expr parameter\index{parameter!expr} and it can be
+used exactly like the expr parameter defined using the ordinary
+$$ \hbox{\tt (expr u)} $$
+syntax.
+
+As the {\tt round} example suggests, a macro can be defined to take a
+\tdescr{secondary}\index{secondary?\tdescr{secondary}},
+\tdescr{tertiary}\index{tertiary?\tdescr{tertiary}}, or an
+\tdescr{expression}\index{expression?\tdescr{expression}} parameter. For example, the
+predefined definition of the {\tt fill} macro is roughly\index{fill?\texttt{fill}}
+$$ \hbox{\tt def fill expr c = addto currentpicture contour c enddef;} $$
+
+It is even possible to define a macro to play the role of
+\tdescr{of operator}\index{of operator?\tdescr{of operator}} in Figure~\ref{syexpr}.
+For example, the {\tt direction of}\index{direction of?\texttt{direction of}} macro has a definition
+of this form:
+$$ \hbox{\tt vardef direction expr t of p =}\, \descr{replacement text}\,
+ \hbox{\tt enddef;}
+$$
+
+Macros can also be defined to behave like binary operators. For instance, the
+definition of the {\tt dotprod} macro has the
+form\index{dotprod?\texttt{dotprod}}\index{primarydef?\texttt{primarydef}}
+$$ \hbox{\tt primarydef w dotprod z =}\, \descr{replacement text}\,
+ \hbox{\tt enddef;}
+$$
+This makes {\tt dotprod} a \tdescr{primary binop}\index{primary binop?\tdescr{primary binop}}.
+Similarly, {\tt secondarydef}\index{secondarydef?\texttt{secondarydef}} and
+{\tt tertiarydef}\index{tertiarydef?\texttt{tertiarydef}} introduce
+\tdescr{secondary binop}\index{secondary binop?\tdescr{secondary binop}} and
+\tdescr{tertiary binop}\index{tertiary binop?\tdescr{tertiary binop}} definitions. These all
+define ordinary macros, not vardef macros; e.g., there is
+no ``{\tt primaryvardef}.''
+
+Thus macro definitions can be introduced by {\tt def}, {\tt vardef},
+{\tt primarydef}, {\tt secondarydef}, or {\tt tertiarydef}.
+A \tdescr{replacement text}\index{replacement text?\tdescr{replacement text}} is any list of tokens
+that is balanced with respect to {\tt def}-{\tt enddef} pairs where all five macro
+definition tokens are treated like {\tt def} for the purpose of
+{\tt def}-{\tt enddef} matching.
+
+The rest of the syntax for macro definitions is summarized in Figure~\ref{symacro}.
+The syntax contains a few surprises. The macro parameters can have a
+\tdescr{delimited part} and an \tdescr{undelimited part}. Normally, one of
+these is \tdescr{empty}, but it is possible to have both parts nonempty:
+$$ \hbox{\tt def foo(text a) expr b =}\, \descr{replacement text}\,
+ \hbox{\tt enddef;}
+$$
+This defines a macro {\tt foo} to take a text parameter in parentheses followed
+by an expression.
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{macro definition} \rightarrow
+ \descr{macro heading} \hbox{\tt =} \descr{replacement text}\, enddef$\\
+$\tt \descr{macro heading} \rightarrow def\, \descr{symbolic token}
+ \descr{delimited part} \descr{undelimited part}$\\
+$\tt \qquad \;|\; vardef\, \descr{generic variable} \descr{delimited part}
+ \descr{undelimited part}$\\
+$\tt \qquad \;|\; vardef\, \descr{generic variable} \hbox{\tt @\#}
+ \descr{delimited part} \descr{undelimited part}$\\
+$\tt \qquad \;|\; \descr{binary def} \descr{parameter}
+ \descr{symbolic token} \descr{parameter}$\\
+$\tt \descr{delimited part} \rightarrow \descr{empty}$\\
+$\tt \qquad \;|\; \descr{delimited part}
+ \hbox{\tt (}\descr{parameter type} \descr{parameter tokens}\hbox{\tt )}$\\
+$\tt \descr{parameter type} \rightarrow expr \;|\; suffix \;|\; text$\\
+$\tt \descr{parameter tokens} \rightarrow \descr{parameter} \;|\;
+ \descr{parameter tokens}\hbox{\tt ,} \descr{parameter}$\\
+$\tt \descr{parameter} \rightarrow \descr{symbolic token}$\\
+$\tt \descr{undelimited part} \rightarrow \descr{empty}$\\
+$\tt \qquad \;|\; \descr{parameter type} \descr{parameter}$\\
+$\tt \qquad \;|\; \descr{precedence level} \descr{parameter}$\\
+$\tt \qquad \;|\; expr\, \descr{parameter}\, of\, \descr{parameter}$\\
+$\tt \descr{precedence level} \rightarrow primary \;|\; secondary \;|\;
+ tertiary$\\
+$\tt \descr{binary def} \rightarrow primarydef \;|\; secondarydef \;|\;
+ tertiatydef$
+\end{ctabbing}
+\caption{The syntax for macro definitions}
+\label{symacro}
+\end{figure}
+
+The syntax also allows the \tdescr{undelimited part} to specify an argument type
+of {\tt suffix}\index{suffix?\texttt{suffix}} or {\tt text}\index{text?\texttt{text}}. An example of
+a macro with an undelimited suffix parameter\index{parameter!suffix}
+is the predefined macro {\tt incr}\index{incr?\texttt{incr}}\label{Dincr} that is actually
+defined like this:
+$$ \hbox{\verb|vardef incr suffix $ = $:=$+1; $ enddef;|} $$
+This makes {\tt incr} a function that takes a variable, increments it, and
+returns the new value. Undelimited suffix parameters may be parenthesized,
+so {\tt incr a} and {\tt incr(a)} are both legal if {\tt a} is a numeric
+variable. There is also a similar predefined macro {\tt decr}\index{decr?\texttt{decr}}
+that subtracts~1.
+
+Undelimited text parameters\index{parameter!text} run to the end of a statement.
+More precisely, an undelimited text parameter is the list of tokens following the
+macro call up to the first ``{\tt ;}\index{semicolon}'' or
+``{\tt endgroup}\index{endgroup?\texttt{endgroup}}'' or ``{\tt end}\index{end?\texttt{end}}''
+except that an argument containing ``{\tt begingroup}'' will always
+include the matching ``{\tt endgroup}.''
+An example of an undelimited text parameter comes from the predefined macro
+{\tt cutdraw}\index{cutdraw?\texttt{cutdraw}}\label{Dctdraw} whose definition is
+roughly\index{linecap?\texttt{linecap}}\index{butt?\texttt{butt}}\index{interim?\texttt{interim}}
+$$\begin{verbatim}
+def cutdraw text t =
+ begingroup interim linecap:=butt; draw t; endgroup enddef;
+\end{verbatim}
+$$
+This makes {\tt cutdraw} synonymous with {\tt draw} except for the {\tt linecap}
+value. (This macro is provided mainly for compatibility with \MF\index{metafont?\MF}.)
+
+
+\section{Loops}
+
+Numerous examples in previous sections have used simple {\tt for} loops of the
+form\index{loops}\index{for?\texttt{for}}\index{endfor?\texttt{endfor}}
+$$ {\tt for}\, \descr{symbolic token}\, \hbox{\tt =}\,
+ \descr{expression}\, {\tt upto}\, \descr{expression}:\
+ \descr{loop text}\, {\tt endfor}
+$$
+It is equally simple to construct a loop that counts downward: just replace
+{\tt upto} by {\tt downto}\index{downto?\texttt{downto}}\label{Ddwnto}
+make the second \tdescr{expression} smaller than the first.
+This section covers more complicated types of progressions, loops where the loop
+counter behaves like a suffix parameter, and ways of exiting from a loop.
+
+The first generalization is suggested by the fact that {\tt upto}\index{upto?\texttt{upto}}
+is a predefined macro for\index{step?\texttt{step}}\index{until?\texttt{until}}
+$$ \hbox{\tt step 1 until} $$
+and {\tt downto}\index{downto?\texttt{downto}} is a macro for {\tt step -1 until}.
+A loop begining
+$$ \hbox{\tt for i=a step b until c} $$
+scans a sequence of {\tt i} values {\tt a}, ${\tt a}+{\tt b}$, ${\tt a}+2{\tt b}$,
+\ldots, stopping before {\tt i} passes {\tt c}; i.e., the loop scans {\tt i} values
+where ${\tt i}\le {\tt c}$ if ${\tt b}>0$ and ${\tt i}\ge {\tt c}$ if ${\tt i}<0$.
+
+It is best to use this feature only when the step size is an integer or some
+number that can be represented exactly in fixed point arithmetic\index{arithmetic}
+as a multiple of $1\over65536$. Otherwise, error will accumulate and the loop
+index might not reach the expected termination value. For instance,
+$$ \hbox{\tt for i=0 step .1 until 1: show i; endfor} $$
+shows ten {\tt i} values the last of which is 0.90005.
+
+The standard way of avoid the problems associated with non-integer step sizes is
+to iterate over integer values and then multiply by a scale factor when using
+the loop index as was done in Figures \ref{fig1} and~\ref{fig40}.
+
+Alternatively, the values to iterate over can be given explicitly. Any sequence
+of zero or more expressions separated by commas can be used in place of
+{\tt a step b upto c}. In fact, the expressions need not all be the same type
+and they need not have known values. Thus
+$$ \hbox{\tt for t=3.14, 2.78, (a,2a), "hello": show a; endfor} $$
+shows the four values listed.
+
+Note that the loop body in the above example is a statement followed by a
+semicolon. It is common for the body of a loop to be one or more statements,
+but this need not be the case. A loop is like a macro definition followed by
+calls to the macro. The loop body can be virtually any sequence of tokens as
+long as they make sense together. Thus, the (ridiculous) statement
+$$ \hbox{\tt draw for p=(3,1),(6,2),(7,5),(4,6),(1,3): p-- endfor cycle;} $$
+is equivalent to
+$$ \hbox{\tt draw (3,1)--(6,2)--(7,5)--(4,6)--(1,3)--cycle;} $$
+(See Figure~\ref{fig17} for a more realistic example of this.)
+
+If a loop is like a macro definition, the loop index is like an
+expr parameter\index{parameter!expr}. It can represent any value, but it is
+not a variable and it cannot be changed by an assignment
+statement\index{assignment}. In order to do that, you need a
+{\tt forsuffixes}\index{forsuffixes?\texttt{forsuffixes}} loop. A {\tt forsuffixes} loop is
+a lot like a {\tt for} loop, except the loop index behaves like a
+suffix parameter\index{parameter!suffix}. The syntax is
+$$ {\tt forsuffixes}\, \descr{symbolic token}\, \hbox{\tt =}\,
+ \descr{suffix list}:\ \descr{loop text}\, {\tt endfor}
+$$
+where a \tdescr{suffix list} is a comma-separated list of suffixes.
+If some of the suffixes are \tdescr{empty}, the \tdescr{loop text} gets executed
+with the loop index parameter set to the empty suffix.
+
+A good example of a {\tt forsuffixes} loop is the definition of the
+{\tt dotlabels}\index{dotlabels?\texttt{dotlabels}} macro\index{str?\texttt{str}}:
+$$\begin{verbatim}
+vardef dotlabels@#(text t) =
+ forsuffixes $=t: dotlabel@#(str$,z$); endfor enddef;
+\end{verbatim}
+$$
+This should make it clear why the parameter to {\tt dotlabels} has to be a
+comma-separated list of suffixes. Most macros that accept variable-length
+comma-separated lists
+use them in {\tt for} or {\tt forsuffixes} loops in this fashion as values to
+iterate over.
+
+When there are no values to iterate over, you can use a
+{\tt forever}\index{forever?\texttt{forever}} loop:
+$$ {\tt forever}\hbox{\tt :}\, \descr{loop text}\, {\tt endfor} $$
+To terminate such a loop when a boolean condition becomes true, use an exit
+clause\index{exitif?\texttt{exitif}}:
+$$ {\tt exitif}\, \descr{boolean expression} \hbox{\tt ;} $$
+When the MetaPost interpreter encounters an exit clause, it evaluates the
+\tdescr{boolean expression} and exits the current loop if the expression is
+true. If it is more convenient to exit the loop when an expression becomes false,
+use the predefined macro {\tt exitunless}\index{exitunless?\texttt{exitunless}}.
+
+Thus MetaPost's version of a {\bf while} loop is
+$$ \hbox{\tt forever: exitunless}\, \descr{boolean expression} \hbox{\tt ;}\,
+ \descr{loop text}\, {\tt endfor}
+$$
+The exit clause could equally well come just before {\tt endfor} or anywhere
+in the \tdescr{loop text}. In fact any {\tt for}, {\tt forever}, or
+{\tt forsuffixes} loop can contain any number of exit clauses.
+
+The summary of loop syntax shown in Figure~\ref{syloop} does not mention
+exit clauses explicitly because a \tdescr{loop text} can be virtually any
+sequence of tokens. The only restriction is that a \tdescr{loop text} must
+be balanced with respect to {\tt for} and {\tt endfor}. Of course this balancing
+process treats {\tt forsuffixes} and {\tt forever} just like {\tt for}.
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{loop} \rightarrow \descr{loop header}\hbox{\tt :}\,
+ \descr{loop text} endfor$\\
+$\tt \descr{loop header} \rightarrow for\, \descr{symbolic token}\,
+ \hbox{\tt =}\, \descr{progression}$\\
+$\tt \qquad \;|\; for\, \descr{symbolic token}\, \hbox{\tt =}\,
+ \descr{for list}$\\
+$\tt \qquad \;|\; forsuffixes\, \descr{symbolic token}\, \hbox{\tt =}\,
+ \descr{suffix list}$\\
+$\tt \qquad \;|\; forever$\\
+$\tt \descr{progression} \rightarrow \descr{numeric expression}\, upto\,
+ \descr{numeric expression}$\\
+$\tt \qquad \;|\; \descr{numeric expression}\, downto\,
+ \descr{numeric expression}$\\
+$\tt \qquad \;|\; \descr{numeric expression}\, step\,
+ \descr{numeric expression}\, until\, \descr{numeric expression} $\\
+$\tt \descr{for list} \rightarrow \descr{expression}
+ \;|\; \descr{for list}\hbox{\tt ,}\, \descr{expression}$\\
+$\tt \descr{suffix list} \rightarrow \descr{suffix}
+ \;|\; \descr{suffix list}\hbox{\tt ,}\, \descr{suffix}$
+\end{ctabbing}
+\caption{The syntax for loops}
+\label{syloop}
+\end{figure}
+
+
+\section{Making Boxes}
+\label{boxessec}
+
+This section describes auxiliary macros not included in Plain MetaPost that
+make it convenient to do things that {\it pic} is good at \cite{ke:pic}. What
+follows is a description of how to use the macros contained in the
+file {\tt boxes.mp}\index{boxes.mp?\texttt{boxes.mp}}. This file is included in a special
+directory reserved for MetaPost macros and support software\footnote{The name
+of this directory is likely to be something like \verb|/usr/lib/mp/lib|, but
+this is system dependent.}
+and can be accessed by giving the MetaPost command {\tt input boxes} before any
+figures that use the box making macros.
+The syntax for the {\tt input} command is \index{input?\texttt{input}}
+$$ {\tt input}\, \descr{file name} $$
+where a final ``{\tt .mp}'' can be omitted from the file name. The {\tt input}
+command looks first in the current directory and then in the special macro
+directory. Users interested in writing macros may want to look at the
+{\tt boxes.mp} file in this directory.
+
+\subsection{Rectangular Boxes}
+
+The main idea of the box-making macros is that one should
+say\index{boxit?\texttt{boxit}}\label{Dboxit}
+$$ {\tt boxit.} \descr{suffix}
+ \hbox{\tt(} \descr{picture expression} \hbox{\tt)}
+$$
+where the \tdescr{suffix} does not start with a subscript.\footnote{Some early
+versions of the box making macros did not allow any subscripts in the
+{\tt boxit} suffix.}
+This creates pair variables \tdescr{suffix}{\tt.c},
+\tdescr{suffix}{\tt.n}, \tdescr{suffix}{\tt.e}, \ldots\ that can then be
+used for positioning the picture before drawing it with a separate command such
+as\index{drawboxed?\texttt{drawboxed}}\label{Ddrbxed}
+$$ \hbox{\tt drawboxed(} \descr{suffix list} \hbox{\tt )} $$
+The argument to {\tt drawboxed} should be a comma-separated list of box names,
+where a box name\index{box name} is a \tdescr{suffix} with which {\tt boxit}
+has been called.
+
+For the command {\tt boxit.bb(pic)}, the box name is {\tt bb} and the contents
+of the box is the picture {\tt pic}. In this case, {\tt bb.c} the position
+where the center of picture {\tt pic} is to be placed, and {\tt bb.sw},
+{\tt bb.se}, {\tt bb.ne}, and {\tt bb.nw} are the corners of a rectangular path
+that will surround the resulting picture. Variables {\tt bb.dx} and {\tt bb.dy}
+give the spacing between the shifted version of {\tt pic} and the surrounding
+rectangle, and {\tt bb.off} is the amount by which {\tt pic} has to be shifted
+to achieve all this.
+
+When the {\tt boxit} macro is called with box name~$b$, it gives linear equations
+that force $b${\tt.sw}, $b${\tt.se}, $b${\tt.ne}, and $b${\tt.nw} to be the
+corners of a rectangle
+aligned on the $x$ and $y$ axes with the box contents centered inside as
+indicated by the gray rectangle in Figure~\ref{fig48}. The values of $b${\tt.dx},
+$b${\tt.dy}, and $b${\tt.c} are left unspecified so that the user can give
+equations for positioning the boxes. If no such equations are given, macros
+such as {\tt drawboxed} can detect this and give default values.
+The default values for {\tt dx} and {\tt dy} variables are controlled by the
+internal variables\index{internal variables}\index{variables!internal}
+{\tt defaultdx}\index{defaultdx?\texttt{defaultdx}}\label{Ddefaultdx} and
+{\tt defaultdy}\index{defaultdy?\texttt{defaultdy}}\label{Ddefaultdy}.
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.48} $$
+\caption[How a {\tt boxit} picture relates to the associated variables]
+ {The relationship between the picture given to {\tt boxit} and the
+ associated variables. The picture is indicated by a gray rectangle.}
+\label{fig48}
+\end{figure}
+
+If $b$ represents a box name, {\tt drawboxed($b$)} draws the rectangular boundary
+of box~$b$ and then the contents of the box. This bounding rectangle can be
+accessed separately as {\tt bpath~b}, or in general\index{bpath?\texttt{bpath}}\label{Dbpath}
+$$ {\tt bpath}\, \descr{box name} $$
+It is useful in combination with operators like
+{\tt cutbefore}\index{cutbefore?\texttt{cutbefore}} and {\tt cutafter}\index{cutafter?\texttt{cutafter}}
+in order to control paths that enter the box.
+For example, if $a$ and $b$ are box names and $p$ is a path from $a${\tt.c}
+to $b${\tt.c},\index{drawarrow?\texttt{drawarrow}}
+$$ \hbox{\tt drawarrow $p$ cutbefore bpath $a$ cutafter bpath $b$} $$
+draws an arrow from the edge of box $a$ to the edge of box $b$.
+
+Figure~\ref{fig49} shows a practical example including some arrows drawn with
+{\tt cutafter bpath} \tdescr{box name}. It is
+instructive to compare Figure~\ref{fig49} to the similar figure in the pic
+manual \cite{ke:pic}. The figure uses a macro\index{boxjoin?\texttt{boxjoin}}\label{Dbxjoin}
+$$ \hbox{\tt boxjoin(} \descr{equation text} \hbox{\tt )} $$
+to control the relationship between consecutive boxes. Within the
+\tdescr{equation text}, {\tt a} and {\tt b} represent the box names given in
+consecutive calls to {\tt boxit} and the \tdescr{equation text} gives equations
+to control the relative sizes and positions of the boxes.
+
+\begin{figure}[htp]
+$$\hbox{$\begin{verbatim}
+input boxes
+beginfig(49);
+boxjoin(a.se=b.sw; a.ne=b.nw);
+boxit.a(btex\strut$\cdots$ etex); boxit.ni(btex\strut$n_i$ etex);
+boxit.di(btex\strut$d_i$ etex); boxit.ni1(btex\strut$n_{i+1}$ etex);
+boxit.di1(btex\strut$d_{i+1}$ etex); boxit.aa(btex\strut$\cdots$ etex);
+boxit.nk(btex\strut$n_k$ etex); boxit.dk(btex\strut$d_k$ etex);
+drawboxed(di,a,ni,ni1,di1,aa,nk,dk); label.lft("ndtable:", a.w);
+interim defaultdy:=7bp;
+boxjoin(a.sw=b.nw; a.se=b.ne);
+boxit.ba(); boxit.bb(); boxit.bc();
+boxit.bd(btex $\vdots$ etex); boxit.be(); boxit.bf();
+bd.dx=8bp; ba.ne=a.sw-(15bp,10bp);
+drawboxed(ba,bb,bc,bd,be,bf); label.lft("hashtab:",ba.w);
+vardef ndblock suffix $ =
+ boxjoin(a.sw=b.nw; a.se=b.ne);
+ forsuffixes $$=$1,$2,$3: boxit$$(); ($$dx,$$dy)=(5.5bp,4bp);
+ endfor; enddef;
+ndblock nda; ndblock ndb; ndblock ndc;
+nda1.c-bb.c = ndb1.c-nda3.c = (whatever,0);
+xpart ndb3.se = xpart ndc1.ne = xpart di.c;
+ndc1.c - be.c = (whatever,0);
+drawboxed(nda1,nda2,nda3, ndb1,ndb2,ndb3, ndc1,ndc2,ndc3);
+drawarrow bb.c -- nda1.w;
+drawarrow be.c -- ndc1.w;
+drawarrow nda3.c -- ndb1.w;
+drawarrow nda1.c{right}..{curl0}ni.c cutafter bpath ni;
+drawarrow nda2.c{right}..{curl0}di.c cutafter bpath di;
+drawarrow ndc1.c{right}..{curl0}ni1.c cutafter bpath ni1;
+drawarrow ndc2.c{right}..{curl0}di1.c cutafter bpath di1;
+drawarrow ndb1.c{right}..nk.c cutafter bpath nk;
+drawarrow ndb2.c{right}..dk.c cutafter bpath dk;
+x.ptr=xpart aa.c; y.ptr=ypart ndc1.ne;
+drawarrow subpath (0,.7) of (z.ptr..{left}ndc3.c) dashed evenly;
+label.rt(btex \strut ndblock etex, z.ptr); endfig;
+\end{verbatim}
+$}
+\atop \vcenter{\vskip8pt\hbox{\epsfbox{manfig.49}}}
+$$
+\caption{MetaPost code and the corresponding figure}
+\label{fig49}
+\end{figure}
+
+For example, the second line of input for the above figure contains
+$$ \hbox{\tt boxjoin(a.se=b.sw; a.ne=b.nw)} $$
+This causes boxes to line up horizontally by giving additional equations that
+are invoked each time some box {\tt a} is followed by some other box~{\tt b}.
+These equations are first invoked on the next line when box~{\tt a} is followed
+by box~{\tt ni}. This yields
+$$ \hbox{\tt a.se=ni.sw; a.ne=ni.nw} $$
+The next pair of boxes is box~{\tt ni} and box~{\tt di}. This time the
+implicitly generated equations are
+$$ \hbox{\tt ni.se=di.sw; ni.ne=di.nw} $$
+This process continues until a new {\tt boxjoin}\index{boxjoin?\texttt{boxjoin}} is given.
+In this case the new declaration is
+$$ \hbox{\tt boxjoin(a.sw=b.nw; a.se=b.ne)} $$
+which causes boxes to be stacked below each other.
+
+After calling {\tt boxit} for the first eight boxes {\tt a} through {\tt dk},
+the box heights are constrained to match but the widths are still unknown.
+Thus the {\tt drawboxed}\index{drawboxed?\texttt{drawboxed}} macro needs to assign default
+values to the \tdescr{box name}{\tt.dx} and \tdescr{box name}{\tt.dy}
+variables. First, {\tt di.dx} and {\tt di.dy} get default values so that all
+the boxes are forced to be large enough to contain the contents of box~{\tt di}.
+
+The macro that actually assigns default values to {\tt dx} and {\tt dy} variables
+is called {\tt fixsize}\index{fixsize?\texttt{fixsize}}\label{Dfixsiz}.
+It takes a list of box names and
+considers them one at a time, making sure that each box has a fixed size and
+shape. A macro called {\tt fixpos}\index{fixpos?\texttt{fixpos}}\label{Dfixpos} then takes
+this same list
+of box names and assigns default values to the \tdescr{box name}{\tt.off}
+variables as needed to fix the position of each box. By using {\tt fixsize}
+to fix the dimensions of each box before assigning default positions to any
+of them, the number of needing default positions can usually be cut to at most
+one.
+
+Since the bounding path for a box cannot be computed until the size, shape, and
+position of the box is determined, the {\tt bpath}\index{bpath?\texttt{bpath}} macro applies
+{\tt fixsize} and {\tt fixpos} to its argument. Other macros that do this
+include\index{pic?\texttt{pic}}\label{Dpic}
+$$ {\tt pic}\, \descr{box name} $$
+where the \tdescr{box name} is a suffix, possibly in parentheses. This returns
+the contents of the named box as a picture positioned so that
+$$ {\tt draw\ pic} \descr{box name} $$
+draws the box contents without the bounding rectangle. This operation can also
+be accomplished by the {\tt drawunboxed}\index{drawunboxed?\texttt{drawunboxed}}\label{Ddrunbx}
+macro that takes a comma-separated list of box names. There is also a
+{\tt drawboxes}\index{drawboxes?\texttt{drawboxes}}\label{Ddrbxes} macro that draws just the
+bounding rectangles.
+
+Another way to draw empty rectangles is by just saying\label{Deboxit}
+$$ {\tt boxit} \descr{box name} \hbox{\tt ()} $$
+with no picture argument as is done several times in Figure~\ref{fig49}.
+This is like calling {\tt boxit} with an empty picture.
+Alternatively the argument can be a string\label{Dsboxit} expression
+instead of a picture
+expression in which case the string is typeset in the default font.
+
+
+\subsection{Circular and Oval Boxes}
+
+Circular and oval boxes are a lot like rectangular boxes except for the shape
+of the bounding path. Such boxes are set up by the
+{\tt circleit}\index{circleit?\texttt{circleit}}\label{Dcircit} macro:
+$$ {\tt circleit} \descr{box name}
+ \hbox{\tt(} \descr{box contents} \hbox{\tt)}
+$$
+where \tdescr{box name} is a suffix and \tdescr{box contents} is either a
+picture expression, a string expression, or \tdescr{empty}.
+
+The {\tt circleit} macro defines pair variable just as {\tt boxit} does, except
+that there are no corner points \tdescr{box name}{\tt.ne},
+\tdescr{box name}{\tt.sw}, etc. A call to
+$$ \hbox{\tt circleit.a(}\ldots \hbox{\tt )} $$
+gives relationships among points {\tt a.c}, {\tt a.s},
+{\tt a.e}, {\tt a.n}, {\tt a.w}
+and distances {\tt a.dx} and {\tt a.dy}. Together with {\tt a.c} and {\tt a.off},
+these variables describe how the picture is centered in an oval as can be seen
+from the Figure~\ref{fig50}.
+
+\begin{figure}[htp]
+$$ \epsfbox{manfig.50} $$
+\caption[How a {\tt circleit} picture relates to the associated variables]
+ {The relationship between the picture given to {\tt circleit} and the
+ associated variables. The picture is indicated by a gray rectangle.}
+\label{fig50}
+\end{figure}
+
+The {\tt drawboxed}\index{drawboxed?\texttt{drawboxed}}, {\tt drawunboxed}\index{drawunboxed?\texttt{drawunboxed}},
+{\tt drawboxes}\index{drawboxes?\texttt{drawboxes}}, {\tt pic}\index{pic?\texttt{pic}}, and
+{\tt bpath}\index{bpath?\texttt{bpath}} macros work for {\tt circleit} boxes just as they do
+for {\tt boxit} boxes. By default, the boundary path for a {\tt circleit} box is
+a circle large enough to surround the box contents with a small safety margin
+controlled by the
+internal variable\index{internal variables}\index{variables!internal}
+{\tt circmargin}\label{Dcmargin}. Figure~\ref{fig51} gives
+a basic example of the use of {\tt bpath} with {\tt circleit} boxes.
+
+\begin{figure}[htbp]
+$$\begin{verbatim}
+vardef drawshadowed(text t) =
+ fixsize(t);
+ forsuffixes s=t:
+ fill bpath.s shifted (1pt,-1pt);
+ unfill bpath.s;
+ drawboxed(s);
+ endfor
+enddef;
+
+beginfig(51)
+circleit.a(btex Box 1 etex);
+circleit.b(btex Box 2 etex);
+b.n = a.s - (0,20pt);
+drawshadowed(a,b);
+drawarrow a.s -- b.n;
+endfig;
+\end{verbatim}
+\qquad \mathcenter{\epsfbox{manfig.51}} $$
+\caption[MetaPost code and the resulting figure.]
+ {MetaPost code and the resulting figure. Note that the {\tt drawshadowed}
+ macro used here is not part of the {\tt boxit.mp} macro package.}
+\label{fig51}
+\index{drawshadowed?\texttt{drawshadowed}}
+\end{figure}
+
+A full example of {\tt circleit} boxes appears in Figure~\ref{fig52}.
+The oval boundary paths around ``Start'' and ``Stop'' are due to the equations
+$$ \hbox{\tt aa.dx=aa.dy;} \quad {\rm and}\quad \hbox{\tt ee.dx=ee.dy} $$
+after
+$$ \hbox{\verb|circleit.ee(btex\strut Stop etex)|}
+ \quad{\rm and}\quad
+ \hbox{\verb|circleit.ee(btex\strut Stop etex)|}.
+$$
+The general rule is that {\tt bpath.}$c$ comes out circular if $c${\tt.dx},
+$c${\tt.dy}, and $c\hbox{\tt.dx}-c\hbox{\tt.dy}$ are all unknown. Otherwise, the
+macros select an oval big enough to contain the given picture with the safety
+margin {\tt circmargin}\index{circmargin?\texttt{circmargin}}.
+
+
+\begin{figure}[htp]
+$$\hbox{$\begin{verbatim}
+vardef cuta(suffix a,b) expr p =
+ drawarrow p cutbefore bpath.a cutafter bpath.b;
+ point .5*length p of p
+enddef;
+
+vardef self@# expr p =
+ cuta(@#,@#) @#.c{curl0}..@#.c+p..{curl0}@#.c enddef;
+
+beginfig(52);
+verbatimtex \def\stk#1#2{$\displaystyle{\matrix{#1\cr#2\cr}}$} etex
+circleit.aa(btex\strut Start etex); aa.dx=aa.dy;
+circleit.bb(btex \stk B{(a|b)^*a} etex);
+circleit.cc(btex \stk C{b^*} etex);
+circleit.dd(btex \stk D{(a|b)^*ab} etex);
+circleit.ee(btex\strut Stop etex); ee.dx=ee.dy;
+numeric hsep;
+bb.c-aa.c = dd.c-bb.c = ee.c-dd.c = (hsep,0);
+cc.c-bb.c = (0,.8hsep);
+xpart(ee.e - aa.w) = 3.8in;
+drawboxed(aa,bb,cc,dd,ee);
+label.ulft(btex$b$etex, cuta(aa,cc) aa.c{dir50}..cc.c);
+label.top(btex$b$etex, self.cc(0,30pt));
+label.rt(btex$a$etex, cuta(cc,bb) cc.c..bb.c);
+label.top(btex$a$etex, cuta(aa,bb) aa.c..bb.c);
+label.llft(btex$a$etex, self.bb(-20pt,-35pt));
+label.top(btex$b$etex, cuta(bb,dd) bb.c..dd.c);
+label.top(btex$b$etex, cuta(dd,ee) dd.c..ee.c);
+label.lrt(btex$a$etex, cuta(dd,bb) dd.c..{dir140}bb.c);
+label.bot(btex$a$etex, cuta(ee,bb) ee.c..tension1.3 ..{dir115}bb.c);
+label.urt(btex$b$etex, cuta(ee,cc) ee.c{(cc.c-ee.c)rotated-15}..cc.c);
+endfig;
+\end{verbatim}
+$}
+\atop \vcenter{\vskip8pt\hbox{\epsfbox{manfig.52}}}
+$$
+\caption{MetaPost code and the corresponding figure}
+\label{fig52}
+\index{self?\texttt{self}}
+\end{figure}
+
+
+\section{Debugging}
+
+MetaPost inherits from \MF\index{metafont?\MF} numerous facilities for interactive
+debugging, most of which can only be mentioned briefly here. Further information
+on error messages, debugging, and generating tracing information can be found in
+{\it The\ \MF book} \cite{kn:c}.
+
+Suppose your input file says
+$$ \hbox{\tt draw z1--z2;} $$
+on line 17 without first giving known values to {\tt z1} and {\tt z2}.
+Figure~\ref{errmsg} shows what the MetaPost interpreter prints on your terminal
+when it finds the error. The actual error message is the line beginning with
+``{\tt !}''; the next six lines give the context that shows exactly what input
+was being read when the error was found; and the ``{\tt ?}'' on last line is a
+prompt for your response. Since the error message talks about an undefined
+$x$~coordinate, this value is printed on the first line after the ``{\tt >>}''.
+In this case the $x$~coordinate of {\tt z1} is just the unknown variable {\tt x1},
+so the interpreter prints the variable name {\tt x1} just as it would if it
+were told to\index{show?\texttt{show}} ``{\tt show x1}'' at this point.
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+>> x1
+! Undefined x coordinate has been replaced by 0.
+<to be read again>
+ {
+--->{
+ curl1}..{curl1}
+l.17 draw z1--
+ z2;
+?
+\end{verbatim}
+$$
+\caption{An example of an error message.}
+\label{errmsg}
+\end{figure}
+
+The context listing may seem a little confusing at first, but it really just
+gives a few lines of text showing how much of each line has been read so far.
+Each line of input is printed on two lines like this:
+\begin{eqnarray*}
+ \descr{descriptor}\ \hbox{Text read so far} \\
+ && \hbox{Text yet to be read}
+\end{eqnarray*}
+The \tdescr{descriptor} identifies the input source. It is either a line number
+like ``{\tt l.17}'' for line 17 of the current file; or it can be a macro name
+followed by ``{\tt ->}''; or it is a descriptive phrase in angle brackets.
+Thus, the meaning of the context listing in Figure~\ref{errmsg} is that the
+interpreter has just read line 17 of the input file up to ``{\tt --},'' the
+expansion of the {\tt --} macro has just started, and the initial
+``\verb|{|'' has been reinserted to allow for user input before scanning
+this token.
+
+Among the possible responses to a {\tt ?} prompt are the following:
+\begin{description}
+\item[x] terminates the run so that you can fix you input file and start over.
+\item[h] prints a help message followed by another {\tt ?} prompt.
+\item[\tdescr{return}] causes the interpreter to proceed as best it can.
+\item[?] prints a listing of the options available, followed by another
+ {\tt ?} prompt.
+\end{description}
+
+Error messages and responses to {\tt show} commands are also written into the
+transcript\index{files!transcript} file whose name is obtained from the name
+of the main input file by changing ``{\tt .mp}'' to ``{\tt .log}''. When the
+internal variable\index{internal variables}\index{variables!internal}
+{\tt tracingonline}\index{tracingonline?\texttt{tracingonline}} is at its default
+value of zero, some {\tt show} commands print their results in full detail only
+in transcript file.
+
+Only one type of {\tt show}\index{show?\texttt{show}} command has been discussed so far:
+{\tt show} followed by a comma-separated list of expressions prints symbolic
+representations of the expressions.
+
+The {\tt showtoken}\index{showtoken?\texttt{showtoken}}\label{Dshtok}
+command can be used to show the
+parameters and replacement text of a macro. It takes a comma-separated list of
+tokens and identifies each one. If the token is a primitive as in
+``\verb|showtoken +|'' it is just identified as being itself:
+$$ \hbox{\verb|> +=+|} $$
+Applying {\tt showtoken} to a variable or a {\tt vardef} macro yields
+$$ \hbox{\tt > } \descr{token}\hbox{\tt =variable} $$
+
+To get more information about a variable, use
+{\tt showvariable}\index{showvariable?\texttt{showvariable}}\label{Dshvar}
+instead of {\tt showtoken}. The
+argument to {\tt showvariable} is a comma-separated list of symbolic tokens
+and the result is a description of all the variables whose names begin with
+one of the listed tokens. This even works for {\tt vardef} macros. For
+example, {\tt showvariable z} yields
+$$ \hbox{\verb|z@#=macro:->begingroup(x(SUFFIX2),y(SUFFIX2))endgroup|} $$
+
+There is also a {\tt showdependencies}\index{showdependencies?\texttt{showdependencies}}\label{Dshdep}
+command that takes no arguments and prints a list of all {\em dependent} variables
+and how the linear equations given so far make them depend on other variables.
+Thus after
+$$ \hbox{\tt z2-z1=(5,10); z1+z2=(a,b);} $$
+{\tt showdependencies} prints what is shown in Figure~\ref{shdep}. This could
+be useful in answering a question like ``What does it mean
+`{\tt !\ Undefined x coordinate}?' I thought the equations given so far would
+determine {\tt x1}.''
+
+\begin{figure}[htp]
+$$\begin{verbatim}
+x2=0.5a+2.5
+y2=0.5b+5
+x1=0.5a-2.5
+y1=0.5b-5
+\end{verbatim}
+$$
+\caption{The result of {\tt z2-z1=(5,10); z1+z2=(a,b); showdependencies;}}
+\label{shdep}
+\end{figure}
+
+When all else fails, the predefined macro
+{\tt tracingall}\index{tracingall?\texttt{tracingall}}\label{Dtall}
+causes the interpreter to print a detailed listing of everything it is doing.
+Since the tracing information is often quite voluminous, it may be better to use
+the {\tt loggingall}\index{loggingall?\texttt{loggingall}}\label{Dlogall}
+macro that produces the same information
+but only writes it in the transcript\index{files!transcript} file. There is also
+a {\tt tracingnone}\index{tracingnone?\texttt{tracingnone}}\label{Dtnone}
+macro that turns off all the tracing output.
+
+Tracing output is controlled by the set of
+internal variables\index{internal variables}\index{variables!internal}
+summarized below.
+When any one of these variables is given a positive value, the corresponding form
+of tracing is turned on. Here is the set of tracing variables and what happens
+when each of them is positive:
+\begin{description}
+\item[{\tt tracingcapsules}]\index{tracingcapsules?\texttt{tracingcapsules}}\label{Dtcapsules}%
+shows the values of temporary quantities (capsules) when they become known.
+%
+\item[{\tt tracingchoices}]\index{tracingchoices?\texttt{tracingchoices}}\label{Dtchoices}%
+shows the B\'ezier control\index{control points} points of each new path when they
+are chosen.
+%
+\item[{\tt tracingcommands}]\index{tracingcommands?\texttt{tracingcommands}}\label{Dtcommands}%
+shows the commands before they are performed. A setting ${}>1$ also shows
+{\tt if}\index{if?\texttt{if}} tests and loops before they are expanded;
+a setting ${}>2$ shows algebraic operations before they are performed.
+%
+\item[{\tt tracingequations}]\index{tracingequations?\texttt{tracingequations}}\label{Dtequations}%
+shows each variable when it becomes known.
+%
+\item[{\tt tracinglostchars}]\index{tracinglostchars?\texttt{tracinglostchars}}\label{Dtlostchars}%
+warns about characters omitted from a picture because they are not in the font
+being used to typeset labels.
+%
+\item[{\tt tracingmacros}]\index{tracingmacros?\texttt{tracingmacros}}\label{Dtmacros}%
+shows macros before they are expanded.
+%
+\item[{\tt tracingoutput}]\index{tracingoutput?\texttt{tracingoutput}}\label{Dtoutput}%
+shows pictures as they are being shipped out as PostScript files.
+%
+\item[{\tt tracingrestores}]\index{tracingrestores?\texttt{tracingrestores}}\label{Dtrestores}%
+shows symbols and internal variables as they are being restored at the end
+of a group.
+%
+\item[{\tt tracingspecs}]\index{tracingspecs?\texttt{tracingspecs}}\label{Dtspecs}%
+shows the outlines generated when drawing with a
+polygonal pen\index{pens!polygonal}.
+%
+\item[{\tt tracingstats}]\index{tracingstats?\texttt{tracingstats}}\label{Dtstats}
+shows in the transcript file at the end of the job how many of the
+MetaPost interpreter's limited resources were used.
+\end{description}
+
+
+\section*{Acknowledgement}
+
+I would like to thank Don Knuth for making this work possible by developing
+\MF\ and placing it in the public domain. I am also indebted to him for helpful
+suggestions, particularly with regard to the treatment of included \TeX\ material.
+
+
+\appendix
+
+\section{Reference Manual}
+
+\let\svtopfrac=\topfraction % prepare to restore values at end of this appendix
+\let\svtxtfrac=\textfraction % grouping would fail because \setcounter is global
+\newcounter{svtopnum}
+\newcounter{svtotnum}
+\setcounter{svtopnum}{\value{topnumber}}
+\setcounter{svtotnum}{\value{totalnumber}}
+
+\renewcommand\topfraction{1.0} % set values to allow *lots* of figures and tables
+\renewcommand\textfraction{0.0}
+\setcounter{topnumber}{10}
+\setcounter{totalnumber}{10}
+
+Tables \ref{ivartab}--\ref{pseudotab} summarize the built-in features of
+Plain MetaPost and the features defined in the {\tt boxes.mp}\index{boxes.mp?\texttt{boxes.mp}}
+macro file. As explained in Section~\ref{boxessec}, the {\tt boxes.mp} macro
+file is not automatically preloaded and the macros defined there are not
+accessible until you ask for them via the command\index{input?\texttt{input}}
+$$ \hbox{\tt input boxes} $$
+
+Features that depend on {\tt boxes.mp} are marked by \bx\ symbols.
+Features from the Plain\index{Plain macros} macro package are marked are marked
+by \pl\ symbols, and MetaPost primitives are not marked by \bx\ or \pl.
+The distinction between primitives and plain macros can be ignored by the casual
+user, but it is important to remember that features marked by a \bx\ can only
+be used after reading in the {\tt boxes.mp}\index{boxes.mp?\texttt{boxes.mp}} macro file.
+
+The tables in this appendix give the name each feature, the page number where
+it is explained, and a short description. A few features are not explained
+elsewhere and have no page number listed. These features exist primarily for
+compatibility with \MF\index{metafont?\MF} and are intended to be self-explanatory.
+Certain other features from \MF\ are omitted entirely because they are of
+limited interest to the MetaPost users and/or would require long explanations.
+All of these are documented in {\it The \MF book} \cite{kn:c} as explained
+in Appendix~\ref{MPvsMF}.
+
+Table~\ref{ivartab} lists internal variables that take on numeric values.
+Table~\ref{pvartab} lists predefined variables of other types.
+Table~\ref{consttab} lists predefined constants. Some of these are implemented
+as variables whose values are intended to be left unchanged.
+
+Tables \ref{optabA}--\ref{optabD} summarize MetaPost operators and list the
+possible argument and result types for each one. A ``--'' entry for the left
+argument indicates a unary operator; ``--'' entries for both arguments indicate a
+nullary operator. Operators that take suffix parameters are not listed in
+these tables because they are treated as ``function-like macros''.
+
+The last two tables are Table~\ref{cmdtab} for commands and Table~\ref{pseudotab}
+macros that behave like functions or procedures. Such macros take parenthesized
+argument lists and/or suffix parameters, returning either a value whose type is
+listed in the table, or nothing. The latter case is for macros that behave
+like procedures. Their return values are listed as ``--''.
+
+The figures in this appendix present the syntax of the MetaPost language
+starting with expressions in Figures \ref{syexpr1}--\ref{sypseudo}.
+Although the productions sometimes specify types for expressions, primaries,
+secondaries, and tertiaries, no attempt is made to give separate syntaxes
+for \tdescr{numeric expression}, \tdescr{pair expression}, etc.
+The simplicity of the productions in Figure~\ref{sytypexpr} is due to this
+lack of type information. Type information
+can be found in Tables \ref{ivartab}--\ref{pseudotab}.
+
+Figures \ref{syprog} and \ref{sycmds} give the syntax for MetaPost programs,
+including statements and commands. They do not mention loops\index{loops}
+and {\tt if}\index{if?\texttt{if}}
+tests because these constructions do not behave like statements. The syntax
+given in Figures \ref{syexpr1}--\ref{pseudotab} applies to the result of
+expanding all conditionals and loops. Conditionals and loops do have a
+syntax, but they deal with almost arbitrary sequences of tokens.
+Figure~\ref{sycondloop} specifies conditionals in terms of
+\tdescr{balanced tokens} and loops in terms of \tdescr{loop text}, where
+\tdescr{balanced tokens} is any sequence of tokens balanced with respect
+to {\tt if} and {\tt fi}, and \tdescr{loop text} is a sequence of tokens
+balanced with respect to {\tt for}, {\tt forsuffixes}, {\tt forever},
+and {\tt endfor}.
+
+\begin{table}[htp]
+\caption{Internal variables with numeric values}
+$$\begin{tabular}{|l|r|l|}
+\hline
+\multicolumn1{|c}{Name}& \multicolumn1{|c}{Page}& \multicolumn1{|c|}{Explanation}\\
+\hline
+\hline
+\pl\tt ahangle& \pageref{Dahangle}&
+ angle for arrowheads in degrees (default: 45)\\\hline
+\pl\tt ahlength& \pageref{Dahlength}&
+ size of arrowheads (default: 4{\tt bp})\\\hline
+\pl\tt bboxmargin& \pageref{Dbbmargin}&
+ extra space allowed by {\tt bbox} (default 2{\tt bp})\\\hline
+\tt charcode& \pageref{Dcharcode}&
+ the number of the next character to be output\\\hline
+\bx\tt circmargin& \pageref{Dcmargin}&
+ clearance around contents of a circular or oval box\\\hline
+\tt day& --&
+ the current day of the month\\\hline
+\bx\tt defaultdx& \pageref{Ddefaultdx}&
+ usual horizontal space around box contents (default 3{\tt bp})\\\hline
+\bx\tt defaultdy& \pageref{Ddefaultdy}&
+ usual vertical space around box contents (default 3{\tt bp})\\\hline
+\pl\tt defaultpen& \pageref{Ddefaultpen}&
+ numeric index used by {\tt pickup} to select default pen\\\hline
+\pl\tt defaultscale& \pageref{Ddfscale}&
+ font scale factor for label strings (default 1)\\\hline
+\pl\tt labeloffset& \pageref{Dlaboff}&
+ offset distance for labels (default 3{\tt bp})\\\hline
+\tt linecap& \pageref{Dlinecap}&
+ 0 for butt, 1 for round, 2 for square\\\hline
+\tt linejoin& \pageref{Dlinejoin}&
+ 0 for mitered, 1 for round, 2 for beveled\\\hline
+\tt miterlimit& \pageref{Dmiterlim}&
+ controls miter length as in PostScript\\\hline
+\tt month& --&
+ the current month (e.g, 3 $\equiv$ March)\\\hline
+\tt pausing& --&
+ ${}>0$ to display lines on the terminal before they are read\\\hline
+\tt prologues& \pageref{Dprologs}&
+ ${}>0$ to output conforming PostScript using built-in fonts\\\hline
+\tt showstopping& --&
+ ${}>0$ to stop after each {\tt show} command\\\hline
+\tt time& --&
+ the number of minutes past midnight when this job started\\\hline
+\tt tracingcapsules& \pageref{Dtcapsules}&
+ ${}>0$ to show capsules too\\\hline
+\tt tracingchoices& \pageref{Dtchoices}&
+ ${}>0$ to show the control points chosen for paths\\\hline
+\tt tracingcommands& \pageref{Dtcommands}&
+ ${}>0$ to show commands and operations as they are performed\\\hline
+\tt tracingequations& \pageref{Dtequations}&
+ ${}>0$ to show each variable when it becomes known\\\hline
+\tt tracinglostchars& \pageref{Dtlostchars}&
+ ${}>0$ to show characters that aren't {\tt infont}\\\hline
+\tt tracingmacros& \pageref{Dtmacros}&
+ ${}>0$ to show macros before they are expanded\\\hline
+\tt tracingonline& \pageref{Dtonline}&
+ ${}>0$ to show long diagnostics on the terminal\\\hline
+\tt tracingoutput& \pageref{Dtoutput}&
+ ${}>0$ to show digitized edges as they are output\\\hline
+\tt tracingrestores& \pageref{Dtrestores}&
+ ${}>0$ to show when a variable or internal is restored\\\hline
+\tt tracingspecs& \pageref{Dtspecs}&
+ ${}>0$ to show path subdivision when using a polygonal a pen\\\hline
+\tt tracingstats& \pageref{Dtstats}&
+ ${}>0$ to show memory usage at end of job\\\hline
+\tt tracingtitles& --&
+ ${}>0$ to show titles online when they appear\\\hline
+\tt truecorners& \pageref{Dtruecorn}&
+ ${}>0$ to make {\tt llcorner} etc. ignore {\tt setbounds}\\\hline
+\tt warningcheck& \pageref{Dwarncheck}&
+ controls error message when variable value is large\\\hline
+\tt year& --&
+ the current year (e.g., 1992)\\\hline
+\end{tabular}
+$$
+\label{ivartab}%
+\index{day?\texttt{day}}\index{month?\texttt{month}}\index{pausing?\texttt{pausing}}\index{showstopping?\texttt{showstopping}}%
+\index{time?\texttt{time}}\index{tracingtitles?\texttt{tracingtitles}}\index{year?\texttt{year}}
+\end{table}
+
+\begin{table}[htp]
+\caption{Other Predefined Variables}
+$$\begin{tabular}{|l|l|r|l|}
+\hline
+\multicolumn1{|c}{Name}& \multicolumn1{|c}{Type}& \multicolumn1{|c}{Page}&
+ \multicolumn1{|c|}{Explanation}\\
+\hline
+\hline
+\pl\tt background& color& \pageref{Dbground}&
+ Color for {\tt unfill} and {\tt undraw} (usually white)\\\hline
+\pl\tt currentpen& pen& \pageref{Dcurpen}&
+ Last pen picked up (for use by the {\tt draw} command)\\\hline
+\pl\tt currentpicture& picture& \pageref{Dcurpic}&
+ Accumulate results of {\tt draw} and {\tt fill} commands\\\hline
+\pl\tt cuttings& path& \pageref{Dcuttings}&
+ subpath cut off by last {\tt cutbefore} or {\tt cutafter}\\\hline
+\pl\tt defaultfont& string& \pageref{Ddffont}&
+ Font used by label commands for typesetting strings\\\hline
+\pl\tt extra\_beginfig& string& \pageref{Dxbfig}&
+ Commands for {\tt beginfig} to scan\\\hline
+\pl\tt extra\_endfig& string& \pageref{Dxefig}&
+ Commands for {\tt endfig} to scan\\\hline
+\end{tabular}
+$$
+\label{pvartab}
+\end{table}
+
+\begin{table}[htp]
+\caption{Predefined Constants}
+$$\begin{tabular}{|l|l|r|l|}
+\hline
+\multicolumn1{|c}{Name}& \multicolumn1{|c}{Type}& \multicolumn1{|c}{Page}&
+ \multicolumn1{|c|}{Explanation}\\
+\hline
+\hline
+\pl\tt beveled& numeric& \pageref{Dbvled}&
+ {\tt linejoin} value for beveled joins [2]\\\hline
+\pl\tt black& color& \pageref{Dblack}&
+ Equivalent to {\tt (0,0,0)}\\\hline
+\pl\tt blue& color& \pageref{Dblue}&
+ Equivalent to {\tt (0,0,1)}\\\hline
+\pl\tt bp& numeric& \pageref{Dbp}&
+ One PostScript point in {\tt bp} units [1]\\\hline
+\pl\tt butt& numeric& \pageref{Dbutt}&
+ {\tt linecap} value for butt end caps [0]\\\hline
+\pl\tt cc& numeric& --&
+ One cicero in {\tt bp} units [12.79213]\\\hline
+\pl\tt cm& numeric& \pageref{Dcm}&
+ One centimeter in {\tt bp} units [28.34645]\\\hline
+\pl\tt dd& numeric& --&
+ One didot point in {\tt bp} units [1.06601]\\\hline
+\pl\tt ditto& string& --&
+ The {\tt "} character as a string of length 1\\\hline
+\pl\tt down& pair& \pageref{Ddown}&
+ Downward direction vector $(0,-1)$\\\hline
+\pl\tt epsilon& numeric& --&
+ Smallest positive MetaPost number [$1\over65536$]\\\hline
+\pl\tt evenly& picture& \pageref{Devenly}&
+ Dash pattern for equal length dashes\\\hline
+\tt false& boolean& \pageref{Dfalse}&
+ The boolean value {\it false\/}\\\hline
+\pl\tt fullcircle& path& \pageref{Dfcirc}&
+ Circle of diameter 1 centered on $(0,0)$\\\hline
+\pl\tt green& color& \pageref{Dgreen}&
+ Equivalent to {\tt (0,1,0)}\\\hline
+\pl\tt halfcircle& path& \pageref{Dhcirc}&
+ Upper half of a circle of diameter 1\\\hline
+\pl\tt identity& transform& \pageref{Dident}&
+ Identity transformation\\\hline
+\pl\tt in& numeric& \pageref{Din}&
+ One inch in {\tt bp} units [72]\\\hline
+\pl\tt infinity& numeric& \pageref{Dinf}&
+ Large positive value [4095.99998]\\\hline
+\pl\tt left& pair& \pageref{Dleft}&
+ Leftward direction $(-1,0)$\\\hline
+\pl\tt mitered& numeric& \pageref{Dmitred}&
+ {\tt linejoin} value for mitered joins [0]\\\hline
+\pl\tt mm& numeric& \pageref{Dmm}&
+ One millimeter in {\tt bp} units [2.83464]\\\hline
+\tt nullpicture& picture& \pageref{Dnlpic}&
+ Empty picture\\\hline
+\pl\tt origin& pair& --&
+ The pair $(0,0)$\\\hline
+\pl\tt pc& numeric& --&
+ One pica in {\tt bp} units [11.95517]\\\hline
+\tt pencircle& pen& \pageref{Dpncirc}&
+ Circular pen of diameter 1\\\hline
+\pl\tt pensquare& pen& \pageref{Dpnsqr}&
+ square pen of height 1 and width 1\\\hline
+\pl\tt pt& numeric& \pageref{Dpt}&
+ One printer's point in {\tt bp} units [0.99626]\\\hline
+\pl\tt quartercircle& path& --&
+ First quadrant of a circle of diameter 1\\\hline
+\pl\tt red& color& \pageref{Dred}&
+ Equivalent to {\tt (1,0,0)}\\\hline
+\pl\tt right& pair& \pageref{Dright}&
+ Rightward direction $(1,0)$\\\hline
+\pl\tt rounded& numeric& \pageref{Drnded}&
+ {\tt linecap} and {\tt linejoin} value for round joins\\
+\tt & & &
+ and end caps [1]\\\hline
+\pl\tt squared& numeric& \pageref{Dsqred}&
+ {\tt linecap} value for square end caps [2]\\\hline
+\tt true& boolean& \pageref{Dtrue}&
+ The boolean value {\tt true}\\\hline
+\pl\tt unitsquare& path& --&
+ The path {\tt (0,0)--(1,0)--(1,1)--(0,1)--cycle}\\\hline
+\pl\tt up& pair& \pageref{Dup}&
+ Upward direction $(0,1)$\\\hline
+\pl\tt white& color& \pageref{Dwhite}&
+ Equivalent to {\tt (1,1,1)}\\\hline
+\pl\tt withdots& picture& \pageref{Dwdots}&
+ Dash pattern that produces dotted lines\\\hline
+\end{tabular}
+$$
+\label{consttab}%
+\index{cc?\texttt{cc}}\index{dd?\texttt{dd}}\index{ditto?\texttt{ditto}}\index{epsilon?\texttt{epsilon}}%
+\index{origin?\texttt{origin}}\index{pc?\texttt{pc}}\index{quartercircle?\texttt{quartercircle}}%
+\index{unitsquare?\texttt{unitsquare}}
+\end{table}
+
+\begin{table}[htp]
+\caption{Operators (Part 1)}
+$$\begin{tabular}{|l|l|l|l|r|l|}
+\hline
+\multicolumn1{|c}{Name}& \multicolumn3{|c}{Argument/result types}&
+ \multicolumn1{|c}{Page}& \multicolumn1{|c|}{Explanation}\\
+\cline{2-4}
+& Left& Right& Result& & \\
+\hline
+\hline
+\tt \verb|&|& string& string& string& \pageref{Damp}&
+ Concatenation---works for paths $l\hbox{\tt\&}r$ if\\
+& path& path& path& &
+ $r$ starts exactly where the $l$ ends\\\hline
+\tt \verb|*|& numeric& color& color& \pageref{Dmldiv}&
+ Multiplication\\
+& & numeric& numeric& &
+ \\
+& & pair& pair& &
+ \\\hline
+\tt \verb|*|& color& numeric& color& \pageref{Dmldiv}&
+ Multiplication\\
+& numeric& & numeric& &
+ \\
+& pair& & pair& &
+ \\\hline
+\tt \verb|**|& numeric& numeric& numeric& \pageref{Dpow}&
+ Exponentiation\\\hline
+\tt \verb|+|& color& color& color& \pageref{Dadd}&
+ Addition\\
+& numeric& numeric& numeric& &
+ \\
+& pair& pair& pair& &
+ \\\hline
+\tt \verb|++|& numeric& numeric& numeric& \pageref{Dpyadd}&
+ Pythagorean addition $\sqrt{l^2+r^2}$\\\hline
+\tt \verb|+-+|& numeric& numeric& numeric& \pageref{Dpysub}&
+ Pythagorean subtraction $\sqrt{l^2-r^2}$\\\hline
+\tt \verb|-|& color& color& color& \pageref{Dadd}&
+ Subtraction\\
+& numeric& numeric& numeric& &
+ \\
+& pair& pair& pair& &
+ \\\hline
+\tt \verb|-|& --& color& color& \pageref{Dneg}&
+ Negation\\
+& & numeric& numeric& &
+ \\
+& & pair& pair& &
+ \\\hline
+\tt \verb|/|& color& numeric& color& \pageref{Dmldiv}&
+ Division\\
+& numeric& & numeric& &
+ \\
+& pair& & pair& &
+ \\\hline
+\tt \verb|< = >|& string& string& boolean& \pageref{Dcmpar}&
+ Comparison operators\\
+\tt \verb|<= >=|& numeric& numeric& & &
+ \\
+\tt \verb|<>|& pair& pair& & &
+ \\
+& color& color& & &
+ \\
+& transform& transform& & &
+ \\\hline
+\pl\tt \verb|abs|& --& numeric& numeric& \pageref{Dabs}&
+ Absolute value\\
+& & pair& & &
+ \\\hline
+\tt \verb|and|& boolean& boolean& boolean& \pageref{Dand}&
+ Logical and\\\hline
+\tt \verb|angle|& --& pair& numeric& \pageref{Dangle}&
+ 2$-$argument arctangent (in degrees)\\\hline
+\tt \verb|arclength|& --& path& numeric& \pageref{Darclng}&
+ Arc length of a path\\\hline
+\tt \verb|arctime|& numeric& path& numeric& \pageref{Darctim}&
+ Time on a path where arclength from\\
+\tt \verb|of|& & & & &
+ the start reaches a given value\\\hline
+\tt \verb|ASCII|& --& string& numeric& --&
+ ASCII value of first character in string\\\hline
+\pl\tt \verb|bbox|& --& picture& path& \pageref{Dbbox}&
+ A rectangular path for the bounding\\
+& & path& & &
+ box\\
+& & pen& & &
+ \\\hline
+\tt \verb|bluepart|& --& color& numeric& \pageref{Drgbprt}&
+ Extracts the third component\\\hline
+\tt \verb|boolean|& --& any& boolean& \pageref{Dboolop}&
+ Is the expression of type boolean?\\\hline
+\tt \verb|bot|& --& numeric& numeric& \pageref{Dbot}&
+ Bottom of current pen when centered\\
+& & pair& pair& &
+ at the given coordinate(s)\\\hline
+\pl\tt \verb|ceiling|& --& numeric& numeric& \pageref{Dceil}&
+ Least integer greater than or equal to\\\hline
+\pl\tt \verb|center|& --& picture& pair& \pageref{Dcenter}&
+ Center of the bounding box\\
+& & path& & &
+ \\
+& & pen& & &
+ \\\hline
+\end{tabular}
+$$
+\index{ASCII?\texttt{ASCII}}%
+\label{optabA}
+\end{table}
+
+\begin{table}[htp]
+\caption{Operators (Part 2)}
+$$\begin{tabular}{|l|l|l|l|r|l|}
+\hline
+\multicolumn1{|c}{Name}& \multicolumn3{|c}{Argument/result types}&
+ \multicolumn1{|c}{Page}& \multicolumn1{|c|}{Explanation}\\
+\cline{2-4}
+& Left& Right& Result& & \\
+\hline
+\hline
+\tt \verb|char|& --& numeric& string& \pageref{Dchar}&
+ Character with a given ASCII code\\\hline
+\tt \verb|color|& --& any& boolean& \pageref{Dcolrop}&
+ Is the expression of type color?\\\hline
+\tt \verb|cosd|& --& numeric& numeric& \pageref{Dcosd}&
+ Cosine of angle in degrees\\\hline
+\pl\tt \verb|cutafter|& path& path& path& \pageref{Dcuta}&
+ Left argument with part after the\\
+& & & & &
+ intersection dropped\\\hline
+\pl\tt \verb|cutbefore|& path& path& path& \pageref{Dcutb}&
+ Left argument with part before the\\
+& & & & &
+ intersection dropped\\\hline
+\tt \verb|cycle|& --& path& boolean& \pageref{Dcycop}&
+ Determines whether a path is cyclic\\\hline
+\tt \verb|decimal|& --& numeric& string& \pageref{Ddecop}&
+ The decimal representation\\\hline
+\pl\tt \verb|dir|& --& numeric& pair& \pageref{Ddirop}&
+ $(\cos\theta,\sin\theta)$ given $\theta$ in degrees\\\hline
+\pl\tt \verb|direction|& numeric& path& pair& \pageref{Ddirof}&
+ The direction of a path at a given\\
+\tt \verb| of|& & & & &
+ `time'\\\hline
+\pl\tt \verb|direction-|& pair& path& numeric& \pageref{Ddpntof}&
+ Point where a path has a given\\
+\tt \verb|point of|& & & & &
+ direction\\\hline
+\tt \verb|direction-|& pair& path& numeric& \pageref{Ddtimof}&
+ `Time' when a path has a given\\
+\tt \verb|time of|& & & & &
+ direction\\\hline
+\pl\tt \verb|div|& numeric& numeric& numeric& --&
+ Integer division $\lfloor l/r\rfloor$\\\hline
+\pl\tt \verb|dotprod|& pair& pair& numeric& \pageref{Ddprod}&
+ vector dot product\\\hline
+\tt \verb|floor|& --& numeric& numeric& \pageref{Dfloor}&
+ Greatest integer less than or equal to\\\hline
+\tt \verb|fontsize|& --& string& numeric& \pageref{Dfntsiz}&
+ The point size of a font\\\hline
+\tt \verb|greenpart|& --& color& numeric& \pageref{Drgbprt}&
+ Extract the second component\\\hline
+\tt \verb|hex|& --& string& numeric& --&
+ Interpret as a hexadecimal number\\\hline
+\tt \verb|infont|& string& string& picture& \pageref{Sinfont}&
+ Typeset string in given font\\\hline
+\pl\tt \verb|intersec-|& path& path& pair& \pageref{Disecpt}&
+ An intersection point\\
+\tt \verb| tionpoint|& & & & &
+ \\\hline
+\tt \verb|intersec-|& path& path& pair& \pageref{Disectt}&
+ Times ($t_l,t_r)$ on paths $l$ and $r$\\
+\tt \verb|tiontimes|& & & & &
+ when the paths intersect\\\hline
+\pl\tt \verb|inverse|& --& transform& transform& \pageref{Dinv}&
+ Invert a transformation\\\hline
+\tt \verb|known|& --& any& boolean& \pageref{Dknown}&
+ Does argument have a known value?\\\hline
+\tt \verb|length|& --& path& numeric& \pageref{Dlength}&
+ Number of arcs in a path\\\hline
+\pl\tt \verb|lft|& --& numeric& numeric& \pageref{Dlft}&
+ Left side of current pen when its\\
+& & pair& pair& &
+ center is at the given coordinate(s)\\\hline
+\tt \verb|llcorner|& --& picture& pair& \pageref{Dcornop}&
+ Lower-left corner of bounding box\\
+& & path& & &
+ \\
+& & pen& & &
+ \\\hline
+\tt \verb|lrcorner|& --& picture& pair& \pageref{Dcornop}&
+ Lower-left corner of bounding box\\
+& & path& & &
+ \\
+& & pen& & &
+ \\\hline
+\tt \verb|makepath|& --& pen& path& \pageref{Dmkpath}&
+ Cyclic path bounding the pen shape\\\hline
+\tt \verb|makepen|& --& path& pen& \pageref{Dmkpen}&
+ A polygonal pen made from the\\
+& & & & &
+ convex hull of the path knots\\\hline
+\tt \verb|mexp|& --& numeric& numeric& --&
+ The function $\exp(x/256)$\\\hline
+\tt \verb|mlog|& --& numeric& numeric& --&
+ The function $256\ln(x)$\\\hline
+\pl\tt \verb|mod|& --& numeric& numeric& --&
+ The remainder function $l-r\lfloor l/r\rfloor$\\\hline
+\tt \verb|normal-|& --& --& numeric& --&
+ Choose a random number with\\
+\tt \verb|deviate|& & & & &
+ mean 0 and standard deviation 1\\\hline
+\end{tabular}
+$$
+\index{div?\texttt{div}}\index{hex?\texttt{hex}}\index{mexp?\texttt{mexp}}\index{mlog?\texttt{mlog}}%
+\index{mod?\texttt{mod}}\index{normaldeviate?\texttt{normaldeviate}}%
+\label{optabB}
+\end{table}
+
+\begin{table}[htp]
+\caption{Operators (Part 3)}
+$$\begin{tabular}{|l|l|l|l|r|l|}
+\hline
+\multicolumn1{|c}{Name}& \multicolumn3{|c}{Argument/result types}&
+ \multicolumn1{|c}{Page}& \multicolumn1{|c|}{Explanation}\\
+\cline{2-4}
+& Left& Right& Result& & \\
+\hline
+\hline
+\tt \verb|not|& --& boolean& boolean& \pageref{Dnot}&
+ Logical negation\\\hline
+\tt \verb|numeric|& --& any& boolean& \pageref{Dnumop}&
+ Is the expression of type numeric?\\\hline
+\tt \verb|oct|& --& string& numeric& --&
+ Interpret a string as an octal number\\\hline
+\tt \verb|odd|& --& numeric& boolean& --&
+ Is the closest integer odd or even?\\\hline
+\tt \verb|or|& boolean& boolean& boolean& \pageref{Dor}&
+ Logical inclusive or\\\hline
+\tt \verb|pair|& --& any& boolean& \pageref{Dpairop}&
+ Is the expression of type pair?\\\hline
+\tt \verb|path|& --& any& boolean& \pageref{Dpathop}&
+ Is the expression of type path?\\\hline
+\tt \verb|pen|& --& any& boolean& \pageref{Dpenop}&
+ Is the expression of type pen?\\\hline
+\tt \verb|penoffset|& pair& pen& pair& --&
+ Point on the pen furthest to the\\
+\tt \verb|of|& & & & &
+ right of the given direction\\\hline
+\tt \verb|picture|& --& any& boolean& \pageref{Dpictop}&
+ Is the expression of type picture?\\\hline
+\tt \verb|point of|& numeric& path& pair& \pageref{Dpntof}&
+ Point on a path given a time value\\\hline
+\tt \verb|postcontrol|& numeric& path& pair& --&
+ First B\'ezier control point on path\\
+\tt \verb|of|& & & & &
+ segment starting at the given time\\\hline
+\tt \verb|precontrol|& numeric& path& pair& --&
+ Last B\'ezier control point on path\\
+\tt \verb|of|& & & & &
+ segment ending at the given time\\\hline
+\tt \verb|redpart|& --& color& numeric& \pageref{Drgbprt}&
+ Extract the first component\\\hline
+\tt \verb|reverse|& --& path& path& \pageref{Drevrse}&
+ `time'-reversed path with beginning\\
+& & & & &
+ swapped with ending\\\hline
+\tt \verb|rotated|& picture& numeric& picture& \pageref{Dtranop}&
+ Rotate counterclockwise a given\\
+& path& & path& &
+ number of degrees\\
+& pair& & pair& &
+ \\
+& pen& & pen& &
+ \\
+& transform& & transform& &
+ \\\hline
+\pl\tt \verb|round|& --& numeric& numeric& \pageref{Dround}&
+ round each component to the nearest\\
+& & pair& pair& &
+ integer\\\hline
+\pl\tt \verb|rt|& --& numeric& numeric& \pageref{Drt}&
+ Right side of current pen when\\
+& & pair& pair& &
+ centered at given coordinate(s)\\\hline
+\tt \verb|scaled|& picture& numeric& picture& \pageref{Dtranop}&
+ Scale all coordinates by the given\\
+& path& & path& &
+ amount\\
+& pair& & pair& &
+ \\
+& pen& & pen& &
+ \\
+& transform& & transform& &
+ \\\hline
+\tt \verb|shifted|& picture& pair& picture& \pageref{Dtranop}&
+ Add the given shift amount to each\\
+& path& & path& &
+ pair of coordinates\\
+& pair& & pair& &
+ \\
+& pen& & pen& &
+ \\
+& transform& & transform& &
+ \\\hline
+\tt \verb|sind|& --& numeric& numeric& \pageref{Dsind}&
+ Sine of an angle in degrees\\\hline
+\tt \verb|slanted|& picture& numeric& picture& \pageref{Dtranop}&
+ Apply the slanting transformation\\
+& path& & path& &
+ that maps $(x,y)$ into $(x+sy,y)$,\\
+& pair& & pair& &
+ where $s$ is the numeric argument\\
+& pen& & pen& &
+ \\
+& transform& & transform& &
+ \\\hline
+\tt \verb|sqrt|& --& numeric& numeric& \pageref{Dsqrt}&
+ Square root\\\hline
+\tt \verb|str|& --& suffix& string& \pageref{Dstr}&
+ String representation for a suffix\\\hline
+\end{tabular}
+$$
+\index{oct?\texttt{oct}}\index{odd?\texttt{odd}}\index{penoffset?\texttt{penoffset}}\index{postcontrol?\texttt{postcontrol}}%
+\index{precontrol?\texttt{precontrol}}%
+\label{optabC}
+\end{table}
+
+\begin{table}[htp]
+\caption{Operators (Part 4)}
+$$\begin{tabular}{|l|l|l|l|r|l|}
+\hline
+\multicolumn1{|c}{Name}& \multicolumn3{|c}{Argument/result types}&
+ \multicolumn1{|c}{Page}& \multicolumn1{|c|}{Explanation}\\
+\cline{2-4}
+& Left& Right& Result& & \\
+\hline
+\hline
+\tt \verb|string|& --& any& boolean& \pageref{Dstrgop}&
+ Is the expression of type string?\\\hline
+\tt \verb|subpath|& pair& path& path& \pageref{Dsubpth}&
+ Portion of a path for given range\\
+\tt \verb|of|& & & & &
+ of time values\\\hline
+\tt \verb|substring|& pair& string& string& \pageref{Dsubstr}&
+ Substring bounded by given indices\\
+\tt \verb|of|& & & & &
+ \\\hline
+\pl\tt \verb|top|& --& numeric& numeric& \pageref{Dtop}&
+ Top of current pen when centered\\
+& & pair& pair& &
+ at the given coordinate(s)\\\hline
+\tt \verb|transform|& --& any& boolean& \pageref{Dtrnfop}&
+ Is the argument of type transform?\\\hline
+\tt \verb|transformed|& picture& transform& picture& \pageref{Dtrfrmd}&
+ Apply the given transform to all\\
+& path& & path& &
+ coordinates\\
+& pair& & pair& &
+ \\
+& pen& & pen& &
+ \\
+& transform& & transform& &
+ \\\hline
+\tt \verb|ulcorner|& --& picture& pair& \pageref{Dcornop}&
+ Upper-left corner of bounding box\\
+& & path& & &
+ \\
+& & pen& & &
+ \\\hline
+\tt \verb|uniform-|& --& numeric& numeric& --&
+ Random number between zero and\\
+\tt \verb|deviate|& & & & &
+ the value of the argument\\\hline
+\pl\tt \verb|unitvector|& --& pair& pair& \pageref{Duvec}&
+ Rescale a vector so its length is 1\\\hline
+\tt \verb|unknown|& --& any& boolean& \pageref{Dunknwn}&
+ Is the value unknown?\\\hline
+\tt \verb|urcorner|& --& picture& pair& \pageref{Dcornop}&
+ Upper-left corner of bounding box\\
+& & path& & &
+ \\
+& & pen& & &
+ \\\hline
+\pl\tt \verb|whatever|& --& --& numeric& \pageref{Dwhatev}&
+ Create a new anonymous unknown\\\hline
+\tt \verb|xpart|& --& pair& number& \pageref{Dxprt}&
+ $x$ or $t_x$ component\\
+& & transform& & &
+ \\\hline
+\tt \verb|xscaled|& picture& numeric& picture& \pageref{Dtranop}&
+ Scale all $x$ coordinates by the\\
+& path& & path& &
+ given amount\\
+& pair& & pair& &
+ \\
+& pen& & pen& &
+ \\
+& transform& & transform& &
+ \\\hline
+\tt \verb|xxpart|& --& transform& number& \pageref{Dtrprt}&
+ $t_{xx}$ entry in transformation matrix\\\hline
+\tt \verb|xypart|& --& transform& number& \pageref{Dtrprt}&
+ $t_{xy}$ entry in transformation matrix\\\hline
+\tt \verb|ypart|& --& pair& number& \pageref{Dyprt}&
+ $y$ or $t_y$ component\\
+& & transform& & &
+ \\\hline
+\tt \verb|yscaled|& picture& numeric& picture& \pageref{Dtranop}&
+ Scale all $y$ coordinates by the\\
+& path& & path& &
+ given amount\\
+& pair& & pair& &
+ \\
+& pen& & pen& &
+ \\
+& transform& & transform& &
+ \\\hline
+\tt \verb|yxpart|& --& transform& number& \pageref{Dtrprt}&
+ $t_{yx}$ entry in transformation matrix\\\hline
+\tt \verb|yypart|& --& transform& number& \pageref{Dtrprt}&
+ $t_{yy}$ entry in transformation matrix\\\hline
+\tt \verb|zscaled|& picture& pair& picture& \pageref{Dtranop}&
+ Rotate and scale all coordinates so\\
+& path& & path& &
+ that $(1,0)$ is mapped into the\\
+& pair& & pair& &
+ given pair; i.e., do complex\\
+& pen& & pen& &
+ multiplication.\\
+& transform& & transform& &
+ \\\hline
+\end{tabular}
+$$
+\index{uniformdeviate?\texttt{uniformdeviate}}%
+\label{optabD}
+\end{table}
+
+\begin{table}[htp]
+\caption{Commands}
+$$\begin{tabular}{|l|r|l|}
+\hline
+\multicolumn1{|c}{Name}& \multicolumn1{|c}{Page}& \multicolumn1{|c|}{Explanation}\\
+\hline
+\hline
+\tt \verb|addto|& \pageref{sydraw}&
+ Low-level command for drawing and filling\\\hline
+\tt \verb|clip|& \pageref{Dclip}&
+ Applies a clipping path to a picture\\\hline
+\pl\tt \verb|cutdraw|& \pageref{Dctdraw}&
+ Draw with butt end caps\\\hline
+\pl\tt \verb|draw|& \pageref{curves}&
+ Draw a line or a picture\\\hline
+\pl\tt \verb|drawarrow|& \pageref{Ddrwarr}&
+ Draw a line with an arrowhead at the end\\\hline
+\pl\tt \verb|drawdblarrow|& \pageref{Ddrwdar}&
+ Draw a line with arrowheads at both ends\\\hline
+\pl\tt \verb|fill|& \pageref{Dfill}&
+ Fill inside a cyclic path\\\hline
+\pl\tt \verb|filldraw|& \pageref{Dfildrw}&
+ Draw a cyclic path and fill inside it\\\hline
+\tt \verb|interim|& \pageref{Dinterm}&
+ Make a local change to an internal variable\\\hline
+\tt \verb|let|& --&
+ Assign one symbolic token the meaning of another\\\hline
+\pl\tt \verb|loggingall|& \pageref{Dlogall}&
+ Turn on all tracing (log file only)\\\hline
+\tt \verb|newinternal|& \pageref{Dnewint}&
+ Declare new internal variables\\\hline
+\pl\tt \verb|pickup|& \pageref{Dpickup}&
+ Specify new pen for line drawing\\\hline
+\tt \verb|save|& \pageref{Dsave}&
+ Make variables local\\\hline
+\tt \verb|setbounds|& \pageref{Dsetbnd}&
+ Make a picture lie about its bounding box\\\hline
+\tt \verb|shipout|& \pageref{Dship}&
+ Low-level command to output a figure\\\hline
+\tt \verb|show|& \pageref{Dshow}&
+ print out expressions symbolically\\\hline
+\tt \verb|showdependencies|& \pageref{Dshdep}&
+ print out all unsolved equations\\\hline
+\tt \verb|showtoken|& \pageref{Dshtok}&
+ print an explanation of what a token is\\\hline
+\tt \verb|showvariable|& \pageref{Dshvar}&
+ print variables symbolically\\\hline
+\tt \verb|special|& \pageref{Dspecl}&
+ print a string directly in the PostScript output file\\\hline
+\pl\tt \verb|tracingall|& \pageref{Dtall}&
+ Turn on all tracing\\\hline
+\pl\tt \verb|tracingnone|& \pageref{Dtnone}&
+ Turn off all tracing\\\hline
+\pl\tt \verb|undraw|& \pageref{Dundraw}&
+ Erase a line or a picture\\\hline
+\pl\tt \verb|unfill|& \pageref{Dunfill}&
+ Erase inside a cyclic path\\\hline
+\pl\tt \verb|unfilldraw|& \pageref{Dunfdrw}&
+ Erase a cyclic path and its inside\\\hline
+\end{tabular}
+$$
+\index{let?\texttt{let}}%
+\label{cmdtab}
+\end{table}
+
+\begin{table}[htp]
+\caption{Function-Like Macros}
+$$\begin{tabular}{|l|l|l|r|l|}
+\hline
+\multicolumn1{|c}{Name}& \multicolumn1{|c}{Arguments}&
+ \multicolumn1{|c}{Result}& \multicolumn1{|c}{Page}&
+ \multicolumn1{|c|}{Explanation}\\
+\hline
+\hline
+\bx\tt \verb|boxit|& suffix, picture& --& \pageref{Dboxit}&
+ Define a box containing the picture\\\hline
+\bx\tt \verb|boxit|& suffix, string& --& \pageref{Dsboxit}&
+ Define a box containing text\\\hline
+\bx\tt \verb|boxit|& suffix, \tdescr{empty}& --& \pageref{Deboxit}&
+ Define an empty box\\\hline
+\bx\tt \verb|boxjoin|& equations& --& \pageref{Dbxjoin}&
+ Give equations for connecting boxes\\\hline
+\bx\tt \verb|bpath|& suffix& path& \pageref{Dbpath}&
+ A box's bounding circle or rectangle\\\hline
+\pl\tt \verb|buildcycle|& list of paths& path& \pageref{buildcy}&
+ Build a cyclic path\\\hline
+\bx\tt \verb|circleit|& suffix, picture& --& \pageref{Dcircit}&
+ Put picture in a circular box\\\hline
+\bx\tt \verb|circleit|& suffix, picture& --& \pageref{Dcircit}&
+ Put a string in a circular box\\\hline
+\bx\tt \verb|circleit|& suffix, \tdescr{empty}& --& \pageref{Dcircit}&
+ Define an empty circular box\\\hline
+\pl\tt \verb|dashpattern|& on/off distances& picture& \pageref{Ddshpat}&
+ Create a pattern for dashed lines\\\hline
+\pl\tt \verb|decr|& numeric variable& numeric& \pageref{Dincr}&
+ Decrement and return new value\\\hline
+\pl\tt \verb|dotlabel|& suffix, picture, pair& --& \pageref{Ddotlab}&
+ Mark point and draw picture nearby\\\hline
+\pl\tt \verb|dotlabel|& suffix, string, pair& --& \pageref{Ddotlab}&
+ Mark point and place text nearby\\\hline
+\pl\tt \verb|dotlabels|& suffix, point numbers& --& \pageref{Ddotlbs}&
+ Mark {\tt z} points with their numbers\\\hline
+\bx\tt \verb|drawboxed|& list of suffixes& --& \pageref{Ddrbxed}&
+ Draw the named boxes and their\\
+& & & &
+ contents\\\hline
+\bx\tt \verb|drawboxes|& list of suffixes& --& \pageref{Ddrbxes}&
+ Draw the named boxes\\\hline
+\pl\tt \verb|drawoptions|& drawing options& --& \pageref{Ddropts}&
+ Set options for drawing commands\\\hline
+\bx\tt \verb|drawunboxed|& list of suffixes& --& \pageref{Ddrunbx}&
+ Draw contents of named boxes\\\hline
+\bx\tt \verb|fixpos|& list of suffixes& --& \pageref{Dfixpos}&
+ Solve for the size and position of the\\
+& & & &
+ named boxes\\\hline
+\bx\tt \verb|fixsize|& list of suffixes& --& \pageref{Dfixsiz}&
+ Solve for size of named boxes\\\hline
+\pl\tt \verb|incr|& numeric variable& numeric& \pageref{Dincr}&
+ Increment and return new value\\\hline
+\pl\tt \verb|label|& suffix, picture, pair& --& \pageref{Dlabel}&
+ Draw picture near given point\\\hline
+\pl\tt \verb|label|& suffix, string, pair& --& \pageref{Dlabel}&
+ Place text near given point\\\hline
+\pl\tt \verb|labels|& suffix, point numbers& --& \pageref{Dlabels}&
+ Draw {\tt z} point numbers; no dots\\\hline
+\pl\tt \verb|max|& list of numerics& numeric& --&
+ Find the maximum\\\hline
+\pl\tt \verb|max|& list of strings& string& --&
+ Find the lexicographically last string\\\hline
+\pl\tt \verb|min|& list of numerics& numeric& --&
+ Find the minimum\\\hline
+\pl\tt \verb|min|& list of strings& string& --&
+ Find the lexicographically first string\\\hline
+\bx\tt \verb|pic|& suffix& picture& \pageref{Dpic}&
+ Box contents shifted into position\\\hline
+\pl\tt \verb|thelabel|& suffix, picture, pair& picture& \pageref{Dthelab}&
+ Picture shifted as if to label a point\\\hline
+\pl\tt \verb|thelabel|& suffix, string, pair& picture& \pageref{Dthelab}&
+ text positioned as if to label a point\\\hline
+\pl\tt \verb|z|& suffix& pair& \pageref{Dzconv}&
+ The pair ${\tt x}\descr{suffix},{\tt y}\descr{suffix})$\\\hline
+\end{tabular}
+$$
+\index{min?\texttt{min}}\index{max?\texttt{max}}%
+\label{pseudotab}
+\end{table}
+
+\clearpage
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{atom} \rightarrow \descr{variable} \;|\; \descr{argument}$\\
+$\tt \qquad \;|\; \descr{number or fraction}$\\
+$\tt \qquad \;|\; \descr{internal variable}$\\
+$\tt \qquad \;|\; \hbox{\tt (}\descr{expression}\hbox{\tt )}$\\
+$\tt \qquad \;|\; begingroup \descr{statement list} \descr{expression} endgroup$\\
+$\tt \qquad \;|\; \descr{nullary op}$\\
+$\tt \qquad \;|\; btex \descr{typesetting commands} etex$\\
+$\tt \qquad \;|\; \descr{pseudo function}$\\
+$\tt \descr{primary} \rightarrow \descr{atom}$\\
+$\tt \qquad \;|\; \hbox{\tt (}\descr{numeric expression}\hbox{\tt ,} \descr{numeric expression}\hbox{\tt )}$\\
+$\tt \qquad \;|\; \hbox{\tt (}\descr{numeric expression}\hbox{\tt ,} \descr{numeric expression}\hbox{\tt ,} \descr{numeric expression}\hbox{\tt )}$\\
+$\tt \qquad \;|\; \descr{of operator} \descr{expression} of \descr{primary}$\\
+$\tt \qquad \;|\; \descr{unary op} \descr{primary}$\\
+$\tt \qquad \;|\; str \descr{suffix}$\\
+$\tt \qquad \;|\; z \descr{suffix}$\\
+$\tt \qquad \;|\; \descr{numeric atom}\hbox{\tt [}\descr{expression}\hbox{\tt ,}\descr{expression}\hbox{\tt ]}$\\
+$\tt \qquad \;|\; \descr{scalar multiplication op} \descr{primary}$\\
+$\tt \descr{secondary} \rightarrow \descr{primary}$\\
+$\tt \qquad \;|\; \descr{secondary} \descr{primary binop} \descr{primary}$\\
+$\tt \qquad \;|\; \descr{secondary} \descr{transformer}$\\
+$\tt \descr{tertiary} \rightarrow \descr{secondary}$\\
+$\tt \qquad \;|\; \descr{tertiary} \descr{secondary binop} \descr{secondary}$\\
+$\tt \descr{subexpression} \rightarrow \descr{tertiary}$\\
+$\tt \qquad \;|\; \descr{path expression} \descr{path join} \descr{path knot}$\\
+$\tt \descr{expression} \rightarrow \descr{subexpression}$\\
+$\tt \qquad \;|\; \descr{expression} \descr{tertiary binop} \descr{tertiary}$\\
+$\tt \qquad \;|\; \descr{path subexpression} \descr{direction specifier}$\\
+$\tt \qquad \;|\; \descr{path subexpression} \descr{path join} cycle$\\
+$\tt $\\
+$\tt \descr{path knot} \rightarrow \descr{tertiary}$\\
+$\tt \descr{path join} \rightarrow --$\\
+$\tt \qquad \;|\; \descr{direction specifier} \descr{basic path join} \descr{direction specifier}$\\
+$\tt \descr{direction specifier} \rightarrow \descr{empty}$\\
+$\tt \qquad \;|\; \char`\{curl \descr{numeric expression}\char`\}$\\
+$\tt \qquad \;|\; \char`\{\descr{pair expression}\char`\}$\\
+$\tt \qquad \;|\; \char`\{\descr{numeric expression}\hbox{\tt ,}\descr{numeric expression}\char`\}$\\
+$\tt \descr{basic path join} \rightarrow \hbox{\tt ..} \;|\; \hbox{\tt ...} \;|\; \hbox{\tt ..}\descr{tension}\hbox{\tt ..} \;|\; \hbox{\tt ..}\descr{controls}\hbox{\tt ..}$\\
+$\tt \descr{tension} \rightarrow tension \descr{numeric primary}$\\
+$\tt \qquad \;|\; tension \descr{numeric primary} and \descr{numeric primary}$\\
+$\tt \descr{controls} \rightarrow controls \descr{pair primary}$\\
+$\tt \qquad \;|\; controls \descr{pair primary} and \descr{pair primary}$\\
+$\tt $\\
+$\tt \descr{argument} \rightarrow \descr{symbolic token}$\\
+$\tt \descr{number or fraction} \rightarrow \descr{number}\hbox{\tt /}\descr{number}$\\
+$\tt \qquad \;|\; \descr{number not followed by `\hbox{\tt /}\tdescr{number}'}$\\
+$\tt \descr{scalar multiplication op} \rightarrow + \;|\; -$\\
+$\tt \qquad \;|\; \descr{`\tdescr{number or fraction}' not followed by `\tdescr{add op}\tdescr{number}'}$
+\end{ctabbing}
+\caption{Part 1 of the syntax for expressions}
+\index{expression?\tdescr{expression}}\index{nullary op?\tdescr{nullary op}}\index{of operator?\tdescr{of operator}}%
+\index{path knot?\tdescr{path knot}}\index{primary?\tdescr{primary}}\index{primary binop?\tdescr{primary binop}}%
+\index{secondary?\tdescr{secondary}}\index{secondary binop?\tdescr{secondary binop}}\index{suffix?\tdescr{suffix}}%
+\index{tertiary?\tdescr{tertiary}}\index{tertiary binop?\tdescr{tertiary binop}}\index{unary op?\tdescr{unary op}}%
+\label{syexpr1}
+\end{figure}
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{transformer} \rightarrow rotated \descr{numeric primary}$\\
+$\tt \qquad \;|\; scaled \descr{numeric primary}$\\
+$\tt \qquad \;|\; shifted \descr{pair primary}$\\
+$\tt \qquad \;|\; slanted \descr{numeric primary}$\\
+$\tt \qquad \;|\; transformed \descr{transform primary}$\\
+$\tt \qquad \;|\; xscaled \descr{numeric primary}$\\
+$\tt \qquad \;|\; yscaled \descr{numeric primary}$\\
+$\tt \qquad \;|\; zscaled \descr{pair primary}$\\
+$\tt \qquad \;|\; reflectedabout\hbox{\tt (}\descr{pair expression}\hbox{\tt ,} \descr{pair expression}\hbox{\tt )}$\\
+$\tt \qquad \;|\; rotatedaround\hbox{\tt (}\descr{pair expression}\hbox{\tt ,} \descr{numeric expression}\hbox{\tt )}$\\
+$\tt $\\
+$\tt \descr{nullary op} \rightarrow false \;|\; normaldeviate \;|\; nullpicture \;|\; pencircle$\\
+$\tt \qquad \;|\; true \;|\; whatever$\\
+$\tt \descr{unary op} \rightarrow \descr{type}$\\
+$\tt \qquad \;|\; abs \;|\; angle \;|\; arclength \;|\; ASCII \;|\; bbox \;|\; bluepart \;|\; bot \;|\; ceiling$\\
+$\tt \qquad \;|\; center \;|\; char \;|\; cosd \;|\; cycle \;|\; decimal \;|\; dir \;|\; floor \;|\; fontsize$\\
+$\tt \qquad \;|\; greenpart \;|\; hex \;|\; inverse \;|\; known \;|\; length \;|\; lft \;|\; llcorner$\\
+$\tt \qquad \;|\; lrcorner\;|\; makepath \;|\; makepen \;|\; mexp \;|\; mlog \;|\; not \;|\; oct \;|\; odd$\\
+$\tt \qquad \;|\; redpart \;|\; reverse \;|\; round \;|\; rt \;|\; sind \;|\; sqrt \;|\; top \;|\; ulcorner$\\
+$\tt \qquad \;|\; uniformdeviate \;|\; unitvector \;|\; unknown \;|\; urcorner \;|\; xpart \;|\; xxpart$\\
+$\tt \qquad \;|\; xypart \;|\; ypart \;|\; yxpart \;|\; yypart$\\
+$\tt \descr{type} \rightarrow boolean \;|\; color \;|\; numeric \;|\; pair$\\
+$\tt \qquad \;|\; path \;|\; pen \;|\; picture \;|\; string \;|\; transform$\\
+$\tt \descr{primary binop} \rightarrow \hbox{\tt *} \;|\; \hbox{\tt /} \;|\; \hbox{\tt **} \;|\; and$\\
+$\tt \qquad \;|\; dotprod \;|\; div \;|\; infont \;|\; mod$\\
+$\tt \descr{secondary binop} \rightarrow + \;|\; - \;|\; ++ \;|\; +-+ \;|\; or$\\
+$\tt \qquad \;|\; intersectionpoint \;|\; intersectiontimes$\\
+$\tt \descr{tertiary binop} \rightarrow \hbox{\tt \&} \;|\; \hbox{\verb|<|} \;|\; \hbox{\verb|<=|} \;|\; \hbox{\verb|<>|} \;|\; \hbox{\tt =} \;|\; \hbox{\verb|>|} \;|\; \hbox{\verb|>=|}$\\
+$\tt \qquad \;|\; cutafter \;|\; cutbefore$\\
+$\tt \descr{of operator} \rightarrow arctime \;|\; direction \;|\; directiontime \;|\; directionpoint$\\
+$\tt \qquad \;|\; penoffset \;|\; point \;|\; postcontrol \;|\; precontrol \;|\; subpath$\\
+$\tt \qquad \;|\; substring$\\
+$\tt $\\
+$\tt \descr{variable} \rightarrow \descr{tag}\descr{suffix}$\\
+$\tt \descr{suffix} \rightarrow \descr{empty} \;|\; \descr{suffix}\descr{subscript} \;|\; \descr{suffix}\descr{tag}$\\
+$\tt \qquad \;|\; \descr{suffix parameter}$\\
+$\tt \descr{subscript} \rightarrow \descr{number} \;|\; \hbox{\tt [}\descr{numeric expression}\hbox{\tt ]}$\\
+$\tt $\\
+$\tt \descr{internal variable} \rightarrow ahangle \;|\; ahlength \;|\; bboxmargin$\\
+$\tt \qquad \;|\; charcode \;|\; day \;|\; defaultpen \;|\; defaultscale \;|\; labeloffset$\\
+$\tt \qquad \;|\; linecap \;|\; linejoin \;|\; miterlimit \;|\; month \;|\; pausing$\\
+$\tt \qquad \;|\; prologues \;|\; showstopping \;|\; time \;|\; tracingoutput$\\
+$\tt \qquad \;|\; tracingcapsules \;|\; tracingchoices \;|\; tracingcommands$\\
+$\tt \qquad \;|\; tracingequations \;|\; tracinglostchars \;|\; tracingmacros$\\
+$\tt \qquad \;|\; tracingonline \;|\; tracingrestores \;|\; tracingspecs$\\
+$\tt \qquad \;|\; tracingstats \;|\; tracingtitles \;|\; truecorners$\\
+$\tt \qquad \;|\; warningcheck \;|\; year$\\
+$\tt \qquad \;|\; \descr{symbolic token defined by {\tt newinternal}}$
+\end{ctabbing}
+\caption{Part 2 of the syntax for expressions}
+\index{nullary op?\tdescr{nullary op}}\index{of operator?\tdescr{of operator}}\index{primary binop?\tdescr{primary binop}}%
+\index{secondary binop?\tdescr{secondary binop}}\index{subscript?\tdescr{subscript}}\index{suffix?\tdescr{suffix}}%
+\index{tertiary binop?\tdescr{tertiary binop}}\index{unary op?\tdescr{unary op}}%
+\label{syexpr2}
+\end{figure}
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{pseudo function} \rightarrow min\hbox{\tt (}\descr{expression list}\hbox{\tt )}$\\
+$\tt \qquad \;|\; max\hbox{\tt (}\descr{expression list}\hbox{\tt )}$\\
+$\tt \qquad \;|\; incr\hbox{\tt (}\descr{numeric variable}\hbox{\tt )}$\\
+$\tt \qquad \;|\; decr\hbox{\tt (}\descr{numeric variable}\hbox{\tt )}$\\
+$\tt \qquad \;|\; dashpattern\hbox{\tt (}\descr{on\hbox{\tt /}off list}\hbox{\tt )}$\\
+$\tt \qquad \;|\; interpath\hbox{\tt (}\descr{numeric expression}\hbox{\tt ,} \descr{path expression}\hbox{\tt ,} \descr{path expression}\hbox{\tt )}$\\
+$\tt \qquad \;|\; buildcycle\hbox{\tt (}\descr{path expression list}\hbox{\tt )}$\\
+$\tt \qquad \;|\; thelabel\descr{label suffix}\hbox{\tt (}\descr{expression}\hbox{\tt ,} \descr{pair expression}\hbox{\tt )}$\\
+$\tt \descr{path expression list} \rightarrow \descr{path expression}$\\
+$\tt \qquad \;|\; \descr{path expression list}\hbox{\tt ,} \descr{path expression}$\\
+$\tt \descr{on\hbox{\tt /}off list} \rightarrow \descr{on\hbox{\tt /}off list}\descr{on\hbox{\tt /}off clause} \;|\; \descr{on\hbox{\tt /}off clause}$\\
+$\tt \descr{on\hbox{\tt /}off clause} \rightarrow on \descr{numeric tertiary} \;|\; off \descr{numeric tertiary}$
+\end{ctabbing}
+\caption{The syntax for function-like macros}
+\index{label suffix?\tdescr{label suffix}}%
+\label{sypseudo}
+\end{figure}
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{boolean expression} \rightarrow \descr{expression}$\\
+$\tt \descr{color expression} \rightarrow \descr{expression}$\\
+$\tt \descr{numeric atom} \rightarrow \descr{atom}$\\
+$\tt \descr{numeric expression} \rightarrow \descr{expression}$\\
+$\tt \descr{numeric primary} \rightarrow \descr{primary}$\\
+$\tt \descr{numeric tertiary} \rightarrow \descr{tertiary}$\\
+$\tt \descr{numeric variable} \rightarrow \descr{variable} \;|\; \descr{internal variable}$\\
+$\tt \descr{pair expression} \rightarrow \descr{expression}$\\
+$\tt \descr{pair primary} \rightarrow \descr{primary}$\\
+$\tt \descr{path expression} \rightarrow \descr{expression}$\\
+$\tt \descr{path subexpression} \rightarrow \descr{subexpression}$\\
+$\tt \descr{pen expression} \rightarrow \descr{expression}$\\
+$\tt \descr{picture expression} \rightarrow \descr{expression}$\\
+$\tt \descr{picture variable} \rightarrow \descr{variable}$\\
+$\tt \descr{string expression} \rightarrow \descr{expression}$\\
+$\tt \descr{suffix parameter} \rightarrow \descr{parameter}$\\
+$\tt \descr{transform primary} \rightarrow \descr{primary}$
+\end{ctabbing}
+\caption{Miscellaneous productions needed to complete the BNF}
+\label{sytypexpr}
+\end{figure}
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{program} \rightarrow \descr{statement list} end$\\
+$\tt \descr{statement list} \rightarrow \descr{empty} \;|\; \descr{statement list} \hbox{\tt ;} \descr{statement}$\\
+$\tt \descr{statement} \rightarrow \descr{empty}$\\
+$\tt \qquad \;|\; \descr{equation} \;|\; \descr{assignment}$\\
+$\tt \qquad \;|\; \descr{declaration} \;|\; \descr{macro definition}$\\
+$\tt \qquad \;|\; \descr{compound} \;|\; \descr{pseudo procedure}$\\
+$\tt \qquad \;|\; \descr{command}$\\
+$\tt \descr{compound} \rightarrow begingroup \descr{statement list} endgroup$\\
+$\tt \qquad \;|\; beginfig\hbox{\tt (}\descr{numeric expression}\hbox{\tt );} \descr{statement list}\hbox{\tt ;} endfig$\\
+$\tt $\\
+$\tt \descr{equation} \rightarrow \descr{expression} \hbox{\tt =} \descr{right-hand side}$\\
+$\tt \descr{assignment} \rightarrow \descr{variable} \hbox{\tt :=} \descr{right-hand side}$\\
+$\tt \qquad \;|\; \descr{internal variable} \hbox{\tt :=} \descr{right-hand side}$\\
+$\tt \descr{right-hand side} \rightarrow \descr{expression} \;|\; \descr{equation} \;|\; \descr{assignment}$\\
+$\tt $\\
+$\tt \descr{declaration} \rightarrow \descr{type} \descr{declaration list}$\\
+$\tt \descr{declaration list} \rightarrow \descr{generic variable}$\\
+$\tt \qquad \;|\; \descr{declaration list}\hbox{\tt ,} \descr{generic variable}$\\
+$\tt \descr{generic variable} \rightarrow \descr{symbolic token} \descr{generic suffix}$\\
+$\tt \descr{generic suffix} \rightarrow \descr{empty} \;|\; \descr{generic suffix} \descr{tag}$\\
+$\tt \qquad \;|\; \descr{generic suffix} \hbox{\tt []}$\\
+$\tt $\\
+$\tt \descr{macro definition} \rightarrow \descr{macro heading} \hbox{\tt =} \descr{replacement text} enddef$\\
+$\tt \descr{macro heading} \rightarrow def \descr{symbolic token} \descr{delimited part} \descr{undelimited part}$\\
+$\tt \qquad \;|\; vardef \descr{generic variable} \descr{delimited part} \descr{undelimited part}$\\
+$\tt \qquad \;|\; vardef \descr{generic variable} \hbox{\verb|@#|} \descr{delimited part} \descr{undelimited part}$\\
+$\tt \qquad \;|\; \descr{binary def} \descr{parameter} \descr{symbolic token} \descr{parameter}$\\
+$\tt \descr{delimited part} \rightarrow \descr{empty}$\\
+$\tt \qquad \;|\; \descr{delimited part}\hbox{\tt (}\descr{parameter type} \descr{parameter tokens}\hbox{\tt )}$\\
+$\tt \descr{parameter type} \rightarrow expr \;|\; suffix \;|\; text$\\
+$\tt \descr{parameter tokens} \rightarrow \descr{parameter} \;|\; \descr{parameter tokens}\hbox{\tt ,} \descr{parameter}$\\
+$\tt \descr{parameter} \rightarrow \descr{symbolic token}$\\
+$\tt \descr{undelimited part} \rightarrow \descr{empty}$\\
+$\tt \qquad \;|\; \descr{parameter type} \descr{parameter}$\\
+$\tt \qquad \;|\; \descr{precedence level} \descr{parameter}$\\
+$\tt \qquad \;|\; expr \descr{parameter} of \descr{parameter}$\\
+$\tt \descr{precedence level} \rightarrow primary \;|\; secondary \;|\; tertiary$\\
+$\tt \descr{binary def} \rightarrow primarydef \;|\; secondarydef \;|\; tertiarydef$\\
+$\tt $\\
+$\tt \descr{pseudo procedure} \rightarrow drawoptions\hbox{\tt (}\descr{option list}\hbox{\tt )}$\\
+$\tt \qquad \;|\; label\descr{label suffix}\hbox{\tt (}\descr{expression}\hbox{\tt ,} \descr{pair expression}\hbox{\tt )}$\\
+$\tt \qquad \;|\; dotlabel\descr{label suffix}\hbox{\tt (}\descr{expression}\hbox{\tt ,} \descr{pair expression}\hbox{\tt )}$\\
+$\tt \qquad \;|\; labels\descr{label suffix}\hbox{\tt (}\descr{point number list}\hbox{\tt )}$\\
+$\tt \qquad \;|\; dotlabels\descr{label suffix}\hbox{\tt (}\descr{point number list}\hbox{\tt )}$\\
+$\tt \descr{point number list} \rightarrow \descr{suffix} \;|\; \descr{point number list}\hbox{\tt ,} \descr{suffix}$\\
+$\tt \descr{label suffix} \rightarrow \descr{empty} \;|\; lft \;|\; rt \;|\; top \;|\; bot \;|\; ulft \;|\; urt \;|\; llft \;|\; lrt$
+\end{ctabbing}
+\caption{Overall syntax for MetaPost programs}
+\index{generic variable?\tdescr{generic variable}}\index{label suffix?\tdescr{label suffix}}\index{replacement text?\tdescr{replacement text}}%
+\index{suffix?\tdescr{suffix}}%
+\label{syprog}
+\end{figure}
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{command} \rightarrow clip \descr{picture variable} to \descr{path expression}$\\
+$\tt \qquad \;|\; interim \descr{internal variable} \hbox{\tt :=} \descr{right-hand side}$\\
+$\tt \qquad \;|\; let \descr{symbolic token} \hbox{\tt =} \descr{symbolic token}$\\
+$\tt \qquad \;|\; newinternal \descr{symbolic token list}$\\
+$\tt \qquad \;|\; pickup \descr{expression}$\\
+$\tt \qquad \;|\; randomseed \hbox{\tt :=} \descr{numeric expression}$\\
+$\tt \qquad \;|\; save \descr{symbolic token list}$\\
+$\tt \qquad \;|\; setbounds \descr{picture variable} to \descr{path expression}$\\
+$\tt \qquad \;|\; shipout \descr{picture expression}$\\
+$\tt \qquad \;|\; special \descr{string expression}$\\
+$\tt \qquad \;|\; \descr{addto command}$\\
+$\tt \qquad \;|\; \descr{drawing command}$\\
+$\tt \qquad \;|\; \descr{font metric command}$\\
+$\tt \qquad \;|\; \descr{show command}$\\
+$\tt \qquad \;|\; \descr{tracing command}$\\
+$\tt $\\
+$\tt \descr{show command} \rightarrow show \descr{expression list}$\\
+$\tt \qquad \;|\; showvariable \descr{symbolic token list}$\\
+$\tt \qquad \;|\; showtoken \descr{symbolic token list}$\\
+$\tt \qquad \;|\; showdependencies$\\
+$\tt $\\
+$\tt \descr{symbolic token list} \rightarrow \descr{symbolic token}$\\
+$\tt \qquad \;|\; \descr{symbolic token}\hbox{\tt ,} \descr{symbolic token list}$\\
+$\tt \descr{expression list} \rightarrow \descr{expression} \;|\; \descr{expression list}\hbox{\tt ,} \descr{expression}$\\
+$\tt $\\
+$\tt \descr{addto command} \rightarrow$\\
+$\tt \qquad addto \descr{picture variable} also \descr{picture expression} \descr{option list}$\\
+$\tt \qquad \;|\; addto \descr{picture variable} contour \descr{path expression} \descr{option list}$\\
+$\tt \qquad \;|\; addto \descr{picture variable} doublepath \descr{path expression} \descr{option list}$\\
+$\tt \descr{option list} \rightarrow \descr{empty} \;|\; \descr{drawing option} \descr{option list}$\\
+$\tt \descr{drawing option} \rightarrow withcolor \descr{color expression}$\\
+$\tt \qquad \;|\; withpen \descr{pen expression} \;|\; dashed \descr{picture expression}$\\
+$\tt $\\
+$\tt \descr{drawing command} \rightarrow draw \descr{picture expression} \descr{option list}$\\
+$\tt \qquad \;|\; \descr{fill type} \descr{path expression} \descr{option list}$\\
+$\tt \descr{fill type} \rightarrow fill \;|\; draw \;|\; filldraw \;|\; unfill \;|\; undraw \;|\; unfilldraw$\\
+$\tt \qquad \;|\; drawarrow \;|\; drawdblarrow \;|\; cutdraw$\\
+$\tt $\\
+$\tt \descr{tracing command} \rightarrow tracingall \;|\; loggingall \;|\; tracingnone$
+\end{ctabbing}
+\caption{The syntax for commands}
+\index{option list?\tdescr{option list}}\index{picture variable?\tdescr{picture variable}}%
+\label{sycmds}
+\end{figure}
+
+\begin{figure}[htp]
+\begin{ctabbing}
+$\tt \descr{if test} \rightarrow if \descr{boolean expression} \hbox{\tt :} \descr{balanced tokens} \descr{alternatives} fi$\\
+$\tt \descr{alternatives} \rightarrow \descr{empty}$\\
+$\tt \qquad \;|\; else\hbox{\tt :} \descr{balanced tokens}$\\
+$\tt \qquad \;|\; elseif \descr{boolean expression} \hbox{\tt :} \descr{balanced tokens} \descr{alternatives}$\\
+$\tt $\\
+$\tt \descr{loop} \rightarrow \descr{loop header}\hbox{\tt :} \descr{loop text} endfor$\\
+$\tt \descr{loop header} \rightarrow for \descr{symbolic token} \hbox{\tt =} \descr{progression}$\\
+$\tt \qquad \;|\; for \descr{symbolic token} \hbox{\tt =} \descr{for list}$\\
+$\tt \qquad \;|\; forsuffixes \descr{symbolic token} \hbox{\tt =} \descr{suffix list}$\\
+$\tt \qquad \;|\; forever$\\
+$\tt \descr{progression} \rightarrow \descr{numeric expression} upto \descr{numeric expression}$\\
+$\tt \qquad \;|\; \descr{numeric expression} downto \descr{numeric expression}$\\
+$\tt \qquad \;|\; \descr{numeric expression} step \descr{numeric expression} until \descr{numeric expression} $\\
+$\tt \descr{for list} \rightarrow \descr{expression} \;|\; \descr{for list}\hbox{\tt ,} \descr{expression}$\\
+$\tt \descr{suffix list} \rightarrow \descr{suffix} \;|\; \descr{suffix list}\hbox{\tt ,} \descr{suffix}$
+\end{ctabbing}
+\caption{The syntax for conditionals and loops}
+\index{balanced tokens?\tdescr{balanced tokens}}\index{suffix?\tdescr{suffix}}%
+\label{sycondloop}
+\end{figure}
+
+\clearpage
+
+\let\topfraction=\svtopfrac % restore values from the start of this appendix
+\let\textfraction=\svtxtfrac
+\setcounter{topnumber}{\value{svtopnum}}
+\setcounter{totalnumber}{\value{svtotnum}}
+
+
+\section{MetaPost Versus METAFONT}
+\label{MPvsMF}
+
+Since the \MF\index{metafont?\MF} and MetaPost languages have so much in common, expert
+users of \MF\ will want to skip most of the explanations in this document and
+concentrate on concepts that are unique to MetaPost. The comparisons in this
+appendix are intended to help experts that are familiar with {\it The\ \MF book}
+as well as other users that want to benefit from Knuth's more detailed
+explanations \cite{kn:c}.
+
+Since \MF\ is intended for making \TeX\ fonts, it has a number of primitives for
+generating the {\tt tfm}\index{tfm file?{\tt tfm} file}\index{files!tfm?{\tt tfm}} files that
+\TeX\ needs for character dimensions, spacing information,
+ligatures\index{ligatures} and kerning\index{kerning}. MetaPost can also be
+used for generating fonts, and it also has \MF's primitives for making
+{\tt tfm} files. These are listed in Table~\ref{tfmprim}. Explanations can be
+found in the \MF\ documentation \cite{kn:c,kn:mf3}
+
+\begin{table}[htp]
+$$\begin{tabular}{|l|l|} \hline
+commands& {\tt charlist}, {\tt extensible},
+ {\tt fontdimen}, {\tt headerbyte} \\
+ & {\tt kern}, {\tt ligtable} \\ \hline
+ligtable operators& \verb!::!, \verb!=:!, \verb!=:|!, \verb!=:|>!,
+ \verb!|=:!, \verb!|=:>!, \\
+ & \verb!|=:|!, \verb!|=:|>!, \verb!|=:|>>!,
+ \verb!||:! \\ \hline
+internal variables\index{internal variables}\index{variables!internal}&
+ {\tt boundarychar}, {\tt chardp},
+ {\tt charext}, {\tt charht}, \\
+ & {\tt charic}, {\tt charwd},
+ {\tt designsize}, {\tt fontmaking} \\ \hline
+other operators& {\tt charexists} \\ \hline
+\end{tabular}
+$$
+\caption{MetaPost primitives for making {\tt tfm} files.}
+\label{tfmprim}
+\end{table}
+
+Even though MetaPost has the primitives for generating fonts, many of the
+font-making primitives and internal variables that are part of Plain
+\MF\index{metafont?\MF} are not defined in Plain MetaPost\index{Plain macros}. Instead,
+there is a separate macro package called {\tt mfplain}\index{mfplain?\texttt{mfplain}} that
+defines the macros required to allow MetaPost to process Knuth's Computer Modern
+fonts as shown in Table~\ref{mfponly} \cite{kn:e}.
+To load these macros, put ``\verb|&mfplain|'' before the name of the
+input file. This can be done at the {\tt **} prompt after invoking the MetaPost
+interpreter with no arguments, or on a command line that looks something like
+this:\footnote{Command line syntax is system dependent. Quotes are needed on
+most Unix\reg systems to protect special characters like {\tt\&}.}
+$$ \hbox{\verb|mp '&mfplain' cmr10|} $$
+The analog of a \MF\ command line like
+$$ \hbox{\verb|mf '\mode=lowres; mag=1.2; input cmr10'|} $$
+is
+$$ \hbox{\verb|mp '&mfplain \mode=lowres; mag=1.2; input cmr10'|} $$
+The result is a set of PostScript files, one for each character in the font.
+Some editing would be required in order to merge them into a downloadable Type~3
+PostScript font \cite{ad:red}.
+
+\begin{table}[htp]
+$$
+\renewcommand{\FancyVerbFormatLine}[1]{\hbox{#1}\strut}
+\begin{tabular}{|l|} \hline
+\multicolumn 1{|c|}
+{Defined in the {\tt mfplain} package} \\ \hline
+\begin{verbatim}
+beginchar font_identifier
+blacker font_normal_shrink
+capsule_def font_normal_space
+change_width font_normal_stretch
+define_blacker_pixels font_quad
+define_corrected_pixels font_size
+define_good_x_pixels font_slant
+define_good_y_pixels font_x_height
+define_horizontal_corrected_pixels italcorr
+define_pixels labelfont
+define_whole_blacker_pixels makebox
+define_whole_pixels makegrid
+define_whole_vertical_blacker_pixels maketicks
+define_whole_vertical_pixels mode_def
+endchar mode_setup
+extra_beginchar o_correction
+extra_endchar proofrule
+extra_setup proofrulethickness
+font_coding_scheme rulepen
+font_extra_space smode
+\end{verbatim}
+ \\ \hline
+\multicolumn 1{|c|}
+{Defined as no-ops in the {\tt mfplain} package}\\ \hline
+\begin{verbatim}
+cullit proofoffset
+currenttransform screenchars
+gfcorners screenrule
+grayfont screenstrokes
+hround showit
+imagerules slantfont
+lowres_fix titlefont
+nodisplays unitpixel
+notransforms vround
+openit
+\end{verbatim}
+ \\ \hline
+\end{tabular}
+\renewcommand{\FancyVerbFormatLine}[1]{#1}
+$$
+\caption{Macros and internal variables defined only in the {\tt mfplain} package.}
+\label{mfponly}
+\end{table}
+
+Another limitation of the {\tt mfplain} package is that certain internal
+variables from Plain \MF\index{metafont?\MF} cannot be given reasonable MetaPost
+definitions. These include {\tt displaying}, {\tt currentwindow},
+\verb|screen_rows|, and \verb|screen_cols| which depend on \MF's ability to
+display images on the computer screen. In addition, \verb|pixels_per_inch| is
+irrelevant since MetaPost uses fixed units of PostScript points.
+
+The reason why some macros and
+internal variables\index{internal variables}\index{variables!internal}
+are not meaningful in MetaPost
+is that \MF\ primitive commands {\tt cull}, {\tt display}, {\tt openwindow},
+{\tt numspecial} and {\tt totalweight} are not implemented in MetaPost. Also not
+implemented are a number of internal variables as well as the
+\tdescr{drawing option} {\tt withweight}. Here is a complete listing of the
+internal variables whose primitive meanings in \MF\ do not make sense in MetaPost:
+$$\begin{verbatim}
+autorounding fillin proofing tracingpens xoffset
+chardx granularity smoothing turningcheck yoffset
+chardy hppp tracingedges vppp
+\end{verbatim}
+$$
+
+There is also one \MF\ primitive that has a slightly different meaning in
+MetaPost. Both languages allow statements of the
+form\index{special?\texttt{special}}\label{Dspecl}
+$$ {\tt special}\, \descr{string expression} \hbox{\tt;} $$
+but \MF\ copies the string into its ``generic font'' output file, while
+MetaPost interprets the string as a sequence of PostScript commands that are
+to be placed at the beginning of the next output file.
+
+All the other differences between \MF\ and MetaPost are features found only in
+MetaPost. These are listed in Table~\ref{mponly}. The only commands listed
+in this table that the preceding sections do not discuss are
+\verb|extra_beginfig|\index{extra_beginfig?\texttt{extra\_beginfig}}\label{Dxbfig},
+\verb|extra_endfig|\index{extra_endfig?\texttt{extra\_endfig}}\label{Dxefig}, and {\tt mpxbreak}.
+The first two are strings that contain extra commands to be processed
+by {\tt beginfig}\index{beginfig?\texttt{beginfig}} and {\tt endfig}\index{endfig?\texttt{endfig}}
+just as \verb|extra_beginchar| and \verb|extra_endchar| are processed by
+{\tt beginchar} and {\tt endchar}.
+(The file {\tt boxes.mp}\index{boxes.mp?\texttt{boxes.mp}} uses these features).
+
+The other new feature listed in Table~\ref{mponly} not listed in the index
+is {\tt mpxbreak}\index{mpxbreak?\texttt{mpxbreak}}. This is used to separate blocks of
+translated \TeX\index{TeX?\TeX} or troff\index{troff} commands in
+{\tt mpx}\index{files!mpx?{\tt mpx}} files. It should be of no concern to
+users since {\tt mpx} files are generated automatically.
+
+\begin{table}[htp]
+$$
+\renewcommand{\FancyVerbFormatLine}[1]{\hbox{#1}\strut}
+\begin{tabular}{|l|} \hline
+\multicolumn 1{|c|}
+{MetaPost primitives not found in \MF} \\ \hline
+$\begin{verbatim}
+bluepart infont redpart
+btex linecap setbounds
+clip linejoin tracinglostchars
+color llcorner truecorners
+dashed lrcorner ulcorner
+etex miterlimit urcorner
+fontsize mpxbreak verbatimtex
+greenpart prologues withcolor
+\end{verbatim}
+$ \\ \hline
+\multicolumn 1{|c|}
+{Variables and Macros defined only in Plain MetaPost}\\ \hline
+$\begin{verbatim}
+ahangle cutbefore extra_beginfig
+ahlength cuttings extra_endfig
+background dashpattern green
+bbox defaultfont label
+bboxmargin defaultpen labeloffset
+beginfig defaultscale mitered
+beveled dotlabel red
+black dotlabels rounded
+blue drawarrow squared
+buildcycle drawdblarrow thelabel
+butt drawoptions white
+center endfig
+cutafter evenly
+\end{verbatim}
+$ \\ \hline
+\end{tabular}
+\renewcommand{\FancyVerbFormatLine}[1]{#1}
+$$
+\caption{Macros and internal variables defined in MetaPost but not \MF.}
+\label{mponly}
+\end{table}
+
+
+
+
+\bibliographystyle{plain}
+\bibliography{mpman}
+
+
+\printindex
+
+
+\end{document}
+
+% Copyright 1990 - 1995 by AT&T Bell Laboratories.
+
+% Permission to use, copy, modify, and distribute this software
+% and its documentation for any purpose and without fee is hereby
+% granted, provided that the above copyright notice appear in all
+% copies and that both that the copyright notice and this
+% permission notice and warranty disclaimer appear in supporting
+% documentation, and that the names of AT&T Bell Laboratories or
+% any of its entities not be used in advertising or publicity
+% pertaining to distribution of the software without specific,
+% written prior permission.
+
+% AT&T disclaims all warranties with regard to this software,
+% including all implied warranties of merchantability and fitness.
+% In no event shall AT&T be liable for any special, indirect or
+% consequential damages or any damages whatsoever resulting from
+% loss of use, data or profits, whether in an action of contract,
+% negligence or other tortious action, arising out of or in
+% connection with the use or performance of this software.
+
+% In addition, John Hobby, the original author of MetaPost and this
+% manual, makes the following requests:
+% - I request that it remain clear that I am the author of
+% "A User's Manual for MetaPost" and "Drawing Graphs with MetaPost".
+% - I request to be consulted before significant changes are made.