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authorNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
committerNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
commite0c6872cf40896c7be36b11dcc744620f10adf1d (patch)
tree60335e10d2f4354b0674ec22d7b53f0f8abee672 /graphics/mfpic/source
Initial commit
Diffstat (limited to 'graphics/mfpic/source')
-rw-r--r--graphics/mfpic/source/grafbase.dtx9710
-rw-r--r--graphics/mfpic/source/mfpic.dtx6735
-rw-r--r--graphics/mfpic/source/mfpic.ins59
3 files changed, 16504 insertions, 0 deletions
diff --git a/graphics/mfpic/source/grafbase.dtx b/graphics/mfpic/source/grafbase.dtx
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+% \iffalse
+% File: grafbase.dtx
+% A part of mfpic 1.10 2012/12/03
+%
+% -------------------------------------------------------------------
+%
+% Copyright 2002--2012, Daniel H. Luecking
+%
+% Mfpic may be distributed and/or modified under the conditions of the
+% LaTeX Project Public License, either version 1.3b of this license or (at
+% your option) any later version. The latest version of this license is in
+% <http://www.latex-project.org/lppl.txt>
+% and version 1.3c or later is part of all distributions of LaTeX version
+% 2008/12/01 or later.
+%
+% Mfpic has maintenance status "author-maintained". The Current Maintainer
+% is Daniel H. Luecking. There are several Base Interpreters: plain TeX, LaTeX,
+% plain Metafont and plain MetaPost.
+%
+%<*driver>
+\ProvidesFile{grafbase.dtx}
+ [2012/12/03 v1.10. Metafont/post macros to interface with mfpic.]%
+\documentclass{ltxdoc}
+\usepackage{docmfp}
+
+\addtolength{\textwidth}{.5878pt}
+
+\def\mytt{\upshape\mdseries\ttfamily}
+\renewcommand\marg[1]{{\mytt \{#1\}}}
+\renewcommand\oarg[1]{{\mytt [#1]}}
+\renewcommand\parg[1]{{\mytt (#1)}}
+\renewcommand{\meta}[1]{{$\langle$\rmfamily\itshape#1$\rangle$}}
+\DeclareRobustCommand\cs[1]{{\mytt\char`\\#1}}
+\def\prog#1{{\mdseries\scshape #1}}
+\def\grafbase{\prog{grafbase}}
+\def\Grafbase{\prog{Grafbase}}
+\def\mfpic{\prog{mfpic}}
+\def\Mfpic{\prog{Mfpic}}
+\def\MF{\prog{meta\-font}}
+\def\MP{\prog{meta\-post}}
+\def\PS{\prog{Post\-Script}}
+\def\CMF{\prog{Meta\-font}}
+\def\CMP{\prog{Meta\-post}}
+\def\opt#1{{\sffamily\upshape#1}}
+\def\mfc#1{{\mytt#1}}
+\let\env\mfc
+\let\file\mfc
+\let\gbc\mfc
+\renewcommand\{{{\mytt\char`\{}}
+\renewcommand\}{{\mytt\char`\}}}
+\renewcommand\|{${}\mathrel{|}{}$}
+
+\makeatletter
+\let\HD@SetMacroIndent\@gobble
+\newcommand\bsl{{\mytt\@backslashchar}}
+% Stupid lists!
+\def\@listi{\leftmargin\leftmargini
+ \parsep \z@ \@plus\p@ \@minus\z@
+ \topsep 4\p@ \@plus\p@ \@minus2\p@
+ \itemsep\parsep}
+\let\@listI\@listi \@listi
+\renewcommand\labelitemi{\normalfont\bfseries \textendash}
+\renewcommand\labelitemii{\textasteriskcentered}
+\renewcommand\labelitemiii{\textperiodcentered}
+\leftmargini\parindent
+% Stupid index!
+\def\usage#1{\textrm{#1}}
+\def\index@prologue{\section*{Index}\markboth{Index}{Index}}
+\def\IndexParms{%
+ \parindent \z@ \columnsep 15pt
+ \parskip 0pt plus 1pt
+ \rightskip 5pt plus2em \mathsurround \z@
+ \parfillskip=-5pt \small
+ % less hanging:
+ \def\@idxitem{\par\hangindent 20pt}%
+ \def\subitem{\@idxitem\hspace*{15pt}}%
+ \def\subsubitem{\@idxitem\hspace*{25pt}}%
+ \def\indexspace{\par\vspace{10pt plus 2pt minus 3pt}}}
+\renewcommand\routinestring{}
+\renewcommand\variablestring{\space(var.)}
+% Why does every command have to be indexed twice?
+\renewcommand\SpecialMfpIndex[3]{\@bsphack
+ \index{%
+ \string#1\actualchar
+ \string\verb\quotechar*\verbatimchar\string#1\verbatimchar
+ #2 \encapchar usage}%
+ \@esphack}
+\def\close@crossref{\SpecialEscapechar{:}}
+\makeatother
+\def\VariableIndex#1{\SpecialMfpIndex{#1}{\variablestring}{}}
+\def\RoutineIndex #1{\SpecialMfpIndex{#1}{}{}}
+
+\def\pdfTeX{\textrm{pdf\kern.04em\TeX}}
+\def\pdfLaTeX{\textrm{pdf\kern.06em\LaTeX}}
+\def\ConTeXt{\textrm{Con\kern-.16em\TeX\kern-0.06em t}}
+\def\PiCTeX{\textrm{P\kern-.13em\lower.3ex\hbox{I}C\TeX}}
+
+\title{The \grafbase{} macros\thanks{This file has version number
+ \fileversion, last revised \filedate. The code described here
+ was developed by several people, notably Thomas Leathrum,
+ Geoffrey Tobin and Dan Luecking. Dan wrote this documentation.}}
+\author{Dan Luecking}
+\date{\filedate}
+\SpecialEscapechar{:}
+\def\bslash{:}
+\DisableCrossrefs
+\CodelineIndex
+\AlsoImplementation
+
+\begin{document}
+ \DeleteShortVerb{\|}
+ \DocInput{grafbase.dtx}
+\end{document}
+%</driver>
+%\fi
+%
+% \CheckSum{1631}
+% \CharacterTable
+% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+% Digits \0\1\2\3\4\5\6\7\8\9
+% Exclamation \! Double quote \" Hash (number) \#
+% Dollar \$ Percent \% Ampersand \&
+% Acute accent \' Left paren \( Right paren \)
+% Asterisk \* Plus \+ Comma \,
+% Minus \- Point \. Solidus \/
+% Colon \: Semicolon \; Less than \<
+% Equals \= Greater than \> Question mark \?
+% Commercial at \@ Left bracket \[ Backslash \\
+% Right bracket \] Circumflex \^ Underscore \_
+% Grave accent \` Left brace \{ Vertical bar \|
+% Right brace \} Tilde \~}
+%
+% \catcode`\_=12
+% \GetFileInfo{grafbase.dtx}
+% \maketitle
+%
+% \begin{abstract}
+% Macros are defined for use with \mfpic{}. The latter is a set of \TeX{}
+% macros which allows a \file{.tex} file to write a \file{.mf} or
+% \file{.mp} file that, with the help of these macros and \MF{} (or \MP),
+% can be used to create pictures in the document, especially mathematical
+% pictures. There are two versions of \grafbase, one for \MF{} and one for
+% \MP{}. As they are more alike than different (95\% of the code is
+% identical), we document both here.
+%
+% This file documents the \grafbase{} source code. The user manual for
+% \mfpic{} is distributed as \file{mfpic-doc.pdf}, produced from
+% \file{mfpic-doc.tex}. An introductory guide to \mfpic{} is available
+% in \file{mfpguide.pdf}, produced from \file{mfpguide.tex}
+% \end{abstract}
+%
+% \StopEventually{\PrintIndex}
+% \tableofcontents
+%
+%
+%
+% \section{Introduction}\label{intro}
+%
+%
+% \subsection{Identification and checks}\label{checks}
+%
+% \DescribeVariable{grafbaseversion} We use \mfc{grafbaseversion} to
+% check if \prog{grafbase} has been previously loaded, later we use it
+% to check a mismatch with the version of \mfpic{} (if used).
+% \gbc{grafbase} was used in previous versions. If either is known, we
+% bail out. The \gbc{grafbase} boolean is really never needed, but it
+% has been around since I took over. It is possible to write files that
+% change behavior when \grafbase{} is loaded, however if they say
+% ``\gbc{if grafbase:}, they can only be called with \gbc{grafbase} a
+% known boolean.
+%
+% \VariableIndex{fileversion} \gbc{fileversion} and
+% \VariableIndex{filedate} \gbc{filedate} provide identifying information.
+%
+% \DescribeRoutine{GBmsg}
+% These are used fairly consistently and identify the source of the
+% message delivered as being `\gbc{Grafbase}'.
+% \DescribeRoutine{GBwarn}
+% Warnings are delivered by \gbc{GBwarn}. The macro \gbc{GBerrmsg} takes
+% care of both the
+% \DescribeRoutine{GBerrmsg}
+% error message and the \mfc{errhelp} string.
+% \begin{macrocode}
+%<*MF|MP>
+if (known grafbaseversion) or (known grafbase):
+ message "Grafbase (" & jobname & "): You have loaded grafbase more "
+ & "than once! Please make sure that it is loaded only once.";
+ endinput;
+fi
+boolean grafbase; grafbase := true;
+
+string fileversion, filedate;
+fileversion := "1.10"; filedate := "2012/12/03";
+
+message " Loading grafbase macros, version " & fileversion & ", " &
+ filedate & ".";
+message " ";
+
+def GBmsg expr s = message "Grafbase (" & jobname & "): " & s;
+enddef;
+def GBwarn expr s = GBmsg "Warning, " & s; enddef;
+def GBerrmsg (expr s) expr t = errhelp t;
+ errmessage "Grafbase (" & jobname & "): " & s; errhelp "";
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeVariable{MFPIC}
+% The \gbc{MFPIC} variable is not yet used at all.
+% It might be possible (at some point) to optimize things for \mfpic{} use
+% whenever \gbc{mfpicversion} is defined, but so far we don't do anything
+% except test the version and set this boolean.
+%
+% \DescribeVariable{mfpicversion}
+% The output file written by the \mfpic{} macros includes a test that
+% \gbc{mfpicversion} and \mfc{grafbaseversion} are the same,
+% but that would fail to catch a new \grafbase{} with an old \mfpic. So
+% we also put a test here with
+% \DescribeRoutine{checkversions}
+% \gbc{checkversions}, though it alone would fail to catch the use
+% of older versions of \grafbase{} with current versions of \mfpic. Newer
+% versions of \mfpic{} signal their version before inputting
+% \file{grafbase}. Unfortunately (for error checking), \grafbase{} can
+% also be used without \mfpic, so if \gbc{mfpicversion} is unknown, we
+% merely write a message.
+% \begin{macrocode}
+boolean MFPIC; MFPIC := false;
+
+def checkversions (expr g)=
+ numeric grafbaseversion; grafbaseversion := g;
+ if unknown mfpicversion: % no mfpic, or < 0.63
+ GBmsg "Recent mfpic not detected.";
+ elseif g = mfpicversion:
+ MFPIC := true;
+ else:
+ message "";
+ GBwarn "Version mismatch: "
+ & "mfpic and grafbase versions do not match.";
+ message "";
+ fi
+enddef;
+
+checkversions (110);
+
+% \end{macrocode}
+%
+% We try to make sure that the macros of \file{plain.mf} or
+% \file{plain.mp} (where \mfc{base_name} is defined to be \mfc{"plain"})
+% are available.
+% \begin{macrocode}
+if unknown base_name : input plain;
+elseif not string base_name: input plain;
+elseif base_name <> "plain": input plain;
+fi
+
+% \end{macrocode}
+%
+% \DescribeRoutine{GBdebug}
+% The \gbc{debug} flag is for developers, who should set it before
+% inputing \file{grafbase}.
+% \DescribeRoutine{GBenddebug}
+% These two routines start and end debug messages.
+%
+% \DescribeRoutine{mftitle}
+% The \gbc{mftitle} macro is useful when debugging.
+% It will put its argument, which should be a string, as a TFM comment,
+% and also print it to the terminal and log file.
+% \begin{macrocode}
+if not boolean debug: boolean debug; debug := false; fi
+
+def GBdebug =
+ begingroup
+ save >>; def >> = message " " & enddef;
+ message "Grafbase DEBUG";
+enddef;
+def GBenddebug =
+ message "End DEBUG";
+ >> "";
+ endgroup
+enddef;
+
+vardef mftitle expr t =
+ if string t: t; message t; fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeVariable{METAPOST}
+% We try to determine which of \MF{} or \MP{} is using these macros.
+% Perhaps one day we'll merge both versions of \grafbase{} in one file and
+% use the following boolean. For now, we only use it to catch cases where
+% the \grafbase{} file is being used by the wrong compiler.
+%
+% Of course, \MP{} natively knows about colors but \MF{} doesn't, so we
+% use that to set a boolean \gbc{METAPOST}. We don't simply check if
+% `\mfc{blue}' is \mfc{known} because `\mfc{blue}' is certainly a legal
+% variable name in \MF. Instead we check `\mfc{known color X}' for
+% some unlikely \gbc{X}. In \MP, `\gbc{color X}' is either true or
+% false (\gbc{X} is a color or it isn't) and therefore always known, so
+% `\gbc{known color X}' is always true.
+%
+% In \MF{} `\mfc{color X}' is an identifier (presumably unknown) with the
+% base name \mfc{color} and suffix \mfc{X}.
+% \begin{macrocode}
+boolean METAFONT, METAPOST;
+METAPOST := known color Carl Philipp Emanuel Bach;
+if METAPOST: METAFONT := false; else: METAFONT := true; fi
+
+%<*MF>
+if METAPOST:
+ GBerrmsg ("wrong compiler.")
+ "This file is for Metafont. For Metapost, use grafbase.mp.";
+fi
+%</MF>
+
+%<*MP>
+if METAFONT:
+ GBerrmsg ("wrong compiler.")
+ "This file is for Metapost. For Metafont, use grafbase.mf.";
+fi
+
+% \end{macrocode}
+%
+% \MP{} now exists in a couple of slightly incompatible versions.
+% Versions 1.000 and later (beta versions 0.9xx also) have native support for
+% \texttt{CMYK} colors with a \mfc{cmykcolor} data type. They also
+% support grayscale colors (i.e., \mfc{withcolor} will accept a numeric
+% expression), and have the alias \mfc{rgbcolor} for \mfc{color}.
+% It also has a means to set the name of the output file: the
+% \mfc{filenametemplate} command. In versions 1.2 and later, this is
+% deprecated in favor of setting the internal string variable
+% \mfc{outputtemplate}.
+% \begin{macrocode}
+numeric metapostversion;
+boolean has_cmyk;
+boolean has_filenametemplate; % e.g., filenametemplate "%j.%n;
+boolean has_outputtemplate; % e.g., outputtemplate := "%j.%n;
+if unknown mpversion:
+ % prior to introduction of cmykcolor and output filename templates.
+ % Latest such version was 0.641.
+ metapostversion := 0.641;
+ has_cmyk := false;
+ has_filenametemplate := false;
+ has_outputtemplate := false;
+else:
+ metapostversion := scantokens (mpversion);
+ has_cmyk := true;
+ has_filenametemplate := true;
+ if metapostversion < 1.200:
+ has_outputtemplate := false;
+ else:
+ has_outputtemplate := true;
+ fi
+fi
+
+%</MP>
+% \end{macrocode}
+%
+%
+% \subsection{Setting up the font, \MF{} only}\label{font}
+%
+% Font-related housekeeping is only for \MF{}. \MF{} only produces
+% fonts, so we have to define the variables it thinks are needed for
+% fonts.
+%
+% \DescribeVariable{GBgeneric}
+% We intercept the \mfc{mode} variable before \mfc{mode_setup} can set
+% \mfc{proof} mode. We used to set \mfc{mode := cx} (and later
+% \mfc{ljfour}) if \mfc{mode} was unknown. For a while we just issued an
+% error message. In this version we define a 600dpi mode called
+% \mfc{GBgeneric} as a fallback (neither \gbc{mode} nor \gbc{localfont}
+% known).
+%
+% The font identifier and coding scheme are just for information and end
+% up as comments in the \file{.tfm} file (in all capitals). The design
+% size just needs to be rather large for graphics, and \mfc{128pt\#} is
+% anyway the default if we didn't set it ourselves.
+% \begin{macrocode}
+%<*MF>
+if unknown mode:
+ GBerrmsg ("Metafont mode is unknown.")
+ "Set mode to a known mode, perhaps ljfour. "
+ & "If you proceed, localfont will be tried. "
+ & "If that is unknown, a generic mode will be tried.";
+ if known localfont: mode := localfont;
+ else:
+ if unknown GBresolution: GBresolution := 600 fi;
+ mode_def GBgeneric =
+ mode_param (pixels_per_inch, GBresolution);
+ mode_param (blacker, 0);
+ mode_param (fillin, 0);
+ mode_param (o_correction, 1);
+ mode_common_setup_;
+ enddef;
+ mode := GBgeneric;
+ fi
+fi
+mode_setup;
+if debug:
+ GBdebug;
+ >> "pixels_per_inch = " & decimal pixels_per_inch;
+ GBenddebug;
+fi
+
+font_identifier := "MFpic graphics";
+font_coding_scheme := "Arbitrary";
+interim designsize := 128pt#;
+
+%</MF>
+% \end{macrocode}
+%
+% \Mfpic-generated files make reference to \mfc{aspect_ratio} and
+% \mfc{pt\#}, while \MP{} has no need for them. Rather than make
+% \mfpic{} write different things, and to make the files intended for
+% \MF{} also work with \MP, we define them in the obvious way. We also
+% add a definition of \mfc{hppp} and \gbc{t_} to simplify maintenance of
+% two versions of the \file{grafbase} files. Then we define
+% \gbc{currenttransform} for \MP{} sake.
+% \begin{macrocode}
+%<MP>pt# := pt; bp# := bp;
+%<MP>def t_ = transformed currenttransform enddef;
+if unknown aspect_ratio: aspect_ratio := 1; fi
+if unknown hppp: hppp := 1 fi;
+if unknown currenttransform:
+ transform currenttransform;
+ currenttransform := identity yscaled aspect_ratio;
+fi
+
+% \end{macrocode}
+%
+% Don't complain when variables get too large. For \MF{} this \emph{must}
+% be after \mfc{mode_setup}. Also don't complain if a clockwise path is
+% filled (only \MF{} worries about this).
+% \begin{macrocode}
+interim warningcheck := 0;
+%<MF>interim turningcheck := 0;
+
+% \end{macrocode}
+%
+%
+% \subsection{Initializations}\label{init}
+%
+% \VariableIndex{unitlen}
+% \VariableIndex{xscale}
+% \VariableIndex{yscale}
+% \VariableIndex{xneg}
+% \VariableIndex{xpos}
+% \VariableIndex{yneg}
+% \VariableIndex{ypos}
+% The following are the various variables determining the extent of a
+% picture. These variables would normally be set by a user for each
+% picture, or by \mfpic, but we give them default values anyway. They
+% give a nominal picture size of one inch with a graph unit corresponding
+% to $1/10$ inch.
+% \begin{macrocode}
+numeric unitlen, xscale, yscale, xneg, xpos, yneg, ypos;
+
+unitlen := 1 bp#;
+xscale := 7.2;
+yscale := 7.2;
+xneg := 0; xpos := 10;
+yneg := 0; ypos := 10;
+
+% \end{macrocode}
+%
+% \DescribeVariable{deg}
+% \VariableIndex{degree}
+% We support both degrees and radians for angles. In \MF, one degree is
+% the unit of angle.
+% \DescribeVariable{radian}
+% One radian is $180/\pi$ degrees. We also define \gbc{pi} so a user can
+% say \gbc{pi/2*radian} for almost the same effect as \gbc{90deg}.
+% \DescribeVariable{pi}
+% But not quite: because of \MF{}'s precision limits, the former is about
+% 90.00025 degrees. \MF{}'s precision is 16 binary places, or slightly
+% under 5 decimals. The accuracy of \gbc{pi} and \gbc{radian} below is the
+% maximum possible. If we \emph{define} \gbc{radian} by its mathematical
+% definition \gbc{radian:=180/pi}, then \gbc{radian} and formulas
+% containing it are even less accurate. (Coincidentally, defining
+% \gbc{radian} as below, and then \gbc{pi := 180/radian} produces exactly
+% the same value for \gbc{pi} as below.)
+% \begin{macrocode}
+newinternal deg, pi, radian;
+deg := 1; pi := 3.14159;
+radian := 57.29578;
+numeric degree; degree := deg;
+
+% \end{macrocode}
+%
+% \VariableIndex{drawpen}
+% \VariableIndex{penwd}
+% \DescribeRoutine{resizedrawpen}
+% Since we need to do this frequently, we define a macro that changes the
+% pen width for subsequent drawing. This enables the file written by
+% \mfpic{} to be less cluttered. At least that was the original reason.
+% Now it gives us the opportunity to localize changes to \mfc{currentpen}
+% and \gbc{drawpen}. (We already had this for different
+% \gbc{beginmfpic}, since that reinitializes drawpen, but now it is local
+% to other groups as well.)
+%
+% \VariableIndex{hatchpen}
+% We could do this for the hatching pen, but it doesn't seem to change as
+% often. The \mfc{pickup} command performs \mfc{yscaled aspect_ratio}, but
+% so does the \gbc{shpath}, the only other place pens are required. In
+% fact, we wouldn't need to \mfc{pickup} the pen at all, except power
+% users may want to rely on \gbc{drawpen} always being the current pen. We
+% make its diameter \mfc{.5pt} for backward compatibility. But many
+% journal publisher (e.g., AMS) recommend no smaller than \mfc{.5bp} for
+% author-supplied drawings.
+%
+% \VariableIndex{hatchwd}
+% The default \gbc{hatchwd} used to be larger, but it seemed ugly to me.
+% (Backward compatibility---what's that?).
+% \begin{macrocode}
+newinternal penwd; penwd := 0.5pt;
+pen drawpen;
+
+def resizedrawpen (expr s) =
+ interim penwd := s;
+ setvariable (pen) (drawpen) pencircle scaled penwd;
+ save currentpen; pen currentpen; pickup drawpen;
+enddef;
+
+numeric hatchwd; hatchwd := 0.5bp;
+pen hatchpen; hatchpen := pencircle scaled hatchwd;
+
+% \end{macrocode}
+%
+% \DescribeVariable{clipall}
+% We have two booleans related to clipping. One, \gbc{clipall} is meant to
+% be turned on just once (per picture), and it causes the \gbc{endmfpic}
+% code to clip the current picture to the boundaries defined by the
+% picture size variables. The other,
+% \DescribeVariable{ClipOn}
+% \gbc{ClipOn}, is meant to be turned on and off. While on, most drawing
+% macros (all?) will clip their result to the current \emph{clipping path
+% array}.
+% \DescribeVariable{ClipPath}
+% The clipping path array is an array of paths: \gbc{ClipPath[\,]} together
+% with a numeric \gbc{ClipPath}. The numeric variable contains the number
+% of clipping paths; the paths are \gbc{ClipPath[1]} through
+% \gbc{ClipPath[ClipPath]}. A macro later on is defined to loop through
+% the array, clipping the current picture to the union of their interiors.
+%
+% \DescribeVariable{truebbox}
+% The \gbc{truebbox} boolean sets the bounding box of the picture to its
+% natural size in \MP. The default behavior of \MP{} is to output a
+% bounding box that is the natural size of the graphic. The \grafbase{}
+% default is to override this default, setting \gbc{truebbox} to
+% \mfc{false}. \CMF's default behavior is to force the user to specify the
+% bounding box, and provides no natural way to obtain any information
+% about the actual extent of the ink. So, for now, this boolean is only
+% for \MP.
+%
+% \DescribeRoutine{DoClip}
+% This is for the frequent conditional code to implement \gbc{ClipOn}.
+% The command \gbc{clipsto} is defined later.
+%
+% \DescribeRoutine{noclip}
+% For debugging we sometimes want to make sure something is drawn
+% without clipping being applied. For this we have \gbc{noclip}.
+% \begin{macrocode}
+boolean clipall; clipall := false;
+boolean ClipOn; ClipOn := false;
+path ClipPath[]; numeric ClipPath; ClipPath = 0;
+boolean truebbox; truebbox := false;
+
+def DoClip (suffix v) =
+ if ClipOn and (ClipPath > 0): clipsto (v, ClipPath); fi
+enddef;
+
+def noclip (text t) =
+ hide ( setboolean (ClipOn) false; t)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeVariable{showbbox}
+% The boolean \gbc{showbbox} is for debugging the \gbc{*bbox} macros.
+% \begin{macrocode}
+boolean showbbox; showbbox := false;
+
+% \end{macrocode}
+%
+%
+% \subsubsection{Colors}\label{colors}
+%
+% Of course colors are only recognized by \MP. The colors \mfc{black},
+% \mfc{white}, \mfc{red}, \mfc{green} and \mfc{blue} are part of
+% \file{plain.mp}. We define other standard colors to get all eight
+% colors where the coordinates are 0 or 1.
+%
+% \DescribeRoutine{color}
+% We define \MF{} replacements for some of the \MP{} color variables and
+% macros. Our point of view will be: make each color variable a numeric in
+% \MF{}. Each will lie between $0$ and $1$ representing shades of gray.
+% For \emph{drawing} commands we will only distinguish between nonwhite
+% (black, ${}<1$) and white (${}\ge1$). For filling commands we will allow
+% levels in between, and fill with an approximation using a version of
+% the \gbc{shade} macro.
+% \begin{macrocode}
+%<*MF>
+let color = numeric; color black, white;
+let rgbcolor = numeric;
+let cmykcolor = numeric;
+black := 0; white := 1;
+def withcolor text t = enddef;
+%</MF>
+%<MP>if not has_cmyk: let rgbcolor = color; let cmykcolor = color; fi
+def _wc_ = withcolor enddef;
+
+% \end{macrocode}
+%
+% \VariableIndex{currentcolor}
+% \VariableIndex{drawcolor}
+% \VariableIndex{fillcolor}
+% \VariableIndex{hatchcolor}
+% \VariableIndex{headcolor}
+% \VariableIndex{pointcolor}
+% \VariableIndex{tlabelcolor}
+% We also define some color variables whose names reflect their use.
+% Thus, \gbc{fillcolor} is used for filling, etc. The color
+% \gbc{currentcolor} isn't used anywhere yet. The color
+% \mfc{background} is used in \MP{} for unfilling a region.
+% \begin{macrocode}
+color currentcolor, drawcolor, fillcolor, hatchcolor,
+ headcolor, pointcolor, tlabelcolor, background;
+currentcolor := fillcolor := drawcolor := hatchcolor :=
+ headcolor := pointcolor := tlabelcolor := black;
+background := white;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{snapto}
+% The \gbc{snapto} macro truncates numerics to the $[0,1]$ range, but also
+% returns a value ($0$) for unknown input. It used to do the same for
+% nonnumeric input, but that should be an error. It would have made at
+% least one of our bugs easier to find if it had produced an error message
+% back then.
+% \begin{macrocode}
+vardef snapto expr t =
+ if numeric t:
+ if unknown t: 0
+ elseif t < 0: 0
+ elseif t > 1: 1
+ else: t
+ fi
+ else:
+ GBerrmsg ("Improper expression type.")
+ "The argument to `snapto' must be a numeric.";
+ fi
+
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{cmykgray}
+% \DescribeRoutine{rgbgray}
+% Now we deal with all the color functions and utilities that enable
+% \mfpic{} users to use colors without knowing what they are doing.
+% Since colors now come in three flavors, we start with gray levels in
+% the three models. In early \MP{}, the \gbc{cmyk} function will return
+% an \opt{rgb} color, so there is will be no difference between these
+% two. In \MF{} \gbc{white} is a numeric and \gbc{cmyk} returns a
+% numeric, so these can be used with \MF{}, and both produce the same
+% result.
+% \begin{macrocode}
+vardef rgbgray (expr g) = (snapto g) * white enddef;
+vardef cmykgray (expr g) = cmyk(0,0,0,1 - snapto g) enddef;
+
+% \end{macrocode}
+%
+%
+% \DescribeRoutine{colorchoice}
+% The \gbc{colorchoice} function (like \cs{mathchoice} in \TeX{}, after
+% which it was named) returns one of four bits of code: \gbc{D} (default)
+% if the first argument is unknown or not one of the recognized color
+% models, \gbc{N} if it is numeric, \gbc{R} if it is \mfc{rgbcolor}, and
+% \gbc{C} if it is \mfc{cmykcolor}. These arguments have to be \mfc{text}:
+% if they were `\mfc{expr}' \MP{} would try to evaluate them, with
+% possible errors since some of them apply functions that are not relevant
+% to the other types.
+%
+% Since this is mostly used to return values inside vardef's, it is
+% important this not be followed by a semicolon. If it is used in
+% another context, semicolons would normally be part of the arguments.
+% \begin{macrocode}
+%<*MP>
+def colorchoice (expr clr) (text D)(text N)(text R)(text C) =
+ if unknown clr: D
+ elseif numeric clr: N
+ elseif rgbcolor clr: R
+ elseif cmykcolor clr: C
+ else: D
+ fi
+enddef;
+
+% \end{macrocode}
+% In recent \MP{}, all the color functions are essentially no-ops. In
+% early \MP{}, they all return an \opt{rgb} color expression. In \MF{}
+% they all return a numeric. It is easiest if we simply separate the three
+% cases (MF, old MP and recent MP) and write the code for each, rather
+% than load all the functions with three-way booleans (often containing
+% nested booleans).
+%
+% \DescribeRoutine{gray}
+% \DescribeRoutine{cmyk}
+% For all three engines we require a definition of the color functions
+% \gbc{gray(g)}, \gbc{rgb(r,g,b)}, and \gbc{cmyk(c,m,y,k)}, as well as
+% conversion functions (see below), and the boolean function
+% \gbc{iscolor}. The first three have to return numerics for \MF{},
+% colors for early \MP{}, and the associated color type for recent \MP{}.
+% We delay the definition of \gbc{rgb} because it only requires
+% distinguishing \MF{} from \MP.
+%
+% \DescribeRoutine{grayscalegray}
+% The grayscale version should return a numeric in recent \MP{}, so
+% it needs a different definition for early \MP{}. Thus, it occurs
+% in the conditional code. Oddly, its definition is the same for \MF{}
+% and recent \MP{}.
+% \begin{macrocode}
+if has_cmyk :
+ vardef grayscalegray (expr g) = snapto g enddef;
+ vardef gray (expr g) = grayscalegray (g) enddef;
+ vardef cmyk (expr c, m, y, k) =
+ (snapto c, snapto m, snapto y, snapto k)
+ enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{makecmyk}
+% \DescribeRoutine{makergb}
+% \DescribeRoutine{makegray}
+% In \gbc{makecmyk} and all the other `\gbc{make}' conversions, the
+% default is to return black in the appropriate model, numerics produce
+% gray in the appropriate model, and cmyk or rgb is either retained
+% unchanged or converted to the appropriate model.
+%
+% \DescribeRoutine{iscolor}
+% A variable is taken to be a color if it can validly follow
+% \mfc{withcolor}. This includes boolean, though we hope no one tries to
+% use that.
+% \begin{macrocode}
+ vardef makecmyk primary clr =
+ colorchoice (clr)(cmykblack)(cmykgray(clr))
+ (rgbtocmyk(redpart clr,greenpart clr,bluepart clr))
+ (clr)
+ enddef;
+ vardef makergb primary clr =
+ colorchoice (clr)(rgbblack)(rgbgray(clr))(clr)
+ (cmyktorgb(cyanpart clr, magentapart clr,
+ yellowpart clr, blackpart clr))
+ enddef;
+ vardef makegray primary clr =
+ colorchoice (clr)(grayscaleblack)(grayscalegray(clr))
+ (rgbtogray (redpart clr, greenpart clr, bluepart clr))
+ (cmyktogray(cyanpart clr, magentapart clr,
+ yellowpart clr, blackpart clr))
+ enddef;
+ vardef iscolor expr clr =
+ (rgbcolor clr) or (cmykcolor clr) or (numeric clr) or (boolean clr)
+ enddef;
+else:
+% \end{macrocode}
+%
+% In early \MP{} \gbc{colorchoice} is only a three-way choice, since
+% \mfc{cmykcolor} is not a data type, but numeric can still be
+% interpreted as a gray. For a numeric or an actual rgbcolor, the first or
+% second branch would be taken. If \gbc{clr} is neither of those then
+% \mfc{cmykcolor}, being equal to \mfc{rgbcolor}, would also be false.
+% Therefore, in the context of early \MP{}, it is irrelevant what goes in
+% the last argument, so we leave it empty.
+%
+% \DescribeRoutine{makecmyk}
+% \DescribeRoutine{makergb}
+% \DescribeRoutine{makegray}
+% The \gbc{make*} functions are simpler in early \MP{}, though not as
+% simple as in \MF{}. Ditto
+% \DescribeRoutine{iscolor}
+% \gbc{iscolor}. The parentheses in \gbc{iscolor} are necessary to force
+% \MP{} to see this \mfc{color} as the start of a boolean expression and
+% not a type declaration.
+% \begin{macrocode}
+ vardef gray (expr g) = rgbgray(g) enddef;
+ vardef grayscalegray (expr g) = rgbgray(g) enddef;
+ vardef cmyk (expr c, m, y, k) = rgb (1-c-k, 1-m-k, 1-y-k) enddef;
+ vardef makergb primary clr =
+ colorchoice (clr)(rgbblack)(rgbgray(clr))(clr)()
+ enddef;
+ vardef makegray primary clr =
+ colorchoice (clr)(rgbblack)(rgbgray(clr))
+ (rgbtogray (redpart clr, greenpart clr, bluepart clr))()
+ enddef;
+ def makecmyk = makergb enddef;
+ vardef iscolor expr clr = (color clr) enddef;
+fi
+
+%</MP>
+% \end{macrocode}
+%
+% \DescribeRoutine{knowncolor}
+% Once we have \gbc{iscolor} all we need to do is add a test for
+% \mfc{known} to get this boolean test.
+% \begin{macrocode}
+vardef knowncolor expr clr = (known clr) and (iscolor clr) enddef;
+
+% \end{macrocode}
+%
+% These are the \MF{} versions. Everything pretty much returns its
+% numeric argument or $0$ (black).
+% \begin{macrocode}
+%<*MF>
+vardef grayscalegray (expr g) = snapto g enddef;
+vardef gray (expr g) = grayscalegray (g) enddef;
+vardef cmyk (expr c, m, y, k) = rgb (1-c-k, 1-m-k, 1-y-k) enddef;
+vardef makegray primary clr =
+ if knowncolor clr: clr else: black fi
+enddef;
+def makergb = makegray enddef;
+def makecmyk = makegray enddef;
+vardef iscolor expr clr = (color clr) enddef;
+
+%</MF>
+% \end{macrocode}
+%
+% \DescribeRoutine{forceclr}
+% \DescribeRoutine{named}
+% This is only used in the \gbc{named} function to force a color. In
+% \MF{} the tests are all `\mfc{if numeric}'. In early \MP{} `\mfc{if
+% cmykcolor}' is the same as `\mfc{if rgbcolor}'.
+% \begin{macrocode}
+vardef forceclr (expr c) =
+ if unknown c :
+ if numeric c: grayscaleblack
+ elseif rgbcolor c: rgbblack
+ elseif cmykcolor c: cmykblack
+ else: black
+ fi
+ elseif numeric c: gray (c)
+ elseif iscolor c: c
+ else: black
+ fi
+enddef;
+vardef named (suffix c) = forceclr (c) enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{togray}
+% \DescribeRoutine{rgbtogray}
+% \DescribeRoutine{cmyktogray}
+% \DescribeRoutine{cmyktorgb}
+% \DescribeRoutine{rgbtocmyk}
+% These are used for the conversions. Strictly speaking they do not
+% `convert' as they all take multiple numeric arguments rather than any
+% sort of color. As \mfc{rgbcolor} exists in both early and recent
+% \MP{} as the same data type, we need only distinguish \MF{} from \MP{}
+% \DescribeRoutine{rgb}
+% in the function \gbc{rgb},
+% \begin{macrocode}
+vardef togray (expr r, g, b) =
+ gray (sqrt((2r*r + 4g*g + b*b)/7))
+enddef;
+
+vardef rgbtogray (expr r, g, b) =
+ togray(snapto r, snapto g, snapto b)
+enddef;
+vardef cmyktogray (expr c, m, y, k) =
+ rgbtogray (1-c-k,1-m-k,1-y-k)
+enddef;
+
+vardef cmyktorgb (expr c,m,y,k) =
+ rgb(1-c-k,1-m-k,1-y-k)
+enddef;
+vardef rgbtocmyk (expr r,g,b) =
+ cmyk(1-r,1-g,1-b,0)
+enddef;
+
+vardef rgb (expr r, g, b) =
+%<MF> togray (snapto r, snapto g, snapto b)
+%<MP> (snapto r, snapto g, snapto b)
+enddef;
+vardef RGB (expr R, G, B) =
+ rgb (R/255, G/255, B/255)
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Arrays}\label{arrays}
+%
+% \gbc{ClipPath} is a typical example of an array. Arrays are based on the
+% fact that a variable can be of a different type from (and can be almost
+% completely unrelated to) the variables formed by putting numeric
+% suffixes on it.
+%
+% \DescribeRoutine{list}
+% The \gbc{list} macro is essentially due to Frank Michielsen, and assigns
+% a \emph{list} (i.e., a comma separated sequence of expressions) to an
+% array. Note that the items in the list have to be the same type, and the
+% same type as \mfc{v[\,]}. But \mfc{v} itself must be numeric.
+%
+% \DescribeRoutine{map}
+% The \gbc{map} macro takes two text parameters. The first is any
+% procedure, the second is a list of expressions. The procedure is applied
+% to each expression and the resulting new expressions are separated by
+% commas, that is, a new list is generated (for use in \mfc{for} loops).
+% This is full of possibilities for errors. One reared its head because
+% the original version started with a comma indicating an empty starting
+% expression (normally it would be ignored and that turn through the loop
+% skipped). However, it managed to produce an error in a reasonable
+% but unforeseen usage (which I've since forgotten) and so I added the
+% \gbc{_map} variable that skips the comma on the first time through the
+% loop. This routine is currently only used in the code that \mfpic's \
+% \cs{plr} writes.
+% \begin{macrocode}
+def list (suffix v) (text lst) =
+ v := 0; for _itm = lst: v[incr v] := _itm; endfor
+ if v = 0:
+ GBerrmsg ("No list to process!")
+ "An attempt was made to produce an array from a "
+ & "list of expressions having no valid entries.";
+ fi
+enddef;
+
+def map (text proc) (text lst) =
+ hide (_map := 0;)
+ for _a = lst:
+ if _map = 0: hide (_map := 1;) else: , fi
+ proc (_a)
+ endfor
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{knownnumericarray}
+% Checks if a suffix is the name of an array. Requires \gbc{arr} to be a
+% known positive integer, and all the variables \gbc{arr[n]} to be known
+% for \gbc{n} from 1 to \gbc{arr}. Since we so far only need it for
+% numeric arrays, we also check if each entry is numeric.
+% \begin{macrocode}
+vardef knownnumericarray suffix arr =
+ setboolean (_kna) (known arr) and (numeric arr);
+ if _kna :
+ _kna := (arr = floor arr) and (arr >= 1);
+ for _idx = 1 upto arr :
+ exitif not _kna;
+ _kna := (known arr[_idx]) and (numeric arr[_idx]);
+ endfor
+ fi
+ _kna
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{copyarray}
+% This makes some code much more readable. It simply steps through an
+% array and copies the values into another array. It is only used for
+% numeric arrays so far, but could be used for any kind.
+% \begin{macrocode}
+def copyarray (suffix src, dest) =
+ for _idx = 1 upto src: dest[_idx] := src[_idx]; endfor
+ dest := src;
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{maparr}
+% The \gbc{maparr} macro applies a procedure \gbc{proc} to each member of
+% array \gbc{p[\,]} with \gbc{p} members. It returns nothing. It is currently
+% unused, although it was once used for things like \gbc{maxpair}.
+% \begin{macrocode}
+def maparr (text proc) (suffix p) =
+ for _idx = 1 upto p: proc (p[_idx]); endfor
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{textpairs}
+% This macro takes a suffix (name of an array to be constructed) and a
+% list of pairs, and assigns them to the array. It is normally called from
+% another macro, which does any necessary \mfc{save}-ing of the variable
+% used for the array name. We used to include \mfc{save} in this macro,
+% but ran into a problem once when the argument had a suffix. You can't
+% apply \mfc{save} to a variable with a suffix. Moreover, `\mfc{save p}'
+% also renders \mfc{p.x} unknown, so I judged it best to let whoever calls
+% this macro decide what to save. Actually, now it expands to the more
+% general command \gbc{gsetarray} with type \gbc{pair}. That command
+% then reads the suffix argument that should follow.
+%
+% Since the above change was made, macros evolved so that \emph{all} uses
+% of \gbc{textpairs} are now preceeded by \gbc{save}. Thus, I have now
+% replaced them all with calls to \gbc{setpairs} (it calls \gbc{setarray},
+% which \emph{does} \gbc{save} the variable). In all those cases, the
+% `\gbc{saved}' variable is a temporary local array.
+%
+% \DescribeRoutine{setuniquepairs}
+% This does the same but omits any pair if it is identical to the previous
+% one. It \mfc{save}\,s the variable, since all its uses are internal
+% and require that.
+% \begin{macrocode}
+def textpairs = gsetarray (pair) enddef;
+
+def setuniquepairs (suffix p) (text t) =
+ save p; pair p[];
+ setpairs (_up) (t);
+ if _up > 0:
+ p := 1; p1 := _up1;
+ for _i = 2 upto _up:
+ if _up[_i] <> p[p]: p[incr p] := _up[_i]; fi
+ endfor
+ else:
+ p := 0;
+ fi
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Utilities}\label{utilities}
+%
+% \DescribeRoutine{chpair}
+% This applies a procedure \gbc{proc} (which maps numeric to numeric) to
+% each part of pair \gbc{p}, and returns the resultant pair. I've decided
+% not to use it (for efficiency), but to leave it defined for backward
+% compatibility.\\
+% \DescribeRoutine{floorpair}
+% \gbc{floorpair} applies \mfc{floor} to both parts of a pair.\\
+% \DescribeRoutine{ceilingpair}
+% \gbc{ceilingpair} does the same with \gbc{ceiling}.\\
+% \DescribeRoutine{hroundpair}
+% \gbc{hroundpair} does the same with \gbc{hround}.
+%
+% All three could use \gbc{chpair} with \gbc{proc} equal to \mfc{floor},
+% \mfc{ceiling} and \mfc{hround}, but I now code them directly.
+%
+% \DescribeRoutine{goodpair}
+% This last one is used (only in \MF{}) to adjust pairs to the pixel grid.
+% It is the only place \gbc{hroundpair} is used. None of these is used in
+% the \MP{} version.
+% \begin{macrocode}
+vardef chpair (text proc) (expr p) =
+ (proc (xpart p), proc (ypart p))
+enddef;
+
+vardef floorpair (expr p) = (floor (xpart p), floor (ypart p))
+enddef;
+vardef ceilingpair (expr p) = (ceiling (xpart p), ceiling (ypart p))
+enddef;
+
+%<*MF>
+def hroundpair (expr p) = (hround (xpart p), hround (ypart p))
+enddef;
+vardef goodpair (expr p) = hroundpair(p.t_) enddef;
+
+%</MF>
+% \end{macrocode}
+%
+% \DescribeRoutine{emin}
+% The macro \gbc{emin} differs from \prog{plain}'s \mfc{min} in that it
+% allows only two values. It can therefore be coded simply, without the
+% overhead of a \mfc{for}-loop.
+% \DescribeRoutine{emax}
+% \gbc{emax} is analogous. Both are needed so often that it is possible a
+% significant amount of time is saved with these versions.
+%
+% \DescribeRoutine{pairmin}
+% The macro \gbc{pairmin} operates on two pairs, returning a pair having
+% the smaller of the two xparts and the smaller of the two yparts. Of
+% course
+% \DescribeRoutine{pairmax}
+% \gbc{pairmax} is analogous, producing the maximum.
+%
+% \DescribeRoutine{minpair}
+% The \gbc{minpair} macro returns the pair comprising the minimum $x$ and
+% minimum $y$ coordinates of all pairs in the array \gbc{p[\,]}.
+% \DescribeRoutine{maxpair}
+% \gbc{maxpair} is analogous. Somehow, both of them have disappeared from
+% \grafbase. They were formerly used only in the \gbc{*bbox} macros.
+% That code used a loop to build an array of control points and these
+% routines would \emph{each} loop through that. The current code uses one
+% loop (instead of three) through the control points, updating both the
+% maximum and minimum at each one.
+% \begin{macrocode}
+vardef emin (expr a, b) = if a < b: a else: b fi enddef;
+vardef emax (expr a, b) = if a > b: a else: b fi enddef;
+
+vardef pairmin (expr z, w) =
+ ( emin (xpart z, xpart w), emin (ypart z, ypart w ) )
+enddef;
+vardef pairmax (expr z, w) =
+ ( emax (xpart z, xpart w), emax (ypart z, ypart w ) )
+enddef;
+
+vardef minpair (suffix p) = setpair (_mp) p1;
+ for _idx = 2 upto p - 1: _mp := pairmin (_mp, p[_idx]); endfor
+ pairmin (_mp, p[p])
+enddef;
+vardef maxpair (suffix p) = setpair (_mp) p1;
+ for _idx = 2 upto p - 1: _mp := pairmax (_mp, p[_idx]); endfor
+ pairmax (_mp, p[p])
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{xprod}
+% A binary operation between pairs $z\sb1$ and $z\sb2$ that returns the
+% cross product $x\sb1 y\sb2 - x\sb2 y\sb1$. This gives, among other
+% things, twice the area of the triangle with two sides $z\sb1$ and
+% $z\sb2$. It is used only in \gbc{mkconvex}.
+% \begin{macrocode}
+primarydef Z xprod W = (xpart Z * ypart W - xpart W * ypart Z)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{force_initial}
+% The command \gbc{force_initial} modifies a path so that it has all the
+% same points and controls as before, except its first point is replaced
+% with \mfc{p}.
+% \DescribeRoutine{force_terminal}
+% The command \gbc{force_terminal} replaces the last point. This is for
+% cases where, theoretically, paths \gbc{f} and \gbc{g} should meet at an
+% endpoint, but do not due to finite precision. Instead of doing
+% \mfc{f..g}, which adds a random tiny segment, we adjust the endpoints to
+% exactly match the other and do \mfc{f\&g}, producing a join without an
+% additional segment.
+%
+% \DescribeRoutine{force_equal_ends}
+% The command \gbc{force_equal_ends} forces the last point of the first
+% path and the first point of the second to equal the average of their
+% original values. It is the only one of these four actually used anywhere
+% else in \grafbase.
+% \DescribeRoutine{replace_ends_of_cycle}
+% The command \gbc{replace_ends_of_cycle} applies something similar to a
+% cycle.
+% \begin{macrocode}
+def force_initial (expr p) (suffix f) =
+ hide( setnumeric (_n) length f;
+ f := p
+ if _n = 0:
+ {0,0}
+ else:
+ ..controls post0 (f) and pre 1 (f).. subpath (1,_n) of f
+ fi;)
+enddef;
+
+def force_terminal (expr p) (suffix f) =
+ hide(setpath (_f) reverse f;
+ force_initial (p) (_f);
+ f := reverse _f;)
+enddef;
+
+def force_equal_ends (suffix f, g) =
+ hide(save _p; pair _p;
+ _p := .5[pnt[length f] (f), pnt0(g)];
+ force_terminal (_p) (f); force_initial (_p) (g);)
+enddef;
+
+def replace_ends_of_cycle (expr p) (suffix f) =
+hide(
+if cycle f:
+ save _n; _n := length f;
+ f := p
+ if _n = 0: &cycle
+ else: .. controls post0 (f) and pre 1 (f) ..
+ if _n = 1: cycle
+ else: subpath (1, _n - 1) of f ..
+ controls post[_n - 1](f) and pre[_n](f) .. cycle
+ fi
+ fi;
+fi)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{intersects}
+% A binary relation, with the precedence level (almost) that of other
+% relations, produces \mfc{true} if \MF{} determines that the paths
+% intersect, false otherwise. It also
+% \DescribeVariable{thetimes}
+% sets the pair variable \gbc{thetimes} and its parts \gbc{_Xtime} and
+% \gbc{_Ytime}. Then
+% \DescribeRoutine{misses}
+% \gbc{misses} is the opposite relation, used when the intersection point
+% is not needed. It only occurs in the (unused) code of \gbc{tightbbox}.
+% \begin{macrocode}
+pair thetimes;
+numeric _Xtime, _Ytime;
+tertiarydef a intersects b =
+ begingroup
+ thetimes := a intersectiontimes b;
+ _Xtime := xpart thetimes;
+ _Ytime := ypart thetimes;
+ (_Xtime > -1)
+ endgroup
+enddef;
+
+tertiarydef a misses b = ((a intersectiontimes b) < origin) enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{makepicture}
+% The \gbc{makepicture} command takes any expression and does what it can
+% to make a picture from it.
+%
+% \DescribeRoutine{onepointpath}
+% The \gbc{onepointpath} command takes a point and forces it to be a path.
+% If a vardef takes a list of points and it \emph{must} return a path that
+% perhaps \emph{must} be cyclic, it can use this as a fallback. If an
+% \mfpic{} command such as \cs{arc} receives an invalid optional
+% parameter, it won't know what command to write to the output file. It
+% can use
+% \DescribeRoutine{fallbackpath}
+% \gbc{fallbackpath} as long as the first parameter is a point.
+%
+% \DescribeRoutine{even}
+% \DescribeRoutine{divides}
+% Of course \gbc{even} means \gbc{not odd}. The relation \gbc{divides}
+% is true if the right side is an integer multiple of the left.
+% \begin{macrocode}
+vardef makepicture (expr s) =
+ if picture s: s
+%<MP> elseif string s: s infont defaultfont scaled defaultscale
+ elseif path s: picpath (s)
+ else: nullpicture
+ fi
+enddef;
+
+vardef onepointpath (expr cyclic, q) =
+ q if cyclic: &cycle else: {0,0} fi
+enddef;
+
+vardef fallbackpath (expr cyclic, p) (text t) =
+ onepointpath (cyclic, p)
+enddef;
+
+def even = not odd enddef;
+
+primarydef a divides b =
+ ((b mod a) = 0)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{image}
+% The \mfc{image} macro exists in \file{plain.mp} but not \file{plain.mf}.
+% The purpose is to just use the \file{plain} \MF{} and \grafbase{} macros
+% as you normally would, but wrap the whole thing in parentheses preceded
+% by \gbc{X := image} to get all those things drawn on the picture
+% variable \gbc{X}.
+%
+% \DescribeRoutine{beginimage}
+% Instead of making lengthy drawing code a parameter, one might prefer an
+% environment-like syntax, writing \gbc{X := beginimage } at the start
+% and
+% \DescribeRoutine{endimage}
+% \gbc{endimage} at the end.
+%
+% \DescribeRoutine{makeimage}
+% This is for the \mfpic{} command \cs{mfpimage}. It takes a suffix
+% parameter (the name of the picture variable) and a coordinate pair (in
+% graph coordinates). The drawing commands, up to the following
+% \gbc{endimage}, draw on this picture variable with the given pair as the
+% reference point.
+% \begin{macrocode}
+%<*MF>
+vardef image (text t) =
+ newpicture (currentpicture);
+ t;
+ currentpicture
+enddef;
+
+%</MF>
+def beginimage =
+ begingroup
+ newpicture (currentpicture);
+enddef;
+def endimage =
+ ; currentpicture
+ endgroup
+enddef;
+
+def makeimage (suffix name) (expr refpt) =
+ setpair (_image_reference_point) zconv (refpt);
+ setpicture (name) beginimage
+enddef;
+def concludeimage =
+ endimage shifted
+%<MF> -goodpair (_image_reference_point)
+%<MP> -_image_reference_point
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{setvariable}
+% This is are mainly to save space in \mfpic-generated files. In \grafbase{}
+% itself the \mfc{save} is often inconvenient, but it turns out there are
+% many cases where it \emph{is} used; enough so that we have abbreviations
+% \RoutineIndex{setnumeric}\gbc{setnumeric},
+% \RoutineIndex{setboolean}\gbc{setboolean},
+% \RoutineIndex{setpair}\gbc{setpair},
+% \RoutineIndex{setpath}\gbc{setpath},
+% \RoutineIndex{setpicture}\gbc{setpicture},
+% and \RoutineIndex{setstring}\gbc{setstring}, together with the
+% common uses \RoutineIndex{newpicture}\gbc{newpicture} and
+% \RoutineIndex{convertpath}\gbc{convertpath}. There is also a
+% \gbc{setcolor}, but that has such a different definition that we reserve
+% it for later.
+%
+% For completeness, we also include the remaining two abbreviations,
+% \RoutineIndex{setpen}\gbc{setpen} and
+% \RoutineIndex{settransform}\gbc{settransform}, even though they are not
+% used anywhere in \grafbase{}.
+%
+% \DescribeRoutine{gsetvariable}
+% The macro \gbc{gsetvariable} is the global version. It has no
+% abbreviations, but it is occasionally needed for \mfpic{}. The only
+% difference between it and the local version is the lack of a
+% \gbc{save}. None of these commands take the value as a parameter. That
+% should follow, and is picked up by the ending \mfc{:=}.
+%
+% \DescribeRoutine{setarray}
+% Then \gbc{setarray} is the array version. It takes the same parameters
+% as \gbc{setvariable}, but what should follow is a list of expressions in
+% parentheses. It calls \gbc{list} to read each item into
+% \gbc{name1}, \gbc{name2}, etc. There is also has a global version
+% \DescribeRoutine{gsetarray}
+% \gbc{gsetarray}.
+% \DescribeRoutine{setpairs}
+% \gbc{setpairs} is an abbreviation for arrays
+% of pairs. Historically, it came first.
+% \begin{macrocode}
+def setvariable (text kind) (suffix name) =
+ save name; kind name; name :=
+enddef;
+def gsetvariable (text kind) (suffix name) = kind name; name :=
+enddef;
+
+def setnumeric (suffix name) = save name; name := enddef;
+def setboolean = setvariable (boolean) enddef;
+def setpair = setvariable (pair) enddef;
+def setpath = setvariable (path) enddef;
+def setpicture = setvariable (picture) enddef;
+def setstring = setvariable (string) enddef;
+def settransform = setvariable (transform) enddef;
+def setpen = setvariable (pen) enddef;
+def settension (suffix tn) expr tens =
+ setnumeric (tn) if tens > 0: tens else: default_tension fi;
+enddef;
+def fixtension (suffix tn) = if tn < .75: tn := .75; fi enddef;
+
+def newpicture (suffix pic) = setpicture (pic) nullpicture; enddef;
+def convertpath (suffix g) expr f = setpath (g) zconv (f); enddef;
+
+def setarray (text kind) (suffix name) =
+ save name; kind name[]; list (name)
+enddef;
+def setpairs = setarray (pair) enddef;
+def gsetarray (text kind) (suffix name) =
+ numeric name; kind name[]; list (name)
+enddef;
+
+% \end{macrocode}
+% The next are slightly different, but seem to belong here.
+
+% \DescribeRoutine{setbbox}
+% In \gbc{setbbox} we save and initialize \emph{two} pair variables and
+% set them to the bounding box of a path that should follow.
+%
+% \DescribeRoutine{setsplit}
+% There are a couple of routines that modify a variable to make sure it is
+% positive and integral. In a couple of places two routine \emph{must} use
+% the same value. Here we isolate the code that does the modification, and
+% then both routines call \gbc{setsplit}.
+% \begin{macrocode}
+def setbbox (suffix ll, ur) =
+ save ll, ur; pair ll, ur; getbbox (ll, ur)
+enddef;
+
+def setsplit (suffix s) expr ss =
+ setnumeric (s) emax (1, ceiling ss);
+enddef;
+
+%<*MP>
+if has_cmyk:
+ def setrgbcolor = setvariable (rgbcolor) enddef;
+ def setcmykcolor = setvariable (cmykcolor) enddef;
+ def setcolor (suffix name) expr val =
+ if boolean val : setboolean
+ elseif numeric val : setnumeric
+ elseif rgbcolor val : setrgbcolor
+ elseif cmykcolor val : setcmykcolor
+ % this should give a suitable error message:
+ else: setvariable (color)
+ fi (name) val;
+ enddef;
+ def gsetcolor (suffix name) expr val =
+ if boolean val : boolean name;
+ elseif numeric val : numeric name;
+ elseif rgbcolor val : rgbcolor name;
+ elseif cmykcolor val : cmykcolor name;
+ else: color name;
+ fi name := val;
+ enddef;
+else:
+ def setrgbcolor = setcolor enddef;
+ def setcmykcolor = setcolor enddef;
+ def setcolor = setvariable (color) enddef;
+ def gsetcolor = gsetvariable (color) enddef;
+fi
+%</MP>
+%<*MF>
+def setrgbcolor = setcolor enddef;
+def setcmykcolor = setcolor enddef;
+def setcolor = setvariable (color) enddef;
+def gsetcolor = gsetvariable (color) enddef;
+%</MF>
+
+% \end{macrocode}
+%
+% And then the standard colors. Using the color functions ensures that
+% they are defined in \MF{} as well as all versions of \MP{}. In early
+% \MP{} they are all \mfc{rgbcolor}, in \MF{} they are all numeric. In
+% recent \MP{}, they have the type correspondimg to the name of the
+% color function, with \gbc{gray()} being numeric.
+% \begin{macrocode}
+setcolor(rgbblack) rgb(0,0,0);
+setcolor(red) rgb(1,0,0);
+setcolor(green) rgb(0,1,0);
+setcolor(blue) rgb(0,0,1);
+setcolor(rgbwhite) rgb(1,1,1);
+setcolor(cmykwhite) cmyk(0,0,0,0);
+setcolor(cyan) cmyk(1,0,0,0); % Maybe these should
+setcolor(magenta) cmyk(0,1,0,0); % be rbg for backward
+setcolor(yellow) cmyk(0,0,1,0); % compatibility?
+setcolor(cmykblack) cmyk(0,0,0,1);
+setcolor(grayscaleblack) gray(0);
+setcolor(grayscalewhite) gray(1);
+
+%<*MP>
+if has_outputtemplate:
+ def setoutputtemplate = outputtemplate := enddef;
+elseif has_filenametemplate:
+ def setoutputtemplate = filenametemplate enddef;
+else:
+ def setoutputtemplate text garbage = enddef;
+fi
+%</MP>
+%<MF>def setoutputtemplate text garbage = enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{GBromannumeral}
+% We will append roman numerals to the ends of a variable name to
+% emulate an array. This will be needed when our `array' consists of
+% colors with different types. \MP{} doesn't permit true arrays to
+% contain different types. We use `\gbc{GB}' in the name because a
+% package exists that defines \mfc{romannumeral} differently
+%
+% \DescribeRoutine{GBromandigit}
+% Roman numerals can conveniently be computed one digit at a time. The
+% algorithm is the same for each digit, differing only in the letters
+% used. Thus we define \gbc{GBromandigit} and call it three times with
+% different sets of letters.
+%
+% \DescribeRoutine{strrepeat}
+% The helper macro \gbc{strrepeat} creates a new string by concatenating
+% \mfc{rep} copies of the string \mfc{str}.
+% \begin{macrocode}
+vardef GBromannumeral (expr X) =
+ save Y, _tmp, U; string U;
+
+ Y.m := X div 1000; % thousands digit
+ _tmp := X - 1000Y.m; % hundreds digits and lower
+ Y.c := _tmp div 100; % hundreds
+ _tmp := _tmp - 100Y.c; % tens and units
+ Y.x := _tmp div 10; % tens
+ Y.i := _tmp - 10Y.x; % units
+
+ strrepeat("m", Y.m) &
+ GBromandigit("c", "d", "m", Y.c) &
+ GBromandigit("x", "l", "c", Y.x) &
+ GBromandigit("i", "v", "x", Y.i)
+enddef;
+
+vardef GBromandigit (expr bot, mid, top, n) =
+ if n > 9 : top & strrepeat(bot, n-10) % shouldn't happen
+ elseif n > 8 :
+ bot & top % "ix"
+ elseif n > 4 : mid & strrepeat (bot, n-5) % "v"--"viii"
+ elseif n > 3 :
+ bot & mid % "iv"
+ else: strrepeat (bot, n) % ""--"iii" for 0--3
+ fi
+enddef;
+
+vardef strrepeat (expr st, rep) =
+ "" for i = 1 upto rep: & st endfor
+enddef;
+
+% \end{macrocode}
+%
+%
+%
+% \section{The \grafbase{} Coordinate System}\label{coordinate}
+%
+% We need to make a distinction between graph units, sharped units, and
+% device units. In \MF, a device unit is 1 pixel. On a LaserJet IV, one
+% inch is 600 pixels. When constructing a character, \MF{} uses the pixel
+% as its unit. Since this differs from one printing device to another,
+% \file{plain.mf} arranges for \emph{sharped} units (the name comes from the
+% convention that they are written using a name that ends in \mfc{\#}). The
+% dimension \mfc{1pt\#} in \MF{} is arbitrarily set to 1, and other
+% units defined by conversion factors (\mfc{in\#=72.27}; neither \MF{}
+% nor \MP{} makes a distinction between distances and numbers: \mfc{2pt}
+% just means \mfc{2} times the value of \mfc{pt}). When one needs to
+% draw something actually \emph{one point long}, then \mfc{1pt} is used.
+% It is defined to equal \mfc{pt\#*hppp}, where \mfc{hppp} stands for
+% ``horizontal pixels per point'' and its value is usually set by
+% \mfc{mode_setup}. So \mfc{1pt} is $600/72.27$ (pixels) if
+% \mfc{mode} is \mfc{ljfour}.
+%
+% Often, when we want numbers not to become too large, we do calculations,
+% define paths, etc., in sharped units, then draw by scaling to device
+% units. In \grafbase{} we take this one step further: a horizontal graph
+% unit (i.e., the difference between the graph points $(0,0)$ and $(1,0)$)
+% represents \gbc{unitlen*xscale} sharped units, and
+% \gbc{unitlen*xscale*hppp} actual pixels. The \grafbase{} macros do much
+% of the calculations in graph units.
+%
+% In \MP, there is no difference between device and sharped units.
+% The \emph{postscript point} or \emph{big point} (1/72 inches) is the
+% unit in \MP: \mfc{bp = 1}.
+%
+% Some things need to be in graph units (for example, positions within a
+% graph defined by the user) or independent of units (standard shapes)
+% that scale appropriately when scales change. Other things (thickness of
+% lines) are a design decision that either should be independent of scale
+% or should scale in a nonobvious way. The diameter of the drawing pen is
+% one of the latter things, so the default pen width is in device units.
+% Also for the hatching pen.
+%
+% When drawing a path we want to use device coordinates. When defining
+% paths, we typically want to use graph coordinates. The macros that do
+% the drawing, therefore, need to convert from one to the other. In
+% addition, for inclusion of the picture in a \TeX{} document, we normally
+% want the lower left corner of the graph space to have device coordinates
+% $(0,0)$.
+%
+%
+% \subsection{The main transforms}\label{ztr}
+%
+% \DescribeVariable{vtr}
+% We therefore have two transforms: \gbc{vtr} is the \emph{vector} or
+% linear transform for pair quantities that remain invariant under shifts,
+% and
+% \DescribeVariable{ztr}
+% \gbc{ztr} is a \emph{point} or affine transformation for pair quantities
+% that change appropriately under shifts.
+%
+% The quantities \gbc{xneg}, \gbc{xpos}, \gbc{yneg}, and \gbc{ypos} are
+% in \emph{graph} coordinates. Shifting by \gbc{(-xneg, -yneg)} transforms
+% the lower left corner to $(0,0)$. Multiplication by \gbc{xscale} and
+% \gbc{yscale} converts to multiples of \gbc{unitlen} and multiplication
+% by \gbc{unitlen} gets us sharped coordinates. For \MF{},
+% multiplication by \mfc{hppp} converts to device coordinates, while for
+% \MP{} sharped and device are the same (the printer's PostScript
+% rasterizing engine---\prog{GhostScript} perhaps---does the final
+% conversion to actual pixels).
+%
+% In \MF{}, \mfc{currenttransform} (via the macro \mfc{.t_}, defined by
+% \mfc{mode_setup}) takes care of the aspect ratio. In \MP{} the final
+% rasterizer should do this.
+%
+% \gbc{charwd} and \gbc{charht} are sharped coordinates defined by the
+% startup code \gbc{beginmfpic}, while \gbc{w_} and \gbc{h_} are the
+% corresponding device (pixel) coordinates
+%
+% \DescribeRoutine{setztr}
+% This macro does the defining of \gbc{ztr} and \gbc{vtr}. It is called
+% by \gbc{beginmfpic}, at which time all the necessary quantities should be
+% known.
+% \begin{macrocode}
+transform ztr, vtr;
+def setztr =
+ if debug:
+ GBdebug;
+%<*MF>
+ >> "charwd = " & decimal charwd & "pt#";
+ >> "charht = " & decimal charht & "pt#";
+ >> "w_ = " & decimal w_ & " pixels";
+ >> "h_ = " & decimal h_ & " pixels";
+ >> "unitlen = " & decimal unitlen & "pt#";
+ >> "hppp = " & decimal hppp;
+%</MF>
+%<*MP>
+ >> "w_ = " & decimal w_ & "bp";
+ >> "h_ = " & decimal h_ & "bp";
+ >> "unitlen = " & decimal unitlen & "bp";
+%</MP>
+ >> "xneg = " & decimal xneg;
+ >> "xpos = " & decimal xpos;
+ >> "yneg = " & decimal yneg;
+ >> "ypos = " & decimal ypos;
+ >> "xscale = " & decimal xscale;
+ >> "yscale = " & decimal yscale;
+ GBenddebug;
+ fi
+ save ztr, vtr;
+ transform ztr, vtr;
+ vtr := identity xscaled xscale yscaled yscale scaled (unitlen*hppp);
+ ztr := identity shifted (-xneg, -yneg) transformed vtr;
+ if debug:
+ GBdebug;
+ >> "ztr is";
+ show ztr;
+ >> "vtr is";
+ show vtr;
+ GBenddebug;
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{zconv}
+% The macro \gbc{zconv} converts a variety of expressions from graph to
+% device coordinates. The expressions include pairs, paths, and transforms.
+% This is an affine transform. The inverse,
+% \DescribeRoutine{invzconv}
+% \gbc{invzconv}, converts a variety of expressions from device to graph
+% coordinates.
+%
+% \DescribeRoutine{vconv}
+% The vector version, \gbc{vconv}, converts a vector \gbc{v} from graph to
+% device coordinates. This is a linear (ie, vector) transform. Also,
+% \DescribeRoutine{invvconv}
+% \gbc{invvconv} converts a vector from device to graph coordinates.
+% \begin{macrocode}
+vardef zconv (expr a) = a transformed ztr enddef;
+vardef invzconv (expr a) = a transformed (inverse ztr) enddef;
+vardef vconv (expr v) = v transformed vtr enddef;
+vardef invvconv (expr v) = v transformed (inverse vtr) enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{The \gbc{mfpic} environment}\label{mfpic}
+%
+% \DescribeRoutine{active_plane}
+% \gbc{active_plane} is the active drawing plane. \mfc{currentpicture} is
+% unknown at this stage (because it's set in \gbc{beginmfpic}). We use a
+% \mfc{def}, and not a picture assignment, partly for this reason but also
+% because we can achieve special effects by redefining it (see the
+% \gbc{tile} macro).
+% \begin{macrocode}
+def active_plane = currentpicture enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{initpic}
+% \gbc{initpic} is called by \gbc{beginmfpic} after \gbc{w_} and
+% \gbc{h_} are defined. At this point \gbc{xneg}, \gbc{xscale}, etc.,
+% have known values and \gbc{setztr} can define the transforms that are
+% based on them. Also, the default \gbc{drawpen} is initialized and the
+% boundary of the graph space is assigned to the clipping array.
+%
+% If \gbc{underlaylabels} is true, we try to make them part of the
+% background, adding them to the picture variable \gbc{background_labels}.
+% Just before shipout, the picture is placed on top of these labels.
+%
+% If \gbc{overlaylabels} is \gbc{true}, we try to make labels in \MP{}
+% behave the same as labels in \TeX{} (for \mfpic) by adding the labels
+% on last. We do this by adding them to the picture variable
+% \gbc{foreground_labels} as they occur, then add that picture onto
+% \gbc{active_plane} just before shipout. For backward compatibility,
+% the default for \gbc{overlaylabels} is \gbc{false}.
+%
+% We initialize \gbc{foreground_labels} and \gbc{background_labels} here. The
+% pair variables \gbc{labelbb.ll} and \gbc{labelbb.ur} keep track of the
+% bounding box of added labels in case \gbc{overlaylabels},
+% \gbc{truebbox}, and \gbc{clipall} are all \gbc{false}.
+% \begin{macrocode}
+%<*MP>
+boolean overlaylabels, underlaylabels, havebackground;
+overlaylabels := false;
+underlaylabels := false;
+havebackground := false;
+
+%</MP>
+def initpic =
+ setztr;
+ resizedrawpen (penwd);
+ if ClipOn: ClipPath := 1;
+ ClipPath1 := rect (origin, (w_, h_));
+ fi
+ if debug:
+ GBdebug;
+ >> "Drawing nominal bounding box around picture";
+ GBenddebug;
+ noclip ( safedraw rect (origin, (w_, h_)) );
+ fi
+%<*MP>
+ newpicture (foreground_labels);
+ newpicture (background_labels);
+ havebackground := false;
+ save labelbb; pair labelbb.ll, labelbb.ur;
+ labelbb.ll := labelbb.ur := origin;
+%</MP>
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{mfpicenv}
+% We define a \gbc{mfpicenv} environment for compatibility with older
+% \file{graphbase.mf} (mainly for \prog{fig2dev}'s \file{genmf.c}).
+% \DescribeRoutine{endmfpicenv}
+% Actually, I have no idea if \prog{fig2dev} even works with the current
+% \mfpic.
+%
+% \DescribeRoutine{bounds}
+% This also used to be unused, for compatibility only, but I decided it was a
+% convenient abbreviation and \mfpic{} uses it again.
+% \begin{macrocode}
+def mfpicenv = enddef;
+def endmfpicenv = enddef;
+def bounds (expr a, b, c, d) =
+ xneg := a; xpos := b;
+ yneg := c; ypos := d;
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{beginmfpic}
+% This is the figure wrapper. \mfpic{} used to begin with figure 1 and
+% progressively increment the number. The current value of \gbc{gcode} was
+% always equal to the current figure number. Now, \mfpic{} explicitly
+% writes the figure number, so we assign \gbc{gcode} to that number in
+% case any old files made use of the current number through the
+% \gbc{gcode} variable.
+%
+% Originally, \gbc{beginmfpic} defined \mfc{w}, \mfc{h} and \mfc{d}, but
+% that caused problems if an \mfpic{} user tried to store a path in a
+% variable named \gbc{h}, etc. So now we use the less obvious names ending
+% in underscore. Apart from this, the code below is a clone of
+% \file{plain.mf}'s \mfc{beginchar} (for \MF). In fact, it used to invoke
+% \mfc{beginchar}. For \MP, we invoke \mfc{beginfig} explicitly. This does
+% the \mfc{clear...} actions and \mfc{charcode} assignment.
+%
+% The `\mfc{extra_...mfpic}' strings provide a compiler-independent way
+% to add to the extra beginning and ending tokens.
+% \begin{macrocode}
+string extra_beginmfpic; extra_beginmfpic := "";
+string extra_endmfpic; extra_endmfpic := "";
+
+def beginmfpic (expr ch) =
+%<MP> beginfig (ch);
+%<MF> begingroup
+ gcode := ch;
+ save w_, h_, d_;
+ charwd := (xpos-xneg)*xscale*unitlen;
+ charht := (ypos-yneg)*yscale*unitlen;
+ chardp := 0;
+%<*MF>
+ charcode := if known ch: byte ch else: 0 fi;
+ w_ := hround (charwd*hppp);
+ h_ := vround (charht*hppp);
+ d_ := vround (chardp*hppp);
+ charic := 0; clearxy; clearit; clearpen;
+ scantokens extra_beginchar;
+%</MF>
+%<*MP>
+ w_ := charwd;
+ h_ := charht;
+ d_ := chardp;
+%</MP>
+ initpic;
+ scantokens extra_beginmfpic;
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{endmfpic}
+% For \MF, we again clone \file{plain.mf}'s \mfc{endchar}, adding support
+% for the \gbc{clipall} (clip to the graph rectangle), and \gbc{ClipOn}
+% (clip to some user specified array of paths), and \gbc{showbbox} (draw
+% the boundary of the graph for debugging purposes).
+% \begin{macrocode}
+def endmfpic =
+ scantokens extra_endmfpic;
+ if debug:
+ GBdebug;
+%<MF> >> "TFM charwd = " & decimal charwd & "pt#";
+%<MF> >> "TFM charht = " & decimal charht & "pt#";
+%<MP> >> "width = " & decimal w_ & "bp";
+%<MP> >> "height = " & decimal h_ & "bp";
+ GBenddebug;
+ fi
+ DoClip (active_plane);
+ if clipall: clipto (active_plane) rect (origin, (w_, h_)); fi
+ if showbbox: noclip ( safedraw rect (origin, (w_, h_)) ); fi
+%<*MF>
+ scantokens extra_endchar;
+ if proofing > 0: makebox (proofrule); fi
+ chardx := w_; % desired width of character in pixels
+ shipit;
+ if displaying > 0: makebox (screenrule); showit; fi
+ endgroup
+%</MF>
+% \end{macrocode}
+%
+% \MP's code is more involved due to the possibility to put typeset text
+% in a picture. In addition to the \gbc{clipall}, \gbc{ClipOn} and
+% \gbc{showbbox} support, we have support for labels and \gbc{truebbox}.
+% \begin{macrocode}
+%<*MP>
+ save _ll, _ur;
+ pair _ll, _ur;
+ if truebbox:
+ _ll := llcorner active_plane;
+ _ur := urcorner active_plane;
+% \end{macrocode}
+% We try to let the bbox include labels, even when they extend beyond the
+% nominal picture boundaries. However, they will have been clipped off if
+% \gbc{clipall} is set. In that case, we just set the bounding box to the
+% coordinates determined by \gbc{w_} and \gbc{h_}, otherwise we expand
+% them to the \gbc{labelbb} values.
+% \begin{macrocode}
+ elseif clipall:
+ _ll := origin;
+ _ur := (w_,h_);
+ else: % expand to accomodate labels
+ _ll := pairmin ((0, 0 ), labelbb.ll);
+ _ur := pairmax ((w_, h_), labelbb.ur);
+ fi
+% \end{macrocode}
+% A bounding box in the output PostScript code can have a side with
+% length 0 (e.g., a picture drawn with \mfpic{} that contains only
+% text placed by \TeX). This can cause division by 0 errors in some
+% cases. That's why we don't just let \MP{} determine the bounding box,
+% but force the upper and lower coordinates to differ.
+% \begin{macrocode}
+ _ur := pairmax (_ur, _ll + eps*(1, 1));
+ setbounds active_plane to rect (_ll, _ur);
+% \end{macrocode}
+% Finally, if \gbc{overlaylabels} or \gbc{underlaylabels} was true during
+% a \gbc{newgblabel} command, then the label was not added to
+% \mfc{currentpicture} but rather to \gbc{foreground_labels} or
+% \gbc{background_labels}. We add those pictures now, the former on top of
+% \mfc{currentpicture}, the latter underneath. This might extend the bbox
+% calculated above, but that is one of the effects we \emph{want} to
+% achieve. Picture variables can consume a lot of memory, so we clear
+% each one after we have added it. Unfortunately, we will temporarily
+% have two copies of the current picture in memory for background text, so
+% we perform this operation only if \gbc{havebackground} is true.
+% \begin{macrocode}
+ if havebackground:
+ addto background_labels also active_plane;
+ active_plane := background_labels;
+ background_labels := nullpicture;
+ fi
+ addto active_plane also foreground_labels;
+ foreground_labels := nullpicture;
+ endfig;
+%</MP>
+enddef;
+
+% \end{macrocode}
+%
+%
+%
+% \section{Text}\label{text}
+%
+% In the \MP{} version, \gbc{label_adjust}, \gbc{label_sep} and
+% \gbc{labelpath_sep} are the equivalent of \mfpic's \cs{tlabeloffset},
+% \cs{tlpointsep} and \cs{tlpathsep}. In the \MF{} version they are still
+% needed (in \gbc{textrect}, etc.) to place the paths that are to surround
+% the text that \TeX{} places.
+%
+% \gbc{label_adjust} is a vector displacement applied to all labels,
+% while \gbc{label_sep} is the distance from the label to
+% the point of placement, when that point is on the edges of the label's
+% bounding box. Both are in device coordinates (e.g., \mfc{3bp}).
+% Finally, \gbc{labelpath_sep} is the separation of a surrounding path
+% from the text.
+% \begin{macrocode}
+pair label_adjust;
+label_adjust := origin;
+numeric label_sep, labelpath_sep ;
+label_sep := 0; labelpath_sep := 0;
+% \end{macrocode}
+%
+% Another aspect of trying to make \mfpic's \file{.mp} and \file{.mf}
+% the same, we here define a version of \mfc{verbatimtex} for \MF. This
+% works only if \mfc{etex} is followed by a semicolon, and no semicolons
+% appear in the \TeX{} material. (There may be other forbidden things, and
+% certainly any parentheses have to be in matching pairs. Not so obvious
+% is that \cs{begingroup} and \cs{endgroup} have to be balanced: \MF{}
+% sees \cs{begingroup} as `\verb$\$' plus \mfc{begingroup}.) We would
+% like the output of \mfpic{} under the \opt{metapost} option to be usable
+% in \MF{} with minimal changes.
+% \begin{macrocode}
+%<MF>def verbatimtex text t = enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Placement of text, \MP{} only}\label{placement}
+%
+% \DescribeRoutine{newgblabel}
+% This is how \mfpic{} places labels when \opt{mplabels} is in effect.
+% Since labels will typically be \mfc{btex...etex}, which are picture
+% expressions, it will actually place any picture, \gbc{s}. If you feed it
+% a string or path, it will convert it to a picture (with the \mfc{infont}
+% operator or the \gbc{picpath} macro).
+%
+% The macro \gbc{newgblabel} takes 6 parameters. The first three
+% parameters could easily be condensed into two if \mfpic{} support were
+% all that was required, however I thought it best to make it general.
+% The parameters \gbc{hf} and \gbc{vf} are numeric, with \gbc{hf}
+% representing the fraction of the text that lies left of the point where
+% the text is placed and \gbc{vf} represents the fraction of
+% text that lies below that point. However, if the third parameter is
+% \mfc{true}, then \gbc{vf} is relative to the baseline (i.e., the depth
+% is ignored). In \mfpic{} this is only used with \gbc{vf = 0} to get
+% placement on the baseline.
+%
+% These three parameters correspond to the optional parameter of
+% \cs{tlabel} in \mfpic{} as follows:
+% \begin{itemize}
+% \item \gbc{hf} determines horizontal position: $0=\mathtt{l}$,
+% $.5=\mathtt{c}$, and $1 = \mathtt{r}$.
+% \item \gbc{vf} and \gbc{BL} determine vertical position. For placement
+% option \texttt{B}, $\mathtt{vf} = 0$ and \gbc{BL} is \mfc{true}. For the
+% rest, \gbc{BL} is \mfc{false} and \gbc{vf} corresponds as follows:
+% $0 = \mathtt{b}$, $.5 = \mathtt{c}$ and $1 = \mathtt{t}$.
+% \end{itemize}
+% The remaining parameters have the following meanings:
+% \begin{itemize}
+% \item \gbc{r} is degrees of rotation about the specified point.
+% \item \gbc{s} is a string or picture expression (typically
+% \mfc{btex ... etex} code)
+% \item \gbc{pts} is a list of pairs in graph coordinates.
+% \end{itemize}
+% First the bounding box of the picture is determined using
+% \gbc{pathdims}. (Why \texttt{\textit{path}dims}? Because it was written
+% for the paths that surround text, and was then incorporated into text
+% placement when \gbc{newgblabel} replace \gbc{gblabel}.) Then
+% \gbc{readjustdims} extends that box by \gbc{label_sep}, a new
+% reference point for the picture is calculated using
+% \DescribeRoutine{ref_shift}
+% \gbc{ref_shift}, and then \gbc{thegblabel} rotates it around the
+% reference point and adds the \gbc{label_adjust}. Finally, for each
+% \gbc{_itm} in \gbc{pts}, the result is shifted by \gbc{_itm}. If
+% \gbc{overlaylabels} is true, the label is placed on the picture
+% \gbc{foreground_labels} and added to \gbc{active_plane} at
+% \gbc{endmfpic}. If \gbc{underlaylabels} is true, it is placed in picture
+% \gbc{background_labels} and \gbc{active_plane} is placed on top of it.
+% Otherwise, it is added directly to \gbc{active_plane} and the
+% \gbc{labelbb} variables are adjusted.
+%
+% We also use \gbc{ref_shift} in \MF{} since the curves that surround text
+% require it.
+%
+% \DescribeRoutine{gblabel}
+% We keep \gbc{gblabel} for backward compatibility with old \mfpic{}
+% files, but it merely calls \gbc{newgblabel}. While the old \gbc{gblabel}
+% had the same flexibility as \gbc{newgblabel}, this one assumes that the
+% parameters are only those that \mfpic{} would write.
+%
+% We provide a null definition of newgblabel for \MF{} to allow \mfpic's
+% \file{.mp} files to be somewhat usable with minimal changes. It
+% requires a text parameter, since \MF{} would be unable to evaluate
+% \mfc{btex} expressions.
+% \begin{macrocode}
+%<*MP>
+vardef newgblabel (expr hf, vf, BL, r) (expr s) (text pts) =
+ save _lab, _ll, _ur; picture _lab; pair _ll, _ur;
+ _lab := makepicture (s);
+ pathdims (origin, _lab) (_ll, _ur);
+ readjustdims (_ll, _ur) (label_sep);
+ _lab := thegblabel (ref_shift (hf, vf, BL, _ll, _ur), r, _lab);
+ save _b; pair _b;
+ for _itm = pts:
+ _b := zconv (_itm);
+ if overlaylabels:
+ addto foreground_labels also _lab shifted _b _wc_ tlabelcolor;
+ elseif underlaylabels:
+ addto background_labels also _lab shifted _b _wc_ tlabelcolor;
+ havebackground := true;
+ else:
+ addto active_plane also _lab shifted _b _wc_ tlabelcolor;
+ labelbb.ll := pairmin (_b + llcorner _lab, labelbb.ll);
+ labelbb.ur := pairmax (_b + urcorner _lab, labelbb.ur);
+ fi
+ endfor
+%</MP>
+%<MF>vardef newgblabel (expr hf, vf, BL, r) (text s) (text pts) =
+enddef;
+
+% Assumes a+b=1 and either c+d=1 or c=d=0:
+%<MP>vardef gblabel (expr a, b, c, d, r) (expr s) (text t) =
+%<MF>vardef gblabel (expr a, b, c, d, r) (text s) (text t) =
+ newgblabel (b, d, (c = 0) and (d = 0), r) (s) (t);
+enddef;
+
+vardef ref_shift (expr hf, vf, BL, ll, ur) =
+ - ( (hf)[xpart ll, xpart ur],
+ (vf)[if BL: 0 else: (ypart ll) fi, ypart ur] )
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{thegblabel}
+% When \gbc{thegblabel} is called by the above, \gbc{p} is a text picture,
+% but it is also called by the \gbc{textrect}, etc., in which case \gbc{p}
+% is a path. This is why it is needed in the \MF{} version.
+% \begin{macrocode}
+vardef thegblabel (expr z, r, p) =
+ ((p shifted z) rotated r) shifted label_adjust
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Decorating the text, \MF{} or \MP{}}\label{decorating}
+%
+% The three macros \gbc{textrect}, \gbc{textoval} and \gbc{textellipse}
+% are designed to surround a bit of text with some curve. These macros
+% return the path in graph coordinates. In
+% \DescribeRoutine{textrect}
+% \gbc{textrect}, the path is a rectangle with optionally rounded corners.
+% The second parameter, \gbc{rad}, is the radius of quarter circles at the
+% corners (in device units). In the other two cases, the path is an
+% ellipse. They differ in the meaning of the second parameter.
+%
+% \DescribeRoutine{textoval}
+% In \gbc{textoval}, the second parameter \emph{multiplies} the ratio of
+% width to height of the text to produce the ratio for the ellipse. Thus,
+% with \gbc{mult}=1, the ratio will be the same as that of the text. In
+% \DescribeRoutine{textellipse}
+% \gbc{textellipse}, the second parameter \gbc{rat} is the actual value of
+% the ratio of width to height of the ellipse and a value of 1 produces a
+% circle. In either macro, if that parameter is 0, we draw a rectangle.
+%
+% The size of each path is determined so that, when the text is placed and
+% the path drawn, it passes through the four corners of the following
+% rectangle: the rectangle which just encloses the text plus the amount of
+% space on all sides determined by \gbc{labelpath_sep}. Note that this means
+% a rectangle with rounded corners will have larger height and width than
+% one without. These versions always center the surrounding path on the
+% the point \gbc{loc}. The extended versions (below) have the same
+% flexibility of placement as the commands that place the label being
+% surrounded.
+%
+% The first parameter \gbc{lbl} is either a pair representing the
+% height and width of the text (only possibility in \MF) or the actual
+% text. These macros are being kept for backward compatibity, but now they
+% call the extended versions that allow the path to follow arbitrary
+% text placement. The parameters \gbc{(.5,.5,false,0)} were those
+% assumed in the past version: centered at the point, with no rotation.
+%
+% The extended versions of \gbc{textoval} and \gbc{textellipse} are both
+% now implemented in a single command \gbc{xellipse}, with a boolean to
+% specify whether the aspect ratio of the text is used to calculate the
+% aspect of the ellipse.
+% \begin{macrocode}
+vardef textrect (expr lbl, rad, loc) =
+ textrectx (.5, .5, false, 0) (origin, lbl, rad, loc)
+enddef;
+vardef textoval (expr lbl, mult, loc) =
+ xellipse (true, .5, .5, false, 0) (origin, lbl, mult, loc)
+enddef;
+vardef textellipse (expr lbl, rat, loc) =
+ xellipse (false, .5, .5, false, 0) (origin, lbl, rat, loc)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{textrectx}
+% Macro \gbc{textrectx} is the extended version of \gbc{textrect} which
+% allows the same adjustments to the rectangle that we can apply to the
+% text it surrounds (via \gbc{newgblabel}). In fact, it calculates the
+% position in exactly the same manner as that macro, and the first 4
+% parameters encode that position in the same way.
+%
+% The placement of each path is: shifted and rotated by the same amount
+% as the text (by \gbc{ref_shift}) according to the first four parameters,
+% then shifted to the point given in the third parameter \gbc{loc}, and
+% finally shifted by the vector specified in \gbc{label_adjust}.
+%
+% \gbc{lbl} is either the upper right corner of the text or the label
+% itself. In the first case \gbc{xy} is the lower left corner, in the
+% second case it is a dummy parameter, the bounding box being obtained (in
+% \gbc{pathdims}) by measuring the label. For these extended macros, the
+% parameters \gbc{lbl}, \gbc{mult}, \gbc{rad}, and \gbc{loc} are as in
+% the unextended versions.
+%
+% \DescribeVariable{roundends}
+% The variable \gbc{roundends} is a boolean. We really only need it to be
+% a type distinguishable from any numeric value. \Mfpic{} users can
+% specify it rather than an explicit radius, and when the code of
+% \gbc{textrectx} detects this, it uses the maximum radius for the corners
+% (making the short side of the `rectangle' a semicircle). That is, if
+% \gbc{rad} is a boolean (and \mfc{true}) then the radius at the corners
+% is so chosen. If \gbc{rad} is \mfc{false} the corners are not rounded at
+% all.
+% \begin{macrocode}
+boolean roundends; roundends := true;
+vardef textrectx (expr a, b, c, rot, xy, lbl, rad, loc) =
+ save ll, ur, _r, f, zz;
+ pair ll, ur, zz; path f;
+ pathdims (xy, lbl) (ll, ur);
+ readjustdims (ll, ur) (labelpath_sep)
+ _r := if numeric rad: rad
+ elseif not boolean rad: 0
+ elseif rad: emin (xpart(ur-ll), ypart (ur-ll))/sqrt(2)
+ else: 0
+ fi;
+ if _r = 0:
+ f := rect (ll, ur);
+ else:
+ save p, q;
+ pair p[]; path q;
+ p1 := ur - _r*dir(45);
+ p3 := ll + _r*dir(45);
+ p2 := (xpart p3, ypart p1);
+ p4 := (xpart p1, ypart p3);
+% \end{macrocode}
+% We allow the rounding radius to be negative and make the corners
+% indented in that case. We no longer reverse the path in this case.
+% \begin{macrocode}
+ q := if _r < 0: reverse fi quartercircle scaled 2_r;
+ f :=
+ (q shifted p1)--(q rotated 90 shifted p2)
+ --(q rotated 180 shifted p3)
+ --(q rotated -90 shifted p4)--cycle;
+
+ fi
+ readjustdims (ll, ur) (label_sep - labelpath_sep);
+ invvconv (thegblabel (ref_shift(a, b, c, ll, ur), rot, f))
+ shifted loc
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{textellipsex}
+% The macro \gbc{textellipsex} is a simlar extension for
+% \gbc{textellipse}. It and the related macro
+% \DescribeRoutine{textovalx}
+% \gbc{textovalx} now call a common macro with different values of a
+% boolean parameter.
+% \begin{macrocode}
+def textovalx = xellipse (true) enddef;
+def textellipsex = xellipse (false) enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{xellipse}
+% In \gbc{xellipse}, \gbc{aa} and \gbc{bb} are the horizontal and
+% vertical radii of the resulting ellipse, while \gbc{ww} and \gbc{hh}
+% are half the width and height size of the text. If the boolean
+% \gbc{aspect} is true, the aspect ratio of the ellipse (i.e., \gbc{aa/bb})
+% equals \gbc{mult*ww/hh}, otherwise it equals \gbc{mult}.
+% \begin{macrocode}
+vardef xellipse (expr aspect, a, b, c, r, xy, lbl, mult, loc) =
+ if mult = 0:
+ textrectx (a, b, c, r) (xy, lbl, 0, loc)
+ else:
+ save ll, ur, cc, ww, hh, f;
+ pair ll, ur, cc; path f;
+ pathdims (xy, lbl) (ll, ur);
+ readjustdims (ll, ur) (labelpath_sep)
+ cc := .5[ll, ur];
+ (ww, hh) = ur - cc;
+ if (ww = 0) or (hh = 0):
+ f = (ll--ur);
+ else:
+ save aa, bb;
+% \end{macrocode}
+%
+% The \gbc{aa} and \gbc{bb} are now calculated in a way that decreases the
+% chance of overflow. As a side effect, negative \gbc{mult} no longer
+% reverses the path.
+% \begin{macrocode}
+ aa := ww ++ if aspect: ww else: hh fi *mult;
+ bb := hh ++ if aspect: hh else: ww fi /mult;
+ f := ellipse (cc, aa, bb, 0);
+ fi
+ readjustdims (ll, ur) (label_sep - labelpath_sep);
+ invvconv (thegblabel (ref_shift(a, b, c, ll, ur), r, f))
+ shifted loc
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{pathdims}
+% This has been changed to make the code of \mfpic{} a bit simpler and
+% to aid in backward compatibility. It takes a couple of pairs (the actual
+% or nominal label bounding box corners) or something visible (picture,
+% string or path) and assigns suitable values to \gbc{ll} and \gbc{ur}.
+%
+% \DescribeRoutine{readjustdims}
+% This is used to add the separations needed to implement the effects of
+% \gbc{label_sep} and \gbc{labelpath_sep}.
+% \begin{macrocode}
+def pathdims (expr xy, lbl) (suffix ll, ur) =
+ if pair lbl:
+ ll := xy; ur := lbl;
+ else:
+%<MF> ll := ur := origin;
+%<*MP>
+ setpicture (_lbl) makepicture (lbl);
+ ll := llcorner _lbl;
+ ur := urcorner _lbl;
+%</MP>
+ fi
+enddef;
+
+def readjustdims (suffix ll, ur) (expr s) =
+ ll := ll - s*(1,1);
+ ur := ur + s*(1,1);
+enddef;
+
+% \end{macrocode}
+%
+%
+%
+% \section{Additional Functions}\label{functions}
+%
+% Complex variable functions are provided, which interpret a pair $(x, y)$
+% as the complex number $z = x + iy$. We also provide for the use of
+% radians, add the standard exponential and logarithms, and add the
+% hyperbolic functions and their inverses.
+%
+% Normally \mfc{infinity = 2**12 - epsilon} is the largest number allowed
+% (as a value involved in actual drawing in \MF). Since we set
+% \mfc{warningcheck=0}, values not assigned to a variable and not
+% written to the \file{.tfm} file (and any value in \MP) can be as high as
+% \mfc{2**15 - epsilon}, which is a speck smaller than \mfc{1/(2epsilon)}.
+% So \gbc{reallysmall} is the smallest number whose reciprocal is a
+% usable number. (\mfc{epsilon} is the smallest possible positive number
+% in \MF.)
+%
+% The value \gbc{eps/2 + epsilon} is the smallest value with
+% reciprocal less than \mfc{infinity}. I set \gbc{nottoosmall} to
+% \gbc{eps/2 + 2epsilon} to ensure that the same is true of
+% \gbc{2*(nottoosmall/2)}. This is probably not necessary as
+% \mfc{epsilon/2} should round up to \mfc{epsilon} and not be lost. But
+% it also ensures that \gbc{nottoosmall} equals \gbc{2*(nottoosmall/2)},
+% which could be useful.
+%
+% We set \gbc{secd x = 1/(cosd x)} unless \gbc{cosd x} is less than
+% \gbc{reallysmall}, then we set it equal to \gbc{1/reallysmall}. We do a
+% similar thing with \gbc{cscd}. (When such a substitution happens
+% \DescribeRoutine{TruncateWarn}
+% \gbc{TruncateWarn} prints a message that a truncation has taken place.)
+%
+% Why not just determine what number will produce arithmetic overflow and
+% test for that? Because I'm lazy: it would require a different number
+% for each of the functions. Instead, since \MF{} has no `arithmetic
+% underflow', I compute something that is guaranteed to work and occurs
+% in the formula for the function as a reciprocal (e.g., $t = e^{-|x|}$
+% for \gbc{cosh x}) and make sure the number is not too small to take its
+% reciprocal.
+%
+% \DescribeRoutine{signof}
+% This expands to a minus sign if its argument is negative, otherwise
+% nothing.
+% \begin{macrocode}
+newinternal reallysmall; reallysmall := 3epsilon;
+newinternal nottoosmall; nottoosmall := eps/2 + 2epsilon;
+def signof (expr X) = if X < 0: - fi enddef;
+def TruncateWarn expr s =
+ GBwarn s & " is too large or undefined, so it will be truncated.";
+enddef;
+
+% \end{macrocode}
+%
+% In addition to \mfc{sind} and \mfc{cosd} which take angles in degrees,
+% we define the remaining trig functions \gbc{tand}, \gbc{cotd},
+% \gbc{secd}, and \gbc{cscd}.
+%
+% We define \RoutineIndex{secd}\gbc{secd}, one of the simplest, to include
+% an out of range test (which also prevents division by 0). Then
+% \RoutineIndex{tand}\gbc{tand} can make use of it without any division.
+% We do the same with \RoutineIndex{cscd}\gbc{cscd} and
+% \RoutineIndex{cotd}\gbc{cotd}.
+% \begin{macrocode}
+vardef secd primary X =
+ setnumeric (temp) cosd(X);
+ if abs(temp) < reallysmall:
+ TruncateWarn "Secant or Tangent";
+ temp := signof (temp) reallysmall;
+ fi
+ 1/temp
+enddef;
+vardef tand primary X = sind(X)*secd(X) enddef;
+
+vardef cscd primary X =
+ setnumeric (temp) sind(X);
+ if abs(temp) < reallysmall:
+ TruncateWarn "Cosecant or Cotangent";
+ temp := signof(temp) reallysmall;
+ fi
+ 1/temp
+enddef;
+vardef cotd primary X = cosd(X)*cscd(X) enddef;
+
+% \end{macrocode}
+%
+% These are the inverse functions, which return an angle in degrees:
+% \RoutineIndex{acos}\gbc{acos}, \RoutineIndex{asin}\gbc{asin} and
+% \RoutineIndex{atan}\gbc{atan}.
+% \begin{macrocode}
+vardef acos primary X =
+ if abs X > 1:
+ TruncateWarn "Argument of arccosine";
+ angle (signof(X) 1, 0)
+ else:
+ angle (X, 1 +-+ X)
+ fi
+enddef;
+vardef asin primary X =
+ if abs X > 1:
+ TruncateWarn "Argument of arcsine";
+ angle (0, signof(X) 1)
+ else:
+ angle (1 +-+ X, X)
+ fi
+enddef;
+vardef atan primary X = angle (1, X) enddef;
+
+% \end{macrocode}
+%
+% Now the trig functions that take angles in radians:
+% \RoutineIndex{sin}\gbc{sin}, \RoutineIndex{cos}\gbc{cos},
+% \RoutineIndex{tan}\gbc{tan}, \RoutineIndex{cot}\gbc{cot},
+% \RoutineIndex{sec}\gbc{sec} and \RoutineIndex{csc}\gbc{csc}.
+% \begin{macrocode}
+vardef sin primary X = sind (X*radian) enddef;
+vardef cos primary X = cosd (X*radian) enddef;
+vardef tan primary X = tand (X*radian) enddef;
+vardef cot primary X = cotd (X*radian) enddef;
+vardef sec primary X = secd (X*radian) enddef;
+vardef csc primary X = cscd (X*radian) enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{degrees}
+% It is useful to have a command to convert from radians to degrees and
+% one to
+% \DescribeRoutine{radians}
+% convert from degrees to radians. For example, \gbc{degrees(pi)} produces
+% (approximately) $180$ and \gbc{radians(180)} is approximately $\pi$.
+% \begin{macrocode}
+vardef degrees (expr t) = t*radian enddef;
+vardef radians (expr t) = t/radian enddef;
+
+% \end{macrocode}
+%
+% And the inverses (\RoutineIndex{invsin}\gbc{invsin},
+% \RoutineIndex{invcos}\gbc{invcos} and \RoutineIndex{invtan}\gbc{invtan})
+% that return angles in radians.
+% \begin{macrocode}
+vardef invcos primary X = radians (acos X) enddef;
+vardef invsin primary X = radians (asin X) enddef;
+vardef invtan primary X = radians (atan X) enddef;
+
+% \end{macrocode}
+%
+% Here we define the standard exponential function
+% \RoutineIndex{exp}\gbc{exp}. (The \MF{} function \mfc{mexp} has the
+% unusual base $e^{1/256}$ to avoid overflow.) The inverse of \gbc{exp} is
+% the natural logarithm (\RoutineIndex{ln}\gbc{ln} or
+% \RoutineIndex{log}\gbc{log}). We also have the general base logarithm
+% \RoutineIndex{logbase}\gbc{logbase} and its two special instances
+% \RoutineIndex{logtwo} \gbc{logtwo} and \RoutineIndex{logten}\gbc{logten}.
+% \begin{macrocode}
+vardef exp primary X = mexp (256 * X) enddef;
+vardef ln primary X = (mlog X) / 256 enddef;
+vardef log primary X = ln (X) enddef;
+vardef logbase (expr B) primary X = (mlog X)/(mlog B) enddef;
+vardef logtwo primary X = logbase( 2) (X) enddef;
+vardef logten primary X = logbase(10) (X) enddef;
+
+% \end{macrocode}
+%
+% The hyperbolic functions: \RoutineIndex{cosh}\gbc{cosh}
+% \RoutineIndex{sinh}\gbc{sinh}, \RoutineIndex{tanh}\gbc{tanh},
+% \RoutineIndex{sech}\gbc{sech}, \RoutineIndex{csch}\gbc{csch} and
+% \RoutineIndex{coth}\gbc{coth}.
+% \begin{macrocode}
+vardef cosh primary X =
+ setnumeric (temp) 2 exp (-abs(X));
+ if temp < reallysmall:
+ TruncateWarn "Cosh";
+ temp := reallysmall;
+ fi
+ 1/temp + temp/4
+enddef;
+
+vardef sinh primary X =
+ setnumeric (temp) 2 exp (-abs(X));
+ if temp < reallysmall:
+ TruncateWarn "Sinh";
+ temp := reallysmall;
+ fi
+ signof (X) (1/temp - temp/4)
+enddef;
+
+vardef sech primary X =
+ setnumeric (temp) exp(-(abs (X)));
+ 2temp/(1 + temp*temp)
+enddef;
+
+vardef tanh primary X =
+ setnumeric (temp) exp(-2(abs (X)));
+ signof (X) (1 - temp)/(1 + temp)
+enddef;
+
+vardef csch primary X =
+ save temp, tempa; temp := exp(-(abs (X)));
+ tempa := (1 - temp*temp)/2;
+ if tempa < reallysmall:
+ TruncateWarn "Csch";
+ tempa := reallysmall;
+ fi
+ signof (X) temp / tempa
+enddef;
+
+vardef coth primary X =
+ setnumeric (temp) tanh(X);
+ if abs(temp) < reallysmall:
+ TruncateWarn "Coth";
+ temp := signof (X) reallysmall;
+ fi
+ 1/temp
+enddef;
+
+% \end{macrocode}
+%
+% The inverses of some of the hyperbolic functions:
+% \RoutineIndex{acosh}\gbc{acosh}, \RoutineIndex{asinh}\gbc{asinh} and
+% \RoutineIndex{atanh}\gbc{atanh}.
+% \begin{macrocode}
+vardef acosh primary y =
+ if y < 1:
+ TruncateWarn "acosh";
+ 0
+ else:
+ ln (y + (y +-+ 1))
+ fi
+enddef;
+
+vardef asinh primary y = ln (y + (y ++ 1)) enddef;
+
+vardef atanh primary y =
+ if abs (y) < 1:
+ (ln (1 + y) - ln (1 - y))/2
+ else:
+ TruncateWarn "atanh";
+ signof (y) infinity
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \CMF's pair variables are a decent replacement for complex variables.
+% These give some of the more basic functions of standard complex
+% analysis: \RoutineIndex{Arg}\gbc{Arg}, \RoutineIndex{Log}\gbc{Log},
+% \RoutineIndex{cis}\gbc{cis}, \RoutineIndex{zexp}\gbc{zexp},
+% \RoutineIndex{sgn}\gbc{sgn}, \RoutineIndex{zsqrt}\gbc{zsqrt} and
+% \RoutineIndex{conj}\gbc{conj}.
+% \begin{macrocode}
+vardef Arg primary Z = (angle Z)/radian enddef;
+vardef Log primary Z = (ln (abs Z), Arg Z) enddef;
+vardef cis primary T = dir (T*radian) enddef;
+vardef zexp primary Z = (exp (xpart Z)) * cis (ypart Z) enddef;
+vardef sgn primary Z = if not (Z = origin): unitvector fi Z
+enddef;
+vardef zsqrt primary Z =
+ if Z = origin: origin else: sqrt(abs(Z)) * dir ((angle Z)/2) fi
+enddef;
+vardef conj primary Z = (xpart Z, -ypart Z) enddef;
+
+% \end{macrocode}
+%
+% DescribeRoutine{zmul}
+% Unfortunately, while \MF{} will happily add and subtract pairs, it
+% will not multiply or divide them without help. We provide alternatives
+% \DescribeRoutine{zdiv} here.
+% \begin{macrocode}
+primarydef Z zmul W = Z zscaled W enddef;
+primarydef Z zdiv W =
+ Z zmul ( unitvector (conj W) / (abs W) )
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{Moebius}
+% A less basic operation: the Moebius shift which takes the disk $|z| <
+% 1$ onto itself. It is a hyperbolic geometry analog of shifting points
+% in Euclidean geometry. Its mathematical definition (all variables are
+% complex numbers):
+% \[
+% M_a(z) = \frac{z + a}{1 + \bar az}
+% \]
+% Its inverse is $M_{-a}$.
+%
+% \DescribeRoutine{pshdist}
+% Related to \gbc{Moebius} is the pseudohyperbolic metric. The distance
+% between $z$ and $w$ in this metric is $|z-w|/|1 - \bar wz|$. There is
+% \DescribeRoutine{pshdist_hp}
+% also a version of this for the upper half-plane: $|z-w|/|z-\bar w|$.
+%
+% Closely related to all this is Kelvin transform. In complex notation
+% it is simply $1/\bar z = z/|z|^2$. The term ``Kelvin transform'' is
+% normally only used in real variables (of any dimension greater than 1).
+% \begin{macrocode}
+vardef Moebius (expr A) primary Z =
+ save _D; pair _D;
+ _D := (1, 0) + (Z zscaled (conj A));
+ (Z + A)/(abs _D) rotated (- angle _D)
+enddef;
+vardef pshdist (expr Z,W) = abs(Moebius(-W)(Z)) enddef;
+vardef pshdist_hp (expr Z,W) = abs(Z-W)/abs(Z-conj(W)) enddef;
+vardef kelvin (expr Z) =
+ save tmp_; tmp_ = abs(Z);
+ if tmp_ = 0:
+ (infinity, infinity)
+ elseif tmp_ < reallysmall:
+ infinity*unitvector Z
+ else:
+ (1/tmp_)*unitvector Z
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{polar}
+% \gbc{polar} converts a polar coordinate pair $(r, \theta)$ to the
+% corresponding rectangular coordinate pair.
+% \DescribeRoutine{id}
+% \gbc{id} returns its argument, which can be any expression of any type.
+% \begin{macrocode}
+vardef polar primary p = (xpart p) * dir (ypart p) enddef;
+def id (expr x) = x enddef;
+
+% \end{macrocode}
+%
+% The definition of powers (\gbc{x**y}) in \prog{plain} \MF{} and \MP{}
+% could be more accurate. In particular \gbc{x=2**10} ought to be an
+% integer (that is, satisfy \mfc{x=floor x}). Here we redefine
+% \prog{plain}'s \mfc{**}, intercepting the case of a positive integer
+% power of an integer.
+%
+% There are some negative powers, and some integer powers of nonintegers
+% that can also be calculated exactly within \MF{}'s limited precision,
+% but it is difficult to determine those cases programmatically. Computing
+% every integer power by repeated multiplication or division might
+% actually reduce accuracy in the nonexact cases, so we limit ourselves to
+% this one special case.
+% \begin{macrocode}
+primarydef x**y =
+ if y=2: x*x
+ elseif (x = floor x) and (abs y = floor y):
+ 1 for n=1 upto y: *x endfor
+ else: takepower y of x
+ fi
+enddef;
+let ^ = **;
+
+% \end{macrocode}
+%
+%
+% \section{Coordinate Systems and Transformations}\label{systems}
+%
+% \DescribeVariable{T_stack}
+% We want to define a localization of the current transform. To do
+% this we define a LIFO stack of transforms \gbc{T_stack[\,]}, and a pair of
+% macros.
+% \DescribeRoutine{T_push}
+% \gbc{T_push} puts its argument (a transform) on the stack, and
+% \DescribeRoutine{T_pop}
+% \gbc{T_pop} pops it off into its argument (a transform variable name).
+% We also define two localizing macros
+% \DescribeRoutine{bcoords}
+% \gbc{bcoords} that pushes our \gbc{ztr}
+% on the stack, and
+% \DescribeRoutine{ecoords}
+% \gbc{ecoords} that pops it off. We no longer put \gbc{vtr} on the stack,
+% since we can recalculate it whenever \gbc{ztr} is changed. \gbc{apply_t}
+% always did this, now \gbc{ecoords} does so as well.
+% \begin{macrocode}
+transform T_stack[];
+numeric T_stack; T_stack := 0;
+
+def T_push (expr T) = T_stack[incr T_stack] := T; enddef;
+def T_pop (suffix $) =
+ if T_stack > 0:
+ $ := T_stack[T_stack];
+ T_stack := T_stack - 1;
+ fi
+enddef;
+
+def bcoords = hide ( T_push (ztr) ) enddef;
+def ecoords = hide ( T_pop (ztr); vtr := vectorpart ztr ) enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Coordinate changes}\label{changes}
+%
+% \DescribeRoutine{apply_t}
+% Here we define a mechanism for changing \gbc{ztr} and \gbc{vtr} by
+% composing them with a new transform. Since a transform can be any affine
+% transform, we get \gbc{ztr} by composing with the transform, but we
+% calculate \gbc{vtr} from \gbc{ztr} by arranging that \mfc{origin
+% transformed vtr} is \mfc{origin}. The syntax is \gbc{apply_t(rotated
+% theta)} or \gbc{apply_t(transformed T)} if \mfc{T} is a variable or
+% expression of type transform. Thus the argument of \gbc{apply_t} is a
+% phrase which, were it to follow a path, would produce a transformed
+% path. Knuth calls such a phrase a \emph{transformer}.
+% \begin{macrocode}
+vardef vectorpart primary T = T shifted -(origin transformed T)
+enddef;
+
+def apply_t (text Transformer) =
+ ztr := identity Transformer transformed ztr;
+ vtr := vectorpart ztr;
+enddef;
+
+% \end{macrocode}
+%
+% And now we define some available transformers.
+% \RoutineIndex{xslant}\gbc{xslant}, \RoutineIndex{yslant}\gbc{yslant},
+% \RoutineIndex{zslant}\gbc{zslant}, \RoutineIndex{xyswap}\gbc{xyswap} and
+% \RoutineIndex{boost}\gbc{boost}. The only two that need comment are
+% \gbc{zslant} and \gbc{boost}. I know that boost comes from special
+% relativity, but I have no idea why zslant is a `slant'.
+% \begin{macrocode}
+def xslant = slanted enddef; % (x+sy, y).
+def yslant primary s = % (x, y+sx).
+ transformed
+ begingroup
+ save T; transform T;
+ origin transformed T = origin;
+ (1, 0) transformed T = (1, s);
+ (0, 1) transformed T = (0, 1);
+ T
+ endgroup
+enddef;
+
+def zslant primary p = % (xu+yv, xv+yu), where p = (u, v).
+ transformed
+ begingroup
+ save T; transform T;
+ xpart T = ypart T = 0;
+ xxpart T = yypart T = xpart p;
+ xypart T = yxpart T = ypart p;
+ T
+ endgroup
+enddef;
+
+def xyswap = zslant (0, 1) enddef;
+def boost primary X = zslant (cosh X, sinh X) enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Path transformation}\label{transformation}
+%
+% These are functions that accept and return a path in graph coordinates.
+% For the most part they are named and defined to apply a similarly named
+% transform to the path and return the result. There are two exceptions.
+% When we draw things, we expect that rotated and reflected objects appear
+% congruent to the originals. If we define a path in graph coordinates,
+% and the $x$ and $y$ directions are scaled differently, then simply
+% rotating the graph coordinates will distort angles. The same is true of
+% reflection. Therefore, we apply \gbc{vtr} (so we are in drawing
+% coordinates) then rotate or reflect, then apply \gbc{inverse vtr}. This
+% may be a mistake, or perhaps we should do it for all of these. For now,
+% I'm sticking with the scheme I inherited. One can always use
+% \gbc{coords} and \gbc{apply_t} if one wants the difference in scales
+% ignored.
+%
+% \DescribeRoutine{transformedpath}
+% This is a vardef that reads an undelimited path expression and returns
+% the path transformed by the text argument. All the others run this,
+% allowing it to grab the path expression.
+%
+% \DescribeRoutine{rotatedpath}
+% This returns the path rotated around point \gbc{p} by angle
+% \gbc{th} in degrees.
+%
+% \DescribeRoutine{reflectedpath}
+% This reflects the path through the line containing points \gbc{p} and
+% \gbc{q}.
+%
+% \DescribeRoutine{scaledpath}
+% This returns the path scaled so that distances from the point
+% \gbc{p} are multiplied by \gbc{s}.
+% \DescribeRoutine{xscaledpath}
+% \gbc{xscaledpath} is similar, but only the horizontal distances from
+% the line $x={}$\gbc{a} are multiplied by \gbc{s}. And with
+% \DescribeRoutine{yscaledpath}
+% \gbc{yscaledpath} the vertical distances from the line $y={}$\gbc{b} are
+% multiplied by \gbc{s}.
+%
+% \DescribeRoutine{xslantedpath}
+% The macro \gbc{xslantedpath} returns the path xslanted with line
+% $y = {}$\gbc{b} being the pivot rather than the $x$-axis.
+% \DescribeRoutine{slantedpath}
+% The command \gbc{slantedpath} is just an alias for \gbc{xslantedpath},
+% while
+% \DescribeRoutine{yslantedpath}
+% \gbc{yslantedpath} is the vertical version, yslanted with line $x =
+% {}$\gbc{a} being the pivot rather than the $y$-axis.
+%
+% \DescribeRoutine{shiftedpath}
+% This returns the path shifted by the vector (pair) \gbc{v}.
+%
+% \DescribeRoutine{xyswappedpath}
+% The command \gbc{xyswappedpath} returns the path in which all points
+% have had the coordinates exchanged $(a, b) \to (b, a)$. Note that this
+% is not the same as \gbc{reflectedpath ((0,0), (1,1))}, as it performs
+% the reflection in graph coordinates, as its name implies. If \gbc{vtr}
+% has not been changed (by \gbc{apply_t}) then \gbc{xyswappedpath} will
+% convert vertical lines to horizontal and vice versa. The
+% \gbc{reflectedpath} version will not when $x$ and $y$ are scaled
+% differently, for then the line \gbc{(0,0)--(1,1)} is not at a 45 degree
+% angle in device coordinates where drawing takes place.
+%
+% \begin{macrocode}
+vardef transformedpath (text Transformer) expr f = f Transformer
+enddef;
+
+def rotatedpath (expr p, th) =
+ transformedpath (
+ transformed vtr
+ rotatedaround (p transformed vtr, th)
+ transformed (inverse vtr)
+ )
+enddef;
+def reflectedpath (expr p, q) =
+ transformedpath (
+ transformed vtr
+ reflectedabout (p transformed vtr, q transformed vtr)
+ transformed (inverse vtr)
+ )
+enddef;
+
+def scaledpath (expr p, s) =
+ transformedpath (shifted -p scaled s shifted p)
+enddef;
+def xscaledpath (expr a, s) =
+ transformedpath (shifted (-a, 0) xscaled s shifted (a, 0))
+enddef;
+def yscaledpath (expr b, s) =
+ transformedpath (shifted (0, -b) yscaled s shifted (0, b))
+enddef;
+
+def slantedpath = xslantedpath enddef;
+def xslantedpath (expr b, s) =
+ transformedpath (shifted (0, -b) slanted s shifted (0, b))
+enddef;
+def yslantedpath (expr a, s) =
+ transformedpath (shifted (-a, 0) yslant s shifted (0, a))
+enddef;
+
+def shiftedpath (expr v) = transformedpath (shifted v) enddef;
+
+def xyswappedpath = transformedpath (xyswap) enddef;
+
+% \end{macrocode}
+%
+% It seems odd, in retrospect, that we got by with a user interface that
+% didn't include any subpath operations. But recently a user asked for the
+% ability to add an arrowhead to the \emph{middle} of a path, and it
+% seemed best to provide a subpath and use existing commands to add an
+% arrowhead on its end.
+%
+% \DescribeRoutine{partialpath}
+% The \gbc{partialpath} macro takes two fractions $\alpha$ and $\beta$
+% between 0 and 1, and a path \gbc{f}, and returns the subpath from
+% $\alpha * {} $\meta{length of \gbc{f}} to $\beta * {}$\meta{length of
+% \gbc{f}} of \gbc{f}. Since the \gbc{gettime} routine was written to
+% find the times for an increasing sequence of lengths, it was optimized
+% to save the index of the previous length and begin from there. Thus it
+% is more efficient to find the smaller of \gbc{a} and \gbc{b} first.
+%
+% Since running \gbc{gettime} would be a very inefficient way to get the
+% first or last point of a path we skip that if either fraction is $0$ or
+% $1$ (a common use is to get the first or last half of a path). We also
+% skip finding the second time if \gbc{a = b} (an unlikely choice, but
+% legal).
+%
+% \DescribeRoutine{gsubpath}
+% \gbc{gsubpath} is the same as \MF's subpath primitive, but follows the
+% prefix macro syntax of accepting a path expression (rather than a
+% primary) and wrapping the result in a \mfc{vardef}.
+% \begin{macrocode}
+vardef partialpath (expr a, b) expr f =
+ save flag, flo, fhi, lo, hi, n;
+ boolean flag; flag = true;
+ convertpath (g) f;
+ n := length f;
+
+ flo := snapto emin(a,b);
+ if flo = 0:
+ lo := 0;
+ elseif flo < 1:
+ setuplengtharray (cum, tot, idx) g;
+ flag := false;
+ lo := gettime (cum, idx) (flo*tot);
+ else:
+ lo := n;
+ fi
+
+ fhi := snapto emax (a,b);
+ if flo = fhi:
+ hi := lo;
+ elseif fhi < 1:
+ if flag: setuplengtharray (cum, tot, idx) g; fi
+ hi := gettime (cum, idx) (fhi*tot);
+ else:
+ hi := n;
+ fi
+
+ if a > b: reverse fi subpath (lo, hi) of f
+enddef;
+
+vardef gsubpath (expr a, b) expr f = subpath (a, b) of f enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{setuplengtharray}
+% This does the frequently repeated saving, rescaling and initializing
+% for those commands that need to convert distance along a path to the
+% corresponding time or point. A path variable should follow, but that
+% is picked up by the \gbc{makelengtharry} at the end.
+% \begin{macrocode}
+def setuplengtharray (suffix cum, tot, idx) =
+ save cum, tot, idx; idx := 0; tot := makelengtharray (cum)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{pathtime}
+% \gbc{pathtime} returns the time \mfc{t} such that \mfc{point t of p} is
+% \gbc{frac} of the distance along \gbc{p} from the start, and
+% \DescribeRoutine{pathpoint}
+% \gbc{pathpoint} returns the point itself. Because the \gbc{gettime}
+% routine requires it anyway, we truncate \gbc{frac} to the interval
+% $[0,1]$ and avoid calling that rather lengthy function at $0$ and $1$.
+%
+% The path in pathtime should be in device coordinates, whereas the
+% user-level command \gbc{pathpoint} expects it in graph coordinates.
+% In fact, since \gbc{pathpoint} would most likely be used in the
+% argument of some figure macro in \mfpic{}, it would require a
+% previously stored path, so we make the path a suffix parameter.
+% \begin{macrocode}
+vardef pathtime@# (suffix p) =
+ if @# <= 0: 0
+ elseif @# >= 1: length p
+ else:
+ setuplengtharray (cum, tot, idx) p;
+ gettime (cum, idx) (@#*tot)
+ fi
+enddef;
+
+vardef pathpoint (expr frac) (suffix p) =
+ convertpath (_pp) p; pnt[pathtime[frac] (_pp)] (p)
+enddef;
+
+% \end{macrocode}
+%
+%
+%
+% \section{Picture-level Operations}\label{picture}
+%
+%
+% \subsection{Bitwise logical operations}\label{logical}
+%
+% None of these operations are available in \MP. Mostly these are used by
+% higher level operations. Those higher level operations are available in
+% \MP, but need to be defined differently.
+%
+% We have two types of operations. One type is a binary operator that
+% takes two picture expressions and returns a picture, the other type
+% returns nothing, but merely modifies a given picture variable. These
+% take the name of a picture and a picture expression and modify the named
+% one. The binary operators are not used elsewhere in graphbase except
+% for \gbc{picsub}, which occurs only in \gbc{shadepic}. They are all
+% rather wasteful of memory.
+%
+% \DescribeRoutine{mono}
+% Here we define the bitwise logical operations: and, or, xor, and
+% difference. These mostly only work if all pixels have values 0 or 1.
+% Since \MF{} allows other integer values, we define a \gbc{mono} operator
+% that converts all pixels with weight ${}\ge 1$ to 1 and all pixels
+% with weight ${}\le 0$ to 0. It is important to note that we can apply
+% \gbc{mono} only to the suffix parameter in such things as \gbc{orto}.
+% The expression parameter needs to be prepared by the routine that calls
+% these. The return result is culled, so it consists only of 0s and 1s.
+% \begin{macrocode}
+%<*MF>
+def mono (suffix u) = cull u keeping (1, infinity); enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{andto, picand}
+% The bitwise and: in the resulting picture, a pixel is \emph{on} if and
+% only if it is \emph{on} in both \gbc{u} and \gbc{v}. \gbc{andto} is
+% only used in \gbc{interior} and \gbc{interiors}, \gbc{picand} is not
+% used at all.
+% \begin{macrocode}
+def andto (suffix u) (expr v) =
+ mono (u); addto u also v; cull u keeping (2, 2);
+enddef;
+primarydef u picand v =
+ begingroup setpicture (t) u; andto (t, v); t endgroup
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{orto, picor}
+% The inclusive or: in the result, a pixel is \emph{on} if and only if it
+% is \emph{on} in \gbc{u} or \gbc{v} or both. I've written these so that
+% it doesn't matter if the expression parameter is not mono. It
+% \emph{is} required that it have only positive pixels. The command
+% \gbc{orto} is only used three places: in \gbc{coloraddto}, which
+% is itself never used, and in \gbc{patcharcs} and \gbc{patchrays}. This
+% lack of use is because a less memory intensive version, \gbc{_orto}, is
+% defined later, and that is what we use. Usually we build a picture
+% in a variable \gbc{src} and add that onto another variable \gbc{dest}.
+% If one used \gbc{orto (dest, src)}, then \MF{} would evaluate \gbc{src}
+% and pass a \emph{copy} of it as the parameter of \gbc{orto}. This
+% doubles the memory used, so mostly we use \gbc{_orto}, which passes both
+% parameters as suffixes. \gbc{picor} is never used.
+% \begin{macrocode}
+def orto (suffix u) (expr v) =
+ mono (u); addto u also v; cull u keeping (1, infinity);
+enddef;
+primarydef u picor v =
+ begingroup setpicture (t) u; orto (t, v); t endgroup
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{xorto, picxor}
+% The exclusive or, also called the symmetric difference:
+% in the result, a pixel is \emph{on} if and only if it is \emph{on} in
+% \gbc{u} or \gbc{v}, but not both. These are not used elsewhere in
+% \grafbase.
+% \begin{macrocode}
+def xorto (suffix u) (expr v) =
+ mono (u); addto u also v; cull u keeping (1, 1);
+enddef;
+primarydef u picxor v =
+ begingroup setpicture (t) u; xorto (t, v); t endgroup
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{subto}
+% The nonsymmetric difference: in the result, a pixel is \emph{on} if
+% and only if it is \emph{on} in \gbc{u} and off in \gbc{v}. It is
+% unclear whether a \gbc{v} with negative weights will ever occur, but
+% if so, subtracting negative pixels ought to be like adding positive
+% ones, so I've changed \mfc{keeping (1,1)} to \gbc{keeping (1,infinity)}.
+% With this understanding, it doesn't matter here whether \gbc{v} is
+% not mono. As with \gbc{orto}, we have a more memory efficient
+% \gbc{_subto} and now use that everywhere. \gbc{subto} is only used in
+% \gbc{coloraddto}, which is not used anymore. The binop version
+% \DescribeRoutine{picsub}\gbc{picsub}
+% is used only in \gbc{shadepic}.
+% \begin{macrocode}
+def subto (suffix u) (expr v) =
+ mono (u); addto u also -v; cull u keeping (1, infinity);
+enddef;
+primarydef u picsub v =
+ begingroup setpicture (t) u; mono (t); subto (t, v); t endgroup
+enddef;
+
+%</MF>
+% \end{macrocode}
+%
+%
+% \subsection{Producing and modifying pictures}\label{pictures}
+%
+% Here we define some slightly higher level commands that make use (in \MF)
+% of the previous bitmap operations. In \MP, they mostly need different
+% definitions, but we have merged most of them by providing a \MP{}
+% alternative for the most frequently used bitmap operation in the
+% previous section, \gbc{orto}. These operations either return a picture
+% or modify a picture variable. They do not draw anything unless
+% \gbc{active_plane} is the modified picture. All curves, points,
+% dimension, etc., are in device coordinates.
+%
+% \DescribeRoutine{coloraddto}
+% This was once a useful abbreviation. In \MF{} it adds when the color
+% is not white, subtracts when it is. Grays are handles in \MF{} by
+% appropriate preparation of \gbc{u} and \gbc{v}. See, for example, the
+% code of \gbc{colorsafefill}. In \MP{} it is an abbreviation for the
+% basic \mfc{addto} operation. It was defined only so that \MP{} and \MF{}
+% can share the same higher level code.
+%
+% When the last parameter \gbc{v} is the name of picture we can save
+% memory if we pass the name rather than the value. Problems with picture
+% memory turned up in the shading macros for \MF{} and the dashing macros
+% for \MP{}.
+% \DescribeRoutine{coloraddon}
+% The macro \gbc{coloraddon} applies this memory-saving trick and has
+% completely replaced \gbc{coloraddto} in \grafbase{} code. Since
+% \gbc{coloraddto} turned out to be used only with \gbc{u} equal to
+% \gbc{active_plane}, we have eliminated that parameter from
+% \gbc{coloraddon}.
+%
+% The command \gbc{_orto} is like \gbc{orto}, but saves memory by passing
+% \emph{both} parameters by name. This also allows the application of
+% \gbc{mono} to both parameters. In addition to \gbc{coloraddon}, it is
+% used in \gbc{shade} and \gbc{tess}.
+% We also have \gbc{_subto}, an analogous version of \gbc{subto}.
+% \begin{macrocode}
+def coloraddto (expr clr) (suffix u) (expr v) =
+%<*MF>
+ if clr < white:
+ orto (u, v);
+ else:
+ subto (u, v);
+ fi;
+%</MF>
+%<MP> addto u also v _wc_ clr;
+enddef;
+
+%<MP>def orto (suffix u) (expr v) = addto u also v; enddef;
+%<MP>
+def coloraddon (expr clr) (suffix v) =
+%<*MF>
+ if clr < white:
+ _orto (active_plane, v);
+ else:
+ _subto (active_plane, v);
+ fi;
+%</MF>
+%<MP> addto active_plane also v _wc_ clr;
+enddef;
+
+def _orto (suffix u, v) =
+%<MF> mono (u); mono (v);
+ addto u also v;
+%<MF>cull u keeping (1, 2);
+enddef;
+
+%<*MF>
+def _subto (suffix u, v) =
+ mono (u); mono (v); addto u also -v; cull u keeping (1, 1);
+enddef;
+
+%</MF>
+% \end{macrocode}
+%
+% \DescribeRoutine{interior}
+% This takes the following expresion, \gbc{c}, which must be a
+% closed path, and returns the picture expression which is that path
+% filled. The cull command (\MF{} only) retains negative pixels
+% (converting them to positive). This way, clockwise contours are filled
+% also. \gbc{interior} is one of the most used commands throughout the
+% rest of \grafbase.
+%
+% We ignore color (new behavior with \mfpic{} version 0.7), since the
+% higher level commands now implement the coloring operations.
+% \begin{macrocode}
+vardef interior expr c =
+ newpicture (v);
+ addto v contour (c.t_);
+%<MF> cull v dropping (0,0);
+ v
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{interiors}
+% This is followed by the name of an array of closed paths and
+% returns the picture of the interiors of those closed paths. It builds
+% the returned picture from \mfc{nullpicture} by successively adding
+% the result of \gbc{interior} applied to each path in the array. This is
+% only used once by \grafbase, in \gbc{clipsto}, which might be a better
+% place to put the \mfc{for}-loop and not use this at all.
+% \begin{macrocode}
+vardef interiors suffix cc =
+ newpicture (_ints);
+ for _idx = 1 upto cc:
+ addto _ints also interior cc[_idx]);
+ endfor
+%<MF> mono (_ints);
+ _ints
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Clipping}\label{basicclipping}
+%
+% \DescribeRoutine{clipto}
+% \gbc{clipto} takes the name of a picture \gbc{vt} and a closed path
+% \gbc{c} and modifies the picture leaving only the part inside the path.
+% In \MP{} we just invoke the \mfc{clip} primitive.
+%
+% \DescribeRoutine{clipsto}
+% This is similar, except it takes an array of paths \gbc{cc} and
+% leaves what is interior to any of the paths. This is one case where
+% \MP{} requires a substantially different point of view. In \MF, we
+% create the interiors and `and' the result to the named picture. In \MP,
+% we have to create the picture which is \gbc{vt} clipped to each separate
+% path, and combine the results. \Grafbase{} only uses this in the
+% \gbc{DoClip} command.
+% \begin{macrocode}
+def clipto (suffix vt) expr c =
+ if path c:
+%<MF> andto (vt, interior c);
+%<MP> clip vt to c;
+ fi
+enddef;
+def clipsto (suffix vt, cc) =
+%<MF> andto (vt, interiors cc);
+%<*MP>
+ begingroup
+ save _cl, _cl_; picture _cl, _cl_; _cl_ := nullpicture;
+ for _idx = 1 upto cc:
+ _cl := vt; clip _cl to cc[_idx]; addto _cl_ also _cl;
+ endfor
+ vt := _cl_;
+ endgroup
+%</MP>
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{Clipped}
+% Here, rather than modify a given picture, \gbc{Clipped} is a vardef
+% returning the picture which is the result of clipping the given picture
+% to the path. This is not used elsewhere in \grafbase{} nor \mfpic.
+%
+% Having found out that \mfc{clipped} is a \MP{} primitive, I've
+% changed the name to the uppercase version.
+% \begin{macrocode}
+vardef Clipped (suffix vt) expr c =
+ setpicture (_Cl) vt; clipto (_Cl) c; _Cl
+enddef;
+%<MF>def clip = Clipped enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{picneg}
+% The reverse video is easy in \MF, where \gbc{picneg} takes a picture
+% name and a closed path, and returns the part of the picture inside the
+% path, but with pixels reversed. In \MP{} we can only approximate this:
+% we clip the given picture and add that (using color \gbc{background})
+% on top of the \gbc{interior} of the curve colored \gbc{fillcolor}. This
+% is not used elsewhere in \file{grafbase.mp} so it may not be really
+% important whether \gbc{fillcolor} and \mfc{background} are the right
+% choices.
+% \begin{macrocode}
+vardef picneg (suffix vt) expr c =
+%<*MF>
+ setpicture (_pn) interior c;
+ _subto (_pn, vt);
+%</MF>
+%<*MP>
+ setpicture (_cl) vt; clip _cl to c;
+ newpicture (_pn);
+ addto _pn also (interior c ) _wc_ fillcolor;
+ addto _pn also _cl _wc_ background;
+%</MP>
+ _pn
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{shpath}
+% \gbc{shpath} does most of the work of drawing curves in \grafbase. It is
+% called by \gbc{safedraw} which is used by almost all the commands that
+% somehow draw a curve. It takes the name of a picture, a pen expression
+% and a path expression. It draws the path on the picture with the pen.
+% Since we use this (ultimately) for almost all drawing of paths, we
+% automatically have the aspect ratio taken care of by the \mfc{.t_}
+% macro.
+%
+% \DescribeRoutine{picpath}
+% \gbc{picpath} accepts a path expression and returns a picture, which is
+% either \gbc{nullpicture} (\gbc{penwd} too small) or the path drawn with
+% \gbc{drawpen}. This is mostly how \gbc{shpath} gets used: curve drawing
+% commands produce a picture with \gbc{picpath} and that gets used.
+%
+% \begin{macrocode}
+def shpath (suffix v) (expr q, f) =
+ addto v doublepath (f.t_) withpen (q.t_);
+enddef;
+
+numeric minpenwd;
+%<MF>minpenwd := 1; % 1 pixel
+%<MP>minpenwd := .05bp; % 1 pixel at 1440dpi
+vardef picpath expr d =
+ newpicture (v);
+ if penwd >= minpenwd:
+ shpath (v, drawpen) (d);
+%<MF> mono (v);
+ fi
+ v
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{picdot}
+% This places a specified picture expression (\gbc{w}) at a specified
+% location (\gbc{p}) in a specified picture variable (\gbc{v}). It is used
+% a number of places. It's \MF{} version takes care of the aspect ratio
+% via \mfc{.t_}. This is how we draw points and symbols and dots along a
+% curve: make the symbol into a picture \gbc{w} and add that picture with
+% \gbc{picdot}.
+% \begin{macrocode}
+def picdot (suffix v) (expr w, p) =
+ addto v also
+%<MP> (w shifted p);
+%<MF> (w shifted goodpair (p));
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{setdot}
+% \gbc{setdot} is named for its use rather than what it does. It takes a
+% path and a scale (numeric expression) and returns a picture which is a
+% drawing of the filled interior of the path (if it is a cycle) or the
+% path itself (not a cycle). In \MF, we ensure that the scale is at least
+% one pixel (assumes that the \gbc{apath} has dimension about 1 and
+% \gbc{minpenwd} is 1). This usually assures that something is drawn. In
+% \MP, \gbc{minpenwd} has the same purpose (though it is probably not
+% necessary). This routine is used a number of times where dots are
+% needed. Not in \gbc{shaded} (just below) but later in \gbc{shade} (an
+% older command taking paths in graph coordinates), \gbc{polkadot} and
+% some grid-making commands.
+% \begin{macrocode}
+vardef setdot (expr apath, sc) =
+ if cycle apath: interior
+ else: picpath
+ fi
+%<MF> (apath scaled emax (ceiling (sc), minpenwd))
+%<MP> (apath scaled emax (sc, minpenwd))
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{shadepic}
+% We want to shade regions with a very regular pattern of black and white
+% pixels for best appearance. Experiments show that symmetric dots
+% (e.g., circles, squares) work better than non-symmetric (e.g.,
+% rectangular). Circular dots are not significantly better than square at
+% the size needed. I believe that the default result of \gbc{shade} looks
+% reasonably good on my system. (That happens to produce two 3-pixel by
+% 3-pixel square dots in a 8-pixel square on a 360dpi printer.) So we try
+% to produce something similar. That is, the shading picture is 1.6bp
+% (8 pixels at 360dpi) square.
+%
+% As a compromise (symmetric dots look better, but rectangular dots give
+% more gray levels) we allow dots to be rectangles $k\times (k+1)$-pixels
+% (assuming the aspect ratio is 1). This produces twice the number of
+% gray levels. In my 360dpi example we get 15 gray levels. The two
+% farthest apart (4 by 4 dots versus 3 by 4 dots) differ by 1/8 in
+% fraction of area of coverage (which we equate to grayness).
+%
+% Why can't we have 64 grey levels in a $8\times 8$ square? Clearly we
+% can in principle turn on any number of the 64 pixels. Unfortunately,
+% spread out patterns (which look best) tend to consume memory, while
+% clumpy patterns are hard to make good-looking. Compensating for aspect
+% ratios unequal to 1 is also pretty hard to do automatically.
+%
+% The parameter \gbc{dims} needs to be a pair variable, and it will be
+% assigned the actual dimensions of the picture returned. These routines
+% are complicated by the fact that we may have an aspect ratio unequal to
+% $1$. When \mfc{aspect_ratio = 1} the basic concept is simple: make an
+% $n\times n$ square with two dots, each nearly $k \times k$ and nearly
+% square, where $2k^2/n^2$ is the gray level needed.
+%
+% The calculations assume a gray level greater than $1/2$, so the final
+% picture will be mostly white (for darker grays, we use the
+% complementary gray level to construct the `reverse video', and then
+% reverse back). Under this assumption, we concentrate all the black
+% pixels into the lower left and upper right quadrant of the picture we
+% are creating, so most of the calculation determines one of these
+% quadrants. The scratch variables \gbc{_hp} and \gbc{_vp} give the number
+% of horizontal and vertical pixels in the lower left quadrant,
+% \gbc{_dotwd} and \gbc{_dotht} do the same for the actual dot. Then
+% \gbc{_shp} is first set equal to one dot; then a copy of itself is added
+% in the upper right quadrant. Finally, the suffix parameter \gbc{dims} is
+% equated to the nominal width and height of the picture, and either
+% \gbc{_shp} or its reverse is returned.
+% \begin{macrocode}
+%<*MF>
+numeric shadepicsize; shadepicsize := 0.8bp;
+vardef shadepic (suffix dims) (expr grparam) =
+ pair dims;
+ setnumeric (_frac) 2*emin (grparam, 1 - grparam);
+ save _hp, _vp, _dotwd, _dotht;
+ if aspect_ratio < 1:
+ _vp := emax (2, hround (shadepicsize.o_));
+ _hp := hround (_vp._o_);
+ _dotwd := hround (_hp*sqrt _frac);
+ _dotht := if _dotwd = 0: 0
+ else: hround (_hp*_vp*_frac/_dotwd)
+ fi;
+ else:
+ _hp := emax (2, hround (shadepicsize));
+ _vp := hround (_hp.o_);
+ _dotht := hround (_vp*sqrt _frac);
+ _dotwd := if _dotht = 0: 0
+ else: hround (_hp*_vp*_frac/_dotht)
+ fi;
+ fi
+ dims := ( _hp, _vp._o_ );
+ newpicture (_shp);
+ addto _shp contour rect (origin, (_dotwd, _dotht));
+ picdot (_shp, _shp, dims);
+ dims := 2dims; mono (_shp);
+ if grparam >= .5: _shp
+ else: (interior (rect (origin, dims))) picsub _shp
+ fi
+enddef;
+
+%</MF>
+% \end{macrocode}
+%
+% \DescribeRoutine{shaded}
+% This fills the interior of a contour (device coordinates) with copies of
+% \gbc{shadepic}. The routine \gbc{fillwith} is defined later, but its
+% name reflects its effect: a bounding rectangle (corners at \gbc{ll} and
+% \gbc{ur}) is filled with copies of a picture (in this case, the result
+% of \gbc{shadepic}), the picture having nominal dimensions \gbc{shdims}
+% in this case.
+%
+% It may seem odd that black and white return the same thing. That is
+% because white is handled in the calling routine by subtracting the
+% black result.
+%
+% The \gbc{setbbox} command was defined earlier, in
+% section~\ref{utilities}. The bounding rectangle it obtains is only
+% approximate in \MF{}, but that is sufficient, since we only use it to
+% produce things that are eventually clipped.
+%
+% We return \gbc{picpath} for non-cycles because I once thought to make
+% \gbc{shaded} a replacement for \gbc{setdot} to get gray dots (in the
+% \gbc{polkadot} routine). That turns out not to work, but this sort of
+% thing is also done in most of the rendering commands that require a
+% closed path.
+% \begin{macrocode}
+vardef shaded (expr clr) expr c =
+ if cycle c:
+%<*MP>
+ newpicture (v);
+ addto v contour c _wc_ clr;
+ v
+%</MP>
+%<*MF>
+ if (clr <= black) or (clr >= white):
+ interior c
+ else:
+ save shdims, shpic;
+ picture shpic; pair shdims;
+ shpic := shadepic (shdims) (clr);
+ setbbox (ll, ur) c;
+ newpicture (vsh);
+ fillwith (vsh) (shpic, shdims, ll, ur);
+ clipto (vsh) c; vsh
+ fi
+%</MF>
+ else: picpath c % should we? or just make it null?
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{fillwith}
+% This is one of the ways we obtain something other than a solid fill. The
+% routines \gbc{polkadot}, \gbc{tess} and (in \MF) \gbc{shade} and
+% \gbc{shaded} all use it.
+%
+% It takes a picture expression \gbc{pic}, along with its dimensions (the
+% pair \gbc{dims}) in device coordinates, plus the opposite corners,
+% \gbc{ll} and \gbc{ur}, of a boundingbox rectangle, and draws that
+% rectangle filled with copies of \gbc{pic}. Starting with \mfpic{}
+% version 0.8, it adds to a predefined picture passed by name. Thus the
+% calling routine must make sure that picture is initialized (it need
+% not be \mfc{nullpicture}).
+%
+% One might do this with one loop nested in another, but it turns out to
+% be much faster (surprisingly much!) to do two separate loops: the second
+% one stacking copies of the row built by the first loop.
+%
+% We try to do any rounding that might have been forgotten. This code
+% takes a mode's aspect ratio into account so that (most) calling routines
+% don't have to. (That is, \gbc{dims} should be measured in horizontal
+% pixels, while \gbc{fwdims} is in actual pixels. This could have been
+% written in terms of \gbc{picdot}, which already handles aspect, but it
+% has got to be more efficient to do the aspect ratio calculations once
+% rather than every time through the loop.)
+% \begin{macrocode}
+vardef fillwith (suffix v) (expr pic, dims, ll, ur) =
+ newpicture (b);
+%<*MF>
+ save fwdims, _ll, _ur; pair fwdims, _ll, _ur;
+ fwdims := goodpair (dims);
+ _ll := floorpair (ll.t_);
+ _ur := ur.t_;
+ for s = xpart _ll step xpart fwdims until xpart _ur:
+ addto b also pic shifted (s, 0);
+ endfor
+ for s = ypart _ll step ypart fwdims until ypart _ur:
+ addto v also b shifted (0, s);
+ endfor
+ mono (v);
+%</MF>
+%<*MP>
+ for s = xpart ll step xpart dims until xpart ur:
+ addto b also pic shifted (s, 0);
+ endfor
+ for s = ypart ll step ypart dims until ypart ur:
+ addto v also b shifted (0, s);
+ endfor
+%</MP>
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Hatching}\label{basichatching}
+%
+% \DescribeRoutine{thatchf}
+% This is the all-purpose macro called by the other macros that
+% fill a region with hatching. It takes the name of a picture \gbc{v},
+% a transform expression \gbc{CT}, a numeric expresion \gbc{sp} giving the
+% space between hatch lines, and two pairs, \gbc{a} and \gbc{b},
+% that represent the lower left and upper right limits of a rectangle.
+% The expression \gbc{sp} must be nonzero. The calling macros should take
+% care of that.
+%
+% It modifies the picture by adding to it the rectangle full of
+% hatching lines spaced \gbc{sp} apart. The rectangle is initially upright
+% and the lines horizontal, but they are drawn transformed by the
+% transform \gbc{CT}. This is how diagonal hatching is accomplished: the
+% transform is a rotation.
+%
+% We guard against \gbc{ypart a} being greater than \gbc{ypart b} or
+% \gbc{sp} being negative: \gbc{_sp} is \gbc{sp} modified to have the same
+% sign as \gbc{ypart (b - a)}. Thus, repeatedly adding it to \gbc{ypart a}
+% gets one to \gbc{ypart b}. We make the starting value an integer
+% multiple of \gbc{_sp} to make sure adjacent regions don't have jarringly
+% misaligned hatch lines. (I guess that's the reason; this algorithm
+% predates my involvement with \mfpic{}.)
+% \begin{macrocode}
+def thatchf (suffix v) (expr CT, sp, a, b) =
+ begingroup
+ setnumeric (_sp) signof (ypart b - ypart a) abs(sp);
+ for _y = _sp*( ceiling ((ypart a)/_sp) ) step _sp until ypart b:
+ shpath (v, hatchpen)
+ ( ( (xpart a, _y)--(xpart b, _y) ) transformed CT );
+ endfor
+%<MF> mono (v);
+ endgroup
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Gradient fills}\label{basicgradient}
+%
+% \CMP{} cannot do true gradients without some external help. Level-3
+% PostScript permits it, so recent \MP{} could do it by inserting
+% appropriate PS prologues and/or \MP{} specials. Doing that runs the
+% risk of introducing code not recognized by post-processors that expect
+% only what \MP\ natively offers. Therefore, we implement gradients by
+% filling a lot of thin regions with a range of different
+% colors.
+%
+% We have to drop down to pretty low-level operations since, before now,
+% we didn't need a command that added a colored region to a named
+% picture.
+%
+% For maximum flexibility, all our gradients pass variation in colors as a
+% function \mfc{clr} which must be previously \gbc{vardef}-ed and must
+% produce a color for each parameter value between $0$ and $1$.
+%
+% \DescribeRoutine{axialgradientf}
+% A linear gradient has colored rectangular strips that vary along a
+% single axis. The function parameter of \gbc{axialgradientf} takes one
+% variable and produces the color of each strip. \gbc{v} is a known
+% picture variable to which the resulting picture will be assigned,
+% \gbc{theta} is an angle, \gbc{a} and \gbc{b} are the opposite corners of
+% a rectangle. What is returned in \gbc{v} is a rectangular picture
+% rotated by \gbc{theta}.
+%
+% Normally, this is called by the \gbc{axialgradient} command which
+% declares the picture variable \gbc{v}, passes its angle parameter
+% \gbc{theta}, and computes the bounding box of a cyclic path for \gbc{a}
+% and \gbc{b}. The calling command will clip the result to the appropriate
+% path.
+%
+% The calculations with \gbc{signof} is for the same reason as in
+% \gbc{thatchf}. The other messy calculations try to cover the rectangle
+% exactly with an integer number of strips, with the first and last having
+% exactly the colors \gbc{clr(0)} and \gbc{clr(1)}.
+%
+% If the thickness of the strip is too small, memory problems might
+% result and appearence might suffer. Nevertheless we make no attempt to
+% enforce a minimum value.
+% \begin{macrocode}
+def axialgradientf (suffix clr, v) (expr theta, sp, a, b) =
+ begingroup
+ save _hh, _sp, _nn, _y;
+ _hh := ypart b - ypart a;
+ _sp := signof (_hh) abs(sp);
+ _nn := emax (1, round (_hh/_sp));
+ _sp := _hh/_nn + signof (_hh) epsilon;
+ _nn := _nn-1;
+ setpath (_p) rect ((xpart a, 0),(xpart b, _sp));
+ _y := ypart a;
+ for _i = 0 upto _nn:
+%<*MF>
+ if (clr(_i/_nn)) < white :
+ addto v also shaded (clr(_i/_nn)) ( _p shifted (0,_y))
+ rotated theta;
+ fi
+%</MF>
+%<*MP>
+ addto v contour (_p shifted (0,_y)) rotated theta
+ withcolor clr(_i/_nn);
+%</MP>
+ _y := _y + _sp;
+ endfor
+%<MF> mono (v);
+ endgroup
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{areagradientf}
+% The command \gbc{areagradientf} fills the rectangle determined by
+% corners \gbc{a} and \gbc{b} with pixels of dimension \gbc{sp} by
+% \gbc{tp}. Each pixel is filled with the color determined by \gbc{clr}.
+% This suffix parameter must be the name of a function taking two
+% parameters.
+%
+% The resulting rectangle is built on the picture variable whose name is
+% passed as the second parameter \gbc{v}. The calling routine is
+% \gbc{areagradient}, which determine the rectangle and initializes
+% the picture variable. It passes its other parameters unchanged.
+% \begin{macrocode}
+def areagradientf (suffix clr, v) (expr sp, tp, a, b) =
+begingroup
+ save _ww, _hh, _sp, _tp, _nn, _mm, _x, _y;
+ _ww := xpart b - xpart a;
+ _hh := ypart b - ypart a;
+ _sp := signof (_ww) abs(sp);
+ _tp := signof (_hh) abs(tp);
+ _nn := emax (1, round (_ww/_sp));
+ _mm := emax (1, round (_hh/_tp));
+ _sp := _ww/_nn + signof (_ww) epsilon;
+ _tp := _hh/_mm + signof (_hh) epsilon;
+ _mm := _mm-1; _nn := _nn-1;
+ setpath (_p) rect (origin,(_sp,_tp));
+ _x := xpart a; y_a := ypart a;
+ for _i = 0 upto _nn:
+ _y := y_a;
+ for _j = 0 upto _mm:
+%<*MF>
+ if (clr(_i/_nn,_j/_mm)) < white:
+ addto v also shaded (clr(_i/_nn,_j/_mm)) (_p shifted (_x,_y));
+ fi
+%</MF>
+%<*MP>
+ addto v contour (_p shifted (_x,_y)) withcolor
+ clr(_i/_nn,_j/_mm);
+%</MP>
+ _y := _y + _tp;
+ endfor
+ _x := _x + _sp;
+ endfor
+%<MF> mono (v);
+endgroup
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{radialgradientf}
+% The command \gbc{radialgradientf} fills the a circle determined by
+% center \gbc{ctr} and radius \gbc{rad} with concentric circular strips of
+% thickness \gbc{sp}. Each strip is filled with the color determined by
+% \gbc{clr}. This suffix parameter must be the name of a function of one
+% parameter.
+%
+% This command is called by \gbc{radialgradient}, which determines the
+% radius of a circle needed to cover a region and clips the picture
+% returned in \gbc{v} to that region.
+% \begin{macrocode}
+path unitcircle;
+unitcircle := fullcircle scaled 2;
+def radialgradientf (suffix clr, v) (expr sp, ctr, rad) =
+ begingroup
+ save _sp, _r, _nn;
+ _nn := emax (1, round (rad/sp));
+ _sp := rad/_nn + epsilon;
+ _nn := _nn - 1;
+ _r := _sp;
+ % fill the small center circle first
+%<*MF>
+ if (clr(0)) < white :
+ addto v also shaded (clr(0)) (unitcircle scaled _r shifted ctr);
+ fi
+%</MF>
+%<*MP>
+ addto v contour (unitcircle scaled _r shifted ctr)
+ withcolor clr(0);
+%</MP>
+ for _i = 1 upto _nn:
+%<*MF>
+ if (clr(_i/_nn)) < white :
+ addto v also shaded (clr(_i/_nn))
+ (unitcircle scaled (_r + _sp) -- reverse unitcircle scaled _r
+ --cycle) shifted ctr;
+ fi
+%</MF>
+%<*MP>
+ addto v contour
+ (unitcircle scaled (_r + _sp) -- reverse unitcircle scaled _r
+ --cycle) shifted ctr withcolor clr(_i/_nn);
+%</MP>
+ _r := _r + _sp;
+ endfor
+%<MF> mono (v);
+ endgroup
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Tiles}\label{tiles}
+%
+% Tesselations are a type of fill in which a rectangular pattern is
+% repeated throughout a region. The repeated rectangle is called a tile.
+% We provide here an environment in which the drawing commands add to a
+% picture variable other than \mfc{currentpicture}. We do this very simply
+% by redefining \gbc{active_plane}, localizing the redefinition between
+% \gbc{tile} and \gbc{endtile}
+%
+% \DescribeRoutine{tile}
+% The macro \gbc{tile} accepts one suffix parameter, the name of the tile,
+% followed by three numeric expressions and a boolean. \gbc{unit} should
+% be a dimension in device units and is the unit of length for all high
+% level drawing commands within the environment. \gbc{width} and
+% \gbc{height} specify the size of the tile in multiples of \gbc{unit}, and
+% \gbc{clipit} is a boolean that determines if the resulting picture is
+% clipped to the rectangle these parameters determine. For example,\\
+% \indent \gbc{tile (fred)(1in, 1, 2, true)} \\
+% starts a tile named \gbc{fred} which will be 1 inch wide and 2 inches
+% tall, and any marks that extend beyond this rectangle are clipped off.
+% The tile is enclosed in a group to delimit these changes to the basic
+% drawing parameters.
+% \DescribeRoutine{endtile}
+% The macro \gbc{endtile} merely implements the clipping and then closes
+% the group.
+%
+% In \MF, the picture should be a whole number of pixels in size, so that
+% the tiles fit perfectly together. The fact that shifts must be integer
+% values is only mildly relevant, because the placement code does the
+% rounding.
+%
+% For tesselation (filling with tiles) we need to know various properties
+% of the tile, so a tile is a composite object consisting of a picture,
+% \gbc{fred.pic} in our example (the actual tile) and a pair
+% \gbc{fred.dims} of the dimensions (in device units). We used to
+% save the \gbc{clipit} parameter in \gbc{atile.clipon}, but it was never
+% used. We also used to have separate numerics \gbc{atile.wd} and
+% \gbc{atile.ht} but they only got used together as a pair.
+% \begin{macrocode}
+def tile (suffix atile) (expr unit, width, height, clipit) =
+ picture atile.pic; atile.pic := nullpicture;
+ pair atile.dims;
+%<MF> atile.dims := round ((width, height)*unit);
+%<MP> atile.dims := (width, height)*unit;
+ begingroup
+% \end{macrocode}
+% We do a subset of what we do in \gbc{beginmfpic}, redefining
+% \gbc{active_plane} so that all drawing commands that add to it will
+% contribute to the tile, and adapting \gbc{ztr} to the tile dimensions.
+% Re also redefine \gbc{xneg}, et al., for the benefit of \gbc{levelset}.
+% \begin{macrocode}
+ save active_plane;
+ def active_plane = atile.pic enddef;
+ save ztr, vtr; transform ztr, vtr;
+ ztr := identity scaled unit; vtr := ztr;
+ save xneg, xpos, yneg, ypos;
+ xneg := 0; xpos := width; yneg := 0; ypos := height;
+% \end{macrocode}
+% To implement \gbc{clipit}, we set the current clipping path array
+% \gbc{ClipPath[\,]} to the boundary of the tile. Note that this turns off
+% user-defined clipping paths, which are unlikely to be correct for the
+% local tile coordinates.
+% \begin{macrocode}
+ save ClipOn; boolean ClipOn;
+ if clipit:
+ ClipOn := true;
+ setarray (path) (ClipPath) (rect(origin, atile.dims));
+ else:
+ ClipOn := false;
+ fi
+enddef;
+def endtile =
+ DoClip (active_plane);
+ endgroup
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{is_tile}
+% To test whether \gbc{atile} is really a tile, just see if the needed
+% components are defined and of the correct type.
+% \begin{macrocode}
+vardef is_tile (suffix atile) =
+ (known atile.pic ) and (picture atile.pic) and
+ (known atile.dims) and (pair atile.dims )
+enddef;
+
+% \end{macrocode}
+%
+%
+%
+% \section{Bounding Boxes of Paths}\label{bboxes}
+%
+% To fill a region with other than a solid fill, we normally fill a
+% rectangle with copies of a picture (or a path) and then clip to the
+% boundary curve. In order not to place too many copies, we try to find a
+% rectangle that is not too much larger than that region. For this we have
+% the macro \gbc{getbbox} which takes two pair variable and a path
+% expression, and sets the pairs to the lower left corner and upper right
+% corner, respectively, of a rectangle enclosing the path. The bounding
+% box macros are used on paths in device coordinates, but there is no
+% intrinsic reason that has to be so: they will return the bounding box in
+% whatever coordinates the supplied path is in.
+%
+% \DescribeRoutine{getbbox}
+% One can get a rather loose bounding rectangle by using the fact that
+% each segment of a path (from \mfc{point j of g} to \mfc{point j+1 of g})
+% is contained in the convex set determined by all 4 control points for
+% that segment. So we get a containing rectangle by getting the smallest
+% and largest values of the $x$- and $y$-coordinates of all those points.
+% We can get a considerably tighter fit if we cut each segment in half
+% (or more) before doing that. A calling routine is expected to save and
+% declare the suffixes \gbc{ll} and \gbc{ur}. Within \grafbase{} commands,
+% \gbc{getbbox} is always called by \gbc{setbbox}, which does this.
+%
+% \DescribeRoutine{ctrlsbbox}
+% There is a difference between ``\mfc{postcontrol 0 of (subpath (j,j+1/2)
+% of p)}'' and ``\mfc{postcontrol j of p}''. To gain the tighter box we have
+% to look at the former. \gbc{ctrlsbbox} just updates the previously found
+% corners \gbc{ll} and \gbc{ur} of the bounding box based on the controls
+% of the path segment \gbc{p}, and the calling routine \gbc{getbbox} passes
+% it half a segment at a time. We don't examine the endpoints of
+% the half-segment: one has already been examined by \gbc{getbbox} and
+% the other (a subdivision point of an original segment) lies on the line
+% segment connecting two control points, and so can't increase the bbox.
+%
+% We've given this potentially unlimited accuracy by allowing the number
+% of subdivisions (\gbc{bbox_split}) to be arbitrary. We choose 2 for the
+% default. The \gbc{setsplit} command (subsection~\ref{utilities}) ensures
+% that \gbc{_s} is integral and positive, just in case \gbc{bbox_split}
+% somehow isn't.
+%
+% This description applies only to \MF, because \MP{} has built-in
+% facilities for determining the bounding box.
+%
+% I have changed \gbc{ctrlsbbox} to have the same syntax as \gbc{getbbox}.
+% I don't know why I defined it differently.
+%
+% \RoutineIndex{pnt}
+% \RoutineIndex{pre}
+% \RoutineIndex{post}
+% I got tired of typing long expressions like ``\gbc{(precontrol length
+% p of p)}'', and now use the following abbreviations.
+% \begin{macrocode}
+vardef pnt@# (expr p) = point @# of p enddef;
+vardef pre@# (expr p) = precontrol @# of p enddef;
+vardef post@# (expr p) = postcontrol @# of p enddef;
+
+numeric bbox_split; bbox_split := 4;
+def getbbox (suffix ll, ur) expr g =
+%<MP> ll := llcorner g; ur := urcorner g;
+%<*MF>
+ setsplit (_s) bbox_split;
+ ur := ll := pnt 0 (g);
+ for _j = 1 upto length g:
+ ll := pairmin (ll, pnt[_j] (g)); ur := pairmax (ur, pnt[_j] (g));
+ endfor
+ for _j = 1 upto _s*(length g):
+ ctrlsbbox (ll, ur) subpath ((_j-1)/_s, _j/_s) of g;
+ endfor
+%</MF>
+ if showbbox: noclip ( safedraw rect (ll, ur) ); fi
+enddef;
+
+%<*MF>
+def ctrlsbbox (suffix ll, ur) expr p =
+ ll := pairmin ( pairmin (ll, post0 (p)), pre 1 (p) );
+ ur := pairmax ( pairmax (ur, post0 (p)), pre 1 (p) );
+enddef;
+
+%</MF>
+% \end{macrocode}
+%
+% \DescribeRoutine{getradius}
+% This is very similar to \gbc{getbbox}, but gets a ``bounding circle''
+% instead of a box. It is used to get nearly the smallest circle with a
+% given center that contains a path. The path is shifted to place the
+% center at the origin and then this function is called. Similarly,
+% \DescribeRoutine{ctrlsradius}
+% \gbc{ctrlsradius} is used like \gbc{ctrlsbbox}.
+% \begin{macrocode}
+def getradius (suffix rad) expr g =
+ setsplit (_s) bbox_split;
+ rad := abs (pnt0 (g));
+ for _j = 1 upto length g:
+ rad := emax(rad, abs(pnt[_j] (g)));
+ endfor
+ for _j = 1 upto _s*(length g):
+ ctrlsradius (rad) subpath ((_j-1)/_s, _j/_s) of g;
+ endfor
+enddef;
+
+def ctrlsradius (suffix rad) expr p =
+ rad := emax( emax (rad, abs(post0 (p))), abs(pre1 (p) ))
+enddef;
+
+% \end{macrocode}
+%
+% We also have \gbc{tightbbox} and \gbc{tbbox} in \MF{} but these are no
+% longer used so we'll omit them from \grafbase, but keep them in the
+% documentation for now.
+%
+% \DescribeRoutine{tightbbox}
+% Calculate tight bounding box points \gbc{ll} and \gbc{ur} for path
+% \gbc{g}. The tight bounding box is accurate to the limits of the
+% \mfc{solve} macro, which is the numeric \mfc{tolerance}, which we set to
+% \mfc{.5} (accurate enough, assuming pixel units). This is only called by
+% \gbc{tbbox}, which is never used.
+%
+% \gbc{xlimit(x)} returns a value of true if the path \gbc{g} doesn't
+% cross the vertical line at \gbc{x}. \gbc{ylimit(y)} is the same for the
+% horizontal line at \gbc{y}.
+% \begin{macrocode}
+%<*unused>
+def tightbbox (expr g) (suffix ll, ur) =
+begingroup
+ interim tolerance := .5;
+ ll := ( (solve _xlimit (-infinity, xpart pnt 0 (g))),
+ (solve _ylimit (-infinity, ypart pnt 0 (g))) );
+ ur := ( (solve _xlimit ( infinity, xpart pnt 0 (g))),
+ (solve _ylimit ( infinity, ypart pnt 0 (g))) );
+endgroup
+ if showbbox:
+ noclip ( safedraw rect (ll, ur) );
+ fi
+enddef;
+
+vardef _xlimit (expr x) =
+ ((x, -infinity)--(x, infinity)) misses g
+enddef;
+
+vardef _ylimit (expr y) =
+ ((-infinity, y)--(infinity, y)) misses g
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{tbbox}
+% \gbc{tbbox} simply calls \gbc{tightbbox} on each of an array of paths
+% and takes the maximum of all the upper right corners and the minimum of
+% all the lower left. Same syntax as \gbc{tightbbox} except that, instead
+% of a path parameter, \gbc{g} must be the name of an array of paths.
+% This macro is never used elsewhere in \grafbase.
+% \begin{macrocode}
+vardef tbbox (suffix g) (suffix ll, ur) =
+ save _gll, _gur; pair _gll, _gur;
+ tightbbox (g1, ll, ur);
+ for _idx = 2 upto g:
+ tightbbox (g[_idx], _gll, _gur);
+ ll := pairmin (ll, _gll); ur := pairmax (ll, _gur);
+ endfor
+ if showbbox: noclip ( safedraw rect (ll, ur) ); fi
+enddef;
+
+%</unused>
+% \end{macrocode}
+%
+%
+%
+% \section{Device Coordinate Rendering Commands}\label{basicrendering}
+%
+% We use the word `rendering' to refer to commands that accept a path
+% expression as one parameter and use it to modify the \gbc{active_plane}.
+% All the commands in this section expect paths, pairs and dimensions in
+% device coordinates.
+%
+%
+% \subsection{Drawing}\label{basicdrawing}
+%
+% \DescribeRoutine{safedraw}
+% \gbc{safedraw} accepts a path expression, and adds the result to
+% \gbc{active_plane}. It is the first drawing command to draw
+% exclusively on \gbc{active_plane}. This is the first of many uses of
+% \gbc{coloraddon}. In \MP{} it is basically the primitives \mfc{addto}
+% and \mfc{withcolor} applied to \gbc{active_plane}, but in \MF{} it adds
+% when the color is less than 1 (gray or black), otherwise it subtracts
+% (white).
+%
+% \RoutineIndex{colorsafedraw}
+% The command \gbc{safedraw} merely calls \gbc{colorsafedraw}, which then
+% calls \gbc{picpath}, which calls \gbc{shpath}. One reason for this
+% roundabout sequence is to support older files (where \gbc{colorsafedraw}
+% was not defined). Another is that color handling in \MF{} requires a
+% picture with pixels of weight 1 or 0 only (\gbc{picpath}). Moreover,
+% \gbc{shpath} guarantees that the mode's aspect ratio is respected.
+% \begin{macrocode}
+def safedraw = colorsafedraw (drawcolor) enddef;
+def colorsafedraw (expr clr) expr d =
+ begingroup
+ setpicture (v) picpath d;
+ DoClip (v); coloraddon (clr, v);
+ endgroup
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Filling}\label{basicfilling}
+%
+% \DescribeRoutine{NoCycle}
+% This is a common warning for all those commands that require a cycle
+% (closed path) but an open path is supplied. In addition to the warning
+% in those commands, we also call \gbc{safedraw} for debugging purposes.
+%
+% \DescribeRoutine{safefill}
+% \RoutineIndex{colorsafefill}
+% The basic \gbc{safefill} simply calls the colored version with the
+% default parameter \gbc{fillcolor}. \gbc{colorsafefill} takes a color as
+% its first parameter and a path expression as second. These commands fill
+% the path in the \gbc{active_plane}. In \MF, when the color is strictly
+% between $0$ and 1, a gray fill is simulated with the \gbc{shaded} macro.
+%
+% To simulate the effect of painting over in gray, the \MF{} version
+% clears the region before adding the shaded fill.
+%
+% \DescribeRoutine{safeunfill}
+% \gbc{safeunfill} is just \gbc{safefill} with the color \mfc{background}.
+% In \MF{}, when \gbc{background = white = 1}, this is detected by
+% \gbc{coloraddon} which then subtracts the picture. We do this inside
+% \gbc{noclip}, just because it seems a user would expect clipping only
+% when things are \emph{added}. In \MP{} the white is indeed added, but
+% conceptually, material is cleared away.
+% \begin{macrocode}
+def NoCycle (expr s) expr p =
+ GBwarn s & " cannot be applied to an open path."
+ & " The path will be drawn instead.";
+ safedraw p;
+enddef;
+
+%<MF>vardef isgray (expr X) = (X > black) and (X < white) enddef;
+%<MF>
+def safefill = colorsafefill (fillcolor) enddef;
+vardef colorsafefill (expr clr) expr c =
+ if cycle c:
+ setpicture (v) interior c;
+ DoClip (v);
+%<*MF>
+ if isgray (clr):
+ _subto (active_plane) (v);
+ v := nullpicture;
+ v := shaded (clr) c;
+ fi
+%</MF>
+ coloraddon (clr, v);
+ else: NoCycle("fill") c;
+ fi
+enddef;
+def safeunfill expr c =
+ if cycle c: noclip (colorsafefill (background) c);
+ else: NoCycle("unfill") c;
+ fi
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Clipping}\label{clipping}
+%
+% \DescribeRoutine{safeclip}
+% This applies \gbc{clipto} to the active drawing plane. It follows the
+% pattern started with \gbc{safefill} where commands that require a cycle
+% will \gbc{safedraw} non-cyclic paths.
+% \begin{macrocode}
+def safeclip expr c =
+ if cycle c: clipto (active_plane) c;
+ else: NoCycle("clip") c;
+ fi
+enddef;
+
+% \end{macrocode}
+%
+%
+%
+% \section{Graph Coordinate Rendering}\label{rendering}
+%
+% \DescribeRoutine{store}
+% Now we come to the highest level rendering operations. These are the
+% commands written to the output file by \mfpic. They accept a path in
+% \emph{graph} coordinates, convert it to device coordinates, rendering
+% the result, and return the original path. This way one can render a
+% path and pass it on to the preceding command for further processing.
+% This is how \mfpic{} implements multiple prefix macros. However, this
+% cannot be kept up because \MF{} abhors an isolated expression. Therefore
+% we provide a command that accepts a path and doesn't pass it on. In
+% theory, it could do nothing, but in \mfpic{} we store the path in
+% \gbc{curpath}, making every \mfpic{} figure a path assigment command
+% and the rendering is `merely' a side-effect.
+%
+% \DescribeRoutine{stored}
+% The macro \gbc{stored} performs \gbc{store}, but passes the same path as
+% its return value. This is used by \mfpic{} to implement the \cs{store}
+% command, allowing it to also be a prefix macro
+%
+% I don't know if \gbc{store} needs to employ \mfc{hide()}, but it seems
+% not to hurt.
+% \begin{macrocode}
+def store (suffix fs) expr f =
+hide (
+ if (not path f) and (not pair f):
+ GBerrmsg ("Improper expression type.")
+ "The second argument to `store' must be a path or pair.";
+ fi
+ if not path fs: path fs; fi
+ fs := f
+)
+enddef;
+vardef stored (suffix fs) expr f = store (fs) f; f enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Drawing}\label{drawing}
+%
+% \DescribeRoutine{drawn}
+% \RoutineIndex{colordrawn}
+% The command \gbc{drawn} merely calls \gbc{colordrawn} with the default
+% color \gbc{drawcolor}. Then \gbc{colordrawn} takes a color \gbc{clr}
+% and a path expression \gbc{f} and returns the same path. In between,
+% \gbc{zconv (f)} is subjected to \gbc{colorsafedraw}.
+% \begin{macrocode}
+def drawn = colordrawn (drawcolor) enddef;
+vardef colordrawn (expr clr) expr f =
+ colorsafedraw (clr) (zconv (f)); f
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{colorwiggle}
+% This is a multi-tasking command that can draw either zigzag or
+% sinewave shapes depending on the boolean first parameter. For \mfc{true}
+% we get smooth wiggles, for \mfc{false} we get jagged ones. In the
+% smooth case, a tension parameter allows an adjustment to the smoothness.
+% The command
+% \DescribeRoutine{zigzag}
+% \RoutineIndex{colorzigzag}
+% \gbc{zigzag} calls it with the value \mfc{false} and an arbitrary
+% value of the tension;
+% \DescribeRoutine{sinewave}
+% \RoutineIndex{colorsinewave}
+% \gbc{sinewave} calls it with \mfc{true}, allowing it to pick up the
+% tension parameter. All expect a quadruple of dimensions to follow
+%
+% The reason for using a loop (at the end) that draws the \gbc{sinewave}
+% path in pieces, is that all the turning can quickly exceed \MF{}'s limit
+% on the ``rounding table size''. I'd never heard of this until I ran
+% this without a loop and received the ``capacity exceeded'' message. This
+% turns out to be a problem mostly when the ratio of \gbc{len} to
+% \gbc{wid} is too small and the `humps' of the sine are more like
+% `bulbs'. However it is always a problem with \gbc{corkscrew} (below).
+%
+% There is no need for the loop in \MP{}, nor in \MF{} if
+% \mfc{autorounding} is set to $0$, but \mfpic's curved paths definitely
+% look better with the default \mfc{autorounding=2}.
+% \begin{macrocode}
+def zigzag = colorzigzag (drawcolor) enddef;
+def colorzigzag (expr clr) = colorwiggle (false, clr, 0) enddef;
+def sinewave = colorsinewave (drawcolor) enddef;
+def colorsinewave = colorwiggle (true) enddef;
+
+vardef colorwiggle (expr smth, clr, tens, blen, elen, len, wid) expr f
+=
+ convertpath (g) f;
+ setuplengtharray (cumlen, totlen, ct) g;
+ save B;
+ if cycle f:
+ B := 0;
+ else:
+ B := abs(blen)/_rescale_factor;
+ totlen := totlen - B - abs(elen)/_rescale_factor;
+ fi
+ setnumeric (n) 2*round (totlen/len*_rescale_factor);
+ if n < 2:
+ colorsafedraw (clr) g;
+ else:
+ save T, U, X, Y, Z, p;
+ pair U, X, Y, Z; path p;
+ T := if cycle f: 0 else: gettime (cumlen, ct) (B) fi;
+ Z := pnt[T] (g);
+ p :=if not cycle f: (subpath (0,T) of g)
+ if smth: {curl 0} ..tension tens.. else: -- fi
+ fi
+ for i = 1 upto n:
+ hide(
+ T := gettime (cumlen, ct) (B+(i/n)*totlen);
+ X := Z; Z := pnt[T] (g);
+ Y := .5[X,Z]; U := sgn (Z-X);
+ )
+ (Y + (U zscaled (0, if even i: - fi wid)))
+ if smth: {U}..tension tens.. else: -- fi
+ endfor
+ if cycle f: cycle
+ else: if smth: {curl 0} fi (subpath (T, length g) of g)
+ fi;
+ newpicture (v);
+%<MP> shpath (v, drawpen) (p);
+%<*MF>
+ if smth:
+ save n, k;
+ n := length p; k = n div 50;
+ for i = 0 step 50 until 50*(k-1):
+ shpath (v, drawpen) (subpath (i,i+50) of p);
+ endfor
+ shpath (v, drawpen) (subpath (50k,n) of p);
+ else: shpath (v, drawpen) (p);
+ fi
+%</MF>
+ DoClip(v); coloraddon (clr, v);
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{corkscrew}
+% \RoutineIndex{colorcorkscrew}
+% The definition of \gbc{corkscrew} shares a lot of code with \gbc{zigzag}
+% and \gbc{sinewave}, but the middle is considerably different, so it is
+% not really possible to make a multipurpose command that can do all
+% three.
+% \begin{macrocode}
+def corkscrew = colorcorkscrew (drawcolor) enddef;
+vardef colorcorkscrew (expr clr, tens, blen, elen, len, wid) expr f =
+ convertpath (g) f;
+ setuplengtharray (cumlen, totlen, ct) g;
+ save B;
+ if cycle f:
+ B := 0;
+ else:
+ B := abs(blen)/_rescale_factor;
+ totlen := totlen - B - abs(elen)/_rescale_factor;
+ fi
+ setnumeric (n) round (totlen/len*_rescale_factor);
+ if n < 2:
+ colorsafedraw (clr) g;
+ else:
+ save T, U, X, Y, Z, p;
+ pair U, X, Y, Z; path p;
+ T := if cycle f: 0 else: gettime (cumlen, ct) (B) fi;
+ Z := pnt[T] (g);
+ p :=if (not cycle f) and (B > 0): (subpath (0,T) of g)-- fi
+ for i = 1 upto n:
+ hide(
+ T := gettime (cumlen, ct) (B+(i/n)*totlen);
+ X := Z; Z := pnt[T] (g);
+ Y := .5[X,Z]; U := sgn (Z-X);
+ )
+ (X + (U zscaled (0,-wid))){ U}..tension tens..
+ (Y + (U zscaled (0, wid))){-U}..tension tens..
+ endfor
+ if cycle f: cycle
+ else:
+ {U}(Z + (U zscaled (0,-wid)))
+ if elen <> 0: --(subpath(T, length g) of g) fi
+ fi;
+ newpicture (v);
+%<MP> shpath (v, drawpen) (p);
+%<*MF>
+ save n, k;
+ n := length p; k = n div 50;
+ for i = 0 step 50 until 50*(k-1):
+ shpath (v, drawpen) (subpath (i,i+50) of p);
+ endfor
+ shpath (v, drawpen) (subpath (50k,n) of p);
+%</MF>
+ DoClip(v); coloraddon (clr, v);
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Filling, unfilling and clipping}\label{filling}
+%
+% \DescribeRoutine{filled}
+% \RoutineIndex{colorfilled}
+% The command \gbc{filled} calls \gbc{colorfilled} with the default color
+% \gbc{fillcolor}. Then \gbc{colorfilled} takes a color \gbc{clr} and a
+% path expression \gbc{c}, returning the same path after subjecting
+% \gbc{zconv (c)} to \gbc{colorsafefill}.
+% \DescribeRoutine{unfilled}
+% The macro \gbc{unfilled} returns the path after running \gbc{safeunfill}.
+%
+% \DescribeRoutine{Clip}
+% Finally, \gbc{Clip} is similar, running \gbc{safeclip}. The name
+% \gbc{clip} (lowercase) is taken: it is a \MP{} primitive.
+% \begin{macrocode}
+def filled = colorfilled (fillcolor) enddef;
+vardef colorfilled (expr clr) expr c =
+ colorsafefill (clr) zconv (c); c
+enddef;
+vardef unfilled expr c = safeunfill zconv (c); c enddef;
+vardef Clip expr c = safeclip zconv (c); c enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Shading}\label{shading}
+%
+% \DescribeRoutine{shade}
+% Shading is accomplished differently in \MP{} from \MF; however, many of
+% the same parameters are used for compatibility (so that \MP{} can be run
+% on a \file{.mf} created for \grafbase{} by \mfpic). In \MP, shading is
+% just filling with some level of gray. In \MF, we place a pattern of
+% small dots with the size and spacing adjustable. For compatibility,
+% \MP{} accepts these size and spacing parameters, but simply uses them to
+% calculate the darkness of gray.
+%
+% Ideally (i.e., for best appearance) one would shade with single pixels
+% placed in a regular pattern. Unfortunately, this is the most memory
+% intensive for \MF, which stores bitmaps by scanning each row of pixels,
+% and records where changes from black to white occur. We do use simple
+% dots, but make them quite a bit larger than one pixel. By default,
+% \gbc{0.5bp} in diameter, spaced (in \mfpic) a default \gbc{1pt} between
+% centers.
+%
+% The shape and size of the dots can be selected by defining
+% \gbc{shadedotpath} and \gbc{shadewd}. A closed path representing the
+% boundary of one dot of unit size, \gbc{shadedotpath} is initialized to a
+% circle.
+%
+% The parameter \gbc{sp} is the distance between the centers of the dots in
+% device coordinates, and \gbc{f} is the path to be filled in \emph{graph
+% coordinates}.
+%
+% As usual, if the path is not closed, we draw the curve instead. If the
+% spacing is too small relative to \gbc{shadewd}, we fill the curve.
+% Otherwise the \gbc{fillwith} macro is used to fill with copies of
+% a dot picture.
+% \begin{macrocode}
+numeric shadewd; shadewd := 0.5bp;
+path shadedotpath;
+shadedotpath := fullcircle;
+
+vardef shade (expr sp) expr f =
+ convertpath (g) f;
+% \end{macrocode}
+% It seems clear that the gray level (\gbc{gr}) should depend
+% quadratically on \gbc{shadewd/sp}. Also, there is a point where the
+% result is essentially black and a fill would be more efficient.
+% The value .88 is arrived at empirically and is a compromise so that
+% \MF{} and \MP{} produce similar levels of gray on both printers
+% available to me.
+% \begin{macrocode}
+ setnumeric (gr) 1 - (.88*abs(shadewd)/sp)**2;
+ if not cycle g: NoCycle("shade") g;
+ elseif gr <= 0: safefill g;
+ else:
+%<*MF>
+ setbbox (ll, ur) g;
+ ll := floorpair (ll);
+% \end{macrocode}
+%
+% What we do is draw a row of dots and stack the rows to fill a rectangle.
+% We call \gbc{fillwith} to draw these copies. We have to produce this
+% on a picture separate from \gbc{active_plane} so we can apply
+% \gbc{DoClip}. Adding one complex picture onto another means there is a
+% time at which the added picture and the result are in memory at the same
+% time. To save a little memory (at the cost of a little speed), we
+% divide the picture into two, each with half the complexity. We add the
+% one, then (I hope) recover the memory by equating it to \mfc{nullpicture}
+% before adding the second.
+%
+% Shifts of pictures need to be by integer number of pixels, but this is
+% ensured by \gbc{fillwith}, using \mfc{ceiling} to define \gbc{dv} is
+% more to ensure it is not rounded down to 0.
+% \begin{macrocode}
+ % setpair (dv) ceiling (sp/(sqrt 2))*(1,1);
+ % test hex spacing:
+ setpair (dv) ( ceiling(.5sp), ceiling(.5sp*sqrt 3) );
+ setpicture (sh) setdot (shadedotpath, abs(shadewd));
+ newpicture (v);
+ fillwith (v) (sh, 2dv, ll, ur);
+ newpicture (w);
+ addto w also v shifted goodpair (dv);
+ DoClip (v); DoClip (w);
+ clipto (v) (g); clipto (w) (g);
+ _orto (active_plane, v);
+ v := nullpicture;
+ _orto (active_plane, w);
+%</MF>
+% \end{macrocode}
+% In \MP{} we just fill with gray. The gray level having been calculated
+% at the beginning.
+% \begin{macrocode}
+%<*MP>
+ colorsafefill (gr*white) g;
+%</MP>
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{polkadot}
+% The macro \gbc{polkadot} is intended to fill a region with \emph{large}
+% dots. The diameter, \gbc{polkadotwd}, is initialized to \mfc{5bp}. The
+% code is similar to that of \gbc{shade}, but here we attempt a hexagonal
+% array: each dot surrounded by 6 equally spaced dots. Because of their
+% larger size and presumably larger spacing, we can be a little less
+% efficient and so we aim for improved visual appearance. We do what we
+% can to avoid unsightly slivers of partial dots, and only draw a dot if
+% its center lies in the bounding box.
+%
+% We also permit the circles to overlap, and only replace the code with a
+% fill if the dots overlap so much that no background can show (this assumes
+% that \gbc{polkadotpath} is a circle).
+%
+% If the space \gbc{sp} and \gbc{polkadotwd} are too small, there will
+% be a great many tiny dots. It is quite easy to overflow \MP{} capacity
+% and tiny paths don't rasterize at all well in \PS. In \MF, we already
+% have \gbc{shade} to place tiny dots. Therefore, we merely fill if
+% \gbc{sp} is less that a certain minimum, even if that minimum is greater
+% than \gbc{polkadotwd}.
+% \begin{macrocode}
+polkadotwd := 5bp;
+mindotspace := 1bp;
+path polkadotpath; polkadotpath := fullcircle;
+
+vardef polkadot (expr sp) expr f =
+ convertpath (g) f;
+ if not cycle g: NoCycle("polkadot") g;
+ elseif sp <= emax (2*polkadotwd/3, mindotspace):
+ safefill g;
+ else:
+ setbbox (ll, ur) g;
+% \end{macrocode}
+% As with \gbc{shade}, we shift alternate rows by half the spacing between
+% dot centers. The vertical shift is slightly larger (relatively speaking)
+% and the horizontal smaller. We apply a further horizontal and vertical
+% shift to avoid small pieces of dots. What it does is take only those dots
+% whose centers lie in the bounding box, and center the whole array
+% relative to that box.
+% \begin{macrocode}
+ save dx, dy, dshift; pair dshift;
+ dx := sp/2; dy := dx*sqrt 3;
+ dshift := (xpart(ur - ll) mod dx, ypart (ur - ll) mod dy)/2;
+% \end{macrocode}
+% Here, \gbc{p} is the center of the first dot in the lower left corner.
+% \begin{macrocode}
+ save p, dims; pair p, dims;
+ p := ll + dshift;
+ dims := 2(dx, dy);
+% \end{macrocode}
+% The extra \MF{} code is to clear what's under the dots in case they
+% are gray dots. And then to `gray' the dots when \gbc{fillcolor} demands
+% it.
+% \begin{macrocode}
+ setpicture (thepolkadot) setdot (polkadotpath, polkadotwd);
+ newpicture (v);
+ fillwith (v) (thepolkadot, dims, p, ur);
+ fillwith (v) (thepolkadot, dims, p + (dx, dy), ur);
+ DoClip (v); clipto (v) g;
+%<*MF>
+ if isgray (fillcolor):
+ _subto (active_plane) (v);
+ v := nullpicture;
+ thepolkadot :=
+ shaded (fillcolor) polkadotpath scaled ceiling (polkadotwd);
+ fillwith (v) (thepolkadot, dims, p, ur);
+ fillwith (v) (thepolkadot, dims, p + (dx, dy), ur);
+ DoClip (v); clipto (v) g;
+ fi
+%</MF>
+ coloraddon (fillcolor, v);
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Hatching}\label{hatching}
+%
+% \DescribeRoutine{thatch}
+% \RoutineIndex{colorthatch}
+% This command hatches the interior of path \gbc{f} (graph coordinates)
+% with lines at angle \gbc{theta}, spaced \gbc{sp} apart (device
+% coordinates). As usual an unclosed path is simply drawn. The thickness
+% of the lines is determined by \gbc{hatchwd}. If \gbc{sp} is not greater
+% than \gbc{abs(hatchwd)}, we simply fill. This will ensure \gbc{thatchf}
+% is called only for positive \gbc{sp}.
+%
+% We find the bounding box of the backward rotated path, so when that box
+% is filled with lines and rotated, it will cover the path. After calling
+% \gbc{thatchf} we add the picture, clipped to the path.
+% \begin{macrocode}
+def thatch = colorthatch (hatchcolor) enddef;
+vardef colorthatch (expr clr) (expr sp, theta) expr f =
+ convertpath (g) f;
+ if not cycle g: NoCycle("hatch") g;
+ elseif sp <= abs(hatchwd): colorsafefill (clr) g;
+ else:
+ newpicture (v);
+ setbbox (ll, ur) g rotated -theta;
+ thatchf (v, identity rotated theta, sp, ll, ur);
+ DoClip (v); clipto (v) (g);
+ coloraddon (clr, v);
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+% We offer some special cases, calling \gbc{thatch} with different angles.
+% These take only the spacing (in device coordinates) and a path
+% expression (in graph coordinates) as parameters.\\
+% \DescribeRoutine{hhatch}
+% \gbc{hhatch} has angle 0 and so produces horizontal lines;\\
+% \DescribeRoutine{vhatch}
+% \gbc{vhatch} produces vertical lines;\\
+% \DescribeRoutine{lhatch}
+% \gbc{lhatch} produces lines tilted to the left (running from upper left
+% to lower right);\\
+% \DescribeRoutine{rhatch}
+% \gbc{rhatch} produces lines running from lower left to upper right;
+% and\\
+% \DescribeRoutine{xhatch}
+% \gbc{xhatch} produces cross-hatching, and essentially runs \gbc{lhatch}
+% and \gbc{rhatch}.
+%
+% Color is a parameter only for \gbc{colorxhatch}. The reason for that
+% is to make code written by \mfpic{} simpler. The \mfpic{} commands for
+% the others actual write calls to \gbc{thatch} or \gbc{colorthatch}.
+%
+% \begin{macrocode}
+def hhatch (expr sp) = thatch (sp, 0) enddef;
+def vhatch (expr sp) = thatch (sp, 90) enddef;
+def lhatch (expr sp) = thatch (sp, -45) enddef;
+def rhatch (expr sp) = thatch (sp, 45) enddef;
+
+def xhatch = colorxhatch (hatchcolor) enddef;
+def colorxhatch (expr clr, sp) =
+ colorthatch (clr) (sp, 45) colorthatch (clr) (sp, -45)
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Gradients}
+%
+% \DescribeRoutine{axialgradient}
+% We pass a \mfc{vardef}-ed function that is to provide the range of
+% colors. It can output colors of different types if desired. Two
+% natural methods are: (1)~interpolate between colors of the same type:\\
+% \indent\mfc{vardef clrgrad (expr t) = (t)[red,blue] enddef}\\
+% and (2)~extract colors from a previously built array of colors:\\
+% \indent\mfc{vardef clrgrad (expr t)= A[round(t*N)]}\\
+% where, \mfc{A0}, \mfc{A1},\dots \mfc{A[N]} are colors (necessarily of
+% the same type).
+%
+% Since we simply fill strips with a single color, \gbc{sp} is the
+% thickness of the strip (in device units) and \gbc{theta} is the angle
+% by which these strips differ from being horizontal.
+% \begin{macrocode}
+vardef axialgradient (suffix clr) (expr sp, theta) expr f =
+ convertpath (g) f;
+ if not cycle g: NoCycle("axialgradient") g;
+ else:
+ newpicture (_grd);
+ setbbox (ll, ur) g rotated -theta;
+ axialgradientf (clr, _grd) (theta, sp, ll, ur);
+ DoClip (_grd); clipto (_grd) (g);
+%<MF> safeunfill g;
+ _orto (active_plane, _grd);
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{areagradient}
+% This fills a cyclic path with colored pixels, with the color
+% determined by the \mfc{vardef}-ed function \gbc{clr} which takes two
+% parameters. The size of the pixels is given in the last two parameters
+% \gbc{sp} and \gbc{tp} which are specified in device units.
+% \begin{macrocode}
+vardef areagradient (suffix clr) (expr sp, tp) expr f =
+ convertpath (g) f;
+ if not cycle g: NoCycle("areagradient") g;
+ else:
+ newpicture (_agr);
+ setbbox (ll, ur) g;
+ areagradientf (clr, _agr) (sp, tp, ll, ur);
+ DoClip (_agr); clipto (_agr) (g);
+%<MF> safeunfill g;
+ _orto (active_plane, _agr);
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{radialgradient}\label{getrad}
+% This fills a cyclic path with colored circular strips, with the color
+% determined by the \mfc{vardef}-ed function \gbc{clr} which takes one
+% parameters. The thickness of the strips is given in the last parameter
+% \gbc{sp} which are specified in device units. The command
+% \gbc{getradius} finds the distance from the center to the farthest point
+% of \gbc{f}. It was added (see section~\ref{bboxes}) solely for this use.
+% \begin{macrocode}
+vardef radialgradient (suffix clr) (expr sp, ctr) expr f =
+ convertpath (g) f;
+ if not cycle g: NoCycle("radialgradient") g;
+ else:
+ setpair (_ctr) zconv (ctr);
+ newpicture (_agr);
+ save _rad;
+ getradius (_rad) g shifted - _ctr;
+ radialgradientf (clr, _agr) (sp, _ctr, _rad);
+ DoClip (_agr); clipto (_agr) (g);
+%<MF> safeunfill g;
+ _orto (active_plane, _agr);
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Tesselations}\label{tess}
+%
+% \DescribeRoutine{tess}
+% Tesselation of the interior of a closed path means filling with copies
+% of a \emph{tile} (see subsection~\ref{tiles}). The path is in graph
+% units, the tile is a suffix parameter and is the name of a previously
+% defined tile. In fact, one can create the picture any way one likes (it
+% doesn't have to be with the \gbc{tile} environment). Thus \gbc{tess
+% (fred) f;} will work as long as \gbc{fred.pic} is a picture and
+% \gbc{fred.dims} is a pair giving its dimensions.
+% \begin{macrocode}
+vardef NoTile (suffix atile) expr g =
+ GBwarn str atile & " is not a valid tile for tess()."
+ & " The path will be drawn instead.";
+ safedraw g;
+enddef;
+vardef tess (suffix atile) expr c =
+ convertpath (_g) c;
+ if not cycle _g: NoCycle("tess") _g;
+ elseif not is_tile (atile): NoTile (atile) _g;
+ else:
+ setbbox (_ll, _ur) _g;
+ newpicture (_ts);
+ fillwith (_ts) (atile.pic, atile.dims, _ll, _ur);
+ DoClip (_ts); clipto (_ts) _g;
+ _orto (active_plane, _ts);
+ fi
+ c
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Dots and dashes}\label{dashes}
+%
+% \MP{} already has commands for drawing a dashed or dotted curve,
+% but \MF{} does not. Considerable effort went into making this possible
+% (before \MP{} even existed). The code is now reasonably fast and the
+% result is actually better quality than \MP{}'s native commands so we use
+% the same code in both versions. It does, however, use pretty much
+% memory in \MP{}.
+%
+% The \grafbase{} dashing code is designed to produce a whole number of
+% dashes on any curve to which it is applied, and (usually) to begin and
+% end with half a dash (so that when dashed curves abut, the result looks
+% decent). \MP{}'s own facilities do neither of these. In addition, the
+% dotting code is flexible enough that copies of any picture (not just a
+% circular dot) can be used to trace a path.
+%
+% The general command is \gbc{gendashed}, which takes a suffix parameter
+% (the name of a \emph{dashing pattern}, see below) and a path expression
+% in graph coordinates.
+%
+% A dashing pattern \gbc{pat} consists of three arrays, \gbc{pat.start},
+% which is used to draw the beginning of the path (half a dash in the
+% default \gbc{dashed} command), \gbc{pat.finish}, which is used to draw
+% the other end, and \gbc{pat.rep}, which is the repeating pattern for
+% drawing the rest of the curve. Each of these is an \emph{array} of
+% numerics. These should be lengths, in device units, and represent the
+% lengths of dashes and spaces.
+%
+% We start with some variables and their defaults, some of which are no
+% longer used. \gbc{segment_split} is used in the code for finding the
+% approximate length of a curve. This is needed to make adjustments in the
+% length of dashes and spaces so that a whole number of repeated patterns
+% are used. \gbc{dashsize} and \gbc{dashgap} are no longer used.
+% Originally they gave the lengths of default dashes and the spaces in
+% between. \gbc{dash_start} and \gbc{dash_finish} are the fractions of a
+% dash length that are used at the start and finish if the command
+% \gbc{dashpat} is used to create the dashing pattern.
+%
+% And \gbc{_rescale_factor} is used to adjust numbers downward and avoid
+% arithmetic overflow. For a 1200dpi \MF{} mode, a curve 4 inches long
+% will be over \mfc{infinity} pixels in length, but only 40 deci-inches.
+% Our default for this variable is just that: 1/10 inch.
+% \begin{macrocode}
+if unknown segment_split: segment_split := 8; fi
+if unknown dashsize: dashsize := 3bp; fi
+if unknown dashgap: dashgap := dashsize + 2penwd; fi
+if unknown dash_finish: dash_finish := .5; fi
+if unknown dash_start: dash_start := .5; fi
+if unknown _rescale_factor: _rescale_factor := 0.1in; fi
+
+% \end{macrocode}
+%
+% \DescribeRoutine{gendashed}
+% The main idea is to have a list of lengths represent the repeating
+% pattern of dashes and dots. These lengths represent a dash length,
+% followed by a gap length, etc., so there are an even number. To start
+% dashing a path, we normally take a fraction (\gbc{dash_start}) of the
+% first dash, then the rest of the pattern. We continue by repeating the
+% pattern as many times as will fit, then we finish off with a fraction
+% (\gbc{dash_finish}) of the first dash. A dash of length 0 is a dot. A
+% gap of length 0 is OK, but useless unless it's between a dot and a dash,
+% and you arrange for the dot's size to be different from \gbc{penwd}.
+%
+% We generalize this so that \gbc{pat.start} and \gbc{pat.finish} can be
+% any patterns, not necessarily related to \gbc{pat.rep}. Also `dots' can be
+% symbols like \gbc{Triangle}.
+%
+% When we tried to deal with arrays of dashing patterns, it became
+% rather a pain to deal with three arrays of arrays. So now we allow the
+% suffix \gbc{pat} to be a single array and call \gbc{mkdasharrays} to
+% produce \gbc{pat.start} and \gbc{pat.finish}. It returns \gbc{true} if
+% all three arrays are successfully produced.
+%
+% The variable \VariableIndex{last_dot_size} \gbc{last_dot_size} is
+% intended to allow the clearing path of arrowhead commands to encompass
+% a final dot larger than \gbc{penwd}.
+% \begin{macrocode}
+numeric last_dot_size; last_dot_size := 0;
+vardef gendashed (suffix pat) expr f =
+ convertpath (_g) f;
+ save _dpat;
+ if not mkdasharrays (pat) (_dpat):
+ GBwarn "Dash pattern " & str pat
+ & " undefined. Path will be drawn instead.";
+ safedraw _g;
+ elseif _dpat.rep < 2:
+ safedraw _g;
+ else:
+% \end{macrocode}
+%
+% After the following loop, \gbc{_dl.s} is the total length of the
+% corresponding \gbc{pat.s} in multiples of \gbc{_rescale_factor}, and
+% \gbc{_dpat.s[i]} has been converted to these units.
+% \begin{macrocode}
+ save _dl;
+ forsuffixes _s = start, rep, finish:
+ _dl._s := 0;
+ for i = 1 upto _dpat._s:
+ _dpat._s[i] := _dpat._s[i]/_rescale_factor;
+ _dl._s := _dl._s + _dpat._s[i];
+ endfor
+ endfor
+
+ if _dl.rep = 0:
+ GBwarn "Dash pattern " & str pat & " has length 0. "
+ & "Path will be drawn instead.";
+ safedraw _g;
+ else:
+% \end{macrocode}
+% Here \gbc{_g} is our path in device units, but \gbc{setuplengtharray}
+% computes lengths in multiples of \gbc{_rescale_factor} to avoid having
+% paths of length \gbc{infinity}.
+%
+% This is how we process a path mathematically: let $f(t)$, $0 \le t \le
+% k$ be the formula for the path \gbc{f}, $k$ being the number of segments
+% of \gbc{f}, we consider the polygon connecting the points $f(0), f(1/s),
+% f(2/s),\ldots,f(k)$ (where $s$ is \gbc{segment_split}) and compute the
+% length of \emph{that} path. Actually, we compute and save the cumulative
+% lengths at each vertex of this polygon, since we use that later to
+% determine `when' (i.e., at what values of $t$) to place a dot or draw
+% a dash. The command \gbc{setuplengtharray} does this, storing the
+% cumulative lengths in the array \gbc{_cumlen} and the total in
+% \gbc{_totlen}. It also initializes \gbc{_ct} the index into that array.
+% \begin{macrocode}
+ setuplengtharray (_cumlen, _totlen, _ct) _g;
+% \end{macrocode}
+% Now we adjust the dashes so that a whole number of patterns make up
+% the lengths of the approximating polygon. \gbc{scale_adjust} returns
+% the scaling factor, equates \gbc{_n} to the total number of
+% \gbc{pat.rep} to use. If the path length is too small compared to the
+% length of the start and finish patterns, this is equated to $-1$ as a
+% flag to draw the path instead. (recall \gbc{_dl.s} holds the length of
+% part \gbc{s}).
+%
+% After this we rescale the dashes and spaces stored in \gbc{_dpat}, and
+% the length of the patterns in \gbc{_dl}.
+% \begin{macrocode}
+ save _n, _sf, _no_dots;
+ boolean _no_dots; _no_dots := true;
+ _sf := scale_adjust (_n, _dl) (_totlen);
+ if _n < 0: safedraw _g;
+ else:
+ forsuffixes _s = start, rep, finish:
+ for _i = 1 upto _dpat._s:
+ if (_dpat._s[_i] = 0) and _no_dots: _no_dots := false;
+ else: _dpat._s[_i] := _dpat._s[_i]*_sf;
+ fi
+ endfor
+ _dl._s := _dl._s*_sf;
+ endfor
+% \end{macrocode}
+% The user has the capability to use something other than a small disk for
+% a dot by defining \gbc{plot_pic} (and preferably also storing its
+% diameter in \gbc{plot_pic.size}). The utility \gbc{makesymbol} is
+% defined later. It examines \gbc{plot_pic} and makes a picture depending
+% on what type of variable it is. The default \gbc{dotpath} is
+% \mfc{fullcircle}, but user may also change that to get different dots.
+% \gbc{makesymbol} scales by \gbc{penwd} \emph{only if the first
+% parameter is a path}. This is how to increase the dot size (the code in
+% \gbc{plot} uses this.)
+% \begin{macrocode}
+ if _no_dots:
+ else:
+ if unknown plot_pic:
+ save plot_pic; path plot_pic;
+ plot_pic := dotpath;
+ fi;
+ last_dot_size :=
+ if known plot_pic.size: plot_pic.size else: penwd fi;
+ setpicture (dashingdot) makesymbol (plot_pic, last_dot_size);
+ fi
+% \end{macrocode}
+% The macro \gbc{dashit} draws the dashes, computing where they go and
+% drawing the appropriate subpaths of \gbc{_g} or placing a dot at the
+% appropriate point. \gbc{dashit} returns nothing and assumes all the
+% information accumulated so far, so it can only be called by
+% \gbc{gendashed}.
+%
+% \gbc{_t} and \gbc{_d} are temporary variables used by
+% \gbc{dashit}, but we declare them here since we initialize them
+% differently for each call. \gbc{_d0} and \gbc{_d1} hold the
+% position along the curve of the ends of a dash in distance from the
+% start; \gbc{_t0} and \gbc{_t1} are the same, but in terms of time.
+% A macro \gbc{gettime} converts the first to the second. It uses the
+% cumulative length array \gbc{_cumlen} for this, and maintains
+% \gbc{_ct} as the current index into that array.
+% \begin{macrocode}
+ save _t, _d, _v;
+ picture _v; _v := nullpicture;
+ _d0 := 0; _t0 := 0;
+ dashit (_dpat.start) (_v);
+% \end{macrocode}
+% The parameters to \gbc{dashit} are the name of the part of the dashing
+% pattern that is being drawn, and a temporary picture variable. The
+% latter holds the picture until \gbc{DoClip} can process it.
+% The code of \gbc{dashit} leaves \gbc{_d0} pointing to the current
+% position on the curve, but for safety and to reduce accumulated
+% round-off error, we initialize it to what it should be before each call.
+%
+% The repeating pattern has the tendency to use lots of memory. Previously
+% I added all the dashes to \gbc{_v} and then added it all at once to
+% \gbc{active_plane}. The purpose was to be able to \gbc{DoClip} it once,
+% and add it once with \gbc{coloraddon} to get it drawn in color under \MF.
+% This was simplest, but a memory hog requiring $O(n)$ in memory, where
+% $n$ is the number of repeated patterns. Then we tried clipping and adding
+% within \gbc{dashit}. This was terribly slow, requiring $O(n)$ in time.
+% Now we use a standard programming trick: accumulate $m < n$ repetitions
+% before adding them, the memory should be $O(m)$ and the time $O(n/m)$.
+% Making $m$ about $\sqrt n$ seems to work well.
+% \begin{macrocode}
+ if _n > 0:
+ save _m; _m := ceiling sqrt(_n);
+ for _j = 0 step _m until _n - 1:
+ for _i = 0 upto _m - 1:
+ exitif (_i + _j) > _n - 1;
+ _d0 := _dl.start + (_j + _i)*_dl.rep;
+ _t0 := gettime (_cumlen, _ct) (_d0);
+ dashit (_dpat.rep) (_v);
+ endfor
+ DoClip (_v);
+ coloraddon (drawcolor, _v);
+ _v := nullpicture;
+ endfor
+ fi
+
+ _d0 := _totlen - _dl.finish;
+ _t0 := gettime (_cumlen, _ct) (_d0);
+ dashit (_dpat.finish) (_v);
+ DoClip (_v);
+ coloraddon (drawcolor, _v);
+ fi
+ fi
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{makelengtharray}
+% This takes an array name and a path expression (which is assumed to be
+% in device coordinates), computes the array of partial lengths (of the
+% polygon approximation), and returns the total length. To avoid numeric
+% overflow we rescale the lengths and so the array elements are in units
+% of \gbc{_rescale_factor}. At one point we used to rescale the path,
+% but that turned out to be unnecessary and made it harder to accomplish
+% several of the things we now do with this. We also save a little
+% memory by making the path a suffix parameter, which avoids the memory
+% used for the `capsule' of an expression parameter. So far all uses apply
+% it to a path variable and we have to remember to keep it that way.
+% \begin{macrocode}
+vardef makelengtharray (suffix clen) suffix p =
+ setsplit (_s) segment_split;
+ numeric clen[];
+ clen := _s * length p; clen0 := 0;
+ for _i = 1 upto clen:
+ clen[_i] := clen[_i-1] + abs (pnt[_i/_s] (p) - pnt[(_i-1)/_s] (p))
+ / _rescale_factor;
+ endfor
+ clen[clen]
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{scale_adjust}
+% Here \gbc{n} is a suffix defined by the calling routine,
+% \gbc{pl.\{start\|rep\|finish\}} are the lengths of corresponding parts
+% of a dashing pattern, \gbc{lngth} is the length of some path (determined
+% by the calling routine). It determines how many times \gbc{pl.rep} goes
+% into \gbc{lngth - pl.start - pl.finish}. If this is negative it remains
+% negative, otherwise it is rounded. \gbc{scale_adjust} then determines
+% and returns the scaling factor \gbc{sf} required to make
+% \gbc{sf*(pl.start + n*pl.rep + pl.finish)} equal to \gbc{lngth}.
+% \begin{macrocode}
+vardef scale_adjust (suffix n, pl) (expr lngth) =
+ n := (lngth - pl.start - pl.finish)/pl.rep;
+ n := if n < 0: -1 else: round(n) fi;
+ lngth/(pl.start + emax (n, 0)*pl.rep + pl.finish)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{gettime}
+% \gbc{arr} is an increasing array of lengths, defined by the calling
+% routine. \gbc{ct} is current index into that array; it will vary with
+% subsequent calls. Calling routine initializes it before the first call,
+% \gbc{gettime} updates it. \gbc{lngth} is a length interpreted as the
+% length along the path associated to the array.
+%
+% Since this array is generated by splitting the segments of the path at
+% times \gbc{i/segment_split} we first determine in which of these splits
+% the given distance is (i.e., find \gbc{ct} so that \gbc{lngth} lies
+% between \gbc{arr[ct-1]} and \gbc{arr[ct]}). To avoid problems with
+% round-off error, bad length parameter, etc., we force \gbc{lngth} to
+% satisfy this for some index between the current value of \gbc{ct} and
+% \gbc{arr} inclusive.
+%
+% Once we know what segment we are in, we determine the time by linear
+% interpolation between the times corresponding to \gbc{ct} and
+% \gbc{ct+1}. Note: in the \mfc{forever} loop, the exit must come before
+% the increment. The function \gbc{inrange} is defined in
+% section~\ref{axes}. It checks if the third argument is between the
+% first two, or equal to one of them.
+% \begin{macrocode}
+vardef gettime (suffix arr, ct) (expr lngth) =
+ setnumeric (_gtl) emax (arr[ct], emin (arr[arr], lngth));
+ setsplit (_s) segment_split;
+ forever: exitif inrange (arr[ct], arr[ct+1]) (_gtl);
+ next ct;
+ endfor
+ if arr[ct] = arr[ct+1]: ct
+ else: ( ct + (_gtl - arr[ct]) / (arr[ct+1] - arr[ct]) )
+ fi /_s
+enddef;
+
+def next suffix X = X := X + 1; enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{dashit}
+% No variables are saved or initialized; \gbc{gendashed} defines array
+% \gbc{_cumlen}, path \gbc{_g}, and initializes \gbc{_d0},
+% \gbc{_t0} and \gbc{_ct}.
+%
+% \gbc{pos} is one of the dashpattern arrays, so it consists of numerics
+% interpreted as lengths of dashes (odd index) and spaces (even index). In
+% the first case \gbc{_d0} and \gbc{_t0} will already be pointing to
+% the beginning of the dash and we get to the end of the dash by adding
+% the length of a dash (\gbc{pos[_j]}) to \gbc{_d0} (getting
+% \gbc{_d1}) and calling \gbc{gettime} (getting \gbc{_t1}). We draw
+% the subpath between those points. Unless \gbc{pos[_j] = 0}, in which case
+% a dot is placed.
+%
+% For even \gbc{j} (a space) we are at \gbc{_d1} and \gbc{_t1} and
+% we increment them to get \gbc{_d0} and \gbc{_t0} for the next
+% iteration.
+% \begin{macrocode}
+def dashit (suffix pos) (suffix pic) =
+ for _k = 1 upto pos:
+ if odd _k:
+ if pos[_k] = 0:
+ _d1 := _d0; _t1 := _t0;
+ picdot (pic, dashingdot, pnt [_t0] (_g));
+ else:
+ _d1 := _d0 + pos[_k];
+ _t1 := gettime (_cumlen, _ct) (_d1);
+ shpath (pic, drawpen) (subpath (_t0, _t1) of _g);
+ fi
+ else:
+ _d0 := _d1 + pos[_k];
+ _t0 := gettime (_cumlen, _ct) (_d0);
+ fi
+ endfor
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{dashpat}
+% This is a utility to convert a list of lengths to an array. It is almost
+% the same as \gbc{list}, but it does make sure the array functions as
+% a dashing pattern. It needs to consist of either $1$ item (the signal to
+% draw a solid line) or an even number of items. So we add a zero length
+% term if the size is $0$ or odd and bigger than $1$.
+%
+% \DescribeRoutine{mkdasharrays}
+% The dashing code in \gbc{gendashed} is written so the the beginning and
+% ending patterns can be different from the repeating patterns. If so,
+% they must be named \gbc{pat.start}, \gbc{pat.rep} and \gbc{pat.finish}.
+% If one of these three is not a known array but \gbc{pat} is, this macro
+% creates the array in a generic way. When created this way, \gbc{pat.rep}
+% is a copy of \gbc{pat}. So is \gbc{pat.start}, except only a fraction of
+% the first dash is used, while \gbc{pat.finish} is just the first dash of
+% \gbc{pat} reduced by the factor \gbc{dash_finish}.
+% \begin{macrocode}
+def dashpat (suffix pat) (text t) =
+ list (pat) (t);
+ if (pat = 0) or (odd (pat) and (pat > 1)):
+ pat[incr pat] := 0;
+ fi
+enddef;
+
+vardef mkdasharrays (suffix src, dest) =
+ save _bad; boolean _bad; _bad := false;
+ forsuffixes _s = start, rep, finish:
+ numeric dest._s, dest._s[];
+ boolean _bad._s;
+ if knownnumericarray src._s:
+ copyarray (src._s) (dest._s);
+ _bad._s := false;
+ else:
+ _bad := _bad._s := true;
+ fi
+ endfor
+ % _bad = one of the three arrays not copied.
+ if _bad:
+ if knownnumericarray src: _bad := false;
+ if _bad.rep: % make dest.rep = src
+ copyarray (src) (dest.rep);
+ fi
+ if _bad.start: % shrink first dash to get dest.start
+ copyarray (src) (dest.start);
+ dest.start1 := dash_start*src1;
+ fi
+ if _bad.finish: % use partial first dash for dest.finish
+ dest.finish := 1;
+ dest.finish1 := dash_finish*src1;
+ fi
+ fi
+ fi
+ not _bad
+enddef;
+
+% \end{macrocode}
+%
+% The \mfpic{} command \cs{dashed} is now implemented by making a
+% dashpattern from the two arguments and calling gendashed. That is the
+% definition of \gbc{DASHED}.
+%
+% \gbc{dashed} takes parameters which are the length and the space (device
+% coordinates) and a path (graph coordinates). It returns the path.
+% \begin{macrocode}
+vardef Dashed (expr dlen, dgap) expr f =
+ save dashes; dashpat (dashes) (dlen, dgap);
+ gendashed (dashes) f
+enddef;
+
+def DASHED = Dashed enddef;
+%<MF> def dashed = Dashed enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{doplot}
+% \gbc{doplot} places symbols at positions along a path determined by
+% \gbc{dgap} (space between symbols), they are scaled by \gbc{sc} and the
+% actual symbol is \gbc{spath}. Currently this may be one of three things:
+% \begin{enumerate}
+% \item A path, giving the shape of the dot, which should be defined in
+% units so that the desired size is obtained under scaling by
+% \gbc{sc}. Normally this means one unit across.
+% \item A picture. This is used unscaled, it being presumed that it has
+% been prepared by a user to the correct size.
+% \item (\MP{} only) a string.
+% \end{enumerate}
+% All these are converted to a picture by the \gbc{makesymbol} command and
+% it is assigned to \gbc{plot_pic}, which \gbc{gendashed} has been
+% trained to use when dots are needed.
+%
+% After this \gbc{gendashed} is called with a pattern where the dashes are
+% 0 length, the signal that dots are to be used.
+% \DescribeRoutine{dotted}
+% The macro \gbc{dotted} is implemented by calling \gbc{doplot} with
+% \gbc{dotpath} as the the symbol.
+% \begin{macrocode}
+vardef doplot (expr spath, sc, dgap) expr f =
+ save dots; dashpat (dots) (0, dgap);
+ setpicture (plot_pic) makesymbol (spath, sc);
+ plot_pic.size := sc;
+ gendashed (dots) f
+enddef;
+
+path dotpath; dotpath := fullcircle;
+def dotted = doplot (dotpath) enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{plotnodes}
+% These are useful little utilities to draw the points on top of the
+% curve through them. \gbc{plotnodes} differs from \gbc{plotsymbol}
+% (defined later) in that it takes a path parameter (rather than a list of
+% points) and returns that path (so it works with \mfpic{} as a prefix
+% macro). It also uses \gbc{drawcolor}. Otherwise it calls the same code.
+%
+% \DescribeRoutine{showcontrols}
+% \gbc{showcontrols} was mainly for debugging; it draws a line segment
+% connecting the control points of each node. It optionally draws a symbol
+% at each control point. We use \mfc{0} for a `symbol' to indicate we
+% don't wish to draw a symbol there. The default color for both the
+% symbols and the segment is \gbc{pointcolor}.
+% \begin{macrocode}
+vardef plotnodes (expr symbol, size) expr f =
+ if size > 0:
+ save pln; pair pln[];
+ pln := 0;
+ for _a = 0 upto (length f) if cycle f: - 1 fi:
+ pln[incr pln] := pnt[_a] (f);
+ endfor
+ dosymbols (drawcolor, symbol, size) (pln);
+ fi
+ f
+enddef;
+
+def showcontrols = colorshowcontrols (pointcolor) enddef;
+vardef colorshowcontrols (expr clr, syma, symb, size) expr f =
+ save shpre, shpost;
+ pair shpre[], shpost[];
+ shpre := 0; shpost := 0;
+ for a = 0 upto (length f) if cycle f: - 1 fi:
+ shpre [incr shpre] := pre [a] (f);
+ shpost[incr shpost] := post[a] (f);
+ colorsafedraw (clr)
+ (zconv (shpre[shpre]--pnt[a](f)--shpost[shpost]));
+ endfor
+ if size > 0:
+ if not numeric syma: dosymbols (clr, syma, size) (shpre) ; fi
+ if not numeric symb: dosymbols (clr, symb, size) (shpost); fi
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Double-line drawing}\label{doubleline}
+%
+% \DescribeRoutine{doubledraw}
+% This invokes \gbc{parapath}, which has issues with smooth but wiggly
+% paths. An easier approach would be to draw a wide line and erase a
+% narrow one in the middle. However, that would not be transparent in
+% \MP{}.
+% \begin{macrocode}
+def doubledraw = colordoubledraw (drawcolor) enddef;
+vardef colordoubledraw (expr clr, sep) expr f =
+ convertpath (g) f;
+ colorsafedraw (clr) (parapath ( sep/2) g);
+ colorsafedraw (clr) (parapath (-sep/2) g);
+ f
+enddef;
+
+% \end{macrocode}
+%
+%
+%
+% \section{Points Symbols and Other Pictures}\label{symbols}
+%
+% \DescribeRoutine{centerit}
+% This accepts a picture and returns the same picture centered. This
+% is close to impossible in \MF, so we only do it in \MP. Actually, we
+% no longer use it, because in the one case where we did
+% (\gbc{makesymbol}), it seemed to restrict the user's choices too much.
+% \begin{macrocode}
+%<*MP>
+vardef centerit (expr pic) =
+ pic shifted -(0.5[urcorner pic, llcorner pic])
+enddef;
+
+%</MP>
+% \end{macrocode}
+%
+% \DescribeRoutine{makesymbol}
+% This utility takes \emph{any} expression and scale and returns a picture.
+% If the expression \gbc{spath} is a cycle it returns the interior, for
+% other paths, a drawing of the path, in either case scaled by \gbc{sc}.
+% If already a picture, it returns it. In \MP, if it is a string, it
+% returns a picture containing that string drawn in the \mfc{defaultfont}.
+% In any other case, the default dot is returned.
+% \begin{macrocode}
+vardef makesymbol (expr spath, sc) =
+ if picture spath :
+%<MF> setpicture (v) spath; mono (v); v
+%<MP> spath
+ elseif path spath: setdot (spath, sc)
+%<MP> elseif string spath:
+%<MP> spath infont defaultfont scaled defaultscale
+ else:
+ GBwarn "Undefined symbol for plotting, "
+ & "dotpath will be used instead.";
+ setdot (dotpath, sc)
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% Points are filled or unfilled circles. They are implemented with
+% \gbc{plotsymbol}, but the code differs in that filled or unfilled
+% circles are determined by a parameter rather than the type of curve.
+% In addition, for unfilled circles, it clears the pixels inside the circle.
+%
+% \DescribeRoutine{bpoint}
+% \gbc{bpoint} is basicly a shorthand for a scaled circle shifted to a
+% point. The scale and the point are in device coordinates. We don't use
+% it anywhere in \grafbase{} anymore.
+% \begin{macrocode}
+vardef bpoint (expr ptwd, b) =
+ fullcircle scaled ptwd shifted b
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{pointd}
+% This draws disks with diameter \gbc{ptwd}, filled or unfilled based on
+% the boolean \gbc{filled}, at the graph coordinate coordinates in the
+% list \gbc{t}. In case \gbc{filled} is true, \gbc{pointd} calls
+% \gbc{plotsymbol (SolidCircle)} otherwise we make \gbc{clearsymbols} true
+% (so that the area where each point is drawn will be cleared before
+% drawing it) and call \gbc{plotsymbol (Circle)}.
+% \begin{macrocode}
+def pointd (expr ptwd, filled) (text t) =
+ if filled:
+ plotsymbol (SolidCircle, ptwd) (t);
+ else:
+ begingroup;
+ setboolean (clearsymbols) true;
+ plotsymbol (Circle, ptwd) (t);
+ endgroup
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{plotsymbol}
+% \RoutineIndex{colorplotsymbol}
+% The \gbc{plotsymbol} command places a symbol centered at each of the
+% graph coordinate points in the list. The symbol placed is the first
+% parameter, which would normally be a path, but can be a picture or, in
+% \MP, a string. Like the \gbc{doplot} command, it calls \gbc{makesymbol}.
+% If \gbc{spath} is of type path, and is cyclic, it is drawn filled. This
+% is because we call \gbc{makesymbol} on it, and that subjects it to
+% \gbc{setdot}, which has that behavior. For other types of symbols, we
+% simply convert them to pictures with \gbc{makesymbol} and then place
+% them. Unlike \gbc{pointd} above, the interior of the path is not erased
+% by default. However, in the special case where the symbol is an open
+% path, if its first point is equal to its last point, and
+% \gbc{clearsymbols} is true, then the interior of the path obtained by
+% \gbc{\& cycle} is cleared before the path itself is drawn. We copy the
+% text list to an array and call \gbc{dosymbols} so that \gbc{plotnodes},
+% \gbc{plotsymbol} and \gbc{showcontrols} can share the code.
+%
+% \DescribeRoutine{dosymbols}
+% Since \gbc{dosymbols} uses identical code twice (once to clear, once to
+% draw), we put that code in \gbc{addsymbols}.
+% \DescribeRoutine{addsymbols}
+% And finally, \gbc{addsymbols} draws copies of a symbol at a given array
+% of points with a given color.
+% \begin{macrocode}
+boolean clearsymbols; clearsymbols := false;
+vardef clearable (expr pth) =
+ if path pth:
+ ( pnt0 (pth) = pnt[length pth] (pth) ) and (not cycle pth)
+ and (length pth > 0)
+ else: false
+ fi
+enddef;
+
+def clearopenpath expr f =
+ if clearable (f): safeunfill f & cycle; fi
+enddef;
+
+def plotsymbol = colorplotsymbol (pointcolor) enddef;
+def colorplotsymbol (expr clr, spath, sc) (text t) =
+ if sc > 0:
+ begingroup
+ setpairs (_cpls) (t);
+ if _cpls > 0: dosymbols (clr, spath, sc) (_cpls); fi
+ endgroup
+ fi
+enddef;
+
+def dosymbols (expr clr, spath, sc) (suffix arr) =
+ if clearsymbols and clearable (spath):
+ addsymbols (background, makesymbol (spath&cycle, sc)) (arr);
+ fi
+ addsymbols (clr, makesymbol (spath, sc)) (arr);
+enddef;
+
+def addsymbols (expr clr, symb) (suffix arr) =
+ newpicture (_pls);
+ for _idx = 1 upto arr:
+ picdot (_pls, symb, zconv (arr[_idx]));
+ endfor
+ DoClip (_pls); coloraddon (clr, _pls);
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{putimage}
+% This is designed to allow \mfpic{} users to make a picture (created with
+% \cs{mfpimage} or \cs{tile} perhaps), and put a copy at several
+% locations. This allows more complex things than \cs{plotsymbol} and
+% more flexibility than \cs{tess}. As the picture is should be completely
+% prepared in advance, there is no color or size parameter. Moreover,
+% \gbc{pic} might be quite complex, so we don't use \gbc{picdot} which
+% takes the added picture as an expression, but rather repeat its code.
+% The indirection of adding to \gbc{_pti} is normal when we want to
+% respect clipping without clipping what is already drawn. Saving memory
+% at some sacrifice of speed, we clip and reset with each addition. (It is
+% uncertain whether adding multiple clipped pictures includes the clipping
+% path for each addition. I'll have to test the memory use of this code
+% versus adding all the copies to \gbc{_pti} and clipping once.)
+% \begin{macrocode}
+def putimage (suffix pic) (text t) =
+ newpicture (_pti);
+ for _itm = t:
+ addto _pti also
+%<MF> (pic shifted goodpair (zconv (_itm)));
+%<MP> (pic shifted zconv (_itm));
+ DoClip (_pti); addto active_plane also _pti;
+ _pti := nullpicture;
+ endfor
+%<MF> mono active_plane
+enddef;
+
+% \end{macrocode}
+%
+%
+%
+% \section{Axes, Tic Marks, and Grids}\label{axes}
+%
+% \DescribeRoutine{arrowdraw}
+% This is used in \gbc{vectorfield} and to draw axes. It returns nothing.
+% This doesn't follow the usual pattern of drawing a path and returning
+% it. This approach makes the old \cs{axes}, \cs{xaxis} and \cs{yaxis}
+% commands in \mfpic{} impossible to dash or dot. The newer axis drawing
+% commands permit this and so use other code.
+%
+% We simply call \gbc{headpath} with default values, but add \gbc{drawn}
+% to make sure the path is drawn, and precede it with \gbc{store} so
+% \MF{} won't complain of an isolated expression. The new \mfpic{}
+% method of drawing an axis is to apply \cs{arrow} to the path
+% \gbc{axisline}. Thus it can also be \gbc{dashed}, \gbc{dotted},
+% etc.
+%
+% The order is significant if axis and head are different colors. This
+% order puts the head on top of the shaft.
+% \begin{macrocode}
+def arrowdraw (expr hlen) (expr f) =
+ store (curpath) headpath (hlen, 0, 0) drawn f;
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{xaxis}
+% The macro \gbc{xaxis} draws the $x$-axis through the point $(0,0)$ in
+% graph coordinates. The only parameter is the length of the arrowhead in
+% device coordinates.
+% \DescribeRoutine{yaxis}
+% The Macro \gbc{yaxis} draws the $y$-axis.
+%
+% \DescribeRoutine{axes}
+% \gbc{axes} draws both axes with the same length of head.
+% \begin{macrocode}
+def xaxis (expr hlen) = arrowdraw (hlen) ((xneg, 0)--(xpos, 0));
+enddef;
+def yaxis (expr hlen) = arrowdraw (hlen) ((0, yneg)--(0, ypos));
+enddef;
+def axes (expr hlen) = xaxis (hlen); yaxis (hlen); enddef;
+
+% \end{macrocode}
+%
+% For axes at the borders of the graph coordinates, we allow for them to
+% be shifted inwards. The amount of the shift is given by \gbc{laxis} for
+% the left side axis, \gbc{baxis} for the bottom axis, etc. They are in
+% graph coordinates.
+%
+% \DescribeRoutine{axisline}
+% The commands \gbc{axisline.x}, etc., return the appropriate straight
+% line at the appropriate location. These are vardefs rather than
+% variables so they can be affected by changing shift values.
+%
+% \DescribeRoutine{axis}
+% Finally, the commands \gbc{axis.x}, etc. examine their suffix and
+% apply \gbc{headpath} to the corresponding axis line. With a recent
+% change in \mfpic{} code, it is no longer used there. Instead, code
+% is written that allows the head to be drawn after the line is.
+% Note it is a vardef, and so returns the line as a path..
+% \begin{macrocode}
+laxis := baxis := raxis := taxis := 0;
+
+vardef xlow = xneg + laxis enddef;
+vardef xhigh = xpos - raxis enddef;
+vardef ylow = yneg + baxis enddef;
+vardef yhigh = ypos - taxis enddef;
+
+vardef axisline.x = (xlow, 0)--(xhigh, 0) enddef;
+vardef axisline.y = (0, ylow)--(0, yhigh) enddef;
+vardef axisline.l = axisline.y shifted (xlow, 0) enddef;
+vardef axisline.b = axisline.x shifted (0, ylow) enddef;
+vardef axisline.r = axisline.y shifted (xhigh, 0) enddef;
+vardef axisline.t = axisline.x shifted (0, yhigh) enddef;
+
+vardef axis@# (expr len) = headpath (len, 0, 0) axisline@# enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{borderrect}
+% These are mostly for the simplification of \mfpic{} and readability of
+% code. The command \gbc{borderrect} produces the border of the picture
+% in graph coordinates, taking into account the four margins.
+%
+% \DescribeRoutine{between}
+% The boolean \gbc{between} checks if the last argument is strictly
+% between the first two (which must be in order).
+%
+% \DescribeRoutine{inrange}
+% The boolean \gbc{inrange} checks if the last argument is in the closed
+% interval determined by the first two (which must be in order).
+%
+% \DescribeRoutine{inbounds}
+% The boolean \gbc{inbounds} checks if the argument (a pair) is in
+% the closed border rectangle (\gbc{borderrect}). It is not yet used in
+% \grafbase{}, though it would seem it ought to be useful.
+% \begin{macrocode}
+vardef borderrect =
+ rect((xlow,ylow),(xhigh,yhigh))
+enddef;
+
+vardef between (expr A, B, X) = (A < X) and (X < B) enddef;
+vardef inrange (expr A, B, X) = (A <= X) and (X <= B) enddef;
+
+vardef inbounds (expr Z) =
+ inrange (xlow, xhigh) (xpart Z) and inrange (ylow, yhigh) (ypart Z)
+enddef;
+
+% \end{macrocode}
+%
+% Possible binary relation versions. The last is just a reversal of the
+% order of the first. These are not yet used in \grafbase{}.
+% \begin{macrocode}
+tertiarydef X isbetween P = between (xpart P, ypart P, X) enddef;
+tertiarydef X isinrange P = inrange (xpart P, ypart P, X) enddef;
+tertiarydef P contains X = between (xpart P, ypart P, X) enddef;
+
+% \end{macrocode}
+%
+% Tick marks can be on the inside or outside of a border axis,
+% above or below any horizontal axes, left or right of any vertical axis
+% or centered on any axis. The following numerics are merely used to
+% convert the names to numeric code that the drawing routine will examine.
+%
+% However, it is no accident that \gbc{onbottom = onright} and that
+% \gbc{centered} is halfway between \gbc{onright} and \gbc{onleft}. The
+% code uses the numeric values to compute a shift, and one can supply an
+% expression like \gbc{.33ontop+.67onbottom]} and then 1/3 of each mark
+% will be above (and 2/3 will be below) the axis.
+%
+% The negative value of \gbc{inside} and \gbc{outside} is a flag that they
+% are to be treated differently. The others have the property that the
+% direction is the direction of the axis rotated a certain way (e.g.,
+% $90$ degrees from \mfc{up} points \mfc{left}, $-90$ points \mfc{right}).
+% But \gbc{inside} is right of the left axis and left of the right axis.
+% \begin{macrocode}
+numeric inside, outside, centered, onleft, onright, ontop, onbottom;
+inside := -2;
+outside := -1;
+onright := 1;
+onleft := 2;
+centered := .5[onright, onleft];
+onbottom := onright;
+ontop := onleft;
+% \end{macrocode}
+%
+% We interact with \mfpic{} by allowing the user to change the value of
+% \gbc{ltick}, for example, with a command like
+% \cs{setaxismarks l}\marg{outside}. Here we set the defaults.
+% \begin{macrocode}
+ltick := rtick := ttick := btick := inside;
+xtick := ytick := centered;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{axismarks}
+% This utility macro draws the tick marks on an arbitrary axis. The
+% different commands \gbc{xmarks}, etc., call this command with particular
+% values of these parameters.
+% \begin{itemize}
+% \item \gbc{inang} is the direction one must rotate the axis to point
+% inside. This is always $\pm90$ degrees. The $x$-axis and $y$-axis
+% are treated just like bottom and left axis in this respect.
+% \item \gbc{tp} is the tick position (e.g., \gbc{inside} or
+% \gbc{ontop}).
+% \item \gbc{loc} is the location of the 0-point of the axis (graph
+% coordinates).
+% \item \gbc{pdir} is \mfc{right} or \mfc{up}, indicating the positive
+% direction on the axis.
+% \item \gbc{len} is the length of a tick mark, supplied as an argument
+% to the individual axis mark commands.
+% \item \gbc{t} is the list of positions, also supplied.
+% \end{itemize}
+% \begin{macrocode}
+vardef axismarks (expr inang, tp, loc, pdir) (expr len) (text t) =
+ save _tp, _U, _P, _tic, _ticang;
+ pair _U, _P; path _tic;
+% \end{macrocode}
+% For \gbc{onleft}, \gbc{onright}, \gbc{ontop} or \gbc{onbottom}, which
+% are positive, don't examine \gbc{inang} but for \gbc{inside/outside}
+% use it to determine what inside means. \gbc{_ticang} will be the angle
+% to rotate \gbc{pdir} to set the direction of the tic mark.
+%
+% Then we shift the numeric value of \gbc{tp} by one, so \gbc{centered}
+% corresponds to $.5$ and the rest to either $0$ or $1$.
+% \begin{macrocode}
+ _ticang := if tp < 0: inang else: 90 fi;
+ _tp := abs(tp) - 1;
+% \end{macrocode}
+% Except, we go through the following shenanigans so that the marks are
+% always perpendicular to the axis, even if a coordinate transform will
+% slant the axis. After this \gbc{_U} should point in direction of inside,
+% onleft or ontop.
+% \begin{macrocode}
+ _U := unitvector (vconv (pdir)) rotated _ticang;
+% \end{macrocode}
+% Next, we use \gbc{_tp} to calculate the mark. For example, if
+% \gbc{tp = inside}, then \gbc{_tp = 1}. Since \gbc{_U} points toward
+% inside, \gbc{_tic} will go from \mfc{(0,0)} to a point a distance
+% \gbc{len} in the direction of \gbc{_U}.
+% \begin{macrocode}
+ _tic := (-_U--(0,0)) shifted (_tp*_U) scaled len;
+% \end{macrocode}
+% Finally, for each numeric value in the list \gbc{t}, draw the tic
+% shifted to the corresponding point on the axis.
+% \begin{macrocode}
+ for _a = t:
+ safedraw (_tic shifted zconv (loc + _a*pdir));
+ endfor
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{xmarks}
+% And now the specialized command for each axis. Inside and outside
+% really make no sense for the $x$- and
+% \DescribeRoutine{ymarks}
+% \RoutineIndex{lmarks}
+% \RoutineIndex{bmarks}
+% \RoutineIndex{rmarks}
+% \RoutineIndex{tmarks}
+% $y$-axis, but since a bottom axis is usually used for $x$ and a left
+% axis for $y$, we give \gbc{xmarks} the same first parameter as
+% \gbc{bmarks} and \gbc{ymarks} the same as \gbc{lmarks}.
+% \begin{macrocode}
+def xmarks = axismarks ( 90, xtick, origin, right) enddef;
+def ymarks = axismarks (-90, ytick, origin, up) enddef;
+def lmarks = axismarks (-90, ltick, (xlow, 0), up) enddef;
+def bmarks = axismarks ( 90, btick, (0, ylow), right) enddef;
+def rmarks = axismarks ( 90, rtick, (xhigh, 0), up) enddef;
+def tmarks = axismarks (-90, ttick, (0, yhigh), right) enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{vargrid}
+% \RoutineIndex{vgrid}
+% Mainly for the purpose of visualising coordinates, \gbc{vargrid}
+% draws a dot of size \gbc{dsize} at every point whose coordinates
+% are are \gbc{(n*xsp, m*ysp)}, \gbc{n} and \gbc{m} being integers.
+% \gbc{dsize} is in device coordinates, the spacings are in graph
+% coordinates.
+% \DescribeRoutine{grid}
+% The macro \gbc{grid} is for backward compatibility, calling
+% \gbc{vargrid} with a default \gbc{dsize} of \mfc{.5bp}. The old name
+% \gbc{vgrid} incorrectly suggests a connection to \gbc{vgridlines}.
+% \begin{macrocode}
+path griddotpath; griddotpath := fullcircle;
+def grid = vargrid (0.5bp) enddef;
+vardef vargrid (expr dsize, xsp, ysp) =
+ save gdot, gridpic; picture gdot, gridpic;
+ gdot := setdot (griddotpath, dsize);
+ gridpic := nullpicture;
+ for n = ceiling ((xlow)/xsp) upto floor ((xhigh)/xsp):
+ for m = ceiling ((ylow)/ysp) upto floor ((yhigh)/ysp):
+ picdot (gridpic, gdot, zconv ((n*xsp, m*ysp)));
+ endfor
+ endfor
+ coloraddon (pointcolor, gridpic);
+enddef;
+def vgrid = vargrid enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{gridlines}
+% This is more what I think of when I hear `grid', but the name was
+% already taken. The macro \gbc{gridlines} draws horizontal and vertical
+% lines through all the points that \gbc{grid} would draw.
+% \DescribeRoutine{hgridlines}
+% The macro \gbc{hgridlines} draws only the horizontal lines through the
+% same points, while
+% \DescribeRoutine{vgridlines}
+% \gbc{vgridlines} draws only vertical lines.
+% \begin{macrocode}
+def hgridlines (expr ysp) =
+ for n = ceiling ((ylow)/ysp) upto floor ((yhigh)/ysp):
+ safedraw zconv ((xlow, n*ysp)--(xhigh, n*ysp));
+ endfor
+enddef;
+def vgridlines (expr xsp) =
+ for n = ceiling ((xlow)/xsp) upto floor ((xhigh)/xsp):
+ safedraw zconv ((n*xsp, ylow)--(n*xsp, yhigh));
+ endfor
+enddef;
+def gridlines (expr xsp, ysp) =
+ vgridlines (xsp); hgridlines (ysp);
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{vectorfield}
+% This command produces a field of arrows from a pair-valued formula (text
+% parameter \gbc{fcn}) in a region described by a boolean-valued
+% expression (text parameter \gbc{cond}). This routine simply makes
+% functions (\mfc{vardef}\,s) out of the expressions and calls
+% \DescribeRoutine{mkvectorfield}
+% \gbc{mkvectorfield}, which steps through the points described by
+% \gbc{xsp} and \gbc{ysp} and places an arrow (actually, any path) at
+% each. The arrow path is given by the function \gbc{vf}. The arrow is
+% placed at the point only if the function \gbc{isOK} returns true. It
+% also omits points that lie in the axis margins.
+%
+% \DescribeRoutine{plrvectorfield}
+% The polar version differs only in the distribution of the arrows. They
+% are placed at regular intervals of $r$ an $\theta$. The text parameters
+% should be expressions in \gbc{r} and \gbc{t}, but are otherwise the
+% same. In particular, \gbc{fcn} should return \MF{} pairs, not polar
+% coordinate pairs. The function \gbc{polar} can be used to convert if
+% necessary. Its code is very similar, except for the boolean code needed
+% to keep the vectors within the bounds of the graph.
+% \DescribeRoutine{mkplrvectorfield}
+% It calls \gbc{mkplrvectorfield}, which is a lot like the non-polar
+% version, except it first calculates the extremes of the polar variables
+% with \gbc{getpolarbounds} and relies on the boolean to keep it out of
+% the axis margins.
+% \begin{macrocode}
+def vectorfield (expr len, xsp, ysp) (text fcn) (text cond) =
+ save _vf, _is_OK;
+ vardef _vf (expr x,y) = ((0,0)--(fcn)) shifted (x,y) enddef;
+ vardef _is_OK (expr x,y) = cond enddef;
+ mkvectorfield (len, xsp, ysp) (_vf, _is_OK);
+enddef;
+
+vardef mkvectorfield (expr len, xsp, ysp) (suffix vf, isOK) =
+ for n = ceiling ((xlow)/xsp) upto floor ((xhigh)/xsp):
+ for m = ceiling ((ylow)/ysp) upto floor ((yhigh)/ysp):
+ if isOK (n*xsp,m*ysp): arrowdraw (len) (vf(n*xsp,m*ysp)); fi
+ endfor
+ endfor
+enddef;
+
+def plrvectorfield (expr len, rsp, tsp) (text fcn) (text cond) =
+ save _vf, _is_OK, _A, _B, _C, _D;
+ _A := xlow; _B := xhigh;
+ _C := ylow; _D := yhigh;
+ vardef _vf (expr r,t) = ((0,0)--(fcn)) shifted (r*dir t) enddef;
+ vardef _is_OK (expr r,t) =
+ save _X, _Y; _X := r*cosd t; _Y := r*sind t;
+ (cond) and between (_A, _B) (_X) and between (_C, _D) (_Y)
+ enddef;
+ mkplrvectorfield (len, rsp, tsp) (_vf, _is_OK);
+enddef;
+
+vardef mkplrvectorfield (expr len, rsp, tsp) (suffix vf, isOK) =
+ save rmin, rmax, tmin, tmax;
+ getpolarbounds;
+ if rmin = 0:
+ if isOK (0,tmin): arrowdraw (len) (vf (0,tmin)); fi
+ rmin := rsp;
+ fi
+ for n = ceiling (rmin/rsp) upto floor (rmax/rsp):
+ for m = ceiling (tmin/tsp) upto floor (tmax/tsp):
+ if isOK (n*rsp,m*tsp): arrowdraw (len) (vf (n*rsp,m*tsp)); fi
+ endfor
+ endfor
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{patcharcs}
+% The macro \gbc{patcharcs} draws on a picture \gbc{X} the arcs
+% \gbc{tstart}${}\le \theta \le{}$\gbc{tstop} with radii starting at
+% \gbc{rstart}, stepping by \gbc{rstep} until \gbc{rstop}.
+% \DescribeRoutine{patchrays}
+% The macro \gbc{patchrays} draws the radial lines with $r$ coordinate
+% varying between \gbc{rstart} and \gbc{rstop} at angles from \gbc{tstart}
+% to \gbc{tstop} stepping by \gbc{tstep}.
+% \DescribeRoutine{plrpatch}
+% And \gbc{plrpatch} simply calls them both, and adds the resulting
+% pictures to \gbc{active_plane}.
+% \begin{macrocode}
+def patcharcs (suffix X) (expr rstart, rstop, rstep, tstart, tstop) =
+ for rad = (if rstart = 0: rstep else: rstart fi)
+ step rstep until rstop:
+ orto (X, picpath zconv (arcplr (origin, tstart, tstop, rad)) );
+ endfor
+enddef;
+def patchrays (suffix X) (expr tstart, tstop, tstep, rstart, rstop) =
+ for _ang = tstart step tstep until tstop:
+ orto (X) (picpath zconv ((rstart*dir _ang)--(rstop*dir _ang)));
+ endfor
+enddef;
+
+def plrpatch (expr rstart, rstop, rstep, tstart, tstop, tstep) =
+begingroup
+ newpicture (v);
+ patcharcs (v) (rstart, rstop, rstep, tstart, tstop);
+ coloraddon (drawcolor, v);
+ v := nullpicture;
+ patchrays (v) (tstart, tstop, tstep, rstart, rstop);
+ coloraddon (drawcolor, v);
+endgroup
+enddef;
+
+% \end{macrocode}
+%
+% Polar coordinate grids are analogous to \gbc{gridlines} and \gbc{grid}.
+% They first draw a grid large enough to cover the whole graph, then clip
+% it to the graph boundaries. Since three of the four require
+% calculating the dimensions of a polar coordinate patch that completely
+% covers the graph rectangle, we isolate that code in
+% \gbc{beginpolargrid}, defined later.
+%
+%
+% \DescribeRoutine{gridarcs}
+% \gbc{gridarcs} creates arcs having radii that are integer multiples of
+% \gbc{rstep} and
+% \DescribeRoutine{gridrays}
+% \gbc{gridrays} draws radial lines at angles that are multiples of
+% \gbc{tstep}.
+% \DescribeRoutine{polargrid}
+% The command \gbc{polargrid}simply calls the first two.
+%
+% \DescribeRoutine{polargridpoints}
+% On the other hand, \gbc{polargridpoints} draws dots at the points where
+% the lines and arcs in \gbc{poloargrid} would intersect. The `step'
+% parameters are in graph coordinates. \gbc{beginpolargrid} also
+% declares the picture variable \gbc{gridpic}, while \gbc{endpolargrid}
+% clips the resulting picture and adds it to \gbc{active_plane}.
+%
+% The \gbc{rmin}, etc., returned by \gbc{beginpolargrid} are modified to
+% fit the grid established by the step sizes. A ray could happen to be one
+% of the graph's sides, so we use \mfc{ceiling} and \mfc{floor} which
+% doesn't change integer values. However, the arc with radius \gbc{rmin}
+% or \gbc{rmax} could touch the graph rectangle in at most 4 points, so we
+% use \mfc{floor (x + 1)} and \mfc{ceiling (x - 1)} to start and stop
+% before the edge of the graph.
+% \begin{macrocode}
+def gridarcs (expr rstep) =
+ beginpolargrid;
+ if rmin = 0:
+ picdot (gridpic, setdot (griddotpath, penwd), zconv (origin));
+ fi
+ rmin := rstep * floor (rmin/rstep + 1);
+ rmax := rstep * ceiling (rmax/rstep - 1);
+ patcharcs (gridpic) (rmin, rmax, rstep, tmin, tmax);
+ endpolargrid (drawcolor, .5penwd);
+enddef;
+
+def gridrays (expr tstep) =
+ beginpolargrid;
+ tmin := tstep * ceiling (tmin/tstep);
+ tmax := tstep * floor (tmax/tstep);
+ patchrays (gridpic) (tmin, tmax, tstep, rmin, rmax);
+ endpolargrid (drawcolor, .5penwd);
+enddef;
+
+def polargrid (expr rstep, tstep) =
+ gridarcs (rstep); gridrays (tstep);
+enddef;
+
+def polargridpoints (expr dsize, rstep, tstep) =
+ beginpolargrid;
+ setpicture (gdot) setdot (griddotpath, dsize);
+ if rmin = 0:
+ picdot (gridpic, gdot, zconv (origin));
+ rmin := rstep;
+ fi
+ for n = ceiling (rmin/rstep) upto floor (rmax/rstep):
+ for m = ceiling (tmin/tstep) upto floor (tmax/tstep):
+ picdot ( gridpic, gdot, zconv ( polar ((n*rstep, m*tstep)) ) );
+ endfor
+ endfor
+ endpolargrid (pointcolor, .5dsize);
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{beginpolargrid}
+% The macro \gbc{beginpolargrid} calls \gbc{getpolarbounds} to compute the
+% bounds (on $r$ and $\theta$) of the smallest polar coordinate patch that
+% covers the graph rectangle.
+% \DescribeRoutine{getpolarbounds}
+% That command leaves the values in \gbc{rmin}, \gbc{rmax}, \gbc{tmin} and
+% \gbc{tmax}. Then \gbc{beginpolargrid} initializes \gbc{gridpic} whereon
+% the grids are drawn.
+% \begin{macrocode}
+def beginpolargrid =
+begingroup;
+ save rmax, rmin, tmax, tmin;
+ getpolarbounds;
+ newpicture (gridpic);
+enddef;
+
+def getpolarbounds =
+ save p, r, t;
+ pair p[];
+ p0 := (xneg, yneg); p1 := (xneg, ypos);
+ p2 := (xpos, ypos); p3 := (xpos, yneg);
+% \end{macrocode}
+% This loop finds the radial coordinate of each corner of the graph and
+% finds the maximum while doing so.
+% \begin{macrocode}
+ r0 := abs(p0); rmax := r0;
+ for j = 1 upto 3:
+ r[j] := abs(p[j]);
+ if rmax < r[j]: rmax := r[j]; fi
+ endfor
+% \end{macrocode}
+% When the origin is inside the graph rectangle we need the full range
+% of $r$ and $\theta$. When the origin is one of the corners, the angles
+% can just be read off. Otherwise, to find the range of $\theta$ we
+% essentially rotate one corner to have angle zero, get the angles to all
+% corners and rotate back. This guarantees that the wedge with
+% \gbc{tmin}${} < \theta < {}$\gbc{tmax} includes the graph.
+% \begin{macrocode}
+ rmin := 0;
+ if between (xneg, xpos) (0) and between (yneg, ypos) (0):
+ tmin := 0; tmax := 360;
+ elseif (p0 = origin): tmin := 0; tmax := 90;
+ elseif (p1 = origin): tmin := -90; tmax := 0;
+ elseif (p2 = origin): tmin := -180; tmax := -90;
+ elseif (p3 = origin): tmin := 90; tmax := 180;
+ else:
+ tmax := tmin := t0 := angle p0;
+ for j = 1 upto 3:
+ t := t0 + anglefromto (p0, p[j]);
+ if tmax < t: tmax := t; fi
+ if tmin > t: tmin := t; fi
+ endfor
+% \end{macrocode}
+% The minimum value of $r$ can be one of 9 possibilities: if the four
+% sides of the graph are extended infinitely far in both directions, the
+% origin can be in any one of the 9 regions formed. We've already disposed
+% of the inside of the graph. This code considers the remaining regions in
+% the following order: (1)~above or below, (2)~left or right, and (3)~one
+% of the four corner regions.
+% \begin{macrocode}
+ if between (xneg, xpos) (0):
+ rmin := emin (abs(yneg), abs(ypos));
+ elseif between (yneg, ypos) (0):
+ rmin := emin (abs(xneg), abs(xpos));
+ else:
+ rmin := min (r0, r1, r2, r3);
+ fi
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{endpolargrid}
+% The \gbc{clr} is \gbc{drawcolor} for line grids, \gbc{pointcolor} for
+% dot grids. The \gbc{size} is half the width of the grid's lines or half
+% the width of the grid's dots. The purpose is to make sure dots and lines
+% on the graph's edge aren't cut off. For dots I should probably put this
+% decision in the code that draws them on \gbc{gridpic}.
+% \begin{macrocode}
+
+def endpolargrid (expr clr, size)=
+ clipto (gridpic) rect ( zconv ((xneg, yneg)) - size*(1,1),
+ zconv ((xpos, ypos)) + size*(1,1) );
+ coloraddon (clr, gridpic);
+ endgroup
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{polarpatch}
+% Finally, this just does \gbc{plrpatch}, but also draws the ending
+% boundaries, in case they are not an integer number of steps from the
+% start.
+% \begin{macrocode}
+vardef polarpatch (expr rstart, rstop, rstep, tstart, tstop, tstep) =
+ plrpatch (rstart, rstop, rstep, tstart, tstop, tstep);
+ safedraw zconv ( arcplr (origin, tstart, tstop, rstop) );
+ safedraw zconv ( ((rstart, 0)--(rstop, 0)) rotated tstop );
+enddef;
+
+% \end{macrocode}
+%
+%
+%
+% \section{Path Construction}\label{pathconstruction}
+%
+% This section is devoted to commands that accept a list or array of
+% points and produce a path, usually (but not necessarily) through those
+% points. In addition there are a few commands that find some of the key
+% points, lines and circles associated with a triangle. No \mfpic{}
+% interface is yet available for the triangle commands.
+%
+%
+% \subsection{Piecewise linear paths}\label{linear}
+%
+% \DescribeRoutine{rect}
+% Most of the macros that only define paths are coordinate independent.
+% The simplest is \gbc{rect}. It accepts two pair expressions and produces
+% the upright rectangle with those points at opposite corners. It might be
+% noted that if the corners really are lower left and upper right, then
+% the path is anticlockwise, If they are on the other diagonal, the
+% path is clockwise. The path is a cycle (closed). The starting/ending
+% point (needed for arrows and the like) is the first point of the two.
+%
+% \DescribeRoutine{triangle}
+% Produces a closed path joining three points with straight lines; first
+% named point \gbc{A} is \mfc{point 0 of triangle (A, B, C)}, etc.
+% \begin{macrocode}
+vardef rect (expr ll, ur) =
+ ll--(xpart ur, ypart ll)--ur--(xpart ll, ypart ur)--cycle
+enddef;
+vardef triangle (expr A, B, C) = A--B--C--cycle enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{regularpolygon}
+% The first argument is the number of sides, the second is an array name
+% to hold the list of vertices. The third argument contains two
+% equations separated by a semicolon, preferably the location of two of
+% the vertices, or the location of the center and one vertex. That plus
+% the equations in the \mfc{for}-loop give \gbc{n+1} equations to
+% determine the \gbc{n} vertices and the center. Note that the vertices
+% are numbered anticlockwise.
+%
+% The equations must \emph{not} be equations that are satisfied by all
+% $n$-gons regardless of size and position. For example:\\
+% \indent \gbc{regulapolygon(4)(Ted)(Ted0 := (0,0);Ted1 + Ted3 = (0,0))}\\
+% The second of these says the middle lies halfway between the extremes,
+% and is already a consequence of the code. It goes without saying that
+% equations that cannot be satisfied by any regular polygon are also out.
+% \begin{macrocode}
+vardef regularpolygon (expr n) (suffix Bob) (text eqns) =
+ pair Bob[]; Bob := emax (round (abs (n)), 2);
+ eqns;
+ for _uncle = 1 upto Bob - 1:
+ (Bob1 - Bob0) rotated (360/Bob*_uncle) = Bob[_uncle+1] - Bob0;
+ endfor
+ mkpoly (true) (Bob)
+enddef;
+
+% \end{macrocode}
+%
+% The following set of commands take a path as argument, but it is
+% intended that it be a triangle. Even then, they work correctly only if
+% it is a cycle.
+%
+% These produce the perpendicular from \,\gbc{point n of t}\, to the
+% (extension of) the opposite side (i.e., the altitude).
+% \DescribeRoutine{altitudept}
+% The first one determines where the altitude meets the opposite side, and
+% the
+% \DescribeRoutine{altitude}
+% second just connects the two points. Since \gbc{altitudept} is always
+% \gbc{point 1 of altitude}, it is actually redundant. However, the
+% command \gbc{medianpt} (defined below) is used outside of the
+% construction of \gbc{median}, so it seemed possible the \gbc{altitudept}
+% might be useful also.
+%
+% We need a cycle so that points $n+1$ and $n+2$ will wrap around to the
+% start of the path when necessary.
+% \begin{macrocode}
+vardef altitudept expr n of t =
+ save A, B, C, zz; pair A, B, C, zz;
+ B := pnt[n + 1] (t);
+ C := pnt[n + 2] (t);
+ zz = whatever[B,C];
+ zz = pnt[n](t) + whatever*((C-B) rotated 90);
+ zz
+enddef;
+
+vardef altitude expr n of t =
+ (pnt[n](t))--(altitudept n of t)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{medianpt}
+% These next two produce the midpoint of the side opposite
+% \,\gbc{point n of t}\, and the
+% \DescribeRoutine{median}
+% line connecting those two points.
+% \begin{macrocode}
+vardef medianpt expr n of t =
+ 0.5[pnt[n + 1] (t), pnt[n + 2] (t)]
+enddef;
+
+vardef median expr n of t =
+ (pnt[n](t))--(medianpt n of t)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{anglebisectorpt}
+% The first produces the point on the side opposite \,\gbc{point n of t}\,
+% where the angle bisector at that corner crosses it and the second
+% produces
+% \DescribeRoutine{anglebisector}
+% the line that bisects that angle.
+% \begin{macrocode}
+vardef anglebisectorpt expr n of t =
+ save A, B, C; pair A, B, C;
+ A := pnt[n ] (t);
+ B := pnt[n + 1] (t);
+ C := pnt[n + 2] (t);
+ save zz; pair zz;
+ zz = whatever[B,C];
+ zz = A + whatever*((B-A) rotated (.5*cornerangle (A,B,C)));
+ zz
+enddef;
+
+vardef anglebisector expr n of t =
+ (pnt[n](t))--(anglebisectorpt n of t)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{cornerangle}
+% This calculates the angle at the corner of a triangle. Specifically,
+% the angle (between $-180$ and $180$) required to rotate the vector
+% \gbc{B-A} into \gbc{C-A}. For degenerate triangles the seemingly
+% arbitrary values 60 and 90 are designed to match the assumptions used
+% in the arc commands. But also to guarantee that the three
+% \gbc{cornerangle}\,s add up to $\pm180$. \gbc{cornerangle (A,B,C)} gives
+% the angle at \gbc{A}, positive if \gbc{A--B--C--cycle} is
+% anticlockwise.
+% \begin{macrocode}
+vardef anglefromto (expr u, v) =
+ if (u = origin) or (v = origin): 0
+ else: angle (v rotated (-angle u))
+ fi
+enddef;
+
+vardef cornerangle (expr A, B, C) =
+ if (A = B) or (A = C) :
+ if (B = C) : 60
+ else: 90
+ fi
+ else: anglefromto (B - A, C - A)
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{mkpath}
+% This accepts the name of an array of pairs and produces a path
+% that connects them. The first and third parameters are booleans. If
+% \gbc{smooth} is \mfc{true} a smooth path is produced, otherwise a
+% polyline. If \gbc{cyclic} is \mfc{true} the path is closed. The work is
+% actually done by \gbc{mksmooth} or \gbc{mkpoly}.
+% \begin{macrocode}
+vardef mkpath (expr smooth, tens, cyclic) (suffix pts) =
+ if smooth: mksmooth (tens)
+ else: mkpoly
+ fi (cyclic, pts)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{mkpoly}
+% This produces the path of line segments connecting \gbc{pts1},
+% \gbc{pts2}, etc., closing it up if the boolean \gbc{cyclic} is true.
+% It can also be used with an array of paths instead of points, connecting
+% the end of each with the beginning of the next. We do this in \mfpic{}'s
+% \cs{connect} \dots\ \cs{endconnect} construct.
+% \begin{macrocode}
+vardef mkpoly (expr cyclic) (suffix pts) =
+ for _i = 1 upto pts-1: pts[_i]-- endfor
+ pts[pts] if cyclic: -- cycle else: {0,0} fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{polyline}
+% This is the \mfpic{} interface. Instead of an array name, it accepts a
+% list of pair expressions, forms an array from them and calls
+% \gbc{mkpoly}.
+% \DescribeRoutine{NoPoints}
+% \mfc{NoPoints} is called when an array of points is defined (using
+% setpairs) that returns $0$ for the number of pairs. It prints a warning
+% and sets the array to a single point, the origin.
+% \begin{macrocode}
+vardef polyline (expr cyclic) (text t) =
+ setpairs (_pl) (t);
+ if _pl=0: NoPoints ("polyline", _pl); fi
+ mkpoly (cyclic, _pl)
+enddef;
+
+def NoPoints (expr s) (suffix pts) =
+ GBwarn s & " attempted with empty list."; pts[incr pts] := origin;
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{turtle}
+% \emph{Turtle graphics} was a teaching tool to get youngsters used to the
+% concept of programming while also teaching geometry. The students fed an
+% Apple II computer a sequence of angles and distances, and a small
+% triangle on the screen (the `turtle') would turn the indicated angle
+% and travel the indicated distance, tracing a polyline on the screen.
+%
+% The argument of \gbc{turtle} is a list of pairs. The first is the
+% starting point, the rest are vector displacements (moves). The
+% distance and incremental angles of the original turtle graphics would
+% require keeping track of the current angle and using the \gbc{polar}
+% command.
+% \begin{macrocode}
+vardef turtle (text t) =
+ setnumeric (_tu) 0;
+ setpair (_tmp) origin;
+ pair _tu[];
+ for _a = t:
+ _tmp := _tmp + _a;
+ _tu[incr _tu] := _tmp;
+ endfor
+ if _tu = 0: NoPoints("turtle", _tu); fi
+ mkpoly (false, _tu)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{brownianpath}
+% I needed the following to illustrate Brownian motion. It takes a given
+% starting point, a given number of steps and a scaling factor. It
+% generates a sequence of random points, each one being chosen randomly
+% using a Gaussian distribution centered at the previous point. The
+% standard deviation of the random distance is the scale factor. Strictly
+% speaking this is a Gaussian random walk, not Brownian motion. A true
+% Brownian motion would be a limit of these, with \gbc{num} tending to
+% $\infty$ and \gbc{sc} tending to 0.
+%
+% \DescribeRoutine{randomwalk}
+% This is like \gbc{brownianpath}, but the distance from one point to
+% the next is always the same, only the direction is random. It takes
+% the same arguments as \gbc{brownianpath}
+%
+% \DescribeRoutine{browniangraph}
+% This command takes a given number of steps \gbc{num} and a scaling
+% factor/step size \gbc{scst}. It generates a sequence of points, each one
+% being chosen right of the previous one by the step size \gbc{scst} and
+% randomly up or down using a Gaussian distribution centered at the
+% previous $y-value$. The Gaussian distribution has standard deviation
+% equal to \gbc{scst}. The path starts at $(0,0)$. One needs to transform
+% the path to get a different start or a scale factor different from the
+% step size.
+%
+% In \MF{} we run into capacity problems when \gbc{num} is greater than
+% 500 or so. This is the \mfc{autorounding} problem again (see the
+% discussion at \gbc{sinewave}. We can't use the same technique we used
+% there since it is the drawing that invokes \mfc{autorounding} and these
+% macros only construct paths; they don't draw them.
+% \begin{macrocode}
+vardef brownianpath (expr start, num, sc) =
+ setnumeric (_brp) 1;
+ setpair (_tmp) start;
+ pair _brp[]; _brp1 := _tmp;
+ for _idx := 1 upto num:
+ _tmp := _tmp + sc/(sqrt 2)*(normaldeviate,normaldeviate);
+ _brp[incr _brp] := _tmp;
+ endfor
+ mkpoly (false, _brp)
+enddef;
+vardef randomwalk (expr start, num, dst) =
+ setnumeric (_rdw) 1;
+ setpair (_tmp) start;
+ pair _rdw[]; _rdw1 := _tmp;
+ for _idx := 1 upto num:
+ _tmp := _tmp + dst*dir(uniformdeviate(360));
+ _rdw[incr _rdw] := _tmp;
+ endfor
+ mkpoly (false, _rdw)
+enddef;
+vardef browniangraph (expr num, scst) =
+ setnumeric (_brg) 1;
+ pair _tmp, _brg[]; _tmp := _brg1 := (0,0);
+ for _idx := 1 upto num:
+ _tmp := _tmp + scst*(1,normaldeviate);
+ _brg[incr _brg] := _tmp;
+ endfor
+ mkpoly (false, _brg)
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Smooth paths}\label{smooth}
+%
+% We added an optional parameter for the tension of smooth curves to
+% \mfpic. It used to be implemented this way: functions that implement a
+% tension parameter set \gbc{cur_tension} and called \gbc{mksmooth}, which
+% used that tension in its formation of a path. Since \gbc{mksmooth} was
+% only ever used in this way, I decided to change its syntax to include a
+% tension parameter. Only the functions \gbc{tcurve} and \gbc{mkpath}
+% actually call \gbc{mksmooth} directly, most other path building commands
+% with tension parameters call \gbc{mkpath} or \gbc{mkfcn} (which calls
+% \gbc{mkpath}).
+%
+% \DescribeRoutine{mksmooth}
+% This takes a tension value, a boolean, and the name of an array of
+% points, draws the curve connecting them and closes it up if the boolean
+% is true. It draws the curve forcing it to have the same direction at a
+% point as the line segment connecting the preceding and following points.
+% This is normally best if the curve direction changes relatively modestly
+% from point to point. For example, if the polyline would be convex, then
+% this smooth version would be pretty close to being convex. If the convex
+% polygon has several consecutive sides that are in the same direction,
+% all but the first and last of these segments in the smooth version would
+% be straight.
+% \begin{macrocode}
+vardef mksmooth (expr tens, cyclic) (suffix pts) =
+ if pts = 1: onepointpath (cyclic, pts1)
+ else:
+ settension (_tn) tens; fixtension (_tn);
+ pts1 if cyclic: {pts[2]-pts[pts]} fi
+ for _i = 2 upto pts-1:
+ ..tension _tn..pts[_i]{pts[_i+1]-pts[_i-1]}
+ endfor
+ ..tension _tn..pts[pts]
+ if cyclic: {pts[1]-pts[pts-1]}..tension _tn..cycle fi
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{mktenser}
+% This is just like \gbc{mksmooth}, except the tension value is preceded
+% by \mfc{atleast}. At this writing only \gbc{mkconvex} uses it (as a
+% fallback when there are three or fewer points to connect).
+% \begin{macrocode}
+vardef mktenser (expr tens, cyclic) (suffix pts) =
+ if pts = 1: onepointpath (cyclic, pts1)
+ else:
+ settension (_tn) tens; fixtension (_tn);
+ pts1 if cyclic: {pts[2]-pts[pts]} fi
+ for _i = 2 upto pts-1:
+ ..tension atleast _tn..pts[_i]{pts[_i+1]-pts[_i-1]}
+ endfor
+ ..tension atleast _tn..pts[pts]
+ if cyclic: {pts[1]-pts[pts-1]}..tension atleast _tn..cycle fi
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{mkconvex}
+% This could have been very much like \gbc{mksmooth}, using
+% \mfc{tension atleast} instead of \mfc{tension} (i.e., exactly
+% \gbc{mktenser} above). Unfortunately This destroys smoothness at the
+% beginning and end of any sequence of three or more points that lie on a
+% straight line. Some geometric situations absolutely prevent smoothness,
+% but this certainly isn't one of them. Two consecutive points
+% identical isn't either, but it does. We let the user or calling command
+% arrange for it not to happen (for example, using \gbc{setuniquepairs}
+% instead of \gbc{setpairs}.
+%
+% What we do is weight the direction to be used at each point by how flat
+% the polygon is on the two sides of each point, the flatter side
+% getting the most weight. If a point is collinear with the next two,
+% the flatness is infinite and the curve is forced in that direction. We
+% measure the flatness using the square root of the area of the triangle
+% made by the given point and the the next two points. Those three points
+% lie on a line just when the area is $0$. Using this measure of flatness
+% can be disputed (two triangles can be equally flat in terms of angles
+% but different in terms of areas), but it has the advantage that if the
+% points are subjected to an affine transformation, the weighting is
+% unchanged and the constructed direction vectors transform the same way.
+% Of course, this is also true of other measures. I've tested only
+% the square root of area and it works well for simple cases.
+%
+% The only way to ensure that the whole path transforms the same as the
+% points is to explicitly calculate the controls (in a manner that
+% transforms the same way as the points). I will do this if I can figure
+% out how it \emph{should} be done. It appears to be a convex programing
+% problem.
+%
+% The end segments of a noncycle are problematic. One could argue that
+% this command doesn't apply to noncycles and just truncate the cyclic
+% path. I decided to treat them specially and hope the result is useful.
+% \begin{macrocode}
+vardef mkconvex (expr tens, cyclic) (suffix pts) =
+ save _B, _d, _tmp; pair _d[];
+ settension (_tn) tens; fixtension (_tn);
+ if pts < 4: mktenser (_tn, cyclic) (pts)
+ else:
+ for _j = 2 upto pts - 1:
+ _B[_j] := sqrt(abs((pts[_j]-pts[_j-1])xprod(pts[_j+1]-pts[_j])));
+ endfor
+ if cyclic:
+ _B1 := sqrt(abs((pts1 - pts[pts])xprod(pts2 - pts1)));
+ _B[pts] := sqrt(abs((pts[pts]-pts[pts-1])xprod(pts1 - pts[pts])));
+ else:
+ _B1 := _B2;
+ _B[pts] := _B[pts-1];
+ fi
+ for _j = 2 upto pts - 1:
+ _tmp := _B[_j-1] + _B[_j+1];
+ _d[_j] :=
+ if _tmp = 0: origin % signal to use curl1
+ else:
+ ( _B[_j+1]*(pts[_j] - pts[_j-1]) +
+ _B[_j-1]*(pts[_j+1] - pts[_j]) )/_tmp
+ fi;
+ endfor
+ if cyclic:
+ _tmp := _B[pts] + _B2;
+ _d1 :=
+ if _tmp = 0: origin
+ else:
+ (_B2*(pts1 - pts[pts]) + _B[pts]*(pts2 - pts1))/_tmp
+ fi;
+ _tmp := _B[pts-1] + _B1;
+ _d[pts] :=
+ if _tmp = 0: origin
+ else:
+ ( _B1*(pts[pts] - pts[pts-1]) +
+ _B[pts-1]*(pts1 - pts[pts]) )/_tmp
+ fi;
+ else:
+ _d1 := origin; _d[pts] := origin;
+ fi
+ pts1
+ for _j = 1 upto pts-1:
+ {if _d[_j] = origin: curl1 else: _d[_j] fi}
+ ..tension atleast _tn..pts[_j+1]
+ endfor
+ {if _d[pts] = origin: curl1 else: _d[pts] fi}
+ if cyclic: ..tension atleast _tn..cycle fi
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% The old \cs{curve} command in \mfpic{} permitted no tension parameter
+% and wrote a \grafbase{} \gbc{curve} command.
+% \DescribeRoutine{curve}
+% For backward compatibility we keep that name, but simply call the
+% \gbc{tcurve} command with the default value for tension.
+% \DescribeRoutine{tcurve}
+% \gbc{tcurve} converts a list of pairs to an array, then calls
+% \gbc{mksmooth} on the array.
+%
+% The next pair call \gbc{mkconvex}, which tries to produce a convex curve
+% when the points form a convex polygon.
+% \DescribeRoutine{ccurve}
+% The first, \gbc{ccurve}, merely calls the second with a default texnsion,
+% while
+% \DescribeRoutine{tccurve}
+% creates an array from the list of pairs and calls \gbc{mkconvex} on it.
+% \begin{macrocode}
+numeric default_tension; default_tension := 1;
+def curve = tcurve (default_tension) enddef;
+vardef tcurve (expr tens, cyclic) (text t) =
+ setpairs (_tc) (t);
+ if _tc=0: NoPoints("curve", _tc); fi
+ mksmooth (tens, cyclic, _tc)
+enddef;
+
+def ccurve = tccurve (default_tension) enddef;
+vardef tccurve (expr tens, cyclic) (text t) =
+ setuniquepairs (_tcc) (t);
+ if _tcc=0: NoPoints("ccurve", _tcc); fi
+ mkconvex (tens, cyclic, _tcc)
+enddef;
+
+% \end{macrocode}
+%
+% It seemed odd that we had no way for an \MF-savvy user to easily get
+% the standard \mfc{p..q..r} kind of path. For such a simple one,
+% \cs{mfobj} with the explicit path expression would work, but when one
+% has to add a tension to it, it is nice to have an abbreviation. That's
+% what these are for.
+%
+% \DescribeRoutine{mkbezier}
+% The command \gbc{mkbezier} takes an array argument and produces either
+% an open or cyclic path with a given tension.
+% \DescribeRoutine{bezier}
+% The macro \gbc{bezier} does nothing more than call \gbc{tbezier} with
+% the default tension,
+% \DescribeRoutine{tbezier}
+% which takes a list of points and creates an array for \gbc{mkbezier} to
+% act on.
+% \begin{macrocode}
+vardef mkbezier (expr tens, cyclic) (suffix pts) =
+ settension (_tn) tens; fixtension (_tn);
+ pts1
+ for _i = 2 upto pts: ..tension _tn..pts[_i] endfor
+ if cyclic: ..tension _tn..cycle else: {0,0} fi
+enddef;
+
+def bezier = tbezier (default_tension) enddef;
+vardef tbezier (expr tens, cyclic) (text t) =
+ setpairs (_tbs) (t);
+ if _tbs=0: NoPoints ("bezier", _tbs); fi
+ mkbezier (tens, cyclic) (_tbs)
+enddef;
+
+% \end{macrocode}
+%
+% It also seemed we ought to allow \mfpic{} users to easily reproduce the
+% effect of a sequence of \LaTeX's \cs{qbezier} commands. That's what
+% these next are for.
+%
+% These commands and the various splines below don't use tension as they
+% have their control points explicitly given, not computed from the
+% tension value by \MF. The \gbc{qbezier} command does not produce a
+% smooth path unless the controls are explicitely chosen for that. The
+% spline commands will almost always produce a smooth path.
+%
+% \DescribeRoutine{mkqbezier}
+% \gbc{mkqbezier} requires an even number of points for a cyclic path,
+% an odd number for a noncyclic path. It does not check for this, but
+% the calling macro \gbc{qbezier} does. If the parity is incorrect, it
+% repeats the last point in the list. This has the effect of making
+% the last link a straight line.
+%
+% \DescribeRoutine{qbezier}
+% The \gbc{qbezier} command takes a list of points and creates an array
+% from then before calling \gbc{mkqbezier}.
+%
+% \DescribeRoutine{mkcbezier}
+% This is like \gbc{mkqbezier}, but needs a multiple of 3 for a closed
+% cubic bezier, one more (the endpoint) for an open cubic bezier.
+%
+% \DescribeRoutine{cbezier}
+% Like \gbc{qbezier}, but calls \gbc{mkcbezier}.
+% \begin{macrocode}
+vardef mkqbezier (expr cyclic) (suffix pts) =
+ pts1
+ if pts=1: {0,0}
+ else:
+ for _i = 2 step 2 until pts - 1:
+ ..controls 1/3[pts[_i], pts[_i-1]]
+ and 1/3[pts[_i], pts[_i+1]].. pts[_i+1]
+ endfor
+ if cyclic:
+ ..controls 1/3[ pts[pts], pts[pts - 1] ]
+ and 1/3[ pts[pts], pts1 ]..cycle
+ fi
+ fi
+enddef;
+
+vardef qbezier (expr cyclic) (text t) =
+ setpairs (_qbz) (t);
+ if _qbz=0: NoPoints ("qbezier", _qbz);
+ else:
+ if (cyclic and odd _qbz) or (not cyclic and even _qbz):
+ _qbz[incr _qbz] := _qbz[_qbz-1];
+ fi
+ mkqbezier (cyclic) (_qbz)
+ fi
+enddef;
+
+vardef mkcbezier (expr cyclic) (suffix pts) =
+ pts1
+ if pts=1: {0,0}
+ else:
+ for _i = 1 step 3 until pts - 3:
+ ..controls pts[_i+1] and pts[_i+2] .. pts[_i+3]
+ endfor
+ if cyclic:
+ ..controls pts[pts - 1] and pts[pts]..cycle
+ fi
+ fi
+enddef;
+
+vardef cbezier (expr cyclic) (text t) =
+ setpairs (_cbz) (t);
+ if _cbz=0: NoPoints ("qbezier", _cbz);
+ else:
+ % Need 0 mod 3 for cyclic, otherwise 1 mod 3
+ setnumeric (_mdt) _cbz mod 3;
+ if cyclic:
+ if _mdt <> 0: _cbz[incr _cbz] := _cbz[_cbz-1]; fi
+ if _mdt = 1 : _cbz[incr _cbz] := _cbz1; fi
+ else: % need 1 more, duplicate next to last
+ if _mdt = 0:
+ _cbz := _cbz + 1;
+ _cbz[_cbz] := _cbz[_cbz-1];
+ _cbz[_cbz-1] := _cbz[_cbz-2];
+ fi
+ if _mdt = 2: % need 2 more, duplicate last 2.
+ _cbz := _cbz + 2; % add 2 slots
+ _cbz[_cbz] := _cbz[_cbz-2]; % fill them
+ _cbz[_cbz-1] := _cbz[_cbz-2]; % with last node
+ _cbz[_cbz-2] := _cbz[_cbz-3]; % orig last slot = orig previous.
+ fi
+ fi
+ mkcbezier (cyclic) (_cbz)
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% When calling \gbc{curve} or \gbc{tcurve} there can be a problem
+% with the resulting path: even with high tension one is not guaranteed
+% that a sequence of points with increasing $x$-coordinate will produce a
+% path with increasing $x$-coordinate. The \gbc{fcnspline} command will do
+% what we want, but we have no control over the path, apart from the
+% equations at the ends.
+%
+% The requirement to guarantee that a path have increasing $x$-coordinates
+% is that the control points of the segment connecting
+% $(x\sb{j}, y\sb{j})$ to the next $(x\sb{j+1}, y\sb{j+1})$ have their
+% $x$-part in the interval $x\sb{j} < x < x\sb{j+1}$.
+%
+% Therefore, if we wish to plot a curve connecting points with increasing
+% $x$-coordinates and believe that the resulting path should be the graph
+% of a function, we pretty much have to select the control points
+% ourselves. By default we choose the two controls so the \mfc{xpart}s
+% divide the $x$-interval into three equal parts. This makes the B\'ezier
+% $f(t)$ linear in the $x$-part and so has the added `advantage' that in
+% each segment, $y$ is a cubic function of $x$. It is not a spline, as
+% the computation of the controls uses only the two nearest points, plus
+% we allow them to be modified by an additional parameter.
+%
+% Another concern is what direction to place the controls. In
+% \gbc{mksmooth} we ask the direction at a given point to be the average
+% of the straight line directions to adjacent points. We do the same
+% here, though it is not clear if this is best.
+%
+% Finally, we permit a tension of sorts by dividing the distance to the
+% controls by a parameter normally equal to \gbc{default_tension}.
+%
+% \DescribeRoutine{fcncontrol}
+% This computes the control point for the points on the path, following
+% the above description. If by chance some $x$ interval is zero, we
+% make the controls equal to the nodes, which gives a straight vertical
+% (the $y$ values differ because we use \gbc{setuniquepairs}. This also
+% abandons smoothness there.
+%
+% The method selecting the controls is new with \mfpic{} version 0.8.
+% Following discussions with Stephan Hennig in \texttt{comp.text.tex} I
+% came to the conclusion that the method used ought to satisfy the
+% following: if the data are xscaled or yscaled, the control vectors ought
+% to scale the same way. The current version does that, the previous one
+% did not.
+%
+% \DescribeRoutine{mkfcnpath}
+% This produces the path, calling \gbc{fcncontrol} to produce the controls.
+%
+% \DescribeRoutine{fcncurve}
+% This is the \mfpic{} interface; \gbc{fcncurve} calls \gbc{functioncurve}
+% with the default tension, and
+% \DescribeRoutine{functioncurve}
+% then takes a list of points, converts it to an array, and calls
+% \gbc{mkfcnpath} to build the path.
+% \begin{macrocode}
+vardef fcncontrol (expr ftens, X, Y, Z) =
+ Y if (xpart(Z-Y) <> 0) and (xpart(Y-X) <> 0):
+ + xpart(Z-Y)/3/xpart(Z-X)*(Z - X)/ftens fi
+enddef;
+
+vardef mkfcnpath (expr ftens) (suffix q) =
+ settension (_tn) ftens;
+ if _tn <= 0: _tn := 1; fi
+ for _i = 1 upto q - 1:
+ q[_i]..controls fcncontrol (_tn) (q[_i-1], q[_i], q[_i+1])
+ and fcncontrol (_tn) (q[_i+2], q[_i+1], q[_i])..
+ endfor
+ q[q]{0,0}
+enddef;
+
+def fcncurve = functioncurve (default_tension) enddef;
+def tfcncurve = functioncurve enddef;
+vardef functioncurve (expr ftens) (text t) =
+ settension (_ftens) ftens; if _ftens < 1/3: _ftens := 1/3; fi
+ setuniquepairs (_fc) (t);
+ if _fc=0: NoPoints ("functioncurve", _fc); fi
+ if _fc > 1: _fc0 := _fc1; _fc[_fc+1] := _fc[_fc]; fi
+ mkfcnpath (_ftens) (_fc)
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Splines with explicit controls}\label{splines}
+%
+% For these quadratic B-splines, a list of pairs representing the control
+% points must be given. The nodes of the path and the cubic Bezi\'er
+% controls required to produce a quadratic B-spline are computed. The
+% nodes are just half way between the one control point and the next.
+%
+% \DescribeRoutine{openqbs}
+% For simplicity, the list is converted to an array \gbc{_oq} first. In
+% the closed version
+% \DescribeRoutine{closedqbs}
+% additional array elements are created at the end, repeating two of the
+% beginning elements. Finally,
+% \DescribeRoutine{mkqbs}
+% \gbc{mkqbs} is called. This draws an open spline based on the points in
+% an array \gbc{b}. The additional array elements defined by
+% \gbc{closedqbs} cause the resulting path to end where it began and a
+% simple \mfc{\&cycle} closes it.
+%
+% \DescribeRoutine{qspline}
+% The \mfpic{} commands \cs{qspline} and \cs{closedqspline} now call
+% \gbc{qspline} with appropriate boolean, for consistency with other
+% commands that have the same argument structure. The commands
+% \gbc{openqbs}, and \gbc{closedqbs} are no longer needed, but are kept
+% for backward compatability. The most efficient setup would be to give
+% \gbc{mkqbs} a boolean argument, but that could break old files.
+% \begin{macrocode}
+def openqbs = qspline (false) enddef;
+def closedqbs = qspline (true) enddef;
+
+vardef mkqbs (suffix b) =
+ 0.5[ b1, b2]
+ if b<3: {0,0}
+ else:
+ for _i = 2 upto b-1:
+ ..controls 1/6[ b[_i], b[_i-1] ] and 1/6[ b[_i], b[_i+1] ]..
+ 0.5[ b[_i], b[_i+1] ]
+ endfor
+ fi
+enddef;
+
+vardef qspline (expr cyclic) (text t) =
+ setpairs (_qs) (t);
+ if _qs=0: NoPoints ("qspline", _qs); fi
+ if _qs=1: _qs[incr _qs] := _qs1; fi
+ if cyclic:
+ _qs[incr _qs] := _qs1; _qs[incr _qs] := _qs2;
+ fi
+ mkqbs (_qs) if cyclic: & cycle fi
+enddef;
+
+% \end{macrocode}
+%
+% These cubic B-splines also require a list of `control' points. Each of
+% the points $Q$, combined with the next one $Q'$, determine two more
+% points that divide the segment from $Q$ to $Q'$ into thirds. These new
+% points become the two control points of a \MF{} B\'ezier segment. The
+% nodes of these segments are half way between the second control of one
+% segment and the first control of the next.
+%
+% \DescribeRoutine{mkcbs}
+% The main code is in \gbc{mkcbs}, which results in an open curve. For
+% backward compatibility, the alias
+% \DescribeRoutine{mkopencbs}
+% \gbc{mkopencbs} is supplied.
+%
+% \DescribeRoutine{mkclosedcbs}
+% Earlier versions of \gbc{mkclosedcbs} would partly redefine its suffix
+% parameter (for example, using \gbc{b[incr b]:=b1}. I decided this
+% shouldn't change the array variable (imagine using two such functions on
+% the same array). Now the code has been rearranged so the cubic case is
+% handled just like the quadratic. We retain \gbc{mkclosedcbs} only for
+% backward compatibility.
+%
+% \DescribeRoutine{opencbs}
+% These are the versions taking a list of points instead of an array name.
+% They create a temporary array and call \gbc{mkcbs}, with
+% \DescribeRoutine{closedcbs}
+% \gbc{closedcbs} extending the array, just like the quadratic versions.
+%
+% \DescribeRoutine{cspline}
+% The \mfpic{} commands \cs{cspline} and \cs{closedcspline} now call
+% \gbc{cspline} with appropriate boolean, for consistency with other
+% commands that have the same argument structure. The commands
+% \gbc{mkopencbs}, \gbc{opencbs}, and \gbc{closedcbs} are no longer
+% needed, but are kept for backward compatability. The most efficient
+% setup would be to give \gbc{mkcbs} a boolean argument, but that could
+% break old files.
+% \begin{macrocode}
+vardef mkcbs (suffix b) =
+ (b[1]+4b[2]+b[3])/6
+ if b < 4: {0,0}
+ else:
+ for _i = 3 upto b-1:
+ ..controls 1/3[ b[_i-1], b[_i] ] and 1/3[ b[_i], b[_i-1] ]
+ .. (b[_i-1] + 4b[_i] + b[_i+1])/6
+ endfor
+ fi
+enddef;
+
+def mkopencbs = mkcbs enddef;
+vardef mkclosedcbs (suffix b) =
+ mkcbs (b) & opencbs (b[b-2],b[b-1],b[b], b1, b2, b3) & cycle
+enddef;
+
+def opencbs = cspline (false) enddef;
+def closedcbs = cspline (true) enddef;
+
+vardef cspline (expr cyclic) (text t) =
+ setpairs (_cs) (t);
+ if _cs=0: NoPoints ("cspline", _cs); fi
+ for _idx = _cs upto 2: _cs[incr _cs] := _cs[_idx]; endfor
+ if cyclic:
+ for _idx = 1 upto 3: _cs[incr _cs] := _cs[_idx]; endfor
+ fi
+ mkcbs (_cs) if cyclic: & cycle fi
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Splines with computed controls}\label{computedsplines}
+%
+% A cubic spline through a set of points is a curve obtained by joining
+% each point to the next with a cubic parametrized curve, where adjoining
+% cubics must have matching first and second derivative at their common
+% point. In the previous section's \gbc{mkcbs}, the control points must be
+% supplied, it being up to the user to arrange (if necessary) that the
+% spline produced passes through given points. If, instead, these points
+% are given, it is possible to compute the necessary controls.
+% Unfortunately, the controls are not uniquely determined unless the curve
+% is required to be closed. For open curves, there is need for two
+% additional conditions at the end points. A `relaxed spline' is produced
+% if we require that the second derivative is $0$ at those points.
+%
+% For a closed curve, the equality of the first and second derivatives at
+% the common beginning/ending point gives the needed additional equations.
+%
+% Note that this equates \emph{time} derivatives, so this works best when
+% points are relatively evenly spaced and so the speed is relatively
+% uniform. If points are differently spaced then the relatively slower
+% speed between closely spaced points allows sharper turns without large
+% second derivatives. Curves produced tend to have a more natural look,
+% and relaxed splines are most suitable for smoothing data that is
+% obtained by taking observations at evenly space times. Still, the
+% technique is somewhat unstable when points are closely spaced, for
+% example when a small change in the position of one point can produce a
+% large change in its direction when viewed from another point.
+%
+% \DescribeRoutine{init_spline_eqns}
+% In this command we generate the equations common to all cubic
+% splines: the equality of derivatives at all interior points.
+% This command accepts a suffix \gbc{pts}, which is the array of points
+% to be connected. It initializes the variables \gbc{_spl_pre[\,]} and
+% \gbc{_spl_post[\,]} to unknown arrays of pairs. These will hold the
+% control points.
+%
+% \DescribeRoutine{closed_spline_eqns}
+% The next two macros contain the additional equations: for a closed
+% spline these are the same as the interior equation, but at the first and
+% last point in the array.
+% \DescribeRoutine{relaxed_spline_eqns}
+% For relaxed splines they force the second derivative to be 0 at the
+% first and last point.
+%
+% The macro \gbc{mksplinepath} simply assembles the previously computed
+% points and controls into a path.
+%
+% \DescribeRoutine{mkspline}
+% The macro \gbc{mkspline} issues the common equations and then either the
+% closed equations (\gbc{closed = true}) or the relaxed equations
+% (\gbc{closed = false}), before calling \gbc{mksplinepath}.
+%
+% The knowledgeable user can call \gbc{init_spline_eqns}, append any
+% choice of equations for the end segments, and then call
+% \gbc{mksplinepath~(false)} to produce any sort of open spline.
+%
+% \DescribeRoutine{dospline}
+% This version accepts a list of pairs and produces a spline through
+% them. It simply stores the list in an array and calls the appropriate
+% version that operates on an array. This is the command passed by \mfpic{}.
+% \begin{macrocode}
+def init_spline_eqns (suffix pts) =
+ save _spl_pre, _spl_post;
+ pair _spl_pre[], _spl_post[];
+ for j= 2 upto pts - 1:
+ _spl_post[j] + _spl_pre[j] = 2pts[j];
+ _spl_pre[j+1]+2_spl_pre[j] = 2_spl_post[j]+_spl_post[j-1];
+ endfor
+enddef;
+
+def closed_spline_eqns (suffix pts) =
+ _spl_post1 + _spl_pre1 = 2pts1;
+ _spl_post[pts] + _spl_pre[pts] = 2pts[pts];
+ _spl_pre2 + 2_spl_pre1 = 2_spl_post1 + _spl_post[pts];
+ _spl_pre1+2_spl_pre[pts] = 2_spl_post[pts]+_spl_post[pts-1];
+enddef;
+
+def relaxed_spline_eqns (suffix pts) =
+ _spl_pre2 + pts1 = 2_spl_post1;
+ pts[pts] + _spl_post[pts-1] = 2_spl_pre[pts];
+enddef;
+
+vardef mksplinepath (expr closed) (suffix pts) =
+ pts1..controls _spl_post1 and
+ for j = 2 upto pts if not closed: -1 fi:
+ _spl_pre[j]..pts[j]..controls _spl_post[j] and
+ endfor
+ if closed: _spl_pre1..cycle else: _spl_pre[pts]..pts[pts] fi
+enddef;
+
+def mkspline (expr closed) (suffix pts) =
+ init_spline_eqns (pts);
+ if closed: closed_spline_eqns (pts);
+ else: relaxed_spline_eqns (pts);
+ fi
+ mksplinepath (closed) (pts)
+enddef;
+
+vardef dospline (expr closed) (text the_list) =
+ setpairs (_sp) (the_list);
+ if _sp=0: NoPoints ("dospline", _sp); fi
+ if _sp=1: _sp[incr _sp] := _sp1; fi
+ mkspline (closed) (_sp)
+enddef;
+
+% \end{macrocode}
+%
+% The above computations produce a $2$-dimensional spline. A $1$-dimensional
+% cubic spline would be a function $f(t)$ with numeric values rather
+% than pair values. Such are often used to interpolate functions. That is,
+% given pairs $(x\sb j,y\sb{j})$, and assuming they lie on the graph of
+% some function (generally unknown), fill in the graph with $y = f(x)$
+% where $f$ is a cubic function of $x$ in each interval $x\sb j \le x
+% \le x\sb {j+1}$, making sure that the resulting graph is as smooth as
+% possible at the points $(x\sb j, y\sb j)$.
+%
+% The requirements on our $2$-dimensional path are the following:
+% \begin{enumerate}
+% \item The $j$th link should connect $(x\sb{j},y\sb{j})$ to $(x\sb{j+1},
+% y\sb{j+1})$.
+% \item The $x$-part of that link should increase linearly from $x\sb{j}$ to
+% $x\sb{j+1}$ as $t$ goes from $0$ to $1$.
+% \item The $y$-part should be a cubic $y = f(x)$.
+% \item The $x$-derivatives $df/dx$ and $d^2f/dx^2$ should match at the
+% connecting points.
+% \end{enumerate}
+%
+% Two necessary equations for converting between $x$ and $t$ coordinates
+% are:
+% \begin{equation}\label{first}
+% x = x\sb{j} + t \Delta x\sb{j}
+% \end{equation}
+% (where $\Delta x\sb{j} = x\sb{j+1} - x\sb{j}$) and
+% \begin{equation}\label{second}
+% \frac{df}{dt} = \frac{dx}{dt}\frac{df}{dx} =
+% \Delta x\sb{j} \frac{df}{dx}.
+% \end{equation}
+% Thus we want to choose controls so that (\ref{first}) is maintained and
+% so that $x$-derivatives match. It turns out that this requires controls
+% at
+% \begin{equation}
+% \begin{array}{c}
+% (x\sb{j}, y\sb{j}) - (\Delta x\sb{j-1}, s\sb{j} \Delta x\sb{j-1})/3\\
+% (x\sb{j}, y\sb{j}) + (\Delta x\sb{j} , s\sb{j} \Delta x\sb{j} )/3
+% \end{array}
+% \end{equation}
+% where $s\sb{j}$ is the slope (derivative) at $x\sb{j}$. This provides
+% matching first derivatives automatically (equation (\ref{second})) and
+% also (\ref{first}). To get matching second derivatives we need the same
+% conditions as in parametric splines. We use these equations simplified to
+% the form:
+% \begin{displaymath}
+% s\sb{j+1} \Delta x\sb{j} - 2s\sb{j} (\Delta x\sb{j} +
+% \Delta x\sb{j-1}) + s\sb{j-1}\Delta x\sb{j-1}
+% = 3y\sb{j+1} - 3y\sb{j-1}.
+% \end{displaymath}
+% There can be almost any equations at the end points. For a relaxed
+% spline we equate the second derivatives to 0. To get a periodic
+% function, we equate the slope and second derivative at beginning to
+% those at the end. This makes it possible to put a shifted copy of the
+% graph with starting point at the end of the original and have the same
+% smoothness at that connection as at the other points.
+%
+% \DescribeRoutine{init_fcnspl_eqns}
+% This declares the temporary arrays \gbc{_dx[\,]} (the set of $dx\sb j$)
+% and \gbc{_sl[\,]} (the desired slopes) and issues the common equations.
+% The parameter \gbc{pts} is the array of $(x,y)$ values.
+%
+% \DescribeRoutine{periodic_fcnspl_eqns}
+% For the periodic case we use \gbc{periodic_fcnspl_eqns} to generate the
+% additional equations and for the
+% \DescribeRoutine{relaxed_fcnspl_eqns}
+% relaxed case we use \gbc{relaxed_fcnspl_eqns}. As before, one can
+% produce custom splines by issuing the common equations and then ones own
+% equations.
+%
+% \DescribeRoutine{mkfcnsplpath}
+% Then we assemble the path from the computed information by calling the
+% command \gbc{mkfcnsplpath}.
+%
+% \DescribeRoutine{mkfcnspline}
+% These commands emits the appropriate equations then assemble the path.
+% The if the first parameter is true it uses the periodic equations,
+% otherwise the relaxed equations.
+%
+% \DescribeRoutine{fcnspline}
+% Finally, this command is the one written by \mfpic{}. It copies a list
+% of pairs into an array and calls the appropriate command to process
+% them.
+% \begin{macrocode}
+def init_fcnspl_eqns (suffix pts) =
+ save _dx, _sl; numeric _dx[], _sl[];
+ _dx1 := xpart (pts2 - pts1);
+ for j = 2 upto pts - 1:
+ _dx[j] := xpart (pts[j+1] - pts[j]);
+ _sl[j + 1]*_dx[j] + _sl[j-1]*_dx[j-1] + 2_sl[j]*(_dx[j] + _dx[j-1])
+ = 3*ypart(pts[j+1] - pts[j-1]);
+ endfor
+enddef;
+
+def periodic_fcnspl_eqns (suffix pts) =
+ _sl1 = _sl[pts];
+ _sl2*_dx1 + 2_sl1*_dx1 + 2_sl[pts]*_dx[pts-1] + _sl[pts-1]*_dx[pts-1]
+ = 3 * ypart (pts[2] - pts[pts-1]);
+enddef;
+
+def relaxed_fcnspl_eqns (suffix pts) =
+ _sl2*_dx1 + 2_sl1*_dx1 = 3 * ypart(pts2 - pts1);
+ _sl[pts-1]*_dx[pts-1] + 2_sl[pts]*_dx[pts-1]
+ = 3 * ypart(pts[pts] - pts[pts-1]);
+enddef;
+
+vardef mkfcnsplpath (suffix pts) =
+ pts1..controls (pts1 + (1, _sl1)/3*_dx1) and
+ for j = 2 upto pts - 1:
+ (pts[j] - (1, _sl[j])/3*_dx[j-1]) ..pts[j]..
+ controls (pts[j] + (1,_sl[j])/3*_dx[j]) and
+ endfor
+ (pts[pts] - (1,_sl[pts])*_dx[pts-1]/3)..pts[pts]
+enddef;
+
+vardef mkfcnspline (expr periodic) (suffix pts) =
+ init_fcnspl_eqns (pts);
+ if periodic: periodic_fcnspl_eqns (pts);
+ else: relaxed_fcnspl_eqns (pts);
+ fi
+ mkfcnsplpath (pts)
+enddef;
+
+vardef fcnspline (expr periodic) (text the_list) =
+ setpairs (_fs) (the_list);
+ if _fs<2:
+ if _fs=0: NoPoints ("fcnspline", _fs); fi
+ onepointpath (false, _fs1)
+ else:
+ mkfcnspline (periodic) (_fs)
+ fi
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Arcs, circles and ellipses}\label{arcs}
+%
+% We have multiple commands that generate circular arcs, differing in
+% how the arc is specified. All are (in part) based on the following
+% \gbc{mkarc}. However, perfectly reasonable arcs can have centers so far
+% away that requiring the center among the parameters can cause numeric
+% overflow.
+%
+% I'd like to use some scheme that avoids this. It is possible, given
+% three reasonably spaced points on an arc with angle less than 90
+% degrees between each, to draw the arc without finding the center.
+% However, I am not sure how to reduce any given format to this
+% information
+%
+% Another problem is that of accuracy. If the angle is small, accuracy is
+% not usually a problem, but if an angle is close to 360, and the
+% endpoints are known, then finding the center (or finding other points on
+% the arc without knowing the center) is unstable.
+%
+% There is really no problem with \gbc{mkarc} itself: if you can express
+% both \gbc{center} and \gbc{begpt} in \MF, then the other values on the
+% arc should normally be no problem. (Of course, if the radius is near
+% \mfc{infinity}, there could be points on the arc with coordinates near
+% \mfc{2infinity}, causing overflow in \MF{}. One hopes this is rare.)
+%
+% Care has been taken that changing the sign of various parameters
+% produces reasonable results. And there should be no more problem for
+% arcs with sweep larger than 360 degrees than with less.
+%
+% \DescribeRoutine{mkarc}
+% This takes the center, starting and ending point (pair expressions) and
+% the angle, and returns the arc defined pretty much the way \file{plain.mf}
+% defines \mfc{quartercircle}.
+%
+% It would be easier to do something like we frequently do with
+% \mfc{fullcircle}: make an arc of unit radius, and then rotate, scale
+% and shift it into place. However, I would like to accomplish at least
+% the following: if an endpoint of the arc is among the parameters, or is
+% straightforwardly implied by them, then the corresponding endpoint of
+% the path created should test equal to that point. Shifting works OK, but
+% scaling and rotating cause roundoff differences.
+%
+% Note that \gbc{mkarc} has parameters that may over-determine the arc.
+% It is only called by arc-making commands that have calculated these
+% parameters and, I hope, ensured they are compatible. \gbc{mkarc}'s job
+% is mainly to ensure that the arc begins at \gbc{begpt} and ends at
+% \gbc{endpt} (exactly). A \gbc{sweep} of $0$ is actually incompatible
+% with any case where \gbc{begpt<>endpt} unless \gbc{center} is
+% literally at $\infty$, but we allow it even though I am pretty sure
+% the other arc commands all filter out that case.
+% \begin{macrocode}
+vardef mkarc (expr center, begpt, endpt, sweep) =
+ if (sweep = 0): begpt--endpt
+ else:
+ setnumeric (n) ceiling (abs(sweep)/45);
+ setpair (d) (begpt - center) rotated (signof (sweep) 90);
+ begpt{d}
+ for j = 1 upto n-1:
+ ..(begpt rotatedabout (center, j/n*sweep)){d rotated (j/n*sweep)}
+ endfor ..endpt{d rotated sweep}
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{arc}
+% The most basic: center of circle, starting point of arc, and angle
+% subtended. Another name for \gbc{arc} is \gbc{arccps}, (``\gbc{cps}'' is
+% for ``center, point, sweep'').
+% \begin{macrocode}
+vardef arc (expr center, begpt, sweep) =
+ if (center = begpt) or (sweep = 0): begpt--begpt
+ else:
+ mkarc (center, begpt, begpt rotatedabout (center, sweep), sweep)
+ fi
+enddef;
+def arccps = arc enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{arcpps}
+% In this form we are given two points and the angle of the arc between
+% them. If the points are equal or the sweep makes the arc undefined, we
+% return a line segment. If the sweep is less than 90 degrees we use the
+% idea from the code of \mfc{quartercircle}, except, when the sweep is
+% greater than 45 degrees we let \MF{} find the midpoint \gbc{m} of the
+% arc. Otherwise, we get the center \gbc{c} of the circle and call
+% \gbc{mkarc}.
+%
+% The code for finding \gbc{c} and \gbc{m} used to be separate commands,
+% \gbc{arccenter} and \gbc{midarc}. However, this is the only place we
+% used them and the several cases that they had to consider are reduced
+% because the \mfc{if} in this command takes care of some of them.
+%
+% The code for finding \gbc{m} uses the fact that the chord and the line
+% from one of its endpoints to the midpoint subtend a circular arc of
+% \gbc{sweep/2} and so the angle between them is half that, \gbc{sweep/4}.
+% The code gets the intersection between the line in that direction and
+% the perpendicular bisector of the chord.
+%
+% We find the center by intersecting two lines. One is the radius from
+% one end of the chord. Then we branch on two cases: if the chord is close
+% to a diameter, use its perpendicular bisector as the other line,
+% otherwise use the radius from the opposite end of the chord. Here
+% \gbc{cd} is a vector in the direction of the chord from \gbc{begpt} to
+% \gbc{endpt}. The angle \gbc{ang} is the amount we have to rotate
+% \gbc{cd} about \gbc{begpt} to make it point toward the center of the
+% circle. This gives the radius mentioned above.
+% \begin{macrocode}
+vardef arcpps (expr begpt, endpt, sweep) =
+ if (begpt = endpt) or (sweep = 0): begpt--endpt
+ else:
+ setpair (cd) unitvector (endpt-begpt);
+ if abs(sweep) <= 45:
+ begpt{cd rotated (-sweep/2)}..endpt{cd rotated (sweep/2)}
+ elseif abs(sweep) <= 90:
+ save m; pair m;
+ m = begpt + whatever*( cd rotated (-sweep/4));
+ m = 0.5[begpt, endpt] + whatever*(cd rotated 90);
+ begpt{cd rotated (-sweep/2)}..m{cd}..endpt{cd rotated (sweep/2)}
+ else:
+ setnumeric (ang) 90 - ((sweep/2) mod 180);
+ if abs(ang) = 90:
+ GBwarn "undefined arc. A line segment will be used instead.";
+ begpt--endpt
+ else:
+ save c; pair c;
+ c = begpt + whatever*(cd rotated ang);
+ c = if abs(ang) < 30:
+ (0.5)[begpt, endpt] + whatever*(cd rotated 90)
+ else:
+ endpt + whatever*(-cd rotated -ang)
+ fi;
+ mkarc (c, begpt, endpt, sweep)
+ fi
+ fi
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{arcpp}
+% In the macro \gbc{arcpp}, two points and the radius of the circle are
+% given. Alone, this would determine two circles and therefore 4 arcs. We
+% reduce the possibilities to two by assuming the arc is anticlockwise
+% from the first point to the second if \gbc{rad} is positive, clockwise
+% if negative. Then \gbc{arcpp} produces the one that has absolute value
+% no more than 180 degrees if \gbc{small} is true, otherwise the other
+% one.
+% \DescribeRoutine{arcppr}
+% The macro \gbc{arcppr} is just \gbc{arcpp} with the boolean argument
+% \gbc{small} last (for compatibility with previous \mfpic{} versions).
+%
+% The code computes the angle of the arc and calls \gbc{arcpps}. If the
+% radius is not larger than half the distance between the points, we make
+% the angle $\pm 180$, which produces a half circle.
+% \begin{macrocode}
+vardef arcpp (expr small, begpt, endpt, rad) =
+ save full, diam, chord, ang;
+ full := signof (rad) 360;
+ diam := 2rad;
+ chord := abs(endpt-begpt);
+ if chord < abs(diam):
+ ang := if not small: full - fi 2*asin (chord/diam);
+ else: ang := signof (rad) 180;
+ fi
+ arcpps (begpt, endpt, ang)
+enddef;
+def arcppr (expr begpt, endpt, rad, small) =
+ arcpp (small, begpt, endpt, rad)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{arcplr}
+% This one takes the center and polar coordinates of the ends relative to
+% the center. We just call \gbc{mkarc} with the obviously computed
+% endpoints and sweep.
+% \begin{macrocode}
+vardef arcplr (expr center, frtheta, totheta, rad) =
+ if rad = 0: center--center
+ else:
+ mkarc (center, center + rad*dir frtheta,
+ center + rad*dir totheta, totheta - frtheta)
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{arcalt}
+% This one is the same as above, but with the same argument order as
+% \gbc{sector}.
+% \begin{macrocode}
+vardef arcalt (expr center, radius, frtheta, totheta) =
+ arcplr (center, frtheta, totheta, radius)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{arcppp}
+% This last one finds the arc connecting three points in the order given.
+% It works by calling \gbc{arcpps} twice, using first the sweep from
+% \gbc{first} to \gbc{second}, and then the sweep from \gbc{second} to
+% \gbc{third}. Each of these is twice the opposite angle of the triangle
+% formed from these points, and calculated by \gbc{cornerangle}.
+% \begin{macrocode}
+vardef arcppp (expr first, second, third) =
+ arcpps (first, second, 2*cornerangle (third, first, second)) &
+ arcpps (second, third, 2*cornerangle (first, second, third))
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{ellipse}
+% We get an ellipse by xscaling and yscaling a unit circle, rotating it
+% and then shifting it into position. All parameters are coordinate
+% independent expressions, with obvious meaning (\gbc{center} is a pair, the
+% rest numeric). \gbc{circle} is similar, but we only scale and shift.
+%
+% If either radius is negative, the sense of the ellipse is reversed and
+% the starting point changes. If both are negative, only the starting
+% point changes.
+% \DescribeRoutine{circle}
+% \gbc{circle} acts like \gbc{ellipse} with both radii the same.
+% \begin{macrocode}
+vardef ellipse (expr center, radx, rady, angle) =
+ fullcircle xscaled (2*radx) yscaled (2*rady) rotated angle
+ shifted center
+enddef;
+
+vardef circle (expr center, rad) =
+ fullcircle scaled (2*rad) shifted center
+enddef;
+
+% \end{macrocode}
+%
+% The next four implement different ways of specifying a circle.
+% \DescribeRoutine{circlecp}
+% The first, \gbc{circlecp}, produces the circle with a given center
+% passing through a given point.
+% \DescribeRoutine{circleppp}
+% The second, \gbc{circleppp}, produces the circle passing through three
+% given points.
+% \DescribeRoutine{circlepps}
+% The third, \gbc{circlepps}, produces the circle passing through two
+% given points in such a way that the arc from the first to the second has
+% a given angle.
+% \DescribeRoutine{circleppr}
+% The fourth, \gbc{circleppr}, produces the circle with the given radius
+% passing through the two points in such a way that the angle from the
+% first point to the second is between $0$ and $180$ degrees if the switch
+% \gbc{small} is true. If \gbc{small} is false, then the clockwise arc
+% from first to second is between $180$ and $360$. If \gbc{rad} is
+% negative, the circles switch and their orientation is reversed.
+% \DescribeRoutine{circlepp}
+% The last, \gbc{circlepp}, is just \gbc{circleppr} with a different order
+% of arguments (for previous \mfpic{} versions).
+%
+% These could be implemented by finding the center and radius and calling
+% \gbc{circle}. However, we call the arc commands so that those points
+% specified in the parameters that lie on the circle will be nodes of the
+% path produced, in the given order.
+% \begin{macrocode}
+vardef circlecp (expr center, point) =
+ mkarc (center, point, point, 360) & cycle
+enddef;
+
+vardef circleppp (expr one, two, three) =
+ arcpps (one, two, 2*cornerangle (three, one, two))
+ & arcpps (two, three, 2*cornerangle (one, two, three))
+ & arcpps (three, one, 2*cornerangle (two, three, one))
+ & cycle
+enddef;
+
+vardef circlepps (expr one, two, sweep) =
+ save ang, full;
+ full := signof (sweep) 360;
+ ang := sweep mod full;
+ arcpps (one, two, ang) & arcpps (two, one, full - ang) & cycle
+enddef;
+
+vardef circlepp (expr small, one, two, rad) =
+ arcpp (small, one, two, rad) & arcpp (not small, two, one, rad)
+ & cycle
+enddef;
+
+def circleppr (expr one, two, rad, small) =
+ circleppr (one, two, rad, small)
+enddef;
+
+% \end{macrocode}
+%
+% Now we implement a different way to specify an ellipse, essentially
+% specifying it by a parallelogram in which it is to be inscribed.
+%
+% \DescribeRoutine{quarterellipse}
+% If an ellipse is inscribed in a parallelogram, tangent to all four
+% sides at the midpoints, this command produces one ``corner'' of that
+% ellipse. The arguments \mfc{A} and \mfc{C} are the midpoints of two
+% adjacent sides and \mfc{B} is the corner between those two sides. This
+% quarter-ellipse starts at \mfc{A} in the direction \mfc{B-A} and ends at
+% \mfc{C} in the direction \mfc{C-B}. As a path \mfc{p} it has two segments, where
+% \mfc{point 0 of p} is \mfc{A}, \mfc{point 2 of p} is \mfc{C}, while
+% \mfc{point 1 of p} lies on the diagonal of the parallelogram through
+% \mfc{B} and has direction there the same as \mfc{C-A}.
+%
+% This was created for the purpose of rounding off corners of a polygonal
+% path.
+% \begin{macrocode}
+vardef quarterellipse(expr A,B,C) =
+ save T_;
+ transform T_;
+ (1,0) transformed T_ = A;
+ (1,1) transformed T_ = B;
+ (0,1) transformed T_ = C;
+ quartercircle scaled 2 transformed T_
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{halfellipse}
+% While \gbc{quarterellipse} is for corners, I don't have much use for
+% \gbc{halfellipse}. Nevertheless, it seems wise (and easy) to provide a
+% definition.
+%
+% The pairs \mfc{A}, \mfc{B}, and \mfc{C} are three midpoints of a
+% parallelogram with \mfc{A} and \mfc{C} on opposite sides and \mfc{B} on
+% a third side. This determines a unique parallelogram, and
+% \gbc{halfellipse} starts at \mfc{A}, passing through \mfc{B} then
+% \mfc{C}, tangent to the respective sides. It makes a point of building
+% it out of two \gbc{quarterellipse}\,s as \mfc{halfcircle} does with
+% \mfc{quartercircle} (at least in \MF{}). We just have to compute their
+% corners.
+% \begin{macrocode}
+vardef halfellipse (expr A,B,C) =
+ save P_; pair P_;
+ P_ = (C - A)/2;
+ quarterellipse (A, B - P_, B) & quarterellipse (B, B + P_, C)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{fullellipse}
+% For \gbc{fullellipse} we specify the center \mfc{C} of the parallelogram
+% and the midpoints \mfc{A} and \mfc{B} of two adjacent sides. We compute
+% the midpoints of the other two sides and draw two \gbc{halfellipse}\,s.
+%
+% Note that the points \gbc{A} and \gbc{B} do not correspond to the
+% usual radii of an ellipse unless the corresponding parallelogram is
+% actually a rectangle (i.e., only if $\angle ACB$ is a right angle).
+% \begin{macrocode}
+vardef fullellipse (expr C, A, B) =
+ save P_; pair P_;
+ P_ := 2[A,C];
+ halfellipse (A,B,P_) & halfellipse (P_,2[B,C],A) & cycle
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{pathcenter}
+% This finds the center of a circle. For other paths, the point found
+% may be meaningless (but it will also obtain the center of an arc or a
+% rectangle). It takes three or four supposedly distinct points on the
+% path and finds the intersection of the perpendicular bisectors of two
+% chords.
+%
+% This code is rather non-robust if applied to an arc that has angular
+% measure very close to either 0 or 360.
+% \begin{macrocode}
+vardef pathcenter expr p =
+ save a, cntr, n; pair cntr, a[];
+ n := length p;
+ a1 = pnt 0 (p);
+ a3 = pnt [n/2] (p);
+ if cycle p:
+ a2 = pnt [ n/4] (p);
+ a4 = pnt [3n/4] (p);
+ else:
+ a2 := a3;
+ a4 := pnt[n] (p);
+ fi
+ cntr = .5[a1, a3] + whatever*((a3 - a1) rotated 90);
+ cntr = .5[a2, a4] + whatever*((a4 - a2) rotated 90);
+ cntr
+enddef;
+
+% \end{macrocode}
+%
+% The next four commands create certain circles associated to
+% triangles. The triangle is specified as a path expression, so they
+% produce results for any path, but make sense only for a cyclic
+% triangular path.
+%
+% \DescribeRoutine{circumcircle}
+% This is just the circle through the three corners.
+%
+% \DescribeRoutine{incircle}
+% The command \gbc{incircle} produces the circle that is tangent to all
+% three sides of the triangle. It makes use of the fact that the two
+% tangent points on the sides adjacent to corner \gbc{A} (for example) are
+% equidistant from \gbc{A}. The three equations then express the fact that
+% the sum of the two distances from the tangent point to the corners on
+% the same side add up to the length of the side.
+%
+% \DescribeRoutine{excircle}
+% In \gbc{excircle}, a corner is given (by number from $0$ to $2$) and the
+% circle is produced that is \emph{outside} the triangle and is tangent to
+% the side opposite the point and tangent to the extensions of the other
+% two sides.
+%
+% \DescribeRoutine{ninepointcircle}
+% The ``nine-point circle'' passes through the following nine points: the
+% midpoint of each side, the point on each side (extended, if necessary)
+% where the altitude from the opposite corner meets it, and the midpoint
+% of the segments connecting each corner to the intersection of the
+% altitudes.
+% \begin{macrocode}
+vardef circumcircle expr t =
+ circleppp (pnt0 (t), pnt1 (t), pnt2 (t))
+enddef;
+
+vardef incircle expr t =
+ save A, B, C; pair A, B, C;
+ A := pnt0 (t);
+ B := pnt1 (t);
+ C := pnt2 (t);
+ save a, b, c, D, E, F;
+ D := abs (B-A) = a + b;
+ E := abs (C-B) = b + c;
+ F := abs (A-C) = a + c;
+ circleppp ((a/D)[A,B], (b/E)[B,C], (c/F)[C,A])
+enddef;
+
+vardef excircle expr n of t =
+ save A, B, C; pair A, B, C;
+ A := pnt[n] (t);
+ B := pnt[n + 1] (t);
+ C := pnt[n + 2] (t);
+ save a, b, c, D, E, F;
+ D := abs (B-A) = a - b;
+ E := abs (C-B) = b + c;
+ F := abs (C-A) = a - c;
+ circleppp ((a/D)[A,B], (b/E)[B,C], (c/F)[A,C])
+enddef;
+
+vardef ninepointcircle expr t =
+ circleppp (medianpt 0 of t, medianpt 1 of t, medianpt 2 of t)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{pshcircle}
+% Here are a couple of circles maybe only I need. They are the
+% pseudohyperbolic circles in the unit disk and upper half-plane.
+% One supplies a point that must be inside the unit circle or above
+% the $x$-axis, and a radius that must be less than $1$. Some degenerate
+% cases will not generate an error. We code this with a boolean that
+% determines whether the disk or the half-plane is to be assumed.
+%
+% If $\alpha=(a,b)$ is the hyperbolic center (the \mfc{ctr} parameter)
+% and $\rho$ is the pseudohyperbolic radius (the \mfc{rad parameter}),
+% the formula for the (Euclidean) center $C$ and radius $R$ of the circle
+% is, for the unit disk:
+% $$
+% C = \frac{ (1 - \rho^2)a }{1 - \rho^2|a|^2},\quad
+% R = \frac{\rho(1 - |a|^2)}{1 - \rho^2|a|^2}
+% $$
+% and for the half-plane:
+% $$
+% C = a + \frac{(1 + \rho^2}{1 - \rho^2}b,\quad
+% R = \frac{2\rho b}{1 - \rho^2}
+% $$
+% \begin{macrocode}
+vardef pshcircle (expr disk, ctr, rad) =
+ if disk:
+ if rad >= 1 :
+ if rad > 1:
+ GBerrmsg ("Impossible radius of pseudohyperbolic circle.")
+ "The radius of a pseudohyperbolic circle can be at most 1.";
+ fi
+ circle ((0,0),1)
+ elseif abs(ctr) >= 1 :
+ if abs(ctr) > 1:
+ GBerrmsg ("Impossible center of pseudohyperbolic circle.")
+ "The center of this pseudohyperbolic circle must be in "
+ & "the unit disk.";
+ fi
+ onepointpath (true,ctr)
+ else:
+ save _r, _dnm;
+ _r := abs(ctr);
+ _dnm := 1 - _r*_r*rad*rad;
+ circle ((1 - rad*rad)/_dnm*ctr, rad*(1 - _r*_r)/_dnm)
+ fi
+ else:
+ if rad >= 1 :
+ GBerrmsg ("Impossible radius of pseudohyperbolic circle.")
+ "The radius of a pseudohyperbolic circle must be less than 1.";
+ onepointpath (true,ctr)
+ elseif ypart ctr <= 0:
+ if ypart ctr < 0:
+ GBerrmsg ("Impossible center of pseudohyperbolic circle.")
+ "The center of this pseudohyperbolic circle must be in "
+ & "the upper half-plane.";
+ fi
+ onepointpath (true,ctr)
+ else:
+ save _y, _dnm;
+ _y := ypart ctr;
+ _dnm := 1 - rad*rad;
+ circle ((xpart ctr, (1 + rad*rad)/_dnm * _y), 2rad/_dnm*_y)
+ fi
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{UHPgeodesic}
+% Here is another arc-producing command. What it produces is the
+% hyperbolic geodesic from one point to another in the \emph{upper
+% half-plane} (UHP). While, theoretically, the points should both be in
+% the UHP, where the hyperbolic geometry is defined, the computations make
+% sense for any pair of points. This could be useful, so I do not enforce
+% this theoretical requirement.
+%
+% Unless two points have the same xpart, there is a unique circle passing
+% through them that meets the $x$-axis at a right angle. The hyperbolic
+% geodesic is an arc of that circle. The path starts at the first listed
+% point and ends at the second. Of the two possible arcs that connect
+% these points, it is the one that doesn't cross the $x$-axis (if there
+% is one). Our computations simply determine the angle of the arc and call
+% \gbc{arcpps}.
+%
+% When the points have the same xpart, the hyperbolic geodesic is the
+% line segment connecting them. When the points have yparts with opposite
+% signs, both arcs cross the $x$-axis. Our code produces the shorter one.
+% If both are $180$ degrees, the one that lies all on the same side of the
+% vertical line through $A$ is produced ($A$ being the first argument).
+%
+% Our method is based on the fact that the reflection $C$ of $A$ (to the
+% other side of the $x$-axis) lies on the circle on which the arc lies.
+% The angle between $A$ and $B$ when viewed from this point is therefore
+% half the angle of the arc. We actually reflect the point farthest from
+% the $x$-axis, as this produces better results.
+%
+% If $A$ and $B$ are on opposite sides of the $x$-axis, then $C$ might
+% coincide with one of the points. In this case $A$ and $B$ would
+% necessarily have equal xparts, a case we will already have processed.
+%
+% If both points lie on the $x$-axis, the computations produce the
+% semicircle from the first to the second in the upper half-plane.
+% \begin{macrocode}
+vardef UHPgeodesic (expr A, B) =
+ if xpart A = xpart B:
+ A--B
+ else:
+ save ang_, C_; pair C_;
+ if abs(ypart A) < abs(ypart B):
+ C_ := conj B;
+ else:
+ C_ := conj A;
+ fi
+ if ypart C_ = 0: % both on x-axis
+ ang_ := anglefromto(up, B - A);
+ else:
+ ang_ := anglefromto(A - C_, B - C_);
+ fi
+ arcpps(A, B, 2ang_)
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{UDgeodesic}
+% There is a hyperbolic geometry defined for any simply connected open
+% set. The standard examples of such are the UHP and the unit disk (UD).
+% This next macro produces the geodesic in the UD. Once again it is the
+% arc of a circle and, if the two points do not lie on the same diameter,
+% that circle is the unique one through the two points that meets the
+% boundary of $UD$ at a right angle. When the two points do lie on the
+% same then the geodesic is the straight line connecting the points.
+%
+% The method we use is also based on reflection, where the `reflection' of
+% a point $A$ is given by $C = A/|A|^2$. Computing this can cause overflow
+% if $|A|$ too near $0$. Unfortunately, overflow can also occur if either
+% point lies are outside the UD. That is because, even for modest sizes of
+% $A$ and $B$, the part of the mentioned circle that lies outside the UD
+% can approach \gbc{infinity} in size, making the arc itself impossible to
+% draw. While it is feasible to compute when this will occur, we try to
+% keep it simple by using an approach that is only guaranteed to work when
+% the points lie in the unit disk. A minor modification allows it to to
+% always work when only one of the points is outside. This is because the
+% geodesic is not unique and we can easily choose one that doesn't
+% overflow.
+%
+% We isolate several special cases: if either point is the origin or if
+% the points have the same angle, a straight line is produced. If either
+% point is on the boundary, the computation is based on the fact that the
+% arc is tangent to the direction of that point. In the remaining cases,
+% we compute two angles based on reflecting both points. In the case where
+% both points lie inside or both lie outside, these angles are
+% theoretically equal, but when one point lies inside and the other
+% outside, these angles have opposite signs and their absolute values sum
+% to 360. They correspond to going opposite ways around the circle. We
+% choose the shorter arc as being more ``geodesic-like''.
+%
+% If $C$ is the point being reflected, but it is close enough to the
+% origin to make overflow a significant problem, we rescale the triangle
+% used to find the angle: we compute the angle between $|C|A$ and $|C|B$
+% as viewed from $C/|C|$.
+% \begin{macrocode}
+vardef UDgeodesic (expr A, B) =
+ save a_, b_;
+ a_ := abs(A); b_ = abs(B);
+ if (a_ = 0) or (b_ = 0):
+ A--B
+ elseif angle A = angle B:
+ A--B
+ else: % note: A, B and B-A are all nonzero from this point
+ save ang_;
+ if a_ = 1:
+ ang_ := anglefromto (if b_>1: A else: -A fi, B-A)
+ elseif b_ = 1:
+ ang_ := anglefromto (A-B, if a_>1: B else: -B fi)
+ else:
+ save C_; pair C_;
+ % reflecting A
+ if a_ < eps:
+ C_ := unitvector A;
+ ang_1 := anglefromto(a_*A - C_, a_*B - C_);
+ else:
+ C_ := (1/a_)*unitvector A;
+ ang_1 := anglefromto(A - C_, B - C_);
+ fi
+ % reflecting B
+ if b_ < eps:
+ C_ := unitvector B;
+ ang_2 := anglefromto(b_*A - C_, b_*B - C_);
+ else:
+ C_ := (1/b_)*unitvector B;
+ ang_2 := anglefromto(A - C_, B - C_);
+ fi
+ ang_ := if abs(ang_1) < abs(ang_2): ang_1 else: ang_2 fi;
+ fi
+ arcpps(A, B, 2ang_)
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{barycenter}
+% This is the average of the three corners of the triangle, or of all the
+% nodes of any path. If \gbc{t} is an open path with length $n$ and the
+% nodes are $x\sb0$ through $x\sb n$, the barycenter is
+% $$ \frac{1}{n+1}\sum\sb{j=0}\sp{n} x\sb j. $$
+% If \gbc{t} is a cycle with $x\sb n = x\sb0$, then it is
+% $$ \frac{1}{n}\sum\sb{j=0}\sp{n-1} x\sb j. $$
+%
+% For a triangle the barycenter is the intersection of the medians. I
+% don't recall if this is the center of any important circle.
+%
+% The centers of the various circles associated with triangles can be
+% found with \gbc{pathcenter}. Or by intersecting various lines: the
+% \emph{incenter} (center of the inscribed circle)is the intersection of
+% the angle bisectors; the \emph{circumcenter} is the intersection of the
+% prependicular bisectors.
+% \begin{macrocode}
+vardef barycenter expr t =
+ save m; m := length t if not cycle t: + 1 fi;
+ pnt0(t)/m for k = 1 upto m - 1: + pnt[k](t)/m endfor
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{sector}
+% \gbc{sector} produces the closed path consisting of a straight line
+% of length \gbc{rad} from \gbc{center} in the direction \gbc{frtheta},
+% thence along an arc of the circle centered at \gbc{center} to angle
+% \gbc{totheta}, and then along the straight line back to \gbc{center}.
+% \begin{macrocode}
+vardef sector (expr center, rad, frtheta, totheta) =
+ center -- arcalt (center, rad, frtheta, totheta) -- cycle
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{mkbrace}
+% Because it doesn't really fit anywhere else, and because it is not
+% really enough to waste a whole subsection on, we put \gbc{mkbrace} here.
+% It is a command to draw a brace (i.e., a ``$\lbrace$'' shape) with its
+% ends and its cusp at given points. The start is at \gbc{S}, the end at
+% \gbc{E} and the cusp at \gbc{C}. \gbc{C} should be close to, but not
+% on, the line from \gbc{S} to \gbc{E}. It should also not be too close to
+% \gbc{S} or \gbc{E}, as we need room to draw two quarter circles on
+% either side of \gbc{C} and one at each of \gbc{S} and \gbc{E}.
+% \begin{macrocode}
+vardef mkbrace (expr S, C, E) =
+ save R_, U_, V_, Z_;
+ pair U_, V_, Z_[];
+ U_ := unitvector (E-S);
+ V_ := U_ rotated 90;
+
+ R_ := 0.5*(C-S) dotprod V_;
+ if R_ = 0:
+ S--C
+ else:
+ if R_ < 0 : V_ := -V_; R_ := -R_; fi
+ V_ := R_*V_; U_ := R_*U_;
+ Z_1 := S + V_ + U_;
+ Z_2 := C - V_ - U_;
+ Z_3 := C - V_ + U_;
+ Z_4 := E + V_ - U_;
+ S{V_}..{U_}Z_1--Z_2{U_}..{V_}C{-V_}..{U_}Z_3--Z_4{U_}..{-V_}E
+ fi
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Plotting of functions}\label{functionplots}
+%
+% In these macros, if the boolean argument \gbc{sm} is true then the
+% path returned will be a B\'ezier, otherwise it will be a polyline. If a
+% \gbc{tens} parameter exists, then the smooth version will have that
+% value of tension, otherwise the value of \gbc{default_tension} is used.
+% These two parameters are simply passed to \gbc{mkpath} by \gbc{mkfcn},
+% and all these macros call \gbc{mkfcn}.
+%
+% \DescribeRoutine{mkfcn}
+% In this command the text parameter \gbc{pf} should be the name of a
+% function of some sort that can take a numeric value in parentheses and
+% return a pair expression. The parameters \gbc{bmin}, \gbc{bmax} and
+% \gbc{bst} determine a sequence of numeric values starting at \gbc{bmin},
+% stepping by \gbc{bst} and ending with \gbc{bmax}. These are fed to
+% \gbc{pf} and the resulting pairs stored in an array. Then either
+% \gbc{mkpoly} or \gbc{mksmooth} is called with the tension \gbc{tens} and
+% the name of the array.
+%
+% For stability, we don't actually step by \gbc{bst}, but round
+% \gbc{(bmax-bmin)/bst} and step that many equal steps. We first adjust
+% the step size upward so the number of steps doesn't exceed
+% \gbc{infinity}. The path is forced to begin at \gbc{pf(bmin)} and
+% end at \gbc{pf(bmax)} even if that is not an integer multiple of
+% \gbc{bst}.
+%
+% \DescribeRoutine{tfcn}
+% The macro \gbc{tfcn} is included for backward compatibility.
+% \begin{macrocode}
+vardef mkfcn (expr sm, tens) (expr bmin, bmax, bst) (text pf) =
+ save _p; pair _p[]; _p := 0;
+ save _dx, _n, _r; numeric _dx, _n, _r;
+ if bmax = bmin: _n := 1;
+ else:
+ _r := bmax - bmin;
+ _dx := max (abs(bst), nottoosmall*abs(_r), epsilon);
+ _n := emax (round(abs(_r)/_dx), 1);
+ fi
+ for _i = 0 upto _n: _p[incr _p] := pf(bmin + _i/_n*_r); endfor
+ mkpath (sm, tens, false, _p)
+enddef;
+
+def tfcn (expr sm) = mkfcn (sm, default_tension) enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{parafcn}
+% This is like \gbc{mkfcn}, but the text argument is not a pair
+% valued function, but rather a text parameter containing code that, when
+% copied literally into a vardef, defines a function in which \gbc{t} is
+% the argument, and which returns a pair.
+%
+% Older files are supported with a definition of \gbc{parafcn} that calls
+% \gbc{tparafcn} with \gbc{default_tension}. I should have made this easier
+% by reversing the smoothness and tension arguments, but for backward
+% compatibility I have to leave it thus. Other commands implement \mfpic's
+% tension options: \gbc{function} and \gbc{plrfcn}. They also have forms
+% that accept a tension argument (\gbc{tfunction} and \gbc{tplrfcn}) and
+% call them with the default tension.
+% \begin{macrocode}
+def parafcn (expr sm) = tparafcn (sm, default_tension) enddef;
+vardef tparafcn (expr sm, tn) (expr bmin, bmax, bst) (text pf) =
+ save _fp; vardef _fp (expr t) = pf enddef;
+ mkfcn (sm, tn) (bmin, bmax, bst) (_fp)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{xfcn}
+% This first converts its final argument, which should be a numeric
+% valued function \gbc{f}, to a pair valued function \gbc{(x, f(x))}, then
+% calls \gbc{mkfcn} to return the path that should be the graph of $f(x)$.
+% \begin{macrocode}
+vardef xfcn (expr sm) (expr xmin, xmax, st) (text _fx) =
+ save _fp; vardef _fp (expr _x) = (_x, _fx(_x)) enddef;
+ mkfcn (sm, default_tension) (xmin, xmax, st) (_fp)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{function}
+% This is like \gbc{xfcn} but its last argument, instead of a function,
+% is a text argument that can be copied literally it into a vardef so as
+% to define a pair valued function with a literal \gbc{x} as the argument.
+%
+% \DescribeRoutine{btwnfcn}
+% This is mainly for the sake of simpler \mfpic{} output, implementing
+% the \cs{btwnfcn} macro. One could code it in \mfpic{} macros with two
+% calls to \gbc{function}.
+%
+% \DescribeRoutine{belowfcn}
+% This is essentially \gbc{btwnfcn} with the first function identically
+% 0, but it more efficiently graphs $0$ with one straight line
+% rather than several end-to-end.
+% \begin{macrocode}
+def function (expr sm) = tfunction (sm, default_tension) enddef;
+vardef tfunction (expr sm, tens, xmin, xmax, st) (text _fx) =
+ save _fp; vardef _fp (expr x) = (x, _fx) enddef;
+ mkfcn (sm, tens) (xmin, xmax, st) (_fp)
+enddef;
+
+def btwnfcn (expr sm) = tbtwnfcn (sm, default_tension) enddef;
+vardef tbtwnfcn (expr sm, tn, xlo, xhi, st)(text _fx)(text _gx) =
+ tfunction (sm, tn) (xlo, xhi, st) (_fx) --
+ ( reverse tfunction (sm, tn) (xlo, xhi, st) (_gx) ) -- cycle
+enddef;
+
+def belowfcn (expr sm) = tbelowfcn (sm, default_tension) enddef;
+vardef tbelowfcn (expr sm, tn, xlo, xhi, st)(text _fx) =
+ (xlo,0)--(xhi,0)--
+ (reverse tfunction (sm, tn, xlo, xhi, st)(_fx))--cycle
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{rfcn}
+% This takes the name of a function \gbc{f} which is a numeric
+% valued function of a numeric parameter. It interprets it as a polar
+% curve $(\theta, f(\theta))$, converts that to a curve in rectangular
+% coordinates and calls \gbc{mkfcn} on it.
+% \begin{macrocode}
+vardef rfcn (expr sm, tmin, tmax, st) (text ft) =
+ save _fq; vardef _fq (expr t) = (ft(t)) * (dir t) enddef;
+ mkfcn (sm, default_tension) (tmin, tmax, st) (_fq)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{plrfcn}
+% This is like \gbc{rfcn}, but with a text argument containing code that
+% can be copied literally into a \mfc{vardef} creating a numeric function
+% with a literal \gbc{t} as the parameter (representing $\theta$).
+%
+% \DescribeRoutine{btwnplrfcn}
+% The macro \gbc{btwnplrfcn} is the polar version of \gbc{btwnfcn}.
+% \begin{macrocode}
+def plrfcn (expr sm) = tplrfcn (sm, default_tension) enddef;
+vardef tplrfcn (expr sm, tens, tmin, tmax, st) (text ft) =
+ save _fq; vardef _fq (expr t) = (ft) * (dir t) enddef;
+ mkfcn (sm, tens) (tmin, tmax, st) (_fq)
+enddef;
+
+def btwnplrfcn (expr sm) = tbtwnplrfcn (sm, default_tension) enddef;
+vardef tbtwnplrfcn (expr sm, tn, tlo, thi, st)(text _ft)(text _gt)=
+ tplrfcn (sm, tn, tlo, thi, st) (_ft) --
+ ( reverse tplrfcn (sm, tn, tlo, thi, st) (_gt) ) -- cycle
+enddef;
+
+def plrregion (expr sm) = tplrregion (sm, default_tension) enddef;
+vardef tplrregion (expr sm, tn, tlo, thi, st) (text _ft) =
+ (0,0)--tplrfcn (sm, tn, tlo, thi, st ) (_ft)--cycle
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{mklevelset}
+% This command assumes \gbc{inside_levelset} has been defined, which
+% should be a boolean-valued function of two variables. It tries to create
+% a path such that the expression is true inside the path and false
+% outside it. The intended application is to obtain a path surrounding a
+% region like $\lbrace (x,y) \mid F(x,y) > 0 \rbrace$.
+%
+% The parameters are
+% \begin{itemize}
+% \item \gbc{sm}: Boolean, smooth path (true) or polyline.
+% \item \gbc{tens}: Numeric, the tension (if \gbc{sm} is true).
+% \item \gbc{X} and \gbc{Y}: A starting point where \gbc{_inside_}
+% should return true.
+% \item \gbc{t}: Numeric, a step size.
+% \item \gbc{a}, \gbc{b}, \gbc{c}, and \gbc{d}: Numeric, the limits
+% beyond which the search routine will not go. This is needed to
+% get a starting range for the binary chop method of \mfc{solve}.
+% In practice, the extent of the \mfpic{} figure will be used.
+% \end{itemize}
+%
+% We use \mfc{solve} to find the first point $z\sb1$ to the right of the
+% given point that is on the edge of the region. Then we find the first
+% point of intersection between the circle at $z\sb1$ with radius \gbc{t}
+% and the edge of the region. Continue from this new point to the next
+% until (one hopes) we are within a distance \gbc{t} of the first point.
+% The radius \gbc{t} should be in coordinates appropriate for the use:
+% graph coordinates when used in \mfpic{} figures.
+%
+% The tolerance used in the first \mfc{solve} is \gbc{t/50}. In later uses
+% it is an angle parameter and is set to a number of degrees sufficient to
+% give a distance tolerance at least that.
+%
+% If the starting point \gbc{(X,Y)} does not actually satisfy the
+% condition, a one point path is returned.
+% \begin{macrocode}
+numeric tolerancefactor;
+tolerancefactor := .02;
+
+vardef mklevelset (expr sm, tens, X, Y, t, a, b, c, d) =
+ save _inside_;
+ vardef _inside_ (expr U, V) =
+ inside_levelset(U, V) and between(a, b)(U) and between(c, d)(V)
+ enddef;
+ if not _inside_ (X, Y):
+ GBwarn "Invalid seed point for levelset.";
+ pairmax((a,c), pairmin((X,Y), (b,d)))&cycle
+ else:
+ save ls, W, A, B, prev, curr, seed;
+ pair ls[], prev, curr, seed;
+ seed := (X,Y);
+ ls := 0; W := 0;
+
+ save _first_, _next_, get_next;
+ vardef _first_ (expr U) = _inside_ (U, Y) enddef;
+ vardef _next_ (expr ang) =
+ _inside_ (X_curr + t * cosd ang, Y_curr + t * sind ang)
+ enddef;
+ def get_next (expr angA, angB) =
+ X_curr := xpart curr; Y_curr := ypart curr;
+ ls[incr ls] := curr + t * dir (solve _next_ (angA, angB));
+ prev := curr; curr := ls[ls];
+ W := W + anglefromto (prev - seed, curr - seed);
+ enddef;
+
+ interim tolerance := t*tolerancefactor;
+ ls[incr ls] := (solve _first_ (X, b), Y);
+ curr := ls[ls];
+ interim tolerance := radian*tolerancefactor;
+ get_next (180, 0);
+ for n = 3 upto max_points:
+ A := angle (curr - prev);
+ get_next (A + 120, A - 120);
+ exitif ((abs(W) > 180) or (ls > 10))
+ and (abs(ls[ls] - ls1) < 1.2t);
+ endfor
+ mkpath (sm, tens, true) (ls)
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{levelset}
+% This is the \mfpic{} interface. It checks the \gbc{t} parameter before
+% passing it to \gbc{mklevelset}, making sure it is not zero, it passes
+% appropriate limits, and defines boolean function \gbc{mklevelset}
+% expects with literal \mfc{x} and \mfc{y} as parameters, using the text
+% parameter \gbc{cond}.
+% \begin{macrocode}
+numeric max_points;
+max_points := 2000;
+
+def levelset (expr s) = tlevelset (s, default_tension) enddef;
+vardef tlevelset (expr smth, tens, seed, seg) (text cond) =
+ save inside_levelset, _t;
+ vardef inside_levelset (expr x, y) = cond enddef;
+ _t := if seg <= 0: emax (xpos-xneg, ypos-yneg)/max_points * 20
+ else: seg fi;
+ mklevelset (smth, tens, xpart seed, ypart seed, _t)
+ (xneg, xpos, yneg, ypos)
+enddef;
+
+% \end{macrocode}
+%
+% Our next set of macros produce approximations to the solutions of
+% differential equations. While we could have several different macros
+% each using a different method (Euler, two-step Runge-Kutta, four-step
+% Runge-Kutta, etc.), our point of view is that we just want to draw a
+% reasonably accurate solution, so we only utilize one method: four-step
+% Runge-Kutta. The variations we allow are the following:
+% \begin{enumerate}
+% \item Drawing the graph of a one-dimensional differential
+% equation,
+% \[ \frac{dy}{dx} = g(x,y)\,.\]
+%
+% \item Drawing the trajectory of a two-dimensional differential
+% equation,
+% \[ \left( \frac{dx}{dt},\frac{dy}{dt} \right) =
+% (f(x,y,t), g(x,y,t))\,.\]
+% \end{enumerate}
+% The first of these is implemented using the second with $f(x,y,t) \equiv
+% 1$ and $g(x,y,t)$ not depending on $t$. The parameters passed include
+% the starting point, the step size, the number of steps and an expression
+% representing the right side of the equation.
+%
+% We do not use exactly the traditional Runge-Kutta method: we use the
+% Runge-Kutta algorithm, but with a variable step size. The time step
+% $\Delta t$ is chosen so that $|\mathbf{F}(x,y,t)|\Delta t$ equals the
+% given step size parameter, and thus the parameter passed is actually a
+% distance step. This makes drawing more stable, especially if the DE is
+% one that produces an infinite path in finite time.
+%
+% This modification is itself unstable if $|\mathbf{F}|$ is very
+% small (and impossible if it is zero), so we never use a $\Delta t$
+% larger than the given step size parameter $\Delta s$. That is, we
+% actually use $\Delta t = \Delta s/\max(1,|\mathbf{F}|)$.
+%
+% As with our other function-like paths, we offer two variants. The basic
+% version has a final text parameter which is the name of a pair-valued
+% function of a numeric (representing $t$) and a pair variable
+% (representing $x$ and $y$). The other version takes a text
+% parameter, which must be a pair-valued expression in \mfc{x}, \mfc{y}
+% and \mfc{t}. This parameter is copied into the definition text of a
+% function and then the first form is called with that function's name.
+%
+% Also like other function-like paths, we offer polygonal or smooth
+% versions controlled by a boolean argument, and the smooth versions make
+% use of a tension parameter.
+% \begin{macrocode}
+def RKIV (expr sm) = tRKIV (sm, default_tension)
+enddef;
+vardef tRKIV (expr sm, tens, zstart, ds, N) (text _RHS_) =
+ save _trj, _ztr, _dz, _ztmp, _ctm;
+ pair _trj[], % The trajectory
+ _ztr, % current point
+ _dz[], % array[4] of displacements
+ _ztmp; % current point for calculating velocity
+%
+ _trj := N+1; % ultimate size of _trj array
+ _trj1 := _ztr := zstart;
+ save _tt, % current time
+ _dt, % current time step
+ _th; % current time plus half a step
+ _tt := 0;
+ for _idx := 2 upto _trj:
+ _dt := ds/emax(1,abs(_RHS_(_tt,_ztr)));
+ _th := _tt + .5_dt;
+ _dz1 := _dt*_RHS_(_tt, _ztr); % displacement for current point
+ _ztmp := _ztr + .5_dz1; % 1st midpoint
+ % use _th instead of twice calculating (_tt + .5_dt)
+ _dz2 := _dt*_RHS_(_th, _ztmp); % displacement for 1st midpoint
+ _ztmp := _ztr + .5_dz2; % 2nd midpoint
+ _dz3 := _dt*_RHS_(_th, _ztmp); % displacement for 2nd midpoint
+ _ztmp := _ztr + _dz3; % temporary end point
+ % get time for next loop now since we need it in the next line:
+ _tt := _tt + _dt;
+ _dz4 := _dt*_RHS_(_tt, _ztmp); % displacement for end point
+ % get next point
+ _ztr := _ztr + (_dz1 + 2_dz2 + 2_dz3 + _dz4)/6;
+ _trj[_idx] := _ztr;
+ endfor
+ mkpath (sm, tens, false, _trj)
+enddef;
+
+def xyRKIV (expr sm) = txyRKIV (sm, default_tension)
+enddef;
+vardef txyRKIV (expr sm, tens, zstart, ds, N) (text _RHS_) =
+ save _fgxy, __fgxy;
+ vardef __fgxy (expr t, x, y) = _RHS_ enddef;
+ vardef _fgxy (expr t, Z) = __fgxy(t, xpart Z, ypart Z) enddef;
+ tRKIV (sm, tens, zstart, ds, N) (_fgxy)
+enddef;
+
+def odeRKIV (expr sm) = todeRKIV (sm, default_tension)
+enddef;
+vardef todeRKIV (expr sm, tens, xstart, ystart, ds, N)
+(text _fxy) =
+ txyRKIV (sm, tens, (xstart, ystart), ds, N) ((1, _fxy))
+enddef;
+
+% \end{macrocode}
+%
+%
+%
+% \section{Modification of Paths}\label{modification}
+%
+%
+% \subsection{Closing a path}\label{closing}
+%
+% In \MF{} one closes a path with any legal path connection between the
+% last point and the keyword \mfc{cycle}. Connecting the last point to the
+% first point is not enough. \Grafbase{} commands provide a few different
+% ways. All the commads take an undelimited path expression as the last
+% parameter and return a cycle (closed path). If the path is already
+% closed, it is returned unchanged.
+%
+% All the closure commands have a version with a tension parameter when
+% that makes sense. Those versions create the join with the supplied
+% tension. The ones where it doesn't make sense are \gbc{lclosed},
+% \gbc{cbclosed} and \gbc{qbclosed}. The first always uses a straight line
+% and the other two require explicit controls.
+%
+% \DescribeRoutine{lclosed}
+% This closes with a line segment. If the first and last point are
+% already equal, we just use the \mfc{\&} path join.
+% \begin{macrocode}
+vardef lclosed expr f =
+ f
+ if not cycle f:
+ if pnt0(f) = pnt[infinity](f): & else: -- fi cycle
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{sclosed}
+% This closes the path in the manner that \gbc{mksmooth} creates a path.
+% This will change the first and last segment of the original path. In
+% particular, if there are fewer than three segments, the whole path can
+% be different.
+% \DescribeRoutine{sclosedt}
+% It has a variant \gbc{sclosedt} that takes a tension argument.
+% \begin{macrocode}
+def sclosed = sclosedt (default_tension) enddef;
+vardef sclosedt (expr t) expr f =
+ if cycle f: f
+ else: save n; n := length f;
+ if n = 0: f&cycle
+ elseif n = 1: pnt0(f)..tension t..pnt1(f)..tension t..cycle
+ else:
+ (pnt0 (f)) { (pnt1(f)) - (pnt[n] (f)) }..tension t
+ ..(subpath (1, n-1) of f)..tension t
+ ..(pnt[n](f)) { pnt0(f) - pnt[n-1](f) }
+ ..tension t..cycle
+ fi
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{bclosed}
+% This closes with the basic default \MF{} Bezi\'er. It is a smooth
+% closure, but it does not have the same direction at the endpoints
+% that \gbc{mksmooth (true)} would have produced. It has a tense variant
+% \DescribeRoutine{bclosedt}
+% \gbc{bclosedt}
+% \begin{macrocode}
+def bclosed = bclosedt (default_tension) enddef;
+vardef bclosedt (expr t) expr f =
+ f
+ if not cycle f:
+ if pnt0(f) = pnt[infinity](f): & else: ..tension t.. fi cycle
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{uclosed}
+% Same as \gbc{bclosed}. Retained for backward compatibility. There is
+% a tense variant only for
+% \DescribeRoutine{uclosedt}
+% consistency.
+% \begin{macrocode}
+def uclosed = bclosed enddef;
+def uclosedt = bclosedt enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{cbcontrols}
+% This utility is for use in \gbc{cbclosed}. It converts Bezier segment
+% key points of a path \gbc{f}, to cubic B-spline control points stored
+% in an array \gbc{b}. The data needed are the first point and first two
+% control points of a path. It is used twice in \gbc{cbclosed} on a path
+% and on its reverse. The appropriate three points are passed in the array
+% \gbc{t}.
+% The B-spline points needed are \gbc{b1} and \gbc{b4}. The extra two
+% points \gbc{b2} and \gbc{b3} divide the line from \gbc{b1} to \gbc{b4}
+% into thirds and will be turned into Bezier control points of a new path
+% segment.
+% \begin{macrocode}
+def cbcontrols (suffix b, t) =
+ b1 := 2[t3, t2];
+ b2 := 2[t2, t1];
+ b3 := 2[b1, b2];
+ b4 := 2[b2, b3];
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{cbclosed}
+% This closes a path with a cubic B-spline. If the path \gbc{f} had been
+% produced by \gbc{opencbs}, then \gbc{q1} and \gbc{q4} would have been the
+% last two points in the argument list, and \gbc{p4} and \gbc{p1} would
+% have been the first two. We just use them and mimic the effect of
+% \gbc{closedcbs}.
+% \begin{macrocode}
+vardef cbclosed expr f =
+ save n; n := length f;
+ if cycle f: f
+ elseif n = 0: f&cycle
+ else:
+ save p, q, t; pair p[], q[], t[];
+ t1 := pnt0(f); t2 := post0(f); t3 := pre1(f);
+ cbcontrols (p, t); % defines p1 to p4
+ t1 := pnt[n](f); t2 := pre[n](f); t3 := post[n-1](f);
+ cbcontrols (q, t); % defines q1 to q4
+ f..controls q2 and q3..opencbs (q1,q4,p4,p1)
+ ..controls p3 and p2..cycle
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{qbclosed}
+% It seemed wrong to be able to close with a cubic B-spline but not a
+% quadratic B-spline; therefore I have add such a possibility. We
+% calculate B-spline controls \gbc{p[n]} that will agree with those of
+% \gbc{f}, if \gbc{f} had been created as a quadratic B-spline. Note
+% that \gbc{cbclosed} required three \MF{} links to close the curve;
+% \gbc{qbclosed} only requires two (\gbc{mkqbs} on an array of $n$
+% points makes $n-2$ links).
+% \begin{macrocode}
+vardef qbclosed expr f =
+ if cycle f: f
+ else: save n; n := length f;
+ if n = 0: f&cycle
+ else:
+ save p; pair p[]; p := 4;
+ p1 := (3/2)[pnt[n](f), pre[n](f)];
+ p2 := 2[p1, pnt[n](f)];
+ p4 := (3/2)[pnt 0 (f), post0 (f)];
+ p3 := 2[p4, pnt 0 (f)];
+ f & mkqbs (p) & cycle
+ fi
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{makesector}
+% This makes sense only if the path being modified is an arc. It closes
+% the arc by connecting its ends to the center of the circle, as
+% computed by \gbc{pathcenter}.
+% \begin{macrocode}
+vardef makesector expr p = (pathcenter p)--p--cycle enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{arccomplement}
+% Getting the complement of an arc is easy if the arc is specified by
+% three points. So we just select three points on the arc and do that.
+% The \gbc{setpairs} statement makes \gbc{pp1}, \gbc{pp2} and \gbc{pp3}
+% three points on the arc \gbc{p} in order. The arc we want goes from
+% \gbc{pp3} to \gbc{pp1} with angle twice that of the corner angle at
+% \gbc{pp2}. This function can be applied to an arbitrary path, and its
+% result will be an arc, but not necessarily a meaningful one.
+% \begin{macrocode}
+vardef arccomplement expr p =
+ if cycle p: onepointpath (false, pnt0(p))
+ else:
+ setnumeric (nn) length p;
+ setpairs (pp) (pnt0(p), pnt[.5nn](p), pnt[nn](p));
+ arcpps (pp3,pp1,2*cornerangle(pp2,pp3,pp1))
+ fi
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Trimming a path}\label{trimming}
+%
+% \DescribeRoutine{cutoffbefore}
+% This is a useful utility operation present in \file{plain.mp} (as
+% \mfc{cutbefore}) but missing from \file{plain.mf}. We write a different
+% version for our purposes: it has the syntax of most of our path
+% modification commands, plus the first loop tries to avoid a bug (or
+% perhaps inaccuracy) in \mfc{intersectiontimes} which can return an
+% intersection time in a later segment of \gbc{f} than the first
+% intersection point.
+%
+% \DescribeRoutine{cutoffafter}
+% What it and \gbc{cutoffafter} do is return the second path with the
+% part before\slash after the first path removed.
+% \begin{macrocode}
+%<MF>path cuttings;
+vardef cutoffbefore (expr b) expr f =
+ save t, n; n := length f;
+ if n > 0:
+ for k = 1 upto n:
+ exitif (subpath (0,k) of f) intersects b;
+ endfor
+ if _Xtime < 0:
+ cuttings := pnt0 (f){0,0};
+ f
+ else:
+ cuttings := subpath (0,_Xtime) of f;
+ subpath (_Xtime, n) of f
+ fi
+ else: f
+ fi
+enddef;
+
+vardef cutoffafter (expr b) expr f =
+ setpath (g) cutoffbefore (b) reverse f;
+ cuttings := reverse cuttings;
+ reverse g
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{trimmedpath}
+% This takes two lengths and a path and trims off the ends of the path
+% that lie within the given lengths of the endpoints. The lengths are in
+% device coordinates, the path in graph coordinates.
+% \begin{macrocode}
+vardef trimmedpath (expr btrim, etrim) expr f =
+ save g, h; path g, h;
+ g := invvconv (fullcircle scaled 2btrim) shifted pnt0(f);
+ h := invvconv (fullcircle scaled 2etrim) shifted pnt[length f] (f);
+ cutoffafter (h) cutoffbefore (g) f
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Creating arrows}\label{arrows}
+%
+% First, some better \mfc{direction} commands. They makes use of the fact
+% (easily proved) that a cubic B\'ezier
+% \[
+% z\sb0(1 - t)^3 + 3z\sb1(1-t)^2t + 3z\sb2(1-t)t^2 + z\sb3t^3
+% \]
+% has a tangent at $z\sb0$ equal to the first one of $z\sb{j} -z\sb0$
+% that is nonzero.
+%
+% \gbc{__dir} gets the direction at point 0 for an arbitrary path.
+% \gbc{postdirection}
+% \DescribeRoutine{postdirection}
+% reduces to this case using \mfc{subpath}. If the postdirection is
+% $(0,0)$, that means the path is trivial from that point to the end so we
+% are effectively at an endpoint (noncyclic path) and we use the incoming
+% direction. If that is $(0,0)$, the path is trivial.
+% \DescribeRoutine{predirection}
+% \gbc{predirection} just runs \gbc{postdirection} on the reversed path.
+%
+% \DescribeRoutine{trivial}
+% This returns \mfc{true} if a path has direction vector $(0,0)$ (as
+% determined by \gbc{__dir}).
+% \begin{macrocode}
+vardef predirection@# (expr p) =
+ - postdirection[length p - @#] (reverse p)
+enddef;
+
+vardef postdirection@# (expr p) =
+ save _n; _n := length (p);
+ setpair (v) __dir (subpath (@#, @# + _n) of p);
+ if v = origin:
+ v := - __dir (subpath (@#, @# - _n) of p);
+ fi
+ v
+enddef;
+
+vardef __dir (expr p) =
+ save v, w; pair v, w; w := pnt0 (p);
+ v := origin;
+ for n = 1 upto length (p):
+ v := post[n-1] (p) - w; exitif v <> origin;
+ v := pre [ n ] (p) - w; exitif v <> origin;
+ v := pnt [ n ] (p) - w; exitif v <> origin;
+ endfor
+ sgn v
+enddef;
+
+vardef trivial expr p = (__dir (p) = origin) enddef;
+
+% \end{macrocode}
+%
+% Arrowheads can be just two straight lines at an angle placed on the end
+% of a curve, or it can be a filled triangle. \grafbase{} permits both,
+% but it also allows the two lines (or the corresponding sides of the
+% triangle) to be gracefully concave and tangent to the path at the
+% endpoint of the path. The parameters controlling the shape of the arrowhead
+% are the two numerics \gbc{hdwdr}, the ratio of the length to width of the
+% arrowhead, and \gbc{hdten}, the tension in the two angled curves. By
+% default, one side of an arrowhead is just the \MF{} path
+% \mfc{a..b\marg{\meta{tangent}}}, where \mfc{a} is the base of the
+% arrowhead (calculated from \gbc{hdwdr}) and \gbc{b} is the end of the
+% path and \meta{tangent} is the direction of the path at that
+% point. The curve can be straightened by increasing \gbc{hdten}, the
+% head widened by increasing \gbc{hdwdr}
+%
+% The arrowhead is drawn by drawing two of the curves described above. If
+% \gbc{hfilled} is \mfc{true}, the two base points (\gbc{a} above) are
+% connected and the three sided region filled.
+% \begin{macrocode}
+newinternal hdwdr, hdten;
+boolean hfilled;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{headshape}
+% The following utility not only adjusts the above parameters, but
+% creates the arrowhead paths as it does so. Call it with two pure numbers
+% \gbc{wr} and \gbc{tens} for the \gbc{hdwdr} and \gbc{hdten}, and a
+% boolean \gbc{fil} for \gbc{hfilled}. The paths include two harpoon
+% tips, an arrowhead, and the paths that give regions that will be erased
+% when requested.
+% \begin{macrocode}
+def headshape (expr wr, tens, fil) =
+ interim hdwdr := wr;
+ interim hdten := if tens>0: tens else: default_tension fi;
+ if hdten < .75: hdten := .75; fi
+ setboolean (hfilled) fil;
+ mkheadpaths;
+enddef;
+def mkheadpaths =
+ save Arrowhead, Leftharpoon, Rightharpoon;
+ path Arrowhead, Leftharpoon, Rightharpoon,
+ Arrowhead.clear, Leftharpoon.clear, Rightharpoon.clear;
+ Rightharpoon := (0,0){down}..tension hdten..(.5hdwdr,-1);
+ Rightharpoon.clear := Rightharpoon--(.5hdwdr,0)--cycle;
+ Leftharpoon := (reverse Rightharpoon) xscaled -1;
+ Leftharpoon.clear := (reverse Rightharpoon.clear) xscaled -1;
+ Arrowhead := Leftharpoon & Rightharpoon;
+ Arrowhead.clear := Leftharpoon.clear & Rightharpoon.clear & cycle;
+ if hfilled:
+ Arrowhead := Arrowhead--cycle;
+ Rightharpoon := Rightharpoon--(0,-1)--cycle;
+ Leftharpoon := Leftharpoon--(0,-1)--cycle;
+ fi
+enddef;
+headshape (1,1,false);
+
+% \end{macrocode}
+%
+% \DescribeRoutine{ahead}
+% This command draws an arrowhead. Current code for arrows no longer uses
+% it. \gbc{front} and \gbc{back} are in device coordinates. They are the
+% point of the arrowhead (\gbc{front}) and the point such that \gbc{front
+% - back} is as long as the arrowhead and points in the direction of the
+% arrow. We use the ratio \gbc{hwr} to compute the other two corners. So
+% \gbc{side} is the vector from \gbc{back} to one of the corners \gbc{p1},
+% and the other corner is on the other side. \gbc{f} is the path of the
+% arrowhead.
+%
+% If \gbc{filled} is true we close the curve draw it and fill it,
+% otherwise we just draw it. (To only fill it, make the pen width $0$.)
+% \gbc{clr} is the color used to draw or fill it.
+%
+% For backward compatibility we define \gbc{head}. In \MF{} \gbc{head}
+% didn't have a color parameter, while in \MP{} it has always had one,
+% in retrospect, this was not a good idea, and I should have followed
+% the pattern of other macros. However, \gbc{head} was never a user-level
+% macro and it didn't seem to mattered at the time.
+% \begin{macrocode}
+%<MF>def head = ahead (headcolor) enddef;
+%<MP>def head = ahead enddef;
+
+vardef ahead (expr clr, front, back, hwr, tens, filled) =
+ settension (_tn) tens; fixtension (_tn);
+ if front <> back:
+ setpair (side) (hwr/2) * ((front-back) rotated 90);
+ setpath (f) (back + side)..tension _tn..
+ {front-back}front{back-front}..tension _tn..(back - side);
+ if clearhead:
+ safeunfill (back - side)--(front-side)--(front+side)--
+ (back+side) & f & cycle;
+ colorsafedraw (background) (back - side)--(front-side)--
+ (front+side)--(back+side) & f & cycle;
+ fi
+ if filled:
+ f := f--cycle;
+ colorsafefill (clr) f;
+ fi
+ colorsafedraw (clr) f;
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% It is a fact of life that, unless the path to which the head is added is
+% a straight line, the arrowhead may appear to point in the wrong
+% direction. But I know of no automatic way of making it always look
+% correct. Therefore \grafbase{} and \mfpic{} have provided a means to
+% micro-adjust the head.
+%
+% The various arrow creation commands take a path expression \gbc{f} in
+% graph coordinates, puts a head/tail/decoration on it and returns
+% \gbc{f}. There are also four parameters determining (i)~the color,
+% (ii)~the size, (iii)~a rotation adjustment, and (iv)~a position
+% adjustment.
+%
+% \DescribeRoutine{headpath}
+% \RoutineIndex{colorheadpath}
+% \RoutineIndex{headpathx}
+% \RoutineIndex{colorheadpathx}
+% \gbc{headpath} calls \gbc{Gheadpath}, a more general command that takes
+% a boolean expression and a shape (path) suffix as arguments. It
+% supplies \mfc{false} for the boolean and \gbc{Arrowhead} for the shape.
+% The \gbc{headpathx} version differs only in that the boolean is
+% \mfc{true}. The \gbc{color...} versions call \gbc{colorGheadpath} in the
+% same way, but require that a color parameter follow.
+% \begin{macrocode}
+def headpath = Gheadpath (false) (Arrowhead) enddef;
+def headpathx = Gheadpath (true) (Arrowhead) enddef;
+
+def colorheadpath = colorGheadpath (false) (Arrowhead) enddef;
+def colorheadpathx = colorGheadpath (true) (Arrowhead) enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{Gheadpath}
+% \RoutineIndex{colorGheadpath}
+% \RoutineIndex{Gheadpathx}
+% \RoutineIndex{colorGheadpathx}
+% For general arrowhead shapes we require two paths; one giving the shape
+% of the head and the other the shape that is cleared when the boolean
+% parameter \gbc{trim} is true. We pass this information by name with a
+% suffix parameter \gbc{ah} that names the head shape. Then \gbc{ah.clear}
+% names the cleared region. If \gbc{ah} is a cycle, the head is filled,
+% otherwise it is drawn with the current \gbc{drawpen}.
+% If one wants it drawn and center erased, one could place first the
+% solid (filled) version with color \mfc{background}, then the outline
+% (drawn only) version.
+%
+% We also need to know which point on the path is the tip, and
+% \gbc{ah.tip} provides that. If \gbc{ah.clear} is undefined, the clearing
+% is silently skipped. If \gbc{ah.tip} is unknown, it is taken to be
+% \mfc{(0,0)}. The head shape is assumed to be initially defined
+% pointing \mfc{up} (to match predefined shapes like \gbc{Triangle} and
+% \gbc{Diamond}).
+%
+% The standard symbols for \gbc{plot} have both the clearing path and
+% the tip defined. Thus one can produce \gbc{Diamond}-tipped arrows. The
+% old arrow heads are given by \gbc{Arrowhead}, which is redefined with
+% every call to \gbc{headshape}.
+%
+% If the \gbc{rot} and \gbc{pos} parameters are zero, the head is placed
+% with its tip at the end of the path, pointing in the direction of the
+% path at that point. Otherwise, the head is rotated around the tip by the
+% amount \gbc{rot} and \emph{then} shifted backward along its new
+% direction by the amount \gbc{pos}.
+%
+% We have a version that takes a color parameter. The simpler version
+% simply calls it with \gbc{clr} equal to \gbc{headcolor}.
+% \begin{macrocode}
+def Gheadpath (expr trim) (suffix ah) =
+ colorGheadpath (trim) (ah) (headcolor)
+enddef;
+vardef colorGheadpath
+(expr trim) (suffix ah) (expr clr, sc, rot, pos) expr f =
+ if (sc <> 0) and (known ah) and (path ah):
+ convertpath (_g) f;
+ setpair (_P) predirection[length _g] (_g);
+ if _P <> origin:
+ _P := _P rotated rot;
+ setnumeric (_ang) anglefromto (up, _P);
+ _P := pnt[length _g] (_g) - pos * _P;
+ setpair (_tip) if known ah.tip: ah.tip else: origin fi;
+
+ if trim:
+ if known ah.clear:
+ safeunfill (ah.clear shifted - _tip)
+ scaled sc
+ rotated _ang
+ shifted _P;
+ fi
+ setnumeric (_ys) max(bp, penwd, last_dot_size);
+ safeunfill cut_path
+%<MP> xscaled sc yscaled _ys
+%<MF> xscaled ceiling sc yscaled ceiling _ys
+ rotated _ang shifted _P;
+ fi
+ if cycle ah: colorsafefill else: colorsafedraw fi (clr)
+ (ah shifted -_tip)
+ scaled sc
+ rotated _ang
+ shifted _P;
+ fi
+ fi
+ f
+enddef;
+
+
+% \end{macrocode}
+%
+% \DescribeVariable{cut_path}
+% Additional clearing path, almost the same as plain.mf's \gbc{cut_} (no
+% \gbc{cut_} in plain.mp) but rotated, and scaled differently. The odd
+% scaling is so that if yscaled by the diameter of a dot, and the dot
+% happens to be digitized to a square shape, then the \gbc{cut_path},
+% centered at the center of the dot and rotated 45 degrees, will encompass
+% the whole square (theoretically).
+% \begin{macrocode}
+path cut_path;
+cut_path := (.5,0)--(.5,.71)--(-.5,.71)--(-.5,0)--cycle;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{tailpath}
+% \RoutineIndex{colortailpath}
+% The macro \gbc{tailpath} places a tail at the start of a path. It is
+% almost like \gbc{Gheadpath} except there is no clearing done and the tip
+% is at the start (point 0) of the path. Also, the position parameter
+% \gbc{pos} is a forward shift.
+%
+% \DescribeRoutine{midpath}
+% \RoutineIndex{colormidpath}
+% The macro \gbc{midpath} is just like \gbc{tailpath} except it puts the
+% given shape somewhere in the middle of the path. Its position parameter
+% indicates the fraction of the length of the path where the shape is to
+% be placed. This works best in two cases: the shape has a definite
+% direction (like the \gbc{Arrowhead}) and the tip is placed at the given
+% position, or the shape has a center of symmetry and that is placed at
+% the given position. We obtain this in most cases by shifting $(0,0)$ to
+% that position. The standard arrowhead has its tip at this point, and the
+% standard symbols (with the exception of \gbc{Circle}) have their center
+% of symmetry there.
+% \begin{macrocode}
+def tailpath (suffix sh) = colortailpath (sh) (headcolor) enddef;
+vardef colortailpath (suffix sh) (expr clr, sc, rot, pos) expr f =
+ if (sc <> 0) and (known sh) and (path sh):
+ convertpath (_g) f;
+ setpair(_P) postdirection0 (_g);
+ if _P <> origin:
+ _P := _P rotated rot;
+ if cycle sh: colorsafefill else: colorsafedraw fi (clr)
+ (sh if known sh.tip: shifted -sh.tip fi)
+ scaled sc
+ rotated anglefromto (up, _P)
+ shifted (pnt0 (_g) + pos * _P);
+ fi
+ fi
+ f
+enddef;
+
+def midpath (suffix sh) = colormidpath (sh) (headcolor) enddef;
+vardef colormidpath (suffix sh) (expr clr, sc, rot, pos) expr f =
+ if (sc <> 0) and (known sh) and (path sh):
+ convertpath (_g) f;
+ setnumeric (_t) pathtime[pos] (_g);
+ setpair (_P) postdirection[_t] (_g);
+ if _P <> origin:
+ _P := _P rotated rot;
+ if cycle sh: colorsafefill else: colorsafedraw fi (clr)
+ sh scaled sc
+ rotated anglefromto (up, _P)
+ shifted (pnt[_t] (_g));
+ fi
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Randomizing a path}
+%
+% In order to randomly change a path, we need to randomly change its
+% points and its controls. If we just apply independent random shifts to
+% every point and control point, it could happen that the direction from a
+% point to a control changes dramatically, introducing a wild change even
+% with a small shift (if point and control were very close to begin with).
+% Also, this method would almost guarantee that a smooth path would
+% randomize into one with all corners. Our solution to these problems is
+% in the following paragraph.
+%
+% If $z\sb0$ and $z\sb3$ are the start and end points, with controls
+% $z\sb1$ and $z\sb2$, then we randomize $z\sb0$ and $z\sb3$ using a
+% random shift with size supplied as a parameter. If there was a
+% preceeding segment, its ending angle and the angle of $z\sb1-z\sb0$
+% determine an angle difference which we multiply by a random factor.
+% This determines the direction to the new control point. If there was no
+% preceeding segment we rotate $z\sb1-z\sb0$ a random amount. Finally we
+% randomly scale $|z\sb1-z\sb0|$.
+%
+% The following `\gbc{deviate}s' are analogous to \MF{}'s
+% \mfc{uniformdeviate}.
+% \DescribeRoutine{signeddeviate}
+% The first, \gbc{signeddeviate X}, produces a random number uniformly
+% distributed in $(-X, X)$. The second,
+% \DescribeRoutine{scaledeviate}
+% \gbc{scaledeviate (W, A)}, produces a pair in a particular direction
+% with length distributed in $(2^{-w}, 2^w)$.
+% \DescribeRoutine{polardeviate}
+% The third, \gbc{polardeviate R} produces a pair whose polar coordinates
+% are separately uniformly distributed, the radius over the interval $(0,
+% R)$ the angle over $(0,360)$.
+% \DescribeRoutine{xydeviate}
+% The last, \gbc{xydeviate (X,Y)}, produces a pair uniformly distributed
+% over the rectangle with corners at $(-X,-Y)$ and $(X,Y)$.
+%
+% \DescribeRoutine{randompair}
+% Finally, \gbc{randompair} runs \gbc{polardeviate} if \gbc{X} is
+% numeric and \gbc{pairdeviate} if it is a pair.
+% \begin{macrocode}
+vardef signeddeviate primary X =
+ (uniformdeviate 1)[-X,X]
+enddef;
+vardef scaledeviate (expr W, A) =
+ 2 ** (signeddeviate W) * dir A
+enddef;
+vardef polardeviate primary R =
+ (uniformdeviate abs(R)) * dir uniformdeviate 360
+enddef;
+vardef xydeviate primary Z =
+ (signeddeviate (xpart Z), signeddeviate (ypart Z))
+enddef;
+
+vardef randompair (expr maxshift) =
+ if numeric maxshift: polardeviate (maxshift)
+ elseif pair maxshift: xydeviate (maxshift)
+ else: (0,0)
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{randompath}
+% The \gbc{randompath} macro returns a path formed from \gbc{f} by
+% shifting each of its point with independent instances of
+% \gbc{randompair (maxshift)}. Its control points are also modified, but
+% the algorithm is not as simple.
+%
+% Let $X$ be one of the points of \gbc{f}, with precontrol $X-U$ and
+% postcontrol $X+V$. We create random $X'$, $U'$ and $V'$ as follows.
+%
+% Let $S$ be the pair that results from \gbc{randompair(maxshift)}, let
+% $w$ be the value of \gbc{weirdness}, let $\eta\sb j(w)$ be the value of
+% the $j$th instance of \gbc{signeddeviate ($w$)}.
+% Then $X' = X + S$, $U'$ is $U$ rotated $\beta=30\eta\sb1(w)$ and scaled
+% $\sigma = 2^{\eta\sb2(w)}$. Let $\alpha$ be the angle between the two
+% vectors $U$ and $V$. Then $V'$ is $\sigma V$ rotated to make the angle
+% between $U'$ and $V'$ equal to $\alpha 2^{\eta\sb3(w)}$.
+% In the new path, the point is $X'$ with precontrol $X'-U'$ and
+% postcontrol $X' + V'$. Note that if the path is smooth at $X$, then
+% $\alpha = 0$ and the new angle is also $0$.
+%
+% \DescribeRoutine{randomlines}
+% This is a simpler version that simply shifts the nodes and connects
+% the results with straight lines. It is intended to be applied to
+% polyline paths.
+%
+% \DescribeRoutine{detrivialized}
+% We start with a routine that strips out trivial segments from a path.
+% This makes some loops a lot easier. It would be weird to differently
+% shift the two (equal) endpoints of a trivial segment.
+% \begin{macrocode}
+vardef detrivialized expr f =
+ save g; path p, g[]; g := 0;
+ for k = 1 upto length f:
+ p := subpath (k-1,k) of f;
+ if not trivial p: g[incr g] := p; fi
+ endfor
+ if g = 0: onepointpath (cycle f, pnt0(f))
+ else: g1 for k = 2 upto g: &g[k] endfor if cycle f: &cycle fi
+ fi
+enddef;
+
+vardef randompath (expr maxshift, weirdness) expr f =
+ save g, n; path g;
+ g := detrivialized f;
+ n := length g;
+ if n = 0:
+ f shifted randompair (maxshift)
+ else:
+ save X, U, V;
+ pair X[], U[], V[];
+ if cycle g: n := n - 1; fi
+ for k = 0 upto n:
+ X[k] := pnt[k](g);
+ U[k] := X[k] - pre[k](g);
+ V[k] := post[k](g) - X[k];
+ endfor
+ save A, B;
+ for k := 0 upto n:
+ X[k] := X[k] shifted randompair (maxshift);
+ A := anglefromto (U[k],V[k]);
+ B := signeddeviate (30weirdness);
+ U[k] := X[k] - (U[k] zscaled scaledeviate (weirdness,B));
+ B := B - A + A * (2 ** signeddeviate weirdness);
+ V[k] := X[k] + (V[k] zscaled scaledeviate (weirdness,B));
+ endfor
+ X0 for k = 1 upto n:
+ .. controls V[k-1] and U[k] .. X[k]
+ endfor
+ if cycle g:
+ .. controls V[n] and U0 .. cycle
+ fi
+ fi
+enddef;
+
+vardef randomlines (expr maxshift) expr f =
+ save g, n; path g;
+ g := detrivialized f;
+ n := length g;
+ if n = 0:
+ f shifted randompair (maxshift)
+ else:
+ if cycle g: n := n - 1; fi
+ (pnt0(g) shifted randompair (maxshift))
+ for k = 1 upto n:
+ -- (pnt[k](g) shifted randompair (maxshift))
+ endfor
+ if cycle g:
+ -- cycle
+ fi
+ fi
+enddef;
+
+% \end{macrocode}
+%
+%
+% \subsection{Interpolating paths}
+%
+% Given two cubic B\'eziers, it is straightforward to create a path that
+% is ``half-way between'' them: just take its control points to be
+% at the midpoint between corresponding control points of the two
+% B\'eziers. Two paths made up of an equal number of B\'ezier are also
+% easily interpolated. However, two paths with different numbers of
+% B\'ezier segments need to be subdivided until they have an equal
+% number.
+%
+% \DescribeRoutine{interpolatedpath}
+% This command accepts a number \gbc{num}, a path or pair \gbc{P} and a
+% path \gbc{Q}. It returns a path which is somewhere ``between'' \gbc{P}
+% and \gbc{Q} if the number is between $0$ and $1$. The case where \gbc{P}
+% or \gbc{Q} is trivial is passed on to another command which is
+% considerably more efficient for that case. In the more general case, the
+% paths are rewritten so that they have equal length. For example, if
+% \gbc{P} has length 2 and \gbc{Q} has length 1, then \gbc{Q} is rewritten
+% as\\
+% \indent \gbc{subpath (0,1/2) of Q \& subpath (1/2,1) of Q}\\
+% which follows the same course as \gbc{Q} but has the same number of
+% B\'ezier parts as \gbc{P}.
+%
+% The splitting of \gbc{Q} shown above can, for reasons unknown to me,
+% produce adjacent subpaths that do not always share an endpoint. One
+% would think that \gbc{subpath (s,t) of Q} and
+% \gbc{subpath (t,u) of Q} would obviously end and start, respectively,
+% at \gbc{point t of Q}. Alas, they don't always. Hence, we employ
+% \gbc{force_equal_ends} to to make them equal, shifting their endpoints a
+% microscopic amount.
+%
+% If \gbc{Q} is a cycle we want the returned path to also be a cycle
+% (but not otherwise). This is possible whenever the ends of \gbc{P} are
+% equal.
+% \begin{macrocode}
+vardef interpolatedpath (expr t, P) expr Q =
+ if not path Q:
+ GBerrmsg ("Improper argument to interpolatedpath.")
+ "The last argument to interpolatedpath must be a path.";
+ if pair P: onepointpath(false, P)
+ else:
+ if path P:
+ P
+ else:
+ onepointpath (false, origin)
+ fi
+ fi
+ elseif pair P:
+ interpolated_pair_path (t, cycle Q, P, Q)
+ elseif not path P:
+ GBerrmsg ("Improper argument to interpolatedpath.")
+ "The second argument to interpolatedpath must be a pair "
+ & "or a path.";
+ Q
+ else:
+ if t=0: Q
+ elseif t=1: P
+ else:
+ save P_, Q_; path P_, Q_;
+ P_ := detrivialized P;
+ Q_ := detrivialized Q;
+ if length P_ = 0:
+ interpolated_pair_path (t, cycle Q, pnt0(P_), Q)
+ elseif length Q_ = 0:
+ interpolated_pair_path (t, cycle Q, pnt0(Q_), P)
+ else:
+ save G, H, n, m, k, r;
+ path G[], H[];
+ G := H := 0;
+ n := length P_; m := length Q_;
+ k := gcd(n, m);
+ r := m/k;
+ for I=0 upto n-1:
+ for J=0 upto r-1:
+ G[incr G] := subpath (I+J/r, I+(J+1)/r) of P_;
+ endfor
+ endfor
+ r := n/k;
+ for I=0 upto m-1:
+ for J=0 upto r-1:
+ H[incr H] := subpath (I+J/r, I+(J+1)/r) of Q_;
+ endfor
+ endfor
+ for N = 1 upto G-1:
+ force_equal_ends(G[N], G[N+1]);
+ force_equal_ends(H[N], H[N+1]);
+ endfor
+ interpolated_segment (t, G1, H1)
+ for N = 2 upto G: & interpolated_segment (t, G[N], H[N])
+ endfor if (pnt0(G1)=pnt1(G[G])) and (cycle Q): & cycle fi
+ fi
+ fi
+ fi
+enddef;
+
+% \end{macrocode}
+% \DescribeRoutine{interpolated_pair_path}
+% Since we cannot rely on the cyclicity of \gbc{Q}, we pass a boolean
+% parameter . That is because the second argument here might actually
+% have been the first argument of \gbc{interpolatedpath}.
+% \begin{macrocode}
+vardef interpolated_pair_path (expr t, cyclic, P, Q) =
+ save N; N := length Q;
+ if N=0: onepointpath (cyclic, (t)[pnt0(Q),P])
+ else:
+ (t)[pnt0(Q),P]..controls (t)[post0(Q),P] and
+ for n=1 upto N - 1:
+ (t)[pre[n](Q),P]..(t)[pnt[n](Q),P]..controls (t)[post[n](Q),P]
+ and
+ endfor
+ (t)[pre[N](Q),P].. if cyclic: cycle else: (t)[pnt[N](Q),P] fi
+ fi
+enddef;
+
+vardef interpolated_segment (expr t, S, T) =
+ (t)[ pnt0(S), pnt0(T)]..controls
+ (t)[ post0(S), post0(T)] and (t)[ pre1(S), pre1(T)]..
+ (t)[ pnt1(S), pnt1(T)]
+enddef;
+
+% \end{macrocode}
+%
+% \subsection{Parallelling a path}
+%
+% \DescribeRoutine{parasegment}
+% This creates a path parallel to a given cubic B\'ezier segment \gbc{f}.
+% It should be called by a command (such as \gbc{parapath}) that makes
+% sure \gbc{f} is nontrivial (meaning the directions are non-zero). It
+% splits the segment into subsegments for accuracy. Its arguments are the
+% distance the original path is shifted, the number of subsegments to
+% split into, and the path.
+% \begin{macrocode}
+vardef parasegment (expr d, segs, f) =
+ if d = 0: f
+ else:
+ save u, v, t; pair u[], v[];
+ for n = 0 upto segs:
+ t := n/segs;
+ u[n] := postdirection [t] (f);
+ v[n] := pnt[t] (f) + (u[n] zscaled (0,d));
+ endfor
+ v0{u0}
+ for n = 1 upto segs: ...v[n]{u[n]} endfor
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{parapath}
+% Attempt to parallel one path with another at a distance \gbc{d}. The
+% algorithm is as follows:
+% \begin{enumerate}
+% \item Create an array of paths that parallel each segment of \gbc{f}.
+% \item Redo the array by adding a half circle at each end. The purpose
+% of the half circle is to force consecutive elements of the array
+% to intersect (I hope).
+% \item Do something like \mfc{buildcycle}: where consecutive array
+% elements intersect, strip the the elements to the subpaths between
+% intersection points, and join them directly.
+% \item If \gbc{f} is a cycle, process the joining at the endpoints in
+% the same way.
+% \end{enumerate}
+% We use \gbc{force_equal_ends} to force exact equality of endpoints so
+% we can join segments with \mfc{\&}. The reason for prefering the
+% \mfc{\&} join is to permit better performance of macros (such as this
+% one) that take a path apart into segments. If we used `\mfc{..}' instead
+% then the returned path would have a great many added segments, nearly
+% all of which are nearly trivial. For similar reasons we drop trivial
+% segments of \gbc{f}.
+% \begin{macrocode}
+vardef parapath (expr d) expr f =
+ if d = 0:
+ f
+ else:
+ save a, g, h, p, q, s, t, u, v, w;
+ path g[], h, p[], q[];
+ numeric a, s, t;
+ pair u, v, w, w[];
+ s := emax(3, emin(segment_split, ceiling(max_points/5/length f)));
+ p := 0;
+ for i = 1 upto length f:
+ h := subpath (i-1, i) of f;
+ if not trivial h:
+ q[incr p] := h;
+ p[p] := parasegment (d, s, h);
+ fi
+ endfor
+ if p = 0:
+ f
+ else:
+ a := if d>0: - fi 180;
+ h := p1;
+ for i = 1 upto p-1:
+ u := predirection 1 (q[i]);
+ v := postdirection 0 (q[i+1]);
+ w1 := pnt 1 (q[i]) - (u zscaled (0,d));
+ w2 := pnt 0 (q[i+1]) - (v zscaled (0,d));
+ w3 := pnt [infinity] (h);
+ w4 := pnt 0 (p[i+1]);
+ g0 := arcpps(w3, w1, a);
+ g1 := h & g0;
+ g2 := arcpps(w2, w4, a) & p[i+1];
+ if (p[i] & g0) intersects reverse g2:
+ s := length g2 - _Ytime;
+ t := length h - length p[i] + _Xtime;
+ g1 := subpath (0, t) of g1;
+ g2 := subpath (s, length g2) of g2;
+ force_equal_ends (g1, g2);
+ h := g1 & g2;
+ else:
+ h := h .. p[i+1];
+ fi
+ endfor
+
+ if cycle f:
+ u := predirection 1 (q[p]);
+ v := postdirection 0 (q[1]);
+ w1 := pnt 1 (q[p]) - (u zscaled (0,d));
+ w2 := pnt 0 (q[1]) - (v zscaled (0,d));
+ w3 := pnt [infinity] (h);
+ w4 := pnt 0 (p[1]);
+ g3 := arcpps(w3, w1, a);
+ g0 := arcpps(w2, w4, a);
+ g1 := g0 & h & g3;
+ g2 := g0 & p[1];
+ if (p[p] & g3) intersects reverse g2:
+ s := length g2 - _Ytime;
+ t := length g0 + length h - length p[p] + _Xtime;
+ g1 := subpath (s, t) of g1;
+ force_equal_ends (g1, g1);
+ h := g1 & cycle;
+ else:
+ h := h..cycle;
+ fi
+ fi
+ h
+ fi
+ fi
+enddef;
+
+vardef turnangle@# (expr f) =
+ anglefromto(predirection@# (f), postdirection@#(f))
+enddef;
+
+% \end{macrocode}
+%
+%
+% \section{Miscellaneous}\label{misc}
+%
+% \subsection{Implementation of \mfpic{}'s \cs{plotdata} command}%
+% \label{plotdata}
+%
+% In \mfpic, the \cs{plotdata} command draws several curves with one
+% command. The curves are drawn with changeable methods of rendering.
+% There are three schemes. The first draws the curves with different dash
+% patterns. Another scheme is to plot the curves with different symbols.
+% Still another is to use different colors (\MP{} only).
+%
+% We implement the changing of patterns (symbols, colors) by defining
+% arrays of such things and changing the index into the array. For
+% example, when the user has selected dashes, the first curve is
+% \gbc{gendashed} with the pattern \gbc{dashtype0}, the next with
+% \gbc{dashtype1}, etc.
+%
+% \DescribeRoutine{setdatadashes}
+% We have this method for users to select their own dash patterns. The
+% \gbc{setdatadashes} command requires a list of suffixes previously
+% defined by the \gbc{dashpat} command. Since a dash pattern need only
+% be an array, we check if it is one and, if it is, we copy it to the
+% next \gbc{dashtype[n]}. We actually copy it to a temporary array and
+% make sure there are at least two patterns before we overwrite
+% \gbc{dashtype}.
+%
+% \DescribeRoutine{getdashpat}
+% We have removed this mod-ing operation from \TeX, where it is
+% cumbersome, to \MF, where it is trivial.
+% \begin{macrocode}
+def setdatadashes (text lst) =
+ save __type; __type := 0;
+ forsuffixes _itm = lst:
+ if knownnumericarray _itm :
+ copyarray (_itm) (__type[__type]);
+ next __type;
+ else: GBwarn "Improper dash pattern in setdatadashes.";
+ fi
+ endfor
+ if __type > 1:
+ save dashtype; dashtype := __type;
+ for _j = 0 upto dashtype - 1:
+ copyarray (__type[_j]) (dashtype[_j]);
+ endfor
+ else:
+ SetdataWarn "dashes";
+ fi
+enddef;
+def getdashpat expr n = dashtype[n mod dashtype] enddef;
+
+def SetdataWarn expr s =
+ GBwarn "command setdata"& s &"() failed. Previous values retained.";
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{defaultdashes}
+% These are the default dash patterns. Their setting is done by a macro
+% so the user may easily restore them. The spaces are apparently larger
+% than the dashes, but taking the thickness of the pen into account
+% (\mfc{.5bp}) the dashes will appear about \mfc{.5bp} larger than stated
+% and the spaces about \mfc{.5bp} smaller (unless the user inexplicably
+% sets \mfc{linecap} to \mfc{butt}).
+% \begin{macrocode}
+numeric Solid, Simpledash, Simpledot, Dotdash, Dotdashdot, Dotdashdash;
+dashpat (Solid) (0);
+dashpat (Simple_dash) (3bp, 4bp);
+dashpat (Simple_dot) (0, 4bp);
+dashpat (Dot_dash) (0, 4bp, 3bp, 4bp);
+dashpat (Dot_dash_dot) (0, 4bp, 3bp, 4bp, 0, 4bp);
+dashpat (Dot_dash_dash) (0, 4bp, 3bp, 4bp, 3bp, 4bp);
+
+numeric dashtype, dashtype[], dashtype[][];
+def defaultdashes =
+ setdatadashes (Solid, Simple_dash, Simple_dot,
+ Dot_dash, Dot_dash_dot, Dot_dash_dash);
+enddef;
+defaultdashes;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{setdatasymbols}
+% This can be used to define the sequence of point plotting styles for
+% \mfpic's \cs{plotdata} command. It is quite similar to
+% \gbc{setdatadashes} above and \gbc{setdatacolors} below.
+%
+% \DescribeRoutine{getsymbol}
+% This is similar to \gbc{getdashpat}. In fact we could write a
+% single macro to do both, but I think we get a more readable \mfpic{}
+% output file if we have separate commands.
+% \begin{macrocode}
+def setdatasymbols (text lst) =
+ save __type; path __type[];
+ __type := 0;
+ for _itm = lst:
+ if (known _itm) and (path _itm):
+ __type[__type] := _itm;
+ next __type;
+ else:
+ GBwarn "Improper symbol in setdatasymbols().";
+ fi
+ endfor
+ if __type > 1:
+ save pointtype; pointtype := __type;
+ path pointtype[];
+ for _j = 0 upto pointtype - 1:
+ pointtype[_j] := __type[_j];
+ endfor
+ else:
+ SetdataWarn "symbols";
+ fi
+enddef;
+def getsymbol expr n := pointtype[n mod pointtype] enddef;
+
+% \end{macrocode}
+%
+% Before we can set the default symbols we need to define some. They need
+% to be paths. The ones below named with `\gbc{Solid}' are closed paths.
+% Since the drawing commands that use them feed the path to \gbc{setdot},
+% they end up filled if they are cyclic, merely drawn if not.
+%
+% All are intended to have roughly the area (when area makes sense) of a
+% circle with diameter 1. The scaling factors are the square root of the
+% ratios of the areas.
+%
+% Associated with each is a another path with the same basename and the
+% suffix \gbc{clear} and a pair with the suffix \gbc{tip}. Moreover,
+% they are (mostly) symmetric about the $y$-axis pointing (where that
+% makes sense) in the direction \mfc{up}. The purpose of all this is so
+% that they can be plugged into code for adding arrowheads/tails of
+% different shapes.\VariableIndex{Triangle}\VariableIndex{Square}
+% \VariableIndex{Circle}\VariableIndex{Diamond}\VariableIndex{Star}
+% \VariableIndex{Plus}\VariableIndex{Cross}\VariableIndex{Asterisk}
+% \VariableIndex{Crossbar}\VariableIndex{Leftbar}\VariableIndex{Rightbar}
+% \VariableIndex{Righthook}\VariableIndex{Lefthook}
+% \VariableIndex{SolidTriangle}\VariableIndex{SolidSquare}
+% \VariableIndex{SolidCircle}\VariableIndex{SolidDiamond}
+% \VariableIndex{SolidStar}
+% \begin{macrocode}
+def DeclareGBSymbols (text S) =
+ forsuffixes _itm = S:
+ path _itm;
+ path _itm.clear;
+ pair _itm.tip;
+ endfor
+enddef;
+DeclareGBSymbols(
+ Triangle, Square, Circle, Diamond, Star, Plus, Cross,
+ Asterisk, Crossbar, Leftbar, Rightbar, Righthook,
+ Lefthook, SolidTriangle, SolidSquare, SolidCircle,
+ SolidDiamond, SolidStar
+);
+
+vardef undo_cycle expr f = subpath (0, length f) of f enddef;
+
+SolidTriangle := (up--(dir 210)--(dir -30)--cycle) scaled .78;
+Triangle := undo_cycle SolidTriangle;
+
+Triangle.clear := SolidTriangle.clear :=
+ ((dir -30)--(cosd 30,1)--(cosd 210,1)--(dir 210)--up--cycle)
+ scaled .78;
+
+SolidSquare := (up--(-1,1)--(-1,-1)--(1,-1)--(1,1)--cycle) scaled .443;
+Square := undo_cycle SolidSquare;
+
+SolidCircle := fullcircle rotated 90;
+Circle := undo_cycle SolidCircle;
+Circle.clear := SolidCircle.clear :=
+ halfcircle--(-.5,.5)--(.5,.5)--cycle;
+
+SolidDiamond := (up--left--down--right--cycle)
+ scaled .522 yscaled 1.44;
+Diamond := undo_cycle SolidDiamond;
+Diamond.clear := SolidDiamond.clear :=
+ (right--(1,1)--(-1,1)--left--up--cycle) scaled .522 yscaled 1.44;
+
+Plus := ((0,0)--up--down--(0,0)--left--right) scaled .65;
+Plus.clear := (right--(1,1)--(-1,1)--(left)--cycle) scaled .65;
+
+Cross := ((0,0)--(dir 45)--(dir -135)--(0,0)--(dir -45)--(dir 135))
+ scaled .65;
+Cross.clear :=
+ ((0,0)--(dir -45)--dir(45)--(dir 135)--(dir -135)--cycle) scaled .65;
+
+Asterisk := ((0,0)--up--down--(0,0)--(dir 30)--(dir -150)
+ --(0,0)--(dir -30)--(dir 150)) scaled .6;
+Asterisk.clear := ((0,0)--(dir -30)--(cosd 30,1)--(cosd 150,1)
+ --(dir -150)--cycle) scaled .6;
+
+Crossbar := ((0,0)--left--right) scaled .65;
+Crossbar.clear := rect (right,(-1,.5)) scaled .65;
+
+Leftbar := ((0,0)--left);
+Rightbar := ((0,0)--right);
+Leftbar.clear := rect((0,0),(-1,.5));
+Rightbar.clear := rect((0,0),(1,.5));
+
+Righthook := arcpps((0,0),(1,0),180);
+Lefthook := Righthook xscaled -1;
+Righthook.clear := Righthook--cycle;
+Lefthook.clear := Lefthook--cycle;
+
+% \end{macrocode}
+%
+% We do some computations to find the vertices of an n-pointed
+% star. We assume that \mfc{A1} is \mfc{up} and the line from there
+% to \mfc{A[1 + m]} determines one side of the top point of the star.
+% We must have $2 \le {}$\gbc{m}${}\le{}$\gbc{n}${}-2$. The rest of the
+% vertices are determined by symmetry. \gbc{Star} is made with \gbc{n=5}
+% and \gbc{m=2}. We store the points in an array so we can use them for
+% the \gbc{Star.clear} path.
+% \begin{macrocode}
+vardef mkstar (expr n, m) (suffix A) =
+ save ang; ang := 360/n;
+ A1 := up; A3 := up rotated ang;
+ A2 = (whatever)[A1, A1 rotated ( ang*m)];
+ A2 = (whatever)[A3, A3 rotated (-ang*m)];
+ for i = 4 upto 2n:
+ A[i] := A[i-2] rotated ang;
+ endfor
+ A := 2n;
+ mkpoly (true, A)
+enddef;
+
+save _A; pair _A[];
+SolidStar := mkstar (5, 2, _A) scaled .84;
+Star := undo_cycle SolidStar;
+Star.clear := polyline (true)
+ (_A9, _A10, _A1, _A2, _A3, (xpart _A3, 1), (xpart _A9, 1))
+ scaled .84;
+SolidStar.clear := Star.clear;
+
+forsuffixes S =
+ Triangle, Square, Circle, Diamond, Star, Plus, Cross,
+ Asterisk, Crossbar, Leftbar, Rightbar, Righthook,
+ Lefthook, SolidTriangle, SolidSquare, SolidCircle,
+ SolidDiamond, SolidStar :
+ S.tip := point 0 of S;
+endfor
+
+% \end{macrocode}
+%
+% \DescribeRoutine{gcd}
+% I thought I was going to use \gbc{gcd} for the \gbc{mkstar} routine
+% above, but went another way. Still, it might have a future use. Once we
+% have it,
+% \DescribeRoutine{lcm}
+% \gbc{lcm} is a snap. Since \gbc{gcd} always returns a positive result,
+% \gbc{lcm} satisfies the rule for signs of products. Note that these both
+% silently accept noninteger arguments, though the results may not be very
+% meaningful.
+% \begin{macrocode}
+vardef gcd (expr n, m) =
+ save a, b, r;
+ a := emax (abs(m), abs(n));
+ b := emin (abs(m), abs(n));
+ if b > 0:
+ forever:
+ r := a mod b;
+ exitif r < 1;
+ a := b; b := r;
+ endfor
+ b
+ else:
+ a
+ fi
+enddef;
+
+vardef lcm (expr n, m) =
+ n/gcd(n, m)*m
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{defaultsymbols}
+% The command for restoring the default symbols.
+% \begin{macrocode}
+numeric pointtype; path pointtype[];
+def defaultsymbols =
+ setdatasymbols( Circle, Cross, SolidDiamond, Square, Plus,
+ Triangle, SolidCircle, Star, SolidTriangle);
+enddef;
+defaultsymbols;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{setdatacolors}
+% Finally, for \MP, we do a similar pair of commands for setting
+% the colors for the \cs{plotdata} command, and for
+% \DescribeRoutine{getcolor}
+% getting the next one. The odd indirection (\gbc{colortype[]} is an array
+% of strings, the names of variables having color values) is because \MP{}
+% now has three different data types for colors. Arrays must be all one
+% type.
+%
+% \DescribeRoutine{defaultcolors}
+% These default colors were tested on screen and on an inkjet printer.
+% The adjustments away from pure colors is based on a compromise between
+% those experiments.
+% \begin{macrocode}
+%<*MP>
+def setdatacolors (text lst) =
+ setnumeric (__type) 0;
+ % First, just count and store the known colors in the list
+ for _itm = lst:
+ if knowncolor _itm :
+ if __type = 0 : def _datacolors = _itm enddef;
+ else:
+ expandafter def
+ expandafter _datacolors
+ expandafter = _datacolors, _itm enddef;
+ fi
+ next __type;
+ else: GBwarn "Improper color in setdatacolors().";
+ fi
+ endfor
+ if __type > 1:
+ save colortype, _tmpstr;
+ colortype := 0;
+ % colortype[] is an array of strings:
+ string colortype[], _tmpstr;
+ for _itm = _datacolors: %
+ % Each string is the name of some color variable
+ _tmpstr := "colortype_" & GBromannumeral(colortype);
+ setcolor (scantokens(_tmpstr)) _itm;
+ colortype[colortype] := _tmpstr;
+ next colortype;
+ endfor
+ else:
+ SetdataWarn "colors";
+ fi
+enddef;
+def getcolor expr n = (scantokens (colortype[n mod colortype])) enddef;
+
+numeric colortype; string colortype[];
+
+setcolor (dRed) (1, 0, 0);
+setcolor (dBlue) (.2,.2,1);
+setcolor (dOrange) (1,.34,0);
+setcolor (dGreen) (0,.80,0);
+setcolor (dBlack) cmykblack;
+if has_cmyk :
+ setcolor (dCyan) cyan;
+ setcolor (dMagenta) magenta;
+ setcolor (dYellow) yellow;
+else: % rgb colors seem to be lighter than the cmyk equivalents.
+ setcolor (dCyan) cmyk(.85,0,0,.15);
+ setcolor (dMagenta) cmyk(0,.85,0,.15);
+ setcolor (dYellow) cmyk(0,0,.85,.15);
+fi
+def defaultcolors =
+ setdatacolors(dBlack, dRed, dBlue, dOrange,
+ dGreen, dMagenta, dCyan, dYellow);
+enddef;
+defaultcolors;
+
+%</MP>
+% \end{macrocode}
+%
+%
+% \subsection{Pie Charts and Bar Charts}\label{charts}
+%
+% \DescribeRoutine{computepie}
+% The \gbc{computepie} command calculates the wedges of a pie from the text
+% parameter \gbc{data}. It should be a list of positive numerics, and the
+% result will be one wedge for each datum, the area of the wedge being
+% proportional to the corresponding datum. The wedge for each datum has
+% its point at \gbc{cent} and the wedge for the first datum begins at
+% angle \gbc{ang}. Each wedge is clockwise from the preceding one if
+% \gbc{sign = -1}, otherwise anticlockwise. The radius of the pie is
+% \gbc{rad}.
+%
+% \DescribeRoutine{piechart}
+% This calls \gbc{computepie} to calculate the angles and store that in
+% the array \gbc{_dat}, then \gbc{mkpiewedges} creates the actual user
+% level arrays. This separation allows us (in a future enhancement) to
+% easily handle named piecharts, so there can be more than one defined at
+% a time.
+% \begin{macrocode}
+def computepie (suffix dat) (expr sign, ang, cent, rad) (text data) =
+begingroup
+ save _tot, _max, _toobig;
+ _max := 0; dat := 0;
+ for _val = data:
+ dat[incr dat] := _val;
+ _max := emax (_max, _val);
+ endfor
+ if dat=0: GBwarn "piechart attempted with empty list.";
+ _toobig := 1;
+ else:
+ _toobig := infinity/dat;
+ fi
+ if _max > _toobig:
+ for _idx = 1 upto dat:
+ dat[_idx] := dat[_idx]/_toobig;
+ endfor
+ fi
+ for _idx = 2 upto dat:
+ dat[_idx] := dat[_idx - 1] + dat[_idx];
+ endfor
+ _tot := dat[dat];
+ for _idx = dat downto 2:
+ dat[_idx] := ang + sign*dat[_idx-1]/_tot*360;
+ endfor
+ dat1 := ang; dat[dat + 1] := ang + 360sign;
+endgroup
+enddef;
+
+def piechart (expr sign, ang, cent, rad) (text data) =
+ save _dat;
+ computepie (_dat) (sign, ang, cent, rad) (data);
+ mkpiewedges (_dat, cent, rad);
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{mkpiewedges}
+% The wedges (closed sectors) are stored in the array \gbc{piewedge[\,]}
+% with the numeric \gbc{piewedge} holding the number of wedges. The center
+% is saved in \gbc{piecenter}, the directions of the wedges (the bisecting
+% rays) are stored in \gbc{piedirection[\,]}, the starting angles of the
+% wedges in \gbc{pieangle[\,]}
+% \begin{macrocode}
+def mkpiewedges (suffix dat) (expr cent, rad) =
+ numeric piewedge, piedirection, pieangle, pieangle[];
+ pair piecenter, piedirection[];
+ path piewedge[];
+ piecenter := cent;
+ piedirection := pieangle := piewedge := dat;
+ for _idx = 1 upto dat:
+ pieangle[_idx] := dat[_idx];
+ piewedge[_idx] := sector (piecenter, rad, dat[_idx], dat[_idx+1]);
+ piedirection[_idx] := dir(0.5[ dat[_idx], dat[_idx+1] ]);
+ endfor
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{namedpiechart}
+% This is a future enhancement. It takes a suffix, the name of the chart
+% which will be the base name of the various arrays and key values, and
+% will be a numeric equal to the number of wedges. The first part is
+% identical to that of \gbc{piechart} above. Note that some elements
+% of the chart are not (yet) directly accessible. Those needed by mfpic
+% are: the paths, the directions and the center. The rest would be easy
+% to add.
+% \begin{macrocode}
+def namedpiechart (suffix nm) (expr sign, ang, cent, rad) (text data) =
+ save _dat;
+ computepie (_dat) (sign, ang, cent, rad) (data);
+ setnumeric (nm) _dat;
+ pair nm.center, nm.direction[];
+ path nm.wedge[];
+ nm.center := cent;
+ for _idx = 1 upto _dat:
+ nm.wedge[_idx] := sector (cent, rad, _dat[_idx], _dat[_idx+1]);
+ nm.direction[_idx] := dir(0.5[ _dat[_idx], _dat[_idx+1] ]);
+ endfor
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{barchart}
+% I was told that there are better ways (than piecharts) to represent
+% quantitative data. Perhaps bar charts are better. \gbc{barchart}
+% calculates the bars from the text parameter, \gbc{data}. These bars are
+% vertical if \gbc{vert} is true, otherwise horizontal.
+%
+% \gbc{firstbar} is the location (on the appropriate axis) of the start of
+% the first bar. \gbc{sep} is the separation between bar centers. \gbc{r}
+% is the ratio of the width of the bars to their separation.
+%
+% After the calculations, the array of paths \gbc{chartbar[\,]} holds the
+% rectangles, \gbc{barend[\,]} holds their rightmost or topmost
+% coordinates (which is just the items in \gbc{data} or their yparts),
+% \gbc{barbegin[\,]} holds their leftmost or bottommost coordinates (either
+% 0 or the xparts of the data), \gbc{barstart[\,]} holds the appropriate
+% coordinate of the leading edge of the bar, and \gbc{barwd = r*sep}.
+%
+% If the data are pair data, this command uses the xpart as the beginning
+% of the bar and the ypart as the end. Thus Gantt diagrams can be
+% created. We keep \gbc{barlength} for backward compatibility (formerly
+% all data had to be numeric and bars went from 0 to \gbc{barlength[\,]}).
+% \gbc{barlength[\,]} was made available to help place some label or symbol
+% at the end of a bar and existing code might break if we omit it.
+% \begin{macrocode}
+def barchart (expr firstbar, sep, r, vert)(text data) =
+ numeric barbegin, barbegin[],
+ barend, barend[],
+ barlength, barlength[],
+ barstart, barstart[],
+ chartbar, barwd;
+ path chartbar[];
+ chartbar := 0; barwd := r*sep;
+ for _itm = data:
+ barend[incr chartbar]
+ := if pair _itm: ypart _itm else: _itm fi;
+ barbegin[chartbar] := if pair _itm: xpart _itm else: 0 fi;
+ endfor
+ barbegin := barend := barlength := barstart := chartbar;
+ for _nn = 1 upto chartbar:
+ barstart[_nn] := firstbar + sep*(_nn-1);
+ barlength[_nn] := barend[_nn];
+ chartbar[_nn] := rect ((barbegin[_nn], 0), ( barend[_nn], barwd))
+ shifted (0, barstart[_nn]) if vert: xyswap fi;
+ endfor
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{namedbarchart}
+% This is a future enhancement. It takes a suffix, the name of the chart,
+% which will be the base name of the path array. Note that the various key
+% elements (ends of the bar, etc.) are not (yet) directly accessible as in
+% the above version, but they would be easy to add.
+% \begin{macrocode}
+def namedbarchart (suffix nm) (expr first, sep, r, vert) (text data) =
+ save nm;
+begingroup
+ save _bb, _ee, _ww;
+ path nm.bar[];
+ nm := 0; _ww := r*sep;
+ for _itm = data:
+ _ee := if pair _itm: ypart _itm else: _itm fi;
+ _bb := if pair _itm: xpart _itm else: 0 fi;
+ nm.bar[incr nm] := rect ((_bb, 0), ( _ee, _ww) )
+ shifted (0, first + sep*(nm-1)) if vert: xyswap fi;
+ endfor
+endgroup
+enddef;
+
+% \end{macrocode}
+%
+%
+%^^A Overlays - taken from MFbook, p 295. (Bruce Leban)
+%
+%
+% \subsection{Overlays}\label{overlays}
+%
+% This final code predates me. When I inherited \mfpic{} it contained no
+% use of \gbc{keepit}. For the \MP{} version I just tried to make sure
+% everything was defined in \MP{} or \file{plain.mp} and otherwise left it
+% alone. One might presumably issue \gbc{keepit} periodically, clearing
+% \mfc{currentpicture} after adding it onto \gbc{totalpicture}. This would
+% save some memory because manipulating \mfc{currentpicture} often
+% requires a couple of copies to be around simultaneously. Right now we
+% treat currentpicture as the place to save things and manipulate local
+% picture variables.
+% \begin{macrocode}
+picture totalpicture;
+boolean totalnull, currentnull;
+def clearit =
+ currentpicture := totalpicture := nullpicture;
+ currentnull := totalnull := true;
+enddef;
+
+def keepit =
+ addto totalpicture also currentpicture;
+%<MF> mono (totalpicture);
+ currentpicture := nullpicture;
+ totalnull := totalnull or currentnull;
+ currentnull := true;
+enddef;
+
+def addto_currentpicture =
+ currentnull := false;
+ addto currentpicture
+enddef;
+
+def mergeit (text do) =
+ if totalnull:
+ do currentpicture
+ elseif currentnull:
+ do totalpicture
+ else:
+ begingroup
+ save _v_; picture _v_;
+ _v_ := currentpicture;
+ addto _v_ also totalpicture;
+ do _v_
+ endgroup
+ fi
+enddef;
+
+% \end{macrocode}
+% This implements \mfpic{}'s \cs{stopshipping} and \cs{resumeshipping}
+% commands. It used to be that those commands wrote a redefinition of
+% \gbc{shipit}, now we write a boolean assignment and the \cs{shipit}
+% command tests it.
+% \begin{macrocode}
+boolean noship; noship := false;
+def shipit = if noship: else: mergeit (shipout) fi enddef;
+
+%<*MF>
+def showit_ = mergeit (show_) enddef;
+def show_ suffix v = display v inwindow currentwindow enddef;
+
+%</MF>
+% \end{macrocode}
+%
+% Here we initialize \gbc{gcode} (which current versions of mfpic do not
+% use) for hacked \mfpic{} files that require it. In \MP{} we load the
+% color definitions. And thats all.
+% \begin{macrocode}
+numeric gcode; gcode := 0;
+
+%<MF>% end grafbase.mf
+%<MP>input dvipsnam.mp;
+%<MP>% end grafbase.mp
+%</MF|MP>
+% \end{macrocode}
+%
+%
+% \subsection{Dvips names for colors}\label{dvipsnam}
+%
+% In order to make \file{dvipsnam.mp} useful outside grafbase, we give
+% here a definition for \gbc{cmyk} when \gbc{grafbaseversion} is unknown.
+% \begin{macrocode}
+%<*dvips>
+if unknown grafbaseversion:
+ if unknown mpversion:
+ let cmykcolor=color;
+ vardef cmyk (expr c, m, y, k) =
+ (max(1-c-k,0), max(1-m-k,0), max(1-y-k,0))
+ enddef;
+ else:
+ vardef cmyk (expr c, m, y, k) = (c, m, y, k) enddef;
+ fi
+fi
+
+% \end{macrocode}
+%
+% The following code was borrowed from the the standard \LaTeX{} graphics
+% package (\file{dvipsname.def} by David Carlisle and Sebastian Rahtz). In
+% fact it was mostly generated automatically by some editor macros that
+% replaced \prog{graphics} package code with the \grafbase{} code.
+%
+%^^A This file may be distributed under the terms of the LaTeX Project Public
+%^^A License, as described in \file{lppl.txt} in the base LaTeX
+%^^A distribution, either version 1.0 or, at your option, any later version.
+%
+% Declare all the dvips color names to be color variables, and define
+% them as in \file{dvipsnam.def}:
+% \begin{macrocode}
+cmykcolor Apricot, Aquamarine, Bittersweet, Black, Blue, BlueGreen,
+BlueViolet, BrickRed, Brown, BurntOrange, CadetBlue, CarnationPink,
+Cerulean, CornflowerBlue, Cyan, Dandelion, DarkOrchid, Emerald,
+ForestGreen, Fuchsia, Goldenrod, Gray, Green, GreenYellow, JungleGreen,
+Lavender, LimeGreen, Magenta, Mahogany, Maroon, Melon, MidnightBlue,
+Mulberry, NavyBlue, OliveGreen, Orange, OrangeRed, Orchid, Peach,
+Periwinkle, PineGreen, Plum, ProcessBlue, Purple, RawSienna, Red,
+RedOrange, RedViolet, Rhodamine, RoyalBlue, RoyalPurple, RubineRed,
+Salmon, SeaGreen, Sepia, SkyBlue, SpringGreen, Tan, TealBlue, Thistle,
+Turquoise, Violet, VioletRed, White, WildStrawberry, Yellow,
+YellowGreen, YellowOrange;
+
+Apricot := cmyk(0,0.32,0.52,0);
+Aquamarine := cmyk(0.82,0,0.30,0);
+Bittersweet := cmyk(0,0.75,1,0.24);
+Black := cmyk(0,0,0,1);
+Blue := cmyk(1,1,0,0);
+BlueGreen := cmyk(0.85,0,0.33,0);
+BlueViolet := cmyk(0.86,0.91,0,0.04);
+BrickRed := cmyk(0,0.89,0.94,0.28);
+Brown := cmyk(0,0.81,1,0.60);
+BurntOrange := cmyk(0,0.51,1,0);
+CadetBlue := cmyk(0.62,0.57,0.23,0);
+CarnationPink := cmyk(0,0.63,0,0);
+Cerulean := cmyk(0.94,0.11,0,0);
+CornflowerBlue := cmyk(0.65,0.13,0,0);
+Cyan := cmyk(1,0,0,0);
+Dandelion := cmyk(0,0.29,0.84,0);
+DarkOrchid := cmyk(0.40,0.80,0.20,0);
+Emerald := cmyk(1,0,0.50,0);
+ForestGreen := cmyk(0.91,0,0.88,0.12);
+Fuchsia := cmyk(0.47,0.91,0,0.08);
+Goldenrod := cmyk(0,0.10,0.84,0);
+Gray := cmyk(0,0,0,0.50);
+Green := cmyk(1,0,1,0);
+GreenYellow := cmyk(0.15,0,0.69,0);
+JungleGreen := cmyk(0.99,0,0.52,0);
+Lavender := cmyk(0,0.48,0,0);
+LimeGreen := cmyk(0.50,0,1,0);
+Magenta := cmyk(0,1,0,0);
+Mahogany := cmyk(0,0.85,0.87,0.35);
+Maroon := cmyk(0,0.87,0.68,0.32);
+Melon := cmyk(0,0.46,0.50,0);
+MidnightBlue := cmyk(0.98,0.13,0,0.43);
+Mulberry := cmyk(0.34,0.90,0,0.02);
+NavyBlue := cmyk(0.94,0.54,0,0);
+OliveGreen := cmyk(0.64,0,0.95,0.40);
+Orange := cmyk(0,0.61,0.87,0);
+OrangeRed := cmyk(0,1,0.50,0);
+Orchid := cmyk(0.32,0.64,0,0);
+Peach := cmyk(0,0.50,0.70,0);
+Periwinkle := cmyk(0.57,0.55,0,0);
+PineGreen := cmyk(0.92,0,0.59,0.25);
+Plum := cmyk(0.50,1,0,0);
+ProcessBlue := cmyk(0.96,0,0,0);
+Purple := cmyk(0.45,0.86,0,0);
+RawSienna := cmyk(0,0.72,1,0.45);
+Red := cmyk(0,1,1,0);
+RedOrange := cmyk(0,0.77,0.87,0);
+RedViolet := cmyk(0.07,0.90,0,0.34);
+Rhodamine := cmyk(0,0.82,0,0);
+RoyalBlue := cmyk(1,0.50,0,0);
+RoyalPurple := cmyk(0.75,0.90,0,0);
+RubineRed := cmyk(0,1,0.13,0);
+Salmon := cmyk(0,0.53,0.38,0);
+SeaGreen := cmyk(0.69,0,0.50,0);
+Sepia := cmyk(0,0.83,1,0.70);
+SkyBlue := cmyk(0.62,0,0.12,0);
+SpringGreen := cmyk(0.26,0,0.76,0);
+Tan := cmyk(0.14,0.42,0.56,0);
+TealBlue := cmyk(0.86,0,0.34,0.02);
+Thistle := cmyk(0.12,0.59,0,0);
+Turquoise := cmyk(0.85,0,0.20,0);
+Violet := cmyk(0.79,0.88,0,0);
+VioletRed := cmyk(0,0.81,0,0);
+White := cmyk(0,0,0,0);
+WildStrawberry := cmyk(0,0.96,0.39,0);
+Yellow := cmyk(0,0,1,0);
+YellowGreen := cmyk(0.44,0,0.74,0);
+YellowOrange := cmyk(0,0.42,1,0);
+
+% End of file `dvipsnam.mp'.
+%</dvips>
+% \end{macrocode}
+% \clearpage
+%\Finale
diff --git a/graphics/mfpic/source/mfpic.dtx b/graphics/mfpic/source/mfpic.dtx
new file mode 100644
index 0000000000..586d051e18
--- /dev/null
+++ b/graphics/mfpic/source/mfpic.dtx
@@ -0,0 +1,6735 @@
+% \iffalse
+% File: mfpic.dtx
+% A part of mfpic 1.10 2012/12/03
+%
+% -------------------------------------------------------------------
+%
+% Copyright 2002--2012, Daniel H. Luecking
+%
+% Mfpic may be distributed and/or modified under the conditions of the
+% LaTeX Project Public License, either version 1.3c of this license or (at
+% your option) any later version. The latest version of this license is in
+% <http://www.latex-project.org/lppl.txt>
+% and version 1.3c or later is part of all distributions of LaTeX version
+% 2008/12/01 or later.
+%
+% Mfpic has maintenance status "author-maintained". The Current Maintainer
+% is Daniel H. Luecking. There are several Base Interpreters: plain TeX, LaTeX,
+% plain Metafont and plain MetaPost.
+%
+%<*driver>
+\ProvidesFile{mfpic.dtx}
+%</driver>
+%<sty>\ProvidesPackage{mfpic}
+%<*sty>
+ [2012/12/03 v1.10. Macros for drawing with Metafont/MetaPost.]%
+%</sty>
+%<*driver>
+\documentclass{ltxdoc}
+
+\addtolength{\textwidth}{.5878pt}
+
+\def\mytt{\upshape\mdseries\ttfamily}
+\renewcommand\marg[1]{{\mytt\{#1\}}}
+\renewcommand\oarg[1]{{\mytt [#1]}}
+\renewcommand\parg[1]{{\mytt (#1)}}
+\renewcommand \arg[1]{{\mytt \##1}}
+\renewcommand\#{\char`\#\relax}
+\renewcommand{\meta}[1]{{$\langle$\rmfamily\itshape#1\/$\rangle$}}
+\newcommand{\mmarg}[1]{\marg{\meta{#1}}}
+\DeclareRobustCommand\cs[1]{{\mytt\char`\\#1}}
+\def\prog#1{{\mdseries\scshape #1}}
+\def\grafbase{\prog{grafbase}}
+\def\Grafbase{\prog{Grafbase}}
+\def\mfpic{\prog{mfpic}}
+\def\Mfpic{\prog{Mfpic}}
+\def\MF{\prog{meta\-font}}
+\def\MP{\prog{meta\-post}}
+\def\CMF{\prog{Meta\-font}}
+\def\CMP{\prog{Meta\-post}}
+\def\opt#1{{\sffamily\upshape#1}}
+\def\mfc#1{{\mytt#1}}
+\let\env\mfc
+\let\file\mfc
+\let\gbc \mfc
+\catcode`\_=12
+\renewcommand\{{\char`\{}
+\renewcommand\}{\char`\}}
+\renewcommand\|{${}\mathrel{|}{}$}
+
+\makeatletter
+\let\HD@SetMacroIndent\@gobble
+\newcommand\bsl{{\mytt\@backslashchar}}
+% Stupid lists!
+\def\@listi{\leftmargin\leftmargini
+ \parsep \z@ \@plus\p@ \@minus\z@
+ \topsep 4\p@ \@plus\p@ \@minus2\p@
+ \itemsep\parsep}
+\let\@listI\@listi \@listi
+\renewcommand\labelitemi{\normalfont\bfseries \textendash}
+\renewcommand\labelitemii{\textasteriskcentered}
+\renewcommand\labelitemiii{\textperiodcentered}
+\leftmargini\parindent
+% Stupid index!
+\def\usage#1{\textrm{#1}}
+\def\index@prologue{\section*{Index}\markboth{Index}{Index}%
+ Numbers refer to the page where the corresponding entry is described.
+}
+\def\IndexParms{%
+ \parindent \z@ \columnsep 15pt
+ \parskip 0pt plus 1pt
+ \rightskip 5pt plus2em \mathsurround \z@
+ \parfillskip-5pt \small
+ % less hanging:
+ \def\@idxitem{\par\hangindent 20pt}%
+ \def\subitem{\@idxitem\hspace*{15pt}}%
+ \def\subsubitem{\@idxitem\hspace*{25pt}}%
+ \def\indexspace{\par\vspace{10pt plus 2pt minus 3pt}}}
+\makeatother
+
+\def\pdfTeX{\textrm{pdf\kern.04em\TeX}}
+\def\pdfLaTeX{\textrm{pdf\kern.06em\LaTeX}}
+\def\ConTeXt{\textrm{Con\kern-.16em\TeX\kern-0.06em t}}
+\def\PiCTeX{\textrm{P\kern-.13em\lower.3ex\hbox{I}C\TeX}}
+
+\title{The \mfpic{} package\thanks{This file has version number
+ \fileversion, last revised \filedate. The code described here
+ was developed by several people, notably Thomas Leathrum,
+ Geoffrey Tobin and Dan Luecking. Dan wrote this documentation.}}
+\author{Dan Luecking}
+\date{\filedate}
+\DisableCrossrefs
+\CodelineIndex
+\AlsoImplementation
+
+\begin{document}
+ \DeleteShortVerb{\|}
+ \DocInput{mfpic.dtx}
+\end{document}
+%</driver>
+%\fi
+% \CheckSum{5188}
+% \CharacterTable
+% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+% Digits \0\1\2\3\4\5\6\7\8\9
+% Exclamation \! Double quote \" Hash (number) \#
+% Dollar \$ Percent \% Ampersand \&
+% Acute accent \' Left paren \( Right paren \)
+% Asterisk \* Plus \+ Comma \,
+% Minus \- Point \. Solidus \/
+% Colon \: Semicolon \; Less than \<
+% Equals \= Greater than \> Question mark \?
+% Commercial at \@ Left bracket \[ Backslash \\
+% Right bracket \] Circumflex \^ Underscore \_
+% Grave accent \` Left brace \{ Vertical bar \|
+% Right brace \} Tilde \~}
+%
+% \GetFileInfo{mfpic.dtx}
+% \maketitle
+%
+% \begin{abstract}
+% An environment, \env{mfpic}, is defined, within which commands can be
+% used for drawing pictures, especially of mathematical objects and
+% functions. These commands actually write code to an external file to be
+% processed through \MF{} or \MP{} (depending on package options) to
+% produce the picture. The picture is then included on a second pass
+% through \TeX.
+%
+% This file documents the \mfpic{} source code. The user manual for
+% \mfpic{} is distributed as \file{mfpic-doc.pdf}, produced from
+% \file{mfpic-doc.tex}. An introductory guide to \mfpic{} is available
+% in \file{mfpguide.pdf}, produced from \file{mfpguide.tex}
+% \end{abstract}
+%
+% \StopEventually{\PrintIndex}
+% \tableofcontents
+%
+%
+%
+% \section{Introductory setup}\label{intro}
+%
+% \Mfpic{} is designed to work with plain \TeX, \LaTeX, \pdfTeX, and
+% \pdfLaTeX. I consistently use it with eplain and AMS\LaTeX{}
+% without problems. Users have reported succesfully using it with \AmSTeX.
+% It has \emph{not} been successfully used with \ConTeXt.
+%
+% \subsection{Versions numbers, category codes, etc.}\label{category}
+%
+% Almost all the various category code settings we do were prompted by
+% an \mfpic{} user who had sent a bug report and had to send along his
+% modified version of \file{mfpic.tex}. It had been modified so that
+% \cs{message}\,s and \cs{write}\,s would do the right thing in latex
+% files where various punctuation were made active (e.g., with
+% \file{french.sty}).
+%
+% We keep track of version numbers here. We store the version and date
+% in macros and later we write this version information to the
+% output file, The \cs{mfpicversion} is 100 times the numeric part of the
+% version. The output file can compare this number with the corresponding
+% value in \file{grafbase} and detect installations where there is a
+% mismatch.
+% \begin{macrocode}
+%<*tex>
+\ifx\mfpfileversion\UndEfInEd\else\expandafter\endinput\fi%
+{%
+ \catcode\lq\.12 \catcode\lq\/12%
+ \gdef\mfpfileversion{1.10}%
+ \gdef\mfpfiledate{2012/12/03}%
+ \gdef\mfpicversion{110}%
+}%
+% \end{macrocode}
+%
+% A bug in older versions of some support files is triggered by the macro
+% \cs{+}, which is an \cs{outer} macro in plain \TeX. Here we save the
+% current definition and then give it a nonouter meaning.
+% \begin{macrocode}
+\let\mfpsaveplus\+%
+\let\+\tabalign%
+% \end{macrocode}
+%
+% We now get rid of unusually catcoded punctuation, space and EOL.
+% We assume that letters, numbers, `\texttt\bsl', `\texttt\{', `\texttt\}',
+% `\texttt\#' and `\texttt\%' have the usual categories. However, styles
+% (e.g., \prog{french}) that make punctuation active foul up \cs{write}\,s
+% of MF/MP code. This can still occur in user-generated \cs{write}\,s, so
+% maybe some of this code should be added in \cs{mfpic} or \cs{mfsrc}
+% command, too.
+% \begin{macrocode}
+{%
+ \catcode61\space12\relax% 61=equal sign
+ \catcode32=10\relax% 32=space
+ \catcode13=5 % 13=EOL
+ \catcode96=12 % 96=left quote
+ \catcode`\@=11
+ \gdef\MFPsavecodes{%
+ \edef\mfp@restoreNLC{\newlinechar\number\newlinechar\relax}%
+ \edef\mfp@restoreELC{\endlinechar\number\endlinechar\relax}%
+ \edef\mfp@restoreSPA{\catcode32=\number\catcode32\relax}%
+ \edef\mfp@restoreEOL{\catcode13=\number\catcode13\relax}%
+ \edef\mfp@restoreALL{\mfp@restorecode{=}\mfp@restorecode{@}%
+ \mfp@restorecode{$}\mfp@restorecode{:}\mfp@restorecode{;}%
+ \mfp@restorecode{.}\mfp@restorecode{,}\mfp@restorecode{!}%
+ \mfp@restorecode{?}\mfp@restorecode{(}\mfp@restorecode{)}%
+ \mfp@restorecode{[}\mfp@restorecode{]}\mfp@restorecode{<}%
+ \mfp@restorecode{>}\mfp@restorecode{*}\mfp@restorecode{/}%
+ \mfp@restorecode{+}\mfp@restorecode{-}\mfp@restorecode{"}%
+ \mfp@restorecode{'}\mfp@restorecode{_}\mfp@restorecode{`}%
+ \mfp@restoreEOL\mfp@restoreSPA\catcode10=\number\catcode10
+ \mfp@restoreNLC\mfp@restoreELC}}%
+ \gdef\mfp@restorecode#1{\catcode`\string#1=\number\catcode`#1\relax}%
+ \ifx\@makeother\UndEfInEd \gdef\@makeother#1{\catcode`#1=12\relax}\fi
+ \gdef\MFPsanitize{\endlinechar13 \newlinechar10
+ \catcode10=12 \catcode13=5 \catcode`\$=3 \catcode`\ =10
+ \@makeother\:\@makeother\;\@makeother\.\@makeother\,%
+ \@makeother\!\@makeother\?\@makeother\(\@makeother\)%
+ \@makeother\[\@makeother\]\@makeother\>\@makeother\<%
+ \@makeother\/\@makeother\*\@makeother\+\@makeother\-%
+ \@makeother\'\@makeother\"\@makeother\_\@makeother\`%
+ \@makeother\=\@makeother\@}%
+}%
+\MFPsavecodes\MFPsanitize
+\catcode`@11
+\chardef\mfp@ctrlJ 10 \chardef\mfp@ctrlM 13
+% \end{macrocode}
+%
+% \cs{MFPsavecodes} saves a bunch of things that must have (mostly)
+% normal settings. \cs{mfp@restoreALL} will be invoked at the end of the
+% file to restore all the values.
+%
+% \cs{MFPsanitize} sets (almost) everything to normal. It can be used to
+% redo these settings inside some other macro, should that ever become
+% necessary. Characters 10 and 13 are \verb|^^J| and \verb|^^M|,
+% control-J and control-M.
+%
+%
+% \subsection{Debugging and messages}\label{debug}
+%
+% \DescribeMacro{\mfpicdebug}
+% To turn on debugging before option processing, the user must make sure
+% \cs{mfpicdebug} is defined. \cs{mfp@DBlog} writes it contents to the
+% log file only if debugging is on.
+% \DescribeMacro{\ifmfpicdebug}
+% The switch \cs {ifmfpicdebug} is normally set to true if the user loads
+% \mfpic{} with the \opt{debug} option, or sets it directly with
+% \cs{mfpicdebugtrue} after loading.
+% \begin{macrocode}
+\newif\ifmfpicdebug
+\ifx\mfpicdebug\UndEfInEd \mfpicdebugfalse
+\else \mfpicdebugtrue
+\fi
+\def\mfp@DBlog#1{%
+ \ifmfpicdebug{\let\@msgbreak\mfp@errbrk\wlog{Mfpic debug: #1}}\fi}%
+% \end{macrocode}
+%
+% We need to write various kinds of messages to the screen and log file.
+% \cs{mfp@msg} is only used in the definition of the others and to write a
+% blank line. We always use the second, which identifies the source of the
+% message, or the third, which adds the word `warning'. The fourth is for
+% errors and provides for help if a user presses `\texttt h' in response
+% to the message. The bit with the spaces is to get a hanging indentation
+% with text properly aligned on screen. The basic \cs{mfp@msg} defines
+% \cs{@nl} to place the \cs{newlinechar}, for multiline messages. We
+% also define \cs{@nl} in \cs{mfp@write} for new lines in writing to a
+% file. We do not define it at all at the outer level.
+% \begin{macrocode}
+\def\mfp@msg#1{%
+ {\newlinechar\mfp@ctrlJ\def\@nl{^^J}\immediate\write16{#1}}}%
+\def\mfp@spaces{\space\space\space\space\space\space}%
+\def\mfp@msgbrk{^^J\mfp@spaces\space}%
+\def\mfp@errbrk{\mfp@msgbrk\mfp@spaces}%
+\def\mfp@warnbrk{\mfp@errbrk\space\space}%
+\def\Mfpic@msg#1{{\let\@msgbreak\mfp@msgbrk\mfp@msg{Mfpic: #1}}}%
+\def\Mfpic@warn#1{%
+ {\let\@msgbreak\mfp@warnbrk\mfp@msg{\@nl Mfpic warning: #1}}}%
+\def\mfp@errmsg#1#2{{\newlinechar\mfp@ctrlJ\let\@msgbreak\mfp@errbrk
+ \errhelp{#2}\errmessage{Mfpic error: #1}}}%
+% \end{macrocode}
+%
+% In \TeX{} 3.x we can identify the line being processed when a message is
+% issued. We do not support \TeX{} 2.x. The command \cs{mfp@lineno} is
+% for information written into the output file.
+% \begin{macrocode}
+\def\mfp@lineno{line \number\inputlineno\space in the TeX source}%
+% \end{macrocode}
+%
+%
+% \subsection{Detecting the format}\label{format}
+%
+% To detect its environment, \mfpic{} needs to test whether certain
+% commands are defined. We usually use the positive version
+% \cs{mfp@ifdefined}. The \cs{@xp} in it is used to `expand away' the rest
+% of a conditional. \cs{@XP} does this from two levels deep in a
+% conditional. Since our check uses \cs{@firstoftwo} and
+% \cs{@secondoftwo}, we cannot use it to check \emph{their} existence (but
+% we could use other means).
+%
+% \cs{mfp@ifundefined} checks its argument to see if it is defined.
+% If so, it gobbles all code up to \cs{mfp@enddef}. Otherwise that code is
+% executed. This avoids reading code prematurely (category code changes).
+%
+% We also copy \LaTeX's \cs{@ifundefined} since we have at least five
+% places where we need to supply the name without a backslash.
+% \begin{macrocode}
+\let\@xp\expandafter
+\def\@XP{\@xp\@xp\@xp}%
+\long\def\@firstoftwo#1#2{#1}%
+\long\def\@secondoftwo#1#2{#2}%
+\def\mfp@ifdefined#1{%
+ \ifx#1\UndEfInEd \@xp\@secondoftwo
+ \else\ifx#1\relax \@XP\@secondoftwo
+ \else \@XP\@firstoftwo
+ \fi\fi}%
+\mfp@ifdefined\@empty{}{\def\@empty{}}%
+\mfp@ifdefined\@gobble{}{\def\@gobble#1{}}%
+\mfp@ifdefined\@firstofone{}{\long\def\@firstofone#1{#1}}%
+\mfp@ifdefined\on@line{}%
+ {\def\on@line{ on input line \number\inputlineno}}%
+\let\mfp@enddef\@empty
+\long\def\mfp@gobbledef#1\mfp@enddef{}%
+\def\mfp@ifundefined#1{\mfp@ifdefined#1\mfp@gobbledef\@empty}%
+\mfp@ifundefined\@ifundefined
+ \def\@ifundefined#1{%
+ \@xp\ifx\csname #1\endcsname\relax \@xp\@firstoftwo
+ \else \@xp\@secondoftwo
+ \fi}%
+\mfp@enddef
+% \end{macrocode}
+%
+% The following might be useful in patching things or adding to
+% \cs{mfp@additions}.
+% \begin{macrocode}
+\mfp@ifundefined\g@addto@macro
+ \long\def\g@addto@macro#1#2{%
+ \begingroup
+ \toks@\@xp{#1#2}%
+ \xdef#1{\the\toks@}%
+ \endgroup}%
+\mfp@enddef
+\mfp@ifundefined\g@prependto@macro
+ \long\def\g@prependto@macro#1#2{%
+ \begingroup
+ \toks@{#2}\toks2\@xp{#1}%
+ \xdef#1{\the\toks@\the\toks2}%
+ \endgroup}%
+\mfp@enddef
+% \end{macrocode}
+%
+% We test for LaTeX. We assume plain\TeX{}, or something compatible,
+% except in a few places where LaTeX{} does thing differently. \Mfpic{}
+% didn't work in \ConTeXt{} the last few times I tried.
+%
+% Here we see if \cs{documentstyle} is defined, and then make sure
+% we are not in \AmSTeX{}. We used to detect \AmSTeX{} by checking
+% \cs{fmtname}. This leads to problems if some other format inputs
+% \file{amstex.tex} and redefined \cs{fmtname}. Thus we will now check
+% \cs{amstexloaded@}. Unfortunately, \file{amstex.tex} sets it equal to
+% \cs{relax}, which is much too easy to be the result of some other
+% attempt to check it. Thus, we add an additional check for
+% \cs{NoBlackBoxes}.
+% \begin{macrocode}
+\newif\ifmfp@latex \mfp@latexfalse
+\mfp@ifdefined\documentstyle
+ {\ifx \amstexloaded@\relax % amstex loaded, or a bad check for it
+ \ifx\NoBlackBoxes\UndEfInEd % amstex not loaded
+ \mfp@DBlog{LaTeX detected.}\mfp@latextrue
+ \else % most likely amstex loaded
+ \mfp@DBlog{AmS-TeX detected.}%
+ \fi
+ \else % amstex not loaded
+ \mfp@DBlog{LaTeX detected.}\mfp@latextrue
+ \fi}%
+ {\mfp@DBlog{Neither LaTeX nor AmS-TeX detected.}}%
+% \end{macrocode}
+%
+% We are in \LaTeXe{}, as opposed to \LaTeX2.09, if \cs{documentclass}
+% is defined. The \file{mfpic.sty} now assumes \LaTeXe{}. It might still
+% be possible to use \LaTeX{}2.09 with \cs{input mfpic}, but I don't
+% care enough to test it.
+% \begin{macrocode}
+\newif\ifmfp@latexe \mfp@latexefalse
+\mfp@ifdefined\documentclass
+ {\mfp@latexetrue \mfp@DBlog{LaTeX2e detected.}}%
+ {\mfp@DBlog{Not LaTeX2e.}}%
+% \end{macrocode}
+%
+% We now test \cs{pdfoutput} to see if pdf support is needed.
+% \begin{macrocode}
+\newif\ifmfp@pdf \mfp@pdffalse
+\mfp@ifdefined\pdfoutput
+ {\mfp@DBlog{PdfTeX detected...}%
+ \ifnum\pdfoutput<1 \mfp@DBlog{...with dvi output.}%
+ \else \mfp@pdftrue \mfp@DBlog{...with PDF output.}%
+ \fi}%
+ {\mfp@DBlog{Not pdfTeX.}}%
+% \end{macrocode}
+%
+%
+% \subsection{Blank line and spacing hacks}\label{spacing}
+%
+% In order for commands to always read their arguments correctly,
+% we need EOLs to have their normal category codes, but then a
+% blank line will end a paragraph, and we would like blank lines to be
+% ignored inside an \env{mfpic} environment. Therefore we turn off
+% \cs{par} inside that environment, and then need(?) to turn it back on
+% inside \cs{tlabel} and \cs{tcaption}. \cs{@@par} (in \LaTeX) and
+% \cs{endgraf} (in plain \TeX) should equal the primitive \cs{par}.
+% \begin{macrocode}
+\ifmfp@latex \else \let\@@par\endgraf \fi
+\def\mfp@restorepar{\let\par\@@par}%
+% \end{macrocode}
+%
+% Some \mfpic{} commands end with \cs{ignorespaces}. For some others a
+% better choice is the \LaTeX{} strategy of conditionally ignoring spaces.
+% \begin{macrocode}
+\mfp@ifdefined\@namedef{}%
+ {\def\@namedef#1{\@xp\def\csname #1\endcsname}}%
+\mfp@ifdefined\@namelet{}%
+ {\def\@namelet#1{\@xp\let\csname #1\endcsname}}%
+\mfp@ifdefined\@nameuse{}%
+ {\def\@nameuse#1{\csname #1\endcsname}}%
+\mfp@ifundefined\@bsphack
+ \@nameuse{newskip}\@savsk
+ \@nameuse{newcount}\@savsf
+ \def\@bsphack{\relax
+ \ifhmode \@savsk\lastskip \@savsf\spacefactor \fi}%
+ \def\@esphack{\relax
+ \ifhmode \spacefactor\@savsf
+ \ifdim\@savsk>\z@ \ignorespaces
+ \fi\fi}%
+\mfp@enddef
+% \end{macrocode}
+%
+%
+% \subsection{Warning if commands are already defined}
+%
+% We sometimes (either by accident or by choice) redefine things defined
+% in some formats. For the accidental redefinitions we want to be told,
+% and since we cannot assume \LaTeX{} with its \cs{newcommand}, we define
+% a simple check and print a warning when it happens.
+% \begin{macrocode}
+\mfp@DBlog{Defining test for previous definitions of macros.}%
+\def\mfp@testdef#1{%
+ \ifx#1\UndEfInEd
+ \else\ifx#1\relax
+ \Mfpic@warn{\string#1 was previouly defined to be \string\relax.}%
+ \else
+ \Mfpic@warn{\string#1 is already defined.}%
+ \fi\fi}%
+% \end{macrocode}
+%
+% Is the proposed definer of new definitions itself not new? Well,
+% we'd like a warning, but go ahead and define it anyway. All
+% \cs{newdef} does is check whether the macro being defined is already
+% defined, raise a warning (but \emph{not} an error) and define it
+% anyway. We do something similar with \cs{let}.
+% \begin{macrocode}
+\mfp@testdef\newdef
+\def\newdef#1{\mfp@testdef#1\def#1}%
+\newdef\newlet#1{\mfp@testdef#1\let#1}%
+% \end{macrocode}
+%
+%
+% \subsection{Error messages and warnings}
+%
+% If one tries to switch on \MP{} support after the output file has been
+% set up, we issue this error message.
+% \begin{macrocode}
+\newdef\mfp@useMP@error{%
+ \mfp@errmsg {Command \mfp@b usemetapost too late}%
+ {It is too late to select the metapost option.^^J%
+ It must be selected before the \opengraphsfile command.^^J%
+ If you proceed, the metafont option will be assumed.}}%
+% \end{macrocode}
+%
+% This one is for macros that need an output file to write to, but are
+% called before one has been opened.
+% \begin{macrocode}
+\def\mfp@nooutput@error#1{%
+ \mfp@errmsg {No output file has been opened}%
+ {You have tried to used a command, #1, that requires an output^^J%
+ file to be opened with the \opengraphsfile command, which^^J%
+ was either not issued or was not successful.}}%
+% \end{macrocode}
+%
+% Some options and commands require the \opt{metapost} option be in
+% effect. We define an error message for the case where it is not in
+% effect. Rarely, we need a similar warning for the \opt{metafont}
+% option.
+% \begin{macrocode}
+\newdef\mfp@noMP@error#1{%
+ \mfp@errmsg {Metafont does not support #1, use MetaPost}%
+ {Metafont doesn't support #1. Perhaps you forgot to turn on^^J%
+ MetaPost support by using the metapost option or issuing the^^J%
+ command \usemetapost. For now, I will ignore it.}}%
+\newdef\mfp@noMF@warn#1{%
+ \Mfpic@warn
+ {Metapost does not support \string#1, it will be ignored.}}%
+% \end{macrocode}
+%
+% This next is like \cs{mfp@useMP@error}, but it is not so specific.
+% It is for other options that also need to be issued before
+% \cs{opengraphsfile}.
+% \begin{macrocode}
+\def\mfp@afteropen@error#1{%
+ \mfp@errmsg{Improper use of command \string#1\on@line}%
+ {You must issue the command #1 prior to \opengraphsfile.
+ It will be ignored now.}}%
+% \end{macrocode}
+%
+% \cs{mfp@end} is defined to be an error message, but that is for
+% debugging purposes. It's actual use is to mark the end when we need to
+% process a list of tokens. The code should simply detect and remove it,
+% and it should never be expanded, so seeing the error message indicates
+% a coding error by me, or incorrect syntax by a user.
+% \begin{macrocode}
+\newdef\mfp@end{%
+ \mfp@errmsg {Unexpected token \string\mfp@end}%
+ {If you get this message but did not misuse the command^^J%
+ \mfp@end, it could mean you have unbalanced braces.}}%
+% \end{macrocode}
+%
+% The next two are error messages for missing or empty data files.
+% \begin{macrocode}
+\newdef\mfp@nofile@error#1{%
+ \mfp@errmsg{No data file: #1}%
+ {The data file you tried to plot was not found.}}%
+\newdef\mfp@nodata@error#1{%
+ \mfp@errmsg{Empty data file: #1}%
+ {The data file contained only empty lines and comments.}}%
+% \end{macrocode}
+%
+% This message is issued when the \cs{tlabeljustify} command (see
+% subsection~\ref{labelsettings}) is issued with an invalid parameter.
+% \begin{macrocode}
+\newdef\mfp@justify@error{%
+ \mfp@errmsg{Invalid justification parameter for text label}%
+ {The optional argument for an mfpic text label must be^^J%
+ one of B, b, c or t, followed by^^J%
+ one of l, r or c, optionally followed by an angle.}}%
+% \end{macrocode}
+%
+% This is the default meaning assigned to \cs{@nountil} at the
+% beginning of \cs{mfploop}. Use of \cs{mfpuntil} cancels the
+% warning, which is otherwise issued by \cs{mfpendloop}.
+% \begin{macrocode}
+\newdef\mfp@untilwarn{%
+ \Mfpic@warn{\mfp@b mfploop without \string\mfpuntil!\@nl}}%
+% \end{macrocode}
+%
+%
+%
+% \section{Configuring \mfpic{} behavior}\label{configuring}
+%
+%
+% \subsection{Options}\label{options}
+%
+% \DescribeMacro{\ifmfpmpost}
+% I have decided to make this a user-level macro so that a user can
+% write code that behaves one way for \MP{} and another for \MF. However
+% the commands to change the setting are private.
+%
+% Our switch to turn \MP{} support on or off needs to be global, so it
+% can't be a normal \cs{newif}.
+% \begin{macrocode}
+\newlet\ifmfpmpost\iffalse
+\newdef\mfp@mposttrue {\global\let\ifmfpmpost\iftrue}%
+\newdef\mfp@mpostfalse{\global\let\ifmfpmpost\iffalse}%
+\newdef\mfp@ifmpost{%
+ \ifmfpmpost\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}%
+% \end{macrocode}
+%
+% \DescribeMacro{\usemetafont}
+% We make user-level commands for switching to the appropriate drawing
+% program. This is necessary for plain\TeX, but also used internally by
+% \mfpic. \CMF{} is the default. This command needs to not only clear the
+% boolean \cs{ifmfpmpost}, but also restore the default graphics inclusion
+% commands. Actually, it doesn't \emph{need} to unless \cs{usemetapost}
+% was previously issued to change the default. This would normally not
+% happen unless the user wanted to close the current graph file and open a
+% new one for \MF. For similar reasons, we clear the \opt{mplabels}
+% option.
+%
+% \MP{} support requires macros for including EPS graphics. What works
+% best depends on the format, so we delay the definition of
+% \cs{usemetapost} until the \LaTeX{} options section.
+%
+% Since \cs{mfpicllx} and \cs{mfpiclly} are globally defined by graphic
+% inclusion, we reset then to 0, in case \opt{metafont} and \opt{metapost}
+% options are mixed in one file.
+% \begin{macrocode}
+\newdef\usemetafont{\mfp@mpostfalse
+ \nomplabels
+ \def\preparemfpicgraphic##1{}%
+ \def\setmfpicgraphic##1{\mfp@graphfont\char\mfp@count}%
+ \def\getmfpicoffset##1{\gdef\mfpicllx{0}\gdef\mfpiclly{0}}}%
+% \end{macrocode}
+%
+% Some options need an open \file{.mf} or \file{.mp} file before they
+% can do anything. This means the \cs{opengraphsfile} command must have
+% been issued but not \cs{closegraphsfile}. Since those commands define and
+% undefine \cs{mfp@filename}, we use that as our test.
+% \begin{macrocode}
+\newdef\mfp@ifopengraphsfile{\mfp@ifdefined\mfp@filename}%
+% \end{macrocode}
+%
+% \DescribeMacro{\mfmode}
+% For use with the \opt{metafont} option, we allow the user to set the
+% \MF{} mode. Example: \verb$\mfmode{ljfour}$
+%
+% \DescribeMacro{\mfresolution}
+% If modes are too esoteric, the user can at least set the resolution.
+% This should be a pure number representing dots per inch. Example:
+% \verb$\mfresolution{600}$
+% \begin{macrocode}
+\newdef\mfmode#1{\edef\mfp@mode{#1}%
+ \mfp@ifopengraphsfile%
+ {\mfp@afteropen@error\mfmode}%
+ {\mfp@ifmpost{\mfp@noMF@warn\mfmode}{}}}%
+\newdef\mfresolution#1{\edef\mfp@resolution{#1}%
+ \mfp@ifopengraphsfile%
+ {\mfp@afteropen@error\mfresolution}%
+ {\mfp@ifmpost{\mfp@noMF@warn\mfresolution}{}}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\setmfvariable}
+% A useful abbreviation for a common requirement: declaring and setting
+% a \MF{} variable. I have decided to change it to a user-level macro for
+% power users.
+% \DescribeMacro{\setmpvariable}
+% For consistency, \cs{setmpvariable} is an alternative name for it. It
+% takes three arguments: the type of variable, the variable itself, and
+% its value. Example:\\
+% \indent \cs{setmfvariable}\marg{numeric}\marg{shadewd}\marg{.7pt}.\\
+% Since the \MF{} code \mfc{save}\,s the variable, there must be no
+% suffix. This is not a big problem since mostly it is for internal use.
+% We document that fact for power users and advise ordinary users on the
+% naming of \MF{} variables.
+%
+% Since \MP{} has three (or four) data types for color, we have to write
+% the command `\gbc{setcolor}' to the output file instead of
+% `\gbc{setvariable (color)}' so that the datatype can be determined
+% before declaring the variable. (Unlike the other data types the
+% \grafbase{} command \gbc{setcolor} is more complicated than
+% \gbc{setvariable(color)}.) For simplicity, we do this with all data
+% types. For example \cs{setmfvariable}\marg{numeric} writes
+% `\gbc{setnumeric}'. This makes spaces significant in the first argument,
+% so we strip them off the ends with \cs{mfp@sp@def} (defined in
+% section~\ref{optional}).
+%
+% We have abbreviations for the most common internal uses. For example
+% \SpecialUsageIndex{\setmfboolean}\cs{setmfboolean} is the equivalent of
+% \cs{setmfvariable} with type \mfc{boolean}. A similar statement is true
+% For \SpecialUsageIndex{\setmfnumeric}\cs{setmfnumeric},
+% \SpecialUsageIndex{\setmfpair}\cs{setmfpair}, and
+% \SpecialUsageIndex{\setmfcolor}\cs{setmfcolor}.
+%
+% \DescribeMacro{\globalsetmfvariable}
+% This is like \cs{setmfvariable}, except it doesn't perform a
+% \mfc{save} and so the variable is globally defined. Also,
+% suffixed variables are permitted (but not numeric suffixes).
+% Here, too, we have to write a different command for setting color.
+% Thus, me need to test whether the first argument is `color'. Here again,
+% spaces are significant.
+% \begin{macrocode}
+\newdef\setmfvariable#1{%
+ \mfp@sp@def\mfp@tempa{#1}\@setmfvariable\mfp@tempa}%
+\newdef\@setmfvariable#1#2#3{%
+ \mfp@ifopengraphsfile%
+ {\mfcmd{set#1 (#2) #3}}%
+ {\mfp@nooutput@error\setmfvariable}}%
+\newdef\setmfboolean{\@setmfvariable{boolean}}%
+\newdef\setmfnumeric{\@setmfvariable{numeric}}%
+\newdef\setmfpair {\@setmfvariable{pair}}%
+\newdef\setmfcolor {\@setmfvariable{color}}%
+\newdef\globalsetmfvariable#1#2#3{%
+ \mfp@sp@def\mfp@tempa{#1}%
+ \mfp@ifopengraphsfile
+ {\mfcmd{\ifx\mfp@tempa\mfp@C gsetcolor \else gsetvariable (#1) \fi
+ (#2) #3}}%
+ {\mfp@nooutput@error\globalsetmfvariable}}%
+\newlet\setmpvariable\setmfvariable
+\newlet\globalsetmpvariable\globalsetmfvariable
+% \end{macrocode}
+%
+% The command \cs{setmfvariable} arranges for the value of
+% that variable to be localized to each figure if set inside one, but
+% it is global otherwise. We have to be careful that \TeX{} and \MF{}
+% stay in synchrony, as they have different understandings as to what is
+% global and what is not. So we define a switch that will be set at the
+% start of an \env{mfpic} environment to test if a command is issued
+% inside one. The command \cs{mfp@local} tests this switch and makes the
+% following assignment global if it fails. Then we have two shortcuts for
+% the more commonly occuring cases.
+%
+% We also define here \texttt{true}, \texttt{false} and \texttt{color} as
+% macros. These are strings we need to write to the output frequently.
+% \begin{macrocode}
+\newlet\ifin@mfpicenv\iffalse
+\newdef\mfp@local{\relax\ifin@mfpicenv\else\global\fi}%
+\newdef\mfp@let{\mfp@local\let}%
+\newdef\mfp@def{\mfp@local\def}%
+\newdef\mfp@T{true}%
+\newdef\mfp@F{false}%
+\newdef\mfp@C{color}%
+% \end{macrocode}
+%
+% \DescribeMacro{\usecenteredcaptions}
+% This (\opt{centeredcaptions}) is the simplest option. Other options
+% follow a similar pattern: a user-level command \cs{usecenteredcaptions}
+% is defined for turning it on and another,
+% \DescribeMacro{\nocenteredcaptions}
+% \cs{nocenteredcaptions}, is defined for turning it off. Selection of
+% centered captions can be turned on or off at will throughout a document.
+% \begin{macrocode}
+\newdef\usecenteredcaptions{\mfp@let\ifmfp@centcapt\iftrue}%
+\newdef\nocenteredcaptions {\mfp@let\ifmfp@centcapt\iffalse}%
+\nocenteredcaptions
+% \end{macrocode}
+%
+% \DescribeMacro{\useraggedcaptions}
+% Here we define the settings for justified and ragged captions.
+% Justified captions are the default. Ragged captions are normally
+% raggedright, but are ragged on both sides if \opt{centeredcaptions} is
+% in effect.
+% \DescribeMacro{\noraggedcaptions}
+% For justified captions, lines are justified on both sides, except the
+% last is centered under \opt{centeredcaptions}.
+% \begin{macrocode}
+\newdef\useraggedcaptions{\mfp@let\mfp@capsettings\mfp@raggedcap}%
+\newdef\noraggedcaptions{\mfp@let\mfp@capsettings\mfp@justifiedcap}%
+\newdef\mfp@raggedcap{%
+ \ifmfp@latex
+ \ifmfp@centcapt \centering \else \raggedright \fi
+ \else
+ \rightskip 0pt plus 1fil\relax
+ \ifmfp@centcapt
+ \leftskip 0pt plus 1fil
+ \parfillskip 0pt plus 1fil\relax
+ \fi
+ \def\\{\unskip\hbox{}\penalty-\@M\ignorespaces}%
+ \fi}%
+\def\mfp@justifiedcap{%
+ \ifmfp@centcapt
+ \leftskip 0pt plus 0.5fil
+ \rightskip 0pt plus -0.5fil
+ \parfillskip 0pt plus 1fil\relax
+ \fi
+ \def\\{\unskip\hbox{}\hfil\penalty-\@M\ignorespaces}}%
+\noraggedcaptions
+% \end{macrocode}
+%
+% This common code is repeated almost verbatim four times, so we define
+% the following abbreviation. The first two arguments are the commands
+% to turn the option on and off. The third and fourth are the \MF{} and
+% \TeX{} booleans to switch. In all cases, the default for the option is
+% \emph{off}, so we issue the second command after it is defined.
+%
+% Since the output file has not been opened while \mfpic{} is loading, the
+% \MF{} boolean is not set, but the \cs{opengraphsfile} code will check
+% the \TeX{} boolean and write the appropriate \MF{} code.
+% \begin{macrocode}
+\newdef\mfp@makeoption#1#2#3#4{%
+ \newdef#1{\mfp@let#4\iftrue
+ \mfp@ifopengraphsfile{\setmfboolean{#3}\mfp@T}{}}%
+ \newdef#2{\mfp@let#4\iffalse
+ \mfp@ifopengraphsfile{\setmfboolean{#3}\mfp@F}{}}%
+ #2}%
+% \end{macrocode}
+%
+% \DescribeMacro{\clipmfpic}
+% The \opt{clip} option shows only what's inside the rectangle given in
+% the \cs{mfpic} command. It can be turned on or off for each figure
+% independently.
+% \DescribeMacro{\noclipmfpic}
+% The default is the old behavior: no clipping.
+% \begin{macrocode}
+\mfp@makeoption\clipmfpic\noclipmfpic{clipall}\ifmfp@clip
+% \end{macrocode}
+%
+% \DescribeMacro{\usetruebbox}
+% The \opt{truebbox} option lets \MP{} set the true bounding box (which
+% may differ from the numbers defined through \cs{mfpic}).
+% \DescribeMacro{\notruebbox}
+% The default is the old behavior, \cs{notruebbox}.
+%
+% We don't need to be too careful with the scope of these option
+% commands. As currently written, \TeX{} never needs to know the setting
+% except to pass the information on to \cs{opengraphsfile} if the option
+% is set before then. However, we treat it like most other settings,
+% turning on or off an appropriate \TeX{} boolean, in case it ever becomes
+% necessary to know the setting. It is local to any \env{mfpic}, global
+% otherwise. The default is to have \opt{truebbox} turned off.
+% \begin{macrocode}
+\mfp@makeoption\usetruebbox\notruebbox{truebbox}\ifmfp@truebbox
+% \end{macrocode}
+%
+% \DescribeMacro{\usemplabels}
+% An option to let \MP{} create labels, \opt{mplabels} mostly just
+% switches between two versions of \cs{tlabel} so no \MP{} booleans
+% are set and so synchronization of scope is not an issue.
+% \DescribeMacro{\nomplabels}
+% The default is the old behavior: labels are placed by \TeX. The actual
+% setting is delayed until \cs{opengraphsfile} so the order of these
+% commands and \cs{usemetapost} is not significant (before
+% \cs{opengraphsfile}).
+%
+% Because of the frequent need to check this setting, we define
+% \cs{@ifmplabels} to execute one of two alternatives based on it.
+% When writing \TeX{} code to the file we need to make sure \texttt\#
+% is not doubled. We do this by issuing \cs{mp@mksharpother} before
+% reading the argument that will be written.
+% \begin{macrocode}
+\newdef\usemplabels{\mfp@let\ifmfp@mplabels\iftrue
+ \mfp@ifopengraphsfile
+ {\mfp@ifmpost{}%
+ {\mfp@noMP@error{mplabels}\global\let\ifmfp@mplabels\iffalse}}{}}%
+\newdef\nomplabels{\mfp@let\ifmfp@mplabels\iffalse}%
+\newdef\@ifmplabels{%
+ \ifmfp@mplabels\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}%
+\nomplabels
+\newdef\mp@mksharpother{\@ifmplabels{\@makeother\#}{}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\overlaylabels}
+% The option \opt{overlaylabels} instructs \MP{} to defer adding labels
+% until the end of the picture. That means they don't get clipped or
+% covered up by any drawing elements.
+% \DescribeMacro{\nooverlaylabels}
+% The default is the old behavior, \cs{nooverlaylabels}. Neither of these
+% commands affect \MF, but they write to the output file anyway if issued
+% after \cs{opengraphsfile}.
+% \begin{macrocode}
+\mfp@makeoption%
+ \overlaylabels\nooverlaylabels{overlaylabels}\ifmfp@overlabels
+% \end{macrocode}
+%
+% \DescribeMacro{\clearsymbols}
+% The option \opt{clearsymbols} causes the symbols drawn by
+% \cs{plotsymbol} and \cs{plotnodes} to have their interiors erased before
+% being drawn.
+% \DescribeMacro{\noclearsymbols}
+% The default is the old behavior: \cs{noclearsymbols}.
+% \begin{macrocode}
+\mfp@makeoption%
+ \clearsymbols\noclearsymbols{clearsymbols}\ifmfp@clearsym
+% \end{macrocode}
+%
+% \DescribeMacro{\stopshipping}
+% This is to turn off character shipping for duration of innermost
+% enclosing \MF{} group (eg, \env{mfpic} environment). As far as I know
+% it has never been tested with \MP. (It hasn't been tested recently even
+% in \MF.) The \MF{} code prevents the figure from being added to the font,
+% the \TeX{} switch \cs{ifmfp@noship} is checked before \cs{endmfpic}
+% attempts to add the (nonexistent) figure to the document.
+% \DescribeMacro{\resumeshipping}
+% The macro \cs{resumeshipping} restores character shipping.
+%
+% \DescribeMacro{\noship}
+% The macro \cs{noship} is the older name for \cs{stopshipping}. Its name
+% indicated that it was intended for use in one picture. This is not
+% actually an option (that is, there is no corresponding
+% \cs{DeclareOption} in \LaTeX), but it is coded the same.
+% \begin{macrocode}
+\mfp@makeoption\stopshipping\resumeshipping{noship}\ifmfp@noship
+\newlet\noship\stopshipping
+% \end{macrocode}
+%
+% \DescribeMacro{\mfpreadlog}
+% This sets the boolean governing the option to examine the \MF{} or
+% \MP{} log file for certain information.
+% \begin{macrocode}
+\newlet\ifmfp@readlog\iffalse
+\newdef\mfpreadlog{%
+ \mfp@ifopengraphsfile
+ {\mfp@afteropen@error\mfpreadlog}{\global\let\ifmfp@readlog\iftrue}}%
+% \end{macrocode}
+%
+%
+% \subsection{Graphic inclusion}\label{graphic}
+%
+% \DescribeMacro{\mfpicllx}
+% The code to include a figure has been divided into three parts. Under
+% \opt{metapost} we need to both include the figure and save the
+% \DescribeMacro{\mfpiclly}
+% bounding box coordinates of the lower left corner. Here we initialize
+% the macros that hold these coordinates.
+% \begin{macrocode}
+\newdef\mfpicllx{0}%
+\newdef\mfpiclly{0}%
+% \end{macrocode}
+%
+% \DescribeMacro{\preparemfpicgraphic}
+% Most EPS inclusion macros make the bounding box information available,
+% and so after the inclusion command we just save it in the above two
+% macros. However, \LaTeX's \cs{includegraphics} does not, so we have to
+% issue a command before the inclusion to force it to become available.
+% That command is \cs{preparemfpicgraphic}. Usually it does nothing,
+% but in the \LaTeXe{} case it locally redefines one of the graphics
+% packages internal macros.
+%
+% \DescribeMacro{\setmfpicgraphic}
+% This command includes the graphic. The \mfpic{} default (for the
+% \opt{metafont} option) is set by running \cs{usemetafont}.
+%
+% \DescribeMacro{\getmfpicoffset}
+% This one saves the lower left corner in the macros \cs{mfpicllx} and
+% \cs{mfpiclly}. In the case described above, that is accomplished by
+% the redefined internal command, so this command does nothing.
+%
+% The command \cs{@setmfpicgraphic} is a wrapper around the inclusion
+% code. It takes the name of the graphic file and passes it along to the
+% above three commands. Those three commands can be redefined by the
+% user for special circumstances. The user is then responsible to make
+% sure \cs{mfpicllx} and \cs{mfpiclly} are properly set when necessary.
+% Redefining one may necessitate redefining one or both of the others.
+%
+% The definition of these three is delayed to the option processing
+% section, except under the \MF{} default: \cs{usemetafont} defines them.
+%
+% We actually define two alternative versions of \cs{@setmfpicgraphic}.
+% The normal one is as described above, and a `nofile' alternative for
+% when the graphic file somehow doesn't exist. Then we just print the
+% filename.
+% \begin{macrocode}
+\newdef\normal@setmfpgraphic#1{%
+ \preparemfpicgraphic{#1}\setmfpicgraphic{#1}\getmfpicoffset{#1}}%
+\newdef\nofile@setmfpgraphic#1{%
+ {\edef\mfp@tempa{#1}%
+ \vbox to 0pt{\vss
+ \rlap{\kern2pt \mfpdraftfont \mfp@sanitized\mfp@tempa}\kern 2pt}}}%
+\mfp@ifdefined\strip@prefix{}{\def\strip@prefix#1>{}}%
+\newdef\mfp@sanitized{\@xp\strip@prefix\meaning}%
+\newlet\@setmfpicgraphic\normal@setmfpgraphic
+\usemetafont
+% \end{macrocode}
+%
+% \DescribeMacro{\setfilename}
+% The user can also change \cs{setfilename} to modify the assumed
+% filenames of the graphics output by \MP. The default is the output
+% file's basename (\arg1) with the picture number as the extension.
+%
+% \DescribeMacro{\setfilenametemplate}
+% The user can also tell \MP{} what kind of filename he wishes it to
+% produce. one uses \verb$\_$ in the template as a placeholder for the
+% \file{.mp} filename (the name used in \cs{opengraphsfile}) and \verb$\#$
+% for the figure number.
+% \begin{macrocode}
+\newdef\setfilename#1#2{#1.#2}%
+\newdef\setfilenametemplate#1{%
+ \begingroup
+ \def\_{\mfp@p j}%
+ \def\#{\mfp@p c}%
+ \let\@nl\relax
+ \xdef\mfp@settemplate{\@nl setoutputtemplate "#1";\@nl}%
+ \let\number\relax
+ \let\mfp@count\relax
+ \let\mfp@filename\relax
+ \def\_{\mfp@filename}%
+ \def\#{\number\mfp@count}%
+ \xdef\setfilename##1##2{#1}%
+ \endgroup
+}%
+% \end{macrocode}
+%
+% The \opt{draft}, \opt{final} and \opt{nowrite} options override
+% \mfpic's attempts to automatically detect whether the pictures should
+% be included. They also work under the \opt{metafont} option, but are
+% less important then (even unnecessary).
+%
+% \DescribeMacro{\mfpicdraft}
+% The \opt{draft} option (\cs{mfpicdraft} command) causes the pictures
+% created by \mfpic{} to be omitted.
+%
+% \DescribeMacro{\mfpicnowrite}
+% The \opt{nowrite} option (\cs{mfpicnowrite} command) turns off creation
+% of the output \file{.mp} file. It is intended to help speed things up
+% after all figures are completed, so it permits the insertion of pictures.
+%
+% \DescribeMacro{\mfpicfinal}
+% The \opt{final} option (\cs{mfpicfinal} command) can be used to
+% override a global draft option. It forces \mfpic{} to attempt to
+% insert the pictures. Under the \opt{metapost} option, missing figures
+% may cause serious slow-downs as a large number of directories may be
+% searched.
+%
+% Each of these sets the user level if-command
+% \SpecialUsageIndex{\ifmfpicdraft}\cs{ifmfpicdraft}, which is retained
+% for backward compatibility. These settings are global since they are
+% intended to be set for the whole document. The four commands
+% \cs{mfp@\{draft\|final\}\{true\|false\}} are used to communicate with
+% the \cs{opengraphsfile} command, in case the user-level commands occur
+% before then.
+% \begin{macrocode}
+\newif\ifmfpicdraft
+\newlet\ifmfp@draft\iffalse
+\newlet\ifmfp@final\iffalse
+\newlet\ifmfp@nowrite\iffalse
+\newdef\mfpicdraft{%
+ \mfp@ifopengraphsfile{\global\mfpicdrafttrue}%
+ {\global\let\ifmfp@draft\iftrue}}%
+\newdef\mfpicfinal{%
+ \mfp@ifopengraphsfile{\global\mfpicdraftfalse}%
+ {\global\let\ifmfp@final\iftrue}}%
+\newdef\mfpicnowrite{%
+ \mfp@ifopengraphsfile{\mfp@afteropen@error\mfpicnowrite}%
+ {\global\let\ifmfp@nowrite\iftrue}}%
+%</tex>
+% \end{macrocode}
+%
+%
+% \subsection{\LaTeX{} options and \MP{} graphics inclusion}\label{latex}
+%
+% Code dependent on \LaTeXe. Mostly this is the option processing
+% facilities. The declaration of options is in the file
+% \file{mfpic.sty}, which will input \file{mfpic.tex}.
+% \begin{macrocode}
+%<*sty>
+\DeclareOption{draft}{\mfpicdraft}%
+\DeclareOption{final}{%
+ \mfpicfinal\PassOptionsToPackage{final}{graphics}}%
+\DeclareOption{nowrite}{\mfpicnowrite}%
+\DeclareOption{metapost}{\mfp@mposttrue}%
+\DeclareOption{metafont}{\usemetafont}%
+\DeclareOption{centeredcaptions}{\usecenteredcaptions}%
+\DeclareOption{justifiedcaptions}{\noraggedcaptions}%
+\DeclareOption{raggedcaptions}{\useraggedcaptions}%
+\DeclareOption{clip}{\clipmfpic}%
+\DeclareOption{truebbox}{\usetruebbox}%
+\DeclareOption{mplabels}{\usemplabels}%
+\DeclareOption{overlaylabels}{\overlaylabels}%
+\DeclareOption{clearsymbols}{\clearsymbols}%
+\DeclareOption{debug}{\mfpicdebugtrue}%
+\DeclareOption{mfpreadlog}{\mfp@readlogtrue}%
+\DeclareOption*{%
+ \@ifpackageloaded{graphics}%
+ {\Mfpic@warn{Unrecognized option \CurrentOption.}}%
+ {\Mfpic@msg{Passing option \CurrentOption\space to %
+ graphics package.}%
+ \PassOptionsToPackage{\CurrentOption}{graphics}}}%
+\newcommand*\mfpdraftfont{\normalfont\ttfamily}%
+% \end{macrocode}
+%
+% Now we define one of three versions of \cs{@usemetapost}. This command
+% will be issued by the user level command \cs{usemetapost}, and consists
+% of the format dependent code. The one here is, of course, for \LaTeX{}
+% and will be in \file{mfpic.sty}.
+%
+% Unlike all the other EPS inclusion macros, \cs{includegraphics} does
+% not make the bounding box coordinates available: \cs{Gin@llx} and the
+% others are defined, but are local to a group surrounding the internals
+% of \cs{includegraphics}. We locally modify one graphics package internal
+% macro to globally save the information we need. Here we save the
+% original definition and define our replacement. For this, we definitely
+% need to make sure the \prog{graphics} package is loaded.
+% \begin{macrocode}
+\def\@usemetapost{%
+ \mfp@ifdefined\includegraphics{}{\RequirePackage{graphics}}%
+ \let\mfp@save@parse@bb\Gread@parse@bb
+ \newdef\mfp@parse@bb##1 ##2 ##3 ##4 ##5\\{%
+ \mfp@save@parse@bb##1 ##2 ##3 ##4 ##5\\%
+ \xdef\mfpicllx{\Gin@llx}\xdef\mfpiclly{\Gin@lly}}%
+% \end{macrocode}
+% The \prog{graphics} package makes decisions about graphic files based on
+% the files' extensions. Since we don't want to force our users to rename
+% all the graphics output by \MP, we add code to \cs{preparemfpicgraphic}
+% to force \cs{includegraphics} to treat the current extension (some
+% number) as if it were `{\tt eps}' or `{\tt mps}' according to the
+% context. We also use this command to substitute the above BoundingBox
+% parsing code for the \prog{graphics} package version.
+% \begin{macrocode}
+ \ifmfp@pdf \gdef\mfp@Gtype{mps}%
+ \else \gdef\mfp@Gtype{eps}%
+ \fi
+ \gdef\preparemfpicgraphic##1{%
+ \@namedef{Gin@rule@.\number\mfp@count}####1{{\mfp@Gtype}%
+ {.\number\mfp@count}{####1}}%
+ \let\Gread@parse@bb\mfp@parse@bb}%
+ \global\let\setmfpicgraphic\includegraphics
+ \gdef\getmfpicoffset##1{}}%
+% \end{macrocode}
+%
+% The \LaTeX{} wrapper ends by inputing \file{mfpic.tex}.
+% \begin{macrocode}
+\input mfpic.tex\relax
+%</sty>
+% \end{macrocode}
+%
+% \DescribeMacro{\usemetapost}
+% The user level command \cs{usemetapost} starts with a few actions common
+% to all formats, then calls \cs{@usemetapost} which is defined
+% differently for different formats.
+% \begin{macrocode}
+%<*tex>
+\newdef\usemetapost{%
+ \mfp@ifopengraphsfile{\mfp@mpostfalse\mfp@useMP@error}%
+ {\mfp@mposttrue \@usemetapost}}%
+% \end{macrocode}
+%
+% Under \LaTeX, if \file{mfpic.cfg} exists, it will be input at this
+% point. Then, we \cs{ProcessOptions}, using the star-form because it allows
+% package options to take precedence over global options. Thus \mfpic's
+% \opt{final} option can counteract the \mfpic-specific side effects of a
+% global \opt{draft} option.
+% \begin{macrocode}
+\ifmfp@latexe
+ \InputIfFileExists{mfpic.cfg}{}{}%
+ \ProcessOptions*\relax
+% \end{macrocode}
+%
+% \cs{ProcessOptions} doesn't allow \cs{RequirePackage}\marg{graphics},
+% so the above and the following were not simply put inside
+% \cs{DeclareOption}\marg{metapost}. The \cs{AtEndDocument} command will
+% close the output file if the user forgot.
+% \begin{macrocode}
+ \mfp@ifmpost{\usemetapost}{}%
+ \AtEndDocument{\mfp@closegraphsfile}%
+% \end{macrocode}
+%
+% Now those cases outside \LaTeXe{} that we try to support.
+% \begin{macrocode}
+\else
+ \mfp@msg{}\Mfpic@msg{\mfpfiledate\space v\mfpfileversion.\@nl}%
+% \end{macrocode}
+%
+% The definition of \cs{@usemetapost} in plain \TeX{} depends on the
+% setting of \cs{ifmfp@pdf}. However both cases are so similar that we
+% define a helper macro \cs{mfp@useMP} that just needs to have the names
+% of three commands and two files fed to it.
+% \begin{macrocode}
+ \newdef\mfp@useMP#1#2#3#4#5{%
+ \mfp@ifdefined#1{}{%
+ \InputIfFileExists{#2}{}{%
+ \InputIfFileExists{#3}{}{%
+ \mfp@errmsg{Files not found: #1, #2}%
+ {One of the two named files is necessary
+ for mfpic, but neither was found.}}}}%
+ \global\let\setmfpicgraphic\mfp@includeMPS
+ \gdef\getmfpicoffset##1{\xdef\mfpicllx{#4}\xdef\mfpiclly{#5}}}%
+% \end{macrocode}
+%
+% Under \pdfTeX{}, the graphics inclusion command is \cs{convertMPtoPDF},
+% defined in \file{supp-pdf.tex}. This command parses the file's PS
+% contents to PDF code, and leaves the bounding box info in macros
+% \cs{MPllx} and \cs{MPlly}.
+% \begin{macrocode}
+ \ifmfp@pdf
+ \newdef\mfp@includeMPS#1{\convertMPtoPDF{#1}{1}{1}}%
+ \newdef\@usemetapost{%
+ \mfp@useMP\convertMPtoPDF{supp-pdf.mkii}{supp-pdf.tex}%
+ \MPllx\MPlly}%
+% \end{macrocode}
+%
+% Under plain\TeX, eplain (still need to test this after recent eplain
+% changes), \AmSTeX, or \LaTeX2.09, the graphics inclusion command is
+% \cs{epsfbox}, defined in \file{epsf.tex}. This command writes a
+% \cs{special} for \prog{dvips} (and other DVI driver programs), and
+% leaves the bounding box info in macros \cs{epsfllx} and \cs{epsflly}.
+% \begin{macrocode}
+ \else
+ \newdef\mfp@includeMPS#1{\epsfbox{#1}}%
+ \newdef\@usemetapost{%
+ \mfp@useMP\epsfbox{epsf.tex}{epsf.sty}\epsfllx\epsflly}%
+ \fi
+\fi
+% \end{macrocode}
+%
+%
+%
+% \section{Optional parameter handling}\label{optional}
+%
+% We borrow \LaTeX's \cs{@ifnextchar}. The first argument is the token
+% being checked. It doesn't have to be a character and we sometimes use
+% \cs{@ifnextchar} to check whether a certain command follows.
+% The next argument is the code to run if that character is present, the
+% last is code to use if the character is not present.
+% \begin{macrocode}
+\mfp@DBlog{Handlers for optional parameters.}%
+\mfp@ifundefined\@ifnextchar
+ \long\def\@ifnextchar#1#2#3{%
+ \let\reserved@d#1%
+ \def\reserved@a{#2}%
+ \def\reserved@b{#3}%
+ \futurelet\@let@token\@ifnch}%
+ \def\@ifnch{%
+ \ifx\@let@token\@sptoken
+ \let\reserved@c\@xifnch
+ \else
+ \ifx\@let@token\reserved@d
+ \let\reserved@c\reserved@a
+ \else
+ \let\reserved@c\reserved@b
+ \fi
+ \fi
+ \reserved@c}%
+ {%
+ \def\:{\global\let\@sptoken= }\: % makes \@sptoken a space token
+ \def\:{\@xifnch}\@xp\gdef\: {\futurelet\@let@token\@ifnch}%
+ }%
+\mfp@enddef
+% \end{macrocode}
+%
+% Here we borrow from \file{keyval.sty}. We need a command that strips
+% spaces off ends of an argument and stores the result in a macro. In
+% \file{keyval}, that is \cs{KV@@sp@def}; we copy it here with name
+% changes. Defining \cs{mfp@tempa}, and then calling it with a space as
+% argument, ensures there will be an actual space token where needed in
+% the definition of \cs{mfp@sp@def}. The first argument is the control
+% sequence to define and the second is its definition text, with possible
+% spaces to be stripped. (I'm wondering now why I didn't just use
+% \cs{zap@space} since there shouldn't be \emph{any} significant spaces.
+% Well, at least I have the code in place for a future key-value
+% interface.)
+% \begin{macrocode}
+\def\mfp@tempa#1{%
+ \def\mfp@sp@def##1##2{%
+ \futurelet\mfp@next\mfp@sp@d##2\@nil\@nil#1\@nil\relax##1}%
+ \def\mfp@sp@d{%
+ \ifx\mfp@next\@sptoken \@xp\mfp@sp@b
+ \else \@xp\mfp@sp@b\@xp#1%
+ \fi}%
+ \def\mfp@sp@b#1##1 \@nil{\mfp@sp@c##1}}%
+\mfp@tempa{ }%
+\def\mfp@sp@c#1\@nil#2\relax#3{\mfp@toks{#1}\edef#3{\the\mfp@toks}}%
+% \end{macrocode}
+%
+% The following save us some typing whenever we need to check if some
+% token follows, and then remove it if it does. That's what
+% \cs{mfp@iftoken} does, being just a generalized form of \LaTeX{}'s
+% \cs{@ifstar}. Almost all our star-form commands differ only in what
+% parameter is passed to some other command. Thus \cs{mfp@ifstar} passes
+% either \arg{2} or \arg{3} to command \arg{1}.
+%
+% The \cs{mfp@ifend} is for use in processing lists, which all end with
+% the token \cs{mfp@end}. \cs{mfp@ifendorcomma} is useful for processing
+% comma-separated lists. If \cs{mfp@end} follows, it is removed and \arg1
+% is executed. Otherwise, if a comma follows, it is removed and \arg2 is
+% executed. When neither follows, nothing is removed and \arg3 is
+% executed. Normally, this last is the main command being applied to each
+% element of the list.
+%
+% We also define \LaTeX's \cs{@ifstar} for the few cases that require it.
+% \begin{macrocode}
+\def\mfp@iftoken#1#2{\@ifnextchar#1{\@firstoftwo{#2}}}%
+\def\mfp@ifstar#1#2#3{\mfp@iftoken*{#1{#2}}{#1{#3}}}%
+\def\mfp@ifend{\mfp@iftoken\mfp@end}%
+\def\mfp@ifendorcomma#1#2#3{\mfp@ifend{#1}{\mfp@iftoken,{#2}{#3}}}%
+\mfp@ifundefined\@ifstar
+ \def\@ifstar#1{\@ifnextchar*{\@firstoftwo{#1}}}%
+\mfp@enddef
+% \end{macrocode}
+%
+% I've taken to making an empty optional argument the default for a lot
+% of commands. I wanted to make it possible to have the same result for
+% empty brackets (i.e., \texttt{[\,]}) as for omitted brackets.
+% \cs{mfp@ifempty} helps check for emptyness. It uses a simple \cs{if} so
+% that \arg{1} will be expanded. I use a non-active \texttt\textasciitilde
+% (so it won't expand) to test for emptiness because, even if
+% a tilde should accidentally occur in a user's \arg1, it will most
+% likely be the usual active version which expands to something else.
+%
+% The shorthand \cs{mfp@ifexist} executes some code if \arg1 is not
+% empty, otherwise it does nothing. And \cs{mfp@emptysub} expands to
+% \arg1 if it isn't empty, otherwise \arg2.
+%
+% The \cs{@ifmtarg} macro is borrowed from \file{ifmtarg.sty}.
+% It is used when a space should count as empty, but expansion isn't
+% needed. It would be nice to have one that counts both as empty, but
+% that seems impossible (or impossibly hard).
+% \begin{macrocode}
+{%
+\catcode`Q3 \catcode`\~12
+\gdef\mfp@ifempty#1{\if~#1~\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}%
+\gdef\mfp@ifexist#1{\if~#1~\@xp\@gobble\else\@xp\@firstofone\fi}%
+\gdef\mfp@emptysub#1#2{\if~#1~#2\else#1\fi}%
+\long\gdef\@ifmtarg#1{\@xifmtarg#1QQ\@secondoftwo\@firstoftwo\@nil}%
+\long\gdef\@xifmtarg#1#2Q#3#4#5\@nil{#4}%
+}%
+% \end{macrocode}
+%
+% These are the two most common methods to parse optional arguments. In
+% \cs{mfp@defaultopt}, if an optional argument is present, we use
+% \cs{mfp@ifemptyopt} to see if it is empty and substitute a default
+% (\arg{2}) if it is. Argument \arg{1} is the command to run
+% on the resulting argument after this check. It needn't be a single
+% command token, but can (and often will) be a series of commands, or a
+% command with additional arguments.
+%
+% \cs{mfp@nullopt} is a convenient abbreviation for a common case: the
+% default option is empty. We could define it in terms of
+% \cs{mfp@defaultopt}, but then we'd have the redundant step of checking
+% for an empty optional argument, only to replace it by an empty argument.
+% That might not make a significant difference, but it just feels wrong.
+%
+% Both commands turn the delimited argument, if present, into an ordinary
+% one. This extra step permits us to avoid many bracket pairs in later
+% code.
+% \begin{macrocode}
+\newdef\mfp@defaultopt#1#2{%
+ \@ifnextchar[{\mfp@ifemptyopt{#1}{#2}}{#1{#2}}}%]
+\newdef\mfp@ifemptyopt#1#2[#3]{\mfp@ifempty{#3}{#1{#2}}{#1{#3}}}%
+\newdef\mfp@nullopt#1{\@ifnextchar[{\mfp@getopt{#1}}{#1{}}}%
+\newdef\mfp@getopt#1[#2]{#1{#2}}%
+% \end{macrocode}
+%
+% Why do we bother to check for an empty optional argument? Because
+% quite a few \mfpic{} commands stuff several parameters into a
+% single optional argument. All but the last are processed as undelimited
+% arguments and the last one as delimited by the right bracket. If only
+% the last parameter is missing, TeX{} takes that as an empty argument and
+% we can proceed gracefully (substituting a default). However, if more
+% than that are missing, \TeX{} takes the closing bracket as the
+% \emph{undelimited} argument and produces pretty inscrutable error
+% mesages. This decision to add extra information in one optional argument
+% was, in retrospect, a bad one. It was made for backward compatibility
+% and (originally) to minimize the changes in existing commands.
+% Unfortunately we are stuck with it for at least the near future.
+%
+% Another reason is that multiple optional arguments generally mean that
+% the to specify a later one, all the previous ones need to be specified.
+% But with the above code, the user can always use an empty bracket pair
+% to indicate the default, minimizing the extra typing.
+%
+%
+%
+% \section{Writing to the \file{.mf} or \file{.mp} file}\label{writing}
+%
+%
+% \subsection{Preserving linebreaks}\label{linebreaks}
+%
+% Some macros (the ones we call list macros) can take quite long
+% arguments, especially if they are automatically generated by some
+% program. While the output file need not ever be looked at, there can be
+% a problem with lines too long for the \MF{} executable's line buffer.
+% Plus, for debugging, we might want to examine the file \mfpic{} writes.
+% We therefore try to get at least the list macros to preserve linebreaks
+% on output.
+%
+% We make use of the \TeX{} property that expandable commands are expanded
+% inside \cs{write} commands, but unexpandable commands and characters are
+% written verbatim (almost). If this were the only consideration, nothing
+% would need to be done. But characters with catagory code 5 (end-of-line)
+% are converted to spaces upon reading, so we have to change the category
+% of the \cs{endlinechar}, or we have change the \cs{endlinechar} to a
+% character of a different category. We do the former, because the line a
+% macro starts on will already have the end-of-line character added, but
+% it will not yet have been tokenized, so its category can be changed. We
+% need a category that \TeX{} will not \emph{ignore}, but that
+% nevertheless has no (or minimal) side effects if encountered in a \TeX{}
+% file. We therefore make it active, but \cs{let} it equal to \cs{relax}.
+% This makes the end-line-character unexpandable since \cs{relax} is not
+% expandable.
+% \begin{macrocode}
+\mfp@DBlog{Direct output to Metafont/Post file.}%
+{%
+ \catcode\mfp@ctrlM\active%
+ \gdef\@activectrlM{^^M}%
+ \gdef\mfp@obeylines{\catcode\mfp@ctrlM\active\let^^M\relax}%
+}%
+% \end{macrocode}
+%
+% The choice \cs{relax} seems to be actually a slightly better choice
+% than \cs{endgraf} (which we used to use, and which equals the
+% unexpandable primative \cs{par}) because \cs{relax} is permitted before
+% the opening brace in the syntax of \meta{general text} (for token
+% variable assignments, \cs{write}, \cs{message}, and the like).
+%
+% The character assigned to \cs{endlinechar} is added to the end of all
+% lines read in by \TeX. The character assigned to \cs{newlinechar}
+% will produce a newline when it occurs in a \cs{write}. Our
+% \cs{mfp@keeplines} ensures these are the same and makes the above
+% changes in category. This will fail in the following case:
+% \cs{endlinechar} is not \verb$^^M$ and the data starts on the same line
+% as the command that issued \cs{mfp@keeplines}. But then the output will
+% lack only that first linebreak.
+% \begin{macrocode}
+\newdef\mfp@keeplines{\newlinechar\mfp@ctrlM
+ \endlinechar\newlinechar \mfp@obeylines}%
+% \end{macrocode}
+%
+% We want our file writes to issue \cs{mfp@keeplines} before the text is
+% read (and category codes assigned). Also we often want to add a few
+% items to the argument (figure macros append a semi-colon, list macros
+% surround with parentheses), so we isolate the common starting code in
+% \cs{mfp@write}, and then invoke the appropriate continuation command.
+%
+% \DescribeMacro{\mfsrc}
+% The macro \cs{mfsrc} is the most basic, merely writing its contents. For
+% figure macros, we ultimately call
+% \DescribeMacro{\mfcmd}
+% \cs{mfcmd}, which appends a semicolon, terminating any pending \MF{}
+% command. For macros which take a comma separated list of values as their
+% arguments, we call
+% \DescribeMacro{\mflist}
+% \cs{mflist} which surrounds the list in parentheses and appends a
+% semicolon
+%
+% The ending code is in \cs{mfp@src}. The \cs{begingroup} in
+% \cs{mfp@write} matches the \cs{endgroup} in \cs{mfp@src}. The group
+% isolates the changes to EOLs. We also define \cs{@nl} to expand to an
+% active control-M so that it can be used for line breaks in cases where
+% it is inconvenient to make EOLs active (in the replacement text of
+% \cs{opengraphsfile} and \cs{@mfpic}, for example).
+% \begin{macrocode}
+\newdef\mfsrc {\mfp@write\mfp@src\mfsrc}%
+\newdef\mfcmd {\mfp@write\mfp@cmd}%
+\newdef\mflist{\mfp@write\mfp@wlist}%
+\newdef\mfp@write{\begingroup\mfp@keeplines \let\@nl\@activectrlM}%
+% \end{macrocode}
+%
+% The macro \cs{mfp@src} does the job of writing to files. The output
+% stream \cs{mfp@out} is declared later. The other two commands invoke
+% \cs{mfp@src} on a slightly modified argument. The first argument to
+% \cs{mfp@src} is the command that called it (for the error message), the
+% second is the code to write.
+% \begin{macrocode}
+\newdef\mfp@src#1#2{%
+ \mfp@ifopengraphsfile{\immediate\write\mfp@out{#2}}%
+ {\mfp@nooutput@error{#1}}\endgroup}%
+\newdef\mfp@cmd#1{\mfp@src\mfcmd{#1;}}%
+\newdef\mfp@wlist#1{\mfp@src\mflist{(#1);}}%
+% \end{macrocode}
+%
+% If \prog{babel} is loaded with some options, certain punctuation will
+% be made active. These will then be expanded when written to the graph
+% file. \prog{Babel} provides a way to turn this off, but it would be an
+% error to run those commands if \prog{babel} is not loaded. Rather than
+% try to detect \prog{babel}, we run these commands only if they are
+% defined.
+% \begin{macrocode}
+\def\mfp@safe@activestrue{%
+ \mfp@ifdefined\@safe@activestrue\@safe@activestrue{}}%
+\def\mfp@safe@activesfalse{%
+ \mfp@ifdefined\@safe@activesfalse\@safe@activesfalse{}}%
+% \end{macrocode}
+%
+%
+% \subsection{Initializing the output file}\label{output}
+%
+% Category 12 characters percent, sharp and backslash signs for the
+% \file{.mf} file. The command \cs{mfp@d} is used to write a line of
+% dashes above and below each figure to make them stand out when debugging
+% the file.
+% \begin{macrocode}
+\edef\mfp@p{\@xp\@gobble\string\%}% percent (%)
+\edef\mfp@s{\@xp\@gobble\string\#}% sharp (#)
+\edef\mfp@b{\@xp\@gobble\string\\}% backslash (\)
+\newdef\mfp@d{--------------}%
+% \end{macrocode}
+%
+% The current date and time stamp will be written to the \file{.mf} file.
+% We calculate it here and store the result in these macros. The commands
+% \cs{normalyear} and \cs{normalmonth} are part of \ConTeXt, where they
+% have been equated to the primitive commands \cs{year} and \cs{month}
+% (which are then redefined).
+% \begin{macrocode}
+\mfp@ifdefined\two@digits{}%
+ {\def\two@digits#1{\ifnum#1<10 0\fi\number#1}}%
+{%
+ \mfp@ifdefined\normalmonth
+ {\let\month\normalmonth\let\year\normalyear}{}%
+ \count@\time \divide\count@ by 60
+ \xdef\mfp@now{\number\year/\two@digits\month/\two@digits\day
+ \space at \two@digits\count@:}%
+ \multiply\count@ by -60 \advance\count@\time
+ \xdef\mfp@now{\mfp@now\two@digits\count@}%
+}%
+% \end{macrocode}
+%
+% These registers are allocated:
+% \begin{itemize}
+% \item \cs{mfp@out} Write stream for the \file{.mf} or \file{.mp} file.
+% \item \cs{mfp@graph} Read stream used to test existence of files and
+% for reading datafiles.
+% \item \cs{mfp@count} Number for next \env{mfpic} environment.
+% \item \cs{mfp@linetype} Number indexing current line style in
+% multi-curve data plots.
+% \item \cs{mfp@sequence} Counter incremented with every datapoint written
+% from a file to implement the feature that a list of numeric data
+% can be plotted against its sequence position.
+% \item \cs{mfp@toks} Temporary token register.
+% \item \cs{mfp@verbtex} A temporary token list used by \cs{mfpverbtex}.
+% \item \cs{ifmfp@switch} Not exactly a register, but a temporary
+% switch for passing information from one macro to another.
+% \end{itemize}
+% \begin{macrocode}
+\mfp@DBlog{Declarations used with data and graphs files.}%
+\newwrite\mfp@out
+\newread\mfp@graph
+\newlet\mfp@data\mfp@graph
+\newcount\mfp@count
+\newcount\mfp@linetype
+\newcount\mfp@sequence
+\newtoks\mfp@toks
+\newtoks\mfp@verbtex
+\newif\ifmfp@switch
+%</tex>
+% \end{macrocode}
+%
+% We store most of the \mfc{verbatimtex} block that \mfpic{} relies on in
+% the file \file{mfpicdef.tex}. This is the code that goes into that
+% file
+%
+% The \cs{mpxshipout} is to cover all three cases of the output of
+% `\texttt{mpto -tex}'. In public versions of \MP{} prior to 0.9 there was
+% no such command in that output file. In 0.9 and 0.901 that command is
+% defined at the start of the file, prior to any verbatimtex, but it is
+% buggy. This code overwrites it and fixes the bugs. In later versions a
+% good definition is used, but it appears only just before the first
+% \mfc{btex} and so will overwrite my definition. Thus all cases are
+% served.
+% \begin{macrocode}
+%<*defs>
+\gdef\mpxshipout{\afterassignment\middlempxshipout\setbox0=\hbox}%
+\gdef\middlempxshipout{\aftergroup\finishmpxshipout}%
+\gdef\finishmpxshipout{\dimen1=\ht0 \dimen2=\dp0
+ \dimen0=\dimen1 \advance\dimen0\dimen2
+ \setbox0=\hbox{\hbox{\box0
+ \ifnum\dimen0>0 \vrule width1sp height\dimen1 depth\dimen2
+ \else \vrule width1sp height1sp depth0sp\relax
+ \fi}}%
+ \ht0=0pt \dp0=0pt \shipout\hbox{\box0}}%
+\def\MFPtext#1{\vbox{\def\\{\cr}\MFPcfont\MFPeverylabel
+ \halign{##\hfil\cr#1\crcr}}}%
+% \end{macrocode}
+%
+% There are vague plans to make the output file equally usable by \MF{}
+% and \MP{}. For this to work, \file{grafbase.mf} defines
+% \mfc{verbatimtex} to gobble everything to the next colon (which we place
+% after each \mfc{etex} we write). This fails if there is a
+% \verb$\begingroup$ without matching \verb$\endgroup$ because \MF{} sees
+% this as the command \verb$\$ (meaning \mfc{relax}) followed by the \MF{}
+% primitive \mfc{begingroup}. But \MF{} doesn't allow gobbled text
+% parameters to contain unbalanced groups. Thus, we use the replacements
+% defined here, hidden from \MF{} in a \TeX{} input file.
+% \begin{macrocode}
+\let\MFPbegingroup\begingroup
+\let\MFPendgroup\endgroup
+%</defs>
+% \end{macrocode}
+%
+% Within \file{mfpic.tex} these commands can be anything unexpandable,
+% since we just want them to be written verbatim. It makes sense to give
+% them the obvious meanings.
+% \begin{macrocode}
+%<*tex>
+\let\MFPbegingroup\begingroup
+\let\MFPendgroup\endgroup
+% \end{macrocode}
+%
+% We store the first several lines that will be written to the output
+% file in a few internal macros, so that changing them will be easier.
+%
+% The first two lines of the output file are comments identifying the
+% file, the \mfpic{} version, the tex source file, and the timestamp.
+% These are stored in \cs{mfp@versioninfo}. We also let \grafbase{} know
+% the \mfpic{} version creating the file. This could catch version
+% mismatches. But since older \grafbase{} didn't test this, some
+% mismatches will be missed.
+%
+% After that (\cs{mfp@init@outfile}) we set the \mfc{mode} and the
+% \mfc{mag} if not known (both ignored by \MP), and arrange for
+% \grafbase{} macros to be loaded.
+% \begin{macrocode}
+\newdef\mfp@versioninfo{%
+ \mfp@p\space\mfp@filename.m\mfp@ifmpost{p}{f}, %
+ generated by Mfpic, v\mfpfileversion\space\mfpfiledate,\@nl
+ \mfp@p\space from TeX source "\jobname" on \mfp@now\@nl
+ numeric mfpicversion; mfpicversion := \mfpicversion;\@nl}%
+\newdef\mfp@init@outfile{%
+ if unknown mode: mode := localfont; fi\@nl
+ if unknown mag: mag := \number\mag/1000; fi\@nl
+ if unknown grafbase: input grafbase; fi\@nl\@nl}%
+% \end{macrocode}
+% Then a comparison of the version of this file, stored in
+% \cs{mfpicversion} with the version stored in the \grafbase{} variable
+% \mfc{grafbaseversion}. This should catch a mismatch caused by an old
+% \grafbase{} with a current \mfpic.
+% \begin{macrocode}
+\newdef\mfp@versioncheck{%
+ if unknown grafbaseversion: grafbaseversion := 0; fi\@nl
+ if grafbaseversion <> mfpicversion :\@nl
+ GBmsg "Bad mfpic installation: mfpic and grafbase %
+ versions do not match.";\@nl
+ fi\@nl}%
+% \end{macrocode}
+%
+% This opens the output file and prints a reminder to the user to run
+% \MF{} or \MP{}.
+% \begin{macrocode}
+\newdef\mfp@openfile{%
+ \ifmfp@nowrite \else \mfp@msg{}%
+ \Mfpic@msg{Initializing \mfp@filename\mfp@ext.}%
+ \immediate\openout\mfp@out=\mfp@filename\mfp@ext\relax
+ \Mfpic@msg{Don't forget to process \mfp@filename\mfp@ext!}%
+ \Mfpic@msg{Apply \mfp@progname\space to \mfp@filename\mfp@ext%
+ \mfp@ifmpost{}{, then gftopk to the resulting gf file}.}%
+ \Mfpic@msg{Then reprocess this file "\jobname".\@nl}%
+ \fi}%
+% \end{macrocode}
+%
+% A little bit of \LaTeX{} for checking file existence.
+% \begin{macrocode}
+\mfp@ifundefined\IfFileExists
+ \def\IfFileExists#1{\openin\mfp@graph=#1\relax
+ \ifeof\mfp@graph \@xp\@secondoftwo
+ \else \immediate\closein\mfp@graph \@xp\@firstoftwo
+ \fi}%
+ \def\InputIfFileExists#1#2{\IfFileExists{#1}{#2\input #1\relax}}%
+\mfp@enddef
+% \end{macrocode}
+%
+% The following is the code involved in opening the output file and
+% checking for the existence of already created figures.
+% \cs{mfp@checkoutput} checks if the output file of \MP{} or \MF{} exists.
+% The first argument is the extension of that output file (\file{tfm} for
+% \MF{}, the starting value of the figure counter for \MP{}). We use the
+% existence of that output as a signal that there are graphics to be
+% included.
+%
+% Based on the file existence, we set the values of the draft switch, and
+% issue a message. Any other actions needed are passed as \arg2 (it
+% exists) and \arg3 (it doesn't). These are nonempty only under the
+% \opt{metafont} option: we load the font only if the \file{.tfm} exists,
+% otherwise we use a the \texttt{dummy} font.
+%
+% We do not try to load a nonexistent font because that might trigger some
+% \prog{maketfm} script which could be disastrous at this point, when the
+% \file{.mf} file is in an uncertain state.
+% \begin{macrocode}
+\newdef\mfp@checkoutput#1#2#3{%
+ \IfFileExists{\setfilename\mfp@filename{#1}}%
+ {\global\mfpicdraftfalse #2}%
+ {\global\mfpicdrafttrue #3\mfp@msg{}%
+ \Mfpic@msg{No file \setfilename\mfp@filename{#1}.}%
+ }}%
+\newdef\mfp@checkMPoutput{\mfp@checkoutput{\number\mfp@count}{}{}}%
+\newdef\mfp@checkMFoutput{\mfp@checkoutput{tfm}%
+ {\global\font\mfp@graphfont\mfp@filename\relax}%
+ {\global\let\mfp@graphfont\mfp@dummy}}%
+% \end{macrocode}
+%
+% For \opt{metapost} the output file needs to load some color macros, and
+% write some setup for text labels (even if \cs{nomplabels}, because
+% \cs{usemplabels could be turned on later}). \MF{} now can also handle
+% the definitions in \file{dvipsnam.mp} but it seems a waste to load all
+% those definitions just to get more names for grayscales (even given
+% that \MF{} can find that file).
+% \begin{macrocode}
+\newdef\mfp@MPextras{\mfp@toks\@xp{\every@tlabel}%
+ \mfsrc{%
+ if unknown Apricot: input dvipsnam.mp; fi\@nl
+ verbatimtex\@nl
+ \ifmfp@verbtex \the\mfp@verbtex\@nl \fi
+ \relax\@nl
+ \noexpand\input mfpicdef.tex\relax\@nl
+ \def\noexpand\MFPeverylabel{\the\mfp@toks}\@nl
+ etex;\@nl
+ \mfp@ifdefined\mfp@settemplate\mfp@settemplate{}%
+ }%
+ \global\mfp@verbtexfalse \global\mfp@verbtex{}%
+% \end{macrocode}
+% Finally, we process the \MP{}-specific options as they are currently
+% known. The \cs{usemplabels} was delayed until now so that the order of
+% it and \cs{usemetapost} before now would not matter.
+% \begin{macrocode}
+ \@ifmplabels\usemplabels{}%
+ \ifmfp@truebbox \usetruebbox \fi
+ \ifmfp@overlabels \overlaylabels \fi}%
+% \end{macrocode}
+%
+% There are a couple of extras that are only needed by \MF{}, the mode
+% and resolution. Unlike the `MPextras' this does not write code, but
+% rather \emph{is} code to write. It will occur inside a \cs{write}
+% command.
+% \begin{macrocode}
+\def\mfp@MFextras{%
+ \mfp@ifdefined\mfp@resolution%
+ {GBresolution := \mfp@resolution;\@nl}{}%
+ \mfp@ifdefined\mfp@mode{mode := \mfp@mode;\@nl}{}}%
+% \end{macrocode}
+%
+% All the characters likely to cause problems in a filename are rendered
+% harmless with this command. We want to allow macros so backslashes,
+% braces and spaces aren't touched.
+% \begin{macrocode}
+\def\mfp@dospecials{\do\$\do\&\do\#\do\^\do\_\do\%\do\~}%
+% \end{macrocode}
+% \DescribeMacro{\opengraphsfile}
+% This is the main command to set up the output file.
+% The name of the output file is its only parameter, and it appends
+% the extension \file{.mf} or \file{.mp} depending on the whether option
+% \opt{metafont} or \opt{metapost} was chosen. We first nullify \TeX{}
+% special characters, then read the graphfile name, then store an
+% expanded version.
+% \begin{macrocode}
+\mfp@DBlog{Macros to open and close graphs files.}%
+\newdef\opengraphsfile{\@bsphack
+ \begingroup
+ \let\do\@makeother
+ \mfp@dospecials\@opengraphsfile}%
+\newdef\@opengraphsfile#1{%
+ \xdef\mfp@tempa{#1}%
+ \endgroup
+% \end{macrocode}
+%
+% With the graphfile name safely stored, we check if a previous one had
+% been opened and close it if so. Then we store in \cs{mfp@filename}
+% a sanitized version of the name.
+%
+% A command can determine if it occurs after \cs{opengraphsfile} but
+% before \cs{closegraphsfile} by testing whether \cs{mfp@filename} is
+% defined. We also save the name of the processing program (\MF{} or \MP)
+% and the file extension (\file{.mf} of \file{.mp}) in macros for use in
+% messages.
+% \begin{macrocode}
+ \mfp@ifopengraphsfile%
+ {\Mfpic@warn
+ {\@nl An attempt has been made to open a graph file, %
+ \mfp@sanitized\mfp@tempa\@msgbreak
+ while another, \mfp@filename, is still open. \@msgbreak
+ Closing \mfp@filename.}\mfp@closegraphsfile}%
+ {}%
+ \xdef\mfp@filename{\mfp@sanitized\mfp@tempa}%
+ \xdef\mfp@progname{meta\mfp@ifmpost{pos}{fon}t}%
+ \xdef\mfp@ext{.m\mfp@ifmpost{p}{f}}%
+% \end{macrocode}
+%
+% These commands test for the existence of the graphics (\file{.tfm} file
+% or first EPS file) and provisionally set \cs{ifmfpicdraft} based on the
+% result. Then the \file{.mf} or \file{.mp} file is opened, and the
+% automatic settings of \cs{ifmfpicdraft}, etc., are overridden if the
+% user has explicitly chosen the \opt{draft} or \opt{final} option. The
+% \opt{nowrite} option merely disables the \cs{mfp@src} command, called
+% by all the commands that write to the file.
+% \begin{macrocode}
+ \mfp@ifmpost\mfp@checkMPoutput\mfp@checkMFoutput
+ \mfp@openfile
+ \ifmfp@draft \global\mfpicdrafttrue \fi
+ \ifmfp@final \global\mfpicdraftfalse \fi
+ \ifmfp@nowrite \gdef\mfp@src##1##2{\endgroup}\fi
+% \end{macrocode}
+%
+% Now we set up the header and preamble of the output file, writing
+% version info, initializations, version checking code and extra code
+% for \MP{}, if required.
+%
+% Finally, we process options as they are currently known. These were
+% delayed until now because they might require a write to the output file.
+% Finally, we read the \prog{meta*o*t} log file, if the option to do that
+% is in effect.
+% \begin{macrocode}
+ \mfsrc{\mfp@versioninfo \mfp@ifmpost{}\mfp@MFextras
+ \mfp@init@outfile \mfp@versioncheck}%
+ \mfp@ifmpost\mfp@MPextras{}%
+ \ifmfp@clip \clipmfpic \fi
+ \ifmfp@clearsym \clearsymbols \fi
+ \ifmfp@noship \stopshipping \fi
+ \ifmfp@readlog \mfp@readlogfile \fi
+ \@esphack}%
+% \end{macrocode}
+%
+% \DescribeMacro{\closegraphsfile}
+% This command finishes off the output file and undefines \cs{mfp@filename}.
+% We define an error message free version \cs{mfp@closegraphsfile} for
+% internal use. We unconditionally undefine \cs{mfp@filename} just in
+% case.
+% \begin{macrocode}
+\newdef\closegraphsfile{%
+ \mfp@ifopengraphsfile{}{\mfp@nooutput@error\closegraphsfile}%
+ \mfp@closegraphsfile}%
+\newdef\mfp@closegraphsfile{%
+ \mfp@ifopengraphsfile
+ {\ifmfp@nowrite \else
+ \mfsrc{\@nl end.}\immediate\closeout\mfp@out
+ \fi \global\let\mfp@filename\UndEfInEd}{}}%
+% \end{macrocode}
+%
+%
+% \subsection{Information from \MF: reading its log file}
+%
+% \DescribeMacro{\assignmfvalue}\SpecialUsageIndex{\globalassignmfvalue}
+% This macro was suggested by Werner Lemberg, who also supplied working
+% code. I have made modifications, so I am to blame if it has bugs. It
+% writes code to the \file{.mf} file that causes \MF{} to write
+% information to its \file{.log} file. If the option \opt{mfpreadlog} has
+% been set, that log file will be examined for that information. The
+% result, for example, of
+% \begin{verbatim}
+% \setmfvariable{numeric}{s}{10}
+% \assignmfvalue\foo{s+2}
+% \end{verbatim}
+% is that \cs{foo} will be a macro that expands to `\texttt{12}' (during
+% the \TeX{} run following the \MF{} pass). Under \opt{mplabels} a label
+% containing \cs{foo} will only be correct in the \file{.mp} file after
+% the second \MP{} pass, and so a third \TeX{} run may be required to
+% insert the correct figure. To implement localization, the code causes
+% the internal command \cs{mfpic-\meta{N}-foo} to be defined to the value,
+% where \meta{N} is the number of the current figure. The \cs{foo} is then
+% defined to the value of that internal command. Outside any \env{mfpic}
+% environment, the internal command is \cs{mfpic--foo}. If the \file{.log}
+% file doesn't exist or doesn't yet contain the information, \cs{foo} is
+% defined to `???'.
+%
+% While \cs{mfpicvalue} is merely a signal and doesn't need a
+% definition, we give it a useful definition for debugging: its own name.
+% It could be accidentally redefined by a user, but that will be harmless;
+% the debugging is mainly for me.
+%
+% It is possible to put \cs{global} in front of \cs{assignmfvalue} and
+% its macro will be defined globally in the \TeX{} run, but I could find
+% no way to make it global in the \file{.mp} file under \opt{mplabels}.
+% Thus, I've created a \cs{globalassignmfvalue}.
+% \begin{macrocode}
+\edef\mfpicvalue{\string\mfpicvalue}%
+\newdef\mfp@cmdname{%
+ mfpic-\ifin@mfpicenv\number\mfp@count\fi-\@xp\@gobble\string}%
+\newdef\assignmfvalue{\mfp@assign\@empty}%
+\newdef\globalassignmfvalue{\mfp@assign\global}%
+\newdef\mfp@assign#1#2#3{%
+ #1\edef#2{%
+ \@ifundefined{\mfp@cmdname#2}{???}{\@nameuse{\mfp@cmdname#2}}}%
+ \@ifundefined{\mfp@cmdname#2}%
+ {\mfp@msg{}\Mfpic@msg{No value for \string#2\ifin@mfpicenv\space
+ in mfpic figure \number\mfp@count\fi.\@nl}}%
+ {}%
+ \mfsrc{%
+ \@ifmplabels{\@nl verbatimtex #1\def\string#2{#2} etex;\@nl}{}%
+ message "\string\mfpicvalue";\@nl
+ message "\mfp@cmdname#2";\@nl
+ show #3;\@nl
+ message "";}}%
+\newlet\assignmpvalue\assignmfvalue
+\newlet\globalassignmpvalue\globalassignmfvalue
+% \end{macrocode}
+%
+% The following code is what really makes \cs{assignmfvalue} work. It is
+% similar to code in the \prog{metatex} package, adapted by Werner Lemberg
+% and then modified by me. It steps through the \mfpic{} output file looking
+% for this signal: a line containing only `\verb$\mfpicvalue$'. It then
+% reads the next line, creates a macro name from it, and assigns to that
+% macro the value it reads on the next line. This is repeated to the end
+% of the file.
+% \begin{macrocode}
+\newdef\strip@gtgt>> #1\mfp@end{#1}%
+\newdef\mfp@readlogfile{%
+ \openin\mfp@data=\mfp@filename.log\relax
+ \ifeof\mfp@data \Mfpic@msg{No file \mfp@filename.log.}%
+ \else
+ {%
+ \mfp@resetwhitespace \endlinechar -1
+ \catcode`\>12
+ \def\@mfpicvalue{\mfpicvalue}%
+ \message{< Mfpic: Reading \mfp@filename.log}%
+ \mfp@readloglines
+ \message{>}%
+ }%
+ \fi \closein\mfp@data}%
+\newdef\mfp@readloglines{%
+ \read\mfp@data to \mfp@tempa
+ \ifeof\mfp@data
+ \else
+ \ifx\mfp@tempa\@mfpicvalue
+ \read\mfp@data to \mfp@tempa
+ \read\mfp@data to \mfp@tempb
+ \@XP\xdef\@nameuse\mfp@tempa{\@xp\strip@gtgt\mfp@tempb\mfp@end}%
+ \fi
+ \@xp\mfp@readloglines
+ \fi}%
+% \end{macrocode}
+%
+%
+%
+% \section{\Mfpic{} dimensions}\label{dimensions}
+%
+% A number of packages, especially figure drawing packages, allocate a
+% great many dimension registers. The most familiar example is \PiCTeX{}
+% which is at least partially responsible for most ``No room for a new
+% \cs{dimen}'' error messages. \Mfpic{} also needs many dimension
+% values, but some are set by macros that simply write the value to the
+% \file{.mf} file and do not use any registers. A few are stored in
+% \TeX{} registers, but most are stored using a trick with font
+% dimensions.
+% \begin{macrocode}
+\mfp@DBlog{Using dummy font's dimensions, to spare TeX registers.}%
+% \end{macrocode}
+%
+% While most fonts have relatively few font dimensions (twenty-two is a
+% common number), it turns out that if one loads a font and immediately
+% assigns a value to \cs{fontdimen}~$n$, Then \TeX{} acts as if the font
+% has \cs{fontdimen}\,s 0 through $n$ available. They can be assigned
+% dimension values just like \TeX{} dimension registers.
+%
+% Virtually all \TeX{} systems have a \file{dummy.tfm}, which contains
+% no characters (it is used for syntax checking by some packages and
+% classes). Since this trick will only work if we are the first to
+% load the font (the \AmSTeX{} document style \file{amsppt.sty} loads
+% the dummy font), we use another trick, which is to load it scaled to
+% some unlikely value, which \TeX{} treats as a new font.
+% \begin{macrocode}
+\font\mfp@dummy=dummy scaled 1042
+\fontdimen100\mfp@dummy0pt
+% \end{macrocode}
+%
+% Some versions of \file{dummy.tfm} have values already assigned to font
+% dimensions 0 through 22. It would probably not hurt to use those, but
+% we'll be cautious and use only font dimensions 23 and up, making 78
+% available.
+%
+% We use an auxiliary count register to keep track of new font dimens.
+% \begin{macrocode}
+\newcount\mfp@fdc
+\mfp@fdc22
+% \end{macrocode}
+%
+% We define a macros that mimics the behavior of \cs{newdimen}, for
+% allocating font dimension numbers. It increments \cs{mfp@fdc} and
+% defines its argument \arg{1} to expand to a reference to the
+% \cs{fontdimen} with that number. The syntax it produces is valid as
+% the left side of a dimension assignment and also as a dimension
+% that may follow a real factor and \cs{the}. For example, later we say
+% \cs{newfdim}\cs{pointsize} and give it the value \texttt{2pt}. Then
+% \verb$\the\pointsize$ prints this value and
+% \verb$\pointsize = 0.5\pointsize$ is a valid assignment.
+% \begin{macrocode}
+\newdef\newfdim#1{%
+ \global\advance\mfp@fdc1
+ \xdef#1{\fontdimen\number\mfp@fdc\mfp@dummy}%
+ \wlog{\string#1=\fontdimen\number\mfp@fdc\mfp@dummy}}%
+% \end{macrocode}
+% \TeX{} will not accept font dimensions in its syntax for the dimension
+% arithmetic commands \cs{advance}, \cs{multiply} and \cs{divide}. In the
+% rare cases where we need to do this we assign the value to a temporary
+% register and calculate with it.
+%
+% We allocate a couple of boxes. At the end of an \mfpic{}
+% environment, \cs{@wholegraph} is loaded with the picture (possibly
+% null) as it exists then. Inside the environment, when text labels are
+% processed, \cs{@textbox} holds this text temporarily so it can be
+% measured, moved about, etc. We also define abbreviations for the
+% frequent commands that specify the dimensions of \cs{@textbox}
+% \begin{macrocode}
+\mfp@DBlog{A box register for whole labeled graph, %
+ and a temporary one.}%
+\newbox\@wholegraph
+\newbox\@textbox
+\newdef\tb@ht{\ht\@textbox}%
+\newdef\tb@dp{\dp\@textbox}%
+\newdef\tb@wd{\wd\@textbox}%
+% \end{macrocode}
+%
+% When labels are processed, they are accumulated in the box
+% \cs{@alltlabels}. We allocate it, and four \cs{newfdim}\,s for the
+% location of three of the sides (We don't need the fourth). The box
+% \cs{@backtext} holds `background text', that is, text accumulated
+% between \cs{startbacktext} and \cs{stopbacktext}.
+% \begin{macrocode}
+\newbox\@alltlabels
+\newbox\@backtext
+\newfdim\@tlabelstop
+\newfdim\@tlabelsbot
+\newfdim\@tlabelsleft
+% \end{macrocode}
+%
+% The following fontdims record the dimensions of the box containing the
+% picture. We need to adjust them when the text labels are added to the
+% picture, because \TeX{} doesn't do it automatically. For example,
+% kerning left outside the picture will not change the picture box's
+% reference point.
+% \begin{macrocode}
+\mfp@DBlog{Internal dimension parameters for graph dimensions...}%
+\newfdim\@graphright
+\newfdim\@graphleft
+\newfdim\@graphtop
+\newfdim\@graphbot
+\mfp@DBlog{...and dimension parameters for tlabel positioning.}%
+\newfdim\tb@raise
+\newfdim\tb@moveright
+\newfdim\tb@totalht
+% \end{macrocode}
+%
+% The following are for label adjustments. The first two are used to
+% implement the feature that all labels can be simultaneously shifted an
+% equal amount. The third is for the feature that the label can be a
+% certain distance separated from the point where it is placed. And the
+% last establishes the distance between a label and a path surrounding it.
+% \begin{macrocode}
+\newfdim\tlabel@hadj
+\newfdim\tlabel@vadj
+\newfdim\tlabel@sep
+\newfdim\tlpath@sep
+\tlabel@hadj0pt
+\tlabel@vadj0pt
+\tlabel@sep0pt
+\tlpath@sep0pt
+% \end{macrocode}
+%
+% It can be argued that dimensions not needed by \TeX{} should be stored
+% in the output file rather than allocating (font) dimensions. These are
+% everything below from \cs{pointsize} through \cs{symbolspace},
+% inclusive. This would make synchrony between \TeX{} and \MF{} easier
+% to maintain. Those that are needed by \TeX{} are then few enough to be
+% regular \cs{dimen} registers, making them automatically local to each
+% \env{mfpic} environment. Unfortunately, changing to such a scheme could
+% break some past \mfpic{} files. It would also require recoding every
+% command that now writes one of these dimensions to the output.
+%
+% The following are default parameters for various graphics commands that
+% take an optional argument for the size of something.
+% \begin{itemize}
+% \item\cs{mfpicunit}\SpecialUsageIndex{mfpicunit}, the unit of length in
+% a picture. \cs{@mfpicunit} is used internally.
+% \item\cs{pointsize}\SpecialUsageIndex{pointsize}, the diameter of a
+% \cs{point}.
+% \item\cs{shadespace}\SpecialUsageIndex{shadespace}, the spacing of dots
+% in \cs{shade}.
+% \item\cs{polkadotspace}\SpecialUsageIndex{polkadotspace}, the spacing of
+% polkadots.
+% \item\cs{hatchspace}\SpecialUsageIndex{hatchspace}, the spacing of hatch
+% lines.
+% \item\cs{headlen}\SpecialUsageIndex{headlen}, the size of arrowheads
+% (except on axes).
+% \item\cs{axisheadlen}\SpecialUsageIndex{axisheadlen}, the size of the
+% arrowhead for $x$- and $y$-axes.
+% \item\cs{sideheadlen}\SpecialUsageIndex{sideheadlen}, the size of the
+% arrowhead for border axes.
+% \item\cs{hashlen}\SpecialUsageIndex{hashlen}, the length of tic marks on
+% axes.
+% \item\cs{dashlen}\SpecialUsageIndex{dashlen}, the length of dashes in
+% \cs{dashed}.
+% \item\cs{dashspace}\SpecialUsageIndex{dashspace}, the space between such
+% dashes.
+% \item\cs{dotsize}\SpecialUsageIndex{dotsize}, the size of dots for
+% \cs{dotted}.
+% \item\cs{griddotsize}\SpecialUsageIndex{griddotsize}, the size of dots
+% for \cs{grid} and \cs{plrgridpoints}.
+% \item\cs{dotspace}\SpecialUsageIndex{dotspace}, the space between such
+% dots.
+% \item\cs{symbolspace}\SpecialUsageIndex{symbolspace}, the space between
+% symbols used in a \cs{plot} command.
+% \end{itemize}
+% We allocate them, and then initialize them.
+% \begin{macrocode}
+\mfp@DBlog{User level dimension parameters, with default settings.}%
+\newfdim\mfpicunit
+\newfdim\@mfpicunit % internal version
+\newfdim\pointsize
+\newfdim\shadespace
+\newfdim\polkadotspace
+\newfdim\hatchspace
+\newfdim\headlen
+\newfdim\axisheadlen
+\newfdim\sideheadlen
+\newfdim\hashlen
+\newfdim\dashlen
+\newfdim\dashspace
+\newfdim\dotsize
+\newfdim\griddotsize
+\newfdim\dotspace
+\newfdim\symbolspace
+\mfpicunit1pt
+\pointsize2pt
+\shadespace1pt
+\polkadotspace10pt
+\hatchspace3pt
+\headlen3pt
+\axisheadlen5pt
+\sideheadlen0pt
+\hashlen4pt
+\dashlen4pt
+\dashspace4pt
+\dotsize0.5pt
+\griddotsize0.5pt
+\dotspace3pt
+\symbolspace5pt
+% \end{macrocode}
+%
+% Font dimensions are global, and so not restricted to an \env{mfpic}
+% environment even when assigned inside one. This would require the
+% user to restore a value every time one picture needed a different
+% value from the rest. To avoid requiring this, the following macro is
+% used to save all values of these dimensions at the start of a picture,
+% and to restore the saved values at the end. Some internal dimensions
+% (all the \cs{@graph...}, \cs{@tlabels...} and \cs{tb@...}) don't need
+% saving since they are reset with each picture.
+% \begin{macrocode}
+\newdef\save@mfpicdimens{%
+ \edef\restore@mfpicdimens{%
+ \mfpicunit \the\mfpicunit
+ \pointsize \the\pointsize
+ \shadespace \the\shadespace
+ \polkadotspace \the\polkadotspace
+ \hatchspace \the\hatchspace
+ \headlen \the\headlen
+ \axisheadlen \the\axisheadlen
+ \sideheadlen \the\sideheadlen
+ \hashlen \the\hashlen
+ \dashlen \the\dashlen
+ \dashspace \the\dashspace
+ \dotsize \the\dotsize
+ \griddotsize \the\griddotsize
+ \dotspace \the\dotspace
+ \symbolspace \the\symbolspace
+ \tlabel@hadj \the\tlabel@hadj
+ \tlabel@vadj \the\tlabel@vadj
+ \tlabel@sep \the\tlabel@sep
+ \tlpath@sep \the\tlpath@sep}}%
+% \end{macrocode}
+%
+% Here are the few dimension registers we do allocate:
+% \SpecialUsageIndex{\mfpframesep}\cs{mfpframesep}, and
+% \SpecialUsageIndex{\mfpframethickness}\cs{mfpframethickness}
+% govern frames.
+% \SpecialUsageIndex{\mfpicheight}\cs{mfpicheight} and
+% \SpecialUsageIndex{\mfpicwidth}\cs{mfpicwidth} are set
+% after each picture to that picture's size. A \cs{vskip} of
+% \SpecialUsageIndex{\mfpiccaptionskip}\cs{mfpiccaptionskip} is placed
+% between figure and caption by \cs{tcaption}. We also allocate a
+% frequently used scratch register
+% \begin{macrocode}
+\newdimen\mfp@scratch
+\newdimen\mfpframesep
+\newdimen\mfpframethickness
+\newdimen\mfpicheight
+\newdimen\mfpicwidth
+\newskip\mfpiccaptionskip
+\mfpframethickness0.4pt
+\mfpframesep2pt
+\mfpiccaptionskip\medskipamount
+% \end{macrocode}
+%
+%
+%
+% \section{Helper macros}\label{helper}
+%
+% \DescribeMacro{\mfpicnumber}
+% We make a user-level macro for setting the number of the next graph.
+% The original purpose of this is so that the font file could have
+% characters in the positions of printable characters. Then creating a
+% composite figure is as easy as changing fonts: \marg{\cs{pic} ABABAAB}.
+% The motivation came from a discussion on \texttt{comp.text.tex} started
+% by someone who wanted to input a sequence of 0's and 1's and get a
+% sequence of black and white squares. For \MP{} users it is not very
+% important.
+% \begin{macrocode}
+\def\mfpicnumber#1{\global\mfp@count#1\relax}%
+% \end{macrocode}
+%
+% \DescribeMacro{\mfpaddto}
+% When placing text at intervals on an axis, we need to automatically
+% calculate coordinates to feed to the \cs{tlabel} command. The
+% following is a utility macro to add two real numbers. It is currently
+% used only in the \cs{axislabels} command, so it needn't be globally
+% defined, but it might be handy in the future.
+% \begin{itemize}
+% \item[] \arg{1} is a \emph{macro} that expands to some number.
+% \item[] \arg{2} is a number to add to it.
+% \end{itemize}
+% The result is stored again in \arg{1}, overwriting the old value.
+%
+% \DescribeMacro{\mfpmultiply}
+% This is the multiplicative version, with the same syntax and behavior.
+% It is not used anywhere yet.
+% \begin{macrocode}
+\newdef\mfpaddto#1#2{\mfp@scratch#1pt \advance\mfp@scratch#2pt
+ \edef#1{\strip@pt\mfp@scratch}}%
+\newdef\mfpmultiply#1#2{\mfp@scratch#1pt \mfp@scratch#2\mfp@scratch
+ \edef#1{\strip@pt\mfp@scratch}}%
+\mfp@ifundefined\strip@pt
+ \begingroup
+ \lccode`\/`\p
+ \lccode`\?`\t
+ \lowercase{%
+ \endgroup
+ \def\rem@pt#1.#2/?}{#1\ifnum#2>\z@.#2\fi}%
+ \def\strip@pt{\@xp\rem@pt\the}%
+\mfp@enddef
+% \end{macrocode}
+%
+% \DescribeMacro{\pointdef}
+% A point defining macro, suggested by a user.
+% \begin{itemize}
+% \item[] \arg{1} is a symbolic name for a point, only letters.
+% \item[] \arg{2} and \arg{3} are the coordinates.
+% \item[] Usage: \cs{pointdef}\marg{A}\parg{2,5} makes \cs{A} expand
+% to \texttt{(2,5)}, \cs{Ax} to \texttt{2}, and \cs{Ay} to
+% \texttt{5}.
+% \end{itemize}
+% \begin{macrocode}
+\newdef\pointdef#1(#2,#3){%
+ \@ifundefined{#1} {}{\Mfpic@warn{\mfp@b#1 already defined.}}%
+ \@ifundefined{#1x}{}{\Mfpic@warn{\mfp@b#1x already defined.}}%
+ \@ifundefined{#1y}{}{\Mfpic@warn{\mfp@b#1y already defined.}}%
+ \@namedef{#1}{(#2,#3)}\@namedef{#1x}{#2}\@namedef{#1y}{#3}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\setmfarray}
+% This enables the simultaneous definition of \MF{} (or \MP) variables
+% such as \mfc{A1}, \mfc{A2}, etc., by writing
+% \cs{setmfarray}\marg{\meta{type}}\marg{\meta{name}}\marg{\meta{list}},
+% the \meta{name} being any \MF{} variable without a suffix, the list being
+% a comma separated list of items of the appropriate \meta{type}.
+% For example, \cs{setmfarray}\marg{pair} would require a list of
+% pairs.
+%
+% \SpecialUsageIndex{\numericarray}
+% As with \cs{setmfvariable}, we provide a few abbreviations.
+% \cs{numericarray} is for arrays of numeric variables,
+% \SpecialUsageIndex{\pairarray} \cs{pairarray} for arrays of pair
+% variables and \SpecialUsageIndex{\colorarray}\cs{colorarray} is for
+% arrays of color variables. Finally, to set arrays globally, there is
+% \SpecialUsageIndex{\globalsetmfarray}\cs{globalsetmfarray}.
+% \begin{macrocode}
+\newdef\setmfarray#1#2{\mfsrc{setarray (#1) (#2)}\mfp@writedata}%
+\newlet\setmparray\setmfarray
+\newdef\pairarray{\setmfarray{pair}}%
+\newdef\numericarray{\setmfarray{numeric}}%
+\newdef\colorarray{\setmfarray{color}}%
+\newdef\rgbcolorarray{\setmfarray{rgbcolor}}%
+\newdef\cmykcolorarray{\setmfarray{cmykcolor}}%
+\newdef\globalsetmfarray#1#2{\mfsrc{gsetarray (#1)(#2)}\mfp@writedata}%
+\newlet\globalsetmparray\globalsetmfarray
+% \end{macrocode}
+%
+% \DescribeMacro{\lightershade}
+% Shade adjustment macros. \cs{lightershade} just multiplies
+% \cs{shadespace} by $1.2$.
+% \DescribeMacro{\darkershade}
+% While \cs{darkershade} divides by $1.2$ (actually, multiplies by
+% $1/1.2$)
+% \begin{macrocode}
+\newdef\lightershade{\shadespace1.2\shadespace}%
+\newdef\darkershade{\shadespace.83333\shadespace}%
+% \end{macrocode}
+%
+% \DescribeMacro{\dashlineset}
+% The following arrange for contrastive dashed lines. \cs{dashlineset}
+% just resets the default parameters that the \cs{dashed} macro uses,
+% \DescribeMacro{\dotlineset}
+% \cs{dotlineset} sets these parameters so that \cs{dashed} produces
+% something almost dotted.
+% \begin{macrocode}
+\newdef\dashlineset{\dashlen4pt \dashspace4pt }%
+\newdef\dotlineset{\dashlen1pt \dashspace2pt }%
+% \end{macrocode}
+%
+% \DescribeMacro{\pointfilltrue}
+% With the commands \cs{pointfilltrue} or \cs{pointfillfalse} the user
+% can specify points drawn with the \cs{point} command to be either
+% filled in or
+% \DescribeMacro{\pointfillfalse}
+% not.
+% \begin{macrocode}
+\let\ifpointfill\iftrue
+\newdef\pointfilltrue{\mfp@let\ifpointfill\iftrue}%
+\newdef\pointfillfalse{\mfp@let\ifpointfill\iffalse}%
+% \end{macrocode}
+%
+% \CMF{} has very flexible looping capabilities. Before version 0.8 there
+% was no \mfpic{} interface because it seemed that \cs{mfsrc} was the
+% simplest way to access this capability. Anything else would have to
+% be either a clone of \cs{mfsrc} or limit the possibilities. However, I
+% now believe it might be a good thing to have a structured way to perform
+% repetitive actions at the \MF{} level.
+%
+% As of version 0.9, \mfpic{} will allow loops outside \env{mfpic}
+% environments. It seems like a reasonable thing to be allowed to do.
+% Fortunately, the macro names are already unique.
+%
+% \DescribeMacro{\mfpfor}
+% This is the simplest: it writes the word \mfc{for} plus the contents of
+% its only mandatory argument, followed by a colon. The loop is ended by
+% \DescribeMacro{\endmfpfor}
+% \cs{endmfpfor} which merely writes the text \mfc{endfor}.
+% \begin{macrocode}
+\newdef\mfpfor#1{\begingroup\mfsrc{for #1:}}%
+\newdef\endmfpfor{\mfsrc{endfor}\endgroup}%
+% \end{macrocode}
+%
+% \DescribeMacro{\mfploop}
+% The macro \cs{mfploop} starts a loop with \mfc{forever}. The user is
+% expected to provide an escape condition with
+% \DescribeMacro{\mfpuntil}
+% \cs{mfpuntil}, which can be placed anywhere among the commands contained
+% in the \env{mfploop} environment.
+% \DescribeMacro{\endmfploop}
+% The macro \cs{endmfploop} is a clone of \cs{endmfpfor} except for the
+% warning. Grouping keeps the warning from being turned off if
+% \cs{mfpuntil} occurs in a nested loop.
+% \begin{macrocode}
+\newdef\mfploop{%
+ \begingroup\mfsrc{forever:}\let\@nountil\mfp@untilwarn}%
+\newdef\mfpuntil#1{\mfcmd{exitif #1}\let\@nountil\@empty}%
+\newdef\endmfploop{\@nountil\endmfpfor}%
+% \end{macrocode}
+%
+% \DescribeMacro{\mfpwhile}
+% This emulates a while-loop, the condition being given in the mandatory
+% argument. It has the same behavior, and writes almost the same code as
+% \cs{mfploop} with an immediate \cs{mfpuntil} using the negative of the
+% condition.
+% \DescribeMacro{\endmfpwhile}
+% The closing, \cs{endmfpwhile}, is again the same as \cs{endmfpfor}.
+% \begin{macrocode}
+\newdef\mfpwhile#1{\begingroup\mfcmd{forever: exitif not(#1)}}%
+\newlet\endmfpwhile\endmfpfor
+% \end{macrocode}
+%
+% \DescribeMacro{\plr}
+% A little utility to convert a list of pairs into another list of
+% pairs, where the given list are assumed to be polar coordinates and the
+% results are their Cartesian counterparts. This is guaranteed to work as
+% expected only in macros with a list argument, or for a single pair.
+% \DescribeMacro{\midpoint}
+% And one to calculate the midpoint, given two points separated by a comma.
+% Can be used in graphics macros where a point is required.
+% \begin{macrocode}
+\newdef\plr#1{map (polar) (#1)}%
+\newdef\midpoint#1{(0.5[#1])}%
+% \end{macrocode}
+%
+%
+%
+% \section{Macros to implement prefix commands}\label{prefix}
+%
+% Some \mfpic{} macros, like \cs{circle} create a path. When used without
+% a prefix, they also draw the path. At the \MF{} level, every figure
+% macro is syntactically a path expression. Other macros are prefix macros
+% and operate on a figure macro that follows. Every prefix accepts a path
+% expression as input and has a path as its return value, so a combination
+% prefix + path expression is again a path expression.
+%
+% Prefixes may also have side effects, such as adding some ink to the
+% current picture. Such prefixes are called \emph{rendering} macros.
+%
+% Rendering macros normally return the same path as their argument, but
+% there are macros that modify the path. Some apply a linear transform,
+% some close an open path, some return a subpath. These we call
+% \emph{modification} macros.
+%
+% This scheme is quite simple, but unfortunately too simple. To avoid
+% having to explicitly type a rendering prefix every time, a figure
+% macro needs to know if a rendering prefix has occured, so prefix macros
+% need to pass information to the figure macro (and to later prefixes).
+%
+% Moreover, \MF{} abhors an `isolated expression', so the first prefix
+% has to write something with a different syntax, and pass the info that
+% it has already done so on to later prefixes, who need to test that
+% info. Thus, every prefix macro needs to know if it is the first or a
+% later one. They test this \cs{if} command and also set it to false. The
+% figure macros, which are last in the chain, close by resetting it true.
+% \begin{macrocode}
+\mfp@DBlog{Tests to control multiple prefix commands.}%
+\newif\if@mfpstart\@mfpstarttrue
+% \end{macrocode}
+%
+% If a prefix is a rendering prefix, it writes its rendering code and sets
+% the following switch to false, letting the figure macro know it needn't
+% invoke the default rendering. Figure macros close by resetting it true.
+% Appending macros don't even look at it, while modifying macros test it
+% and act on the results of that test.
+% \begin{macrocode}
+\newif\if@mfprend\@mfprendtrue
+% \end{macrocode}
+%
+% The following macro is always (and only) invoked by the first prefix
+% macro (or the figure macro, if there are no prefixes). It writes the
+% code that prevents an isolated expression. It stores the following path
+% expression in the variable \gbc{curpath}. Thus every combination of
+% prefixes followed by a figure macro is syntactically a \MF{} variable
+% assignment command.
+% \begin{macrocode}
+\newdef\mfp@storepath{\mfsrc{\@nl store (curpath)}}%
+% \end{macrocode}
+%
+% Every prefix and figure macro has to invoke \cs{mfp@first}, since each
+% is potentially the first in the sequence. If \cs{@mfpstart} is true, it
+% writes the storing command, sets \cs{@mfpstartfalse}, and says that
+% rendering is needed. (The rendering macros immediately cancel that
+% switch.)
+% \begin{macrocode}
+\newdef\mfp@first{\relax
+ \if@mfpstart \mfp@storepath \@mfprendtrue \@mfpstartfalse\fi}%
+% \end{macrocode}
+%
+% Every rendering macro invokes this next command. After the first-stage
+% code it writes its rendering command (via \cs{mfsrc}) and passes on the
+% info that rendering is no longer required of the figure macro.
+% \begin{macrocode}
+\newdef\mfp@rendmac{\mfp@first \@mfprendfalse \mfsrc}%
+% \end{macrocode}
+%
+% The \emph{figure} macros proceed as follows: (i)~they invoke the
+% \cs{mfp@first} code, then (ii)~write a rendering macro if necessary
+% (\cs{mfp@autorend}), (iii)~then make \cs{@mfpstarttrue} (for the next
+% figure) and write \MF{} code that defines the path. The \cs{mfp@modmac}
+% macros do only the first two, then write their modification code,
+% counting on the following figure macro to do (iii). The idea is that
+% [modifier + figure] should act the same as if the path it defines were
+% encoded as a single figure command.
+% \begin{macrocode}
+\newdef\mfp@modmac{\mfp@first\mfp@autorend\mfsrc}%
+\newdef\mfp@autorend{\if@mfprend\@xp\mfp@render\fi}%
+% \end{macrocode}
+%
+% Only axis and arrow commands (plus \cs{reverse}) use \cs{mfp@addmac}.
+% As the name suggests, something is added onto an existing figure. That
+% is, it is designed to ignore the rendering state and not invoke implicit
+% rendering. If no explicit rendering occurs, the figure macro that
+% follows invokes a rendering. For arrows, this causes the arrowhead to be
+% drawn on top of the drawing of the shaft.
+%
+% It would seem \cs{reverse} doesn't fit this name, but it also needs to
+% be transparent to the rendering state so that it will work in
+% combination with arrows.
+%
+% \cs{mfp@addmac} simply invokes the common first-stage code and writes
+% its \MF{} code.
+% \begin{macrocode}
+\newdef\mfp@addmac{\mfp@first \mfsrc}%
+% \end{macrocode}
+%
+% The \cs{mfp@addmac} and \cs{mfp@modmac} macros differ when there are no
+% preceeding rendering macros. In that case it is as if \cs{draw} (or
+% more generally \cs{mfp@render}) precedes the \cs{mfp@modmac} macro. That
+% drawing happens \emph{in addition to} any rendering macros that follow.
+% This implicitly added \cs{draw} is not present for \cs{mfp@addmac}
+% macros; it is added by the following figure macro. The effect of this is
+% the \cs{mfp@addmac} macros perform their additions on top of an already
+% rendered figure.
+%
+% \DescribeMacro{\setrender}
+% As mentioned above, when a figure macro is invoked without a rendering
+% prefix, it will invoke some default. That default is stored in a macro
+% and can be changed for special effects. Initially we define it as
+% \cs{draw} (with empty optional argument) to draw solid paths in the
+% default color, but we provide a command for the user to change it.
+% \begin{macrocode}
+\newdef\mfp@render{\draw[]}%
+\newdef\setrender{\mfp@def\mfp@render}%
+% \end{macrocode}
+%
+% A figure macro will invoke the common first-stage code and the rendering
+% check. It also resets the \cs{@mfpstart} switch. Because a figure macro
+% completes an assignment command in \MF, it writes its code with
+% \cs{mfcmd}, which places a semicolon at the end. We define the first
+% part (all but the writing) as a separate command because there are a few
+% cases where we need to delay that writing.
+% \begin{macrocode}
+\newdef\mfp@figmacbase{\mfp@first \mfp@autorend \@mfpstarttrue}%
+\newdef\mfp@figmac{\mfp@figmacbase\mfcmd}%
+% \end{macrocode}
+%
+% The list macros take as their last argument a list of $(x,y)$ pairs (or
+% occasionally a list of numbers). That list can either be typed directly
+% into the \TeX{} source file (inside braces, separated by commas) or be
+% present in an external datafile. Either way, the data is written as a
+% list within an enclosing pair of parentheses. Since these are all
+% figure macros, \cs{mfp@listmac} starts out just like \cs{mfp@figmac},
+% but it writes only part of its arguments, the last one (the data list)
+% being handled by \cs{mfp@writedata}.
+%
+% \cs{mfp@writedata} looks ahead to see if the token \cs{datafile}
+% follows, if it does \cs{mfp@writefile} is invoked, which gobbles
+% the \cs{datafile} token and processes the filename that should follow it.
+% Otherwise, \cs{mflist} is invoked to write the list in braces
+% that should follow.
+%
+% \cs{mfp@writedata} is also used by non-figure macros such as the axis
+% marks macros, plus \cs{piechart} and \cs{barchart}.
+% \begin{macrocode}
+\newdef\mfp@listmac#1{\mfp@figmacbase\mfsrc{#1}\mfp@writedata}%
+\newdef\mfp@writedata{\mfp@iftoken\datafile{\mfp@writefile}{\mflist}}%
+\newdef\mfp@writefile{%
+ \mfp@ifopengraphsfile
+ {\mfp@processfile{\mfp@rwdata}}{\mfp@nooutput@error\datafile}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\mftitle}
+% This is mainly for debugging the example files. \cs{mfp@title} takes
+% two arguments: the command that called it (for the error message of
+% \cs{mfp@src}), and some \TeX{} code. That code is written into the string
+% argument of the \grafbase{} command \gbc{mftitle} and ends up in the
+% \MF{} \file{.log} file and as a comment in the GF file. \cs{mftitle}
+% merely calls \cs{mfp@title}, while
+% \DescribeMacro{\tmtitle}
+% \cs{tmtitle} also writes the argument to the \TeX{} \file{.log} file and
+% typesets it in the document.
+%
+% We write the argument as a token list because that is the the easiest
+% verbatim-like way to do it.
+% \begin{macrocode}
+\newdef\mftitle#1{\@bsphack\mfp@title\mftitle{#1}\@esphack}%
+\newdef\tmtitle#1{\mfp@title\tmtitle{#1}\wlog{\the\mfp@toks}{#1}}%
+\newdef\mfp@title#1#2{\mfp@toks{#2}%
+ \mfp@write\mfp@src#1{mftitle "\the\mfp@toks";}}%
+% \end{macrocode}
+%
+%
+%
+% \section{Macros for getting data from files}\label{rwdata}
+%
+% The following command contains the common code to open a data file for
+% reading. The second parameter is the filename, the first is what to do
+% if it exists and is nonempty. We first try to \cs{openin} the file and
+% issue an error message if that fails. Then we skip any initial blank
+% lines and comments and complain if that takes us to the end of the
+% file. Otherwise we execute the command specified. Afterwards, since
+% each \cs{plotdata} command in a given \env{mfpic} environment
+% starts with the linetype at which the previous one ended, we allow the
+% value of \cs{mfp@linetype} to survive this group but not any enclosing
+% group.
+% \begin{macrocode}
+\newdef\mfp@processfile#1#2{%
+ \openin\mfp@data=#2\relax
+ \ifeof\mfp@data \mfp@nofile@error{#2}%
+ \else
+ \mfp@DBlog{Opening file #2 for data.}%
+ {%
+ \mfp@resetwhitespace \mfp@setcomment
+ \mfp@skipblanklines
+ \ifeof\mfp@data
+ \mfp@nodata@error{#2}%
+ \else
+ \message{< Mfpic: Reading #2}%
+ #1%
+ \message{>}%
+ \fi
+ \@xp
+ }\@xp\mfp@linetype\number\mfp@linetype\relax
+ \mfp@DBlog{Closing file #2.}%
+ \fi
+ \closein\mfp@data}%
+% \end{macrocode}
+%
+% When reading files it is important that white space in the files
+% always have the same meaning for \TeX{}. This utility resets spaces,
+% tabs and EOLs to the standard categories.
+% \begin{macrocode}
+\newdef\mfp@resetwhitespace{\mfp@restorepar
+ \endlinechar`\^^M \catcode`\^^M5 \catcode`\ 10 \catcode`\^^I10 }%
+% \end{macrocode}
+%
+% \cs{mfp@skipcomments}
+% does what it says, skipping comments in a data file. It reads a
+% line from a previously opened file (input stream \cs{mfp@data}) and
+% compares it to \cs{@empty}. It calls itself when they are equal (a
+% comment line, a blank line would put a \cs{par} token in \cs{mfp@data}).
+% When this macro finishes, \cs{mfp@temp} contains either \cs{par} (blank
+% line or EOF) or real data.
+% \begin{macrocode}
+\newdef\mfp@skipcomments{\read\mfp@data to \mfp@temp
+ \ifx\mfp@temp\@empty \@xp\mfp@skipcomments\fi}%
+% \end{macrocode}
+%
+% \cs{mfp@skipblanklines}
+% first skips comments. After that \cs{mfp@temp} can contain \cs{par},
+% indicating either EOF or a blank line, or something else. In the first
+% case (EOF) the macro terminates, in the second case (blank) it calls
+% itself again, in the other case (something else) it also terminates,
+% having presumably found real data.
+%
+% When this is called, a file should already be open using the handle
+% \cs{mfp@data}.
+% \begin{macrocode}
+\newdef\mfp@skipblanklines{\mfp@skipcomments
+ \ifeof\mfp@data
+ \else\ifx\mfp@temp\mfp@par \@XP\mfp@skipblanklines
+ \fi\fi}%
+% \end{macrocode}
+%
+% The following two commands, \cs{mfp@rwdata} and \cs{mfp@rwdataloop}
+% perform the work of writing the file (read stream \cs{mfp@data} opened
+% by \cs{mfp@processfile}).
+%
+% The figure counter \cs{mfp@count} is reused (this should all be inside
+% a group) to keep track of how many lines of the data file have been
+% written to one line of the output. The token \cs{sequence} in a
+% \cs{using} specification will expand to the value of the counter
+% \cs{mfp@sequence}.
+%
+% When \cs{mfp@rwdata} is called, a non-blank line has been read to the
+% macro \cs{mfp@temp}, to which we now add a space. The macro
+% \cs{mfp@buffer} is what will be written to the output file as soon as it
+% is long enough. We initialize it by defining it to be an open
+% parenthesis followed by the result of applying \cs{parse@data} to the
+% contents of \cs{mfp@temp}. Then we call \cs{mfp@rwdataloop}, which does
+% the remaining reading and all the writing.
+% \begin{macrocode}
+\newdef\mfp@nextdatum{\@xp\parse@data\mfp@temp\mfp@end}%
+\newdef\mfp@rwdata{\mfp@count1 \mfp@sequence1
+ \def\sequence{\number\mfp@sequence}%
+ \edef\mfp@temp{\mfp@temp\space}%
+ \edef\mfp@buffer{(\mfp@nextdatum}\mfp@rwdataloop}%
+% \end{macrocode}
+%
+% At this point, the buffer has been started with the first datum, so we
+% read another line. If that is blank, we write the buffer (and the closing
+% parenthesis and a semicolon) and terminate the loop. Otherwise we have
+% read another datum and then we:
+% \begin{enumerate}
+% \item increment the sequence counter,
+% \item add a comma to the buffer,
+% \item test whether the buffer is `full', and if so, write the buffer
+% to the output and clear it, then
+% \item add the datum to the buffer.
+% \end{enumerate}
+% The buffer is `full' when \cs{mfp@count} (the number of data in the
+% buffer) exceeds \cs{mfp@dataperline}.
+%
+% \cs{mfp@rwdataloop} loops by calling itself at the end. Whenever
+% \cs{mfp@temp} is empty (comment line read), (1)--(4) are skipped and
+% another line read.
+% \begin{macrocode}
+\newdef\mfp@rwdataloop{%
+ \read\mfp@data to \mfp@temp
+ \ifx\mfp@temp\mfp@par
+ \mfcmd{\mfp@buffer)}%
+ \else
+ \ifx\mfp@temp\@empty
+ \else
+ \edef\mfp@temp{\mfp@temp\space}%
+ \advance\mfp@sequence1
+ \edef\mfp@buffer{\mfp@buffer\mfp@join}%
+ \ifnum\mfp@count<\mfpdataperline\relax \advance\mfp@count1
+ \else \mfsrc{\mfp@buffer}\mfp@count1 \def\mfp@buffer{}%
+ \fi
+ \edef\mfp@buffer{\mfp@buffer\mfp@nextdatum}%
+ \fi
+ \@xp\mfp@rwdataloop
+ \fi}%
+% \end{macrocode}
+%
+% \section{Various \mfpic{} Settings.}\label{mfpsettings}
+%
+% \subsection {\CMF{} settings}\label{MFsettings}
+%
+% \DescribeMacro{\penwd}
+% Macros that write changes in default \grafbase{} variables now use the
+% construct ``\gbc{save x; <type> x; x := <value>;}'' (via a call to
+% \cs{setmfvariable}). This makes changes local if inside a given \env{mfpic}
+% environment, global if outside. An exception is \gbc{penwd} which, as an
+% internal variable uses `\gbc{interim}' instead (via the \grafbase{}
+% command \gbc{resizedrawpen}).
+%
+% \SpecialUsageIndex{\drawpen}
+% \cs{drawpen} was the original command used to set the pen diameter
+% (prior to version 0.2.13). At some point it was replaced by
+% \SpecialUsageIndex{\pen}\cs{pen}. However \cs{penwd} seems to be more
+% consistent with other commands (e.g., \cs{shadewd}), and we keep all three.
+% \begin{macrocode}
+\newdef\pen#1{\mfcmd{\@nl resizedrawpen (#1)}}%
+\newlet\penwd\pen \newlet\drawpen\pen
+% \end{macrocode}
+%
+% \DescribeMacro{\shadewd}
+% The \cs{shade} macro works in \MF{} by adding many copies of the small
+% picture \gbc{shadedot} to the figure, in \MP{} the parameters
+% \gbc{shadewd} and \cs{shadespace} are simply used to calculate a level
+% of gray and a color fill is used.
+%
+% \DescribeMacro{\polkadotwd}
+% However, both \MF{} and \MP{} handle polkadots the same. The only
+% difference is \MF's need for fitting to a pixel grid. But that is
+% handled in \grafbase.
+% \begin{macrocode}
+\newdef\shadewd#1{\setmfnumeric{shadewd}{#1}}%
+\newdef\polkadotwd#1{\setmfnumeric{polkadotwd}{#1}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\hatchwd}
+% The thickness of lines in the hatching fill is set by \cs{hatchwd}.
+% \begin{macrocode}
+\newdef\hatchwd#1{\setmfnumeric{hatchwd}{#1}%
+ \setmfvariable{pen}{hatchpen}{pencircle scaled hatchwd}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\dashpattern}
+% A dash pattern consists (internal to \MF) of an array of lengths. The
+% \grafbase{} command \gbc{dashpat} accepts a name, and a sequence of
+% numbers representing the sizes of dashes and spaces, and builds this
+% array. It differs from \cs{numericarray} in that a little additional
+% checking is done (at the \grafbase{} level).
+% \begin{macrocode}
+\newdef\dashpattern#1{\mfsrc{save #1; dashpat (#1)}\mfp@writedata}%
+% \end{macrocode}
+%
+% \DescribeMacro{\settension}
+% This sets the default tension for any command (for example, \cs{curve})
+% that takes an optional argument that contains a tension value. This
+% includes also \cs{function}, \cs{plrfcn}, \cs{fcncurve}, \cs{datafile},
+% \cs{plotdata}. It also sets the default tension for \cs{sclosed} and
+% \cs{bclosed}. The value should be at least $3/4$.
+% \begin{macrocode}
+\newdef\settension#1{\setmfnumeric{default_tension}{#1}}%
+% \end{macrocode}
+%
+% \subsection{Settings for text label placement in figures}
+% \label{labelsettings}
+%
+% \cs{tlabel}\,s are `justified' by placing a particular point of the
+% text at the location specified. The \cs{tlabel} command accepts an
+% optional argument containing two letters. There are 4 possibilities
+% for the first and three for the second. It turns out to be easier to
+% process these if we have a command that converts these to numbers, and
+% then \cs{ifcase} can be used. There is a third part of the optional
+% argument usable only in \MP{}: a number indicating degrees of rotation.
+%
+% \DescribeMacro{\tlabeljustify}
+% The multi-part definition of \cs{tlabeljustify} is to guard
+% against confusing error messages if some part of this optional argument
+% is empty. The code assumes that if there are only 2, 1 or no parts, then
+% later parts should be left unchanged. The rotation part is empty by
+% default (rather than 0), because some commands will emit a warning if
+% a nonempty rotation exists without \opt{mplabels}.
+%
+% The result of \cs{tlabeljustify} is that \cs{tl@vpos} is set equal to
+% $-1$ if the first letter is (default) \texttt{B} (for Baseline), and
+% 0--2 if it is \texttt{b} (bottom), \texttt{c} (center) or \texttt{t}
+% (top). Similarly, \cs{tl@hpos} is set equal to $-1$ if the second
+% letter is (default) \texttt{l} (for left), and 0 or 1 if it is
+% \texttt{c} (center) or \texttt{r} (right). Making the defaults
+% negative means they must be handled by the \cs{else} portion of any
+% \cs{ifcase}.
+%
+% The rotation part is just saved in \cs{tl@rot}. This is true even
+% without \opt{mplabels}, in case it is turned on later. It is ignored by
+% \cs{tlabel} without \opt{mplabels}, but a warning is issued here anyway.
+% \begin{macrocode}
+\newcount\tl@vpos \newcount\tl@hpos
+\newdef\mfp@tlh@l{-1}%
+\newdef\mfp@tlh@c{0}%
+\newdef\mfp@tlh@r{1}%
+\newdef\mfp@tlv@B{-1}%
+\newdef\mfp@tlv@b{0}%
+\newdef\mfp@tlv@c{1}%
+\newdef\mfp@tlv@t{2}%
+\newdef\tl@rot{}\tl@vpos-1 \tl@hpos-1
+\newdef\tlabeljustify#1{\mfp@ifexist{#1}{\mfp@justify#1\mfp@end}}%
+\newdef\mfp@justify#1#2\mfp@end{\mfp@local\mfp@vconv{#1}%
+ \mfp@ifexist{#2}{\mfp@@justify#2\mfp@end}}%
+\newdef\mfp@@justify#1#2\mfp@end{\mfp@local\mfp@hconv{#1}%
+ \mfp@local\edef\tl@rot{#2}%
+ \@ifmplabels{}%
+ {\mfp@ifexist{#2}%
+ {\Mfpic@warn{Rotation of labels is not supported without the %
+ mplabels\@msgbreak option. Rotation setting\on@line\space
+ may be ignored.}}%
+ }}%
+\newdef\mfp@vconv#1{%
+ \@ifundefined{mfp@tlv@#1}%
+ {\tl@vpos-1 \mfp@justify@error}%
+ {\tl@vpos\@nameuse{mfp@tlv@#1}\relax}}%
+\newdef\mfp@hconv#1{%
+ \@ifundefined{mfp@tlh@#1}%
+ {\tl@hpos-1 \mfp@justify@error}%
+ {\tl@hpos\@nameuse{mfp@tlh@#1}\relax}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\tlpathjustify}
+% This acts just like \cs{tlabeljustify} from the user's point of view.
+% Since it is only for users, and not used internally, it merely defines
+% three macros that are passed to \cs{tlabeljustify} when no justification
+% parameter is present for a \cs{tlabel}-path command.
+% \begin{macrocode}
+\newdef\tlpathjustify#1{%
+ \mfp@ifexist{#1}{\tlp@justify#1\mfp@end}}%
+\newdef\tlp@justify#1#2\mfp@end{\mfp@def\tlpath@v{#1}%
+ \mfp@ifexist{#2}{\tlp@@justify#2\mfp@end}}%
+\newdef\tlp@@justify#1#2\mfp@end{%
+ \mfp@def\tlpath@h{#1}\mfp@def\tlpath@r{#2}}%
+\tlpathjustify{cc}%
+% \end{macrocode}
+%
+% \DescribeMacro{\tlpointsep}
+% This takes an absolute length as its argument and sets parameters that
+% determine by how much a tlabel is separated from the point it is placed
+% at. This is in case something is drawn at that point, such as a dot or
+% a hash mark on an axis.
+%
+% \DescribeMacro{\tlpathsep}
+% This takes an absolute length and sets parameters that determine by how
+% much a path created by one of the \cs{tlabel\meta{path}} macros is
+% separated from the label it surrounds.
+%
+% \DescribeMacro{\tlabelsep}
+% This simply issues both the above commands. This is for compatibility
+% with \mfpic{} versions 0.7 or earlier, where both these separations were
+% determined by the same parameters.
+%
+% \DescribeMacro{\tlabeloffset}
+% \cs{tlabeloffset}, on the other hand, takes two arguments: dimensions,
+% representing offsets of labels from their point of placement. This is
+% mainly so that, if labels seem to be off, they can all be shifted with
+% one command.
+%
+% Both these values are written to the \MF{} file for the sake of the
+% commands that draw rectangles or ovals around labels.
+% \begin{macrocode}
+\newdef\tlpointsep#1{\mfp@local\tlabel@sep#1\relax
+ \setmfnumeric{label_sep}{\the\tlabel@sep}}%
+\newdef\tlpathsep#1{\mfp@local\tlpath@sep#1\relax
+ \setmfnumeric{labelpath_sep}{\the\tlpath@sep}}%
+\newdef\tlabelsep#1{\tlpointsep{#1}\tlpathsep{#1}}%
+\newdef\tlabeloffset#1#2{%
+ \mfp@local\tlabel@hadj#1\relax \mfp@local\tlabel@vadj#2\relax
+ \setmfpair{label_adjust}%
+ {(\the\tlabel@hadj,\the\tlabel@vadj)}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\everytlabel}
+% Every line of a multiline label is written inside a group, so
+% adjustments to baselineskip and other things are difficult or
+% impossible. Therefore, \cs{everytlabel} is provided to define code to be
+% inserted before the label code, outside these groups. The messing around
+% with the token register \cs{mfp@toks} is so one can put definitions in
+% the code without having to double \arg{} characters.
+%
+% The command \cs{extra@endtlabel} is called at the end of every
+% \cs{tlabel}. Normally it does nothing, but the \cs{tlabels} command
+% defines it to repeat the start of a loop.
+% \begin{macrocode}
+\newdef\everytlabel{\afterassignment\mfp@everytlabel\mfp@toks}%
+\newdef\mfp@everytlabel{\mfp@local\edef\every@tlabel{\the\mfp@toks}%
+ \mfp@ifmpost
+ {\mfp@ifopengraphsfile
+ {\mfcmd{verbatimtex\@nl
+ \def\noexpand\MFPeverylabel{\the\mfp@toks} etex}}%
+ {}}%
+ {}}%
+\newdef\every@tlabel{}%
+\newdef\extra@endtlabel{}%
+% \end{macrocode}
+%
+% Without \opt{mplabels} it is impossible to use \MP{} variables in
+% a text label placement parameter. I have occasionally found it useful
+% to have a \TeX{} version of the numeric variable \gbc{pi}.
+% \begin{macrocode}
+\newdef\pinumber{3.14159}%
+% \end{macrocode}
+%
+% \subsection{Other settings}\label{othersettings}
+%
+% \DescribeMacro{\headshape}
+% This sets the arrowhead shape. The parameters are: the ratio of width
+% to length, the tension (see \file{grafbase.dtx}) and a boolean
+% (\gbc{true} or \gbc{false}) to say whether the the head is filled, or
+% only the barbs are drawn. It would be more consistent with other
+% commands to have one \TeX{} parameter with commas separating the
+% components, but this is legacy code.
+% \begin{macrocode}
+\newdef\headshape#1#2#3{\mfcmd{\@nl headshape (#1, #2, #3)}}%
+% \end{macrocode}
+%
+% There are up to 4 optional arguments to any given \cs{arrow} command,
+% and these can take any one of 6 special indicators ({\mytt b}, {\mytt c},
+% {\mytt f}, {\mytt l}, {\mytt r} or {\mytt s}). Here we define macros
+% to handle these options. Each macro is fed a \emph{value}. For example,
+% the indicator {\mytt b} must be followed by a length, and that length is
+% the argument of \cs{mfp@arr@b}.
+% \begin{macrocode}
+\newdef\mfp@arr@b#1{\edef\mfp@hlocate{#1}}%
+\newdef\mfp@arr@c#1{\edef\mfp@hcolour{#1}}%
+\newdef\mfp@arr@l#1{\edef\mfp@hlength{#1}}%
+\newdef\mfp@arr@r#1{\edef\mfp@hrotate{#1}}%
+\newlet\mfp@arr@f\mfp@arr@b
+\newlet\mfp@arr@s\mfp@arr@l
+% \end{macrocode}
+%
+% The \cs{arc} and \cs{circle} commands take an optional argument that
+% indicates how the arc or circle is defined (and therefore the format of
+% the non-optional argument). These are passed to different \grafbase{}
+% commands whose names start with \gbc{arc} or \gbc{circle}. The commands
+% below store the rest of that name. The switch is set true by the
+% \cs{arc} command and false by \cs{circle}.
+% \begin{macrocode}
+\newdef\mfp@arc@s{pps}%
+\newdef\mfp@arc@t{ppp}%
+\newdef\mfp@arc@p{\ifmfp@switch plr\fi}%
+\newdef\mfp@arc@a{\ifmfp@switch alt\fi}%
+\newdef\mfp@arc@c{cp\ifmfp@switch s\fi}%
+\newdef\mfp@arc@r{pp (\mfp@T)}%
+\newdef\mfp@arc@q{pp (\mfp@F)}%
+% \end{macrocode}
+%
+% \DescribeMacro{\mfpdefinecolor}
+% \prog{Mfpic} supports color under \MP{} by writing all color
+% definitions to the output \file{.mp} file. In order to support a
+% \LaTeX-like color syntax, all the color models allowed by the color
+% package correspond to a \grafbase{} function that returns a \MP{}
+% color value. For example, \gbc{cmyk(.3,.3,0,.7)} returns \gbc{(0,0,.3)},
+% a very dark blue. This can be set by
+% \cs{mfpdefinecolor}\marg{vdblue}\marg{cmyk}\marg{.3,.3,0,.7}.
+%
+% Several commands are provided to set certain special colors. Currently
+% we have the following commands, setting the color of the same name as
+% the command (except \cs{backgroundcolor} which sets a color named
+% \gbc{background}):
+% \cs{drawcolor}\SpecialUsageIndex{\drawcolor},\\
+% \cs{fillcolor}\SpecialUsageIndex{\fillcolor},
+% \cs{hatchcolor}\SpecialUsageIndex{\hatchcolor},
+% \cs{headcolor}\SpecialUsageIndex{\headcolor},
+% \cs{pointcolor}\SpecialUsageIndex{\pointcolor},
+% \cs{tlabelcolor}\SpecialUsageIndex{\tlabelcolor}, and\\
+% \cs{backgroundcolor}\SpecialUsageIndex{\backgroundcolor}.
+%
+% These commands support the \LaTeX-like syntax:\\
+% \indent\cs{fillcolor}\oarg{\meta{model}}\marg{\meta{parameters}}.\\
+% The code for \cs{backgroundcolor} has to be spelled out since the
+% command name has the string `\texttt{color}', but the color does not.
+% \begin{macrocode}
+\newdef\mfpdefinecolor#1#2#3{\setmfcolor{#1}{#2(#3)}}%
+\newdef\mfp@defclr#1#2#3{%
+ \setmfcolor{#1}{\mfp@ifempty{#2}{#3}{#2(#3)}}}%
+\newdef\mfp@newcolorcmd#1{%
+ \@namedef{#1\mfp@C}{\mfp@nullopt{\mfp@defclr{#1\mfp@C}}}}%
+\mfp@newcolorcmd{draw}\mfp@newcolorcmd{fill}\mfp@newcolorcmd{tlabel}%
+\mfp@newcolorcmd{hatch}\mfp@newcolorcmd{head}\mfp@newcolorcmd{point}%
+\def\backgroundcolor{\mfp@nullopt{\mfp@defclr{background}}}%
+% \end{macrocode}
+%
+% When \cs{plotdata} is used to draw several graphs from one file, each
+% sequence of non-blank lines corresponds to a different curve with a
+% different style of rendering. The user has the option to cycle through
+% different dashing patterns, different colors, or different symbols.
+%
+% Internally, \cs{plotdata} issues the command \cs{mfp@setstyle} (which
+% can be defined to be any rendering macro), followed by a path defining
+% command such as \cs{curve} or \cs{polyline}, followed by a command
+% that reads the data and writes it in a suitable format. The following
+% commands essentially allow the user to select the rendering macro.
+% They start by resetting \cs{mfp@linetype}, the count register that keeps
+% track of how many curves have been drawn, then they define
+% \cs{mfp@setstyle}.
+%
+% \DescribeMacro{\dashedlines}
+% \cs{dashedlines} uses the rendering prefix \cs{gendashed} with 6
+% different dashing patterns \gbc{dashtype0} through \gbc{dashtype5}.
+% These are actually selected in \MF{} through the command
+% \gbc{getdashpat}. This takes care of cycling back to
+% \gbc{dashtype0} when the number of available dash patterns is exceeded.
+% Dashed lines is the default.
+% \begin{macrocode}
+\newdef\dashedlines{\mfp@local\mfp@linetype0
+ \mfp@def\mfp@setstyle{\gendashed{getdashpat \number\mfp@linetype}}}%
+\dashedlines
+% \end{macrocode}
+%
+% \DescribeMacro{\coloredlines}
+% This uses \cs{draw} with an optional color parameter set to
+% \gbc{colortype0} through (by default) \gbc{colortype7}. These are color
+% variables defined by \grafbase{}. It also issues a warning if \MF{} is
+% being used. As with \cs{dashedlines}, a \MP{} command \gbc{getcolor}
+% selects these colors.
+% \begin{macrocode}
+\newdef\coloredlines{%
+ \mfp@ifmpost
+ {\mfp@local\mfp@linetype0
+ \mfp@def\mfp@setstyle{\draw[getcolor \number\mfp@linetype]}}%
+ {\Mfpic@warn{You can't use \mfp@b coloredlines in %
+ Metafont.\@msgbreak Using \dashedlines instead}%
+ \dashedlines}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\pointedlines}
+% This makes a sort of dotted line, except the dots are actually different
+% symbols. It uses the rendering macro \cs{plot} with a parameter cycling
+% through 9 different symbols.
+% \begin{macrocode}
+\newdef\pointedlines{\mfp@local\mfp@linetype0
+ \mfp@def\mfp@setstyle{\plot{getsymbol \number\mfp@linetype}}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\datapointsonly}
+% \cs{datapointsonly} uses symbols like \cs{pointedlines}, but places
+% them only at the data points from the file. It uses the rendering macro
+% \cs{plotnodes} with a parameter cycling through the same 9 symbols.
+% \begin{macrocode}
+\newdef\datapointsonly{\mfp@local\mfp@linetype0
+ \mfp@def\mfp@setstyle{\relax
+ \plotnodes{getsymbol \number\mfp@linetype}}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\reconfigureplot}
+% This command allows the user to set the dashes, colors and symbols used
+% by the \cs{plotdata} command. The first argument is the plotting method
+% to be changed (\texttt{dashes}, \texttt{colors}, or \texttt{symbols})
+% and the second argument is a comma separated list of appropriate type
+% variables for \cs{plotdata} to cycle through. For example,
+% \cs{reconfigureplot}\marg{dashes} takes a list of dashing patterns
+% (previously defined by the \cs{dashpattern} command) separated by
+% commas. It writes a command instructing \MF{} to redefine the arrays
+% \mfc{dashtype0}, \mfc{dashtype1}, etc., used with \cs{gendashed} when
+% \cs{dashedlines} is in effect. The number of patterns is counted by \MF,
+% which also handles the mod-ing operation now.
+%
+% \cs{reconfigureplot}\marg{colors} takes a list of colors (expressions or
+% color names previously defined) separated by commas. It redefines
+% \gbc{colortype0}, etc., used with \cs{plotdata} when \cs{coloredlines}
+% is in effect.
+%
+% \cs{reconfigureplot}\marg{symbols} takes a list of symbols (\MF{} path
+% expressions, one of the symbols available to \cs{plotsymbol}, or path
+% names defined by \cs{store}) separated by commas. It redefines
+% \gbc{pointtype0}, etc., used by \cs{plotdata} when \cs{pointedlines} or
+% \cs{datapointsonly} is in effect.
+%
+% \DescribeMacro{\defaultplot}
+% This restores \mfpic's predefined defaults for dash patterns, colors and
+% symbols in \cs{plotdata}. The one argument is one of the words
+% \texttt{dashes}, \texttt{colors}, or \texttt{symbols}, indicating which
+% plotting method is to be restored.
+% \begin{macrocode}
+\newdef\reconfigureplot#1{\mfsrc{setdata#1}\mfp@writedata}%
+\newdef\defaultplot#1{\mfcmd{default#1}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\mfplinetype}
+% The user-level command \cs{mfplinetype} allows the user to set the
+% starting value of \cs{mfp@linetype}. The default is to start at $0$.
+% \DescribeMacro{\mfplinestyle}
+% The macro \cs{mfplinestyle} is an alias for this same command. The names
+% `linestyle' and `linetype' come from an analogous system of changing
+% rendering in \prog{gnuplot}. (The reason for two names is that
+% \prog{gnuplot} documentation was not consistent in its terminology.)
+% \begin{macrocode}
+\newdef\mfplinetype#1{\mfp@local\mfp@linetype#1\relax}%
+\newlet\mfplinestyle\mfplinetype
+\mfplinetype{0}%
+% \end{macrocode}
+%
+% \prog{Mfpic} has 6 possible axes, the $x$- and $y$-axes pass through the
+% origin, $(0,0)$, the four border axes are at the edges of the graph
+% space, possibly offset from it by some amount. These four commands hold
+% the amount of that offset in graph units, the default being 0.
+% \begin{macrocode}
+\newdef\mfp@lshift{0}\newdef\mfp@bshift{0}%
+\newdef\mfp@rshift{0}\newdef\mfp@tshift{0}%
+% \end{macrocode}
+%
+% \DescribeMacro{\axismargin}
+% The main user-level command for setting the above offsets is
+% \cs{axismargin}. Its first argument is one of the letters l, b, r, or t
+% to select the axis and the second is the offset in graph units. A
+% positive offset represents an inward shift, which is why we call them
+% \emph{margins}.
+% \DescribeMacro{\setaxismargins}
+% The arguments of \cs{setaxismargins} are four offsets for the axes (in
+% the above order, which is anticlockwise, starting with the left).
+% \DescribeMacro{\setallaxismargins}
+% \cs{setallaxismargins} takes one offset and applies it to all axes.
+%
+% The values are passed on to \MF{} in the variables \gbc{laxis},
+% \gbc{baxis}, \gbc{raxis}, and \gbc{taxis}. They are also needed by the
+% \cs{axislabels} command so they are saved in macros.
+% \begin{macrocode}
+\newdef\axismargin#1#2{\setmfnumeric{#1axis}{#2}%
+ \mfp@local\@namedef{mfp@#1shift}{#2}}%
+\newdef\setaxismargins#1#2#3#4{\axismargin l{#1}\axismargin b{#2}%
+ \axismargin r{#3}\axismargin t{#4}}%
+\newdef\setallaxismargins#1{\setaxismargins{#1}{#1}{#1}{#1}}%
+% \end{macrocode}
+%
+% The following converts axis letters to a numeric, making conditionals for
+% axes simpler to code (\cs{ifcase} versus six nested \cs{if}\,s).
+% Plus we get to define the error message only once. The first argument
+% of \cs{mfp@getaxis} is the letter and the third argument is the calling
+% command (for the error message). We allow the letter to be instead a word,
+% using only the first letter and discarding the rest (\arg2).
+% \begin{macrocode}
+\def\mfp@axis@x{0}%
+\def\mfp@axis@y{1}%
+\def\mfp@axis@l{2}%
+\def\mfp@axis@b{3}%
+\def\mfp@axis@r{4}%
+\def\mfp@axis@t{5}%
+\def\mfp@getaxis#1#2\mfp@end#3{%
+ \@ifundefined{mfp@axis@#1}%
+ {\mfp@errmsg {Invalid axis letter "#1"}%
+ {The command #3 requires an axis to be designated by one of^^J%
+ the following letters: x, y, l, b, r, or t. If you proceed, ^^J%
+ x will be assumed.}\def\mfp@axisnum{0}\def\mfp@axisletter{x}}%
+ {\edef\mfp@axisnum{\@nameuse{mfp@axis@#1}}\def\mfp@axisletter{#1}}%
+ \edef\mfp@defaultheadlen{%
+ \the\ifnum\mfp@axisnum>1 \sideheadlen\else \axisheadlen\fi}}%
+% \end{macrocode}
+%
+% The tick marks (or hash marks) on the axes can be in three possible
+% positions: centered on the axes, on one side, or on the other side.
+% The default for the $x$- and $y$-axes is centered, for the border axes it
+% is inside.
+% \DescribeMacro{\setaxismarks}
+% For changing this we provide the \cs{setaxismarks} command, whose first
+% argument is the axis letter, and whose second argument is one of the
+% words \texttt{inside}, \texttt{outside}, \texttt{centered},
+% \texttt{ontop}, \texttt{onbottom}, \texttt{onleft}, or \texttt{onright}.
+%
+% \DescribeMacro{\setxmarks}
+% \cs{setxmarks} takes one argument and sets the position for the $x$ axis
+% only, while
+% \DescribeMacro{\setymarks}
+% \cs{setymarks} does the same for the $y$ axis. For the border axes we
+% provide
+% \DescribeMacro{\setbordermarks}
+% \cs{setbordermarks}, which takes four arguments: the positions for the
+% left, bottom, right and top axis (in that order).
+% \DescribeMacro{\setallbordermarks}
+% The abbreviation \cs{setallbordermarks} takes one argument and sets the
+% position for all border axis marks to that.
+% \begin{macrocode}
+\newdef\setaxismarks#1#2{\setmfnumeric{#1tick}{#2}}%
+\newdef\setxmarks#1{\setaxismarks x{#1}}%
+\newdef\setymarks#1{\setaxismarks y{#1}}%
+\newdef\setbordermarks#1#2#3#4{\setaxismarks l{#1}\setaxismarks b{#2}%
+ \setaxismarks r{#3}\setaxismarks t{#4}}%
+\newdef\setallbordermarks#1{\setbordermarks{#1}{#1}{#1}{#1}}%
+% \end{macrocode}
+%
+% The \cs{datafile} and \cs{plotdata} commands originally just created
+% polygonal paths, but users requested the ability to draw smooth ones.
+% Originally only the commands below (\cs{smoothdata} and
+% \cs{unsmoothdata}) were written to do this. Later, these file
+% manipulation commands were given optional arguments to achieve the same
+% effect, so now the commands below just set the defaults for the optional
+% parameter, storing them in the macros \cs{mfp@smoothness} and
+% \cs{mfp@tension}. The latter can be used to adjust the tension of the
+% \MF{} path if smoothness is `\texttt{s}'.
+%
+% \DescribeMacro{\smoothdata}
+% This takes an option tension argument. It defines \cs{mfp@smoothness} to
+% be `\texttt{s}' and \cs{mfp@tension} to be the optional argument.
+% Default tension is empty rather than 1 so we can implement a scheme to
+% change the actual default used.
+% \DescribeMacro{\unsmoothdata}
+% The macro \cs{unsmoothdata} defines \cs{mfp@smoothness} to be
+% `\texttt{p}' and \cs{mfp@tension} to be empty. The latter might not be
+% necessary.
+% \begin{macrocode}
+\newdef\mfp@smdata#1{\mfp@def\mfp@smoothness{#1}%
+ \mfp@nullopt{\mfp@def\mfp@tension}}%
+\newdef\smoothdata{\mfp@smdata{s}}%
+\newdef\unsmoothdata{\mfp@smdata{p}\@empty}%
+\unsmoothdata% default
+% \end{macrocode}
+%
+% \DescribeMacro{\using}
+% The command \cs{using} is named after a keyword in \prog{gnuplot}. It
+% controls which datum is plotted against which by \cs{plotdata} and
+% \cs{datafile}.
+%
+% \DescribeMacro{\usingpairdefault}
+% The macro \cs{usingpairdefault} sets the default, which is to read the
+% first two space separated words on a line as the $x$ and $y$ coordinate
+% of a point. To allow that there might be more data on a line, it is also
+% assumed that the second word is followed by a space and the rest of the
+% line is read as a third parameter that is not used. In case there are
+% only two words, \mfpic{} will always add a space on the end and then the
+% third parameter will be empty.
+%
+% \DescribeMacro{\usingnumericdefault}
+% For the occasional command that needs numeric data,
+% \cs{usingnumericdefault} sets \cs{using} so that the first
+% space-separated word on a line is selected and the remainder of the line
+% discarded.
+%
+% The \cs{parse@data} command reads its argument as delimited
+% parameters, and the \cs{using} command tells what those delimiters are.
+% For example, in the first part of
+% \cs{using}\marg{\#1 \#2 \#3}\marg{(\#1,\#2)}, we are telling
+% \cs{parse@data} that everything on a line up to the first space is the
+% first parameter, and everything up to the second space is the second
+% parameter and the rest of the line is the third parameter. In the second
+% part we are telling it that what we write to the output is a pair in
+% parentheses with the first two parameters separated by a comma. This
+% example is the default as we see here in \cs{usingpairdefault}. This
+% command can be issued by a user to return to that default. The user can
+% also issue \cs{usingnumericdefault} to write the first word as a single
+% numeric for commands like \cs{piechart}.
+%
+% \DescribeMacro{\mfpdataperline}
+% \cs{mfpdataperline} is a user level macro that controls the maximum
+% number of points that will be written to a single line in the output.
+% \begin{macrocode}
+\newdef\using#1#2{\mfp@def\parse@data#1\mfp@end{#2}}%
+\newdef\usingpairdefault{\using{##1 ##2 ##3}{(##1,##2)}}%
+\newdef\usingnumericdefault{\using{##1 ##2}{##1}}%
+\usingpairdefault
+\newdef\mfpdataperline{5}%
+% \end{macrocode}
+%
+% The following macros are used internally by \cs{datafile} and
+% \cs{plotdata}. A blank line is read as \cs{par} (not the primitive
+% \cs{par}, but the literal token \cs{par}, so it tests as equal to
+% \cs{mfp@par}. Early versions of \cs{plotdata} wrote a literal \MF{} path
+% expression to the output file and \cs{mfp@join} created the \mfc{--}
+% between points. Now we write one of the \grafbase{} list-to-path
+% commands followed by a comma separated list of points, so \cs{mfp@join}
+% has morphed into a comma.
+% \begin{macrocode}
+\newdef\mfp@par{\par}\newdef\mfp@join{,}%
+% \end{macrocode}
+%
+% \DescribeMacro{\mfpdatacomment}
+% The user can change the comment character for data files with
+% something like \cs{mfpdatacomment}\marg{\cs{\#}}. This also changes the
+% percent character to category `other'. We can also use the percent sign
+% as part of the numeric data by placing
+% \DescribeMacro{\makepercentother}
+% \cs{makepercentother} before the \cs{using} command to turn `\texttt\%'
+% into an ordinary character.
+% \DescribeMacro{\makepercentcomment}
+% \cs{makepercentcomment} returns it to its usual role as a comment. Every
+% file reading command issues \cs{mfp@setcomment} before reading. The
+% count register \cs{mfp@commentchar} holds the ASCII code of the current
+% comment character, the default being the percent sign.
+% \begin{macrocode}
+\newcount\mfp@commentchar \mfp@commentchar`\%%
+\newdef\makepercentother{\@makeother\%}%
+\newdef\makepercentcomment{\catcode`\%14 }%
+\newdef\mfpdatacomment#1{\mfp@local\mfp@commentchar`#1}%
+\newdef\mfp@setcomment{\makepercentother\catcode\mfp@commentchar14 }%
+% \end{macrocode}
+%
+% \DescribeMacro{\fdef}
+% The following is for defining functions to be used in \cs{function}
+% through \cs{plrregion}. The first argument is the name of the function,
+% the second is its variable, the last is the \MF{} code. Obviously its
+% use requires a knowledge of the \MF{} language. The function produced is
+% a \mfc{vardef}, and can contain almost arbitrary \MF{} code, but must
+% end with an expression. Normally it must be a numerical expression, to
+% be use in \cs{function}, or a pair expression so it can be used alone in
+% the last argument of \cs{parafcn}.
+%
+% We apply \cs{@makeother} to \arg{} so it doesn't get doubled. The
+% \arg{} is frequent in vardefs among the \MF{} gurus.
+% \begin{macrocode}
+\newdef\fdef{\begingroup\@makeother\#\mfp@keeplines\mfp@fdef}%
+\newdef\mfp@fdef#1#2#3{\mfcmd{\@nl save #1;\@nl
+ vardef #1 (expr #2) = \@nl #3\@nl enddef}\endgroup}%
+% \end{macrocode}
+%
+% The following is executed right after \cs{mfp@grafmacs}. It is a hook
+% for adding new definitions inside and \env{mfpic} environment. It is
+% initialized to empty, and add-on packages can add to it.
+% \begin{macrocode}
+\newdef\mfp@additions{}%
+% \end{macrocode}
+%
+%
+%
+% \section{The user level graphics macros}
+%
+% This is the start of the definition of \cs{mfp@grafmacs}. It contains
+% the definition of every user command that has no purpose outside an
+% \env{mfpic} environment. It will be expanded inside every \env{mfpic}
+% environment, localizing all these definitions. We do this so we can
+% avoid name clashes without giving them all names beginning with \cs{mfp}.
+% \begin{macrocode}
+\newdef\mfp@grafmacs{%
+% \end{macrocode}
+%
+%
+% \subsection{Figure macros}\label{figure}
+%
+% These are the figure macros. These are implemented as \MF{} paths that
+% can be variously rendered, transformed, etc., by the prefix macros of
+% subsection~\ref{prefixcommands}.
+%
+% Most of the \mfpic{} graphic commands take optional arguments. Most of
+% them follow the scheme \cs{mfp@command} takes mandatory arguments
+% enclosed in braces, and does the work. \cs{command} is simply defined to
+% test for an optional argument (with \cs{mfp@defaultopt}) and pass it or a
+% default to \cs{mfp@command}. A typical example is \cs{ellipse} defined
+% to be \verb$\mfp@defaultopt\mfp@ellipse0$. In some cases, the default
+% option is signaled by passing an empty argument via \cs{mfp@nullopt}.
+%
+% Of the commands that have no optional argument, most simply write
+% the appropriate commands to the output file, and require no
+% \cs{mfp@} version.
+%
+% This section contains a description of the behavior of each command and
+% its syntax in use. For those that call an \cs{mfp@} version, the
+% implementation is in the next section.
+%
+%
+% \subsubsection{Common geometric objects}\label{geometric}
+%
+% The first few (\cs{rect} through \cs{sector}) are closed paths and
+% need not be prefixed with a closing macro before filling, etc.
+%
+% \DescribeMacro{\rect}
+% This takes one mandatory argument, the two diagonally opposite
+% corners of the rectangle separated by a comma, and draws the rectangle
+% with horizontal and vertical sides having these as two corners. The
+% direction of the path (which seldom makes a difference) is such that it
+% starts at the first specified point and moves horizontally first.
+%
+% This is an example of a command that requires no further processing.
+% The \cs{mfp@figmac} command sets the appropriate switches and writes its
+% argument to the output file, followed by a semicolon.
+% \begin{macrocode}
+ \newdef\rect##1{\mfp@figmac{rect (##1)}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\regpolygon}
+% There are four arguments. The first is the number of sides. The second
+% is the name of a variable to represent the vertices. The other
+% two should be equations locating two vertices or the center and one
+% vertex. If the name used is \gbc{Ver}, then the vertices will be
+% \gbc{Ver1}, \gbc{Ver2}, etc., numbering anticlockwise. The center is
+% \gbc{Ver0}. Case is significant. After the command, the points can be
+% referred to in other \mfpic{} commands (except text label commands
+% when \opt{mplabels} is not in effect).
+% \begin{macrocode}
+ \newdef\regpolygon##1##2##3##4{%
+ \mfp@figmac{regularpolygon (##1) (##2) (##3; ##4;)}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\circle}
+% This has one optional argument containing one of the letters
+% \texttt{t}, \texttt{p} (the default), \texttt{c}, \texttt{s},
+% \texttt{r}, or \texttt{q}. Its purpose is to specify what kind of data
+% is in the single mandatory argument. These are a subset of the optional
+% arguments that the \cs{arc} commad (below) takes, and have almost the
+% same meaning:
+% \begin{itemize}
+% \item \oarg{t}, the three point form needs three points (of course).
+% \item \oarg{p}, the polar form needs the center point and the radius.
+% This is the default when the optional argument is absent.
+% \item \oarg{c}, the center/point form needs the center and one point on
+% the circle.
+% \item \oarg{s}, the point-sweep form requires two points and the angle of
+% arc between them. The angle may be positive or negative. A
+% positive angle indicates the number of degrees anticlockwise from
+% the first point to the second. Negative means clockwise.
+% \item \oarg{r}, the point-radius form requires two points and the
+% radius of the circle. Of the two possible circles, it produces the
+% one where the anticlockwise arc from the first point to the second
+% is not more than 180 degrees.
+% \item \oarg{q}, an alternative point-radius with the same parameters.
+% It produces the circle
+% where the anticlockwise arc from the first point to the second
+% is not less than 180 degrees.
+% \end{itemize}
+%
+% Supplying \cs{arc} with the same parameters, or the appropriate superset
+% in the case of \oarg{p} and \oarg{c}, would produce a sub-arc of the
+% circle that \cs{circle} produces. This is the reason for giving
+% \cs{circle} these options: to make diagrams which show a circle and an
+% arc on it.
+%
+% We use \cs{def} instead of \cs{newdef} because we know \LaTeX{} has
+% already defined a \cs{circle} command. We now use the same command,
+% \cs{mfp@arccircle}, for arcs and circles, setting a switch to tell it
+% which to produce.
+% \begin{macrocode}
+ \def\circle{\mfp@switchfalse\mfp@defaultopt\mfp@arccircle{p}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\ellipse}
+% \cs{ellipse} has one optional argument, an angle of rotation and one
+% mandatory argument, containing the center, the horizontal and vertical
+% radii (before rotation), separated by commas.
+% \begin{macrocode}
+ \newdef\ellipse{\mfp@defaultopt\mfp@ellipse{0}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\quarterellipse}
+% \DescribeMacro{\halfellipse}
+% \DescribeMacro{\fullellipse}
+% The next three commands simply call \grafbase{} commands of the same
+% name. They draw part or all of an ellipse inscribed in a parallelogram.
+% Each requires for its argument three points, separated by commas. For
+% \cs{quarterellipse} these points are, respectively, the midpoint of one
+% side of a parallelogram, a corner point on that side, and the midpoint
+% of the side adjacent to the first at that corner. For \cs{halfellipse}
+% they are the midpoints of three successive sides. For \cs{fullellipse}
+% they are the center and the midpoints of two adjacent sides. They are
+% all simple figmacs with no optional arguments and no star forms.
+%
+% It would be possible to give all of them the same three arguments,
+% for example those of \cs{fullellipse}, but I wanted to put the
+% endpoints of the paths into the argument list. The reason
+% \cs{quarterellipse} is passed the corner point instead of the center is
+% because of the intended use: rounding off an asymmetric corner without
+% having to compute the center of the parallelogram.
+% \begin{macrocode}
+ \newdef\quarterellipse##1{\mfp@figmac{quarterellipse (##1)}}%
+ \newdef\halfellipse##1{\mfp@figmac{halfellipse (##1)}}%
+ \newdef\fullellipse##1{\mfp@figmac{fullellipse (##1)}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\sector}
+% This is the closed curve starting at the center of a circle, out
+% along a radius to the circumference of the circle, along an arc of the
+% circle, then back to the center. It has one mandatory argument
+% containing the center, the radius, and the two angles, separated by
+% commas.
+% \begin{macrocode}
+ \newdef\sector##1{\mfp@figmac{sector (##1)}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\arc}
+% We have seven (actually 14) ways to specify a circular arc. \cs{arc}
+% takes an optional argument: a letter to indicate which of the 7
+% specifications. It has one mandatory argument containing the data for
+% that specification, separated by commas. It also has a star-form,
+% which produces the complementary arc. That is, instead of the arc that
+% is specified (as described below) it produces the rest of the circle
+% from the end of the specified arc to its start.
+%
+% The available letters and the corresponding required data are listed
+% below. The default optional argument is \oarg{s}.
+% \begin{itemize}
+% \item \oarg{t}, the three point form needs three ordered pairs.
+% \item \oarg{p}, the polar form needs the center point, two angles and
+% the radius (in that order).
+% \item \oarg{a}, the alternative polar form needs the center, the radius
+% and two angles (in that order).
+% \item \oarg{c}, the center-point-sweep form needs the center, the
+% starting point, and the angle.
+% \item \oarg{s}, the points-sweep form requires the starting point, the
+% ending point and the angle.
+% \item \oarg{r}, the points-radius form requires the starting point, the
+% ending point and the radius of the arc. There are two circles with
+% the given radius that pass through the given points, and hence
+% four possible arcs. One arc on each circle is anticlockwise from
+% the first point to the second. Of those two arcs, only one is
+% less than $180$ degrees. That one is produced.
+% \item \oarg{q} Same as \oarg{r}, except the arc is the larger of two
+% anticlockwise arcs.
+% \end{itemize}
+% \begin{macrocode}
+ \newdef\arc{\mfp@iftoken*{\arccomplement\mfp@arc}{\mfp@arc}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\pshcircle}
+% This final circle is for us complex function theorists. It makes a
+% `pseudohyperbolic circle'. This requires a point $z$ inside the circle
+% with center $(0,0)$ and radius $1$, and a radius $R < 1$. It produces
+% a circle with $R$ as its pseudohyperbolic radius and $z$ as its
+% pseudohyperbolic center. The star form switches contexts to the upper
+% half-plane.
+% \begin{macrocode}
+ \newdef\pshcircle{\mfp@ifstar\mfp@pshcircle\mfp@F\mfp@T}%
+% \end{macrocode}
+%
+% \DescribeMacro{\hypergeodesic}
+% And this final arc is also for we complex function theorists. It makes a
+% `hyperbolic geodesic'. This requires two points $z$ and $w$ inside the
+% circle $T$ with center $(0,0)$ and radius $1$. It produces a circular arc
+% from the first point to the second on the unique circle that passes
+% through these two points and meets $T$ at right angles. The star form
+% switches contexts to the upper half-plane.
+% \begin{macrocode}
+ \newdef\hypergeodesic{\mfp@ifstar\mfp@hypergeodesic{UHP}{UD}}%
+% \end{macrocode}
+%
+%
+% \subsubsection{List macros}\label{list}
+%
+% The figure macros of this section are all list macros. They produce a
+% path from a list of ordered pairs. The list can be explicitly given as a
+% mandatory argument containing a comma-separated list of pair
+% expressions, or it can be taken from an external datafile. This is true
+% for \cs{polyline} through \cs{turtle}. These macros differ only in how
+% the data determine the path. (Some of the splines don't connect given
+% points, but rather use the points in a different way to determine a
+% curve.) The list of pairs is not an argument of any of these. It should
+% follow, and is ultimately written to output by the ending code of
+% \cs{mfp@listmac}. This is basic to list macros, and will not be repeated
+% for each one.
+%
+% Commands in this section are grouped by the argument structure of the
+% command written to the output file. The first twelve come in pairs
+% that are distinguished by a boolean parameter, which usually determines
+% whether the path produced is closed, followed by the list.
+%
+% \DescribeMacro{\polyline}
+% The most basic, a polyline or polygonal path, \cs{polyline} draws line
+% segments from each point to the next.
+% \DescribeMacro{\lines}
+% \cs{lines} is an alias.
+%
+% \DescribeMacro{\closedpolyline}
+% The macro \cs{closedpolyline} does the same, except it produces a closed
+% path, connecting the last point to the first.
+% \DescribeMacro{polygon}
+% \cs{polygon} is an alias.
+%
+% \DescribeMacro{\computedspline}
+% The \cs{computedspline} command takes a list of points and computes the
+% controls of a cubic spline that connects the points. Adjacent segments
+% have matching first and second derivatives at the common endpoint. It is
+% limited to what is called a \emph{relaxed} spline, which has zero
+% curvature at the beginning and ending points.
+%
+% \DescribeMacro{\closedcomputedspline}
+% This is similar, but computes the unique \emph{closed} cubic spline that
+% connects a list of points.
+%
+% The next four commands also produce types of splines. The list of pairs
+% is not, however, the points passed through but rather are control points,
+% of a sort.
+%
+% \DescribeMacro{\qspline}
+% This produces quadratic splines. The points create the path as follows:
+% for successive points $p\sb i, p\sb {i+1}$ in the list, the path passes
+% through the midpoint of the line segment connecting $p\sb i$ and
+% $p\sb {i+1}$, tangent to that line segment. This causes adjacent
+% B\'ezier segments to have matching derivatives at their common
+% endpoint. Each three successive data points determines one segment of
+% the path.
+%
+% \DescribeMacro{\closedqspline}
+% The closed variant works by creating two additional segments, viewing
+% the triples $p\sb{n-1}, p\sb{n}, p\sb1$ and $p\sb{n}, p\sb1, p\sb2$ as
+% being `successive'.
+%
+% \DescribeMacro{\cspline}
+% The macro \cs{cspline} produces cubic splines. The points determine the
+% curve as follows: for successive points $p\sb i, p\sb {i+1}$ in the
+% list, the line segment connecting them is divided into thirds with two
+% points $q\sb i$ and $q\sb i'$. The curve then passes through the
+% midpoint of the segment from $q\sb {i-1}'$ to $q\sb i$ and tangent to
+% that segment. This causes adjacent B\'ezier segments to have matching
+% first and second derivatives at their common endpoint. Each four
+% successive data points determines one segment of the path.
+%
+% \DescribeMacro{\closedcspline}
+% The closed variant works by viewing the first point as a succesor of the
+% last point, creating three additional path segments.
+%
+% \DescribeMacro{\qbeziers}
+% This produces the equivalent of a sequence of \LaTeX{} \cs{qbezier}
+% commands. The mandatory argument will be a list of points alternating
+% between nodes and control points, ending with the last node.
+% \DescribeMacro{\closedqbeziers}
+% The closed version ends with the control point between the last node and
+% the first. Therefore, the \cs{qbeziers} command needs an odd number of
+% points in the list, while \cs{closedqbeziers} needs an even number. If
+% this is not the case, the last point in the list is repeated, causing
+% the final segment to be a straight line (closed case) or trivial.
+%
+% Note that quadratic B\'eziers need not be smooth at the nodes unless
+% the control points line up. Our plural name distinguishes it from the
+% \LaTeX{} command.
+%
+% \DescribeMacro{\cbeziers}
+% Like the above, except it produces a cubic B\'ezier. It requires a
+% list of $3n+1$ points in the pattern node-control-control, ending with
+% an extra node.
+% \DescribeMacro{closedcbeziers}
+% The closed version requires only $3n$ points, taking the first for the
+% final node.
+%
+% \DescribeMacro{\fcnspline}
+% The above splines are two dimensional splines, that is, functions of
+% $t$ with pair values $(x,y)$. One dimensional splines are functions of
+% $x$ with numeric values $y$. This command produces the \emph{graph} of
+% such a function computed from a supplied list of points. It is required
+% that the $x$-values of the points are in increasing order, although it
+% will produce some sort of curve as long as adjacet $x$-values are
+% distinct. It produces the graph of a \emph{relaxed} spline, in the sense
+% that the second derivative $d^2y/dx^2$ is zero at the beginning and end.
+%
+% \DescribeMacro{\periodicfcnspline}
+% This is the same, but instead of requiring the second derivative to be
+% zero at both ends, it produces a graph such that the first and second
+% derivatives at the end match the ones at the beginning. If a copy of the
+% graph is placed with its starting point at the original graph's ending
+% point, then there is a smooth join. If the $y$-values of the
+% beginning and ending points match, a periodic graph is obtained. For
+% these last two commands the paths cannot be closed, and the boolean
+% determines the behavior at the endpoints.
+%
+% The command \cs{mfp@listi} that implements these writes the command
+% name and the argument and ends with code to write the list of pairs.
+% \begin{macrocode}
+ \newdef\lines{\mfp@listi{polyline}\mfp@F}%
+ \newdef\polygon{\mfp@listi{polyline}\mfp@T}%
+ \newdef\computedspline{\mfp@listi{dospline}\mfp@F}%
+ \newdef\closedcomputedspline{\mfp@listi{dospline}\mfp@T}%
+ \newdef\qspline{\mfp@listi{qspline}\mfp@F}%
+ \newdef\closedqspline{\mfp@listi{qspline}\mfp@T}%
+ \newdef\cspline{\mfp@listi{cspline}\mfp@F}%
+ \newdef\closedcspline{\mfp@listi{cspline}\mfp@T}%
+ \newdef\qbeziers{\mfp@listi{qbezier}\mfp@F}%
+ \newdef\closedqbeziers{\mfp@listi{qbezier}\mfp@T}%
+ \newdef\cbeziers{\mfp@listi{cbezier}\mfp@F}%
+ \newdef\closedcbeziers{\mfp@listi{cbezier}\mfp@T}%
+ \newdef\fcnspline{\mfp@listi{fcnspline}\mfp@F}%
+ \newdef\periodicfcnspline{\mfp@listi{fcnspline}\mfp@T}%
+ \newlet\closedpolyline\polygon
+ \newlet\polyline\lines
+% \end{macrocode}
+%
+% The next six macros have an optional argument called `tension', a
+% numerical value that controls how tightly the path turns at each point
+% in the list. If the optional argument is absent (or empty), the
+% \grafbase{} command that is written takes only a boolean argument
+% (closed or not closed) followed by the list of points. If the tension
+% argument is present, a different command is written that also has a tension
+% argument. The command name is almost the same, but has a `\gbc{t}'
+% prepended. The default tension is the \cs{grafbase}{} variable
+% \gbc{default_tension}, set with \cs{settension}, initialized to 1.
+% Tension of 0 is ignored (ultimately tension is divided by) and may
+% produce error messages from \MF{}.
+%
+% \DescribeMacro{\curve}
+% \cs{curve} draws a smooth curve connecting a list of points. it arranges
+% for the curve, at point $p\sb n$ to be traveling parallel to the direction
+% from $p\sb {n-1}$ to $p\sb {n+1}$ (except at the first and last point).
+%
+% \DescribeMacro{\closedcurve}
+% The closed version \cs{closedcurve} treats the
+% first point and last point the same as the rest. It has the alias
+% \DescribeMacro{\cyclic}
+% \cs{cyclic}.
+%
+% \DescribeMacro{\convexcurve}
+% This convex version produces a smooth curve that is convex in those
+% places where \cs{polyline} (with the same list of points) would produce
+% a convex shape.
+% \DescribeMacro{\closedconvexcurve}
+% It also comes in a closed version, with the alias
+% \DescribeMacro{\convexcyclic}
+% \cs{convexcyclic}.
+%
+% \DescribeMacro{\mfbezier}
+% This produces the standard \MF{} cubic B\'ezier using the \MF{} path
+% join operator (\mfc{..}) with tension given by an optional argument.
+% \DescribeMacro{\closedmfbezier}
+% The closed version connects the last point to the first in the same way.
+%
+% The command \cs{mfp@curve} examines the optional argument and writes
+% the command name (first argument, possibly modified) with the
+% appropriate arguments (the boolean or tension plus boolean).
+% \begin{macrocode}
+ \newdef\curve{\mfp@curve{curve}\mfp@F}%
+ \newdef\cyclic{\mfp@curve{curve}\mfp@T}%
+ \newdef\convexcurve{\mfp@curve{ccurve}\mfp@F}%
+ \newdef\convexcyclic{\mfp@curve{ccurve}\mfp@T}%
+ \newdef\mfbezier{\mfp@curve{bezier}\mfp@F}%
+ \newdef\closedmfbezier{\mfp@curve{bezier}\mfp@T}%
+ \newlet\closedcurve\cyclic
+ \newlet\closedconvexcurve\convexcyclic
+% \end{macrocode}
+%
+% \DescribeMacro{\fcncurve}
+% A `function curve' is the graph of a function of $x$. It should always
+% travel left to right. This is not guaranteed by the \cs{curve} macro,
+% above, even if the $x$ coordinates are listed in order. \cs{fcncurve} is
+% guaranteed to do this if the list of points has the $x$ coordinates so
+% ordered.
+%
+% While the optional argument is called tension, it is not quite the
+% same as the tension argument to \cs{curve}, but it has a similar effect.
+% The default value is the sane as for \cs{curve}. The above guarantee is
+% voided if the value used is ${}\le1/3$.
+% \begin{macrocode}
+ \newdef\fcncurve{\mfp@nullopt\mfp@fcncurve}%
+% \end{macrocode}
+%
+% \DescribeMacro{\turtle}
+% Many years ago there was a tool for teaching programming concepts to
+% young children. They could draw a figure on a screen by programming a
+% `turtle' with a sequence of turns and distances. The following is
+% similar. It has a mandatory argument containing a list of ordered pairs.
+% The first is the coordinates of the starting point, and each of the rest
+% is interpreted as a displacement from the current position, along which
+% a line segment is to be drawn. The terminal point of that segment is the
+% new current position
+% \begin{macrocode}
+ \newdef\turtle{\mfp@listmac{turtle}}%
+% \end{macrocode}
+%
+%
+% \subsubsection{Graphing functions}\label{functions}
+%
+% \DescribeMacro{\function}
+% \prog{Mfpic} provides the ability to plot the graph of any function that
+% can be defined in \MF. This is done with the \cs{function} command, a
+% figure macro which takes one optional parameter that selects
+% the smoothness of the \MF{} path produced. It can be \oarg{p}, which
+% draws a polyline path through selected points (much like \cs{polyline}),
+% or \oarg{s\meta{tension}} which selects a smooth path (much like
+% \cs{curve}\oarg{\meta{tension}}). The tension may be omitted and then
+% will be 1 (or the value set by \cs{settension}). The default optional
+% argument is \oarg{s}, a smooth curve with the default tension.
+%
+% There are two mandatory arguments. The second is the function, which
+% must be written as a formula (strictly speaking, a \MF{} numerical
+% expression) containing only one variable, \texttt{x}. The first
+% mandatory argument contains three numbers separated by commas. They are,
+% in order, the starting value of \texttt{x}, the ending value, and the
+% step size.
+%
+% \DescribeMacro{\parafcn}
+% \cs{parafcn} is similar, except its formula must be an ordered pair of
+% formulas (strictly speaking, a \MF{} pair expression) in which
+% \texttt{t} is the only variable.
+%
+% \DescribeMacro{\plrfcn}
+% \cs{plrfcn} is similar, except its argument is a formula (a \MF{}
+% numeric expression) representing the radius, in which \texttt{t} is the
+% only variable, interpreted as the angle coordinate, $\theta$, in degrees.
+%
+% These three commands are so alike that they call the same macro
+% with the name to be written as the first argument and the default
+% optional argument as the second. The latter is so that we can reuse it
+% for two other commands where the default is different.
+% \begin{macrocode}
+ \newdef\function{\mfp@fcn{function}s}%
+ \newdef\parafcn{\mfp@fcn{parafcn}s}%
+ \newdef\plrfcn{\mfp@fcn{plrfcn}s}%
+% \end{macrocode}
+%
+% \DescribeMacro{\btwnfcn}
+% The region between two functions \cs{btwnfcn} is a figure macro. It is
+% mostly intended for shading or other filling operations so it is already
+% closed and, by default, it is not `smooth'. Its parameters are the
+% same as those of \cs{function} (above), with two exceptions: the default
+% optional argument is \oarg{p} and it has an additional mandatory
+% argument, the other function. The resulting \MF{} path travels along the
+% graph of the first function from the starting $x$ to the ending $x$,
+% thence vertically to the graph of the second function, along it (in
+% reverse) to the starting $x$ and finally, vertically to the start of the
+% first function's graph.
+%
+% \DescribeMacro{\btwnplrfcn}
+% We also have the polar function version, where the two function
+% arguments are as in \cs{plrfcn}. The default option is also \oarg{p}.
+% As with \cs{function}, these are implemented by calling a common
+% command with the basename of the \grafbase{} command as argument.
+%
+% \DescribeMacro{\plrregion}
+% This is like \cs{btwnplrfcn} with 0 as the first function, but it
+% is more efficient. It takes only the other function as an argument.
+% Again, the default option is \oarg{p}.
+%
+% \DescribeMacro{\belowfcn}
+% This is like \cs{btwnfcn} with 0 as the first function, but it
+% is more efficient. It takes only the other function as an argument.
+% The default option is \oarg{p}.
+% \begin{macrocode}
+ \newdef\btwnfcn{\mfp@btwn{btwnfcn}}%
+ \newdef\btwnplrfcn{\mfp@btwn{btwnplrfcn}}%
+ \newdef\belowfcn{\mfp@fcn{belowfcn}p}%
+ \newdef\plrregion{\mfp@fcn{plrregion}p}%
+% \end{macrocode}
+%
+% \DescribeMacro{\levelcurve}
+% One way to graph a function of two variables is to draw level curves.
+% The figure macro \cs{levelcurve} takes an optional smoothness
+% argument (\texttt{[s\meta{tension}]} or \texttt{[p]} with \texttt{p}
+% being the default), and two mandatory arguments. The first
+% mandatory argument is a `seed' point which is known to lie inside the
+% level curve, followed by a comma and a step size. The second is an
+% inequality between the function and the level value that is true inside
+% the level curve. Strictly speaking, the last argument can be any \MF{}
+% boolean valued expression with variables \mfc{x} and \mfc{y}. The simple
+% example,\\
+% \indent\verb$\levelcurve[p]{(0,0),0.1}{x*x + y*y < 3}$\\
+% will draw approximately the circle with radius $\sqrt3$ as a sequence of
+% line segments with length $0.1$. That will require about 109 segments.
+% Its parameters are similar enough to those of \cs{function} that we
+% can use the same interface.
+% \begin{macrocode}
+ \def\levelcurve{\mfp@fcn{levelset}p}%
+% \end{macrocode}
+%
+% We now define two macros for depicting the solution of an
+% ordinary differential equation. The first of these, \cs{DEgraph},
+% produces the graph of the solution for equations of the form
+% $$
+% \frac{dy}{dx} = f(x,y),\quad y(x\sb0) = y\sb0.
+% $$
+% The data required are the \emph{initial values} (the two numbers $x\sb0$
+% and $y\sb0$), the formula $f(x,y)$ (an explicit \MF{} numerical
+% expression with literal variables \mfc{x} and \mfc{y} as the only
+% unknowns. Since \MF{} can only handle paths connecting discrete points,
+% also required is a step size which, roughly speaking, is the distance
+% between points in graph coordinates, and the number of steps $N$ to use.
+% The \MF{} macros then calculate $N$ additional points (the first is
+% $(x\sb0,y\sb0)$) and join them together to produce the graph.
+%
+% The second macro, \cs{DEtrajectory}, draws the trajectory of a
+% two-dimentional differential equation of the form
+% $$
+% \left( \frac{dx}{dt}, \frac{dy}{dt}\right) = (f(x,y,t),g(x,y,t)\,,
+% \quad (x(0),y(0)) = (x\sb0,y\sb0)
+% $$
+% This is not, strictly speaking, a \emph{graph} since that would
+% require three dimensions to indicate the relation between $x$, $y$, and
+% $t$. Mathematicians call it a trajectory, and it only shows the
+% two-dimensional path followed by the moving object. The required data
+% are an initial point $(x\sb0,y\sb0)$, the formula for the right-hand side
+% (an explicit \MF{} pair-valued expression or a pair of numerical
+% expressions in parentheses, with literal variables \mfc{x}, \mfc{y} and
+% \mfc{t} as the only unknowns. Also required are a step size and the
+% number of steps, as in \cs{DEgraph}.
+%
+% The method used to calculate the points to connect is a modified version
+% of the four-step Runge-Kutta method. The modifications are intended to
+% avoid numerical overflow. That is, infinite blow-up in finite time. An
+% example is the simple equation
+% $$
+% \frac{dy}{dx} = y^2 \quad y(0) = 1,
+% $$
+% whose solution is $y = 1/(1-x)$. The usual definition of
+% Runge-Kutta-IV would have one process equally spaced $x$-values. If a
+% user had no idea of the character of the solution, and asked for $N$
+% $x$-values with separation $h$ to get a graph on $0\le x \le Nh$, and if
+% $Nh > 1$, then the result would be either quite different from the actual
+% solution (especially around $x=1$), or would reach a point of numerical
+% overflow.
+%
+% Thus our macros require the user to supply a parameter that will be
+% interpreted as a distance step: if $h$ is this parameter, instead of
+% finding values of $y$ at $x\sb0$, $x\sb{0+h}$, $x\sb{0+2h}$, etc., it
+% finds points $(x\sb{n},y\sb{n})$ on the graph so that the
+% two-dimensional distance from $(x\sb0,y\sb0)$ to $(x\sb1,y\sb1)$ is $h$,
+% then the distance from $(x\sb1,y\sb1)$ to $(x\sb2,y\sb2)$ is also $h$,
+% etc. The resulting graph therefore simply follow the graph of $y =
+% 1/(1-x)$ a distance roughly $Nh$ and numerical overflow is impossible
+% unless the value of $Nh$ exceeds \MF{}'s size limits. There is an
+% optional smoothness parameter as in \cs{function}.
+% \begin{macrocode}
+ \newdef\DEgraph{\mfp@fcn{odeRKIV}s}%
+ \newdef\DEtrajectory{\mfp@fcn{xyRKIV}s}%
+% \end{macrocode}
+%
+% \DescribeMacro{\brownianmotion}
+% Used by me once to ilustrate Brownian motion. The argument consists of
+% a starting point, the number of steps, and a scale factor, separated
+% by commas. It draws a polyline starting with the starting point moving
+% in a random direction a random distance, then repeating that from the
+% new point, etc., for the number of steps given. The size of each step is
+% random, but the scale factor gives the average size.
+%
+% \DescribeMacro{\browniangraph}
+% This approximates the graph of one-dimensional Brownian motion. The
+% argument consists of the number of steps and a scale factor/step size,
+% separated by commas. It draws a polyline starting at $(0,0)$, moving
+% right by the step size and up or down (randomly chosen) by a random
+% amount, then repeating that from the new point, etc., for the number of
+% steps given.
+%
+% \DescribeMacro{\randomwalk}
+% This is a two dimensional random walk. The argument consists of a
+% starting point, the number of steps, and a distance, separated by
+% commas. Like \cs{brownianmotion}, it draws a polyline starting at the
+% starting point, moving in a random direction, but for this command it
+% always moves a fixed distance.
+% \begin{macrocode}
+ \newdef\brownianmotion##1{\mfp@figmac{brownianpath (##1)}}%
+ \newdef\browniangraph##1{\mfp@figmac{browniangraph (##1)}}%
+ \newdef\randomwalk##1{\mfp@figmac{randomwalk(##1)}}%
+% \end{macrocode}
+%
+%
+% \subsection{Plotting data from files}\label{datafiles}
+%
+% \DescribeMacro{\plotdata}
+% The optional parameter is the smoothness, which may be
+% \oarg{s\meta{tension}} for a smooth path or \oarg{p} for a polygonal
+% path, or empty to select the default. The default is \oarg{p} when
+% \mfpic{} is loaded, but can be changed with \cs{smoothdata} and restored
+% with \cs{unsmoothdata}.
+% \begin{macrocode}
+ \newdef\plotdata{\mfp@nullopt\mfp@plotdata}%
+% \end{macrocode}
+%
+% \DescribeMacro{\datafile}
+% \cs{datafile} defines a path connecting the points in a datafile. It has
+% an optional parameter for the smoothness, as in \cs{plotdata}.
+% \begin{macrocode}
+ \newdef\datafile{\mfp@nullopt\mfp@datafile}%
+% \end{macrocode}
+%
+%
+% \subsection{The prefix commands}\label{prefixcommands}
+%
+%
+% \subsubsection{Storing and reusing a figure}\label{storing}
+%
+% \DescribeMacro{\store}
+% This stores an \mfpic{} path in a \MF{} path variable. The argument is
+% the variable name. It should be followed by an \mfpic{} figure macro
+% (such as \cs{rect}\marg{...}). The stored path can be reused via the
+% \cs{mfobj} command. This command is written as a \cs{mfp@rendmac} even
+% though it does no rendering. This is so (1)~it will turn off implicit
+% rendering (any rendering must be explicitly written) and (2) it can be
+% placed anywhere within the prefix area. This is new behavior with
+% version 0.7. Prior to this it was required to place the path to be
+% stored as a second argument. Though that argument could contain other
+% prefixes in addition to the path, it was required that no other prefixes
+% precede \cs{store}. Note that \cs{store} takes a second argument but
+% it merely puts it back. This is so that the old syntax (with the figure
+% macro in braces) will continue to work.
+%
+% \DescribeMacro{\norender}
+% Occasionally it is useful to turn off rendering (say we want to add an
+% arrowhead to a subpath of a path that is already rendered). One could
+% put \cs{store} at the start with an unused name, but as every path is
+% already stored in \gbc{curpath}, that seems redundant. So here is a
+% prefix macro with null rendering and no side effects. It is exactly
+% like \cs{mfp@rendmac}, except it does not end with \cs{mfsrc}.
+% \begin{macrocode}
+ \newdef\store##1##2{\mfp@rendmac{stored (##1)}##2}%
+ \newdef\norender{\mfp@first\@mfprendfalse}%
+% \end{macrocode}
+%
+% \DescribeMacro{\mfobj}
+% This is the interface to use a \cs{store}-d object.
+% \cs{mfobj}\marg{\meta{name}} is a figure macro, and should act exactly
+% the same as if the figure macro that had been stored in the variable
+% were typed in its place. Its one argument is the variable name.
+%
+% Instead of the variable name, the argument can contain any reasonable
+% \MF{} path expression. In this context ``reasonable'' means it should
+% contain no special \TeX{} characters that might expand upon being
+% written to the output file. In particular, the backslash and `\verb$~$'
+% should especially be avoided.
+%
+% \DescribeMacro{\mpobj}
+% The macro \cs{mpobj} is just another name for the same command.
+% \begin{macrocode}
+ \newlet\mfobj\mfp@figmac \newlet\mpobj\mfobj
+% \end{macrocode}
+%
+% \DescribeMacro{\mfpimage}
+% The command \cs{mfpimage} starts a group in which drawing commands
+% work just as usual, except they draw in a picture variable, whose name
+% is the only mandatory argument to \cs{mfpimage}.
+% \DescribeMacro{\endmfpimage}
+% The command \cs{endmfpimage} ends that group. There is an optional
+% argument to set the reference point of the created picture, the default
+% being \gbc{(0,0)}. The resulting picture can then be placed using
+% \cs{putmfpimage}.
+% \begin{macrocode}
+ \newdef\mfpimage{\mfp@defaultopt\mfp@image{(0,0)}}%
+ \newdef\endmfpimage{\mfcmd{\@nl concludeimage}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\putmfpimage}
+% This takes the name of a picture variable created with the
+% \cs{mfpimage} environment (or even with the \cs{tile} environment), and
+% places it with its reference point at the points in a list.
+% \begin{macrocode}
+ \newdef\putmfpimage##1{\mfsrc{\@nl putimage (##1)}\mfp@writedata}%
+% \end{macrocode}
+%
+%
+% \subsubsection{Subpaths}
+%
+% \DescribeMacro{\cutoffbefore}
+% This macro take one mandatory argument which is an object stored with
+% \cs{store}. It modifies the path it is prefixed to so that it starts
+% at its first point of intersection with the object, cutting off any
+% earlier part of the path.
+%
+% \DescribeMacro{\cutoffafter}
+% This is similar, except the modified path ends at the last point of
+% intersection between the following path and the object, cutting off
+% any later part of the path. It is internally the same as applying
+% \cs{cutoffbefore} to the reversed path and reversing the result.
+%
+% \DescribeMacro{\trimpath}
+% This trims off the specified absolute amounts from the start and end of
+% the path. The amounts can be specified as two absolute dimensions
+% separated by a comma, or a single dimension. In the second case, that
+% dimension is used at both the start and end.
+%
+% The implementation uses \cs{mfp@modi} which simply reads the argument
+% and writes it (via \cs{mfp@modmac}) as a set of parameters to the
+% appropriate \grafbase{} command. With \cs{trimpath}, however, we have to
+% parse that argument with \cs{mfp@getdimens}.
+% \begin{macrocode}
+ \newdef\cutoffafter {\mfp@modi{cutoffafter}}%
+ \newdef\cutoffbefore{\mfp@modi{cutoffbefore}}%
+ \newdef\trimpath##1{\mfp@getdimens##1,,\mfp@end
+ \mfp@modi{trimmedpath}\mfp@thedimens}%
+% \end{macrocode}
+%
+% \DescribeMacro{\partpath}
+% This produces a subpath of the following path. Its mandatory argument
+% contains two numbers separated by a comma. The numbers should be between
+% 0 and 1 and produce a fraction of the following path.
+% \DescribeMacro{\subpath}
+% The macro \cs{subpath} has a similar argument, but the numbers are
+% between $0$ and the number of segments in the path, and it produces the
+% equivalent of \MF's \mfc{subpath} primitive.
+% \begin{macrocode}
+ \newdef\partpath{\mfp@modi{partialpath}}%
+ \newdef\subpath{\mfp@modi{gsubpath}}%
+% \end{macrocode}
+%
+% \subsubsection{Drawing a curve}\label{drawing}
+%
+% These are rendering macros that draw the curve in some fashion: solid,
+% dashed, colored, etc.
+%
+% \DescribeMacro{\draw}
+% \cs{draw} takes one optional argument, a color. In \MF{}, where the
+% color commands return a numeric, if it is ${}\ge 1$ (representing white)
+% the path is subtracted from the picture, otherwise it is drawn in black
+% (the default). The default in \MP{} is \gbc{drawcolor}.
+%
+% We signal the use of the default color by passing a null optional
+% argument. Because several commands handle color the same way, and all
+% are rendering macros, we have a common macro that only needs the name of
+% the \grafbase{} command.
+% \begin{macrocode}
+ \newdef\draw{\mfp@nullopt{\mfp@clroptrend{drawn}}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\doubledraw}
+% Behaves something like \cs{draw}, except it draws a double line. It
+% takes two optional arguments; the first is the distance between the
+% centers of the two lines (default \gbc{2penwd}) and the second is a
+% color.
+% \begin{macrocode}
+ \newdef\doubledraw{\mfp@defaultopt\mfp@doubledraw{2penwd}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\dashed}
+% \cs{dashed} and \cs{dotted} take one optional argument, which should
+% contain the length of the dashes (diameter of
+% \DescribeMacro{\dotted}
+% dots) and the length of the spaces between, separated by a comma.
+%
+% Several rendering macros take an optional argument which is written as
+% the first parameter of a \grafbase{} command. Thus all are handled with
+% a single macro whose arguments are the name of the command and the
+% default for the optional argument.
+% \begin{macrocode}
+ \newdef\dashed{\mfp@optrendi{DASHED}{\the\dashlen,\the\dashspace}}%
+ \newdef\dotted{\mfp@optrendi{dotted}{\the\dotsize,\the\dotspace}}%
+% \end{macrocode}
+%
+% \cs{gendashed} takes one mandatory argument, the name of a dash pattern
+% previously defined by \cs{dashpattern}. The implementing macro,
+% \cs{mfp@rendi}, is like the above, except it doesn't need to perform the
+% option processing.
+% \begin{macrocode}
+ \newdef\gendashed{\mfp@rendi{gendashed}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\zigzag}
+% This draws a zigzag, crossing from one side of the following path to
+% the other. It has one required argument containing four dimensions,
+% \meta{dim$\sb1$} through \meta{dim$\sb4$}, separated by commas. What is
+% drawn depends on whether the path is closed or open. For an open path,
+% this draws the start of the path for a distance \meta{dim$\sb1$},
+% followed by zigzags, followed by the end of the path for a distance
+% \meta{dim$\sb2$}. For a closed path the first two dimensions are
+% ignored and the zigzags follow the entire path.
+%
+% The last two dimensions define the size and slant of each `zig'. If we
+% think of the zigzags as representing a periodic wave, \meta{dim$\sb3$}
+% is the wavelength and \meta{dim$\sb4$} is the amplitude (distance from
+% the true path to the sharp points).
+%
+% \DescribeMacro{\sinewave}
+% This is similar, but the result is a smooth sinewave shaped rendering.
+%
+% \DescribeMacro{\coil}
+% This creates a coil, or corkscrew-shaped rendering. In this case
+% \meta{dim$\sb3$} is the distance from one loop to the next and
+% \meta{dim$\sb4$} is the maximum distance to each side of the path.
+% An alias,
+% \DescribeMacro{\corkscrew}
+% \cs{corkscrew}, is provided.
+%
+% Both \cs{sinewave} and \cs{corkscrew} take an optional `tension'
+% argument that affects the smoothness of the result. The default is $1$
+% and the higher the value the closer they become to a zigzag.
+% \begin{macrocode}
+ \newdef\zigzag{\mfp@rendi{zigzag}}%
+ \newdef\sinewave {\mfp@defaultopt{\mfp@wiggle{sinewave}}1}%
+ \newdef\coil{\mfp@defaultopt{\mfp@wiggle{corkscrew}}1}%
+ \newlet\corkscrew\coil
+% \end{macrocode}
+%
+% \cs{plot} is like \cs{dotted}, except the dots are variously shaped
+% symbols. It takes one optional argument containing the size of the
+% symbol and the spacing, separated by a comma, and one mandatory
+% argument, the name of the symbol. This can in principal be any
+% expression of type \mfc{picture} or \mfc{path}. (In \MP, type
+% \mfc{string} is also permitted). For the size parameter to work
+% correctly, the symbol should be a path, and it should have 1
+% as it's natural size. \Mfpic{} provides the predefined paths
+% \texttt{Triangle}, \texttt{Square}, \texttt{Circle}, \texttt{Star},
+% \texttt{Diamond}, \texttt{Plus}, \texttt{Cross} and \texttt{Asterisk}.
+% The first 5 have filled-in versions: \texttt{SolidTriangle}, etc.
+%
+% The utility \cs{mfp@optrendii} is similar \cs{mfp@optrendi} except it
+% takes an additional argument, in this case, the first mandatory argument
+% of \cs{plot}.
+% \begin{macrocode}
+ \newdef\plot{%
+ \mfp@optrendii{doplot}{\the\pointsize,\the\symbolspace}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\plotnodes}
+% Sometimes, rather than equally spaced dots or symbols, we would rather
+% have a symbol at each `node' of the following path. If the path is
+% created by a list macro, then usually each point in the list is a node
+% (but some of the spline paths are exceptions). For other figures, the
+% nodes can seem pretty random. For example, a \cs{circle} command without
+% optional argument produces a curve with 8 equally spaced nodes, but with
+% optional argument the nodes depend on how it is drawn and the size of
+% any angle parameter.
+%
+% \cs{plotnodes} has almost the same syntax as \cs{plot}, but since it
+% places the symbols only at the nodes of the path, there is no spacing
+% parameter in the optional argument.
+% \begin{macrocode}
+ \newdef\plotnodes{\mfp@optrendii{plotnodes}{\the\pointsize}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\showcontrols}
+% Mainly for debugging, this places a symbol at the precontrol and
+% postcontrol of each node, and draws a line segment from each symbol to
+% the node. The names of two symbols are given in a mandatory argument,
+% separated by commas. As a special case, if either symbol is just the
+% number `\mfc{0}', it is not drawn. There is an optional argument for
+% the size of these symbols. If the size is zero, neither symbol is drawn.
+% That is useful if one only wants to see the line segments.
+% \begin{macrocode}
+ \newdef\showcontrols{\mfp@optrendii{showcontrols}{\the\pointsize}}%
+% \end{macrocode}
+%
+%
+% \subsubsection{Closing a curve}\label{closing}
+%
+% There are several commands used to close paths (by prefixing them to
+% figure macros). The \grafbase{} commands that implement closing have a
+% common naming pattern. They all contain the string `\texttt{closed},
+% to which is added a distinguishing prefix. If the command takes a
+% tension argument, the letter `\texttt{t}' is appended. The corresponding
+% command without the `\texttt{t}' calls it with a default tension value.
+%
+% These macros all call \cs{mfp@close}, feeding it the proper prefix and
+% the tension value (empty if there is no tension option).
+%
+% \DescribeMacro{\sclosed}
+% The first, \cs{sclosed}, closes smoothly in the same manner that
+% \cs{curve} creates a smooth path.
+%
+% \DescribeMacro{\bclosed}
+% The second, \cs{bclosed}, uses an ordinary \MF{} B\'ezier. These two
+% have an optional argument: the amount of tension to put in the
+% connecting link. This makes no sense with the rest.
+%
+% \DescribeMacro{\lclosed}
+% The macro \cs{lclosed} always draws a straight line from the end of a
+% path to its start.
+%
+% \DescribeMacro{\cbclosed}
+% The macro \cs{cbclosed} computes a closure by calculating cubic B-spline
+% control points from the path data and then generating a connecting
+% spline (see \cs{cspline}).
+%
+% \DescribeMacro{\qbclosed}
+% The macro \cs{qbclosed} is similar, but quadratic B-splines are used.
+%
+% \DescribeMacro{\uclosed}
+% At one time \cs{uclosed} was defined in a more complicated manner than
+% necessary, but actually produced exactly the same path as \cs{bclosed}.
+% So now they are just made to be equal.
+% \begin{macrocode}
+ \newdef\bclosed{\mfp@nullopt{\mfp@close b}}%
+ \newdef\sclosed{\mfp@nullopt{\mfp@close s}}%
+ \newdef\lclosed{\mfp@close l{}}%
+ \newdef\cbclosed{\mfp@close{cb}{}}%
+ \newdef\qbclosed{\mfp@close{qb}{}}%
+ \newlet\uclosed\bclosed
+% \end{macrocode}
+%
+% \DescribeMacro{\makesector}
+% Takes a following arc and draws the sector (even if the center point
+% is not known). It really only works if prefixed to one of the
+% \cs{arc} commands, but it will attempt to create something even for
+% other paths. It calculates the center of the circle containing the arc
+% and connects that to the endpoints with two straight lines.
+% \begin{macrocode}
+ \newdef\makesector{\mfp@modmac{makesector}}%
+% \end{macrocode}
+%
+%
+% \subsubsection{Filling a closed curve}\label{filling}
+%
+% The next few prefix macros fill the interior of the closed path that
+% follows with some sort of pattern. The path must be closed; if not,
+% the path will simply be drawn with a solid line.
+%
+% \DescribeMacro{\tess}
+% \cs{tess} takes one mandatory argument, the name of a previously defined
+% tile (see subsection~\ref{composite}), and fills with shifted copies of
+% the tile.
+% \begin{macrocode}
+ \newdef\tess{\mfp@rendi{tess}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\shade}
+% \cs{shade} is implemented in \MF{} by filling a region with a pattern
+% of dots, simulating gray. It take one optional parameter, the separation
+% between dots, defaulting to \cs{shadespace}. The size of the dots is
+% set by issuing the command \cs{shadewd}. In \MP{} a gray fill is
+% used but the syntax is the same. It would be better to just use
+% \cs{gfill} instead (described later).
+%
+% \DescribeMacro{\polkadot}
+% This command is actually meant to fill with a pattern of large dots.
+% It also takes the separation between dots (default \cs{polkadotspace})
+% as an optional argument. The command \cs{polkadotwd} sets the size of
+% the dots.
+% \begin{macrocode}
+ \newdef\shade{\mfp@optrendi{shade}{\the\shadespace}}%
+ \newdef\polkadot{\mfp@optrendi{polkadot}{\the\polkadotspace}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\thatch}
+% The hatching macros mostly take two optional parameters, the
+% separation of the hatching lines (default \cs{hatchspace}) and their
+% color. One slight exception is \cs{thatch}, which is called by all the
+% others. Its first optional argument is the separation \emph{and} the
+% angle (default $0$) of the lines, separated by a comma. The other
+% hatching macros have a fixed angle.
+% \DescribeMacro{\lhatch}
+% The macro \cs{lhatch} has lines at a $-45$ degree angle (upper left to
+% lower right),
+% \DescribeMacro{\rhatch}
+% \cs{rhatch} produces the opposite diagonal (angle $45$ degrees), and
+% \DescribeMacro{\xhatch}
+% \cs{xhatch} does both sets of lines (cross hatching).
+% \begin{macrocode}
+ \newdef\thatch{\mfp@defaultopt\mfp@thatch{\the\hatchspace,0}}%
+ \newdef\lhatch{\mfp@defaultopt\mfp@lhatch{\the\hatchspace}}%
+ \newdef\rhatch{\mfp@defaultopt\mfp@rhatch{\the\hatchspace}}%
+ \newdef\xhatch{\mfp@defaultopt\mfp@xhatch{\the\hatchspace}}%
+ \newlet\hatch\xhatch
+% \end{macrocode}
+%
+% \DescribeMacro{\gradient}
+% The \cs{gradient} command approximates a linear gradient fill using
+% adjacent strips of different colors. It takes one mandatory argument
+% containing three items separated by commas. The first is a
+% \mfc{vardef}-ed function that produces the colors, the second is the
+% thickness of the color strips and the third is the angle these strips
+% make with the horizontal.
+%
+% \DescribeMacro{\areagradient}
+% This allows more general shading. It produces a fill consisting of
+% differently colored ``pixels''. It also takes one mandatory argument
+% containing three items separated by commas. The first is a
+% function of two variables that produces the colors, the second is the
+% width of the pixels and the third is their height.
+%
+% \DescribeMacro{\radialgradient}
+% This allows shading in a concentric circular pattern. It produces a fill
+% consisting of differently colored circular strips. It also takes one
+% mandatory argument containing three items separated by commas. The first
+% is a function of one variable that produces the colors, the second is
+% the thickness of the circular strips, and the third is an ordered pair
+% (the center the circles).
+% \begin{macrocode}
+ \newdef\gradient{\mfp@rendi{axialgradient}}%
+ \newdef\areagradient{\mfp@rendi{areagradient}}%
+ \newdef\radialgradient{\mfp@rendi{radialgradient}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\gfill}
+% \cs{gfill} take one optional argument, the color with which to fill.
+% Under \MF{} the color must be a numeric (representing a level of gray)
+% or one of the color functions, which are defined in \file{grafbase.mf}
+% to return numeric values.
+%
+% \DescribeMacro{\gclear}
+% \cs{gclear} will actually fill with the color \texttt{background} in \MP.
+% This merely covers what originally was there.
+%
+% \DescribeMacro{\gclip}
+% \cs{gclip} clears the outside of a region, but it is a true clip
+% operation, completely different from \cs{gclear}.
+% \begin{macrocode}
+ \newdef\gfill{\mfp@nullopt{\mfp@clroptrend{filled}}}%
+ \newdef\gclear{\mfp@rendmac{unfilled}}%
+ \newdef\gclip{\mfp@rendmac{Clip}}%
+% \end{macrocode}
+%
+%
+% \subsubsection{Modifying a curve}\label{transforming}
+%
+% The following are transformation prefixes. They are all (except
+% \cs{reverse}, for reasons covered in section~\ref{prefix}) implemented
+% with \cs{mfp@modmac}. Most have a mandatory argument, the parameters of
+% the transformation. Because of the implementation, one can insert
+% rendering macros between a transformation macro and the path being
+% transformed, and they will apply to the untransformed path.
+%
+% If the \env{mfpic} environment has different $x$ and $y$ scales, the
+% \cs{rotatepath} and \cs{reflectpath} compensates so that the result is
+% congruent to the original. This is what users seem to expect. The rest
+% do not compensate and users don't seem to mind.
+% \begin{itemize}
+% \item \cs{reverse} and \cs{xyswappath} have no argument.
+% \item \cs{reversepath} = \cs{reverse}.
+% \item \cs{rotatepath} takes \marg{\meta{point},\meta{angle}}.
+% \item \cs{shiftpath} takes \marg{\meta{pair}}.
+% \item \cs{scalepath} takes \marg{\meta{center of scaling},\meta{factor}}.
+% \item \cs{xscalepath} takes \marg{\meta{x-center},\meta{factor}}.
+% \item \cs{yscalepath} takes \marg{\meta{y-center},\meta{factor}}.
+% \item \cs{slantpath} takes \marg{\meta{y-pivot},\meta{factor}}.
+% \item \cs{xslantpath} = \cs{slantpath}.
+% \item \cs{yslantpath} takes \marg{\meta{x-pivot},\meta{factor}}.
+% \item \cs{reflectpath} takes \marg{\meta{point1},\meta{point2}}, the
+% endpoints of a `mirror'.
+% \item \cs{transformpath} takes \marg{\meta{transformer}}, a \MF{}
+% `transformer' (see ``The \MF book'', page~212).
+% \end{itemize}
+%
+% All of them simply write the \grafbase{} command of nearly the same
+% name, with the same arguments, via
+% \cs{mfp@modi}.\SpecialUsageIndex{\reversepath}\SpecialUsageIndex{\reverse}
+% \SpecialUsageIndex{\xyswappath}\SpecialUsageIndex{\rotatepath}
+% \SpecialUsageIndex{\shiftpath}\SpecialUsageIndex{\scalepath}
+% \SpecialUsageIndex{\xscalepath}\SpecialUsageIndex{\yscalepath}
+% \SpecialUsageIndex{\slantpath}\SpecialUsageIndex{\xslantpath}
+% \SpecialUsageIndex{\yslantpath}\SpecialUsageIndex{\reflectpath}
+% \SpecialUsageIndex{\transformpath}
+%
+% \begin{macrocode}
+ \newdef\reversepath{\mfp@addmac{reverse}}%
+ \newlet\reverse\reversepath
+ \newdef\xyswappath{\mfp@modmac{xyswappedpath}}%
+ \newdef\rotatepath{\mfp@modi{rotatedpath}}%
+ \newdef\shiftpath{\mfp@modi{shiftedpath}}%
+ \newdef\scalepath{\mfp@modi{scaledpath}}%
+ \newdef\xscalepath{\mfp@modi{xscaledpath}}%
+ \newdef\yscalepath{\mfp@modi{yscaledpath}}%
+ \newdef\xslantpath{\mfp@modi{xslantedpath}}%
+ \newlet\slantpath\xslantpath
+ \newdef\yslantpath{\mfp@modi{yslantedpath}}%
+ \newdef\reflectpath{\mfp@modi{reflectedpath}}%
+ \newdef\transformpath{\mfp@modi{transformedpath}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\parallelpath}
+% This is not an affine transformation, but is a transformation of sorts.
+% It tries to define a curve that parallels the following curve at a fixed
+% distance. The mandatory argument is the separation, in graph units. It
+% is sort of experimental and will probably fail if the path that follows
+% is too wild. Since \cs{doubledraw} shares code with \cs{parallelpath},
+% that too will probably fail in similar circumstances.
+% \begin{macrocode}
+ \newdef\parallelpath{\mfp@modi{parapath}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\randomizepath}
+% This, too, is a nonaffine transformation. It tries to define a curve
+% that remains smooth if the original is, but applies random shifts to the
+% nodes and random changes to the direction at each node.
+%
+% The only argument is mandatory and consists of a maximum shift (in graph
+% units) and a `weirdness' (a pure number) separated by a comma.
+%
+% The maximum shift can be a number, in which case it is the maximum
+% possible distance and there is no preferred direction, or it can be an
+% ordered pair, in which case it gives maximum horizontal and vertical
+% shift amounts.
+%
+% The `weirdness' is applied to path directions. If it is 0, there is no
+% change in directions of the path at nodes. If it is a positive number
+% $w$, a random angle of up to $\pm 30 w$ is added to the directions at
+% each node. Also a random scaling between $2^{-w}$ and $2^w$ is applied
+% to the control vectors. Also, if a path has a corner, the angle between
+% the two sides is randomly adjusted to be between $2^{-w}$ and $2^w$
+% times its original size.
+%
+% \DescribeMacro{\randomlines}
+% This is a simpler version whose mandatory argument contains only the
+% maximum shift parameter. It shifts the nodes and connects the results
+% with straight lines.
+% \begin{macrocode}
+ \newdef\randomizepath{\mfp@modi{randompath}}%
+ \newdef\randomlines{\mfp@modi{randomlines}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\interpolatepath}
+% This prefix macro has one mandatory argument containing a number and a path,
+% separated by a comma. The number can be any \MF{} numeric expression and
+% the path can be any \MF{} path expression. The number should normally be
+% in the range $0$ to $1$ and the path normally a variable (e.g, defined using
+% \cs{store}). It returns a path which is between the one in its argument
+% and the one that follows. If the number is $0$, the following path is
+% returned unchanged, if the number is $1$ the path in the argument is
+% returned.
+% \begin{macrocode}
+ \newdef\interpolatepath{\mfp@modi{interpolatedpath}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\arccomplement}
+% This can precede an arc and then returns its complement. It starts at
+% the last point of the following arc and ends at the beginning of that
+% arc. The sense of the resulting arc (clockwise or anticlockwise) is the
+% same as the arc it is applied to. If the following path is not an arc,
+% the result is an arc that connects the end of the path to the beginning,
+% but may bear no obvious relation to that path.
+% \begin{macrocode}
+ \newdef\arccomplement{\mfp@modmac{arccomplement}}%
+% \end{macrocode}
+%
+%
+% \subsubsection{Adding arrowheads (and tails)}\label{arrows}
+%
+% Paths can now have heads and tails and something placed in the
+% middle. Also the head, tail or the `something' can be any symbol. In
+% addition to the \cs{plotsymbol} possibilities, there is \gbc{Arrowhead},
+% which is the arrowhead of previous \mfpic{} versions, as well as
+% \gbc{Leftharpoon} and \gbc{Rightharpoon}, which are the left and right
+% halves.
+%
+% Intended for middle or tail of an arrow are \gbc{Crossbar}, \gbc{Leftbar}
+% and \gbc{Rightbar}, and for the tail there are \gbc{Lefthook} and
+% \gbc{Righthook}. \gbc{Crossbar} cuts right across the shaft
+% (perpendicular unless a rotation option is present) while \gbc{Leftbar}
+% extends to the left (from the point of view of someone facing in the
+% direction of the path. Similarly \gbc{Lefthook} curls to the left of the
+% arrow's shaft.
+%
+% All the new arrow commands take one mandatory argument, the name of the
+% symbol, followed by up to four optional arguments. Three of the optional
+% arguments are the same for all commands: these set the color, the size
+% and the amount of rotation of the symbol. Normally the symbol is
+% adjusted to `point' in the direction of the path (\gbc{Crossbar} is
+% actually perpendicular to it); with the rotation option it can be
+% adjusted.
+%
+% For example, \oarg{c red} makes the symbol red, \oarg{l 5pt} or \oarg{s
+% 5pt} scales the symbol so its nominal size or length is 5 points,
+% \oarg{r 10} rotates the symbol 10 degrees (anticlockwise) from its
+% default position.
+%
+% The other optional argument depends on the command: for heads it is an
+% amount to shift the head backward, for tails the amount to shift it
+% forward, and for things placed in the middle, it is the fraction of the
+% length of the path along which the symbol is placed.
+%
+% For example, \oarg{b 2pt} shifts a head two points backward, \oarg{f
+% 2pt} shifts a tail 2 points forward and \oarg{f .5} puts the symbol
+% right in the middle.
+%
+% The letters are mnemonics, being the initial letters of the words
+% `color', `length' or `size', `rotate', `back', and `forward' or
+% `fraction'. Actually, the letters \texttt{s} and \texttt{l} have
+% identical effects since I couldn't decide which mnemonic was more
+% natural. Oddly, \texttt{b} and \texttt{f} have identical effects also.
+% So if you accidentally use \oarg{b 2pt} for a tail, it will still be
+% shifted \emph{forward}! The reason for this is that these \mfpic{}
+% commands all correspond to \grafbase{} commands that have the same
+% argument structure. Both \texttt{b} and \texttt{f} simply cause the
+% accompanying value to be written into the fourth argument, while the
+% different commands use that argument differently. (You can shift a
+% tail backwards with a negative argument, if you really need that.)
+%
+% \DescribeMacro{\arrowhead}\SpecialUsageIndex{\arrow}
+% The \cs{arrowhead} command has a star-form which allows part of the
+% path beyond the tip of the arrowhead to be erased. The presence of a
+% star ultimately changes the \MF{} command passed on to later macros.
+% It does so by passing to \cs{mfp@arrowhead} the word `\texttt{true}' for
+% the star-form and `\texttt{false}' otherwise.
+%
+% The old \cs{arrow} command first checks for a star, and passes to
+% \cs{mfp@arrow} either an `\texttt{x}' or an empty parameter. This is
+% used to construct the \grafbase{} command name. We use \cs{def}
+% because \prog{eplain} defines one also.
+% \begin{macrocode}
+ \newdef\arrowhead{\mfp@ifstar\mfp@arrowhead\mfp@T\mfp@F}%
+ \def\arrow{\mfp@ifstar\mfp@arrow{x}{}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\arrowtail}
+% Tails are just like heads except that there is no star-form and the
+% \grafbase{} command name (second parameter of \cs{mfp@arr}) is
+% different.
+% \DescribeMacro{\arrowmid}
+% And \cs{arrowmid} is just like \cs{arrowtail} except for the command
+% name and the different default for the position optional argument.
+% \begin{macrocode}
+ \newdef\arrowtail##1{\mfp@arr 0{tailpath (##1)}}%
+ \newdef\arrowmid##1{\mfp@arr{0.5}{midpath (##1)}}%
+% \end{macrocode}
+%
+% This ends the prefix macros.
+%
+%
+% \subsection{Transforming the coordinate system}\label{coordinate}
+%
+% \DescribeMacro{\coords}
+% The following apply transforms to the \MF{} coordinate system, and
+% these can be localized with the \env{coords} environment. In plain
+% \TeX{} \cs{coords} starts the environment and
+% \DescribeMacro{\endcoords}
+% \cs{endcoords} closes it. The transforms apply when the curve is
+% \emph{rendered} and do not affect what paths are defined. Thus, for
+% example, \cs{store} will store the same path whatever the state of the
+% coordinate system. In terms of \grafbase{} commands, only \gbc{vtr} and
+% \gbc{ztr} are changed, and these are only applied when rendering.
+%
+% \DescribeMacro{\applyT}
+% These are implemented by \cs{applyT} which takes as its argument a \MF{}
+% `transformer'.
+%
+% These commands (except \cs{turn}, kept for backward compatibility) have
+% no optional arguments, only mandatory arguments containing the
+% parameters of the transformation. \cs{rotate} rotates around the origin,
+% \cs{rotatearound} rotates around a given point, \cs{turn} takes the point
+% as an optional argument, defaulting to $(0,0)$.
+%
+% Transformations are applied in reverse of the order they appear in the
+% file (i.e., the most recent first). This applies even if \env{coords}
+% environments are nested, except that at \cs{endcoords} the state at
+% the matching \cs{coords} is restored.
+% \begin{macrocode}
+ \newdef\coords{\mfsrc{\@nl bcoords\@nl}}%
+ \newdef\endcoords{\mfsrc{\@nl ecoords\@nl}}%
+ \newdef\applyT##1{\mfsrc{\@nl hide(apply_t (##1))}}%
+ \def\rotate##1{\applyT{rotated ##1}}%
+ \newdef\rotatearound##1##2{%
+ \applyT{rotatedaround (##1, ##2)}}%
+ \newdef\turn{\mfp@defaultopt\mfp@turn{(0,0)}}%
+ \newdef\reflectabout##1##2{\applyT{reflectedabout (##1, ##2)}}%
+ \let\mirror\reflectabout
+ \newdef\shift##1{\applyT{shifted ##1}}% pair.
+ \def\scale##1{\applyT{scaled ##1}}% same scaling
+ \newdef\xscale##1{\applyT{xscaled ##1}}%
+ \newdef\yscale##1{\applyT{yscaled ##1}}%
+ \newdef\zscale##1{\applyT{zscaled ##1}}%
+ \newdef\xslant##1{\applyT{xslant ##1}}%
+ \newdef\yslant##1{\applyT{yslant ##1}}%
+ \newdef\zslant##1{\applyT{zslant ##1}}%
+ \newdef\boost##1{\applyT{boost ##1}}%
+ \newdef\xyswap{\applyT{xyswap}}%
+% \end{macrocode}
+%
+%
+% \subsection{Axes and grids}\label{axes}
+%
+% An axis would normally not be thought of as a figure macro, because one
+% would not want to position them arbitrarily nor transform them. One
+% would also not want the hassle of having always to specify the origin,
+% etc. However, one would want to occasionally make them dotted or dashed
+% or colored.
+%
+% \DescribeMacro{\axes}
+% The original \cs{axes} was therefore not a figure macro. It always drew
+% both axes with solid lines.
+% \DescribeMacro{\xaxis}
+% Later \cs{xaxis} and \cs{yaxis} were introduced, which mimicked the
+% behavior of \cs{axes}. For backward
+% \DescribeMacro{\yaxis}
+% compatibility, the old behavior is supported with the old commands. The
+% optional argument to \cs{axes}, \cs{xaxis} and \cs{yaxis} is the length
+% of the arrowhead, defaulting to \cs{the}\cs{axisheadlen}.
+% \begin{macrocode}
+ \newdef\xaxis{\mfp@defaultopt{\mfp@simple{xaxis}}{\the\axisheadlen}}%
+ \newdef\yaxis{\mfp@defaultopt{\mfp@simple{yaxis}}{\the\axisheadlen}}%
+ \newdef\axes{\mfp@defaultopt {\mfp@simple{axes}}{\the\axisheadlen}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\axis}
+% When border axes were added, all the code was revamped, and now we
+% create axes as figure macros. It also became clear there was so
+% much common code that it would be better to implement the syntax
+% \cs{axis}\marg{l}, etc., rather than have separate commands \cs{laxis},
+% etc. for all six possible axes. Therefore, only the $x$- and $y$-axes
+% have their own commands. The \cs{axis} command has one optional argument
+% (the arrowhead length) and one mandatory argument, a single letter
+% designating the axis to draw.
+%
+% \DescribeMacro{\axisline}
+% The \cs{axisline} command is provided for users who want more control.
+% It is a figure macro that produces the corresponding line, and the user
+% can use all the flexibility of the \cs{arrowhead} command if necessary.
+% \begin{macrocode}
+ \newdef\axis{\mfp@nullopt\mfp@axis}%
+ \newdef\axisline##1{\mfp@figmac{axisline.##1}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\doaxes}
+% \cs{doaxes} takes one optional argument, the arrowhead length, and one
+% mandatory argument, all the axes to draw. These should be just letters
+% with nothing separating them except optional spaces. \cs{doaxes} is
+% implemented via the self-looping macro \cs{mfp@doaxis}.
+%
+% \DescribeMacro{\border}
+% The \cs{border} command is almost equivalent to \cs{doaxes}\marg{lbrt},
+% except it is a figure macro. With it, the whole picture can be filled
+% with a background color, or outlined in any available rendering.
+% \begin{macrocode}
+ \newdef\doaxes{\mfp@nullopt\mfp@doaxes}%
+ \newdef\border{\mfp@figmac{borderrect}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\axismarks}
+% Hash marks (or tick marks) on the axes are provided by the
+% \cs{axismarks} command (\cs{marks} was the original choice, but that
+% turned out to conflict with one of e\kern-.06em\TeX's primitives).
+% For brevity, therefore, we also have separate commands
+% \DescribeMacro{\xmarks}
+% \cs{xmarks},
+% \cs{ymarks}\SpecialUsageIndex\ymarks,
+% \cs{lmarks}\SpecialUsageIndex\lmarks,
+% \cs{bmarks}\SpecialUsageIndex\bmarks,
+% \cs{rmarks}\SpecialUsageIndex\rmarks, and
+% \cs{tmarks}\SpecialUsageIndex\tmarks.
+%
+% \cs{axismarks} has two mandatory arguments: the letter for the axis, and
+% a comma separated list of numbers giving the positions on the axis. It
+% functions as a list macro (see previous discussion of types of macros)
+% and so the list of numbers can be prescribed by an external file
+% \cs{datafile}\marg{\meta{file}}. One must be sure to issue an
+% appropriate \cs{using} command to produce a numeric list in the output.
+% This is the main reason for the \cs{usingnumericdefault} command.
+%
+% I had a change of heart after the \cs{axis} command, and I made things
+% simpler to code here by putting the optional argument of \cs{axismarks}
+% after the first mandatory argument. I could justify this because I
+% expect most users will use the shorter versions that don't require the
+% first argument (that is, the expectation that optional arguments come
+% first is still satisfied). The optional argument is the length of the
+% marks. The position of the marks is controlled separately by the
+% \cs{setaxismarks} command, described in subsection~\ref{othersettings}.
+% \begin{macrocode}
+ \newdef\axismarks##1{\mfp@defaultopt{\mfp@marks{##1}}{\the\hashlen}}%
+ \newdef\xmarks{\axismarks x}%
+ \newdef\ymarks{\axismarks y}%
+ \newdef\lmarks{\axismarks l}%
+ \newdef\bmarks{\axismarks b}%
+ \newdef\rmarks{\axismarks r}%
+ \newdef\tmarks{\axismarks t}%
+% \end{macrocode}
+%
+% \DescribeMacro{\grid}
+% The command \cs{grid} is, I think, inaccurately named: the word `grid'
+% suggests a set of lines, but this actually produces dots. Its function
+% is to allow a visualization of the coordinates. The dots are located at
+% the points whose coordinates are integer multiples of two numbers, which
+% are given in the single mandatory argument separated by commas. The
+% optional first argument is the diameter of the dots. This defaults to
+% \cs{griddotsize}, initially \texttt{0.5pt}.
+%
+% Because of my concerns about the name, I have provided two aliases:
+% \cs{gridpoints}\SpecialUsageIndex{\gridpoints} (for consistency with
+% \cs{gridlines}, below) and \cs{lattice}\SpecialUsageIndex{\lattice}.
+%
+% \DescribeMacro{\gridlines}
+% Macro \cs{gridlines} has no optional argument and takes the same
+% mandatory argument as \cs{grid}. It draws horizontal and vertical lines
+% intersecting at all the dots that \cs{grid} would draw. Some packages
+% (and \ConTeXt) define a \cs{grid} command so we use \cs{def} here
+% instead of \cs{newdef}.
+%
+% \DescribeMacro{\hgridlines}
+% This draws only the horizontal lines and its argument is the separation
+% (in graph units) between successive lines.
+% \DescribeMacro{\vgridlines}
+% \cs{vgridlines} is the vertical version.
+%
+% No arguments are shown because \cs{mfp@simple} or \cs{mfp@simplei} will
+% read what it needs.
+% \begin{macrocode}
+ \def\grid{\mfp@defaultopt{\mfp@simplei{vargrid}}{\the\griddotsize}}%
+ \newlet\gridpoints\grid \newlet\lattice\grid
+ \newdef\gridlines{\mfp@simple{gridlines}}%
+ \newdef\vgridlines{\mfp@simple{vgridlines}}%
+ \newdef\hgridlines{\mfp@simple{hgridlines}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\plrgrid}
+% The command \cs{plrgrid} takes one mandatory argument which contains two
+% numbers separated by a comma. The first is a radius (in graph
+% coordinates) and the second an angle. It draws circular arcs at integer
+% multiples of the radius, and radial lines at integer multiples of the
+% angle. It fills the picture rectangle and is clipped to that rectangle.
+%
+% \DescribeMacro{\plrgridpoints}
+% \cs{plrgridpoints} is similar but places only dots at the intersections
+% of these arcs and rays. It takes an optional argument for the size of
+% the dots, default \cs{griddotsize}.
+% \DescribeMacro{\gridarcs}
+% The macro \cs{gridarcs} draws only the arcs. Its only parameter is the
+% distance between them (in graph units).
+% \DescribeMacro{\gridrays}
+% The macro \cs{gridrays} draws only the rays and its parameter is the
+% angle separating the rays.
+%
+% \DescribeMacro{\plrpatch}
+% \cs{plrpatch} draws: a circular arc at a starting radius, then
+% increments that radius by a step size and draws another arc, etc., and
+% finally draws an arc at the ending radius. It does a similar thing for
+% radial lines and angles. It takes one mandatory argument, which contains
+% $6$ numbers separated by commas: starting radius, ending radius, radius
+% step, starting angle, ending angle, angle step.
+% \begin{macrocode}
+ \newdef\plrgrid{\mfp@simple{polargrid}}%
+ \newdef\plrgridpoints{%
+ \mfp@defaultopt{\mfp@simplei{polargridpoints}}{\the\griddotsize}}%
+ \newdef\gridarcs{\mfp@simple{gridarcs}}%
+ \newdef\gridrays{\mfp@simple{gridrays}}%
+ \newdef\plrpatch{\mfp@simple{polarpatch}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\vectorfield}The \cs{vectorfield} command draws arrows to
+% represent a vector field. It takes one optional argument, the length of
+% the arrowhead (default \cs{headlen}), and three mandatory arguments. The
+% first is like \cs{grid}: it contains two numbers, separated by commas.
+% The arrows are located at the points whose coordinates are integer
+% multiples of two numbers. The second argument is the formula for the
+% vector field. It should consist of a pair expression in which the only
+% unknowns are the variables \texttt{x} and \texttt{y}. The last argument
+% should contain a boolean expression in which the only unknowns are
+% \texttt{x} and \texttt{y}. The macro steps through the relevant set of
+% points and draws the vector only if the boolean is true at that point.
+% The set of points considered includes only those inside the axis margins
+% (set by \cs{axismargin} and related commands). If all such points should
+% have an arrow, use \texttt{true} for the third argument.
+% \begin{macrocode}
+ \newdef\vectorfield{%
+ \mfp@defaultopt{\mfp@vectorfield{}}{\the\headlen}}%
+ \newdef\plrvectorfield{%
+ \mfp@defaultopt{\mfp@vectorfield{plr}}{\the\headlen}}%
+% \end{macrocode}
+%
+%
+% \subsection{Visualizing points}\label{points}
+%
+% \DescribeMacro{\point}
+% \cs{point} draws a dot or a small circle. It takes one optional
+% argument, the diameter of the dot, and one mandatory argument, the ordered
+% pair (or comma separated list of them) at which to draw a dot. After
+% \cs{pointfillfalse}, the dots will be unfilled (i.e., small circles),
+% after \cs{pointfilltrue} (the default), they will be solid. Like the
+% grid macros, it is not a figure macro, but merely does its drawing and
+% is not subject to any prefix macros.
+%
+% \DescribeMacro{\plotsymbol}
+% In order to mark points with other than circles, we have the
+% \cs{plotsymbol} command, so-called because it uses the same symbols as
+% the \cs{plot} command (previously discussed). There is an optional
+% argument that gives the size of the symbol, a mandatory argument with
+% the name of the symbol, and a second mandatory argument which
+% is a comma separated list of ordered pairs for the locations.
+%
+% For both these commands, the default optional argument is
+% \cs{the}\cs{pointsize}. These are list macros: in place of a list of
+% points in braces, one can use the \cs{datafile} construct.
+% \begin{macrocode}
+ \newdef\point{\mfp@defaultopt\mfp@point{\the\pointsize}}%
+ \newdef\plotsymbol{\mfp@defaultopt\mfp@plotsymbol{\the\pointsize}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\plottext}
+% We keep \cs{plottext} for backward compatability. It is equivalent
+% to a \cs{tlabel} command with a different default for the justification
+% (\oarg{cc}) and the arguments rearranged. The order for \cs{plottext}
+% is: optional justification, then \TeX{} label, then list of points.
+% Because of the complication involved in making it work, in this case we
+% do not allow a datafile to replace the list of points. All this
+% command now does is perform a simplified version of the start of the
+% \cs{tlabel} command and jump to one of its internals, \cs{mfp@thelabel}.
+% \begin{macrocode}
+ \newdef\plottext{\begingroup\mfp@resetwhitespace\mp@mksharpother
+ \mfp@defaultopt\mfp@plottext{cc}}%
+% \end{macrocode}
+%
+%
+% \subsection{Some composite objects}\label{composite}
+%
+% \DescribeMacro{\tile}
+% A tile is syntactically a \MF{} picture variable. The commands between
+% \cs{tile} and \cs{endtile} (or \cs{begin}\marg{tile} and
+% \cs{end}\marg{tile}) should be any sequence of \mfpic{} figure macros
+% with possible prefix macros. They will be drawn not on the main picture
+% (\gbc{currentpicture}), but on the picture given by the name in the
+% \cs{tile} command's argument.
+%
+% The argument has five parts separated by commas, the first is the
+% name, the second is the unit of length (some absolute dimension like
+% \mfc{1pt}), the next two are the width and height (pure numbers, the
+% actual dimensions being these numbers times the unit of length), the
+% last is one of the words \mfc{true} or \mfc{false}, indicating whether
+% tile should be clipped to these dimensions (true) or allowed to extend
+% beyond them (false).
+% \begin{macrocode}
+ \newdef\tile{\mfp@simple{tile}}%
+ \newdef\endtile{\mfcmd{\@nl endtile}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\patharr}
+% A path array is a set of \MF{} path variables each having the same base
+% name followed by a different numeric suffix. This command turns off
+% rendering and simply assigns these variables in order to the path
+% expressions written by the figure macros between \cs{patharr} and
+% \cs{endpatharr}. It works by redefining \cs{mfp@storepath}, issued by all
+% the figure macros. These variables can be accessed in \mfpic{} by using
+% the \cs{mfobj} command. The name of the path array \emph{without} a
+% numeric suffix is a \MF{} numeric variable whose value is the number of
+% array elements.
+%
+% \emph{Explicit} rendering commands will work in the sense that the
+% affected path will be rendered in addition to being assigned to the
+% variable. The only parameter is the common base name of the variables.
+% \begin{macrocode}
+ \newdef\patharr##1{\begingroup
+ \mfsrc{hide(numeric ##1; path ##1[]; ##1 = 0)}%
+ \def\mfp@storepath{\mfsrc{\@nl store (##1[incr ##1])}}%
+ \setrender{}}%
+ \newdef\endpatharr{\endgroup}%
+% \end{macrocode}
+%
+% \DescribeMacro{\connect}
+% This starts an environment that can contain any sequence of figure
+% macros. The environment functions as a single large \cs{mfp@figmac}, the
+% figure being the path obtained by connecting all the paths
+% \DescribeMacro{\endconnect}
+% (up to the following \cs{endconnect} command) by straight lines from the
+% end of one to the beginning of the next. It is implemented via a path
+% array, so the individual figures within the environment have implicit
+% rendering turned off. Explicit rendering commands within the
+% environment will have the expected effect.
+%
+% The \cs{mfp@checklatexenv} at the end allows \cs{connect} to be used as
+% a \LaTeX{} environment.
+% \begin{macrocode}
+ \newdef\connect{\mfp@figmac{begingroup; save mfpicnexus}%
+ \patharr{mfpicnexus}}%
+ \newdef\endconnect{\endpatharr
+ \mfcmd{\@nl mkpoly (\mfp@F, mfpicnexus)\@nl endgroup}%
+ \mfp@checklatexenv{connect}{\aftergroup\@mfpstarttrue}{}}%
+% \end{macrocode}
+%
+%
+% \subsection{Other graphical features}\label{otherfeatures}
+%
+% And now for something completely different.
+%
+% \DescribeMacro{\piechart}
+% Actually, nothing is drawn by the \cs{piechart} command; it only defines
+% a path array and a few other internal variables. Since nothing is
+% drawn by this command, it makes sense to allow it outside the drawing
+% environment, so it has been made available outside \env{mfpic}
+% environments with the command name \cs{mfppiechart}(page
+% \pageref{charts}).
+%
+% The \cs{piechart} command takes one optional argument and two mandatory
+% argument. The first mandatory argument contains the center and radius of
+% the chart separated by a comma; the second is the list of data. The
+% optional argument contains a letter and an optional angle. The letter
+% can be `\texttt{c}' for clockwise or `\texttt{a}' for anticlockwise and
+% determines the direction around the pie the different wedges will be drawn.
+% The angle determines the angle of the first edge (in the given
+% direction) of the first wedge. The default letter is \texttt{c} and the
+% default angle is $90$ degrees, which starts the first wedge at
+% 12~o'clock and proceeds from there either clockwise or anticlockwise..
+%
+% \cs{piechart} acts like a list macro in that the above arguments should
+% be followed by either a list of numbers in braces or a \cs{datafile}
+% command (with numbers in a datafile). If this feature is used, an
+% appropriate \cs{using} command is needed. If the numbers are listed one
+% per line, then \cs{usingnumericdefault} should work.
+%
+% \DescribeMacro{\piewedge}
+% Each individual wedge (elements of the above mentioned path array) may
+% be drawn using the \cs{piewedge} command, which is a figure macro. That
+% command takes one optional argument that determines how the wedge is
+% drawn (described below) and one mandatory argument, the number of the
+% wedge. Without the optional argument, \cs{piewedge}\marg{1} draws the
+% first wedge in the position given by the arguments of the \cs{piechart}
+% command, as described above.
+%
+% The optional arguments can be:
+% \begin{itemize}
+% \item \oarg{x\meta{distance}}, the wedge is `exploded': drawn
+% shifted radially outward from the center of the pie chart by
+% \meta{distance} (in graph units).
+% \item \oarg{s(\meta{dx},\meta{dy})}, the wedge is shifted from its
+% position by \meta{dx} horizontally and \meta{dy} vertically (in
+% graph units).
+% \item \oarg{m(\meta{x},\meta{y})}, the wedge is moved so that its
+% point is at coordinates \parg{\meta{x},\meta{y}}.
+% \end{itemize}
+% \begin{macrocode}
+ \newlet\piechart\mfppiechart
+ \newdef\piewedge{\mfp@defaultopt\mfp@piewedge{d}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\barchart}
+% As with \cs{piechart}, the \cs{barchart} command does not actually draw
+% anything, but defines a path array, so we make available a version for
+% use outside \cs{mfpic} environments: \cs{mfpbarchart} (page
+% \pageref{charts}).
+%
+% The \cs{barchart} command takes one optional and a mandatory
+% arguments. The data follow the mandatory argument. Like \cs{piechart},
+% the data can be a list in braces or an external file. The data determine
+% the length of the bars in graph units. Since version 0.7, the data can
+% be a list of numbers or pairs or both. It is difficult to give a
+% \cs{using} specification that allows both in an external file, but a
+% list in braces is simply written as is and \MF{} has no problems mixing
+% them.
+%
+% Pairs are interpreted as giving the start and end of a bar. Numbers are
+% interpreted as giving only the end, with the start at $0$. If a datafile
+% is used, an appropriate \cs{using} command should be issued beforehand.
+% In that case probably all the data should be pairs or all numeric.
+%
+% The optional argument determines the position and width of the bars and
+% the first mandatory argument determines the orientation. The mandatory
+% argument can be \marg{v} for vertical bars with base on the $x$ axis, or
+% \marg{h} for horizontal bars with base on the $y$ axis.
+% The optional argument contains three numbers separated by commas. The
+% first is the coordinate of the leading edge of the first bar, the second
+% is the distance (in graph coordinates) between the leading edge of one
+% bar and that of the next, and the third is the fraction of that distance
+% that is occupied by the bar. The default is \oarg{0,1,1}.
+%
+% We provide the aliases \cs{bargraph}, \cs{histogram} and \cs{gantt}.
+%
+% \DescribeMacro{\chartbar}
+% The \cs{chartbar} command can then be used to draw a bar (one of the
+% members of the just mentioned array). It is a figure macro and take the
+% number of the bar as its mandatory argument. Aliases \cs{graphbar},
+% \cs{histobar} and \cs{ganttbar} are provided. We avoid \cs{bar} because
+% it is already a math accent command.
+% \begin{macrocode}
+ \newlet\barchart\mfpbarchart
+ \newlet\bargraph\barchart
+ \newlet\histogram\barchart
+ \newlet\gantt\barchart
+ \newdef\chartbar##1{\mfp@figmac{chartbar[##1]}}%
+ \newlet\graphbar\chartbar
+ \newlet\histobar\chartbar
+ \newlet\ganttbar\chartbar
+% \end{macrocode}
+%
+% \DescribeMacro{\gbrace}
+% Finally, a command to create a brace shape with given ends and a given
+% location for the cusp. We name it with a `g' so as not to overwrite a
+% possible text command with a similar name. The argument is simply passed
+% to the corresponding \grafbase{} command; it consists of three points
+% separated by commas denoting, respectively, the start of the brace,
+% the cusp, and the end. Distance from cusp to line connecting start and
+% end must be less than about 3/4 the distance to either start or end.
+%
+% This ends \cs{mfp@grafmacs}.
+% \begin{macrocode}
+ \newdef\gbrace##1{\mfp@figmac{mkbrace(##1)}}%
+}% end \mfp@grafmacs
+% \end{macrocode}
+%\toks0={^^A What's this all about?
+\def\mfp@abc#1#2#3#4#5#6#7#8/{#6#3#1#2#6#7#6#4#5}%
+\@namedef{\mfp@abc STAGGER LEE/}{%
+\mfpic[20]{-2}{2}{-3}{3}%
+ \tile{peter, 1pt, 10, 10, false}%
+ \penwd{1pt}%
+ \draw[rgb(1,0,0)]\lines{(0,0), (5,5), (10,0)}%
+ \pointcolor{rgb(0,.8,0)}%
+ \point[3pt]{(2.5,7.5)}%
+ \endtile
+ \draw\tess{peter}\gfill[rgb(1,1,0)]\ellipse{(0,0),2,3}%
+\endmfpic}%
+%}
+%
+% \DescribeMacro{\mfppiechart}
+% We make \cs{mfppiechart} and \cs{mfpbarchart}\label{charts} global,
+% since all they do is define variables
+% \DescribeMacro{\mfpbarchart}
+% and arrays.
+% \begin{macrocode}
+\newdef\mfppiechart{\mfp@defaultopt\mfp@piechart{c}}%
+\newdef\mfpbarchart{\mfp@defaultopt\mfp@barchart{0,1,1}}%
+\newlet\mfpbargraph\mfpbarchart
+\newlet\mfphistogram\mfpbarchart
+\newlet\mfpgantt\mfpbarchart
+% \end{macrocode}
+%
+%
+%
+% \section{Implementing the graphics macros}
+%
+% To keep \cs{mfp@grafmacs} from becoming unweildy, we define here all the
+% internal commands that are called by the user level macros. We've
+% given most of them names that begin \cs{mfp} to keep from overwriting
+% other package commands
+%
+% \cs{arc} and \cs{circle} are defined identically except that the first
+% sets a switch to true, the second to false. This selects the spelling of
+% the corresponding \grafbase{} command and the value of the first
+% argument of \gbc{fallbackpath}. Both \cs{arc} and \cs{circle} then
+% call \cs{mfp@arccircle} which takes the optional argument as \arg1,
+% and the data required for that option as \arg2. Actually, \cs{arc}
+% first checks for a * and then calls \cs{mfp@arc}, which follows the
+% above description.
+%
+% The \gbc{fallbackpath} is a $0$-length path at whatever point is listed
+% first in \arg2, it is cyclic in the case of a circle (not that it
+% matters for a one-point path).
+% \begin{macrocode}
+\newdef\mfp@arc{\mfp@switchtrue\mfp@defaultopt\mfp@arccircle{s}}%
+\newdef\arc@or@circle{\ifmfp@switch arc\else circle\fi}%
+\newdef\mfp@arccircle#1#2{%
+ \@ifundefined{mfp@arc@#1}%
+ {\Mfpic@warn{Invalid option; ignoring \mfp@b\arc@or@circle\space
+ command.}%
+ \mfp@figmac{fallbackpath(\ifmfp@switch\mfp@F\else\mfp@T\fi) (#2)}}%
+ {\mfp@figmac{\arc@or@circle\@nameuse{mfp@arc@#1} (#2)}}}%
+\newdef\mfp@ellipse#1#2{\mfp@figmac{ellipse (#2, #1)}}%
+% \end{macrocode}
+%
+% The first argument of \cs{mfp@pshcircle} is a boolean: \texttt{true}
+% in the disk case, \texttt{false} for the half-plane. The second argument
+% should be \marg{\meta{center},\meta{radius}}.
+% \begin{macrocode}
+\newdef\mfp@pshcircle#1#2{\mfp@figmac{pshcircle (#1, #2)}}%
+% \end{macrocode}
+%
+% The first argument of \cs{mfp@hypergeodesic} is a string that sets the
+% context: \texttt{UD} for the unit disk, \texttt{UHP} for the upper
+% half-plane. The second argument should be the two points to join.
+% \begin{macrocode}
+\newdef\mfp@hypergeodesic#1#2{\mfp@figmac{#1geodesic (#2)}}%
+% \end{macrocode}
+%
+% Six graphics macros call \cs{mfp@curve}: \cs{curve},
+% \cs{convexcurve}, \cs{mfbezier} and their closed variants . Its two arguments
+% are the name of a \grafbase{} command, and \mfc{true} or \mfc{false}.
+% This works for any \grafbase{} command that creates a smooth path from a
+% list of points, provided the command has two versions: one that begins
+% with \texttt{t} and takes a tension as the first argument and the other
+% without a \texttt{t} that takes the same arguments \emph{except} the
+% tension. The second argument of \cs{mfp@curve} determines whether the
+% curve is to be closed. These are passed along to \cs{mfp@@curve}, which
+% gets the tension as the third argument.
+% \begin{macrocode}
+\newdef\mfp@curve#1#2{\mfp@nullopt{\mfp@@curve{#1}{#2}}}%
+\newdef\mfp@@curve#1#2#3{%
+ \mfp@listmac{\mfp@ifempty{#3}{#1}{t#1 (#3)} (#2)}}%
+% \end{macrocode}
+%
+% The argument of \cs{mfp@fcncurve} is the optional argument of
+% \cs{fcncurve} (the tension).
+% \begin{macrocode}
+\newdef\mfp@fcncurve#1{%
+ \mfp@listmac{\mfp@ifempty{#1}{fcncurve}{functioncurve (#1)}}}%
+% \end{macrocode}
+%
+% \cs{mfp@fcn} is called by eight commands: \cs{levelcurve},
+% \cs{plrregion}, \cs{belowfcn}, \cs{function}, \cs{plrfcn},
+% \cs{parafcn}, \cs{DEgraph} and \cs{DEtrajectory}. They have a common
+% argument structure for both the \mfpic{} and \grafbase{} command.
+% \cs{mfp@fcn} calls \cs{mfp@dotension} to parse the optional tension
+% argument (the second argument) leaving the results in the macros
+% \cs{mfp@t} (a \texttt{t} for command name if a tension value was
+% present) and \cs{mfp@sval} (\texttt{true} or \texttt{false} and the
+% tension value if it was present). The first argument is the base name of
+% a \grafbase{} command.
+%
+% Ultimately, \cs{mfp@@fcn} is called, and only then the two mandatory
+% arguments of the calling command are read.
+% \begin{macrocode}
+\newdef\mfp@fcn#1#2{%
+ \mfp@defaultopt{\mfp@dotension{\mfp@@fcn{#1}}}{#2}}%
+\newdef\mfp@@fcn#1#2#3{%
+ \mfp@figmac{\mfp@t#1 (\mfp@sval) (#2) (#3)}}%
+% \end{macrocode}
+%
+% \cs{mfp@btwn} is called by \cs{btwnfcn} and \cs{btwnplrfcn}, with the
+% name of a \grafbase{} command as its argument. It does the same tension
+% processing as \cs{mfp@fcn} and calls \cs{mfp@@btwn}, which reads the
+% three mandatory arguments of the calling command.
+% \begin{macrocode}
+\newdef\mfp@btwn#1{%
+ \mfp@defaultopt{\mfp@dotension{\mfp@@btwn{#1}}}p}%
+\newdef\mfp@@btwn#1#2#3#4{%
+ \mfp@figmac{\mfp@t#1 (\mfp@sval) (#2) (#3) (#4)}}%
+% \end{macrocode}
+%
+% The tension parsing is done by \cs{mfp@@dotension}. It causes
+% \cs{mfp@t} to be empty or \texttt{t} depending on the existence of
+% the tension parameter, and causes \cs{mfp@sval} to contain either the
+% smoothness boolean alone or the boolean followed by the tension value,
+% separated by a comma. After this, \cs{mfp@dotension} calls the code it
+% is passed in its first argument.
+% \begin{macrocode}
+\newdef\mfp@dotension#1#2{\mfp@@dotension#2\mfp@end#1}%
+\newdef\mfp@@dotension#1#2\mfp@end{\def\mfp@t{}%
+ \if p#1\relax \let\mfp@sval\mfp@F
+ \else \let\mfp@sval\mfp@T
+ \mfp@ifexist{#2}{\def\mfp@t{t}\edef\mfp@sval{\mfp@sval, #2}}%
+ \fi}%
+% \end{macrocode}
+%
+% \cs{mfp@plotdata} is called by \cs{plotdata}. I follows the scheme of
+% most of the data file processing commands: it passes another command (in
+% this case \cs{mfp@doplots}) to \cs{mfp@processfile}, which reads the
+% filename as its second argument. The argument of \cs{mfp@plotdata},
+% which is passed to \cs{mfp@doplots}, is the original optional parameter
+% of \cs{plotdata}.
+% \begin{macrocode}
+\newdef\mfp@plotdata#1{\mfp@processfile{\mfp@doplots{#1}}}%
+% \end{macrocode}
+%
+% \cs{mfp@doplots} is a self-looping macro. It issues the rendering
+% prefix (in \cs{mfp@setstyle}), increments the linetype, and writes the
+% path expression defined by the first set of data in the file (via
+% \cs{mfp@@datafile}, which stops at a blank line). That part, when prefixed
+% by \cs{mfp@figmacbase}, functions as a \cs{mfp@figmac}. When finished with
+% one figure, \cs{mfp@ifenddata} checks if there is another set of data in
+% the file. If so, \cs{mfp@doplots} calls itself again.
+%
+% The argument is the smoothness parameter(s), which it merely passes on.
+% There is no filename argument: \cs{mfp@plotdata} has already opened
+% the input stream \cs{mfp@data} and subsequent macros will read from it.
+% \begin{macrocode}
+\newdef\mfp@doplots#1{\mfp@setstyle
+ \advance\mfp@linetype1 \mfp@figmacbase
+ \mfp@@datafile{#1}\mfp@ifenddata{}{\mfp@doplots{#1}}}%
+% \end{macrocode}
+%
+% When two blank lines occur in succession, \cs{plotdata} is suppose to
+% stop. \cs{mfp@ifenddata} is called when \cs{mfp@@datafile} ends (one blank
+% line was found), it checks for another blank line (or end-of-file)
+% and, if it finds one, selects the first of the two alternatives that
+% follow.
+% \begin{macrocode}
+\newdef\mfp@ifenddata{%
+ \ifeof\mfp@data \@xp\@firstoftwo
+ \else \mfp@skipcomments
+ \ifx\mfp@temp\mfp@par \@XP\@firstoftwo
+ \else \@XP\@secondoftwo
+ \fi
+ \fi}%
+% \end{macrocode}
+%
+% The \cs{datafile} command, which calls \cs{mfp@datafile} is really
+% obsolete, as one can now use \cs{curve} or \cs{polyline} (or any other
+% list-to-path command) followed by the \cs{datafile} construct.
+% However, it requires only two extra commands and then calls a link in
+% the \cs{plotdata} chain of commands, so we keep it.
+%
+% \cs{mfp@datafile} is called by \cs{datafile}, which passes the optional
+% smoothness argument. It does the figure macro stuff, then calls
+% \cs{mfp@processfile} to open the data file and run \cs{mfp@@datafile}.
+% \begin{macrocode}
+\newdef\mfp@datafile#1{%
+ \mfp@figmacbase\mfp@processfile{\mfp@@datafile{#1}}}%
+% \end{macrocode}
+%
+% \cs{mfp@@datafile} checks for an empty optional parameter and then calls
+% \cs{mfp@dodatafile}. We could handle the options earlier in the chain,
+% but we would have to do that for both \cs{plotdata} and \cs{datafile}
+% since both call on it. It is somewhat more efficient to reproduce a bit
+% of the internals of \cs{mfp@defaultopt} just once.
+% \begin{macrocode}
+\newdef\mfp@@datafile#1{%
+ \mfp@ifempty{#1}%
+ {\mfp@dodatafile\mfp@smoothness\mfp@tension}%
+ {\mfp@dodatafile#1}\mfp@end}%
+% \end{macrocode}
+%
+% And \cs{mfp@dodatafile} is mainly just a wrapper, processing the optional
+% parameter (passed by either \cs{plotdata} or \cs{datafile}) which
+% selects what figure making code should be written. For
+% \oarg{s\meta{tension}} a \gbc{curve} command is used. For \oarg{p}, a
+% \gbc{polyline}. (The `\cs{mfp@F}' means the curve is not closed).
+%
+% After that, \cs{mfp@rwdata} is called, which initiates the reading and
+% writing of the the data.
+% \begin{macrocode}
+\newdef\mfp@dodatafile#1#2\mfp@end{%
+ \mfsrc{%
+ \if s#1 \mfp@ifempty{#2}{curve }{tcurve (#2) }%
+ \else polyline
+ \fi (\mfp@F)}\mfp@rwdata}%
+% \end{macrocode}
+%
+% \cs{mfp@image} is called by \cs{mfpimage} with the optional argument in
+% \arg1 and the mandatory argument in \arg2.
+% \begin{macrocode}
+\newdef\mfp@image#1#2{\mfsrc{\@nl makeimage (#2, #1)}}%
+% \end{macrocode}
+%
+% The command \cs{mfp@getdimens} is only called by \cs{trimpath}. Its
+% first argument is the comma separated pair of dimensions from that
+% command. It defines \cs{mfp@thedimens} to expand to exactly that,
+% except: if there is only one dimension and no comma (empty \arg2 and
+% \arg3), the result is a pair of dimensions both equal to it; if there is
+% a comma, but nothing on one side, it returns \texttt{0pt} for the empty
+% side. If a \TeX{} dimension command is used in either position, its
+% value is returned.
+%
+% The only time \arg3 is nonempty is when the original argument of
+% \cs{trimpath} contains a comma (then \arg3 is a comma). In that case we
+% let \cs{mfp@scratch} keep its previous value.
+% \begin{macrocode}
+\newdef\mfp@getdimens#1,#2,#3\mfp@end{%
+ \mfp@scratch \@ifmtarg{#1}{\z@}{#1}\relax
+ \edef\mfp@thedimens{\the\mfp@scratch}%
+ \@ifmtarg{#3}{}{\mfp@scratch \@ifmtarg{#2}{\z@}{#2}\relax}%
+ \edef\mfp@thedimens{\mfp@thedimens,\the\mfp@scratch}}%
+% \end{macrocode}
+%
+% The argument of \cs{mfp@doubledraw} is the optional separation parameter
+% passed by \cs{doubledraw}. Then the optional color is processed.
+% \begin{macrocode}
+\newdef\mfp@doubledraw#1{%
+ \mfp@nullopt{\mfp@clroptrendi{doubledraw}{#1}}}%
+% \end{macrocode}
+%
+% Parameter \arg1 of \cs{mfp@wiggle} is the name, \gbc{sinewave} or
+% \gbc{corkscrew}, \arg2 is the optional argument (tension). The four
+% dimension values should follow as the mandatory argument of the calling
+% command.
+% \begin{macrocode}
+\newdef\mfp@wiggle#1#2{\mfp@rendi{#1 (#2)}}%
+% \end{macrocode}
+%
+% The utility macros, \cs{mfp@rendi}, \cs{mfp@modi} and \cs{mfp@listi}
+% write a command name and one set of parameters. They differ in the setup
+% required. The first writes via \cs{mfp@rendmac}, the second via
+% \cs{mfp@modmac} and the last via \cs{mfp@listmac}. \cs{mfp@rendii}
+% writes two sets of parameters, reversing them. \cs{mfp@optrendi} runs
+% \cs{mfp@rendi}, but feeds it the result of processing a optional
+% argument. \cs{mfp@optrendii} is similar, running \cs{mfp@rendii}.
+% \begin{macrocode}
+\newdef\mfp@rendi#1#2{\mfp@rendmac{#1 (#2)}}%
+\newdef\mfp@rendii#1#2#3{\mfp@rendmac{#1 (#3, #2)}}%
+\newdef\mfp@optrendi#1{\mfp@defaultopt{\mfp@rendi{#1}}}%
+\newdef\mfp@optrendii#1{\mfp@defaultopt{\mfp@rendii{#1}}}%
+\newdef\mfp@modi#1#2{\mfp@modmac{#1 (#2)}}%
+\newdef\mfp@listi#1#2{\mfp@listmac{#1 (#2)}}%
+% \end{macrocode}
+%
+% A great many \grafbase{} rendering commands come in two parts: one
+% command takes a color argument and the other command simply calls the
+% color version with a default color. In all these commands, the name of
+% the first is just the second with `\texttt{color}' prepended. This means
+% that the corresponding \mfpic{} rendering commands can share a common
+% macro, \cs{mfp@clropt}, which is passed the optional color and the name of
+% the second command.
+%
+% To simplify the output file, if a command's color option is empty
+% (signaling the default color) the `\texttt{color}'-free \grafbase{}
+% command name is written, otherwise the command name beginning with
+% `\texttt{color}' is written along with the color parameter.
+% \cs{mfp@clropt} makes the test and writes the command. \arg1 is the
+% color argument and \arg2 is the command's base name. The commands
+% \cs{mfp@clroptrend} and \cs{mfp@clroptrendi} provide useful
+% abbreviations when the command is a rendering command and the
+% \grafbase{} command takes no argument or one argument beside the color.
+%
+% There are a number of commands that are not subject to prefixes. For
+% this we have \cs{mfp@simple}. It reads a command name and its
+% arguments and writes the command, followed by the arguments wrapped in
+% parentheses, followed by a semicolon. We also have \cs{mfp@simplei}
+% which reads an additional argument.
+% \begin{macrocode}
+\newdef\mfp@clropt#1#2{\mfp@ifempty{#1}{#2}{\mfp@C#2 (#1)}}%
+\newdef\mfp@clroptrend#1#2{\mfp@rendmac{\mfp@clropt{#2}{#1}}}%
+\newdef\mfp@clroptrendi#1#2#3{\mfp@rendmac{\mfp@clropt{#3}{#1} (#2)}}%
+\newdef\mfp@simple#1#2{\mfcmd{\@nl #1 (#2)}}%
+\newdef\mfp@simplei#1#2#3{\mfcmd{\@nl #1 (#2, #3)}}%
+% \end{macrocode}
+%
+% All the path closing commands are implemented with \cs{mfp@close}. It
+% writes the closing command determined by the first argument, with a
+% `\texttt{t}' plus the second argument as a parameter, if that is not
+% empty.
+% \begin{macrocode}
+\newdef\mfp@close#1#2{\mfp@modmac{#1closed\mfp@ifexist{#2}{t (#2)}}}%
+% \end{macrocode}
+%
+% These next implement hatching. Except for \cs{mfp@xhatch}, they all call
+% \cs{mfp@thatch}. The argument of these is the optional argument of the
+% calling command. The second optional argument of that command (the
+% color) is processed by \cs{mfp@clroptrendi}.
+% \begin{macrocode}
+\newdef\mfp@thatch#1{\mfp@nullopt{\mfp@clroptrendi{thatch}{#1}}}%
+\newdef\mfp@lhatch#1{\mfp@thatch{#1,-45}}%
+\newdef\mfp@rhatch#1{\mfp@thatch{#1,45}}%
+\newdef\mfp@xhatch#1{\mfp@nullopt{\mfp@clroptrendi{xhatch}{#1}}}%
+% \end{macrocode}
+%
+% The all-purpose \cs{mfp@arr} sets the default values for the
+% \cs{mfp@h...} commands (two of which are passed as parameters). This
+% is accomplished by the \cs{mfp@arr@\meta{x}} commands. The
+% next two commands constitute a loop that ends when no more options
+% follow. Each turn through the loop the initial letter is checked and the
+% value that follows it is stored in one of those macro (possibly
+% replacing some default). Ultimately, \cs{mfp@doarrow} writes the
+% code to the output file.
+% \begin{macrocode}
+\newdef\mfp@arrowhead#1#2{\mfp@arr 0{Gheadpath (#1) (#2)}}%
+\newdef\mfp@arrow#1{\mfp@arr 0{headpath#1}}%
+\newdef\mfp@arr#1#2{\mfp@arr@c{}\mfp@arr@l{\the\headlen}%
+ \mfp@arr@b{#1}\edef\mfp@hcommand{#2}\mfp@arr@r{0}\mfp@@arr}%
+\newdef\mfp@@arr{\@ifnextchar[{\mfp@arropt}{\mfp@doarrow}}%
+\newdef\mfp@arropt[#1]{%
+ \mfp@ifexist{#1}{\mfp@@arropt#1\mfp@end}\mfp@@arr}%
+\newdef\mfp@@arropt#1#2\mfp@end{%
+ \mfp@ifempty{#2}%
+ {\Mfpic@warn{Arrow option #1 has no value\on@line; %
+ it will be ignored.}}%
+ {\@ifundefined{mfp@arr@#1}%
+ {\Mfpic@warn{Arrow option #1 invalid \on@line; %
+ it will be ignored}}%
+ {\@nameuse{mfp@arr@#1}{#2}}%
+ }}%
+\newdef\mfp@doarrow{%
+ \mfp@addmac{\mfp@clropt{\mfp@hcolour}{\mfp@hcommand} (\mfp@hlength, %
+ \mfp@hrotate, \mfp@hlocate)}}%
+% \end{macrocode}
+%
+% Implement the optional argument of \cs{turn} with \cs{mfp@turn}.
+% \begin{macrocode}
+\newdef\mfp@turn#1#2{\rotatearound{#1}{#2}}%
+% \end{macrocode}
+%
+% The first argument of \cs{mfp@axis} is the optional argument of
+% \cs{axis}. Unfortunately, the $x$ and $y$ axis have one default head
+% length, \cs{axisheadlen} (initially \texttt{5pt}), while the side
+% axes have a different default, \cs{sideheadlen} (initially \texttt{0pt}).
+% Therefore we don't know which default to use until the mandatory argument
+% is read. In retrospect, it would have been better to put the optional
+% argument after the mandatory one. So if there is no optional argument,
+% the first argument of \cs{mfp@axis} is empty rather than a default.
+%
+% We use \cs{mfp@getaxis} to check the mandatory argument and return the
+% default head length in \cs{mfp@defaultheadlen}. It also stores the
+% letter in \cs{mfp@axisletter}, but returns `\texttt{x}' for an invalid
+% one.
+%
+% The letter argument \arg{2} is permitted to be a word, provided it
+% begins with one of the allowed letters. The `\marg{(none)}' is there to
+% provide an argument (and an informative error message) in the unlikely
+% case that this argument is empty.
+%
+% The \cs{mfp@addmac} command is the internal code that \cs{arrow[l \#1]}
+% would produce. This is slightly more efficient than calling \cs{arrow}.
+% \begin{macrocode}
+\newdef\mfp@axis#1#2{\mfp@getaxis#2{(none)}\mfp@end\axis
+ \mfp@addmac{headpath (\mfp@emptysub{#1}{\mfp@defaultheadlen}, 0, 0)}%
+ \axisline{\mfp@axisletter}}%
+\newdef\mfp@doaxes#1#2{%
+ \def\mfp@@axis##1{\mfp@axis{#1}{##1}\mfp@@doaxis}%
+ \mfp@@doaxis#2\mfp@end}%
+\newdef\mfp@@doaxis{\mfp@ifend{}\mfp@@axis}%
+% \end{macrocode}
+%
+% The six commands for putting hashmarks on an axis call \cs{mfp@marks}
+% with the axis letter and the optional length of the marks. Note that
+% the location of the marks is a list of numbers, so it is picked up by
+% \cs{mfp@writedata} just as in \cs{mfp@listmac}. But it is not a figure
+% macro, so we cannot use \cs{mfp@listmac} itself.
+% \begin{macrocode}
+\newdef\mfp@marks#1#2{\mfsrc{\@nl #1marks (#2)}\mfp@writedata}%
+% \end{macrocode}
+%
+% Called by \cs{vectorfield} or \cs{plrvectorfield} after processing the
+% optional argument, the following simply writes the appropriate
+% \grafbase{} code. The first argument is either empty or the string
+% `\texttt{plr}'.
+% \begin{macrocode}
+\newdef\mfp@vectorfield#1#2#3#4#5{%
+ \mfcmd{#1vectorfield (#2, #3) (#4) (#5)}}%
+% \end{macrocode}
+% The first argument of both these is the optional size argument of the
+% calling commands (\cs{point} and \cs{plotsymbol}). For
+% \cs{mfp@plotsymbol} the second argument is the symbol name.
+% \begin{macrocode}
+\newdef\mfp@point#1{%
+ \mfsrc{\@nl pointd (#1, \ifpointfill\mfp@T\else\mfp@F\fi)}%
+ \mfp@writedata}%
+\newdef\mfp@plotsymbol#1#2{%
+ \mfsrc{\@nl plotsymbol (#2, #1)}\mfp@writedata}%
+% \end{macrocode}
+%
+% \cs{mfp@plottext} now simply calls the \cs{tlabel} code, since that can
+% now handle a list of points.
+% \begin{macrocode}
+\newdef\mfp@plottext#1#2#3{\tlabeljustify{#1}%
+ \def\tl@points{#3}\mfp@toks{#2}\mfp@thelabel\endgroup}%
+% \end{macrocode}
+%
+% The first argument of \cs{mfp@piechart} is the optional argument of
+% \cs{piechart}. It comes in two parts: a letter and an optional angle,
+% so \cs{mfp@@piechart} is used to parse it. A similar description
+% applies to \cs{piewedge}.
+% \begin{macrocode}
+\newdef\mfp@piechart#1{\mfp@@piechart#1\mfp@end}%
+\newdef\mfp@@piechart#1#2\mfp@end#3{% #3 = center,radius
+ \mfsrc{\@nl piechart(\if a#1 \else-\fi1, %
+ \mfp@emptysub{#2}{90}, #3)}\mfp@writedata}%
+\newdef\mfp@piewedge#1{\mfp@@piewedge#1\mfp@end}%
+\newdef\mfp@@piewedge#1#2\mfp@end#3{% #3=number
+ \mfp@figmac{(piewedge[#3]%
+ \mfp@ifexist{#2}%
+ {\if #1x shifted (#2*piedirection[#3])%
+ \else\if #1s shifted (#2)%
+ \else\if #1m shifted (#2 - piecenter)%
+ \fi\fi\fi})}}%
+% \end{macrocode}
+%
+% The optional argument of \cs{barchart} becomes \arg{1} and the data is
+% \arg{2}.
+% \begin{macrocode}
+\newdef\mfp@barchart#1#2{%
+ \mfsrc{\@nl barchart (#1, \if h#2\mfp@F\else\mfp@T\fi)}%
+ \mfp@writedata}%
+% \end{macrocode}
+%
+%
+%
+% \section{The \env{mfpic} environment}\label{mfpicenv}
+%
+%
+% \subsection{Initializing the environment}\label{mfpicinitializations}
+%
+% For mostly historical reasons, the macros that create labels are
+% redefined anew with each \env{mfpic} environment. These have gradually
+% grown in number and size. We are still doing this, but the internal
+% part of these commands have now been moved outside the environment,
+% and only the user lever commands are localized.
+%
+% \DescribeMacro{\everymfpic}
+% This takes a token list and saves it in \cs{every@mfpic} to be issued
+% at the very end of the \cs{mfpic} command. There is a companion macro
+% \DescribeMacro{\everyendmfpic}
+% that is issued at the very beginning of \cs{endmfpic}. These are always
+% global, otherwise they are defined much like \cs{everytlabel}.
+% \begin{macrocode}
+\newdef\everymfpic{\afterassignment\@everymfpic\mfp@toks}%
+\newdef\everyendmfpic{\afterassignment\@everyendmfpic\mfp@toks}%
+\newdef\@everymfpic{\edef\every@mfpic{\the\mfp@toks}}%
+\newdef\@everyendmfpic{\edef\every@endmfpic{\the\mfp@toks}}%
+\everymfpic{}\everyendmfpic{}%
+% \end{macrocode}
+%
+% \DescribeMacro{\mfpic}
+% This command (or the \LaTeX{} environment \cs{begin}\marg{mfpic}) takes
+% two optional arguments: the xscale and the yscale. When both are
+% omitted, they are both set to 1, if the yscale is omitted, it is set
+% equal to the xscale.
+%
+% There are also 4 mandatory arguments: the first two give the minimum and
+% maximum extent of the $x$ coordinates and the last two do the same for the
+% $y$ coordinates. The coordinates used in the figure commands within an
+% mfpic environment are multiplied by the corresponding scales, and that
+% is the multiple of \cs{mfpicunit} used in positioning, drawing, etc.
+%
+% The optional arguments are converted by the option handling code to
+% mandatory arguments for \cs{@mfpic}, which contains the actual
+% definition.
+% \begin{macrocode}
+\newdef\mfpic{\mfp@defaultopt\@mfpicA{1}}%
+\newdef\@mfpicA#1{\mfp@defaultopt{\@mfpic{#1}}{#1}}%
+\newdef\@mfpic#1#2#3#4#5#6{%
+% \end{macrocode}
+%
+% We start by saving \mfpic's extra fontdimen values. They will be
+% restored by \cs{endmfpic}. This makes the values local (font dimensions
+% are normally always global). Also, we save \cs{mfpicunit} using an
+% internal name to prevent users from changing it midpicture.
+%
+% Then we begin a group to localize ordinary
+% \TeX{} definitions and assignments, disable writes to the output if
+% \cs{opengraphsfile} hasn't occurred yet (an error), save the current
+% font in \cs{@tcurr} for restoration inside text labels and captions, set
+% the switch that says we are in an \cs{mfpic} environment, disable
+% \cs{par} so that blank lines have (little or) no effect, and switch to
+% \cs{nullfont} so that stray characters (always an error) and spaces (not
+% an error) have no effect. I don't know whether \cs{relax} is a better
+% disabling definition for \cs{par} than \cs{@empty}.
+% \begin{macrocode}
+ \save@mfpicdimens
+ \begingroup % ended near end of \endmfpic
+ \@mfpicunit\mfpicunit
+ \mfp@safe@activestrue
+ \mfp@ifopengraphsfile{}%
+ {\ifmfp@nowrite\else \mfp@nooutput@error\mfpic
+ \def\mfp@src##1##2{\endgroup}%
+ \fi}%
+ \mfp@DBlog{Starting mfpic \number\mfp@count \on@line.}%
+ \@xp\let\@xp\@tcurr\the\font
+ \let\ifin@mfpicenv\iftrue
+ \let\par\@empty
+ \nullfont
+% \end{macrocode}
+%
+% The graphics macros are undefined outside the \env{mfpic} environment.
+% They are defined now by invoking \cs{mfp@grafmacs}.
+%
+% One can add to \cs{mfp@additions} to extend \mfpic. Currently it
+% is empty.
+% \begin{macrocode}
+ \mfp@grafmacs
+ \mfp@additions
+% \end{macrocode}
+%
+% We also define the coordinate conversion macros \cs{@xconv} and
+% \cs{@yconv} via \cs{mfp@setconv}. These are used to calculate the
+% nominal dimensions of the picture and to position \TeX{} labels.
+%
+% We make all the arguments of \cs{@mfpic} available to the user. Macro
+% \cs{mfp@setconv} now also makes use of them.
+% \begin{itemize}
+% \item the x-scaling factor.
+% \DescribeMacro{\xfactor}
+% \item the y-scaling factor.
+% \DescribeMacro{\yfactor}
+% \item the leftmost x-coordinate.
+% \DescribeMacro{\xmin}
+% \item the rightmost x-coordinate.
+% \DescribeMacro{\xmax}
+% \item the leftmost y-coordinate.
+% \DescribeMacro{\ymin}
+% \item the rightmost y-coordinate.
+% \DescribeMacro{\ymax}
+% \end{itemize}
+%
+% We are going to add all text labels to a box \cs{@alltlabels} and put
+% that whole box on top of the graph during \cs{endmfpic}.
+% We initialize the box here, as well as the dimensions \cs{@tlabelsbot},
+% etc.
+% \begin{macrocode}
+ \edef\xfactor{#1}\edef\yfactor{#2}%
+ \edef\xmin{#3}\edef\xmax{#4}\edef\ymin{#5}\edef\ymax{#6}%
+ \mfp@setconv
+ \mfp@inittlabels
+% \end{macrocode}
+%
+% The graph is loaded into a box register at the end of the \env{mfpic}
+% environment. The size of that box register may have to correspond to the
+% arguments of the \cs{mfpic} command. Since \cs{endmfpic} doesn't receive
+% these arguments we process them now.
+%
+% Dimensions \cs{@graphright}, etc., represent the extent of the graph
+% box. We increase them when the text is added if it extends beyond the
+% edges of the graph. These are all font dimensions, so they are global
+% assignments.
+% \begin{macrocode}
+ \@xconv\xmax\@graphright
+ \@yconv\ymax\@graphtop
+ \@graphleft 0pt
+ \@graphbot 0pt
+% \end{macrocode}
+%
+%
+% \subsection{Initializing the \MF{} picture}
+% \label{current}
+%
+% Each \env{mfpic} environment produces a \gbc{beginmfpic...endmfpic}
+% environment in the output file. Here we initialize the variables that
+% correspond there to the 6 arguments and \cs{mfpicunit}, then begin the
+% \MF{} environment. \gbc{beginmfpic} takes the number of the current
+% figure as its only argument. For debugging purposes we write a \MF{}
+% comment the contains the line number in the \TeX{} source file where the
+% \cs{mfpic} command occurred. We also write a line of dashes
+% (\cs{mfp@d}) to make it easier to find the start of the environment.
+%
+% When \opt{mplabels} is in effect we need to set the font for the labels
+% to the saved \cs{@tcurr} by writing some \mfc{verbatimtex} code. Since
+% \opt{mplabels} can be turned on at any time, we write this whenever the
+% \opt{metapost} option is in effect. It is a common misconception that
+% \mfc{verbatimtex} can be used only once at the start of a \MP{} file. In
+% fact it can be used any number of times. It's purpose is to be at an
+% outer level, while \mfc{btex} is localized inside a box
+% \begin{macrocode}
+ \mfsrc{\@nl \mfp@p\mfp@d\mfp@d<\number\mfp@count>\@nl
+ unitlen := \the\@mfpicunit\mfp@s;\@nl
+ xscale := #1; yscale := #2;\@nl
+ bounds (#3, #4, #5, #6);\@nl
+ \@nl
+ beginmfpic(\number\mfp@count); \mfp@p\space\mfp@lineno.
+ \mfp@ifmpost
+ {\@nl verbatimtex \relax
+ \MFPbegingroup\font\noexpand\MFPcfont=\fontname\@tcurr\relax
+ etex;}{}}%
+% \end{macrocode}
+%
+%
+% \subsection{Placing text labels}\label{textlabels}
+%
+% All the rest of \cs{@mfpic} defines commands that in some way or another
+% produce labels. The easiest part is defining \cs{tlabel} under
+% \opt{mplabels} where most of the work takes place in \file{grafbase.mp}.
+% The most involved is the definition of \cs{tlabel} without
+% \opt{mplabels}. All the rest (except \cs{tcaption}) add labels by
+% calling \cs{tlabel} or one of its internal commands. Some of those could
+% be simpler under \opt{mplabels} but then we'd need two versions of each.
+%
+% \DescribeMacro{\tlabel}
+% This is the common code, processing optional arguments, etc. for
+% \cs{tlabel}. It, and the first several commands it calls, set switches
+% and macros based on the arguments. The first argument is optional: the
+% justification (the default is that set with the \cs{tlabeljustify}
+% command), the second is a location or locations, and the last is the
+% \TeX{} code that constitutes the label. All these are ultimately stored
+% as macros and then either \cs{mp@tlabel} or \cs{tex@tlabel} is called
+% without arguments.
+%
+% The location can be in the form \parg{\meta{x},\meta{y}}, but if \MP{}
+% is processing labels we would like to be able to use \MP{} expressions,
+% which can contain parentheses and commas. This would mess up \TeX's
+% token matching for reading the two coordinates. Therefore we allow the
+% whole point to be enclosed in braces. Under \opt{mplabels} the contents
+% of the braces will be written to the output file without any further
+% parsing. Without \opt{mplabels}, the braces should contain a comma
+% separated list of locations in the previously described syntax.
+%
+% We start things off by calling \cs{mfp@label} to process the optional
+% argument, feeding it an empty one if none is present.
+% \begin{macrocode}
+ \newdef\tlabel{\mfp@nullopt\mfp@label}%
+% \end{macrocode}
+%
+% We allow some \cs{tlabels} to be placed behind the picture rather than
+% on top. This is easy under \opt{mplabels}: just use \cs{tlabel} before
+% any rendering commands (making sure \opt{overlaylabels} is turned off.)
+% However, \MP{} labels can contain only fonts and one would often want
+% to do this with an included graphic (as a background). After
+% \DescribeMacro{\startbacktext}
+% \cs{startbacktext}, labels are saved in a special place to be included
+% behind the \mfpic{} graphic. This continues until
+% \DescribeMacro{\stopbacktext}
+% \cs{stopbacktext}.
+%
+% We simply turn off \gbc{overlaylabels} and turn on \gbc{backtextlabels}
+% for \MP{}. We also save the box \cs{@alltlabels} in \cs{@wholegraph}
+% (which will not be needed until \cs{endmfpic}) and then reinitialize it.
+% At \cs{endbacktext} we copy the box \cs{@alltlables} into \cs{@backtext},
+% and then restore what we changed. We let the values of \cs{@tlabelsleft},
+% etc., be updated normally as labels are added to the backtext.
+% \begin{macrocode}
+ \newdef\startbacktext{\let\mfp@restoreOL\@empty
+ \mfp@ifmpost
+ {\ifmfp@overlabels
+ \let\mfp@restoreOL\overlaylabels\nooverlaylabels
+ \fi \setmfboolean{underlaylabels}{\mfp@T}}{}%
+ \setbox\@wholegraph \box\@alltlabels
+ \setbox\@alltlabels \vbox{\hbox{}}}%
+ \newdef\stopbacktext{%
+ \setbox\@backtext \box\@alltlabels
+ \setbox\@alltlabels\box\@wholegraph
+ \mfp@ifmpost{\setmfboolean{underlaylabels}{\mfp@F}}{}%
+ \mfp@restoreOL}%
+% \end{macrocode}
+%
+%
+% \subsection{Surrounding the label with a path}\label{surrounding}
+%
+% These next macros place labels, but also define a path surrounding the
+% label. This allows one to clear a space for the label, create a
+% background, draw a box around it, etc., all in one command (with the
+% appropriate sequence of prefixes). They take two optional arguments.
+% The first optional argument depends on the type of path being created.
+% It is described below. The second one is the justification as in the
+% \cs{tlabel} command. There are two mandatory arguments, the location
+% (same syntax as for \cs{tlabel} except it cannot be a list) and the text.
+% They also have star-forms, which create the path but don't place the
+% label.
+%
+% All the curves are sized to just contain what I will call the
+% \emph{label strip}. Imagine a rectangle that just fits around the text.
+% Now extend it on all sides by the value defined with \cs{tlpathsep}.
+% This rectangle is the label strip.
+%
+% \DescribeMacro{\tlabelrect}
+% \cs{tlabelrect} creates the rectangular boundaries of the label strip by
+% default. With the optional argument a rectangle with rounded corners is
+% produced; the optional argument is the radius of the quarter circles used.
+% These quarter circles pass through the corners of the label strip and are
+% joined together with straight lines to create the rest of the rectangle.
+% As a special case, the optional argument can contain the keyword
+% `\texttt{roundends}' to produce a rectangle with the short sides
+% replaced by a semicircle. If the radius is negative, the corners are
+% concave.
+%
+% \DescribeMacro{\tlabeloval}
+% \cs{tlabeloval} produces an ellipse which, by default has the same
+% aspect ratio as the label strip. It's size just includes the label strip,
+% and it passes through the four corners. The optional argument contains a
+% numerical factor $m$ that changes the width and the height so that the
+% width-to-height ratio is multiplied by $m$, but the ellipse still
+% passes through those four corners. The default value of $m$ is 1.
+% Negative $m$ has the same effect as $|m|$. If $m=0$ a rectangle is
+% produced.
+%
+% \DescribeMacro{\tlabelellipse}
+% \cs{tlabelellipse} produces an ellipse, where the optional argument $a$
+% is the ratio of its width to its height. The default for $a$ is 1, which
+% produces a circle. Negative $a$ has the same effect as $|a|$. If $a =
+% 0$, a rectangle is produced.
+%
+% \DescribeMacro{\tlabelcircle}
+% To get a circle one can use \cs{tlabelellipse} without the optional
+% argument, or with argument \oarg{1}. However, it makes for a more
+% readable source if the command reflects the shape. \cs{tlabelcircle} is
+% defined to be essentially \cs{tlabelellipse}\oarg{1}, but it first
+% checks for the \texttt{*} and passes it along in front of the \oarg{1}.
+%
+% We implement the other three by calling \cs{mfp@labelpath} with the name
+% of the curve (\texttt{rect}, \texttt{oval}, or \texttt{ellipse}) and the
+% default value of the optional argument (\texttt{0}, \texttt{1}, and
+% \texttt{1}, respectively) as arguments.
+% \begin{macrocode}
+ \newdef\tlabelrect{\mfp@labelpath{rect}{0}}%
+ \newdef\tlabeloval{\mfp@labelpath{oval}{1}}%
+ \newdef\tlabelellipse{\mfp@labelpath{ellipse}{1}}%
+ \newdef\tlabelcircle{%
+ \mfp@iftoken*{\tlabelellipse*[1]}{\tlabelellipse[1]}}%
+% \end{macrocode}
+%
+%
+% \subsection{Placing multiple text labels}\label{multiple}
+%
+% One of the more onerous things about labels in \env{mfpic} environments
+% used to be the shear amount of typing required. The following commands
+% are intended to reduce the work by having to repeat only the arguments.
+%
+% \DescribeMacro{\tlabels}
+% This command takes only one argument containing any sequence of valid
+% arguments for the \cs{tlabel} command, optionally separated by
+% whitespace (but not blank lines).
+%
+% We begin a group, reset whitespace to normal, and call the command that
+% reads the argument.
+% \begin{macrocode}
+ \newdef\tlabels{\begingroup
+ \mfp@resetwhitespace\mp@mksharpother \mfp@labels}%
+% \end{macrocode}
+%
+% Labelling axes generates a lot of \cs{tlabel} commands, or a single
+% \cs{tlabels} command with a lot of arguments. Given that such labels
+% have much in common (e.g., for the $x$ axis the $y$ coordinate and the
+% justification are usually the same for all the labels) we can do still
+% better at abbreviating this task.
+%
+% \DescribeMacro{\axislabels}
+% The \cs{axislabels} command takes a mandatory argument (the axis)
+% followed by an optional argument (the justification, as in the
+% \cs{tlabel} command), followed by another mandatory argument, a comma
+% separated list of items of the form \marg{\meta{label}}\meta{coord}. The
+% label is placed at that coordinate on the given axis.
+%
+% Here we call \cs{mfp@axislabel}, feeding it the arguments of the current
+% \cs{@mfpic} command.
+% \begin{macrocode}
+ \newlet\axislabels\mfp@axislabels
+% \end{macrocode}
+%
+%
+% \subsection{Captions}\label{captions}
+%
+% Captions are not added immediately by the \cs{tcaption} command. That
+% command redefines \cs{mfp@docaption} to add it, and \cs{endmfpic}
+% executes \cs{mfp@docaption}. For now, \cs{mfp@docaption} does nothing.
+%
+% \DescribeMacro{\tcaption}
+% \cs{tcaption} takes an optional argument that holds two numbers,
+% separated by a comma. The first number determines the maximum width of a
+% one line caption. Captions larger than that multiple of the width of the
+% graphic will be reset as a paragraph, whose lines have length the second
+% number times the width of the graph. The defaults are \oarg{1.2,1}.
+% \begin{macrocode}
+ \let\mfp@docaption\@empty
+ \newdef\tcaption{\mfp@defaultopt\mfp@caption{1.2,1.0}}%
+% \end{macrocode}
+%
+% Finally, invoke the \cs{everymfpic} tokens.
+% \begin{macrocode}
+ \every@mfpic}% End of \@mfpic
+% \end{macrocode}
+%
+%
+%
+% \section{Implementing the label and caption macros}
+%
+% \cs{mfp@setconv} uses the stored arguments of the surrounding
+% \env{mfpic} environment: \cs{xfactor} and \cs{yfactor}
+% are the scale factors; \cs{xmin} and \cs{ymin} are the coordinates of
+% the lower left corner. It defines two macros used to convert graph
+% coordinates into actual dimensions relative to the lower left corner of
+% the picture. The second argument of \cs{@xconv} and \cs{@yconv} is a
+% dimension register in which the converted result is returned. We make
+% the final assignment global for consistency (font dimension assignments
+% are always global). We don't really need to define \cs{mfp@setconv}
+% and then call it. We could just define \cs{@xconv} and \cs{@yconv}.
+% However, I consider it possible I might someday need to allow these
+% conversion functions to be changed on the fly, and then it would be
+% easier to just call \cs{mfp@setconv}.
+% \begin{macrocode}
+\newdef\mfp@setconv{%
+ \def\@xconv##1##2{% ##1 = coord. ##2 = dimension register or fdim
+ {\mfp@scratch=##1\@mfpicunit
+ \advance\mfp@scratch by -\xmin\@mfpicunit
+ \mfp@scratch=\xfactor\mfp@scratch
+ \global ##2=\mfp@scratch}}%
+ \def\@yconv##1##2{% ditto
+ {\mfp@scratch=##1\@mfpicunit
+ \advance\mfp@scratch by -\ymin\@mfpicunit
+ \mfp@scratch=\yfactor\mfp@scratch
+ \global ##2=\mfp@scratch}}}%
+\newdef\mfp@inittlabels{%
+ \setbox\@alltlabels\vbox{\hbox{}}%
+ \setbox\@backtext\vbox{\hbox{}}%
+ \@tlabelsbot \maxdimen
+ \@tlabelstop -\maxdimen
+ \@tlabelsleft \maxdimen}%
+% \end{macrocode}
+%
+% The first argument of \cs{mfp@label} is the justification option. After
+% starting a group (it is ended in \cs{mfp@label@@}, we restore the
+% meaning of whitespace and give \texttt\# a safe catcode for writing to the
+% output, if necessary (\opt{mplabels}). Then we run \cs{tlabeljustify} on
+% the optional argument. An empty argument means any previously set values
+% won't be changed. Then we check for a left parenthesis and assume the
+% \parg{\meta{x},\meta{y}} syntax in that case, calling \cs{mfp@label@}
+% to process it. All it does is pack it in braces and call
+% \cs{mfp@label@@}. If no parentheses we skip to \cs{mfp@label@@}.
+%
+% Then \cs{mfp@label@@} gets a point or list of points as its first argument
+% and the label code as the second. It defines the macro \cs{tl@points} to
+% equal the point (or list) and puts the label into a token register.
+%
+% After all the data are stored away, we call \cs{mfp@thelabel} with no
+% parameters. Then the group (begun by \cs{mfp@label}) is ended, and the
+% \cs{extra@endtlabel} command invoked.
+% \begin{macrocode}
+\newdef\mfp@label#1{\begingroup\mfp@resetwhitespace\mp@mksharpother
+ \tlabeljustify{#1}\@ifnextchar({\mfp@label@}{\mfp@label@@}}%
+\newdef\mfp@label@(#1,#2){\mfp@label@@{(#1,#2)}}%
+\newdef\mfp@label@@#1#2{\def\tl@points{#1}\mfp@toks{#2}\mfp@thelabel
+ \endgroup\extra@endtlabel}%
+% \end{macrocode}
+%
+% At this point all the necessary information has been stored and
+% \cs{mfp@thelabel} merely branches depending on \opt{mplabels}.
+%
+% Clearly we could have put this code at the end of \cs{mfp@label@@} instead
+% of invoking this command. The point of doing things this way is so that
+% other commands too can store away the necessary info while doing other
+% things and simply call \cs{mfp@thelabel} to get the label placed.
+% \begin{macrocode}
+\newdef\mfp@thelabel{\@ifmplabels\mp@tlabel\tex@tlabel}%
+% \end{macrocode}
+%
+% We now come to the main code under \opt{mplabels}, which should just
+% arrange to write the correct commands to the output file. All labels are
+% passed as \mfc{btex ... etex} structures via the command \cs{mfp@btex}
+% below. The definition of \cs{MFPtext} will have been written earlier by
+% \cs{opengraphsfile}.
+% \begin{macrocode}
+\newdef\mfp@btex#1{btex \noexpand\MFPtext{#1} etex}%
+% \end{macrocode}
+%
+% The command \gbc{newgblabel}, defined in \file{graphbase.mp} for placing
+% labels, uses three parameters to justify the text. If we didn't have to
+% support the \texttt{B} vertical placement option, there would be only
+% two. All the other options involve placement \emph{relative} to the
+% bounding box. But the baseline is absolute: \MP{} always puts it at
+% $y$ coordinate 0 in the picture returned by \gbc{btex}. The first two
+% parameters are numerical and the third is boolean. The
+% first represents the fraction of the text that lies left of given
+% point. The second is the fraction of the text below the point. If the
+% third parameter is \mfc{true}, this fraction ignores the depth of the
+% text (so $0$ indicates baseline placement); if it is false, the depth
+% is accounted for (so $0$ indicates the very bottom of the text).
+%
+% By this point the letters of the optional parameter have been converted
+% to integer values for \cs{tl@hpos} and \cs{tl@vpos} by
+% \cs{tlabeljustify}. The following converts these integers to comma
+% separated parameters as described above using \cs{ifcase}.
+%
+% The fourth parameter of \gbc{newgblabel} is the angle of rotation. Empty
+% rotation is converted to the angle 0, otherwise it is written as is.
+% All four parameters are stored in \cs{mfp@MPpos} by the invocation of
+% \cs{mfp@setMPpos}.
+% \begin{macrocode}
+\newdef\mfp@setMPpos{%
+ \edef\mfp@MPpos{%
+ \ifcase\tl@hpos .5\or 1\else 0\fi, % horizontal
+ \ifcase\tl@vpos0\or .5\or 1\else 0\fi, % vertical
+ \ifnum \tl@vpos<0 \mfp@T\else\mfp@F\fi, % true=Baseline
+ \mfp@emptysub{\tl@rot}{0}}}%
+% \end{macrocode}
+%
+% \cs{mp@tlabel} is the code called by \cs{tlabel} when \opt{mplabels} is
+% in effect. At this point all the data needed are stored in macros and
+% we just have to put it together. Thus, it has no parameters and other
+% commands can prepare the data and simply invoke it (or, more usually,
+% invoke \cs{mfp@thelabel} which invokes \cs{tex@tlabel} or \cs{mp@tlabel} as
+% appropriate). The \gbc{newgblabel} command is written with the
+% justification and two more parameters: the label tokens, and the point
+% (or list of points).
+% \begin{macrocode}
+\newdef\mp@tlabel{\mfp@setMPpos
+ \mfcmd{\@nl newgblabel(\mfp@MPpos)(\mfp@btex{\the\mfp@toks})\@nl
+ (\tl@points)}}%
+% \end{macrocode}
+%
+% When \opt{mplabels} is not in effect, we use \TeX{} to position the
+% label. We put the label in a box (\cs{@textbox}) so we can measure and
+% position it. It is a \cs{vbox} so we can have multiline labels. This is
+% effected by wrapping it in a simple, left-aligned, one column
+% \cs{halign}, defining \cs{\bsl} to be \cs{cr}. We make this procedure a
+% separate macro, since we need to do the identical thing in the
+% \cs{tlabelpath} code.
+% \begin{macrocode}
+\newdef\set@textbox#1{%
+ \setbox\@textbox\vbox{\def\\{\cr}\mfp@restorepar
+ \mfp@safe@activesfalse\@tcurr\every@tlabel
+ \halign{##\hfil\cr#1\crcr}}}%
+% \end{macrocode}
+%
+% We call \cs{tex@tlabel} to perform the necessary positioning. Again, all
+% the necessary data has been stored so \cs{tex@tlabel} takes no parameters.
+% Before calling the (rather extensive) positioning code, we have to
+% process the (possible) list of points into separate points. Spaces
+% outside parentheses are consumed by \cs{@ifnextchar} and separating
+% commas (even if there are too many) are gobbled by the
+% \cs{mfp@ifendorcomma}. Oddly, this code works even if the separating commas
+% are omitted, but the \opt{mplabels} code does not.
+% \begin{macrocode}
+\newdef\tex@tlabel{\@xp\@textlabel\tl@points\mfp@end}%
+\newdef\@textlabel{\mfp@ifendorcomma{}\@textlabel\@@textlabel}%
+\newdef\@@textlabel(#1,#2){\tex@@tlabel{#1}{#2}\@textlabel}%
+% \end{macrocode}
+%
+% This is the code that sets and positions the label. We compute its total
+% height to simplify later code. The macro \cs{tb@ht} expands to
+% \cs{ht}\cs{@textbox} and \cs{tb@dp} to \cs{dp}\cs{@textbox}.
+% \begin{macrocode}
+\newdef\tex@@tlabel#1#2{%
+ \set@textbox{\the\mfp@toks}%
+ \mfp@scratch\tb@ht \advance\mfp@scratch\tb@dp
+ \tb@totalht\mfp@scratch
+ \@xconv{#1}\mfp@scratch
+% \end{macrocode}
+%
+% The following computes whether the leftmost extent of the picture will
+% be increased by the addition of the label. The macro \cs{tb@wd} expands
+% to \cs{wd}\cs{@textbox}.
+%
+% We have convert the $x$ coordinate of the label's location to a distance
+% from the left edge of the picture. We then subtract from that the amount
+% of the label extending to the left of this location. Then we add or
+% subtract the separation of the label from its location. Finally, we
+% simply add the horizontal adjustment defined by the user with
+% \cs{tlabeloffset}. The value of \cs{@tlabelsleft} is set equal to this
+% if it was greater. We save the calculated value in \cs{tb@moveright},
+% because it is exactly the amount we'll need to \cs{kern} later to
+% position the label.
+% \begin{macrocode}
+ \begingroup
+ \ifcase\tl@hpos \advance\mfp@scratch -0.5\tb@wd % center
+ \or \advance\mfp@scratch -\tb@wd % right
+ \advance\mfp@scratch -\tlabel@sep
+ \else \advance\mfp@scratch \tlabel@sep % left (default)
+ \fi
+ \advance\mfp@scratch \tlabel@hadj
+ \ifdim \mfp@scratch<\@tlabelsleft
+ \@tlabelsleft\mfp@scratch
+ \fi
+ \tb@moveright\mfp@scratch
+ \endgroup
+% \end{macrocode}
+%
+% Now we base the calculations on the uppermost extent of picture and
+% label. Otherwise the idea is the same, and we may increase
+% \cs{@tlabelstop}. Again we save the calculated result (in \cs{tb@raise})
+% for later positioning of the label.
+% \begin{macrocode}
+ \@yconv{#2}\mfp@scratch
+ \begingroup
+ \ifcase\tl@vpos \advance\mfp@scratch \tb@totalht
+ \advance\mfp@scratch \tlabel@sep % bottom
+ \or \advance\mfp@scratch 0.5\tb@totalht % center
+ \or \advance\mfp@scratch -\tlabel@sep % top
+ \else \advance\mfp@scratch \tb@ht % baseline (default)
+ \fi
+ \advance\mfp@scratch \tlabel@vadj
+ \ifdim \mfp@scratch>\@tlabelstop
+ \@tlabelstop\mfp@scratch
+ \fi
+ \tb@raise\mfp@scratch
+ \endgroup
+% \end{macrocode}
+%
+% Finally we compute the lowermost extent, and save it in \cs{@tlabelsbot}.
+% \begin{macrocode}
+ \begingroup
+ \ifcase\tl@vpos \advance\mfp@scratch \tlabel@sep % bottom
+ \or \advance\mfp@scratch -0.5\tb@totalht % center
+ \or \advance\mfp@scratch -\tb@totalht
+ \advance\mfp@scratch -\tlabel@sep % top
+ \else \advance\mfp@scratch -\tb@dp % baseline (default)
+ \fi
+ \advance\mfp@scratch \tlabel@vadj
+ \ifdim \mfp@scratch<\@tlabelsbot
+ \@tlabelsbot\mfp@scratch
+ \fi
+ \endgroup
+% \end{macrocode}
+%
+% Now we've computed the new size of \cs{@alltlabels} and we put the
+% new label on top of it with the above calculated kerns.
+% \begin{macrocode}
+ \global \setbox\@alltlabels
+ \vtop{\unvbox\@alltlabels
+ \vbox to 0pt{\kern-\tb@raise
+ \hbox{\kern\tb@moveright\box\@textbox}\vss}}%
+}% End \tex@@tlabel
+% \end{macrocode}
+%
+% The argument of \cs{mfp@labels} is the ostensible argument of
+% \cs{tlabels}. Looping is accomplished by defining \cs{extra@endtlabel}
+% to repeat \cs{mfp@dolabel}.
+%
+% If \cs{mfp@dolabel} sees the end-of-loop delimiter it ends the group and
+% gobbles the delimiter. Otherwise it calls \cs{tlabel}, which sees the
+% next set of arguments, places the label, and repeats the loop.
+% \begin{macrocode}
+\newdef\mfp@labels#1{\let\extra@endtlabel\mfp@dolabel
+ \mfp@dolabel#1\mfp@end}%
+\newdef\mfp@dolabel{\mfp@ifend\endgroup\tlabel}%
+% \end{macrocode}
+%
+% \cs{mfp@labelpath} stores the name of the path in \cs{mfp@name} and the
+% default argument in \cs{mfp@param}, then check for a \texttt{*} and set
+% a switch based on it before calling \cs{mfp@tlpath}. The
+% \cs{mfp@figmacbase} must be outside the group so the switches it sets
+% will survive for later commands to detect.
+% \begin{macrocode}
+\newdef\mfp@labelpath#1#2{\mfp@figmacbase
+ \begingroup % ended in \tlabelpath@
+ \def\mfp@name{#1}\def\mfp@param{#2}%
+ \mfp@ifstar\mfp@tlpath\mfp@switchtrue\mfp@switchfalse}%
+% \end{macrocode}
+%
+% Process the first optional argument, which has value of \cs{mfp@param}
+% as default. The second one is a justification with a default set by the
+% user with \cs{tlpathjustify} or the initial \texttt{[cc]} setting. Once
+% these are processed, we check whether a parentheses follows, calling one
+% of two different handlers for the location parameter(s).
+% \begin{macrocode}
+\newdef\mfp@tlpath#1{#1\mfp@defaultopt\mfp@@tlpath\mfp@param}%
+\newdef\mfp@@tlpath#1{\edef\mfp@param{#1}%
+ \mfp@defaultopt\mfp@tlpath@{\tlpath@v\tlpath@h\tlpath@r}}%
+\newdef\mfp@tlpath@#1{\mfp@resetwhitespace\mp@mksharpother
+ \tlabeljustify{#1}\@ifnextchar({\@tlabelpath}{\@@tlabelpath}}%
+% \end{macrocode}
+%
+% These handlers just stuff the point into a macro and call the command
+% that reads the label code. The name \cs{tl@points} is plural (even
+% though only one point is supported) because the internal command that
+% places the label expects this macro.
+% \begin{macrocode}
+\newdef\@tlabelpath(#1,#2){\@@tlabelpath{(#1,#2)}}%
+\newdef\@@tlabelpath#1#2{%
+ \def\tl@points{#1}\mfp@toks{#2}\tlabelpath@}%
+% \end{macrocode}
+%
+% By this point, all the elements we need are gathered. For the purposes
+% of the label we have \cs{mfp@toks} holding the label code,
+% \cs{tl@points} holding the location, and \cs{tlabeljustify} encoding
+% the adjustments. For the path, we have the type in \cs{mfp@name}, and the
+% one optional parameter parameter in \cs{mfp@param}.
+%
+% Under \opt{mplabels} we expect \MP{} to measure the text. Otherwise we
+% measure the text in \TeX{} and feed the bounding box corners to \MF{}
+% for making the path. If \cs{ifmfp@switch} is false we place the text by
+% calling \cs{mfp@thelabel}. All the data needed by \cs{mfp@thelabel} have
+% already been stored in macros by previous commands in the chain. Then we
+% complete the command by ending the group that \cs{mfp@labelpath}
+% started.
+% \begin{macrocode}
+\newdef\tlabelpath@{\@ifmplabels{}{\measure@textbox{\the\mfp@toks}}%
+ \mfp@setMPpos
+ \mfcmd{text\mfp@name x (\mfp@MPpos)\@nl
+ (\@ifmplabels{origin, \mfp@btex{\the\mfp@toks}}{\tb@ll, \tb@ur}, %
+ \mfp@param) (\tl@points)}%
+ \ifmfp@switch\else \mfp@thelabel\fi \endgroup}%
+% \end{macrocode}
+%
+% Here we define the command that measures the text. It is almost a clone
+% of a similar code sequence in \cs{tex@tlabel}, but it produces the \gbc{ll}
+% and \gbc{ur} (corners of the boundingbox) that the \grafbase{} code
+% needs.
+% \begin{macrocode}
+\newdef\measure@textbox#1{\set@textbox{#1}%
+ \mfp@scratch-\tb@dp
+ \edef\tb@ll{(0, \the\mfp@scratch)}%
+ \edef\tb@ur{(\the\tb@wd, \the\tb@ht)}}%
+% \end{macrocode}
+%
+% The argument of \cs{mfp@axislabels} is the axis letter.
+%
+% We start by determining the default justification. which is different
+% for each axis (for example, for the left axis it is \oarg{cr} but for
+% the bottom axis it is \oarg{tc}). For all of them, one of the parts is
+% \texttt{c}, so after resetting whitespace we initialize them both to
+% this and later change only the one that needs it.
+%
+% Also, one of the coordinates of a label will be given by the argument,
+% while the other is the location of the axis. However, which coordinate
+% is which is different for different axes. What we do is feed this
+% position to \cs{mfp@xcoord} and \cs{mfp@ycoord}. Both are initialized to
+% the identity macro, but one is changed to expand to just the axis
+% position. The axis position for $x$ and $y$ axis is 0. For border axes
+% it is calculated by adding the border axis shift to the border
+% coordinate.
+% \begin{macrocode}
+\newdef\mfp@axislabels#1{%
+ \begingroup % ended in \mfp@(@)doaxislabel
+ \mfp@resetwhitespace
+ \tl@vpos1 \tl@hpos0
+ \let\mfp@xcoord\@firstofone \let\mfp@ycoord\@firstofone
+ \mfp@getaxis#1\relax\mfp@end\axislabels
+ \ifcase\mfp@axisnum\relax
+ \tl@vpos2 \def\mfp@ycoord{0}\or
+ \tl@hpos1 \def\mfp@xcoord{0}\or
+ \tl@hpos1\edef\mfp@xcoord{\mfp@lshift}%
+ \mfpaddto\mfp@xcoord\xmin\or
+ \tl@vpos2\edef\mfp@ycoord{\mfp@bshift}%
+ \mfpaddto\mfp@ycoord\ymin\or
+ \tl@hpos-1 \edef\mfp@xcoord{-\mfp@rshift}%
+ \mfpaddto\mfp@xcoord\xmax\or
+ \tl@vpos0 \edef\mfp@ycoord{-\mfp@tshift}%
+ \mfpaddto\mfp@ycoord\ymax
+ \fi
+% \end{macrocode}
+%
+% Now one of \cs{mfp@xcoord} or \cs{mfp@ycoord} expands to the axis
+% position. We change it so that it does this while gobbling the
+% parameter, and then call \cs{mfp@@axislabels}, feeding it an empty
+% argument as the default optional argument. We also give \texttt\# a safe
+% catcode for writing to a \file{.mp} file.
+% \begin{macrocode}
+ \ifx\@firstofone\mfp@xcoord
+ \edef\mfp@ycoord##1{\mfp@ycoord}%
+ \else
+ \edef\mfp@xcoord##1{\mfp@xcoord}%
+ \fi \mp@mksharpother\mfp@nullopt\mfp@@axislabels}%
+% \end{macrocode}
+%
+% \cs{mfp@@axislabels} processes the contents of the optional parameter
+% and calls \cs{mfp@doaxislabel} on the list of labels.
+% \begin{macrocode}
+\newdef\mfp@@axislabels#1#2{%
+ \tlabeljustify{#1}\mfp@doaxislabel#2,\mfp@end}%
+% \end{macrocode}
+%
+% \cs{mfp@doaxislabel} checks for the end delimiter or a comma (the latter
+% is an error). In the first case it ends all label processing, in the
+% second it repeats itself, otherwise it continues with
+% \cs{mfp@@doaxislabel}.
+%
+% The argument of \cs{mfp@@doaxislabel} should be the label. This command
+% stores the label in \cs{mfp@toks}, then again checks for the delimiter
+% or an comma (both are errors). In the first case it ends the process; in
+% the second it goes back to \cs{mfp@doaxislabel} for another try. In both
+% cases the label is discarded as we don't know where to put it. If
+% neither the delimiter nor a comma occurs, \cs{mfp@mkaxislabel} is run.
+%
+% The argument of \cs{mfp@mkaxislabel} (everything to the next comma) is the
+% location. The command \cs{mfp@mkaxislabel} does nothing if the location
+% parameter is empty, otherwise it stores the location as a pair and calls
+% \cs{mfp@thelabel} to place the label. Finally, it repeats
+% \cs{mfp@doaxislabel} for the next iteration.
+% \begin{macrocode}
+\newdef\mfp@doaxislabel{%
+ \mfp@ifendorcomma\endgroup\mfp@doaxislabel\mfp@@doaxislabel}%
+\newdef\mfp@@doaxislabel#1{\mfp@toks{#1}%
+ \mfp@ifendorcomma\endgroup\mfp@doaxislabel\mfp@mkaxislabel}%
+\newdef\mfp@mkaxislabel#1,{%
+ \@ifmtarg{#1}{}%
+ {\edef\tl@points{(\mfp@xcoord{#1},\mfp@ycoord{#1})}\mfp@thelabel}%
+ \mfp@doaxislabel}%
+% \end{macrocode}
+%
+% \cs{mfp@caption} has one mandatory argument, the caption's text. In it, the
+% \cs{\bsl} command forces line breaks. This works by forcing the width to
+% be greater than any reasonable setting of the maximum line width. When
+% this happens, the caption is reset in a \cs{vbox} and \cs{\bsl} is
+% redefined like a normal line break.
+% \begin{macrocode}
+\newdef\mfp@caption#1{\begingroup\mfp@resetwhitespace
+ \mfp@@caption#1\mfp@end}%
+\long\def\mfp@@caption#1,#2\mfp@end#3{\endgroup
+ \def\mfp@docaption{%
+ \setbox\@textbox
+ \hbox{\def\\{\hskip\@M\p@\let\\\relax}\mfp@restorepar
+ \mfp@safe@activesfalse\@tcurr#3}%
+% \end{macrocode}
+%
+% If the above \cs{hbox} is too wide, we re-assign \cs{@textbox} to a
+% \cs{vbox} with \cs{hsize} determined by the second part of the optional
+% argument. \cs{mfp@capsettings} implements the \opt{centeredcaptions}
+% and \opt{raggedcaptions} options.
+% \begin{macrocode}
+ \mfp@scratch\wd\@wholegraph
+ \ifdim\tb@wd>#1\mfp@scratch
+ \setbox\@textbox
+ \hbox{%
+ \vbox{%
+ \hsize#2\mfp@scratch \mfp@restorepar
+ \leftskip0pt \rightskip0pt\relax
+ \mfp@capsettings
+ \mfp@safe@activesfalse
+ \noindent\@tcurr #3%
+ }% End vbox.
+ }% End hbox.
+ \fi
+% \end{macrocode}
+%
+% Here we determine the wider of the graph or the caption, then put the
+% graph and caption together by centering them in \cs{hbox}es with this
+% width, stacked one atop the other with \cs{mfpiccaptionskip} (default:
+% \cs{medskipamount}) in between.
+% \begin{macrocode}
+ \ifdim \mfp@scratch<\tb@wd \mfp@scratch\tb@wd \fi
+ \global\setbox\@wholegraph
+ \vbox{\hbox to \mfp@scratch{\hss\box\@wholegraph\hss}%
+ \nointerlineskip \vskip\mfpiccaptionskip
+ \hbox to \mfp@scratch{\hss\box\@textbox\hss}}% End vbox.
+ }% End \mfp@docaption.
+}% End \mfp@@caption.
+% \end{macrocode}
+%
+% The following is used in \cs{endmfpic}. The first argument of
+% \cs{mfp@adjustdim} is a border dimension of the box containing all the
+% labels, the second is a comparison between that and the corresponding
+% dimension of the graph box, which is the third argument. The last is the
+% macro holding the adjustments needed when \opt{truebbox} is used. The
+% first and last line compensate for the fact that the bbox need not start
+% at $(0,0)$. The second line checks if the label box extends past the
+% graph's edge and adjusts the graph's dimension is it does.
+% \begin{macrocode}
+\def\mfp@adjustdim#1#2#3#4{%
+ \mfp@scratch#1\relax \advance\mfp@scratch-#4 bp
+ \ifdim\mfp@scratch#2#3\relax #3\mfp@scratch \fi
+ #1\mfp@scratch}%
+% \end{macrocode}
+%
+%
+%
+% \section{Placing the picture: \cs{endmfpic}}\label{endmfpic}
+%
+% \DescribeMacro{\endmfpic}
+% By the time \cs{endmfpic} is reached, the \MF{} code for the figure has
+% all been written to the output file. The labels have all been
+% accumulated and arrangements have been made to add the caption.
+%
+% Under \opt{metapost}, write a right brace to match the left written
+% by \cs{mfpic} Then write the \cs{grafbase} command \gbc{endmfpic},
+% followed by a comment telling which figure number it is, and the line
+% number in the \TeX{} source where the \cs{endmfpic} occurred.
+% \begin{macrocode}
+\mfp@DBlog{Definition of closure of mfpic environment.}%
+\newdef\endmfpic{\every@endmfpic
+ \mfsrc{%
+ \mfp@ifmpost{\@nl verbatimtex \relax\MFPendgroup\space etex;}{}\@nl
+ endmfpic; \mfp@p\space(\number\mfp@count) \mfp@lineno.\@nl
+ \mfp@p\mfp@d\mfp@d\mfp@d\@nl}%
+ \ifmfpicdebug\wlog{}\wlog{Mfpic: ENTERED endmfpic.}\wlog{}\fi
+% \end{macrocode}
+%
+% The code to load and place the box is skipped if figure shipping was
+% turned off.
+% \begin{macrocode}
+ \ifmfp@noship \else
+% \end{macrocode}
+%
+%
+% \subsection{Loading and measuring the graphic}\label{loading}
+%
+% Now we load the graphic. It would be at this point that we could add a
+% \cs{write18} to process the figure on-the-fly. Of course that would
+% require re-initializing the output file with each graphic (i.e., the
+% equivalent of issuing \cs{opengraphsfile} at every \cs{mfpic} and
+% \cs{closegraphsfile} at every \cs{endmfpic}.)
+% \begin{macrocode}
+ \global\setbox\@wholegraph
+ \vbox{%
+% \end{macrocode}
+%
+% If draft mode has been detected or set by the user, we just make a box
+% of the size specified with the figure number inside it.
+% \begin{macrocode}
+ \ifmfpicdraft
+ \vbox to \@graphtop{\vss
+ \hbox to \@graphright{%
+ \kern2pt \mfpdraftfont\#\number\mfp@count\hss}%
+ \kern2pt}%
+ \else
+% \end{macrocode}
+%
+% The \opt{truebbox} option works at the \MP{} level. Without it the
+% height of the EPS bounding box will actually be the calculated value
+% of \cs{@graphtop}. With it, the box will be its natural height. But
+% in either case, the value will actually be increased by up to
+% \texttt{2bp} because of rounding, so the bounding box height is what we
+% use in the \MP{} case. For \MF{} we use the nominal height calculated in
+% \cs{mfpic}.
+% \begin{macrocode}
+ \mfp@ifmpost{\vbox}{\vbox to \@graphtop}%
+ {\vss
+% \end{macrocode}
+%
+% \cs{@graphfilename} holds the name of the \MP{} output figure. (It will be
+% ignored by the \opt{metafont} option.) We test if the file exists before
+% doing the graphic inclusion because of the large number of confusing error
+% messages many graphic inclusion packages generate otherwise. When absent
+% we substitute the figure filename for \cs{@setmfpicgraphic}.
+% \begin{macrocode}
+ \edef\@graphfilename{%
+ \setfilename\mfp@filename{\number\mfp@count}}%
+ \mfp@ifmpost
+ {\IfFileExists{\@graphfilename}{}%
+ {\Mfpic@msg{No graph file: \@graphfilename\space.}%
+ \let\@setmfpicgraphic\nofile@setmfpgraphic}%
+% \end{macrocode}
+%
+% We put the graphic in an \cs{hbox} of the required width and close off
+% the boxes.
+% \begin{macrocode}
+ \hbox}%
+ {\hbox to \@graphright}%
+ {\@xp\@setmfpicgraphic\@xp{\@graphfilename}\hss}%
+ \kern0pt
+ }% End vbox
+ \fi
+ }%% End \setbox\@wholegraph \vbox
+% \end{macrocode}
+%
+% We change the horizontal and vertical dimensions under \opt{metapost} in
+% case \opt{truebbox} was in effect, but also because they are almost always
+% rounded up to whole numbers of \texttt{bp} and therefore, even without
+% \opt{truebbox}, the calculated values needn't be correct.
+% \begin{macrocode}
+ \mfp@ifmpost
+ {\@graphright\wd\@wholegraph
+ \@graphtop\ht\@wholegraph}{}%
+ \ifmfpicdebug
+ \wlog{Mfpic debug: Dimensions of wholegraph before labels %
+ are added, and before truebbox adjustments:}%
+ \wlog{Mfpic: graphleft = \the\@graphleft}%
+ \wlog{Mfpic: graphright = \the\@graphright}%
+ \wlog{Mfpic: graphtop = \the\@graphtop}%
+ \wlog{Mfpic: graphbot = \the\@graphbot}%
+ \wlog{Mfpic: graphwd = \the\wd\@wholegraph}%
+ \wlog{Mfpic: graphht = \the\ht\@wholegraph}%
+ \wlog{Mfpic: graphdp = \the\dp\@wholegraph}%
+ \wlog{}%
+ \fi
+% \end{macrocode}
+%
+%
+% \subsection{Adding labels and caption}\label{addlabel}
+%
+% We now add the boxes \cs{@alltlabels} and \cs{@backtext} onto the graph
+% box. But first we have to adjust for \cs{mfpicllx} and \cs{mfpiclly}
+% which can only be known after the graphics inclusion. We reset those
+% boxes so their contents are shifted by these values. We only need to do
+% this if labels have have actually been added. The signal that this has
+% happened is that the label dimensions are no longer $\pm$\cs{maxdimen}.
+% This doesn't indicate whether it is \cs{@alltlabels} or \cs{@backtext}
+% (or both) that is nonempty, so one of the box resets may be redundant.
+% \begin{macrocode}
+ \ifmfpicdebug
+ \wlog{Mfpic debug: Dimensions of alltlabels, before %
+ truebbox adjustments:}%
+ \mfp@scratch\mfpicllx bp\relax
+ \wlog{Mfpic: mfpicllx = \the\mfp@scratch}%
+ \mfp@scratch\mfpiclly bp\relax
+ \wlog{Mfpic: mfpiclly = \the\mfp@scratch}%
+ \wlog{Mfpic: tlabelsleft = \the\@tlabelsleft}%
+ \wlog{Mfpic: tlabelstop = \the\@tlabelstop}%
+ \wlog{Mfpic: tlabelsbot = \the\@tlabelsbot}%
+ \wlog{}%
+ \fi
+ \ifdim\@tlabelsleft < \maxdimen
+ \setbox\@alltlabels\vtop to 0pt{%
+ \kern \mfpiclly bp
+ \moveleft \mfpicllx bp \box\@alltlabels\vss}%
+ \setbox\@backtext \vtop to 0pt{%
+ \kern \mfpiclly bp
+ \moveleft \mfpicllx bp \box\@backtext \vss}%
+% \end{macrocode}
+% Then we adjust \cs{@tlabelsleft}, etc. for the boundingbox offset.
+% The graph dimensions are also adjusted if the tlabel dimensions extend
+% past the edges.
+% \begin{macrocode}
+ \mfp@adjustdim\@tlabelsleft <\@graphleft \mfpicllx
+ \mfp@adjustdim\@tlabelstop >\@graphtop \mfpiclly
+ \mfp@adjustdim\@tlabelsbot <\@graphbot \mfpiclly
+ \ifmfpicdraft\else
+ \setbox\@wholegraph
+ \vtop{\offinterlineskip
+ \box\@backtext\kern-\ht\@wholegraph
+ \unvbox\@wholegraph\box\@alltlabels}%
+ \fi
+ \fi
+% \end{macrocode}
+%
+% The method for putting labels in \cs{@alltlabels} does not add any width
+% on the left, nor any height to the top or depth to the bottom of that box
+% (that \TeX{} can detect), but we have saved their values and now we can
+% use them. For example, if \cs{@graphleft} is negative after the above,
+% then a label sticks out to the left, but \TeX{} thinks it has no width.
+% Therefore put \cs{kern}\texttt{-}\cs{@graphleft} to move the graph
+% rightward. The height and depth are taken care of by putting it in
+% \cs{vbox} with the calculated total height and moving the graph upward
+% with \cs{kern}\texttt{-}\cs{@graphbot}. The rightward extensions do get
+% detected by \TeX.
+% \begin{macrocode}
+ \ifmfpicdebug
+ \wlog{Mfpic debug: Dimensions of alltlabels after %
+ truebbox adjustments:}%
+ \wlog{Mfpic: tlabelsleft = \the\@tlabelsleft}%
+ \wlog{Mfpic: tlabelstop = \the\@tlabelstop}%
+ \wlog{Mfpic: tlabelsbot = \the\@tlabelsbot}%
+ \wlog{}%
+ \fi
+ \mfp@scratch\@graphtop
+ \advance\mfp@scratch by -\@graphbot
+ \global \setbox\@wholegraph
+ \vbox to \mfp@scratch{\vss
+ \hbox{\kern-\@graphleft\box\@wholegraph}%
+ \kern-\@graphbot}% End vbox.
+% \end{macrocode}
+%
+% Now add the caption, then set the parameters \cs{mfpicheight} and
+% \cs{mfpicwidth} for user to examine. The debug info now should show
+% graphwd${}={}$graphright${}-{}$graphleft,
+% graphht${}={}$graphtop${}-{}$graphbot, and graphdp${}= 0$.
+% \begin{macrocode}
+ \ifmfpicdebug
+ \wlog{Mfpic debug: Dimensions of wholegraph after %
+ labels are added:}%
+ \wlog{Mfpic: graphleft = \the\@graphleft}%
+ \wlog{Mfpic: graphright = \the\@graphright}%
+ \wlog{Mfpic: graphtop = \the\@graphtop}%
+ \wlog{Mfpic: graphbot = \the\@graphbot}%
+ \wlog{Mfpic: graphwd = \the\wd\@wholegraph}%
+ \wlog{Mfpic: graphht = \the\ht\@wholegraph}%
+ \wlog{Mfpic: graphdp = \the\dp\@wholegraph}%
+ \wlog{}%
+ \fi
+ \mfp@docaption
+ \ifmfpicdebug
+ \wlog{Mfpic debug: Values after tcaption added:}%
+ \wlog{Mfpic: graphleft = \the\@graphleft}%
+ \wlog{Mfpic: graphright = \the\@graphright}%
+ \wlog{Mfpic: graphtop = \the\@graphtop}%
+ \wlog{Mfpic: graphbot = \the\@graphbot}%
+ \wlog{Mfpic: graphwd = \the\wd\@wholegraph}%
+ \wlog{Mfpic: graphht = \the\ht\@wholegraph}%
+ \wlog{Mfpic: graphdp = \the\dp\@wholegraph}%
+ \wlog{}%
+ \fi
+ \global\mfpicheight\ht\@wholegraph
+ \global\mfpicwidth\wd\@wholegraph
+% \end{macrocode}
+%
+%
+% \subsection{Placing the graphic}\label{placing}
+%
+% Add a frame around the box \cs{@wholegraph} in draft mode (so its size
+% can be seen) since there is practically nothing in it except the picture
+% number and perhaps some tlabels.
+% \begin{macrocode}
+ \ifmfpicdraft
+ \setbox\@wholegraph
+ \hbox{\@mfpframed{-\mfpframethickness}{\box\@wholegraph}}%
+ \fi
+% \end{macrocode}
+% Now we save the graph globally, for placement after \cs{endgroup}. This
+% is so that tokens placed by \cs{everypar} will be typeset in the current
+% font (instead of \cs{nullfont}). Then we advance the figure count and
+% end the group started by the \cs{mfpic} command.
+% \begin{macrocode}
+ \global\setbox\@wholegraph\box\@wholegraph
+ \global \advance\mfp@count1
+ \fi % \ifmfp@noship
+ \endgroup % Begun near start of \@mfpic
+% \end{macrocode}
+% If the \cs{savepic} command occurred prior to this, then \cs{s@vemfpic}
+% is defined and we save this picture in the box named in that command and
+% stored in the macro \cs{s@vemfpic}. Otherwise, \cs{s@vemfpic} is
+% undefined, and we place the picture (\cs{@wholegraph}) in the document.
+% Afterward, we undefine \cs{s@vemfpic}. Note: if \opt{noship} is in
+% force, it remains defined, so the next picture will be stored.
+% \begin{macrocode}
+ \ifmfp@noship \else
+ \mfp@ifdefined\s@vemfpic
+ {\global\setbox\s@vemfpic\box\@wholegraph}%
+ {\leavevmode\box\@wholegraph}%
+ \global\let\s@vemfpic\UndEfInEd
+ \fi
+% \end{macrocode}
+% Then we restore the values of extra font dimensions that were saved.
+% Just in case, we unset the switch \cs{in@mfpicenv} and then arrange for
+% spaces to be ignored in case this is a \LaTeX{} environment. (I think we
+% shouldn't ignore spaces, but I was overruled. The argument was that
+% \cs{end}\marg{mfpic} should behave like \cs{endmfpic}. I don't buy it,
+% but I will keep things the way they were since version 0.4.)
+% \begin{macrocode}
+ \restore@mfpicdimens
+ \mfp@checklatexenv{mfpic}{\@ignoretrue}{}%
+ \global\let\in@mfpicenv\iffalse
+ \ignorespaces
+}% endmfpic
+% \end{macrocode}
+%
+% The following is a little utility to run at \cs{foo} or \cs{endfoo} to
+% see if was initiated as a \LaTeX{} environment or as a plain command.
+% The first argument is the environment name, the second is what to do
+% if that is the current environment, and the third is what to do
+% otherwise. In the two cases we now use it, the third argument is
+% empty.
+% \begin{macrocode}
+\ifmfp@latex
+ \newdef\mfp@checklatexenv#1#2#3{%
+ \def\mfp@tempa{#1}\ifx\mfp@tempa\@currenvir #2\else #3\fi}%
+\else
+ \newdef\mfp@checklatexenv#1#2#3{#3}%
+\fi
+% \end{macrocode}
+%
+%
+%
+% \section{Additional features}\label{additional}
+%
+%
+% \subsection{Saving a copy of the picture}\label{saving}
+%
+% \DescribeMacro{\newsavepic}
+% One saves a picture by first allocating a save box with
+% \cs{newsavepic}\cs{mypic}, then saying \cs{savepic}\cs{mypic}
+% beforehand. The \cs{endmfpic} code detects that this has been done and
+% saves the picture in \cs{mypic}. It can be placed with
+% \cs{usepic}\cs{mypic}.
+%
+% \cs{newsavepic} is just a clone of \LaTeX's \cs{newsavebox}. The one
+% argument is a control sequence.
+% \begin{macrocode}
+\ifmfp@latex
+ \newlet\newsavepic\newsavebox
+\else
+ \newdef\newsavepic#1{%
+ \mfp@ifdefined{#1}%
+ {\mfp@errmsg{Command \string #1 already defined}%
+ {You have used \newsavepic with an already defined or an^^J%
+ improper control sequence. Replace #1 with another name.^^J%
+ If you proceed, \newsavepic will be ignored.}}%
+ {\@nameuse{newbox}#1}}%
+\fi
+\newlet\newpic\newsavepic
+% \end{macrocode}
+%
+% \DescribeMacro{\savepic}
+% All \cs{savepic} really does is define \cs{s@vepic} to its argument,
+% which should be a control sequence defined by \cs{newsavepic} (or
+% \cs{newbox}).
+% \begin{macrocode}
+\newdef\savepic#1{\relax
+ \mfp@ifdefined{#1}{\gdef\s@vemfpic{#1}}%
+ {\mfp@errmsg{Box \string#1 undefined}%
+ {You tried to save a picture in a box which had not been^^J%
+ previously allocated. Use \newsavepic to allocate a box.}}}%
+% \end{macrocode}
+%
+% \DescribeMacro{\usepic}
+% \cs{usepic} takes one argument, a control sequence that should have
+% been previously used in a \cs{savepic} comamnd. It just places the box
+% with the primitive \cs{copy} command.
+% \begin{macrocode}
+\newdef\usepic#1{\leavevmode \copy#1\relax}%
+% \end{macrocode}
+%
+%
+% \subsection{Putting a frame around it}\label{frame}
+%
+% This is essentially code I wrote years ago to have a framing command
+% that had two properties not enjoyed by some example `\cs{boxit}' commands
+% for plain\TeX: It should have a baseline the same as the baseline of the
+% contents, and it should work as an environment, not by reading an
+% argument (so verbatim material could be included).
+%
+% \DescribeMacro{\mfpframed}
+% \cs{mfpframed} takes an optional argument (the separation) and a
+% mandatory argument, the contents.
+% \DescribeMacro{\mfpframe}
+% \cs{mfpframe}$\ldots$\cs{endmfpframe} is an environment and
+% \cs{mfpframe} also takes the separation as an optional argument.
+%
+% \DescribeMacro{\framed}
+% \cs{framed} is a common command (\ConTeXt{} has one, a few packages also
+% define it), therefore we only define it if it is not already defined.
+% \begin{macrocode}
+\newdef\mfpframed{\mfp@defaultopt\@mfpframed\mfpframesep}%
+\newdef\mfpframe{\mfp@defaultopt\@mfpframe\mfpframesep}%
+\newdef\@mfpframed#1#2{\@mfpframe{#1}#2\endmfpframe}%
+\newdef\@mfpframe#1{%
+ \leavevmode\hbox\bgroup
+ \mfpframesep#1\relax
+ \vrule width\mfpframethickness
+ \vtop\bgroup
+ \vbox\bgroup
+ \hrule height\mfpframethickness
+ \kern\mfpframesep
+ \hbox\bgroup \kern\mfpframesep \ignorespaces}%
+\newdef\endmfpframe{\unskip
+ \kern\mfpframesep
+ \egroup % end hbox
+ \egroup % end vbox
+ \kern\mfpframesep
+ \hrule height\mfpframethickness
+ \egroup % end vtop
+ \vrule width\mfpframethickness
+ \egroup}% end hbox
+\mfp@ifdefined\framed{}{\let\framed\mfpframed}%
+% \end{macrocode}
+%
+%
+% \subsection{Adding \mfc{verbatimtex} to the \file{.mp}
+% file}\label{verbatimtex}
+%
+% \DescribeMacro{\mfpverbtex}
+% This is a user level utility for putting some \mfc{verbatimtex} material
+% in the output file for \MP. We allow the user to do this even before
+% \cs{opengraphsfile} by appending it to a token register and setting a
+% flag for that command to test.
+%
+% We now want \texttt\% signs to be written to the output, but also
+% want to allow the construct
+% \begin{verbatim}
+% \mfpverbtex% Comment
+% {<tex code>}\end{verbatim}
+% to function as users might expect. We therefore perform a
+% \cs{futurelet} (which should set \cs{mfp@next} equal to the opening
+% brace) but we do it solely to get \TeX's scanning mechanism past the
+% comment before we change the category of \texttt\%.
+% \begin{macrocode}
+\def\mfpverbtex{\futurelet\mfp@next\@mfpverbtex}%
+% \end{macrocode}
+%
+% We start a group, change \texttt\% and \texttt\# to `other' so they
+% can be written normally, then continue after loading the code into the
+% token register \cs{mfp@toks}.
+% \begin{macrocode}
+\def\@mfpverbtex{%
+ \begingroup
+ \@makeother\#\@makeother\%\mfp@keeplines
+ \afterassignment\mfp@writetex\global\mfp@toks}%
+% \end{macrocode}
+% We end the group and then either write it to the
+% output, between the commands \gbc{verbatimtex} and \cs{etex},
+% or append it to \cs{mfp@verbtex} and set the switch to tell
+% \cs{opengraphsfile} that there is material to write when the output file
+% is opened.
+% \begin{macrocode}
+\newif\ifmfp@verbtex
+\def\mfp@writetex{%
+ \mfp@ifopengraphsfile
+ {\mfp@ifmpost
+ {\mfcmd{verbatimtex\@nl\the\mfp@toks\@nl\mfp@p\@nl etex}}%
+ {\mfp@noMP@error\mfpverbtex}%
+ \global\mfp@verbtexfalse\global\mfp@verbtex{}}%
+ {\edef\mfp@tempa{\the\mfp@verbtex\the\mfp@toks}%
+ \global\mfp@verbtex\@xp{\mfp@tempa}%
+ \global\mfp@verbtextrue}%
+ \endgroup
+}%
+% \end{macrocode}
+%
+%
+%
+% \section{Finale}\label{finale}
+%
+% Set the default starting graphic number. We used to start at 0 and
+% increment it at the start of each picture, but now we start at 1 and
+% increment it at the end of each picture.
+% \begin{macrocode}
+\mfpicnumber{1}%
+% \end{macrocode}
+%
+% Set the font for missing or draft mode figures (for the file name or
+% figure number). This will already have been defined if \file{mfpic.sty}
+% was used, so here we can probably assume \LaTeX{} wasn't used, and
+% \cs{tt} ought to be available.
+% \begin{macrocode}
+\mfp@ifdefined\mfpdraftfont{}{\newdef\mfpdraftfont{\tt}}%
+% \end{macrocode}
+%
+% Restore the meaning of \cs{+}, and execute the command that restores all
+% the saved category codes and \cs{newlinechar}. Input user defaults in
+% the file \file{mfpic.usr}.
+% \begin{macrocode}
+\mfp@DBlog{Punctuation, etc., will revert to old catcodes now.}%
+\let\+\mfpsaveplus
+\mfp@restoreALL%
+\InputIfFileExists{mfpic.usr}{}{}%
+%</tex>
+% \end{macrocode}
+%
+%\clearpage
+%\Finale
+%\endinput
diff --git a/graphics/mfpic/source/mfpic.ins b/graphics/mfpic/source/mfpic.ins
new file mode 100644
index 0000000000..a66d283223
--- /dev/null
+++ b/graphics/mfpic/source/mfpic.ins
@@ -0,0 +1,59 @@
+% File: mfpic.ins
+% A part of mfpic 1.10 2012/12/03
+%
+\input docstrip
+\keepsilent
+\preamble
+
+-------------------------------------------------------------------
+
+Copyright 2002--2012, Daniel H. Luecking
+
+Mfpic may be distributed and/or modified under the conditions of the
+LaTeX Project Public License, either version 1.3c of this license or (at
+your option) any later version. The latest version of this license is in
+ <http://www.latex-project.org/lppl.txt>
+and version 1.3c or later is part of all distributions of LaTeX version
+2008/12/01 or later.
+
+Mfpic has maintenance status "author-maintained". The Current Maintainer
+is Daniel H. Luecking. There are several Base Interpreters associated
+with mfpic: plain TeX, LaTeX, plain Metafont and plain MetaPost.
+
+\endpreamble
+\edef\zyx{%
+endinput.^^J%
+\DoubleperCent^^J%
+\DoubleperCent\space End of file `\outFileName'.}
+
+\askforoverwritefalse
+
+\generate{\file{mfpic.tex}{\from{mfpic.dtx}{tex}}
+ \file{mfpic.sty}{\from{mfpic.dtx}{sty}}
+ \file{mfpicdef.tex}{\from{mfpic.dtx}{defs}}
+ \usepostamble\zyx
+ \file{grafbase.mf}{\from{grafbase.dtx}{MF}}
+ \file{grafbase.mp}{\from{grafbase.dtx}{MP}}
+ \file{dvipsnam.mp}{\from{grafbase.dtx}{dvips}}}
+
+\def\sJ{^^J \space\space\space}
+\immediate\write16{%
+^^J*******************************************************************
+^^J
+^^J To finish the installation:
+^^J
+^^J -- Copy the files mfpic.tex, mfpic.sty and mfppatch.tex to a
+\sJ location where TeX and LaTeX will find them. For example, in a
+\sJ TDS compliant system, the directory TEXMF/tex/generic/mfpic/ .
+^^J
+^^J -- Copy the file grafbase.mf to a location where Metafont
+\sJ will find it, for example TEXMF/metafont/mfpic/ .
+^^J
+^^J -- Copy the files grafbase.mp and dvipsnam.mp to a location where
+\sJ MetaPost will find them, for example TEXMF/metapost/mfpic/ .
+^^J
+^^J -- Refresh the filename database if your TeX system requires it.
+^^J
+^^J*******************************************************************}
+
+\endbatchfile