% \iffalse
% -------------------------------------------------------------------
%
% Copyright 2002--2006, 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 1.3b or later is part of all distributions of LaTeX version
% 2003/12/01 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}
  [2006/05/26 v0.9. Metafont/post macros to interface with mfpic.]%
\documentclass[draft]{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
\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{1369}
% \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 (96\% 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{mfpman.pdf} produced from
% \file{mfpman.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 takes care of both the
% \DescribeRoutine{GBerrmsg}\gbc{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 := "0.9";  filedate := "2006/05/26";

message " Loading grafbase macros, version " & fileversion & " " &
   filedate & ".";

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 (90);

%    \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}
%
% \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{known blue}' is \mfc{true} 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 METAPOST;

if known color Geamparalele din Babadag:
  METAPOST := true;
else:
  METAPOST := false;
fi

%<*MF>
if METAPOST:
  GBerrmsg ("wrong compiler.")
    "This file is for Metafont. For Metapost use grafbase.mp.";
fi
%</MF>
%<*MP>
if not METAPOST:
  GBerrmsg ("wrong compiler.")
    "This file is for Metapost. For Metafont use grafbase.mf.";
fi
%</MP>

%    \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;
    >> "Grafbase DEBUG";
enddef;
def GBenddebug =
    >> "End DEBUG";
  endgroup
enddef;

vardef mftitle expr t =
  if string t:  t;  message t;  fi
enddef;

%    \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 it 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 localfont or ljfour. "
  & "If you proceed, a generic 600dpi mode will be used.";
  if known localfont:   mode := localfont;
  else:
    mode_def GBgeneric =
      proofing := 0;
      fontmaking := 1;
      tracingtitles := 0;
      if unknown pixels_per_inch:
        pixels_per_inch := 600;
      fi
      blacker := 0;
      fillin := 0;
      o_correction := 1;
    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{} does 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{90} or \gbc{90deg} or \gbc{pi/2*radian}
% \DescribeVariable{pi} for the same effect.
% Actually, not quite: because of \MF{}'s precision limits, the latter 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} is
% the maximum possible. If we \emph{define} \gbc{radian} to be
% \gbc{90/(pi/2)} or \gbc{180/pi} the value of \gbc{pi/2*radian} is even
% less accurate.
%    \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;
black := 0;  white := 1;
def withcolor text t =  enddef;
%</MF>
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}
% This truncates numerics to the $[0,1]$ range, but also returns a value
% ($0$) for unknown and non-numeric input.
%    \begin{macrocode}
vardef snapto expr t =
  if unknown t:            0
  elseif not (numeric t):  0
  elseif t < 0:            0
  elseif t > 1:            1
  else:                    t
  fi
enddef;

%    \end{macrocode}
%
% The \mfpic{} handling of \LaTeX-like color models relies on being able
% to convert those models to \MP's \opt{rgb} system. Because of the use of
% \gbc{snapto}, the following color functions will return \mfc{black} for
% unknown parameters. In the \MF{} case, they are all converted to
% numerics through \gbc{makeclr}.
%
% \DescribeRoutine{gray}
% The simplest is \gbc{gray} which converts a numeric to a multiple of
% white. In \MF, \gbc{white} is a numeric and equal to $1$ so this is
% almost redundant except for handling unknowns and out of range values.
%
% \DescribeRoutine{makeclr}
% This is defined to convert a triple of numerics to a color, mainly for
% \MF. The formula has three desirable properties: it weights the
% different color coordinates approximately like some color luminescence
% models do, it assigns different graylevels to the eight colors that have
% components 0 or 1 only, and it is biased toward lighter grays. Of course
% it takes \mfc{black} to 0 and \mfc{white} to 1. In \MP, it simply turns
% three numeric parameters to a color triple in the obvious way. It does
% \emph{not} truncate the parameters, so if that is necessary, use
% \gbc{rgb}.
%
% \DescribeRoutine{rgb}
% To simplify \mfpic, we have the nearly redundant \gbc{rgb} which
% converts a triple of numeric arguments to \opt{rgb}. Rather than make
% it formally the identity function under \MP, we define it to handle
% unknowns, and truncate out of range values.
%    \begin{macrocode}
vardef gray (expr g) =  (snapto g)*white  enddef;

vardef makeclr (expr r, g, b) =
%<MF>  gray (sqrt((2r*r + 4g*g + b*b)/7))
%<MP>  (r, g, b)
enddef;

vardef rgb (expr r, g, b) =
  makeclr (snapto r, snapto g, snapto b)
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{cmyk}
% This algorithm for converting \opt{cmyk} values to \opt{rgb} values is
% the one used in the PostScript header file \file{color.pro} (distributed
% with \prog{dvips}).
%    \begin{macrocode}
vardef cmyk (expr c, m, y, k) =
  rgb (1-c-k, 1-m-k, 1-y-k)
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{RGB}
% This merely rescales numbers in the range 0--255 to the range 0--1.
%
% \DescribeRoutine{named}
% These last two, like \gbc{rgb}, are nearly redundant, but they convert
% numerics to gray, and convert other non-color variables and unknown
% color variables to black.
% \DescribeRoutine{forceclr}
% The difference between \gbc{named} and \gbc{forceclr} is that the
% former requires a suffix parameter (that is, a \emph{name}), while the
% latter takes an expression. It may be that the latter will never be
% needed, but for a time it seemed there were cases where we ought to use
% it to force an expression to be a color.
%    \begin{macrocode}
vardef RGB (expr R, G, B) =
  rgb (R/255, G/255, B/255)
enddef;

vardef named (suffix c) =
  if unknown c:  black  else:  forceclr (c)  fi
enddef;
vardef forceclr (expr c) =
  if numeric c:    gray (c)
  elseif color c:  c
  else:            black
  fi
enddef;

%    \end{macrocode}
% And then the standard colors. Using \gbc{rgb} ensures they are defined
% in \MF{} as well as \MP.
%    \begin{macrocode}
color red, green, blue, cyan, magenta, yellow;
red     := rgb (1, 0, 0);
green   := rgb (0, 1, 0);
blue    := rgb (0, 0, 1);
cyan    := rgb (0, 1, 1);
magenta := rgb (1, 0, 1);
yellow  := rgb (1, 1, 0);

%    \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}
% \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 omit 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}
% \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}
% \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 $x\sb2$ 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}
% \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}\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}
% This 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}\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}
% Takes any expression and does what it can to make a picture from it.
%
% \DescribeRoutine{onepointpath}
% 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}
% Of course \gbc{even} means \gbc{not odd}.
%    \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{setpicture}\gbc{setpicture},
% \RoutineIndex{setpath}\gbc{setpath}, \RoutineIndex{setpair}\gbc{setpair}
% and \RoutineIndex{setboolean}\gbc{setboolean}, together with the
% common uses \RoutineIndex{newpicture}\gbc{newpicture} and
% \RoutineIndex{convertpath}\gbc{convertpath}.
% \DescribeRoutine{gsetvariable}\gbc{gsetvariable} is the global
% version. It has no abbreviations, but it is occasionally needed for
% \mfpic{}. The only difference 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.
% \DescribeRoutine{setpairs}\gbc{setpairs} is an abbreviation for arrays
% of pairs. There is also has a global version
% \DescribeRoutine{gsetarray}\gbc{gsetarray}.
%    \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 setcolor     =  setvariable (color)  enddef;
def setpicture   =  setvariable (picture)  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;

%    \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---or \prog{GhostScript}---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.
%
% 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 used to be for compatibility also, but I decided it was a
% convenient abbreviation, so \mfpic{} uses it now.
%    \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.
%
% In \DescribeRoutine{textoval}\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}\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}\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} a speck
% bigger to ensure that the same is true of \gbc{2*(nottoosmall/2)}.
%
% 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}
%
% It is useful to have a \DescribeRoutine{degrees}command to convert
% from radians to degrees and one to \DescribeRoutine{radians}convert
% from degrees to radians. Thus \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 =  (acos X)/radian  enddef;
vardef invsin primary X =  (asin X)/radian  enddef;
vardef invtan primary X =  (atan X)/radian  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;
def log =  ln  enddef;
vardef logbase (expr B) primary X =  (mlog X)/(mlog B)  enddef;
def logtwo =  logbase( 2)  enddef;
def logten =  logbase(10)  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} and
% \RoutineIndex{sgn}\gbc{sgn}.
%    \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;

%    \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}
%
% \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.
%    \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}
% This 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}
% This 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}\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 eliminate that parameter from
% \gbc{coloraddon}.
%
% \DescribeRoutine{_orto}
% This version of \gbc{orto} 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}. \DescribeRoutine{_subto}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
% 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 two nested loops, 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 me.)
%    \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{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}\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.
%
% \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;

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 (subpath ((_j-1)/_s, _j/_s) of g) (ll, ur);
  endfor
%</MF>
  if showbbox:  noclip ( safedraw rect (ll, ur) );  fi
enddef;

%<*MF>
numeric bbox_split;  bbox_split := 2;
def ctrlsbbox (expr p) (suffix ll, ur) =
  ll := pairmin ( pairmin (ll, post0 (p)), pre 1 (p) );
  ur := pairmax ( pairmax (ur, post0 (p)), pre 1 (p) );
enddef;

%</MF>
%    \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.
%
% \DescribeRoutine{xlimit}
% \gbc{xlimit(x)} returns a value of true if the path \gbc{g} doesn't
% cross the vertical line at \gbc{x}.
% \DescribeRoutine{ylimit}\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 implements 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 autorounding stack. I'd never heard of this stack 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).
%    \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}
% This 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}\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}
% Hatch 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{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.
%    \begin{macrocode}
vardef gettime (suffix arr, ct) (expr lngth) =
  setnumeric (_gtl) emax (arr[ct], emin (arr[arr], lngth));
  setsplit (_s) segment_split;
  forever:  exitif ( (arr[ct] <= _gtl) and (_gtl <= arr[ct+1]) );
    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}
% \gbc{dotted} is implemented by calling \gbc{doplot} with \gbc{dotpath}
% 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}
% These place 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}
% \gbc{dosymbols} uses identical code twice (once to clear, once to draw),
% so 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}
% \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}\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 axisline.x =  (xneg + laxis, 0)--(xpos - raxis, 0)  enddef;
vardef axisline.y =  (0, yneg + baxis)--(0, ypos - taxis)  enddef;
vardef axisline.l =  axisline.y shifted (xneg + laxis, 0)  enddef;
vardef axisline.b =  axisline.x shifted (0, yneg + baxis)  enddef;
vardef axisline.r =  axisline.y shifted (xpos - raxis, 0)  enddef;
vardef axisline.t =  axisline.x shifted (0, ypos - taxis)  enddef;

vardef axis@# (expr len) =  headpath (len, 0, 0) axisline@#  enddef;

vardef borderrect =
  rect((xneg+laxis,yneg+baxis),(xpos-raxis,ypos-taxis))
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}
% \RoutineIndex{lmarks}
% \RoutineIndex{bmarks}
% \RoutineIndex{rmarks}
% \RoutineIndex{tmarks}
% And now the specialized command for each axis. Inside and outside
% really make no sense for the $x$- and \DescribeRoutine{ymarks}$y$-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, (xneg + laxis, 0), up)    enddef;
def bmarks =  axismarks ( 90, btick, (0, yneg + baxis), right)  enddef;
def rmarks =  axismarks ( 90, rtick, (xpos - raxis, 0), up)    enddef;
def tmarks =  axismarks (-90, ttick, (0, ypos - taxis), right)  enddef;

%    \end{macrocode}
%
% \DescribeRoutine{vargrid, 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}\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 ((xneg + laxis)/xsp) upto floor ((xpos - raxis)/xsp):
  for m = ceiling ((yneg + baxis)/ysp) upto floor ((ypos - taxis)/ysp):
    picdot (gridpic, gdot, zconv ((n*xsp, m*ysp)));
  endfor
  endfor
  coloraddon (pointcolor, gridpic);
enddef;
def vgrid =  vargrid  enddef;

%    \end{macrocode}
%
% This is more what I think of when I hear `grid', but the name was
% already taken. \DescribeRoutine{hgridlines}\gbc{hgridlines} draws
% horizontal lines through the same points where \gbc{grid} would draw
% dots, and \DescribeRoutine{vgridlines}\gbc{vgridlines} draws only
% vertical lines through the same points. Finally,
% \DescribeRoutine{gridlines}\gbc{gridlines} draws both.
%    \begin{macrocode}
def hgridlines (expr ysp) =
  for n = ceiling ((yneg + baxis)/ysp) upto floor ((ypos - taxis)/ysp):
    safedraw zconv ((xneg + laxis, n*ysp)--(xpos - raxis, n*ysp));
  endfor
enddef;
def vgridlines (expr xsp) =
  for n = ceiling ((xneg + laxis)/xsp) upto floor ((xpos - raxis)/xsp):
    safedraw zconv ((n*xsp, yneg + baxis)--(n*xsp, ypos - taxis));
  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 ((xneg + laxis)/xsp) upto floor ((xpos - raxis)/xsp):
  for m = ceiling ((yneg + baxis)/ysp) upto floor ((ypos - taxis)/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 := xneg + laxis;  _B := xpos + raxis;
  _C := yneg + baxis;  _D := ypos + taxis;
  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 (_A < _X) and (_X < _B) and (_C < _Y) and (_Y < _D)
  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}
% \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}\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}.
% And \DescribeRoutine{plrpatch}\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}. The command
% \DescribeRoutine{polargrid}\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}
% This 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 it
% 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 (xneg < 0) and (xpos > 0) and (yneg < 0) and (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 (xneg < 0) and (0 < xpos):
        rmin := emin (abs(yneg), abs(ypos));
      elseif (yneg < 0) and (0 < ypos):
        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).
%
% \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 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) and (B = C) :  60
  elseif (A=B) or (A=C) :  90
  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 is also used with an array of paths instead of points.
%    \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} prints a warning and sets the
% array 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;
  pair _tu[];  _tu0 := origin;
  for _a = t:  _tu[incr _tu] := _tu[_tu - 1] + _a;  endfor
  if _tu = 0:  NoPoints("turtle", _tu);  fi
  mkpoly (false, _tu)
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
% uses 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 fltness
% 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. The first,
% \DescribeRoutine{ccurve}\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);  mksmooth (tens, cyclic, _tc)
enddef;

def ccurve =  tccurve (default_tension)  enddef;
vardef tccurve (expr tens, cyclic) (text t) =
  setuniquepairs (_tcc) (t);  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}
% \gbc{mkbezier} takes an array argument and produces either an open or
% cyclic path with a given tension. \DescribeRoutine{bezier}\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. The \mfc{for}-loop ends on the last
% odd index. If \gbc{pts} is odd and \gbc{cyclic} is true we need to
% manufacture another control point for the connection back to the
% start. We take that to be a repetition of the last point. If \gbc{pts}
% is even and \gbc{cyclic} is false then the loop ends at \gbc{pts-1}.
% Again we need to pick a control to connect to the last point, and again
% we use that last point as the control. In these two cases the last
% link is a straight line.
%
% \DescribeRoutine{qbezier}
% The \gbc{qbezier} command takes a list of points and creates an array
% from then before calling \gbc{mkqbezier}.
%    \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);  fi
  if _qbz=1:  onepointpath (cyclic, _qbz1)
  else:
    if (cyclic and odd _qbz) or (not cyclic and even _qbz):
      _qbz[incr _qbz] := _qbz[_qbz-1];
    fi
    mkqbezier (cyclic) (_qbz)
  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 we
% compute the controls locally and 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 now 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.
%
% \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, which \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 > 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. For
% \DescribeRoutine{relaxed_spline_eqns}relaxed splines they force
% the second derivative to be 0 at the first and last point.
%
% The macro \gbc{mksplinepath} simply assembles the points and controls
% previously computed into a path.
%
% \DescribeRoutine{mkspline}\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 $x\sb j < x < x\sb
% {j+1}$, making sure that the resulting graph is as smooth as possible at
% the points $x\sb j$.
%
% 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 be no problem.
%
% 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 infinity, 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 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:  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 this, 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}\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, ang;  full := signof (rad) 360;
  if 2*abs(rad) > abs(begpt - endpt):
    ang := if not small:  full -  fi  2*asin (abs(begpt-endpt)/(2rad));
  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 produces the circle with a given center passing through a
% given point.
% \DescribeRoutine{circleppp}
% The second produces the circle passing through three given points.
% \DescribeRoutine{circlepps}
% The third 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 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}\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}
%
% \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 supposedly distinct points on the path and
% finds the intersection of the perpendicular bisectors of two chords.
%    \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}
% This produces the circle inside the triangle that is tangent to all
% three sides. 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}
% This 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{barycenter}
% This is the average of the three corners of the triangle, or 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 vasious 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}
%
%
% \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}This 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} This 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 (a < U) and (U < b)
      and (c < V) and (V < d)
  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}
%
% \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. It has a variant \DescribeRoutine{sclosedt}\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}.
%    \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.
%
% \DescribeRoutine{__dir}
% \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}
%
% \DescribeRoutine{tailpath}\RoutineIndex{colortailpath}
% This 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}
% This 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}. The first, \DescribeRoutine{signeddeviate}
% \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 $(2^{-w}, 2^w)$. The third,
% \DescribeRoutine{polardeviate}\gbc{polardeviate R} produces a pair
% whose polar coordinates are separately uniformly distributed, the radius
% over the interval $(0, R)$ the angle over $(0,360)$. The last,
% \DescribeRoutine{xydeviate}\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. We wouldn't want to differently
% shift the two (equal) endpoints of a trivial segment, making it
% nontrivial.
%    \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{Parallelling a path}
%
% \DescribeRoutine{parasegment}
% This creates a path parallel to a given cubic Bezier 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]);
%      __type := __type + 1;
      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 path 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;

% As arrow heads, these have their "tips" at their center.
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 usual rule for
% signs of products.
%    \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*m/gcd(n, m)
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{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.
%
% \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}
def setdatacolors (text lst) =
  save __type;  color __type[];
  __type := 0;
  for _itm = lst:
    if (known _itm) and (color _itm):
      __type[__type] := _itm;
      next __type;
    else:  GBwarn "Improper color in setdatacolors().";
    fi
  endfor
  if __type > 1:
    save colortype;  colortype := __type;
    color colortype[];
    for _j = 0 upto colortype - 1:
      colortype[_j] := __type[_j];
    endfor
  else:
    SetdataWarn "colors";
  fi
enddef;
def getcolor expr n =  colortype[n mod colortype]  enddef;

color dBlue, dOrange, dGreen, dMagenta, dCyan, dYellow;
dBlue := 0.80blue + .2white;
dOrange := 0.66yellow + .34red;
dGreen := 0.80green;
dMagenta := 0.85magenta;
dCyan := 0.85cyan;
dYellow := 0.85yellow;

numeric colortype;  color colortype[];
def defaultcolors =
  setdatacolors(black, red, dBlue, dOrange, dGreen,
    dMagenta, dCyan, dYellow);
enddef;
defaultcolors;

%    \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 repeat
% the definition of \gbc{cmyk} here.
%    \begin{macrocode}
%<*dvips>
if unknown grafbaseversion:
  vardef snapto expr t =
    if unknown t:            0
    elseif not (numeric t):  0
    elseif t < 0:            0
    elseif t > 1:            1
    else:                    t
    fi
  enddef;
  vardef cmyk (expr c, m, y, k) =
    (snapto 1-c-k, snapto 1-m-k, snapto1-y-k)
  enddef;
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}
color 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
