% \iffalse
% Copyright 2002--2003, Daniel H. Luecking
%
% Mfpic consists of the 3 files mfpic.dtx, grafbase.dtx and mfpic.ins
% and the 5 files they generate: mfpic.tex, mfpic.sty, grafbase.mf,
% grafbase.mp, and dvipsnam.mp.
%
% Mfpic may be distributed and/or modified under the conditions of the
% LaTeX Project Public License, either version 1.2 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 1.2 or later is part of all distributions of LaTeX version
% 1999/12/01 or later.
%
% Documentation, examples, and ancillary files are separate and not
% covered by this license. See readme.1st for a complete list. See
% individual files for any copyright and license restrictions.
%
% With respect to the proposed draft LPPL-1.3: mfpic has maintenance
% status "maintained". The Current Maintainer is Daniel H. Luecking. There
% are several Base Interpreters: TeX, LaTeX, MetaPost and Metafont.
%
%<*driver>
\documentclass[draft]{ltxdoc}
\usepackage{docmfp}
\def\fileversion{0.7a beta}
\def\filedate{2004/04/16}

\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{metafont}}
\def\MP{\prog{metapost}}
\def\CMF{\prog{Metafont}}
\def\CMP{\prog{Metapost}}
\def\opt#1{{\sffamily\upshape#1}}
\def\env#1{{\mytt#1}}
\let\file\env
\let\mfc\env
\let\gbc\env
\renewcommand\{{\char`\{}
\renewcommand\}{\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\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}
\makeatother

\def\pdfTeX{\textrm{pdf\kern.04em\TeX}}
\def\pdfLaTeX{\textrm{pdf\kern.06em\LaTeX}}
\def\ConTeXt{\textrm{Con\kern-.16em\TeX\kern-0.06em t}}
\def\PiCTeX{\textrm{P\kern-.13em\lower.3ex\hbox{I}C\TeX}}

\title{The \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}
\DisableCrossrefs
\CodelineIndex
\AlsoImplementation

\begin{document}
  \DeleteShortVerb{\|}
  \DocInput{grafbase.dtx}
\end{document}
%</driver>
%\fi
%
% \CheckSum{1}
% \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
% \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 alike more than unlike, we document both here.
%
% This file documents the \grafbase{} source code. The user manual for
% \mfpic{} is \file{mfpicdoc.tex}.
% \end{abstract}
%
% \StopEventually{\PrintIndex}
% \tableofcontents
%
%
% \section{Introduction}\label{intro}
%
% \subsection{Identification and checks}\label{checks}
%
% We start with identifying information. Then we detect if grafbase was
% loaded already, but we don't do anything about it except write a
% message.
%
% The \gbc{grafbaseversion} is required to match the definition of
% \cs{mfpicversion} in \file{mfpic.tex}. The output file written by the
% \mfpic{} macros includes a test that these are the same, but that would
% fail to catch a new \grafbase{} with an old \mfpic. So we also put a test
% here, and 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.
%
% 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.
%
% \DescribeRoutine{GBmsg}
% \DescribeRoutine{GBerrmsg}
% These are used fairly consistently and identify the source of the
% message delivered as being \gbc{"Grafbase"}. The \gbc{GBerrmsg} takes
% care of both the error message and the \mfc{errhelp} string.
%    \begin{macrocode}
%<*MF|MP>
string fileversion, filedate;
fileversion := "0.7a beta"; filedate := "2004/04/16";
def GBmsg expr s = message "Grafbase: " & s; enddef;
def GBerrmsg (expr s) expr t =
  errhelp t;
  errmessage "Grafbase: "& s;
  errhelp "";
enddef;
if (known grafbaseversion) or (known grafbase) :
  GBmsg "You have loaded grafbase more than once! " &
    "Please make sure that it is loaded only once.";
  endinput;
fi

boolean grafbase; grafbase := true;
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:
    GBerrmsg ("version mismatch")
      "The installation may be broken: mfpic and grafbase " &
      "versions do not match.";
  fi
enddef;

checkversions (70);

%    \end{macrocode}
% We try to make sure that the macros of \file{plain.mf} or
% \file{plain.mp} 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}
%
% 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 Maurits Cornelis Escher :  METAPOST := true;
else:  METAPOST := false;
fi

%<*MF>
if METAPOST :
  GBerrmsg ("wrong compiler")
    "You may have input to Metapost a file designed for Metafont. " &
    "Instead of the file grafbase.mf, Metapost should be using " &
    "grafbase.mp. Make sure Metapost can find it.";
fi
%</MF>
%<*MP>
if not METAPOST :
  GBerrmsg ("wrong compiler")
    "You may have input to Metafont a file designed for Metapost. " &
    "Instead of the file grafbase.mp, Metafont should be using " &
    "grafbase.mf.  Make sure the extension was not changed.";
fi
%</MP>

%    \end{macrocode}
%
% \DescribeRoutine{GBdebug}
% \DescribeRoutine{GBenddebug}
% The \gbc{debug} flag is for developers, who should set it before
% inputing \file{grafbase}. 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 (unknown debug) or (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 =
  t; message t;
enddef;

%    \end{macrocode}
%
% \subsection{Setting up the font}\label{font}
%
% Font-related housekeeping is for \MF{} only. \MF{} only produces
% fonts, so we have to define the variables it thinks are needed for
% fonts.
%
% We intercept the \mfc{mode} variable before \mfc{mode_setup} can set
% \mfc{proof} mode. We used to set \mfc{mode := cx} if it was unknown,
% then for a while we just issued an error message; in this version we
% try \mfc{ljfour}.
%
% 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 ("unknown Metafont mode")
    "Please use \mode:=localfont; or a mode known on your " &
    "system. If you continue, ljfour mode will be tried.";
  mode := lfjour;
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# := 1pt;
%<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 has to be
% after \mfc{mode_setup}, which sets \mfc{warningcheck := 1}. Also don't
% complain if a clockwise path is filled (\MF).
%    \begin{macrocode}
%<MF>interim turningcheck := 0;
interim warningcheck := 0;
%    \end{macrocode}
%
% \subsection{Initializations}\label{init}
%
% Picture size variables would normally be set by a user for each
% picture, or by \mfpic, but we give them default values anyway.
%    \begin{macrocode}
numeric unitlen, xscale, yscale, xneg, xpos, yneg, ypos;

%<MF>unitlen := 1 bp#;
%<MP>unitlen := 1 bp;
xscale := 7.2;  % (xscale * unitlen) = 1/10 inch
yscale := 7.2;  % (yscale * unitlen) = 1/10 inch
xneg := 0; xpos := 10;
yneg := 0; ypos := 10;

%    \end{macrocode}
%
% We support both degrees and radians for angles.  In \MF, one degree is
% the unit of angle. One radian is $180/\pi$ degrees. A user can say
% \gbc{90} or \gbc{90deg} or \gbc{pi/2*radian} for the same effect.
%    \begin{macrocode}
newinternal radian, pi, deg;
deg := 1; pi := 3.14159;
radian := 180/pi;

%    \end{macrocode}
%
% \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.)

% 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.
%
% 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}
%
% 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, \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}. 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.
%
% 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( save ClipOn; boolean ClipOn; ClipOn := false; t)
enddef;

%    \end{macrocode}
%
% 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 begin trying to merge the format of mfpic output files by defining
% \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 between nonwhite (black, ${}<1$) or
% white (${}=1$). For filling commands we will allow levels in between,
% and fill with an approximation using a version of \gbc{shade}
%    \begin{macrocode}
%<*MF>
let color = numeric; color black, white;
black := 0; white := 1;
def _wc_ = killtext enddef;
%</MF>
%<MP>def _wc_ = withcolor enddef;

%    \end{macrocode}
%
% 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, fillcolor, drawcolor, 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.
%
% \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}
% \DescribeRoutine{forceclr}
% These, too, are nearly redundant, but they convert numerics to gray, and
% convert other non-color variables and unknown color variables to
% black. The difference between \gbc{named} and \gbc{forceclr} is that the
% former requires a suffix parameter, 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) = forceclr (c) enddef;
vardef forceclr (expr c) =
  if unknown c     :  black
  elseif 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 standard example of an array. It is 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 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 \mfpic's \cs{plr} writes.
%    \begin{macrocode}
vardef list (suffix v) (text lst) =
  v := 0;  for _itm = lst: v[incr v] := _itm; endfor
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{knownarray}
% 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}.
%    \begin{macrocode}
vardef knownarray suffix arr =
  save _kna; boolean _kna;
  _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];
    endfor
  fi
  _kna
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{copyarray}
% We only need this once, but it makes the code much more readable. It
% simply steps through an array and copies the values into another array.
%    \begin{macrocode}
def copyarray(suffix from, to) =
  to := 0;
  for _idx = 1 upto from:
    to[incr to] := from[_idx];
  endfor
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, and currently
% is unused. Though it could have been 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 p;}'' in this
% macro, but ran into a problem once when \mfc{p} 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.
%    \begin{macrocode}
def textpairs (suffix p) (text t) =
  numeric p; pair p[]; list (p) (t);
enddef;

%    \end{macrocode}
% \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.
%
% \DescribeRoutine{floorpair}
% \DescribeRoutine{ceilingpair}
% \DescribeRoutine{hroundpair}
% \gbc{floorpair}, \gbc{ceilingpair} and \gbc{hroundpair} use it with
% \gbc{proc} equal to \mfc{floor}, \mfc{ceiling} and \mfc{hround}.
% The last one is not defined in the \MP{} version of \grafbase{}
% because \mfc{hround} is not defined (only \mfc{round}, which already
% works on pairs). Actually, none of these is used any longer in
% \file{grafbase.mp}.
%    \begin{macrocode}
vardef chpair (text proc) (expr p) =
  (proc (xpart p), proc (ypart p))
enddef;

vardef floorpair (expr p) = chpair (floor) (p) enddef;
vardef ceilingpair (expr p) = chpair (ceiling) (p) enddef;
%<MF>vardef hroundpair (expr p) = chpair (hround) (p) enddef;

%    \end{macrocode}
%
% \DescribeRoutine{emin}
% \DescribeRoutine{emax}
% These are more efficient versions of \prog{plain}'s \mfc{min} and
% \mfc{max}: avoiding a \mfc{for} loop when only two values are compared.
%
% \DescribeRoutine{pairmin}
% \DescribeRoutine{pairmax}
% \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
% \gbc{pairmax} is analogous, producing the maximum.
%
% \DescribeRoutine{minpair}
% \DescribeRoutine{maxpair}
% The \gbc{minpair} macro returns the pair comprising the minimum $x$ and
% minimum $y$ coordinates of all pairs in the array \gbc{p[]}, where
% \gbc{p} itself is a numeric count of the members in \gbc{p[]}.
% \gbc{maxpair} is analogous. These operate by repeatedly calling
% \gbc{pairmin} or \gbc{pairmax}.
%    \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) =
  save _mp; pair _mp; _mp := p1;
  for _idx = 2 upto p - 1 :
    _mp := pairmin(_mp, p[_idx]);
  endfor
  pairmin (_mp, p[p])
enddef;
vardef maxpair (suffix p) =
  save _mp; pair _mp; _mp := p1;
  for _idx = 2 upto p - 1: _mp := pairmax(_mp, p[_idx]); endfor
  pairmax (_mp, p[p])
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 be 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 is either independent of scale, or
% scales in a non-obvious 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).
%
% We therefore have two transforms: \gbc{vtr} is the \emph{vector} or
% linear transform for pair quantities that remain invariant under shifts,
% and \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).
%
% \mfc{currenttransform} (via the macro \mfc{.t_}, defined by
% \mfc{mode_setup}) takes care of the aspect ratio.
%
% \gbc{charwd} and \gbc{charht} are sharped coordinates defined by the
% startup code \gbc{beginmfpic}, and \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: ";
      show ztr;
      >> "vtr: ";
      show vtr;
    GBenddebug;
  fi
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{zconv}
% \DescribeRoutine{invzconv}
% The macro \gbc{zconv} converts a variety of expressions from graph to
% device coords. The expressions include pairs, paths, and transforms.
% This is an affine transform. The inverse, \gbc{invzconv}, converts a
% variety of expressions from device to graph coordinates.
%
% \DescribeRoutine{vconv}
% \DescribeRoutine{invvconv}
% The vector version, \gbc{vconv}, converts a vector \gbc{v} from graph to
% device coordinates. This is a linear (ie, vector) transform. Finally,
% \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}
%
% \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 (see \gbc{image} below, and the
% \gbc{tile} macro) by redefining it.
%
% \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}.
%    \begin{macrocode}
def active_plane = currentpicture enddef;

%<*MF>
vardef image(text t) =
  save currentpicture; picture currentpicture;
  currentpicture := nullpicture;
  t;
  currentpicture
enddef;

%</MF>
%    \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{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{current_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{current_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;
overlaylabels = 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>
  save current_labels; picture current_labels;
  current_labels := nullpicture;
  save labelbb; pair labelbb.ll, labelbb.ur;
  labelbb.ll := labelbb.ur := origin;
%</MP>
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{mfpicenv}
% \DescribeRoutine{endmfpicenv}
% For compatibility with older \file{graphbase.mf} (for
% \prog{fig2dev}'s \file{genmf.c}). Actually, I have no idea if
% \prog{fig2dev} even works with the current \mfpic.
%    \begin{macrocode}
def mfpicenv = enddef;
def endmfpicenv = enddef;
%    \end{macrocode}
% \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 bounds (expr a, b, c, d) =
  xneg := a; xpos := b;
  yneg := c; ypos := d;
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{setvariable}
% This is mainly to save space in \mfpic-generated files.
def setvariable (text kind) (suffix name) (expr value) =
  save name; kind name; name := value;
enddef;

%    \begin{macrocode}
%    \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_; numeric 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
%<*MF>
  if proofing>0 : makebox(proofrule); fi
  chardx := w_;     % desired width of character in pixels
%</MF>
  DoClip (active_plane);
  if clipall : clipto (active_plane) rect(origin, (w_, h_)); fi
  if showbbox : noclip ( safedraw rect (origin, (w_, h_)) ); fi
%<*MF>
  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, just set the bounding box to the
% coordinates determined by \gbc{w_} and \gbc{h_}
%    \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
% contexts. 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}
% If \gbc{overlaylabels} was true during a \gbc{newgblabel} command,
% then \gbc{current_labels} contains that labels. We add them now, on top
% of the picture. This might also extend the bbox, but that is an effect
% we want to achieve.
%    \begin{macrocode}
    addto active_plane also current_labels;
  endfig;
%</MP>
enddef;

%    \end{macrocode}
%
%
% \section{Text}\label{text}
%
% In the \MP{} version, \gbc{label_adjust} and \gbc{label_sep} are the
% equivalent of \mfpic's \cs{tlabeloffset} and \cs{tlabelsep}. 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 for the label,
% 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}).
%    \begin{macrocode}
pair label_adjust; label_adjust := (0, 0);
numeric label_sep;    label_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 fillowed 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.)  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 = killtext enddef;

%    \end{macrocode}
%
% \subsection{Placement of text, \MP{} only}\label{placement}
%
% \DescribeRoutine{newgblabel}
% \DescribeRoutine{gblabel}
% 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 first two parameters could easily
% be condensed into one if \mfpic{} support were all that was required,
% however I thought it best to generalize. The first two parameters
% \gbc{hf} and \gbc{vf} are numeric. The \gbc{hf} represent the fraction of
% the text that lies left of the point where the text is placed. Normally,
% \gbc{vf} represents the fraction of text that lies below the point, but
% if the third parameter is \mfc{true}, this fraction is relative to the
% baseline (i.e., the depth is ignored). Currently we only use this with
% \gbc{vf = 0} to get placement on the baseline. (Actually, \mfpic{} only
% ever uses values of 0, .5 and 1 for \gbc{hf} or \gbc{vf}).
%
% The macro \gbc{newgblabel} takes 6 parameters. The first 3, as explained
% above, effect the justification of the text (location of the point of
% placement relative to the label). They correspond to the optional
% parameter of \cs{tlabel} in \mfpic{} as follows:
% \begin{itemize}
% \item \gbc{hf} determines horizontal position: 0 = \texttt{l},
%   .5 = \texttt{c}, and  1 = \texttt{r}.
% \item \gbc{vf} and \gbc{BL} determine vertical position. For placement
%   option \texttt{B}, \gbc{vf} = 0 and \gbc{BL} is \mfc{true}. For the
%   rest, \gbc{BL} is \mfc{false} and \gbc{vf} corresponds as follows:
%   0 = \texttt{b}, .5 = \texttt{c} and 1 = \texttt{t}.
% \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 extended by \gbc{label_sep} in
% all directions by \gbc{labeldims}, then 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{current_labels} and
% added to \gbc{active_plane} at \gbc{endmfpic}. Otherwise, it is added
% directly to \gbc{active_plane} and and the \gbc{labelbb} are adjusted.
%
% We also use \gbc{ref_shift} in \MF{} since the curves that surround text
% require it.
%
% 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 for \MF{} to allow \mfpic's \file{.mp} files to be usable with
% minimal changes.
%    \begin{macrocode}
%<MF>def newgblabel (expr hf, vf, BL, r) (text s) (text pts) = enddef;
%<*MP>
vardef newgblabel (expr hf, vf, BL, r) (expr s) (text pts) =
  save _lab, _ll, _ur; picture _lab; pair _ll, _ur;
  _lab :=
    if picture s    :  s
    elseif string s :  s infont defaultfont scaled defaultscale
    elseif path s   :  picpath (s)
    else            :  nullpicture
    fi;
  labeldims (origin, _lab) (_ll, _ur);
  _lab := thegblabel(ref_shift (hf, vf, BL, _ll, _ur), r, _lab);
  save _b; pair _b;
  for _itm = pts :
    _b := zconv(_itm);
    if overlaylabels :
      addto current_labels also _lab shifted _b _wc_ tlabelcolor;
    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
enddef;

% Assumes a+b=1 and either c+d=1 or c=d=0:
vardef gblabel (expr a, b, c, d, r) (expr s) (text t) =
  newgblabel (b, d, (c = 0) and (d = 0), r) (s) (t);
enddef;

%</MP>
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}\label{decorating}
%
% \DescribeRoutine{textrect}
% \DescribeRoutine{textoval}
% \DescribeRoutine{textellipse}
% 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 the first one, the path is a
% rectangle with 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 \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
% \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{label_sep}. Note that this means
% a rectangle with rounded corners will have larger height and width than
% one without.
%
% The placement of each path is: centered at the point given in the third
% parameter \gbc{loc}, shifted by the vector specified in \gbc{label_adjust}.
%
% 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}
% \DescribeRoutine{textovalx}
% \DescribeRoutine{textellipsex}
% These are extended versions of the previous three. They will now be
% able adjust the position of the path in the same manner as \gbc{newgblabel}
% does the text. In fact, they calculate the position in exactly the same
% manner as that macro, and the first 4 parameters encode that in the same
% way.
%
% \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{labeldims}) 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.
%
% \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;
  labeldims (xy, lbl) (ll, ur);
  _r :=
    if boolean rad :
      if rad :  emin (xpart (ur-ll), ypart (ur-ll))/sqrt(2)
      else:     0
      fi
    elseif numeric rad :  rad
    else:                 0
    fi;
  if _r = 0 :
    f :=  rect(ll, ur);
  else:
    save p, q;
    pair p[];  path q;
    p1 := ur - _r*dir(45);    % center of upper right arc
    p3 := ll + _r*dir(45);    % lower left
    p2 := (xpart p3, ypart p1); % upper left
    p4 := (xpart p1, ypart p3); % lower right
    q  := quartercircle scaled 2_r;
    if _r > 0:
      f  := (q shifted p1) -- (q rotated 90 shifted p2)
            -- (q rotated 180 shifted p3)
            -- (q rotated -90 shifted p4)
            -- cycle;
    else:
      f  := (q shifted p1) -- (q rotated -90 shifted p4)
            -- (q rotated 180 shifted p3)
            -- (q rotated 90 shifted p2)
            -- cycle;
    fi
  fi
  invvconv(thegblabel(ref_shift(a, b, c, ll, ur), rot, f)) shifted loc
enddef;

%    \end{macrocode}
%
% As the coding of \gbc{textoval} and \gbc{textellipse} was refined, it
% turned out that each refinement in one suggested a similar change in the
% other. In the end the two differed only in two lines, so now both call
% another macro \gbc{xellipse}.
%    \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*hh/ww}, 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;
    labeldims (xy, lbl) (ll, ur);
    cc := .5[ll, ur]; % center
    (ww, hh) = ur - cc;
    if (ww = 0) or (hh = 0) : % make a line:
      f = (ll--ur);
    else:
      save aa, bb, mm;
      mm := if aspect : ww/hh*mult else: mult fi;
      aa := ww ++ hh*mm;
      bb := aa/mm;
      f := ellipse(cc, aa, bb, 0);
    fi
    invvconv(thegblabel (ref_shift (a, b, c, ll, ur), r, f)) shifted loc
  fi
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{labeldims}
% 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}
% then expands the bbox by \gbc{label_sep}.
%    \begin{macrocode}

def labeldims (expr xy, lbl) (suffix ll, ur) =
  if pair lbl :
    ll := xy; ur := lbl;
  else:
%<MF>    ll := ur := (0, 0);
%<*MP>
    save _lbl; picture _lbl;
    _lbl :=
      if picture lbl    :  lbl
      elseif string lbl :
        lbl infont defaultfont scaled defaultscale
      elseif path lbl   :  picpath (lbl)
      else              :  nullpicture
      fi;
    ll := llcorner _lbl;
    ur := urcorner _lbl;
%</MP>
  fi
  ll := ll - label_sep*(1, 1);
  ur := ur + label_sep*(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.
%
% 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)}.
%
% 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 to be the smallest number whose reciprocal is a
% usable number. (\mfc{epsilon} is the smallest possible positive number
% in \MF.)
%
% We set \gbc{secd x = 1/(cosd x)} unless \gbc{cosd x} is less than ``really
% small'', then we set it equal to \gbc{1/reallysmall}. We do a similar
% thing with \gbc{cscd}.
%^^A
% \DescribeRoutine{TruncateWarn}
% (When such a substitution happens \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 nottoosmall; nottoosmall := eps/2 + 2epsilon;
newinternal reallysmall; reallysmall := 3epsilon;
def signof (expr X) = if X < 0 : - fi enddef;
def TruncateWarn expr s =  GBmsg s & " too large; truncating";  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}.
%
% \DescribeRoutine{secd}
% \DescribeRoutine{tand}
% \DescribeRoutine{cscd}
% \DescribeRoutine{cotd}
% We define \gbc{secd}, one of the simplest, to include an ``out of range''
% test (which also prevents division by 0). Then \gbc{tand} can make use
% of it without any division. We do the same with \gbc{cscd} and
% \gbc{cotd}.
%    \begin{macrocode}
vardef secd primary X =
  save temp; temp := cosd(X);
  if abs(temp) < reallysmall :
    TruncateWarn "Secant";
    temp := signof (temp) reallysmall;
  fi
  1/temp
enddef;
vardef tand primary X =  sind(X)*secd(X)  enddef;

vardef cscd primary X =
  save temp; temp := sind(X);
  if abs(temp) < reallysmall :
    TruncateWarn "Cosecant";
    temp := signof(temp) reallysmall;
  fi
  1/temp
enddef;
vardef cotd primary X =  cosd(X)*cscd(X)  enddef;

%    \end{macrocode}
% \DescribeRoutine{acos}
% \DescribeRoutine{asin}
% \DescribeRoutine{atan}
% These are the inverse functions, which return an angle in degrees.
%    \begin{macrocode}
vardef acos primary X =  angle (X, 1 +-+ X)  enddef;
vardef asin primary X =  angle (1 +-+ X, X)  enddef;
vardef atan primary X =  angle (1, X)  enddef;

%    \end{macrocode}
% \DescribeRoutine{sin}
% \DescribeRoutine{cos}
% \DescribeRoutine{tan}
% \DescribeRoutine{cot}
% \DescribeRoutine{sec}
% \DescribeRoutine{csc}
% Now the trig functions that take angles in radians.
%    \begin{macrocode}
vardef sin primary X =  sind (X*radian)  enddef;
vardef cos primary X =  cosd (X*radian)  enddef;
vardef tan primary X =  tand (X*radian)  enddef;
vardef cot primary X =  cotd (X*radian)  enddef;
vardef sec primary X =  secd (X*radian)  enddef;
vardef csc primary X =  cscd (X*radian)  enddef;

%    \end{macrocode}
% \DescribeRoutine{invsin}
% \DescribeRoutine{invcos}
% \DescribeRoutine{invtan}
% And the inverses 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}
% \DescribeRoutine{exp}
% \DescribeRoutine{ln}
% \DescribeRoutine{log}
% \DescribeRoutine{logbase}
% \DescribeRoutine{logtwo}
% \DescribeRoutine{logten}
% Here we define the standard exponential function. (The \MF{} function
% \mfc{mexp} has the unusual base $e^{1/256}$ to avoid overflow.) The
% inverse of \gbc{exp} is the natural logarithm (\gbc{ln} or \gbc{log}).
% We also have the general base logarithm \gbc{logbase} and its two
% special instances \gbc{logtwo} and \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}
% \DescribeRoutine{Arg}
% \DescribeRoutine{Log}
% \DescribeRoutine{cis}
% \DescribeRoutine{zexp}
% \DescribeRoutine{sgn}
% \CMF's pair variables are a decent replacement for complex variables.
% These give some of the more basic functions of standard complex
% analysis.
%    \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(radian*T) enddef;
vardef zexp primary Z = (exp (xpart Z))*(cis(ypart Z)) enddef;
vardef sgn primary Z =
  if Z = origin :  origin  else:  unitvector Z  fi
enddef;

%    \end{macrocode}
% \DescribeRoutine{cosh}
% \DescribeRoutine{sinh}
% \DescribeRoutine{tanh}
% \DescribeRoutine{sech}
% \DescribeRoutine{csch}
% \DescribeRoutine{coth}
% The hyperbolic functions.
%    \begin{macrocode}
vardef cosh primary X =
  save temp; temp := 2 exp (-abs(X));
  if temp < reallysmall :
    TruncateWarn "Cosh";
    temp := reallysmall;
  fi
  1/temp + temp/4
enddef;

vardef sinh primary X =
  save temp; temp := 2 exp (-abs(X));
  if temp < reallysmall :
    TruncateWarn "Sinh";
    temp := reallysmall;
  fi
  signof (X) (1/temp - temp/4)
enddef;

vardef sech primary X =
  save temp; temp := exp(-(abs (X)));
  2temp/(1 + temp*temp)
enddef;

vardef tanh primary X =
  save temp; temp := exp(-2(abs (X)));
  signof (X) (1 - temp)/(1 + temp)
enddef;

vardef csch primary X =
  save temp; temp := exp(-(abs (X)));
  if abs(1 - temp*temp) < reallysmall :
    TruncateWarn "Csch";
    signof (X) 2temp / reallysmall
  else:
    signof (X) 2temp / (1 - temp*temp)
  fi
enddef;

vardef coth primary X =
  save temp; temp := tanh(X);
  if abs(temp) < reallysmall :
    TruncateWarn "Coth";
    temp := signof (temp) reallysmall;
  fi
  1/temp
enddef;

%    \end{macrocode}
% \DescribeRoutine{acosh}
% \DescribeRoutine{asinh}
% \DescribeRoutine{atanh}
% The inverses of some of the hyperbolic functions.
%    \begin{macrocode}
vardef acosh primary y =
  if y < 1 :
    GBerrmsg ("Undefined function: acosh " & decimal y)
      "If you proceed, a value of 0 will be used. " &
      "Expect more errors later.";
    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:
    GBerrmsg ("Undefined function: atanh " & decimal y)
      "If you proceed, a value of plus or minus infinity " &
      "will be used. Expect more errors later.";
    signof (y) infinity
  fi
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{polar}
% \DescribeRoutine{id}
% \gbc{polar} converts a polar coordinate pair $(r, \theta)$ to the
% corresponding rectangular coordinate pair.
%
% \gbc{id} returns its argument, which can be any expression of any type.
%    \begin{macrocode}
vardef polar (expr p) = (xpart p) * dir (ypart p) enddef;
def id (expr x) = x enddef;

%    \end{macrocode}
%
%
% \section{Coordinate systems and transformations}\label{systems}
%
% \DescribeRoutine{T_push}
% \DescribeRoutine{T_pop}
% \DescribeRoutine{bcoords}
% \DescribeRoutine{ecoords}
% We want to define a localization of the ``current transform''. To do
% this we define a LIFO stack of transforms \gbc{T_stack[]}, a pair of
% macros \gbc{T_push} puts its argument (a transform) on the stack, and
% \gbc{T_pop} pops it off into its argument (a transform variable name).
% We also define two localizing macros \gbc{bcoords} that pushes our
% \gbc{ztr} and \gbc{vtr} on the stack, and \gbc{ecoords} that pops them
% off.
%    \begin{macrocode}
transform 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); T_push (vtr) ) enddef;
def ecoords = hide ( T_pop (vtr); T_pop (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
% phrase which, were it to follow a path, would produce a transformed
% path. Knuth calls such a phrase a \emph{transformer}.
%    \begin{macrocode}
def apply_t (text Transformer) =
  ztr := identity Transformer transformed ztr;
  vtr := ztr shifted - zconv(origin);
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{xslant}
% \DescribeRoutine{yslant}
% \DescribeRoutine{zslant}
% \DescribeRoutine{xyswap}
% \DescribeRoutine{boost}
% And now we define some available transformers. 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 what zslant is about.
%    \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 a path 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, 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{rotatedpath}
% This returns the path rotated around point \gbc{p} by angle
% \gbc{th} in degrees.
%
% \DescribeRoutine{scaledpath}
% This returns the path scaled so that distances from the point
% \gbc{p} are multiplied by \gbc{s}.
%
% \DescribeRoutine{xslantedpath}
% This returns the path x-slanted with line $y = {}$\gbc{b}
% being the pivot rather than the x-axis.
%
% \DescribeRoutine{yslantedpath}
% This returns the path y-slanted with line $x = {}$\gbc{a}
% being the pivot rather than the y-axis.
%
% \DescribeRoutine{xscaledpath}
% This returns the path scaled so that vertical distances
% from the line $y={}$\gbc{a} are multiplied by \gbc{s}.
%
% \DescribeRoutine{yscaledpath}
% This returns the path scaled so that vertical distances
% from the line $x={}$\gbc{b} are multiplied by \gbc{s}.
%
% \DescribeRoutine{shiftedpath}
% This returns the path shifted by the vector (pair) \gbc{v}.
%
% \DescribeRoutine{reflectedpath}
% This returns the path relected about the line through the
% points \gbc{p} and \gbc{q}.
%
% \DescribeRoutine{xyswappedpath}
% Finally, this returns the path in which all coordinates
% 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 rotatedpath (expr p, th) expr f =
  f transformed vtr rotatedaround (p transformed vtr, th)
    transformed (inverse vtr)
enddef;
vardef scaledpath (expr p, s) expr f =
  f shifted -p scaled s shifted p
enddef;
vardef xslantedpath (expr b, s) expr f =
  f shifted (0, -b) slanted s shifted (0, b)
enddef;
def slantedpath = xslantedpath enddef;
vardef yslantedpath (expr a, s) expr f =
  f shifted (-a, 0) yslant s shifted (0, a)
enddef;
vardef xscaledpath (expr a, s) expr f =
  f shifted (-a, 0) xscaled s shifted (a, 0)
enddef;
vardef yscaledpath (expr b, s) expr f =
  f shifted (0, -b) yscaled s shifted (0, b)
enddef;
vardef shiftedpath (expr v) expr f = f shifted v enddef;
vardef reflectedpath (expr p, q) expr f =
  f transformed vtr
    reflectedabout (p transformed vtr, q transformed vtr)
    transformed (inverse vtr)
enddef;
vardef xyswappedpath expr f = f xyswap enddef;
vardef transformedpath (text Transformer) expr f =
    f Transformer
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{partialpath}
% \DescribeRoutine{gsubpath}
% 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. This 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}. \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 p; path p;
  p := zconv (f) scaled (1/unit_of_length);
  save cumlen, totlen, idx, ta, tb;
  totlen := makelengtharray(cumlen) p;
%    \end{macrocode}
% \gbc{idx} holds the current index into the array \gbc{cumlen[]}. The
% code of \gbc{gettime} is optimized for sorted lengths. If we always
% found \gbc{ta} first, we'd have to re-initialize \gbc{idx} in case
% \gbc{tb < ta}, i.e., search from the beginning again.
%    \begin{macrocode}
  idx := 0;
  if a <= b:
    ta := gettime (cumlen, idx) (a*totlen);
    tb := gettime (cumlen, idx) (b*totlen);
  else:
    tb := gettime (cumlen, idx) (b*totlen);
    ta := gettime (cumlen, idx) (a*totlen);
  fi
  subpath (ta, tb) of f
enddef;
vardef gsubpath (expr a, b) expr f = subpath (a, b) of f enddef;

%    \end{macrocode}
%
%
% \section{Picture level operations}\label{picture}
%
% 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.
%
% \subsection{Bitwise logical operations}\label{logical}
%
% 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 in \gbc{picneg} and \gbc{shadepic}.
%
% \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.
%    \begin{macrocode}
%<*MF>
def mono (suffix u) = cull u keeping (1, infinity); enddef;

%    \end{macrocode}
% \DescribeRoutine{andto}
% \DescribeRoutine{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}.
%    \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  save t; picture t;
    t := u; andto (t, v); t
  endgroup
enddef;

%    \end{macrocode}
% \DescribeRoutine{orto}
% \DescribeRoutine{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.
%    \begin{macrocode}
def orto (suffix u) (expr v) =
  mono (u); addto u also v; cull u keeping (1, 2);
enddef;
primarydef u picor v =
  begingroup  save t; picture t;
    t := u; orto (t, v); t
  endgroup
enddef;

%    \end{macrocode}
% \DescribeRoutine{xorto}
% \DescribeRoutine{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  save t; picture t;
    t := u; xorto (t, v); t
  endgroup
enddef;

%    \end{macrocode}
% \DescribeRoutine{subto}
% \DescribeRoutine{picsub}
% 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}.
%    \begin{macrocode}
def subto (suffix u) (expr v) =
  mono (u); addto u also -v; cull u keeping (1, 1);
enddef;
primarydef u picsub v =
  begingroup save t; picture t;
    t := u; mono (t); subto (t, v); t
  endgroup
enddef;

%</MF>
%    \end{macrocode}
%
% \subsection{Producing and modifying 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 has become 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, and is defined only so that \MP{} and \MF{}
% can share the same higher level code.
%    \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>
%    \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), since the higher level commands now
% implement the coloring operations.
%    \begin{macrocode}
vardef interior expr c =
  save v; picture v; v := nullpicture;
  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 =
  save _ints; picture _ints; _ints := nullpicture;
  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 =
%<MF>  andto (vt, interior c);
%<MP>  clip vt to c;
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.
%
% Having found out that \mfc{clipped} is a \MP{} primitive, I've
% changed the name to the uppercase version, but keep the lowercase
% version for now (backward compatibility). We save the primitive
% meaning in \gbc{clipped_}. We also define \gbc{clip} in \MF{} for
% backward compatibility.
%    \begin{macrocode}
vardef Clipped (suffix vt) expr c =
  save _Cl; picture _Cl; _Cl := vt; clipto (_Cl) c; _Cl
enddef;
%<MP>let clipped_ = clipped;
def clipped = Clipped 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. This is not used elsewhere
% in \file{grafbase.mp} so it is not really important if \gbc{fillcolor}
% or \mfc{black} is the right choice.
%    \begin{macrocode}
vardef picneg (suffix vt) expr c =
%<*MF>
  mono (vt);
  (interior c) picsub vt
%</MF>
%<*MP>
  save _pn; picture _pn; _pn := nullpicture;
  addto _pn (interior c) _wc_ fillcolor;
  addto _pn also (Clipped (vt) c) _wc_ background;
  _pn
%</MP>
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 =
  save v; picture v; v := nullpicture;
  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 hroundpair (p.t_));
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.
%    \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 $k\times (k+1)$-pixels (assuming
% the aspect ratio is 1) rectangles. 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).
%
% 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.
%
% First \gbc{dims} is equated to half the size needed, the dot is created
% (\gbc{_shp}) and then the dot is repeated and \gbc{dims} is doubled.
% Finally, for dark gray (gray levels less than .5) we calculate the shade
% picture for \gbc{1 - greylevel} and subtract it from a black square.
%    \begin{macrocode}
%<*MF>
numeric shadepicsize; shadepicsize := 0.8bp;
vardef shadepic (suffix dims) (expr grparam) =
  pair dims;
  save _frac; _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_ );
  save _shp; picture _shp; _shp := nullpicture;
  addto _shp contour rect ((0,0), (_dotwd, _dotht));
  picdot (_shp, _shp, dims);
  dims := 2dims; mono (_shp);
  if grparam >= .5 :  _shp
  else : (interior (rect ((0,0), 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{filledwith} 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 (\gbc{shpic} in this case),
% 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{gbbox} command is defined in the next section. 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.
%
% I am not sure why we return \gbc{picpath} for non-cycles, but I think
% I once thought to make \gbc{shaded} a replacement for setdot to get gray
% dots (in the \gbc{polkadot} routine).
%    \begin{macrocode}
vardef shaded (expr clr) expr c =
  if cycle c :
%<*MP>
    save v; picture v;
    v := nullpicture;
    addto v contour c _wc_ clr;
    v
%</MP>
%<*MF>
    if (clr <= black) or (clr >= white) :
      interior c
    else:
      save vsh, shpic, shdims, ll, ur;
      picture vsh, shpic; pair shdims, ll, ur;
      shpic := shadepic (shdims) (clr);
      gbbox (c, ll, ur);
      vsh := filledwith (shpic) (shdims, ll, ur);
      Clipped (vsh) c
    fi
%</MF>
  else: picpath c % should we? or just make it null?
  fi
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{filledwith}
% 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, 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 returns the picture which
% is that rectangle filled with copies of the picture.
%
% One might do this with two nested loops, but it turns out to be much
% (much!) faster to do two separate loops: the second one stacking 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. I would have written
% this 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 filledwith (expr pic, dims, ll, ur) =
  save b, v; picture b, v;
  b := v := nullpicture;
%<*MF>
  save fwdims, _ll, _ur; pair fwdims, _ll, _ur;
  fwdims := hroundpair (dims.t_);
  _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>
  v
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}
vardef thatchf (suffix v) (expr CT, sp, a, b) =
  save _sp;
  _sp = signof (ypart(b - 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);
enddef;

%    \end{macrocode}
%
% \subsection{Tiles}\label{tiles}
%
% Tesselations are a generalization type of fill in which a rectangular
% pattern is repeated throughout a region. The rectangular pattern 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}
% \DescribeRoutine{endtile}
% \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{height} and \gbc{width}
% 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.
%
% 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, in fact, a tile is a composite object consisting of a
% picture, \gbc{fred.pic} in our example (the actual tile), two numerics
% \gbc{fred.wd} and \gbc{fred.ht} (the device dimensions) and a boolean
% \gbc{fred.clipon}.
%    \begin{macrocode}
def tile (suffix atile) (expr unit, width, height, clipit) =
  picture atile.pic; atile.pic := nullpicture;
  numeric atile.wd, atile.ht;
%<MF>  (atile.wd, atile.ht) = round ((width, height)*unit);
%<MP>  (atile.wd, atile.ht) = (width, height)*unit;
  boolean atile.clipon; atile.clipon := clipit;
  begingroup
%    \end{macrocode}
% We simply 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.
%    \begin{macrocode}
%    save active_plane;
    save active_plane;
    def active_plane = atile.pic enddef;
    save ztr, vtr; transform ztr, vtr;
    ztr := identity scaled unit; vtr := ztr;
%    \end{macrocode}
% And if clipit is true we set the \gbc{ClipPath} to the bounding
% rectangle so that all commands that respect \gbc{ClipOn} will draw only
% inside the tile. And in case some don't, \gbc{endtile} clips it all
% anyway.
%    \begin{macrocode}
    save ClipOn; boolean ClipOn;
    if clipit :
      ClipOn := true;
      save ClipPath; path ClipPath[];
      ClipPath = 1;
      ClipPath[1] = rect(origin, (atile.wd, atile.ht));
    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 all the
% 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.wd    ) and (numeric atile.wd    ) and
  (known atile.ht    ) and (numeric atile.ht    ) and
  (known atile.clipon) and (boolean atile.clipon)
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{gbbox} which takes a path expression and two pair
% variables, and sets the pairs to the lower left corner and upper right
% corner, respectively, of a rectangle enclosing the path. In \MF, this
% finds a rather loose box, the smallest rectangle containing all the
% control points and all the nodes of the path. Or rather it used to do
% that. Now we break the path into twice as many nodes and use the control
% points relative to that, which gives a tighter box. Both algorithms make
% use of the fact that the convex hull of the points and controls for a
% path segment contains the segment.
%
% 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.
%
% 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} (pixel). This is only called by \gbc{tbbox}, which is never
% used.
%
% \DescribeRoutine{xlimit}
% \DescribeRoutine{ylimit}
% \gbc{xlimit(x)} returns a value of true if the path \gbc{g} doesn't
% cross the vertical line at \gbc{x}. \gbc{ylimit(y)} is the same for
% the horizontal line at \gbc{y}.
%    \begin{macrocode}
%<*unused>
vardef tightbbox (expr g) (suffix ll, ur) =
  % true iff horizontal at y does not intersect g:
  vardef xlimit (expr x) =
    (((x, -infinity)--(x, infinity)) intersectiontimes g) < origin
  enddef;
  % true iff vertical at x does not intersect g:
  vardef ylimit (expr y) =
     (((-infinity, y)--(infinity, y)) intersectiontimes g) < origin
  enddef;
  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))) );
  if showbbox :
    noclip ( safedraw rect (ll, ur) );
  fi
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 lowerleft. 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. Changed recently to
% use the new \gbc{pairmin} and \gbc{pairmax} macros. This avoids two
% \mfc{for} loops (which seem to be something of a \MF{} bottleneck).
%    \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}
%
% \DescribeRoutine{gbbox}
% 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.
%
% \DescribeRoutine{ctrlsbbox}
% There is a difference between `\mfc{postcontrol j 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{gbbox} passes
% it half a segment at a time. We don't actually examine both endpoints of
% the half-segment, only those that are nodes of the original path. The
% reason is that the subdivision points are already on the line segment
% connecting the controls of the subpaths on either side. 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.
%
% This description applies only to \MF, because \MP{} has primitive
% facilities for determining the bounding box.
%
% \DescribeRoutine{pnt}
% \DescribeRoutine{pre}
% \DescribeRoutine{post}
% I got tired of typing long expressions like `\gbc{precontrol length p of p}',
% and now use the abbreviation `\gbc{pre[length p](p)}'.
%    \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;

vardef gbbox (expr g) (suffix ll, ur) =
%<MP>  ll := llcorner g; ur := urcorner g;
%<*MF>
  save _s; _s := emax(1, ceiling bbox_split);
  ur := ll := pnt 0 (g);
  if (length g) > 0 :
    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
  fi
%</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 (ll, post0 (p)); ll := pairmin (ll, pre 1 (p));
  ur := pairmax (ur, post0 (p)); ur := pairmax (ur, pre 1 (p));
enddef;

%</MF>
%    \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{coloraddto}. In \MP{} it is basically the primitive \mfc{addto
% ... also ... withcolor ...}, but in \MF{} it adds when the color is less
% than 1 (gray or black), otherwise it subtracts (white).
%
% \DescribeRoutine{colorsafedraw}
% \gbc{safedraw} merely calls colorsafedraw, which calls \gbc{picpath},
% which calls \gbc{shpath}. One reason for this roundabout path
% is to support older files (\gbc{colorsafedraw} 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;
vardef colorsafedraw (expr clr) expr d =
  save v; picture v; v := picpath d;
  DoClip (v);
  coloraddto (clr) (active_plane, v);
enddef;

%    \end{macrocode}
%
% \subsection{Filling}\label{basicfilling}
%
% \DescribeRoutine{NoCycleWarn}
% 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.
% We make no attempt to color the path, but maybe we should.
%
% \DescribeRoutine{safefill}
% \DescribeRoutine{colorsafefill}
% \DescribeRoutine{safeunfill}
% These three take one parameter that is a path expression, and
% \gbc{colorsafefill} takes another that is a color. These commands fill
% (or unfill) it 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.
%
% \gbc{safeunfill} is just \gbc{safefill} with the color \mfc{background}.
% In \MF{}, when \gbc{background = white = 1}, this is detected by
% \gbc{coloraddto} which then subtracts the picture.
%    \begin{macrocode}
def NoCycleWarn expr s =
  GBmsg  s & " cannot be applied to an open path. "
           & "The path will be drawn instead.";
enddef;

def safefill = colorsafefill (fillcolor) enddef;
vardef colorsafefill (expr clr) expr c =
  if cycle c :
    save v; picture v; v := interior c;
    DoClip (v);
%<*MF>
    if (clr > black) and (clr < white) :
      subto (active_plane) (v);
      v := nullpicture;
      v := shaded (clr) c;
    fi
%</MF>
    coloraddto (clr) (active_plane, v);
  else: NoCycleWarn "fill"; safedraw c;
  fi
enddef;
def safeunfill expr c =
  if cycle c : noclip (colorsafefill (background) c);
  else: NoCycleWarn "unfill"; safedraw 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: NoCycleWarn "clip"; safedraw c;
  fi
enddef;

%    \end{macrocode}
%
% \section{Rendering: the highest level commands}\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 ("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}
% \DescribeRoutine{colordrawn}
% \gbc{drawn} and \gbc{colordrawn} accept a path
% expression \gbc{f} and return the same. 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}
%
% \subsection{Filling, unfilling and clipping}\label{filling}
%
% \DescribeRoutine{filled}
% \DescribeRoutine{colorfilled}
% \DescribeRoutine{unfilled}
% \DescribeRoutine{Clip}
% These subject \gbc{zconv(f)} to \gbc{colorsafefill}, \gbc{safeunfill} or
% \gbc{safeclip}. The name \gbc{clip} (lowercase) is taken: it is a \MP{}
% primitive and an old \file{grafbase.mf} command we keep for compatibility.
%
%    \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}
%
% 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.
%
% \DescribeRoutine{shade}
% This is the old \gbc{shade} macro, filling a contour with small dots.
% The shape and size of the dot used 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{filledwith} macro is used to fill with copies of
% a dot picture. For speed, it actually fills with a two-dot picture.
%    \begin{macrocode}
numeric shadewd; shadewd := 0.5bp;
path shadedotpath;
shadedotpath := fullcircle; % unitsquare;

vardef shade (expr sp) expr f =
  save g; path g; g := zconv (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. Theoretically, no white will appear when
% \gbc{sqrt(2)*shadewd/sp >= 1}
%    \begin{macrocode}
  save gr; numeric gr;
  gr :=  1 - (.88*abs(shadewd)/sp)**2;
  if not cycle g : NoCycleWarn "shade"; safedraw g;
  elseif gr <= 0 : safefill g;
  else:
%<*MF>
    save ll, ur; pair ll, ur;
    gbbox (g, ll, ur);
    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{filledwith} to draw these copies. To save a little memory
% we do this twice with half the dots each time. The second set
% interleaves the first and is staggered from the first by half the
% horizontal spacing. I wonder which which uses less memory, rendering
% \gbc{v} unknown with \mfc{picture v}, or null with \mfc{v := nullpicture}?
%
% Shifts of pictures need to be by integer number of pixels, but this is
% ensured by \gbc{filledwith}, using \mfc{ceiling} to define \gbc{dx} is
% more to ensure it is not rounded down to 0.
%    \begin{macrocode}
    save sh, v; picture sh, v;
    save dx; dx := ceiling(sp/(sqrt 2));
    sh := setdot (shadedotpath, abs(shadewd));
    v := filledwith (sh, 2(dx, dx), ll, ur);
    DoClip(v);
    orto (active_plane, Clipped (v) g);
    sh := sh shifted hroundpair ((dx, dx).t_);
    v := nullpicture;
    v := filledwith (sh, 2(dx, dx), ll, ur);
    DoClip(v);
    orto (active_plane, Clipped (v) g);
%</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 the dots are really ugly. 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 =
  save g; path g; g := zconv (f);
  if not cycle g : NoCycleWarn "polkadot"; safedraw g;
  elseif sp <= emax (2*polkadotwd/3, mindotspace) :
    safefill g;
  else:
    save ll, ur; pair ll, ur;
    gbbox (g, ll, ur);
%    \end{macrocode}
% As with \gbc{shade}, we shift alternate rows by half the spacing between
% dot centers, \gbc{dx}. The vertical shift \gbc{dy} is slightly larger
% (relatively speaking) and the horizontal smaller. We also apply a
% 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; dx := sp/2; dy := dx*(sqrt 3);
    hshift := ((xpart (ur - ll)) mod dx)/2;
    vshift := ((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 + (hshift, vshift);
    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 fillcolor demands it.
%    \begin{macrocode}
    save v, thepolkadot; picture v, thepolkadot;
    thepolkadot := setdot (polkadotpath, polkadotwd);
    v := filledwith (thepolkadot, dims, p, ur);
    p := p + (dx, dy);
    orto (v, filledwith (thepolkadot, dims, p, ur));
    DoClip (v); clipto (v) g;
%<*MF>
    if (fillcolor > black) and (fillcolor < white): % gray
      subto (active_plane) (v);
      thepolkadot :=
        shaded (fillcolor) polkadotpath scaled ceiling(polkadotwd);
      v := filledwith (thepolkadot, dims, p, ur);
      p := p - (dx, dy);
      orto (v, filledwith (thepolkadot, dims, p, ur));
      DoClip (v); clipto (v) g;
    fi
%</MF>
    coloraddto (fillcolor) (active_plane) (v);
  fi
  f
enddef;

%    \end{macrocode}
%
% \subsection{Hatching}\label{hatching}
%
% \DescribeRoutine{thatch}
% \DescribeRoutine{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 =
  save g; path g; g := zconv (f);
  if not cycle g : NoCycleWarn "hatch"; safedraw g;
  elseif sp <= abs(hatchwd) : colorsafefill (clr) g;
  else:
    save v; picture v; v := nullpicture;
    save CT; transform CT; CT := identity rotated theta;
    save ll, ur; pair ll, ur;
    gbbox (g transformed inverse CT, ll, ur);
    thatchf (v, CT, sp, ll, ur);
    DoClip(v);
    coloraddto (clr) (active_plane) (Clipped (v) g);
  fi
  f
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{hhatch}
% \DescribeRoutine{vhatch}
% \DescribeRoutine{lhatch}
% \DescribeRoutine{rhatch}
% \DescribeRoutine{xhatch}
% 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. \gbc{hhatch} has angle
% 0 and so produces horizontal lines; \gbc{vhatch} produces vertical
% lines; \gbc{lhatch} produces lines tilted to the left (running from
% upper left to lower right), and \gbc{rhatch} produces lines running from
% lower left to upper right. \gbc{xhatch} produces cross-hatching, and
% essentially runss \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;
vardef colorxhatch (expr clr, sp) expr f =
  colorthatch (clr) (sp, 45) colorthatch (clr) (sp, -45) f
enddef;

%    \end{macrocode}
%
% \subsection{Tesselations}\label{tess}
%
% \DescribeRoutine{tess}
% Tesselation of interior of closed path is filling with copies of a 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 \gbc{fred.wd} is its width, etc.
%    \begin{macrocode}
vardef tess (suffix atile) expr c =
  save _g; path _g; _g := zconv (c);
  if not is_tile (atile) :
    GBerrmsg ("Tile parameter " & str atile & " of tess() is invalid")
      "This tile may be undefined or incorrectly defined. " &
      "If you proceed, tess() will be abandoned and the curve " &
      "merely drawn."; safedraw _g;
  elseif not cycle _g : NoCycleWarn "tess"; safedraw _g;
  else:
    save _ll, _ur; pair _ll, _ur;
    gbbox (_g, _ll, _ur);
    save _ts; picture _ts;
    _ts := filledwith (atile.pic, (atile.wd, atile.ht), _ll, _ur);
    DoClip (_ts);
    orto (active_plane, Clipped (_ts) _g);
  fi
  c
enddef;

%    \end{macrocode}
%
% \subsection{Dots and Dashes}\label{dashes}
%
% \MP{} has some builtin 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 reasonably fast and the result
% is actually better quality than the builtin commands of \MP{} so we use
% the same code in both versions. 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). The built-in 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 so that adjustments
% can be made 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{unit_of_length} 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 unit_of_length : unit_of_length := 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}. We also supply (later) the \gbc{dashpat}
% command which takes a list of lengths, equates \gbc{pat.rep} to them,
% and generates \gbc{pat.start} and \gbc{pat.finish} according to the
% description above.
%    \begin{macrocode}
vardef gendashed (suffix pat) expr f =
  save _g; path _g; _g := zconv(f);
  if (unknown pat.rep) : % no "pattern"
    GBmsg "Dash pattern " & str pat & " undefined. " &
      "Path will be drawn instead.";
    safedraw _g;
  elseif pat.rep < 2 : % no "spaces"
    safedraw _g;
  else:
%    \end{macrocode}
% We want to manipulate the values of \gbc{pat} so that a whole number of
% repetitions are used. So we copy \gbc{pat} to \gbc{_tmppat}.
%
% After this loop, \gbc{_dl.s} is the total length of the corresponding
% \gbc{pat.s} in multiples of \gbc{unit_of_length}, and \gbc{_tmppat.s[i]}
% is \gbc{pat.s[i]} converted to these units.
%    \begin{macrocode}
    save _dl, _tmppat;
    forsuffixes _s = start, rep, finish :
      _dl._s := 0; _tmppat._s := pat._s;
      for i = 1 upto pat._s :
        _tmppat._s[i] := pat._s[i]/unit_of_length;
        _dl._s := _dl._s + _tmppat._s[i];
      endfor
    endfor
    if _dl.rep = 0 :
      GBmsg "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 we convert that to our
% unit of length 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 $f(0), f(1/8),
% f(2/8),\ldots,f(k)$ and compute the length of \emph{that} path (assuming
% \gbc{segment_split = 8}). 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{makelengtharray} does this, storing the
% cumulative lengths in the array \gbc{_cumlen} and returning the total
% length.
%    \begin{macrocode}
      save _p; path _p;
      _p := _g scaled (1/unit_of_length);
      save _cumlen, _totlen, _n, _sf;
      _totlen := makelengtharray(_cumlen) _p;
%    \end{macrocode}
% Now we scale 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 already less than 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{_tmppat}, and
% the length of the patterns in \gbc{_dl}.
%    \begin{macrocode}
      _sf := scale_adjust (_n, _dl)(_totlen);
      if _n < 0 : safedraw _g;
      else:
        forsuffixes _s = start, rep, finish :
          for _i = 1 upto _tmppat._s :
            _tmppat._s[_i] := _tmppat._s[_i]*_sf;
          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}. 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}
        save dashingdot; picture dashingdot;
        if known plot_pic :  dashingdot := makesymbol(plot_pic, penwd);
        else:  dashingdot := makesymbol(dotpath,  penwd);
        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. 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, then it is added to \gbc{active_plane}.
% 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.
%    \begin{macrocode}
        save _ct, _t, _d, _v;
        picture _v; _v := nullpicture;
        _ct := 0;
        %   Begin with pat.start
        _d0 := 0; _t0 := 0;
        dashit (_tmppat.start) (_v);
%    \end{macrocode}
% 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{coloraddto} 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 I'm going to try a standard programming trick: accumulate $m < n$
% repetitions before adding them, the memory should be $O(m)$ and the
% time $O(n/m)$. As a first try, we make $m$ about $\sqrt n$.
%    \begin{macrocode}
        % then pat.rep
        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 (_tmppat.rep) (_v);
            endfor
            % add _m patterns and reset.
            DoClip(_v);
%<MF>            mono (_v)
            coloraddto (drawcolor) (active_plane, _v);
            _v := nullpicture;
          endfor
        fi
        %   and finally, pat.finish
        _d0 := _totlen - _dl.finish;
        _t0 := gettime(_cumlen, _ct) (_d0);
        dashit (_tmppat.finish) (_v);
        DoClip(_v);
%<MF>        mono(_v)
        coloraddto (drawcolor) (active_plane, _v);
      fi
    fi
  fi
  f
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{makelengtharray}
% This takes an array name and a path expression (any coordinates),
% computes the array of partial lengths (of the polygon approximation),
% and returns the total length.
%    \begin{macrocode}
vardef makelengtharray (suffix clen) expr p =
  save _s; _s := emax (1, ceiling segment_split);
  clen := _s*length p; clen[0] := 0;
  for _i = 1 upto clen :
    clen[_i] := clen[_i-1] + abs (pnt[_i/_s] (p) - pnt[(_i-1)/_s] (p));
  endfor
  clen[clen]
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{scale_adjust}
% Here \gbc{n} is a suffix defined by the calling routine,
% \gbc{pl.\{start\|ref\|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}.
%    \begin{macrocode}
vardef gettime (suffix arr, ct) (expr lngth) =
  save _gtl, _s;
  _s := emax(1, ceiling segment_split);
  _gtl := emax (arr[ct], emin (arr[arr], lngth));
  forever:
    exitif ( (arr[ct] <= _gtl) and (_gtl <= arr[ct+1]) );
    ct := ct + 1; % need to exit *before* incrementing
  endfor
  if arr[ct] = arr[ct+1] :  ct
  else: ( ct + (_gtl - arr[ct]) / (arr[ct+1] - arr[ct]) )
  fi /_s
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 thos 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 : % draw a dash of length pos[_k]
      if pos[_k] = 0 : % point required
        _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: % find the start of the next dash:
      _d0 := _d1 + pos[_k];
      _t0 := gettime(_cumlen, _ct) (_d0);
    fi
  endfor
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{dashpat}
% This is a utility to convert a text list of lengths to the three dash
% pattern arrays required by \gbc{gendashed}. We first simply copy the
% list to array \gbc{pat.rep}. If the number is odd we add a 0-length
% item (a nonspacing space). Unless the number is 1, which we use as a
% signal that a curve should be solid and not dashed at all.
%
% \gbc{pat.start} is the same as \gbc{pat.rep} except the first dash is
% reduced by the factor \gbc{dash_start}. \gbc{pat.finish} is just the
% first dash of \gbc{pat.rep} reduced by the factor \gbc{dash_finish}.
%    \begin{macrocode}
def dashpat (suffix pat) (text t) =
  pat.rep := 0;
  for _itm = t:
    pat.rep[incr pat.rep] := _itm;
  endfor;
  if odd (pat.rep) and (pat.rep > 1):
    pat.rep[incr pat.rep] := 0;
  fi
  pat.start := 1;
  pat.start[1] := pat.rep[1]*dash_start;
  for _idx = 2 upto pat.rep :
    pat.start[incr pat.start] := pat.rep[_idx];
  endfor
  pat.finish := 1;
  pat.finish[1] := pat.rep[1]*dash_finish;
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{dashed}
% \DescribeRoutine{DASHED}
% The old \gbc{dashed} is now implemented by making a dashpattern from the
% two arguments and calling gendashed. When \MP{} support was added, I
% thought it best not to overwrite the \MP{} primitive \gbc{dashed} and
% the command was named \gbc{DASHED}, but then later it seemed better to
% keep \MF/\MP{} compatibility so \gbc{dashed} was used. Now for backward
% compatibility we maintain both. In \MP, we save the primitive
% \mfc{dashed} as \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;

%<MP>let dashed_ = dashed;
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, signalling that dots are used. \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);
  save plot_pic; picture plot_pic;
  plot_pic := makesymbol (spath, sc);
  gendashed (dots) f
enddef;

path dotpath; dotpath := fullcircle;
vardef dotted (expr dsize, dgap) expr f =
  doplot (dotpath, dsize, dgap) f
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{plotnodes}
% This is a useful little utility to draw the points on top of the
% curve through them. It 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.
%    \begin{macrocode}
vardef plotnodes (expr symbol, size) expr f =
  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);
  f
enddef;

%    \end{macrocode}
%
% \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 it draws 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 path spath        : setdot (spath, sc)
  elseif picture spath :
%<MF>    save v; picture v; v:= spath; mono (v); v
%<*MP>
  spath
  elseif string spath  :
    spath infont defaultfont scaled defaultscale
%</MP>
  else:
    GBmsg "Undefined symbol for plotting, using dotpath instead.";
    setdot (dotpath, sc)
  fi
enddef;

%    \end{macrocode}
%
% These are some symbols to be used by \gbc{doplot} and \gbc{plotsymbol}.
% They are paths. The ones named with ``\gbc{Solid}'' are closed paths.
% Since these two drawing commands 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.
%    \begin{macrocode}
path Triangle, Square, Circle, Diamond, Star, Plus, Cross,
    Asterisk, SolidTriangle, SolidSquare, SolidCircle,
    SolidDiamond, SolidStar;

Triangle := (for n = 0 upto 2:
  (up rotated 120n)-- endfor up) scaled .78;
SolidTriangle := Triangle & cycle;

Square := (for n = 0 upto 3:
  dir (90n + 45)-- endfor dir 45) scaled .63;
SolidSquare := Square & cycle;

Circle := halfcircle & halfcircle rotated 180;
SolidCircle := Circle & cycle;

Diamond := (Square rotated 45) xscaled (1/1.2) yscaled 1.2;
SolidDiamond := Diamond & cycle;

Plus := (origin for n = 0 upto 3:
    --(up rotated 90n)--origin endfor) scaled .65;
Cross := Plus rotated 45;
Asterisk := (origin for n = 0 upto 5:
    --(up rotated 60n)--origin endfor) scaled .6;

%    \end{macrocode}
%
% We do some computations to find the vertices of a standard 5-pointed
% star (pentagram). The first equation says the indented vertex at the
% ``left shoulder'' is on the line from the top of the ``head'' to the
% ``left foot'', and the second says it is on the line from the ``left
% hand'' to the ``right hand''. That point determined, we get the rest by
% rotaing 72 degrees.
%    \begin{macrocode}
pair zz;
zz = (whatever)[up, up rotated 144];
zz = (whatever)[up rotated 72, up rotated -72];
Star := (for n = 0 upto 4:
    (up rotated 72n)--(zz rotated 72n)-- endfor up) scaled .84;
SolidStar := Star & cycle;
save zz;

%    \end{macrocode}
%
% 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{defaultdashes}
% These are the usual 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.
%    \begin{macrocode}
numeric dashtype;
forsuffixes s = start, rep, finish :
  numeric dashtype[].s, dashtype[].s[];
endfor
def defaultdashes =
  dashpat (dashtype0) (0);                           % solid
  dashpat (dashtype1) (3bp, 4bp);                    % dashed
  dashpat (dashtype2) (0, 4bp);                      % dotted
  dashpat (dashtype3) (0, 4bp, 3bp, 4bp);          % dot-dash
  dashpat (dashtype4) (0, 4bp, 3bp, 4bp, 0, 4bp);% dot-dash-dot
  dashpat (dashtype5) (0, 4bp, 3bp, 4bp, 3bp, 4bp);% dot-dash-dash
  dashtype := 6;
enddef;
defaultdashes;

%    \end{macrocode}
%
% \DescribeRoutine{isdashpat}
% Checks, for the given variable \gbc{pat}, if the three arrays that
% make up a dash pattern are know arrays. It does not check if they are
% numeric arrays, but one hardly thinks they could all three exist
% accidentally if they hadn't been created by \gbc{dashpat}.
%
% \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.
%
% \DescribeRoutine{getdashpat}
% And finally, we remove the mod-ing operation from \TeX, where it is
% cumbersome, to \MF, where it is trivial, with this command.
%    \begin{macrocode}
vardef isdashpat suffix pat =
  (knownarray pat.start)  and
  (knownarray pat.finish) and
  (knownarray pat.rep)
enddef;

def setdatadashes (text lst) =
  save dashtype; dashtype := 0;
  forsuffixes _itm = lst :
    if isdashpat _itm :
      forsuffixes _s = start, rep, finish :
        copyarray (_itm._s, dashtype[dashtype]._s);
      endfor
      dashtype := dashtype + 1;
    else: GBmsg "Improper dash pattern in setdatadashes.";
    fi
  endfor
  if dashtype = 0 :
    SetdataWarn "dashes";
    defaultdashes;
  fi
enddef;
def getdashpat expr n = dashtype[n mod dashtype] enddef;

def SetdataWarn expr s =
  GBmsg "Command setdata"& s &"() failed; using defaults."
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{setdatasymbols}
% This can be used to define the sequence of point plotting styles for
% \mfpic's \cs{plotdata} command. We could use it to set the default
% symbols, but I worry about the difficulty chasing down bugs if
% \gbc{defaultpoints} calls \gbc{setdatasymbols} which can again call
% \gbc{defaultpoints}.
%
% \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 pointtype; path pointtype[]; pointtype := 0;
  for _itm = lst :
    if (known _itm) and (path _itm):
      pointtype[pointtype] := _itm;
      pointtype := pointtype + 1;
    else:
      GBmsg "Improper path in setdatasymbols().";
    fi
  endfor
  if pointtype = 0:
    SetdataWarn "symbols";
    defaultsymbols;
  fi
enddef;
def getsymbol expr n := pointtype[n mod pointtype] enddef;

%    \end{macrocode}
%
% \DescribeRoutine{defaultsymbols}
% We store the default definitions in a macro so the user can restore
% them easily.
%    \begin{macrocode}
numeric pointtype; path pointtype[];
def defaultsymbols =
  pointtype0 := Circle;
  pointtype1 := Cross;
  pointtype2 := SolidDiamond;
  pointtype3 := Square;
  pointtype4 := Plus;
  pointtype5 := Triangle;
  pointtype6 := SolidCircle;
  pointtype7 := Star;
  pointtype8 := SolidTriangle;
  pointtype := 9;
enddef;
defaultsymbols;

%    \end{macrocode}
%
% \DescribeRoutine{setdatacolors}
% \DescribeRoutine{getcolor}
% Finally, for \MP, we do a similar pair of commands for setting and
% getting the colors for the \cs{plotdata} command. The 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.
%
% \DescribeRoutine{defaultcolors}
% We store the default definitions in a macro so the user can restore
% them easily.
%    \begin{macrocode}
%<*MP>
def setdatacolors (text lst) =
  save colortype; color colortype[]; colortype := 0;
  for _itm = lst :
    if (known _itm) and (color _itm) :
      colortype[colortype] := _itm;
      colortype := colortype + 1;
    else:  GBmsg "Improper color in setdatacolors().";
    fi
  endfor
  if colortype = 0 :
    SetdataWarm "colors";
    defaultcolors;
  fi
enddef;
def getcolor expr n = colortype[n mod colortype] enddef;

numeric colortype; color colortype[];
def defaultcolors =
  colortype0 := black;
  colortype1 := red;
  colortype2 := 0.80blue + .2white;  % blue
  colortype3 := 0.66yellow + .34red; % orange
  colortype4 := 0.80green;           % green
  colortype5 := 0.85magenta;         % magenta
  colortype6 := 0.85cyan;            % cyan
  colortype7 := 0.85yellow;          % yellow
  colortype := 8;
enddef;
defaultcolors;

%</MP>
%    \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;
      save clearsymbols; boolean clearsymbols; clearsymbols := true;
      plotsymbol (Circle, ptwd) (t);
    endgroup
  fi
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{plotsymbol}
% \DescribeRoutine{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.
%
% \DescribeRoutine{dosymbols}
% \DescribeRoutine{addsymbols}
% We copy the text list to an array and call \gbc{dosymbols} so that
% \gbc{plotnodes} can share the code. Also, since \gbc{dosymbols} uses
% identical code twice (once to clear, once to draw), we put that code
% in \gbc{addsymbols}
%    \begin{macrocode}
boolean clearsymbols; clearsymbols := false;
vardef clearable (expr pth) =
  false
  if path pth :
  if (not cycle pth) and (length pth > 0):
  if ( pnt0 (pth) = pnt[length pth] (pth) ) :
    or true
  fi fi fi
enddef;

def plotsymbol = colorplotsymbol (pointcolor) enddef;
vardef colorplotsymbol (expr clr, spath, sc) (text t) =
  save _cpls;
  textpairs (_cpls) (t);
  dosymbols (clr, spath, sc) (_cpls);
enddef;

vardef dosymbols (expr clr, spath, sc) (suffix arr) =
  save one_symbol, _pls; picture one_symbol, _pls;
  if clearsymbols and  clearable (spath):
    addsymbols (background, spath&cycle, sc) (arr);
  fi
  addsymbols (clr, spath, sc) (arr);
enddef;

def addsymbols (expr clr, spath, sc) (suffix arr) =
  one_symbol := makesymbol (spath, sc);
  _pls := nullpicture;
  for _idx = 1 upto arr:
    picdot (_pls, one_symbol, zconv(arr[_idx]));
  endfor
  DoClip (_pls);
%<MF>  mono (_pls);
  coloraddto (clr) (active_plane) (_pls);
enddef;

%    \end{macrocode}
%
%
% \section{Modification of paths}\label{modification}
%
% \subsection{Closing a path}\label{closing}
%
% In \MF{} one can close a path by 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 closure commands have a version with a tension parameter
% when that makes sense. These version make the connection 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.
%    \begin{macrocode}
vardef lclosed expr f = f  if not cycle f : --cycle fi enddef;

%    \end{macrocode}
% \DescribeRoutine{sclosed}
% \DescribeRoutine{sclosedt}
% 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 is
% different.
%    \begin{macrocode}
numeric default_tension;
default_tension := 1;

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 : 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.
%    \begin{macrocode}
def bclosed = bclosedt (default_tension) enddef;
vardef bclosedt (expr t) expr f =
  f  if not cycle f : ..tension t..cycle fi
enddef;

%    \end{macrocode}
% \DescribeRoutine{uclosed}
% This is now just a renaming of \gbc{bclosed}. It turns out (contrary
% to my earlier belief) that just adding \mfc{..cycle} does \emph{not}
% change the original curve. \gbc{sclosed} will do that because it
% takes the curve apart and redoes its end segments.
%    \begin{macrocode}
def uclosed  = bclosed enddef;

%    \end{macrocode}
% \DescribeRoutine{bsplinecontrols}
% 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 \gbc{f}. 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 bsplinecontrols (suffix b) expr f =
  b  := 4;
  b1 := 2[pre 1(f), post0(f)];
  b2 := 2[post0(f), pnt 0(f)];
  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 =
  if cycle f : f
  elseif (length f)=0 : f&cycle
  else:
    save p, q; pair p[], q[];
    bsplinecontrols (p) f;         % defines p1 to p4
    bsplinecontrols (q) reverse f; % 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 seems wrong to be able to close with a cubic B-spline but not a
% quadratic B-spline. Therefore I will 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.
%    \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}
% \subsection{Trimming a path}\label{trimming}
%
% \DescribeRoutine{cutoffbefore}
% \DescribeRoutine{cutoffafter}
% This is a useful utility operation present in \file{plain.mp} 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.
% If I can learn the actual method used to find intersection times, I'll
% put in some \mfc{solve} code to get the first \gbc{t} rather than the
% minimal `shuffled binary' of the pair \gbc{w}.
%    \begin{macrocode}
%<MF>path cuttings;
vardef cutoffbefore (expr b) expr f =
  save w, t, u, n; n:= length f;
  pair w;
  for k = 1 upto n :
    w := (subpath (0,k) of f) intersectiontimes b;
    exitif w > left;
  endfor
  if debug :
    GBdebug;
      >> "Intersectiontimes:";
      show w;
    GBenddebug;
  fi
  t := xpart w;
  if t < 0:
    cuttings := pnt0 (f);
    f
  else:
    cuttings := subpath (0,t) of f;
    subpath (t, n) of f
  fi
enddef;

vardef cutoffafter (expr b) expr f =
  save g; path g;
  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{Appending an arrowhead}
%
%
% \DescribeRoutine{predirection}
% \DescribeRoutine{postdirection}
% \DescribeRoutine{__dir}
% First, some better \mfc{direction} commands. They makes use of the fact
% (easily proved) that a cubic B\'ezier $z\sb0(1 - t)^3 + 3z\sb1(1-t)^2t +
% z\sb2(1-t)t^2 + z\sb3t^3$ has a direction at $z\sb0$ equal to the first
% one of $z\sb{j} -z\sb0$ that is nonzero.
%
% \gbc{__dir} gets the direction at point 0 for an arbitrary path.
% \gbc{postdirection} reduces to this case using \mfc{subpath}. If the
% postdirection is 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. \gbc{predirection} just runs
% \gbc{postdirection} on the reversed path.
%    \begin{macrocode}
vardef predirection@# (expr p) =
  - postdirection[length p - @#] (reverse p)
enddef;

vardef postdirection@# (expr p) =
  save _n; _n := length (p);
  save v; pair v; 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;
    if v = origin :
      v := pre[n] (p) - w;
      if v = origin :
        v := pnt[n] (p) - w;
      fi
    fi
    exitif v <> origin;
  endfor
  v
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 curve 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;
interim hdwdr := 1; interim hdten := 1;
boolean hfilled; hfilled := false;

%    \end{macrocode}
%
% \DescribeRoutine{headshape}
% The following little utility adjusts the above parameters, 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}.
%    \begin{macrocode}
def headshape (expr wr, tens, fil) =
 interim hdwdr := wr; interim hdten := tens;
 save hfilled; boolean hfilled; hfilled := fil;
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{ahead}
% \DescribeRoutine{colorhead}
% This command draws an arrowhead. \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} 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 and fill it, otherwise we
% draw it. \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 we should have followed
% the pattern of other macros. However, \gbc{head} was never a user-level
% macro and I didn't think it mattered.
%    \begin{macrocode}
%<MF>def head = ahead (headcolor) enddef;
%<MP>def head = ahead enddef;

vardef ahead (expr clr, front, back, hwr, tens, filled) =
  if front <> back :
    save side; pair side;
    side := (hwr/2) * ((front-back) rotated 90);
    save f; path f;
    f := (back + side)..tension tens..
         {front-back}front{back-front}..tension tens..
         (back - side)
      if filled : --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 above described arrowhead looks ``off''. But I know
% of no automatic way of making it look good. Therefore \grafbase{} and
% \mfpic{} have provided a means to micro-adjust the head. (Actually, I
% think the best looking arrowhead for small heads and paths of modest
% curvature is obtained by taking a secant for the direction of the head
% head rather than a tangent.)
%
% \DescribeRoutine{headpath}
% \DescribeRoutine{colorheadpath}
% This takes a path expression \gbc{f} in graph coordinates, puts an
% arrowhead on it and returns \gbc{f}. The arrowhead is placed according
% to the first four parameters. \gbc{hlen} is the length of the head in
% device coordinates, the width being determined by \gbc{hdwdr}, and
% and \gbc{hrot} is a rotation adjustment. \gbc{hback} is a distance (in
% device coordinates) by which it is set back from the point of placement.
% It is set back in the direction determined after the rotation.
%
% If the length of the head is 0, we just skip everything and return
% \gbc{f}.
%
% \gbc{headpath} calls \gbc{colorheadpath} with the color set to
% \gbc{headcolor}.
%    \begin{macrocode}
def headpath = colorheadpath (headcolor) enddef;
vardef colorheadpath (expr clr, hlen, hrot, hback) expr f =
  if hlen <> 0 :
    save g; path g; g := zconv (f);
    save P; pair P[];
    P2 := pnt[length g] (g);
    P1 := predirection[length g] (g);
    if P1 <> (0, 0) :
      P3 := (unitvector P1) rotated hrot;
      P4 := P2 - (hback * P3);
      P5 := P4 - (hlen  * P3);
      ahead (clr, P4, P5, hdwdr, hdten, hfilled);
    fi
  fi
  f
enddef;

%    \end{macrocode}
%
% \section{Axes, Axis Tic Marks, and Grids}
%
% \DescribeRoutine{arrowdraw}
% This is used elsewhere only to draw axes. It returns nothing. This
% doesn't follow the usual pattern of drawing something and returning the
% same path. This 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 axis commands
% just call \gbc{headpath}, and the \mfpic{} code makes sure it is drawn,
% but it can also be \gbc{dashed}, \gbc{dotted}, and \gbc{doplot}\,ed.
%
% The order is important 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}
% \DescribeRoutine{yaxis}
% \DescribeRoutine{axes}
% These draw the obvious things: the corresponding axis or axes through
% the point (0, 0) in graph coordinates. The only parameter is the length
% of the arrowhead in device coordinates. \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.x}
% \DescribeRoutine{axisline.y}
% \DescribeRoutine{axisline.l}
% \DescribeRoutine{axisline.b}
% \DescribeRoutine{axisline.r}
% \DescribeRoutine{axisline.t}
% The commands \gbc{axisline.l}, etc., return the straight line along the
% corresponding edge shifted the appropriate amount. 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 no longer gets used, as we now simply apply
% \cs{arrow} to the appropriate \gbc{axisline}. The change was made for
% consistency: so that the head would be drawn on top of the shaft.
%    \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;

%    \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 the positive direction on the axis (right or up).
% \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, _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 ends of the mark. For example,
% if \gbc{tp = inside}, then \gbc{_tp = 1}. Since \gbc{_U} points toward
% inside, \gbc{_tic2} will be \gbc{len} toward the inside and \gbc{_tic1 =
% (0, 0)}.
%    \begin{macrocode}
 _tic1 := (_tp - 1) * len * _U; % start of mark
 _tic2 := _tp * len * _U;       % end of mark
%    \end{macrocode}
% Finally, convert each numeric position to a point on the axis, a
% multiple of \gbc{pdir} from the 0 point of the axis, and than draw the
% tic.
%    \begin{macrocode}
 for _a = t:
   safedraw ((_tic1--_tic2) shifted zconv (loc + _a*pdir));
 endfor
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{xmarks}
% \DescribeRoutine{ymarks}
% \DescribeRoutine{lmarks}
% \DescribeRoutine{bmarks}
% \DescribeRoutine{rmarks}
% \DescribeRoutine{tmarks}
% And now the specialized command for each axis. Inside and outside
% really make no sense for the x- and 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, (0, 0), right) enddef;
def ymarks =  axismarks (-90, ytick, (0, 0), 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{vgrid}
% \DescribeRoutine{grid}
% This is mainly for the purpose of visualising coordinates. \gbc{vgrid}
% draws a dot of size \gbc{dsize} at every point whose coordinates
% are are \gbc{(n*xspace, m*yspace)}, \gbc{n} and \gbc{m} being integers.
% \gbc{dsize} is in device coordinates, the spacings are in graph
% coordinates. \gbc{grid} is for backward compatibility, calling vgrid
% with a default \gbc{dsize} of \mfc{.5bp}.
%
%    \begin{macrocode}
path griddotpath; griddotpath := fullcircle;
def grid = vgrid (0.5bp) enddef;
vardef vgrid (expr dsize, xspace, yspace) =
  save gdot, gridpic; picture gdot, gridpic;
  gdot := setdot (griddotpath, dsize);
  gridpic := nullpicture;
  for n = ceiling(xneg/xspace) upto floor(xpos/xspace):
  for m = ceiling(yneg/yspace) upto floor(ypos/yspace):
      picdot (gridpic, gdot, zconv((n*xspace, m*yspace)));
  endfor
  endfor
%<MF>  mono (gridpic);
  coloraddto (pointcolor) (active_plane) (gridpic);
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{hgridlines}
% \DescribeRoutine{vgridlines}
% \DescribeRoutine{gridlines}
% This is more what I think of when I hear ``grid'', but the name was already
% taken. \gbc{gridlines} draws horizontal and vertical lines through the
% same points where \gbc{grid} would draw a dot. To draw only horizontal
% or only vertical lines use \gbc{hgridlines} or \gbc{vgridlines}.
%    \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{plrpatch}
% \DescribeRoutine{patcharcs}
% \DescribeRoutine{patchrays}
% Polar grids can be drawn two ways. \gbc{patcharcs} draws the arcs
% \gbc{tstart}${}\le \theta \le{}$\gbc{tstop} with $r = {}$\gbc{rstart},
% stepping by \gbc{rstep} until \gbc{rstop}. \gbc{patchrays} draws the lines
% \gbc{rstart}${}\le r \le{}$\gbc{rstop} with $\theta = {}$\gbc{tstart}
% stepping by \gbc{tstep} until \gbc{tstop}. \gbc{plrpatch} then calls
%
% They are utilities that draw on a picture variable \gbc{X}, and then a
% calling command like \gbc{plrpatch} adds them to \gbc{active_plane}.
% \gbc{plrpatch} used to be called by \gbc{polarpatch}, but now it is
% not called at all.
%    \begin{macrocode}
vardef plrpatch (expr rstart, rstop, rstep, tstart, tstop, tstep) =
  save v; picture v; v := nullpicture;
  patcharcs (v) (rstart, rstop, rstep, tstart, tstop);
  coloraddto (drawcolor) (active_plane, v);
  v := nullpicture;
  patchrays (v) (tstart, tstop, tstep, rstart, rstop);
  coloraddto (drawcolor) (active_plane, v);
enddef;
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 ((0, 0), 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;

%    \end{macrocode}
%
% \DescribeRoutine{polargrid}
% \DescribeRoutine{polargridpoints}
% \DescribeRoutine{gridarcs}
% \DescribeRoutine{gridrays}
% These 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. The arcs have radii that are multiples of \gbc{rstep} and
% radial lines have angles that are multiples of \gbc{tstep}. The command
% \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} computes the bounds for the patch and
% 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 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 (1
% + x)} and \mfc{ceiling(x - 1)} to start and stop before the edge of the
% graph.
%    \begin{macrocode}
def polargrid (expr rstep, tstep) =
  gridarcs (rstep); gridrays (tstep);
enddef;

def polargridpoints (expr dsize, rstep, tstep) =
  beginpolargrid;
    save gdot; picture gdot; 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;

def gridarcs (expr rstep) =
  beginpolargrid;
    if rmin = 0 :  % add "circle" of radius 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;

%    \end{macrocode}
% \DescribeRoutine{beginpolargrid}
% This computes the bounds (on $r$ and $\theta$) of the smallest polar
% coordinate patch that covers the graph rectangle. It leaving the values
% in \gbc{rmin}, \gbc{rmax}, \gbc{tmin} and \gbc{tmax}. It is only for use
% in \gbc{polargrid}, \gbc{gridarcs} and \gbc{gridrays}.
%    \begin{macrocode}
def beginpolargrid =
  begingroup;
    save p, r, t, rmax, rmin, tmax, tmin;
    pair p[];
    % Four corners:
    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
% rotate one corner to have angle zero (so now we are guaranteed all
% angles are between $-180$ and $180$) and get the largest and smallest of
% the angles to all the corners.
%    \begin{macrocode}
    rmin := 0;
    if (xneg < 0) and (xpos > 0) and (yneg < 0) and (ypos > 0) :
      tmin := 0; tmax := 360;
    elseif (p0 = (0,0)) : tmin :=    0; tmax :=  90;
    elseif (p1 = (0,0)) : tmin :=  -90; tmax :=   0;
    elseif (p2 = (0,0)) : tmin := -180; tmax := -90;
    elseif (p3 = (0,0)) : tmin :=   90; tmax := 180;
    else :
      tmax := tmin := t0 := angle p0;
      for j = 1 upto 3:
        t := t0 + angle (p[j] rotated -t0);
        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 (xpos > 0) :        % (1)
        rmin := emin(abs(yneg), abs(ypos));
      elseif (yneg < 0) and (ypos > 0) :    % (2)
        rmin := emin(abs(xneg), abs(xpos));
      else :                                % (3)
        rmin := min(r0, r1, r2, r3);
      fi
    fi
    save gridpic; picture gridpic; gridpic := nullpicture;
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{endpolargrid}
% The \gbc{clr} is \gbc{drawcolor} for line grids, \gbc{pointcolor} for
% dot grids. The 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) );
    coloraddto (clr) (active_plane) (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 ((0, 0), tstart, tstop, rstop) );
  safedraw zconv ( ((rstart, 0)--(rstop, 0)) rotated tstop );
enddef;

%    \end{macrocode}
%
% \section{Path construction}
%
% \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, 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.
%    \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*_uncle/Bob) = Bob[_uncle+1] - Bob0;
  endfor
  mkpoly (true) (Bob)
enddef;

%    \end{macrocode}
%
% The following set of commands accept any path as argument, but it is
% intended that it be a triangle. Even then, they work correctly only if
% it is a cycle.
%
% \DescribeRoutine{altitudept}
% \DescribeRoutine{altitude}
% These first two produce the perpendicular from \gbc{point n of t}
% to the (extension of) the opposite side (i.e., the altitude). The
% first one determines where the altitude meets the opposite side, and the
% second just connects the two points
%    \begin{macrocode}
vardef altitudept expr n of t =
  save A, B, C, zz; pair A, B, C, zz;
  A := pnt[n]     (t);
  B := pnt[n + 1] (t); % wraps around a cyclic path
  C := pnt[n + 2] (t);
  zz = whatever[B,C];
  zz = A + 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}
% \DescribeRoutine{median}
% These two produce the line from \gbc{point n of t} to the midpoint of
% the opposite side.
%    \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}
% \DescribeRoutine{anglebisector}
% These two produce the line from \gbc{point n of t} to the opposite side
% that bisects the angle there.
%    \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 0, 60 and 90 are designed to match the assumptions used
% in the \gbc{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 cornerangle (expr A, B, C) =
  if (A = B) and (B = C)    : 60
  elseif (B = C)            :  0
  elseif (A = B) or (A = C) : 90
  else:  angle ((C - A) rotated (-angle (B - 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 two 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, cyclic, pts)
  else      :  mkpoly (cyclic, pts)
  fi
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.
%    \begin{macrocode}
vardef mkpoly (expr cyclic) (suffix pts) =
  for _i = 1 upto pts-1:  pts[_i]-- endfor
  pts[pts] if cyclic : -- cycle 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}.
%    \begin{macrocode}
vardef polyline (expr cyclic) (text t) =
  save _pl; textpairs (_pl) (t);  mkpoly (cyclic, _pl)
enddef;

%    \end{macrocode}
%
% 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, other path building commands with
% tension parameters call \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. We should experiment with ``\mfc{tension atleast}'' here
% to see what difference it makes.
%    \begin{macrocode}
vardef mksmooth (expr tens, cyclic) (suffix pts) =
  pts1
  if pts = 1 :
    if cyclic : &cycle fi
  else:
    if cyclic :
      {pts[2]-pts[pts]}
    fi
    for _i = 2 upto pts-1:
      ..tension tens..pts[_i]{pts[_i+1]-pts[_i-1]}
    endfor
    ..tension tens..pts[pts]
    if cyclic :
      {pts[1]-pts[pts-1]}..tension tens..cycle
    fi
  fi
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{curve}
% \DescribeRoutine{tcurve}
% The old \cs{curve} command in \mfpic{} permitted no tension parameter
% and wrote a \grafbase{} \gbc{curve} command. For backward compatibility
% we keep that name, but simply call the \gbc{tcurve} command with the
% default value for tension. \gbc{tcurve} converts a list of pairs to an
% array, then calls \gbc{mksmooth} on the array.
%    \begin{macrocode}
def curve = tcurve (default_tension) enddef;
vardef tcurve  (expr tens, cyclic) (text t) =
  save _tc;  textpairs (_tc) (t);  mksmooth (tens, cyclic, _tc)
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{mkbezier}
% \DescribeRoutine{bezier}
% \DescribeRoutine{tbezier}
% 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.
%    \begin{macrocode}
vardef mkbezier (expr tens, cyclic) (suffix pts) =
  for _i = 1 upto pts-1 : pts[_i]..tension tens.. endfor
    pts[pts] if cyclic : ..tension tens..cycle fi
enddef;

def bezier = tbezier (default_tension) enddef;
vardef tbezier (expr tens, cyclic) (text t) =
  save _tsb;  textpairs (_tsb) (t);  mkbezier (tens, cyclic) (_tsb)
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{qbezier}
% \DescribeRoutine{mkqbezier}
% It also semed we ought to allow \mfpic{} users to easily reproduce the
% effect of a sequence of \LaTeX's  \cs{qbezier} commands. That's what
% these are for.
%
% These commands and the various splines 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.
%
% \gbc{mkqbezier} requires an even number of points for a cyclic path,
% an oddnumber for a noncyclic path. If \gbc{pts} has the wrong parity,
% the last point in the list is repeated. This makes the last segment
% either trivial or a straight line. We increment \gbc{_mqb} instead of
% \gbc{pts} so a user's suffix doesn't unexpectedly change.
%    \begin{macrocode}
vardef mkqbezier (expr cyclic) (suffix pts) =
  save _mqb;  _mqb := pts;
  if (cyclic and odd pts) or not (cyclic or odd pts):
    pts[incr _mqb] := pts[pts];
  fi
  if cyclic : pts[incr _mqb] := pts1; fi
  pts1
  for _i = 2 step 2 until _mqb - 1 :
    ..controls  1/3[pts[_i],pts[_i-1]] and 1/3[pts[_i], pts[_i+1]]
    ..pts[_i+1]
  endfor
  if cyclic : &cycle fi
enddef;

vardef qbezier (expr cyclic) (text t) =
  save _qbz;  textpairs (_qbz) (t);  mkqbezier (cyclic) (_qbz)
enddef;

%    \end{macrocode}
%
% For 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.
%
% \DescribeRoutine{mkqbs}
% For simplicity, the list is converted to an array \gbc{_oq} first and
% \gbc{mkqbs} is called. This draws an open spline based on the points in
% an array \gbc{b}.
%    \begin{macrocode}
vardef openqbs (text t) =
  save _oq;  textpairs (_oq) (t);  mkqbs (_oq)
enddef;

vardef closedqbs (text t) =
  save _cq; textpairs (_cq) (t);
  _cq[incr _cq] := _cq1;  _cq[incr _cq] := _cq2;
  mkqbs (_cq) & cycle
enddef;

vardef mkqbs (suffix b) =
  for _i = 1 upto b-2:
    0.5[b[_i], b[_i+1]]
      ..controls 1/6[b[_i+1], b[_i]] and 1/6[b[_i+1], b[_i+2]]..
  endfor
  0.5[b[b-1], b[b]]
enddef;

%    \end{macrocode}
% As for cubic B-splines, I'll have to trust the previous coder, as I
% didn't even know what a cubic B-spline was until I deduced it from his
% code. Earlier versions of \gbc{mkclosedcbs} would define
% \gbc{b[incr b]:=b1} and \gbc{b[incr b]:=b2}. I decided we shouldn't
% change the values of variables associated with the given suffix \gbc{b}
% and so now we use \gbc{mkopencbs} to get most of the way around and then
% fill in the gap with an explicit call to \gbc{opencbs}).
%    \begin{macrocode}
vardef mkopencbs (suffix b) =
  for _i = 1 upto b-3:
    (b[_i]+4b[_i+1]+b[_i+2])/6
      ..controls 1/3[b[_i+1], b[_i+2]] and 2/3[b[_i+1], b[_i+2]]..
  endfor
  (b[b-2]+4b[b-1]+b[b])/6
enddef;
vardef mkclosedcbs (suffix b) =
  mkopencbs (b) & opencbs (b[b-2],b[b-1],b[b], b1, b2, b3) & cycle
enddef;

vardef opencbs (text t) =
  save _oc;  textpairs (_oc) (t);  mkopencbs (_oc)
enddef;
vardef closedcbs (text t) =
  save _clc; textpairs (_clc) (t);  mkclosedcbs (_clc)
enddef;

%    \end{macrocode}
% When calling \gbc{curve} or \gbc{tcurve} there 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 requirement to guarantee this is
% that the control points of the segment connecting $(x\sb1, y\sb1)$ to the
% next $(x\sb2, y\sb2)$ have their \gbc{xpart} in the interval $x\sb1 < x
% < x\sb2$.
%
% 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.
% A related problem is to keep the path under control. That is, the
% segment of the curve connecting $(x\sb1, y\sb1)$ to the next $(x\sb2,
% y\sb2)$ should have \gbc{ypart} within an interval not too much larger
% than the interval $y\sb1 < y < y\sb2$.
%
% We accomplish both these tasks at once by making the vector from
% $(x\sb n, y\sb n)$ to its \mfc{postcontrol} have length less than
% $|x\sb{n+1} - x\sb n|$, and the same for the vector from $(x\sb{n+1},
% y\sb{n+1})$ to its \mfc{precontrol}
%
% 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. For graphing
% functions, we average the slopes instead. An added refinement is that
% this is a weighted average, with the nearer x-coordinate being weighted
% more.
%
% Finally, we permit a tension of sorts by dividing the distance to the
% controls by a parameter \gbc{fcn_tension}.
%
% \DescribeRoutine{fcncontrol}
% This computes the control point for the points on the path. The
% parameters \gbc{X, Y, Z} are three successive points of the path to be
% constructed. If they are given in order, it gives the postcontrol of
% \gbc{Y}. If they are in reverse order, the precontrol is obtained.
% Oddly enough, the addition of the trap for \gbc{dl=0} or \gbc{dr=0}
% made it possible to trivially extend the array (in
% \gbc{functioncurve}) and get better looking results than either method
% used before this.
%
% \DescribeRoutine{mkfcnpath}
% This produces the path, calling \gbc{fcncontrol} to produce the controls.
%
% \DescribeRoutine{functioncurve}
% \DescribeRoutine{fcncurve}
% This is the interface; \gbc{fcncurve} calls \gbc{functioncurve} with the
% default tension, which 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) =
  save dl, dr, before, after;  pair before, after;
  before := Y - X;  after := Z - Y;
  dl := xpart (before);  dr := xpart (after);
  if (dr = 0) or (dl = 0):
    Y + abs(dr)/ftens * sgn before
  else:
    Y + abs(dr)/ftens * unitvector (before*dr/dl + after*dl/dr)
  fi
enddef;

vardef  mkfcnpath (expr ftens) (suffix q) =
  for _i = 1 upto q - 1:
    q[_i]..controls fcncontrol (ftens) (q[_i-1], q[_i],  q[_i+1])
           and fcncontrol (ftens) (q[_i+2], q[_i+1], q[_i])..
  endfor
  q[q]
enddef;

def fcncurve = functioncurve (emax(1.2default_tension, eps)) enddef;
vardef functioncurve (expr ftens) (text t) =
  save _fc; textpairs (_fc) (t);
  if _fc > 1 :  _fc0 :=  _fc1;  _fc[_fc+1] := _fc[_fc]; fi
  mkfcnpath (ftens)(_fc)
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 amount
% 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 seems
% to have been abandoned at some point in the development of \grafbase.
%    \begin{macrocode}
vardef turtle (text t) =
  save _tu; pair _tu[]; _tu := 0; _tu0 := (0, 0);
  for _a = t: _tu[incr _tu] := _tu[_tu - 1] + _a; endfor
  mkpoly (false, _tu)
enddef;

%    \end{macrocode}
%
%
% \section{Arcs, Circles and Ellipses}
%
% 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{from} in \MF, then the other values on the
% arc should be no problem.
%
% \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 over-determine the arc. It
% is only called by arc making commands that have calculated these
% parameters. \gbc{mkarc}'s job is to ensure that the arc begins at
% \gbc{from} and ends at \gbc{to} (exactly).
%    \begin{macrocode}
vardef mkarc (expr center, from, to, sweep) =
  save n, d; pair d;
  n := ceiling (abs(sweep)/45);
  d := (from - center) rotated (signof (sweep) 90);
  from{d}
  for j = 1 upto n-1 :
    ..(from rotatedabout (center, j/n*sweep)){d rotated (j/n*sweep)}
  endfor ..to{d rotated sweep}
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, from, sweep) =
  if (center = from) or (sweep = 0) :
    from--from
  else:
    save to; pair to;
    to := from rotatedabout (center, sweep);
    mkarc (center, from, to, sweep)
  fi
enddef;
def arccps = arc enddef;

%    \end{macrocode}
%
% \DescribeRoutine{arccenter}
% For arcs greater than 90 degrees we will convert to the above basic
% \gbc{mkarc}. Since two of the methods don't provide the center among the
% parameters, we use the utility \gbc{arccenter} to locate that center,
% given two points and the angle. This can fail (arithmetic overflow) if
% the angle is too small relative to the distance between the points.
% Therefore, we try not to call it for small angles (or angles near
% multiples of 360).
%
% We find the center by solving equations representing two lines which
% must be perpendicular to the circle. Which two lines we use depends on
% the sweep. For accurate solutions we want the angle between the two
% lines to be closer to 90 than to 0. \gbc{ang} is the angle we need to
% rotate the chord \gbc{(to - from)} to be perpendicular to the circle at
% \gbc{from}. \gbc{cd} is a vector in the direction of the chord. When
% \gbc{ang} is close to $\pm90$, we use the lines perpendicular to the circle
% at \gbc{from} and \gbc{to}. Otherwise we use the lines perpendicular
% to the circle at \gbc{from} and perpendicular to the chord at its
% midpoint. The latter is better when \gbc{ang} is between $-30$ and $30$
% degrees.
%    \begin{macrocode}
vardef arccenter (expr from, to, sweep) =
  save ang, c;
  pair c;
  ang := 90 - (sweep mod 360)/2; %  -90 < ang <= 90
  if (abs(ang) = 90) or (from = to) :
    GBmsg "The central point of this arc is undefined. " &
      "Using midpoint of chord instead.";
    0.5[from, to]
  else:
    save cd; pair cd;  cd := to - from;
    c = from + whatever*(cd rotated ang);
    if abs(ang) < 30 :
      c = (0.5)[from, to] + whatever*(cd rotated 90);
    else:
      c = to + whatever*(-cd rotated -ang);
    fi
    c
  fi
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{midarc}
% This finds the midpoint of the arc determined by two points and an
% angle. It work because the angle subtended at \gbc{from} by an arc of
% length \gbc{sweep/2} is \gbc{sweep/4}. We use it for small angles, as
% we can then draw the arc without having to find its center.
%    \begin{macrocode}
vardef midarc (expr from, to, sweep) =
  save m, cd; pair m, cd;
  cd := to - from;
  m = from + whatever*( cd rotated (-sweep/4));
  m = 0.5[from, to] + whatever*(cd rotated 90);
  m
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 midpint \gbc{m} of the
% arc. Otherwise, we get the center of the circle and call \gbc{mkarc}.
%    \begin{macrocode}
vardef arcpps (expr from, to, sweep) =
  if ((sweep mod 360) = 0) or (from = to) :
    GBmsg "Undefined arc. A line segment will be used instead.";
    from--to
  elseif abs(sweep) <= 90 :
    save cd; pair cd; cd := to - from;
    if abs(sweep) <= 45 :
      from{cd rotated (-sweep/2)}..to{cd rotated (sweep/2)}
    else:
      from{cd rotated (-sweep/2)}..midarc(from, to, sweep){cd}
        ..to{cd rotated (sweep/2)}
    fi
  else:
    save center; pair center;
    center := arccenter (from, to, sweep);
    mkarc (center, from, to, sweep)
  fi
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:
    save from, to; pair from, to;
    from := center + rad*dir frtheta;
    to   := center + rad*dir totheta;
    if frtheta = totheta :
      from--to
    else:
      mkarc (center, from, to, totheta - frtheta)
    fi
  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, anglefrom, angleto) =
  arcplr (center, anglefrom, angleto, 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}
% \DescribeRoutine{circle}
% We get an ellipse by x-scaling and y-scaling 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.
%    \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}
% \DescribeRoutine{circlecp}
% \DescribeRoutine{circleppp}
% \DescribeRoutine{circlepps}
% The next three implement different ways of specifying a circle. The
% first produces the circle with a given center passing through a given
% point. The second produces the circle passing through three given
% points. 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.
%    \begin{macrocode}
vardef circlecp (expr center, point) =
  mkarc (center, point, point, 360) & cycle
enddef;
vardef circleppp (expr one, two, three) =
  save ang; numeric ang[];
  ang0 := cornerangle(three, one, two);
  ang1 := cornerangle(one, two, three);
  ang2 := cornerangle(two, three, one);
  arcpps (one, two, 2ang0) & arcpps (two, three, 2ang1) &
    arcpps (three, one, 2ang2) & cycle
enddef;
vardef circlepps (expr one, two, sweep) =
  save ang, full; numeric ang[], full;
  full := signof (sweep) 360;
  ang1 := sweep mod (full);
  ang2 := full - ang1;
  arcpps (one, two, ang1) & arcpps (two, one, ang2) & cycle
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{pathcenter}
% This finds the center of a circle. For other paths, the point found
% may be meaningless (but it will also obtain the center of an arc or a
% rectangle). It takes three 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}
% \DescribeRoutine{circumcircle}
% \DescribeRoutine{incircle}
% \DescribeRoutine{excircle}
% \DescribeRoutine{ninepointcircle}
% These four create the relevant circles from a given triangle. The
% triangle is specified as a path expression, so they produce results for
% any path, but make sense only for a cyclic triangular path.
%    \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);
  % Find the tangent points on the sides. E.g., a is the common
  % distance from A to the tangent points on the adjacent sides.
  save a, b, c;
  a + b     = abs (B-A);
      b + c = abs (C-B);
  a     + c = abs (A-C);
  circleppp (A + a*unitvector (B-A),
             B + b*unitvector (C-B),
             C + c*unitvector (A-C))
enddef;

vardef excircle expr n of t =
  save A, B, C; pair A, B, C;
  A := pnt[n]     (t);
  B := pnt[n + 1] (t); % wraps around
  C := pnt[n + 2] (t);
  save a, b, c;
  a - b     = abs (B-A);
      b + c = abs (C-B);
  a     - c = abs (C-A);
  circleppp (A + a*unitvector(B-A),
             B + b*unitvector(C-B),
             C + c*unitvector(C-A))
enddef;

vardef ninepointcircle expr t =
  circleppp (medianpt 0 of t, medianpt 1 of t, medianpt 2 of t)
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{circumcenter}
% \DescribeRoutine{incenter}
% \DescribeRoutine{excenter}
% \DescribeRoutine{ninepointcenter}
% \DescribeRoutine{barycenter}
% These find various centers associated with a triangle. The last one is
% made to work for any path.
%    \begin{macrocode}
vardef circumcenter expr t = pathcenter circumcircle t enddef;
vardef incenter expr t = pathcenter incircle t enddef;
vardef excenter expr n of t = pathcenter excircle n of t enddef;
vardef ninepointcenter expr t = pathcenter ninepointcircle t enddef;

% Make this work for any path.
% Divide as we go,
% decrease chance of overflow.
vardef barycenter expr t =
  save n, m; n := length t; m := n + 1;
  save xxx;
  xxx : = pnt0 (t)/m  for k = 1 upto n-1 :  + pnt[k] (t)/m  endfor;
  if cycle t:  xxx*(1 + 1/n)
  else:  xxx + pnt[n] (t)/m
  fi
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}
%
%
% \section{Plotting of functions}
%
% In these macros, if the boolean argument \gbc{smooth} is true then the
% path returned will be a B\'ezier, otherwise it will be a polyline. The
% parameter is simply passed to \gbc{mkpath}. 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.
%
% All of 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
% \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)}
%    \begin{macrocode}
vardef mkfcn (expr smooth, 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 (smooth, tens, false, _p)
enddef;
% compatibility:
def tfcn (expr smooth) = mkfcn (smooth, 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 smooth) = tparafcn (smooth, 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 smooth) (expr xmin, xmax, st) (text _fx) =
  save _fp;  vardef _fp (expr _x) = (_x, _fx(_x)) enddef;
  mkfcn (smooth, default_tension) (xmin, xmax, st) (_fp)
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{function}
% This is to \gbc{xfcn} as \gbc{parafcn} is to \gbc{mkfcn}: it
% takes a text argument and copies 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.
%    \begin{macrocode}
def function (expr smooth) = tfunction (smooth, default_tension) enddef;
vardef tfunction (expr smooth, tens) (expr xmin, xmax, st) (text _fx) =
  save _fp;  vardef _fp (expr x) = (x, _fx) enddef;
  mkfcn (smooth, tens) (xmin, xmax, st) (_fp)
enddef;

def btwnfcn (expr sm) = tbtwnfcn (sm, default_tension) enddef;
vardef tbtwnfcn (expr sm, tn)(expr xlo, xhi, st)(text _fx)(text _gx) =
  tfunction (sm, tn) (xlo, xhi, st) (_fx) --
  ( reverse tfunction (sm, tn) (xlo, xhi, st) (_gx) ) -- 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 smooth) (expr tmin, tmax, st) (text ft) =
  save _fq;  vardef _fq (expr t) = (ft(t)) * (dir t) enddef;
  mkfcn (smooth, default_tension) (tmin, tmax, st) (_fq)
enddef;

%    \end{macrocode}
%
% \DescribeRoutine{plrfcn}
% This is to \gbc{rfcn} as \gbc{parafcn} is to \gbc{mkfcn}: the
% text argument should be code that can be copied literally into a
% \mfc{vardef} creating a numeric function with a literal \gbc{t} as the
% parameter (representing $\theta$).
%    \begin{macrocode}
def plrfcn (expr smooth) = tplrfcn (smooth, default_tension) enddef;
vardef tplrfcn (expr smooth, tens) (expr tmin, tmax, st) (text ft) =
  save _fq;  vardef _fq (expr t) = (ft) * (dir t) enddef;
  mkfcn (smooth, tens) (tmin, tmax, st) (_fq)
enddef;

%    \end{macrocode}
%
%
% \section{Pie charts and bar charts}
%
% \DescribeRoutine{piechart}
% The \gbc{piechart} 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}.
%
% After the calculations, 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}
vardef piechart (expr sign, ang, cent, rad) (text data) =
  save _sum, _tot;
  numeric piewedge; piewedge := 0;
  numeric pieangle, pieangle[]; pieangle0 := 0;
  for _val = data :
    pieangle[incr piewedge] := pieangle[piewedge - 1] + _val;
  endfor
  _tot := pieangle[piewedge];
  pair piecenter; piecenter := cent;
  path piewedge[];
  numeric piedirection; pair piedirection[];
  pieangle[piewedge + 1] = ang + sign*360;
  for _n = piewedge downto 1 :
    pieangle[_n] := ang + sign*pieangle[_n - 1]/_tot*360;
    piewedge[_n] =
        sector(cent, rad, pieangle[_n], pieangle[_n+1]);
    piedirection[_n] := dir(0.5[ pieangle[_n], pieangle[_n+1] ]);
  endfor
  piedirection := pieangle := piewedge;
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 \gbc{vert} is true, otherwise horizontal.
%
% \gbc{start} 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 y-parts),
% \gbc{barbegin[]} holds their leftmost or bottommost coordinates (either
% 0 or the x-parts 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 x-part as the beginning
% of the bar and the y-part 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 omitted it.
%    \begin{macrocode}
def barchart (expr start, 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 _n = 1 upto chartbar :
    barstart[_n]  := start + sep*(_n-1);
    barlength[_n] := barend[_n];
    chartbar[_n]  :=  rect ((barbegin[_n], 0), ( barend[_n], barwd) )
          shifted (0, barstart[_n]) if vert: xyswap fi;
  endfor
enddef;

%    \end{macrocode}
%
%^^A Overlays - taken from MFbook, p 295.  (Bruce Leban)
%
% \section{Overlays}
%
% This final code predates me. I've never seen it used and don't know what
% its for. For the \MP{} version I just tried to make sure everything was
% defined in \MP{} or \file{plain.mp} and otherwise left it alone.
%    \begin{macrocode}
picture totalpicture;
boolean totalnull, currentnull;
def clearit =
  currentpicture := totalpicture := nullpicture;
  currentnull := totalnull := true;
enddef;

def keepit =
%<MF>  mono (currentpicture);
  addto totalpicture also currentpicture;
  currentpicture := nullpicture;
  totalnull := 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;
%<MF>      mono (_v_);
      addto _v_ also totalpicture;
      do _v_
    endgroup
  fi
enddef;

%    \end{macrocode}
% This apparently redundant definition (\gbc{shipit} = \gbc{shipit_}) is
% so that \mfpic{} can turn shipping off and back on by redefining
% \gbc{shipit} to either \gbc{shipit_} or \mfc{relax}.
%    \begin{macrocode}
def shipit_ =
  mergeit (shipout)
enddef;
def shipit = shipit_ 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. And that's all.
%    \begin{macrocode}
numeric gcode; gcode := 0;

%<MF>%  end grafbase.mf
%<MP>%  end grafbase.mp
%</MF|MP>
%    \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:
%    \begin{macrocode}
%<*dvips>
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;

%    \end{macrocode}
% The function \gbc{cmyk} (which converts a CMYK quadruple to \MP's rgb
% triple) is defined in \file{grafbase.mp}, which should be input before
% \file{dvipsnam.mp}:
%    \begin{macrocode}
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