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% MetaPost graph doc, by John Hobby.  License at end.
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\author{John D. Hobby}
\title{Drawing Graphs with {MetaPost}}
\date{}


\newcommand\myabstract{%
This paper describes a graph-drawing package that has been implemented
as an extension to the MetaPost graphics language.  MetaPost has
a powerful macro facility for implementing such extensions.  There are
also some new language features that support the graph macros.
Existing features for generating and manipulating pictures
allow the user to do things that would be difficult to achieve
in a stand-alone graph package.}

\newcommand\mykeywords{%
    typesetting; graphs; MetaPost}

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\usepackage{hyperref}
\hypersetup{
  pdftitle={Drawing Graphs with MetaPost},
  pdfauthor={John D. Hobby and the MetaPost development team},
  pdfkeywords={typesetting, graphs, MetaPost, TeX}
}
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\begin{document}
  \maketitle
  \begin{abstract} \myabstract \end{abstract}
  \ifx\keywords\undefined \else
    \begin{keywords} \mykeywords \end{keywords}
  \fi

\setlength{\columnsep}{2.5em}
\begin{multicols}{2}
\tableofcontents
\end{multicols}

\section{Introduction}
\label{intro}

MetaPost is a batch-oriented graphics language based on Knuth's \MF\footnote{\MF\
is a trademark of Addison Wesley Publishing Company.}, but with
PostScript\footnote{PostScript is a registered trademark of Adobe Systems Inc.}
output and numerous features for integrating text and graphics.
The author has tried to make this paper as independent as possible of the
user's manual~\cite{ho:mp3}, but fully appreciating all the material requires
some knowledge of the MetaPost language.

We concentrate on the mechanics of producing particular kinds of graphs
because the question of what type of graph is best in a given situation
is covered elsewhere; e.g., Cleveland~\cite{Cleve85,Cleve93,Cleve93a} and
Tufte~\cite{Tufte83}.
The goal is to provide at least the power of UNIX\footnote{UNIX is a registered
trademark of UNIX System Laboratories, Inc.} {\it
grap\/}~\cite{BenKer90}, but within the MetaPost language.
Hence the package is implemented using MetaPost's powerful macro facility.

The graph macros provide the following functionality:
\begin{enumerate}
\item Automatic scaling
\item Automatic generation and labeling of tick marks or grid lines
\item Multiple coordinate systems
\item Linear and logarithmic scales
\item Separate data files
\item Ability to handle numbers outside the usual range
\item Arbitrary plotting symbols
\item Drawing, filling, and labeling commands for graphs
\end{enumerate}
In addition to these items, the user also has access to all the features
described in the MetaPost user's manual~\cite{ho:mp3}.
These include access to almost all the features of PostScript,
ability to use and manipulate typeset text,
ability to solve linear equations,
and data types for points, curves, pictures, and coordinate transformations.

Section~\ref{grmac} describes the graph macros from a user's perspective
and presents several examples.  Sections \ref{nummac} and~\ref{formsec} discuss
auxiliary packages for manipulating and typesetting numbers and Section~\ref{concl}
gives some concluding remarks.  Appendix~\ref{summsec} summarizes the graph-drawing
macros.


\section{Using the Graph Macros}
\label{grmac}

A MetaPost input file that uses the graph macros should begin with
$$ \hbox{\tt input graph} $$
This reads a macro file {\tt graph.mp} and defines the graph-drawing commands
explained below.  The rest of the file should be one or more instances of
$$ \vbox{\hbox{\tt beginfig($\descr{figure number}$);}
        \hbox{\tt $\descr{graphics commands}$ endfig;}}
$$
followed by {\tt end}.

The following \tdescr{graphics commands} suffice to generate the graph in
Figure~\ref{fig1} from the data file {\tt agepop91.d}:
$$\begin{verbatim}
draw begingraph(3in,2in);
  gdraw "agepop91.d";
  endgraph;
\end{verbatim}
$$
(Each line of {\tt agepop91.d} gives an age followed the estimated number of
Americans of that age in 1991 \cite{Census92}.)

\begin{figure}[htp]
$$ \includegraphics{mpgraph-1.mps} $$
\caption{A graph of the 1991 age distribution in the United States}
\label{fig1}
\end{figure}

\subsection{Basic Graph-Drawing Commands}

% begingraph, endgraph, gdraw

All graphs should begin with
$$ \hbox{\tt begingraph($\descr{width}$,$\descr{height}$);} $$
and end with {\tt endgraph}.  This is syntactically a \tdescr{picture expression},
so it should be preceded by {\tt draw} and followed by a semicolon as in the
example.\footnote{See the User's Manual~\cite{ho:mp3} for explanations of {\tt draw}
commands and syntactic elements like \tdescr{picture expression}.}
The \tdescr{width} and \tdescr{height} give the dimensions of the
graph itself without the axis labels.

The command
$$ {\tt gdraw}\ \descr{expression}\ \descr{option list} $$
draws a graph line.  If the \tdescr{expression} is of type string, it names a data
file; otherwise it is a path that gives the function to draw.  The
\tdescr{option list} is zero or more drawing options
$$ {\tt withpen} \descr{pen expression}
   \mid {\tt withcolor} \descr{color expression}
   \mid {\tt dashed} \descr{picture expression}
$$
that give the line width, color, or dash pattern as explained in the User's
Manual~\cite{ho:mp3}.

% plot <picture>

In addition to the standard drawing options, the \tdescr{option list} in a
{\tt gdraw} statement can contain
$$ {\tt plot}\ \descr{picture expression} $$
The \tdescr{picture expression} gives a plotting symbol to be drawn at each
path knot.  The {\tt plot} option suppresses line drawing so that%
\footnote{Troff users should replace {\tt btex \$\string\bullet\$ etex} with
{\tt btex \string\(bu etex}.}
\Ignore{\)}% Help delatex so spell will work
$$ \hbox{\verb|gdraw "agepop91.d" plot btex $\bullet$ etex|} $$
generates only bullets as shown in Figure~\ref{fig2}.
(Following the {\tt plot} option with a {\tt withpen} option would
cause the line to reappear superimposed on the plotting symbols.)

\begin{figure}[htp]
$$ \includegraphics{mpgraph-2.mps} $$
\caption{The 1991 age distribution plotted with bullets}
\label{fig2}
\end{figure}

Watch out for the following: the \tdescr{picture expression} is placed
with the lower-left corner at the path knot, not its center. If you
want it to be dead-center, you have to correct the placement
yourself. For the example above, you need something like this instead:

\medskip
\begin{verbatim}
def MPbullet = 
   btex \lower\fontdimen22\cmsy \hbox to 0pt{\hss\cmsy\char15\hss} etex 
enddef;
\end{verbatim}

\medskip\noindent
followed by:
$$ \hbox{\verb|gdraw "agepop91.d" plot MPbullet|} $$


% glabel, gdotlabel, OUT

The {\tt glabel} and {\tt gdotlabel} commands add labels to a graph.  The
syntax for {\tt glabel} is
$$ {\tt glabel.}\ \descr{label suffix}
    \hbox{\tt ($\descr{string or picture expression}$, $\descr{location}$)}
    \ \descr{option list}
$$
where \tdescr{location} identifies the location being labeled and
\tdescr{label suffix} tells how the label is offset relative to that location.
The {\tt gdotlabel} command is identical, except it marks the location with a
dot.  A \tdescr{label suffix} is as in plain MetaPost:
\tdescr{empty} centers the label on the location;
{\tt lft}, {\tt rt}, {\tt top}, {\tt bot} offset the label horizontally or
vertically; and {\tt ulft}, {\tt urt}, {\tt llft}, {\tt lrt} give diagonal
offsets.  The \tdescr{location} can be a pair of graph coordinates, a knot
number on the last {\tt gdraw} path, or the special location {\tt OUT}.
Thus
$$ \hbox{\verb|gdotlabel.top(btex $(50,0)$ etex, 50,0)|} $$
would put a dot at graph coordinates {\tt(50,0)} and place the typeset text
``$(50,0)$'' above it.  Alternatively,
$$ \hbox{\verb|glabel.ulft("Knot3", 3)|} $$
typesets the string {\tt "Knot3"} and places it above and to the left of
Knot~3 of the last {\tt gdraw} path.  (The knot number 3 the path's ``time''
parameter~\cite[Section 8.2]{ho:mp3}.)

The \tdescr{location} {\tt OUT} places a label relative to the whole graph.
For example, replacing ``{\tt gdraw "agepop91.d"}'' with
$$\begin{verbatim}
glabel.lft(btex \vbox{\hbox{Population} \hbox{in millions}} etex, OUT);
glabel.bot(btex Age in years etex, OUT);
gdraw "agepopm.d";
\end{verbatim}
$$
in the input for Figure~\ref{fig1} generates Figure~\ref{fig3}.
This improves the graph by adding axis labels and using a new data file
{\tt agepopm.d} where the populations have been divided by one million to avoid
large numbers.  We shall see later that simple transformations such as this can
be achieved without generating new data files.

\begin{figure}[htp]
$$ \includegraphics{mpgraph-3.mps} $$
\caption{An improved version of the 1991 age distribution graph}
\label{fig3}
\end{figure}

All flavors of \TeX\ can handle multi-line labels via the \verb|\hbox| within
\verb|\vbox| arrangement used above, but \LaTeX\ users will find it more natural
to use the {\tt tabular} environment~\cite{LaTeXman}.
Troff user's can use nofill mode:
$$\begin{verbatim}
btex .nf
Population
in millions etex
\end{verbatim}
$$


\subsection{Coordinate Systems}
\label{coords}

The graph macros automatically shift and rescale coordinates from data files,
{\tt gdraw} paths, and {\tt glabel} locations to fit the graph.  Whether the
range of $y$~coordinates is 0.64 to 4.6 or 640,000 to 4,600,000, they
get scaled to fill about 88\% of the height specified in the {\tt begingraph}
statement.  Of course line widths, labels, and plotting symbols are not
rescaled.

% setrange

The {\tt setrange} command controls the shifting and rescaling process by
specifying the minimum and maximum graph coordinates:
$$ \hbox{\tt setrange($\descr{coordinates}$,\,$\descr{coordinates}$)} $$
where
\begin{ctabbing}
$\tt \descr{coordinates} \rightarrow \descr{pair expression}$\\
$\tt \qquad \;|\; \descr{numeric or string expression}\hbox{\tt ,}
        \descr{numeric or string expression}$
\end{ctabbing}
The first \tdescr{coordinates} give $(x_{\rm min},y_{\rm min})$ and the second give
$(x_{\rm max},y_{\rm max})$.  The lines $x=x_{\rm min}$, $x=x_{\rm max}$,
$y=y_{\rm min}$, and $y=y_{\rm max}$ define the rectangular frame around the graph
in Figures \ref{fig1}--\ref{fig3}.  For example, an adding a statement
$$ \hbox{\tt setrange(origin, whatever,\,whatever)} $$
to the input for Figure~\ref{fig3} yields Figure~\ref{fig4}.
The first \tdescr{coordinates} are given by the predefined pair constant
{\tt origin}, and the other coordinates are left unspecified.  Any unknown
value would work as well, but {\tt whatever} is the standard MetaPost
representation for an anonymous unknown value.

\begin{figure}[htp]
$$ \begin{verbatim}
draw begingraph(3in,2in);
 glabel.lft(btex \vbox{\hbox{Population} \hbox{in millions}} etex, OUT);
 glabel.bot(btex Age in years etex, OUT);
 setrange(origin, whatever,whatever);
 gdraw "agepopm.d";
 endgraph;
\end{verbatim}
\atop
\includegraphics{mpgraph-4.mps}
$$
\caption{The 1991 age distribution graph and the input that creates it.}
\label{fig4}
\end{figure}

% strings for big coordinates

Notice that the syntax for {\tt setrange} allows coordinate values to be given
as strings.  Many commands in the graph package allow this option.  It is
provided because the MetaPost language uses fixed point numbers that must be
less than 32768.  This limitation is not as serious as it sounds because good
graph design dictates that coordinate values should be ``of reasonable
magnitude''~\cite{Cleve85,Tufte83}.  If you really want $x$ and $y$
to range from 0 to 1,000,000,
$$ \hbox{\tt setrange(origin, "1e6",\,"1e6")} $$
does the job.  Any fixed or floating point representation is acceptable
as long as the exponent is introduced by the letter ``{\tt e}''.

% setcoords

Coordinate systems need not be linear.  The {\tt setcoords} command allows
either or both axes to have logarithmic spacing:
\begin{ctabbing}
$\tt \descr{coordinate setting} \rightarrow setcoords\hbox{\tt (}
        \descr{coordinate type}\hbox{\tt ,}\,\descr{coordinate type}\hbox{\tt )}$\\
$\tt \descr{coordinate type} \rightarrow
        log \;|\; linear \;|\; \hbox{\tt -}log \;|\; \hbox{\tt -}linear$
\end{ctabbing}
A negative \tdescr{coordinate type} makes $x$ (or $y$) run backwards so it is
largest on the left side (or bottom) of the graph.

Figure~\ref{fig5} graphs execution times for two matrix multiplication algorithms
using
$$ \hbox{\tt setcoords(log,log)} $$
to specify logarithmic spacing on both axes.  The data file {\tt matmul.d} gives
timings for both algorithms:
$$ \hbox to\hsize{\footnotesize\hfil$
\begin{verbatim}
20      .007861  standard MM: size, seconds
30      .022051
40      .050391
60      .15922
80      .4031
120     1.53
160     3.915
240     18.55
320     78.28
480     279.24

20      .006611  Strassen: size, seconds
30      .020820
40      .049219
60      .163281
80      .3975
120     1.3125
160     3.04
240     9.95
320     22.17
480     72.60
\end{verbatim}
\hfil$}
$$
A blank line in a data file ends a data set.  Subsequent {\tt gdraw} commands
access additional data sets by just naming the same data file again.
Since each line gives one $x$~coordinate and one $y$~coordinate, commentary
material after the second data field on a line is ignored.

\begin{figure}[htp]
$$ \mathcenter{\includegraphics{mpgraph-5.mps}}
 \quad
\begin{BVerbatim}[baseline=c]
draw begingraph(2.3in,2in);
 setcoords(log,log);
 glabel.lft(btex Seconds etex,OUT);
 glabel.bot(btex Matrix size etex,
   OUT);
 gdraw "matmul.d" dashed evenly;
 glabel.ulft(btex Standard etex,8);
 gdraw "matmul.d";
 glabel.lrt(btex Strassen etex,7);
 endgraph;
\end{BVerbatim}
$$
\caption{Timings for two matrix multiplication algorithms with the corresponding
        MetaPost input.}
\label{fig5}
\end{figure}

Placing a {\tt setcoords} command between two {\tt gdraw} commands graphs two
functions in different coordinate systems as shown in Figure~\ref{fig6}.
Whenever you give a {\tt setcoords} command, the interpreter examines what has
been drawn, selects appropriate $x$ and $y$ ranges, and scales everything to
fit.  Everything drawn afterward is in  a new coordinate system that need not
have anything in common with the old coordinates unless {\tt setrange} commands
enforce similar coordinate ranges.  For instance, the two {\tt setrange}
commands force both coordinate systems to have $x$ ranging from 80 to~90 and
$y$~starting at 0.

\begin{figure}[htp]
$$ \begin{verbatim}
draw begingraph(6.5cm,4.5cm);
 setrange(80,0, 90,whatever);
 glabel.bot(btex Year etex, OUT);
 glabel.lft(btex \vbox{\hbox{Emissions in} \hbox{thousands of}
   \hbox{metric tons} \hbox{(heavy line)}}etex, OUT);
 gdraw "lead.d" withpen pencircle scaled 1.5pt;
 autogrid(,otick.lft);
 setcoords(linear,linear);
 setrange(80,0, 90,whatever);
 glabel.rt(btex \vbox{\hbox{Micrograms} \hbox{per cubic}
   \hbox{meter of air} \hbox{(thin line)}}etex, OUT);
 gdraw "lead.d";
 autogrid(otick.bot,otick.rt);
 endgraph;
\end{verbatim}
  \atop
\mathcenter{\includegraphics{mpgraph-6.mps}}
$$
\caption{Annual lead emissions and average level at atmospheric monitoring
        stations in the United States.  The MetaPost input is shown above
        the graph.}
\label{fig6}
\end{figure}

% autogrid

When you use multiple coordinate systems, you have to specify where the axis
labels go.  The default is to put tick marks on the bottom and the left side of
the frame using the coordinate system in effect when the {\tt endgraph} command
is interpreted.  Figure~\ref{fig6} uses the
$$ \hbox{\tt autogrid(,otick.lft)} $$
to label the left side of the graph with the $y$~coordinates in effect before
the {\tt setcoords} command.  This suppresses the default axis labels, so another
{\tt autogrid} command is needed to label the bottom and right sides of the
graph using the new coordinate system.  The general syntax is
$$ \hbox{$\tt autogrid\hbox{\tt (}\descr{axis label command}\hbox{\tt ,}\,
        \descr{axis label command}\hbox{\tt )}\ \descr{option list}$}
$$
where
\begin{ctabbing}
$\tt \descr{axis label command} \rightarrow \descr{empty}
        \;|\; \descr{grid or tick}\, \descr{label suffix}$\\
$\tt \descr{grid or tick} \rightarrow grid \;|\; itick \;|\; otick$
\end{ctabbing}
The \tdescr{label suffix} should be {\tt lft}, {\tt rt}, {\tt top},
or {\tt bot}.

The first argument to {\tt autogrid} tells how to label the $x$~axis and the
second argument does the same for~$y$.  An \tdescr{empty} argument suppresses
labeling for that axis.  Otherwise, the \tdescr{label suffix} tells which side
of the graph gets the numeric label.  Be careful to use {\tt bot} or {\tt top}
for the $x$~axis and {\tt lft} or {\tt rt} for the $y$~axis.  Use {\tt otick}
for outward tick marks, {\tt itick} for inward tick marks, and {\tt grid} for
grid lines.  The \tdescr{option list} tells how to draw the tick marks or grid
lines.  Grid lines tend to be a little overpowering, so it is a good idea to
give a {\tt withcolor} option to make them light gray so they do not make the
graph too busy.


\subsection{Explicit Grids and Framing}

% otick, itick, grid, (format)

In case {\tt autogrid} is not flexible enough, axis label commands generate
grid lines or tick marks one at a time.  The syntax is
$$ \descr{grid or tick}\hbox{\tt .}\descr{label suffix}
        \hbox{\tt ($\descr{label format}$,\,$\descr{numeric or string expression}$)}
        \ \descr{option list}
$$
where \tdescr{grid or tick} and \tdescr{label suffix} are as in {\tt autogrid},
and \tdescr{label format} is either a format string like \verb|"%g"| or a
picture containing the typeset numeric label.

The axis label commands use a macro
$$ \hbox{\tt format($
        \descr{format string}$,\,$\descr{numeric or string expression}$)}
$$
to typeset numeric labels.  Full details appear in Section~\ref{formsec},
but when the \tdescr{format string} is \verb|"%g"|, it uses decimal notation
unless the number is large enough or small enough to require scientific notation.

The example in Figure~\ref{fig7} invokes
$$ \hbox{\verb|format("%g",y)|} $$
explicitly so that grid lines can be placed at transformed coordinates.
It defines the transformation ${\tt newy}(y)=y/75+\ln y$ and shows that
this function increases almost linearly.\footnote{The manual~\cite{ho:mp3}
explains how {\tt vardef} defines functions and {\tt mlog} computes logarithms.}
This is a little like using logarithmic $y$-coordinates, except that $y$ is
mapped to $y/75+\ln y$ instead of just $\ln y$.

\begin{figure}[htp]
$$ \begin{verbatim}
vardef newy(expr y) = (256/75)*y + mlog y enddef;
draw begingraph(3in,2in);
 glabel.lft(btex \vbox{\hbox{Population} \hbox{in millions}} etex, OUT);
 path p;
 gdata("timepop.d", $, augment.p($1, newy(Scvnum $2)); );
 gdraw p withpen nullpen;
 for y=5,10,20,50,100,150,200,250:
   grid.lft(format("%g",y), newy(y)) withcolor .85white;
 endfor
 autogrid(grid.bot,) withcolor .85white;
 gdraw p;
 frame.llft;
 endgraph;
\end{verbatim}
   \atop
   \includegraphics{mpgraph-7.mps}
$$
\caption{Population of the United States in millions versus time with the
        population re-expressed as $p/75+\ln p$.  The MetaPost input shown
        above the graph assumes a data file {\tt ttimepop.d} that gives
        (year, $p/75+\ln p$) pairs.}
\label{fig7}
\end{figure}

% frame

Figure~\ref{fig7} uses the command
$$ \hbox{\tt frame.} \descr{label suffix}\ \descr{option list} $$
to draw a special frame around the graph.  In this case the \tdescr{label suffix}
is {\tt llft} to draw just the bottom and left sides of the frame.  Suffixes
{\tt lrt}, {\tt ulft}, and {\tt urt} draw other combinations of two sides;
suffixes {\tt lft}, {\tt rt}, {\tt top}, {\tt bot} draw one side, and \tdescr{empty}
draws the whole frame.  For example
$$ \hbox{\verb|frame dashed evenly|} $$
draws all four sides with dashed lines.  The default four-sided frame is drawn
only when there is no explicit {\tt frame} command.

% auto, (sarith.mp)

To label an axis as {\tt autogrid} does but with the labels transformed
somehow, use
$$ \hbox{{\tt auto.x}\quad or\quad {\tt auto.y}} $$
for positioning tick marks or grid lines.  These macros produce comma-separated
lists for use in {\tt for} loops.  Any $x$ or $y$ values in these lists that
cannot be represented accurately within MetaPost's fixed-point number system
are given as strings.  A standard macro package that is loaded via
$$ \hbox{\tt input sarith}  $$
defines arithmetic operators that work on numbers or strings.  Binary operators
{\tt Sadd}, {\tt Ssub}, {\tt Smul}, and {\tt Sdiv} do addition, subtraction
multiplication, and division.

One possible application is rescaling data.
Figure~\ref{fig4} used a special data file {\tt agepopm.d} that had $y$~values
divided by one million.  This could be avoided by replacing
``{\tt gdraw "agepopm.d"}'' by
$$\begin{verbatim}
gdraw "agepop91.d";
for u=auto.y: otick.lft(format("%g",u Sdiv "1e6"), u); endfor
autogrid(otick.bot,)
\end{verbatim}
$$


\subsection{Processing Data Files}
\label{dfilesec}

% gdata

The most general tool for processing data files is the {\tt gdata} command:
$$ \hbox{\tt gdata(} \descr{string expression} \hbox{\tt,}\, \descr{variable}
        \hbox{\tt,}\, \descr{commands} \hbox{\tt )}
$$
It takes a file name, a variable~$v$, and a list of commands to be executed for
each line of the data file.  The commands are executed with {\tt i} set to the
input line number and strings $v${\tt 1}, $v${\tt 2}, $v${\tt 3}, \ldots\ set
to the input fields on the current line.  A null string marks the end of the
$v$ array.

Using a {\tt glabel} command inside of {\tt gdata} generates a scatter plot
as shown in Figure~\ref{fig8}.  The data file {\tt countries.d} begins
$$\begin{verbatim}
20.910 75.7 US
 1.831 66.7 Alg
\end{verbatim}
$$
where the last field in each line gives the label to be plotted.  Setting
{\tt defaultfont} in the first line of input selects a small font for these
labels.  Without these labels, no {\tt gdata} command would be needed.
Replacing the {\tt gdata} command with
$$ \hbox{\verb|gdraw "countries.d" plot btex$\circ$etex|} $$
would change the abbreviated country names to open circles.

\begin{figure}[htp]
$$\begin{verbatim}
defaultfont:="cmr7";
draw begingraph(3in,2in);
  glabel.lft(btex \vbox{\hbox{Life}\hbox{expectancy}} etex, OUT);
  glabel.bot(btex Per capita G.N.P. (thousands of dollars) etex, OUT);
  setcoords(log,linear);
  gdata("countries.d", s,
    glabel(s3, s1, s2);
  )
  endgraph;
\end{verbatim}
 \atop
 \includegraphics{mpgraph-8.mps}
$$
\caption{A scatter plot and the commands that generated it}
\label{fig8}
\end{figure}

Both {\tt gdraw} and {\tt gdata} ignore an optional initial `\%' on each input
line, parse data fields separated by white space, and stop if they encounter an
input line with no data fields.  Leading percent signs make graph data
look like MetaPost comments so that numeric data can be placed at the beginning
of a MetaPost input file.

% augment

It is often useful to construct one or more paths when reading a data file
with {\tt gdata}.  The {\tt augment} command is designed for this:
$$ \hbox{\tt augment.} \descr{path variable} \hbox{\tt(}\descr{coordinates}\hbox{\tt)}
$$
If the path variable does not have a known value, it becomes a path of length
zero at the given coordinates; otherwise a line segment to the given coordinates
is appended to the path.  The \tdescr{coordinates} may be a pair expression or
any combination of strings and numerics as explained at the beginning of
Section~\ref{coords}.

If a file {\tt timepop.d} gives $t$,~$p$ pairs, {\tt augment} can be used like
this to graph {\tt newy(}$p${\tt)} versus~$t$:
$$\begin{verbatim}
path p;
gdata("timepop.d", s, augment.p(s1, newy(scantokens s2)); );
gdraw p;
\end{verbatim}
$$
(MetaPost's {\tt scantokens} primitive interprets a string as if it were the
contents of an input file.  This finds the numeric value of data field
{\tt s2}.)

% gfill (energy.d)

Figure~\ref{fig9} shows how to use {\tt augment} to read multiple column data
and make multiple paths.  Paths {\tt p2}, {\tt p3}, {\tt p4}, {\tt p5} give
cumulative totals for columns 2 through 5 and pictures {\tt lab2} through
{\tt lab5} give corresponding labels.  The expression
$$ \hbox{\verb|image(unfill bbox lab[j]; draw lab[j])|} $$
executes the given drawing commands and returns the resulting picture:
``{\tt unfill bbox lab[j]}'' puts down a white background and ``{\tt draw
lab[j]}'' puts the label on the background.
The {\tt gfill} command is just like {\tt gdraw}, except it takes a cyclic
path and fills the interior with a solid color.  The color is black unless
a {\tt withcolor} clause specifies another color.
See the manual~\cite{ho:mp3} for
explanations of {\tt for} loops, arrays, colors, and path construction
operators like \verb|--|, {\tt cycle}, and {\tt reverse}.

\begin{figure}[htp]
$$\begin{verbatim}
draw begingraph(3in,2in);
  glabel.lft(btex \vbox{\hbox{Quadrillions}\hbox{of BTU}} etex, OUT);
  path p[];
  numeric t;
  gdata("energy.d", $,
    t:=0; augment.p1($1,0);
    for j=2 upto 5:
       t:=t+scantokens $[j]; augment.p[j]($1,t);
    endfor)
  picture lab[];
  lab2=btex coal etex; lab3=btex crude oil etex;
  lab4=btex natural gas etex; lab5=btex hydroelectric etex;
  for j=5 downto 2:
    gfill p[j]--reverse p[j-1]--cycle withcolor .16j*white;
    glabel.lft(image(unfill bbox lab[j]; draw lab[j]), .7+length p[j]);
  endfor
  endgraph;
\end{verbatim}
 \atop
 \includegraphics{mpgraph-9.mps}
$$
\caption{A graph of U.S. annual energy production
        and the commands that generated it}
\label{fig9}
\end{figure}


\section{Manipulating Big Numbers}
\label{nummac}

MetaPost inherits a fixed-point number system from Knuth's \MF~\cite{kn:d}.
Numbers are expressed in multiples of $2^{-16}$ and they must have absolute
value less than 32768.  Knuth chose this system because it is perfectly adequate
for font design, and it guaranteed to give identical results on all types of
computers.  Fixed-point numbers are seldom a problem in MetaPost because all
computations are based on coordinates that are limited by the size the paper on
which the output is to be printed.  This does not hold for the input data in a
graph-drawing application.  Although graphs look best when coordinate axes are
labeled with numbers of reasonable magnitude, the strict limits of fixed-point
arithmetic would be inconvenient.

% Sadd, Ssub, Smul, Sdiv

A simple way to handle large numbers is to include the line
$$ \hbox{\tt input sarith} $$
and then use binary operators {\tt Sadd}, {\tt Ssub}, {\tt Smul}, and {\tt Sdiv}
in place of \verb|+|, \verb|-|, \verb|*|, and \verb|/|.  These operators are
inefficient but very flexible.  They accept numbers or strings and return
strings in exponential notation with the exponent marked by ``{\tt e}'';
e.g., \verb|"6.7e-11"| means $6.7\times10^{-11}$.

% Sabs, Sleq, Sneq

The unary operator\footnote{The argument to a unary operator need not be
parenthesized unless it is an expression involving binary operators.}
$$ {\tt Sabs}\ \descr{string} $$
finds a string the represents the absolute value.  Binary operators {\tt Sleq}
and {\tt Sneq} perform numeric comparisons on strings and return boolean
results.

% Scvnum

The operation
$$ {\tt Scvnum}\ \descr{string} $$
finds the numeric value for a string if this can be done without overflowing
MetaPost's fixed-point number system.  If the string does not contain
``{\tt e}'', it is much more efficient to use the primitive operation
$$ {\tt scantokens}\ \descr{string} $$

% Mten

The above operators are based on a low-level package that manipulates numbers
in ``{\tt Mlog} form.''  A number $x$ in {\tt Mlog} form represents
$$ \mu^{2^{16}x}, \quad {\rm where\ } \mu=-e^{2^{-24}}. $$
Any value between $1.61\times10^{-28}$ and $3.88\times10^{55}$ can be
represented this way.  (There is a constant {\tt Mten} such that $k*{\tt Mten}$
represents $10^k$ for any integer~$k$ in the interval $[-29,55]$.)

% Mreadpath, Gpaths

The main reason for mentioning {\tt Mlog} form is that it allows graph data
to be manipulated as a MetaPost path.  The function
$$ \hbox{\tt Mreadpath($\descr{file name}$)} $$
reads a data file and returns a path where all the coordinates are in
{\tt Mlog} form.  An internal variable {\tt Gpaths} determines whether
{\tt gdraw} and {\tt gfill} expect paths to be given in {\tt Mlog} form.
For example, this graphs the data in {\tt agepop91.d} with $y$ coordinates
divided by one million:
$$\begin{verbatim}
interim Gpaths:=log;
gdraw Mreadpath("agepop91.d") shifted (0,-6*Mten);
\end{verbatim}
$$


\section{Typesetting Numbers}
\label{formsec}

% format

The graph package needs to compute axis labels and then typeset them.
The macro
$$ \hbox{\tt format($\descr{string expression}$,\,%
        $\descr{numeric or string expression}$)} $$
does this.  You must first {\tt input graph} or {\tt input format} to load the
macro file.  The macro takes a format string and a number to typeset and returns
a picture containing the typeset result.  Thus
$$ \hbox{\verb|format("%g",2+2)|}\quad {\rm yields}\quad \includegraphics{mpgraph-10.mps} $$
and
$$ \hbox{\verb|format("%3g","6.022e23")|}
        \quad {\rm yields}\quad \includegraphics{mpgraph-11.mps}
$$

A format string consists of
\begin{itemize}
\item an optional initial string not containing a percent sign,
\item a percent sign,
\item an optional numeric precision $p$,
\item one of the conversion letters {\tt e}, {\tt f}, {\tt g}, {\tt G},
\item an optional final string $\beta$.
\end{itemize}
The initial and final strings are typeset in the default font (usually
{\tt cmr10}), and the typeset number is placed between them.  For the {\tt e}
and {\tt g} formats, the precision~$p$ is the number of significant digits
allowed after rounding; for {\tt f} and {\tt G}, the number is rounded to the
nearest multiple of $10^{-p}$.  If the precision is not specified, the default
is $p=3$.  The {\tt e} format always uses scientific
notation and the {\tt f} format uses ordinary decimal notation but reverts to
scientific notation if the number is at least 10000.  The {\tt g} and {\tt G}
formats also revert to scientific notation for non-zero numbers of magnitude
less than 0.001.

% init_numbers

The {\tt format} macro needs a set of templates to determine what font to use,
how to position the exponent, etc.  The templates are normally initialized
automatically, but it is possible to set them explicitly by passing five picture
expressions to {\tt init\_numbers}.  For instance, the default definition for
\TeX\ users is
$$\begin{verbatim}
init_numbers(btex$-$etex, btex$1$etex, btex${\times}10$etex,
    btex${}^-$etex, btex${}^2$etex)
\end{verbatim}
$$
The first argument tells how to typeset a leading minus sign;
the second argument is an example of a 1-digit mantissa;
third comes whatever to put after the mantissa in scientific notation;
next come a leading minus sign for the exponent and a sample 1-digit exponent.

% Fe_plus, Fe_base

Picture variable \verb|Fe_plus| gives a leading plus sign for positive numbers,
and \verb|Fe_base| gives whatever should precede the exponent when typesetting
a power of ten.  Calling \verb|init_numbers| initializes \verb|Fe_plus| to an
empty picture and constructs \verb|Fe_base| from its second and third arguments.



\section{Conclusion}
\label{concl}

The graph package makes it convenient to generate graphs from within the MetaPost
language.  The primary benefits are the power of the MetaPost language and its
ability to interact with \TeX\ or troff for typesetting labels.  Typeset labels
can be stored in picture variables and manipulated in various ways such measuring
the bounding box and providing a white background.

We have seen how to generate shaded regions and control line width, color, and
styles of dashed lines.  Numerous other variations are possible.
The full MetaPost language~\cite{ho:mp3} provides many other potentially useful
features.  It also has enough computing power to be useful for generating and
processing data.



\appendix
\section{Summary of the Graph Package}
\label{summsec}

% Not described elsewhere: gdrawarrow, gdrawdblarrow, Gmarks, Gminlog, Autoform

In the following descriptions, italic letters such as $w$ and $h$ denote
expression parameters and words in angle brackets denote other syntactic
elements.  Unless specified otherwise, expression parameters can be either
numerics or strings.  An \tdescr{option list} is a list of drawing options such
as {\tt withcolor .5white} or {\tt dashed evenly};
a \tdescr{label suffix} is one of {\tt lft}, {\tt rt}, {\tt top}, {\tt bot},
{\tt ulft}, {\tt urt}, {\tt llft}, {\tt lrt}.

\subsection{Graph Administration}

\begin{description}
\item[{\tt begingraph($w$,$h$)}]
        Begin a new graph with the frame width and height given by numeric
        parameters $w$ and $h$.
\item[{\tt endgraph}]
        End a graph and return the resulting picture.
\item[{\tt setcoords($t_x$,\,$t_y$)}]
        Set up a new coordinate system as specified by numeric flags $t_x$,
        $t_y$.  Flag values are $\pm{\tt linear}$ and $\pm{\tt log}$.
\item[{\tt setrange($\descr{coordinates}$,\,$\descr{coordinates}$)}]
        Set the lower and upper limits for the current coordinate system.
        Each \tdescr{coordinates} can be a single pair expression or two
        numeric or string expressions.
\end{description}

\subsection{Drawing and Labeling}

All of the drawing and labeling commands can be followed by an
\tdescr{option list}.  In addition to the usual MetaPost drawing options, the
list can contain a {\tt plot} \tdescr{picture} clause to plot a specified
picture at each data point.

The drawing and labeling commands are closely related to a set of similarly
named commands in plain MetaPost.  The {\tt gdrawarrow} and {\tt gdrawdblarrow}
commands are included to maintain this relationship.

\begin{description}
\item[{\tt gdotlabel.$\descr{label suffix}$($p$,\,$\descr{location}$)}]
        This is like {\tt glabel} except it also puts a dot at the location
        being labeled.
\item[{\tt gdraw $p$}]
        Draw path $p$, or if $p$ is a string, read coordinate pairs from
        file~$p$ and draw a polygonal line through them.
\item[{\tt gdrawarrow $p$}]
        This is like {\tt dgraw} $p$ except it adds an arrowhead at the end of
        the path.
\item[{\tt gdrawdblarrow $p$}]
        This is like {\tt dgraw} $p$ except it adds an arrowheads at each end of
        the path.
\item[{\tt gfill $p$}]
        Fill cyclic path~$p$ or read coordinates from the file named by
        string~$p$ and fill the resulting polygonal outline.
\item[{\tt glabel.$\descr{label suffix}$($p$,\,$\descr{location}$)}]
        If $p$ is not a picture, it should be a string.  Typeset it using
        {\tt defaultfont}, then place it near the given location and offset as
        specified by the \tdescr{label suffix}.  The \tdescr{location} can be
        $x$~and $y$ coordinates, a pair giving $x$~and $y$, a numerc value giving
        a time on the last path drawn, or {\tt OUT} to label the outside of the
        graph.
\end{description}

\subsection{Grids, Tick Marks, and Framing}

\begin{description}
\item[{\tt auto.}$\descr{{\tt x} or {\tt y}}$]
        Generate default $x$ or $y$ coordinates for tick marks.
\item[{\tt autogrid($\descr{axis label command}$,\,$\descr{axis label command}$)}]
        Draw default axis labels using the specified commands for the $x$ and
        $y$ axes.  An \tdescr{axis label command} may be \tdescr{empty} or it
        may be {\tt itick}, {\tt otick}, or {\tt grid} followed by a
        \tdescr{label suffix}.
\item[{\tt frame.}$\descr{label suffix}$\ $\descr{option list}$]
        Draw a frame around the graph, or draw the part of the frame specified
        by the \tdescr{label suffix}.
\item[{\tt grid.}$\descr{label suffix}$($f$,$z$)]
        Draw a grid line across the graph from the side specified by the
        \tdescr{label suffix}, and label it there using format string~$f$ and
        coordinate value~$z$.  If $f$ is a picture, it gives the label.
\item[{\tt itick.}$\descr{label suffix}$($f$,$z$)]
        This is like {\tt grid} except it draws an inward tick mark.
\item[{\tt otick.}$\descr{label suffix}$($f$,$z$)]
        This is like {\tt grid} except it draws an outward tick mark.
\end{description}

\subsection{Miscellaneous Commands}

\begin{description}
\item[{\tt augment.$\descr{variable}$($\descr{coordinates}$)}]
        Append \tdescr{coordinates} to the path stored in \tdescr{variable}.
\item[{\tt format($f$,\,$x$)}]
        Typeset $x$ according to format string~$f$ and return the resulting
        picture.
\item[{\tt gdata($f$,\,$\descr{variable}$,\,$\descr{commands}$)}]
        Read the file named by string~$f$ and execute \tdescr{commands} for each
        input line using the \tdescr{variable} as an array to store data fields.
\item[{\tt init\_numbers($s$,\,$m$,\,$x$,\,$t$,\,$e$)}]
        Provide five pictures as templates for future {\tt format} operations:
        $s$ is a leading minus; $m$ is a sample mantissa; $x$ follows the
        mantissa; $t$ is a leading minus for the exponent~$e$.
\item[{\tt Mreadpath($f$)}]
        Read a path for the data file named by string~$f$ and return it in
        ``{\tt Mlog} form''.
\end{description}

\subsection{Arithmetic on Numeric Strings}

It is necessary to {\tt input sarith} before using the following macros:
\begin{description}
\item[{\tt Sabs $x$}]
        Compute $|x|$ and return a numeric string.
\item[{\tt $x$ Sadd $y$}]
        Compute $x+y$ and return a numeric string.
\item[{\tt Scvnum $x$}]
        Return the numeric value for string $x$.
\item[{\tt $x$ Sdiv $y$}]
        Compute $x/y$ and return a numeric string.
\item[{\tt $x$ Sleq $y$}]
        Return the boolean result of the comparison $x\leq y$.
\item[{\tt $x$ Smul $y$}]
        Compute $x*y$ and return a numeric string.
\item[{\tt $x$ Sneq $y$}]
        Return the boolean result of the comparison $x\neq y$.
\item[{\tt $x$ Ssub $y$}]
        Compute $x-y$ and return a numeric string.
\end{description}

\subsection{Internal Variables and Constants}

\begin{description}
\item[{\tt Autoform}]
        Format string used by {\tt autogrid}.  Default: \verb|"%g"|.
\item[{\tt Fe\_base}]
        What precedes the exponent when typesetting a power of ten.
\item[{\tt Fe\_plus}]
        Picture of the leading plus sign for positive exponents.
\item[{\tt Gmarks}]
        Minimum number of tick marks per axis for {\tt auto} and {\tt autogrid}.
        Default: 4.
\item[{\tt Gminlog}]
        Minimum largest/smallest ratio for logarithmic spacing with {\tt auto}
        and {\tt autogrid}.  Default: 3.0.
\item[{\tt Gpaths}]
        Code for coordinates used in {\tt gdraw} and {\tt gfill} paths:
        {\tt linear} for standard form, {log} for ``{\tt Mlog} form''.
\item[{\tt Mten}]
        The ``{\tt Mlog} form'' for 10.0
\end{description}


\bibliographystyle{plain}
\bibliography{mpgraph}

\end{document}



% Copyright 1990 - 1995 by AT&T Bell Laboratories.

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