path: root/Master/texmf-dist/source/latex/bez123/bez123.dtx

% \iffalse meta-comment
%
% bez123.dtx
% Author: Peter Wilson (CUA and NIST)
%         peter.r.wilson@boeing.com
% 
% Copyright 1998 Peter R. Wilson
%
% This program is provided under the terms of the
% LaTeX Project Public License distributed from CTAN
% archives in directory macros/latex/base/lppl.txt.
%
% 
%<*driver>
\documentclass{ltxdoc}
\usepackage{bez123}
\EnableCrossrefs
\CodelineIndex
\setcounter{StandardModuleDepth}{1}
\begin{document}
  \DocInput{bez123.dtx}
\end{document}
%</driver>
%
% \fi
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% \def\filedate{1998/10/14}
% \newcommand*{\Lpack}[1]{\textsf {#1}}        ^^A typest a package
% \newcommand*{\Lopt}[1]{\textsf {#1}}         ^^A typeset an option
% \newcommand*{\file}[1]{\texttt {#1}}         ^^A typeset a file
% \newcommand*{\Lcount}[1]{\textsl {\small#1}} ^^A typeset a counter
% \newcommand*{\pstyle}[1]{\textsl {#1}}       ^^A typeset a pagestyle
% \newcommand*{\Lenv}[1]{\texttt {#1}}         ^^A typeset an environment
% \newcommand{\eqref}[1]{equation~(\ref{#1})}  ^^A typeset ref to an equation
%
% \title{The \Lpack{bez123} and \Lpack{multiply} packages\thanks{This
%        file has version number \fileversion, last revised
%        \filedate.}}
%
% \author{%
% Peter Wilson\\
% Catholic University of America\thanks{This work was originally
% performed as
% a Guest Researcher at the National Institute of Standards and Technology.} \\
% Now at \texttt{peter.r.wilson@boeing.com}
% }
% \date{\filedate}
% \maketitle
% \begin{abstract}
%    The \Lpack{bez123} package provides for the drawing of linear, cubic,
% and rational quadratic Bezier curves. The \Lpack{multiply} package
% provides a command to multiply a length without numerical overflow.
% \end{abstract}
% \tableofcontents
% \listoftables
% \listoffigures
%
% \StopEventually{}
%
% 
%
% \section{Introduction}
%
% This document provides the commented source for a \LaTeX{}
% package file that extends the \LaTeX{} facilities for drawing
% Bezier curves. The package was originally developed as part of 
% a suite designed for the typesetting of
% documents according to the rules for ISO international 
% standards~\cite{PRW96i}.
% This manual is typeset according to the conventions of the
% \LaTeX{} \textsc{docstrip} utility which enables the automatic
% extraction of the \LaTeX{} macro source files~\cite{GOOSSENS94}.
%
%    Drawing a non-rational quadratic Bezier curve is provided as part 
% of the standard \LaTeX{} system.
%    Section~\ref{sec:usage} provides the user manual for the new commands
% supplied by this package for drawing a variety of Bezier curves.
% These include commands for drawing linear and cubic non-rational Bezier 
% curves and rational quadratic curves.
%
%    Section~\ref{sec:bez} describes the implementation of the package.
% As a side-effect of the implementation, a facility is also provided
% for performing multiplication in \TeX{} without overflow. This is
% described in Section~\ref{sec:mnoflow}.
%
%
%
% \section{Usage} \label{sec:usage}
%
%    Leslie Lamport provided the means of drawing a quadratic Bezier curve
% \emph{via} the \LaTeXe{} |\qbezier|~\cite[pp. 125--126]{LAMPORT94} command.
% This package
% extends the Bezier facility by providing commands to draw linear,
% rational quadratic, and cubic Bezier curves.
%
%    Bezier curves are named after Pierre Bezier who invented them. They
% are widely used within Computer Aided Design (CAD) programs and other
% graphics systems; descriptions can be found in many places, with varying
% degrees of mathematical complexity, such 
% as~\cite{FandP,MORTENSON85,FARIN90}.
%
%    The Bezier curve is a parameterized curve of degree $n$ and can 
% therefore be specified by $(n+1)$ points 
% (i.e., point $p_{0}$ through $p_{n}$).
% Among its other properties, a Bezier curve of degree $n$ passes through 
% through the points $p_{0}$ and $p_{n}$ and passes close to the other 
% defining points. The general equation for a Bezier curve of degree $n$ with
% parameter $t$ is
% \begin{equation}
%  p(t) = a_{0} + a_{1}t + a_{2}t^{2} + \cdots + a_{n}t^{n} \label{eq:gen}
% \end{equation}
% where the coefficients $a_{i}$ depend on the defining points, and 
% traditionally $0 \leq t \leq 1$.
%
%    For a linear (degree $1$) curve, the equation is
% \begin{equation}
%  p(t) = p_{0} + (p_{1} - p_{0})t  \label{eq:lin}
% \end{equation}
% By inspection, $p(0) = p_{0}$ and $p(1) = p_{1}$.
%
%    Rearranging \eqref{eq:gen} slightly we get
% \begin{equation}
%  p(t) - p_{0} = (p_{1} - p_{0})t  \label{eq:lin2}
% \end{equation}
% In other words, we can march along the curve from the starting point to
% the ending point by evaluating the right hand side of 
% \eqref{eq:lin2} for increasing values of the parameter $t$.
%
%   In order to shorten the equations slightly, and also make them more
% convenient to work with numerically, we will use the notation
% \begin{displaymath}
% l_{pq} = p_{p} - p_{q}
% \end{displaymath}
% Thus, the final form for the linear Bezier curve is
% \begin{equation}
% p(t) - p_{0} = l_{10}t  \label{eq:lin3}
% \end{equation}
% 
% \DescribeMacro{\lbezier}
% The command |\lbezier[|\meta{N}|](|\meta{p0}|)(|\meta{p1}|)| draws a
% linear Bezier curve with \meta{N} plotted points from the point \meta{p0}
% (with coordinates \meta{x0,y0}) to the point \meta{p1} (with
% coordinates \meta{x1,y1}). \meta{N} is an optional argument. If it is
% either not given or is given with a value of zero, 
% then the command will calculate the number of points to be 
% plotted, subject to a maximum number. 
% There must be no spaces between the arguments to the 
% |\lbezier| command; this restriction also applies to the other Bezier
% drawing commands provided by the \Lpack{bez123} package.
%
%    Figure~\ref{fig:beta} shows an example of a dotted line drawn using
% the |\lbezier| command. The actual code used is:
% \begin{verbatim}
% \lbezier[50](15,30)(30,0)
% \end{verbatim}
% thus drawing a straight line consisting of 50 points.
%
% \DescribeMacro{\qbeziermax}
%    The standard \LaTeX{} command |\qbeziermax| sets a maximum limit
% on the number of points used to draw any of the Bezier curves.
%
% \DescribeMacro{\thinlines}
% \DescribeMacro{\thicklines}
% \DescribeMacro{\linethickness}
%    The `points' used in drawing the Bezier curves are small squares. The
% size of these squares are controlled by the standard \LaTeX{}
% |\thinlines|, |\thicklines| and/or |\linethickness| commands.
% Consult Lamport~\cite{LAMPORT94} for descriptions of these, and 
% |\qbeziermax|, commands.
%
%
%   It is convenient to introduce some general properties of Bezier curves
% at this point.
% \begin{itemize}
% \item A degree $n$ Bezier curve is defined by $(n+1)$ points which we
% will label as $p_{0}$ through $p_{n}$. The lines joining the points
% $p_{0}, p_{1}, \ldots , p_{n}$ are called the \emph{control polygon}.
% The Bezier curve is parameterized by a variable we will call $t$, with
% $0 \leq t \leq 1$.
% \item A degree $n$ Bezier curve starts at point $p_{0}$ and ends at
% point $p_{n}$.
% \item At $t=0$ the curve passes through $p_{0}$ and is tangent to the
% line $l_{10} = p_{1}-p_{0}$.
% \item At $t=1$ the curve passes through $p_{n}$ and is tangent to the
% line $l_{(n)(n-1)} = p_{n}-p_{(n-1)}$.
% \item A \emph{non-rational} Bezier curve lies within the \emph{convex 
% hull}\footnote{The convex hull can be thought of as the shape that a rubber
% band will take if it is stretched around pins placed at each point.}
% of the points $p_{0}$ through $p_{n}$. For examples of convex hulls see
% figure~\ref{fig:ch}. Note that the shape of a convex hull is independant
% of the ordering of the points.
% \end{itemize}
%
% \begin{figure}
% \centering
% \setlength{\unitlength}{1mm}
% \begin{picture}(70,80)
%  ^^A degree 3
% \put(0,5){\begin{picture}(30,30)
%   \thinlines
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{0}}
%   \put(10,30){\circle{2}}
%   \put(8,30){\makebox(0,0)[br]{1}}
%   \put(20,0){\circle{2}}
%   \put(22,0){\makebox(0,0)[bl]{2}}
%   \put(30,30){\circle{2}}
%   \put(32,30){\makebox(0,0)[bl]{3}}
%   ^^A convex hull
%   \put(0,0){\line(1,0){20}}
%   \put(20,0){\line(1,3){10}}
%   \put(30,30){\line(-1,0){20}}
%   \put(10,30){\line(-1,-3){10}}
%   ^^A control polygon
%   ^^A \put(0,0){\vector(1,3){10}}
%   ^^A \put(10,30){\vector(1,-3){10}}
%   ^^A \put(20,0){\vector(1,3){10}}
%   \thicklines
%  ^^A   \cbezier[30](0,0)(10,30)(20,0)(30,30)
%   \thinlines
%   \end{picture}}
%  ^^A degree 3
% \put(0,45){\begin{picture}(30,30)
%   \thinlines
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{0, 3}}
%   \put(30,0){\circle{2}}
%   \put(32,0){\makebox(0,0)[bl]{1}}
%   \put(0,30){\circle{2}}
%   \put(2,30){\makebox(0,0)[bl]{2}}
%   ^^A convex hull
%   \put(0,0){\line(1,0){30}}
%   \put(30,0){\line(-1,1){30}}
%   \put(0,30){\line(0,-1){30}}
%   ^^A control polygon
%   ^^A \put(0,0){\vector(1,0){30}}
%   ^^A \put(30,0){\vector(-1,1){30}}
%   ^^A \put(0,30){\vector(0,-1){30}}
%   \thicklines
%  ^^A \cbezier[30](0,0)(30,0)(0,30)(0,0)
%   \thinlines
%   \end{picture}}
%  ^^A degree 3
% \put(45,0){\begin{picture}(30,30)
%   \thinlines
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{0}}
%   \put(10,30){\circle{2}}
%   \put(8,30){\makebox(0,0)[br]{2}}
%   \put(20,0){\circle{2}}
%   \put(22,0){\makebox(0,0)[bl]{3}}
%   \put(30,30){\circle{2}}
%   \put(32,30){\makebox(0,0)[bl]{1}}
%   ^^A convex hull
%   \put(0,0){\line(1,0){20}}
%   \put(20,0){\line(1,3){10}}
%   \put(30,30){\line(-1,0){20}}
%   \put(10,30){\line(-1,-3){10}}
%   ^^A control polygon
%   ^^A \put(0,0){\vector(1,1){30}}
%   ^^A \put(30,30){\vector(-1,0){20}}
%   ^^A \put(10,30){\vector(1,-3){10}}
%   \thicklines
%   ^^A \cbezier(0,0)(30,30)(10,30)(20,0)
%   \thinlines
%   \end{picture}}
% \put(45,45){\begin{picture}(30,30)
%   \thinlines
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{0}}
%   \put(30,0){\circle{2}}
%   \put(32,0){\makebox(0,0)[bl]{1}}
%   \put(0,30){\circle{2}}
%   \put(2,30){\makebox(0,0)[bl]{2}}
%   \put(10,10){\circle{2}}
%   \put(8,10){\makebox(0,0)[br]{3}}
%   ^^A convex hull
%   \put(0,0){\line(1,0){30}}
%   \put(30,0){\line(-1,1){30}}
%   \put(0,30){\line(0,-1){30}}
%   ^^A control polygon
%   ^^A \put(0,0){\vector(1,0){30}}
%   ^^A \put(30,0){\vector(-1,1){30}}
%   ^^A \put(0,30){\vector(1,-2){10}}
%   \thicklines
%   ^^A \cbezier(0,0)(30,0)(0,30)(10,10)
%   \thinlines
%   \end{picture}}
% \end{picture}
% \setlength{\unitlength}{1pt}
% \caption{Four sets of points and their convex hulls} \label{fig:ch} 
% \end{figure}
%
%    The equation for cubic Bezier curves is
% \begin{equation}
% p(t) - p_{0} = 3l_{10}t + 3(l_{21} - l_{10})t^{2} + (l_{30} - 3l_{21})t^{3}
%  \label{eq:cubic}
% \end{equation}
%
% \DescribeMacro{\cbezier}
%    The command
% |\cbezier[|\meta{N}|](|\meta{p0}|)(|\meta{p1}|)(|\meta{p2}|)(|\meta{p3}|)|
% draws a cubic Bezier curve, as defined by \eqref{eq:cubic},
% from point \meta{p0} to point \meta{p3}, where \meta{p1} and \meta{p2}
% are the intermediate points defining the control polygon.
% 
% \begin{figure}
% \centering
% \setlength{\unitlength}{1mm}
% \begin{picture}(70,80)
%  ^^A degree 3
% \put(0,5){\begin{picture}(30,30)
%   \thinlines
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{0}}
%   \put(10,30){\circle{2}}
%   \put(8,30){\makebox(0,0)[br]{1}}
%   \put(20,0){\circle{2}}
%   \put(22,0){\makebox(0,0)[bl]{2}}
%   \put(30,30){\circle{2}}
%   \put(32,30){\makebox(0,0)[bl]{3}}
%   ^^A convex hull
%   ^^A \put(0,0){\line(1,0){20}}
%   ^^A \put(20,0){\line(1,3){10}}
%   ^^A \put(30,30){\line(-1,0){20}}
%   ^^A \put(10,30){\line(-1,-3){10}}
%   ^^A control polygon
%   \put(0,0){\vector(1,3){10}}
%   \put(10,30){\vector(1,-3){10}}
%   \put(20,0){\vector(1,3){10}}
%   \thicklines
%   \cbezier[30](0,0)(10,30)(20,0)(30,30)
%   \thinlines
%   \end{picture}}
%  ^^A degree 3
% \put(0,45){\begin{picture}(30,30)
%   \thinlines
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{0, 3}}
%   \put(30,0){\circle{2}}
%   \put(32,0){\makebox(0,0)[bl]{1}}
%   \put(0,30){\circle{2}}
%   \put(2,30){\makebox(0,0)[bl]{2}}
%   ^^A convex hull
%   ^^A \put(0,0){\line(1,0){30}}
%   ^^A \put(30,0){\line(-1,1){30}}
%   ^^A \put(0,30){\line(0,-1){30}}
%   ^^A control polygon
%   \put(0,0){\vector(1,0){30}}
%   \put(30,0){\vector(-1,1){30}}
%   \put(0,30){\vector(0,-1){30}}
%   \thicklines
%   \cbezier[30](0,0)(30,0)(0,30)(0,0)
%   \thinlines
%   \end{picture}}
%  ^^A degree 3
% \put(45,0){\begin{picture}(30,30)
%   \thinlines
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{0}}
%   \put(10,30){\circle{2}}
%   \put(8,30){\makebox(0,0)[br]{2}}
%   \put(20,0){\circle{2}}
%   \put(22,0){\makebox(0,0)[bl]{3}}
%   \put(30,30){\circle{2}}
%   \put(32,30){\makebox(0,0)[bl]{1}}
%   ^^A convex hull
%   ^^A \put(0,0){\line(1,0){20}}
%   ^^A \put(20,0){\line(1,3){10}}
%   ^^A \put(30,30){\line(-1,0){20}}
%   ^^A \put(10,30){\line(-1,-3){10}}
%   ^^A control polygon
%   \put(0,0){\vector(1,1){30}}
%   \put(30,30){\vector(-1,0){20}}
%   \put(10,30){\vector(1,-3){10}}
%   \thicklines
%   \cbezier(0,0)(30,30)(10,30)(20,0)
%   \thinlines
%   \end{picture}}
% \put(45,45){\begin{picture}(30,30)
%   \thinlines
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{0}}
%   \put(30,0){\circle{2}}
%   \put(32,0){\makebox(0,0)[bl]{1}}
%   \put(0,30){\circle{2}}
%   \put(2,30){\makebox(0,0)[bl]{2}}
%   \put(10,10){\circle{2}}
%   \put(8,10){\makebox(0,0)[br]{3}}
%   ^^A convex hull
%   ^^A \put(0,0){\line(1,0){30}}
%   ^^A \put(30,0){\line(-1,1){30}}
%   ^^A \put(0,30){\line(0,-1){30}}
%   ^^A control polygon
%   \put(0,0){\vector(1,0){30}}
%   \put(30,0){\vector(-1,1){30}}
%   \put(0,30){\vector(1,-2){10}}
%   \thicklines
%   \cbezier(0,0)(30,0)(0,30)(10,10)
%   \thinlines
%   \end{picture}}
% \end{picture}
% \setlength{\unitlength}{1pt}
% \caption{Four sets of points, the cubic Bezier curves and their control 
%          polygons. Left --- curves plotted with $N=30$; Right ---
%          curves plotted with $N=0$} \label{fig:cp} 
% \end{figure}
% 
%    Figure~\ref{fig:cp} shows four such cubic Bezier curves, their
% defining points and their control polygons. These are the same points
% that were used in figure~\ref{fig:ch} to illustrate convex hulls. It is
% easy to verify that a cubic Bezier curve does indeed lie within the convex
% hull of its defining points. The curves on the left of the figure were
% specified with a value of 30 for the argument \meta{N}, while those
% on the right had no value given for \meta{N} and thus were drawn with
% the number of plotted points calculated by the drawing algorithm.
% The actual drawing commands used were:
% \begin{verbatim}
% \cbezier[30](0,0)(10,30)(20,0)(30,30)
% \cbezier[30](0,0)(30,0)(0,30)(0,0)
% \cbezier(0,0)(30,30)(10,30)(20,0)
% \cbezier(0,0)(30,0)(0,30)(10,10)
% \end{verbatim}
% Note that points are plotted along the curve at equidistant values of the
% of the parameter $t$. However, as the relationship between the actual
% distance in $(x,y)$ coordinate space is a non-linear function of $t$,
% the seperation between the plotted points is non-uniform.
%
% The equation for a \emph{non-rational} quadratic Bezier curve is
% \begin{equation}
% p(t) - p_{0} = 2l_{10}t + (l_{20} - 2l_{10})t^{2} \label{eq:quad}
% \end{equation}
% Using standard \LaTeX{} this can be drawn by the |\qbezier| command.
% There is another form of a quadratic Bezier curve called a \emph{rational}
% quadratic Bezier curve. Its equation is
% \begin{equation}
% p(t) - p_{0} =  \frac^^A
% {2w_{1}l_{10}t + (w_{2}l_{20} - 2w_{1}l_{10})t^{2}}^^A
% {w_{0} + 2\omega_{10}t + (\omega_{20} - \omega_{10})t^{2}}
% \label{eq:rqfull}
% \end{equation}
% where the $w_{i}$ are the \emph{weights} corresponding to the 
% points $p_{i}$ and $\omega_{pq} = w_{p} - w_{q}$. The shape of a 
% non-rational curve can be changed by changing the positions of the defining
% points. The shape of a rational curve can also be modified by changing 
% the values of the weights. A rational curve
% has the same general properties, outlined above, as a non-rational curve
% with the exception that the curve may lie outside the convex hull of the
% control polygon.
%
%    For the purposes at hand, we use a more restricted form of a
% rational quadratic Bezier curve, obtained by putting 
% $W = w_{1}/w_{0}$ and then making
% $w_{0} = w_{2} = 1$ in \eqref{eq:rqfull}. Performing these
% substitutions we end up with
% \begin{equation}
% p(t) - p_{0} =  \frac^^A
% {2Wl_{10}t + (l_{20} - 2Wl_{10})t^{2}}^^A
% {1 + 2(1 - W)t + 2(1 - W)t^{2}}
% \label{eq:rqfinal}
% \end{equation}
% Note that when $W=1$, (\ref{eq:rqfinal}) reduces to \eqref{eq:quad}
% and when $W=0$ it effectively reduces to \eqref{eq:lin3}.
%
%    It turns out that a non-rational quadratic Bezier curve is an arc of
% a parabola, which is one of the conic curves. All the other conic curves
% can be represented by the rational quadratic Bezier curve described
% by \eqref{eq:rqfinal} by suitable choices for the value of $W$.
% From now on, we will call $W$ the \emph{weight} of the rational quadratic
% Bezier curve. Table~\ref{tab:rq} lists the value, or value range,
% of $W$ for the various forms of the conic curve.\footnote{We do not deal
% with the degenerate cases.} For the case of a circle, $\beta$ is the
% angle between the lines $l_{10} = (p_{1} - p_{0})$ and
% $l_{20} = (p_{2} - p_{0})$, as shown in figure~\ref{fig:beta}.
%
% \begin{table}
% \centering
% \caption{Conic forms of the rational quadratic Bezier curve} \label{tab:rq}
% \begin{tabular}{lc} \hline
% Conic form & Weight ($W$) \\ \hline
% Hyperbola & $\|W\| > 1$ \\
% Parabola  & $\|W\| = 1$ \\
% Ellipse   & $0 < \|W\| < 1$ \\
% Circle    & $\|l_{10}\| = \|l_{21}\|$ and $W = \cos \beta$ \\
% Straight line & $W = 0$ \\ \hline
% \end{tabular}
% \end{table}
%
%
% \begin{figure}
% \centering
% \setlength{\unitlength}{1mm}
% \begin{picture}(30,40)
% \put(0,5){\begin{picture}(30,30)
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{1}}
%   \put(15,30){\circle{2}}
%   \put(13,30){\makebox(0,0)[br]{0}}
%   \put(30,0){\circle{2}}
%   \put(32,0){\makebox(0,0)[bl]{2}}
%   ^^A polygon
%   \thicklines
%   \put(15,30){\vector(-1,-2){15}}
%   \put(0,0){\vector(1,0){30}}
%   ^^A dashed line from 0 to 2
%   ^^A \lbezier(15,30)(30,0)
%   \lbezier[50](15,30)(30,0)
%   \thinlines
%   \put(15,26){\makebox(0,0){$\beta$}}
%   \end{picture}}
% \end{picture}
% \setlength{\unitlength}{1pt}
% \caption{The angle $\beta$} \label{fig:beta}
% \end{figure}
%
% \DescribeMacro{\rqbezier}
% The command
% |\rqbezier[|\meta{N}|](|\meta{p0}|)(|\meta{p1}|)(|\meta{p2}|)(|\meta{W}|)|
% draws a rational quadratic Bezier curve from \meta{p0} to \meta{p2} with
% weight \meta{W}, according to \eqref{eq:rqfinal}. As in the
% other Bezier commands, \meta{N} is optional and controls the number
% of plotted points along the curve. Figure~\ref{fig:qrb} shows several
% rational quadratic curves, all with the same control polygon but with
% differing values for the weight $W$. The code is:
% \begin{verbatim}
%   \rqbezier[100](15,30)(0,0)(30,0)(4)
%   \rqbezier[100](15,30)(0,0)(30,0)(2)
%   \rqbezier(15,30)(0,0)(30,0)(1)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.75)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.5)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.25)
%   \rqbezier(15,30)(0,0)(30,0)(0)
% \end{verbatim}
% When $W > 1$ the curve is pulled toward the point $p_{1}$. Conversely,
% when $W < 1$ the curve is pushed away from the point $p_{1}$. In all
% cases, though, the curve starts and stops at $p_{0}$ and $p_{2}$
% respectively.
%
% \begin{figure}
% \centering
% \setlength{\unitlength}{1mm}
% \begin{picture}(30,40)
% \put(0,5){\begin{picture}(30,30)
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{1}}
%   \put(15,30){\circle{2}}
%   \put(13,30){\makebox(0,0)[br]{0}}
%   \put(30,0){\circle{2}}
%   \put(32,0){\makebox(0,0)[bl]{2}}
%   ^^A polygon
%   \thinlines
%   \put(15,30){\vector(-1,-2){15}}
%   \put(0,0){\vector(1,0){30}}
%   ^^A dashed line from 0 to 2
%   ^^A \lbezier(15,30)(30,0)
%   ^^A \lbezier[50](15,30)(30,0)
%   ^^A \thinlines
%   ^^A \put(15,26){\makebox(0,0){$\beta$}}
%   \thicklines
%   \rqbezier[100](15,30)(0,0)(30,0)(4)
%   \rqbezier[100](15,30)(0,0)(30,0)(2)
%   \rqbezier(15,30)(0,0)(30,0)(1)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.75)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.5)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.25)
%   \rqbezier(15,30)(0,0)(30,0)(0)
%   \end{picture}}
% \end{picture}
% \setlength{\unitlength}{1pt}
% \caption{The effect of weight variation ($W \geq 0$)
%          on rational quadratic Bezier curves
%          (\texttt{weightscale = \theweightscale} (the default)) } 
% \label{fig:qrb}
% \end{figure}
%
% Like the case of the cubic curve, points are plotted at equidistant
% values of the parameter $t$. The relationship between parameter value
% and coordinate positions in the rational case are highly non-linear.
% Thus the distance between the plotted points can vary quite remarkably.
% This is an inherent disadvantage with this type of curve. The user's remedy
% is to increase the number of points to be plotted, but this can lead to
% \TeX{} running out of memory, not to mention the increased time to
% generate the drawing.
%
% \DescribeMacro{\setweightscale}
% \DescribeMacro{\resetweightscale}
%    Because of the way in which \TeX{} performs arithmetic, and especially
% division, it 
% is necessary to perform some scaling operations on the divisor when
% evaluating \eqref{eq:rqfinal}. The optimum value for the
% scaling is a complex function of the weight and the size and orientation
% of the control polygon. The algorithm uses a heuristic approach to
% calculate a `good' value but is not always successful. The 
% |\setweightscale{|\meta{number}|}| command can be used to specify
% a scale factor. \meta{number} must be a positive integer. The
% |\resetweightscale| command resets the scale factor to its default
% value, which is currently 10000 (ten thousand).
%
% \begin{figure}
% \centering
% \setlength{\unitlength}{1mm}
% \begin{picture}(70,80)
% \put(0,5){\begin{picture}(30,30)
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{1}}
%   \put(15,30){\circle{2}}
%   \put(13,30){\makebox(0,0)[br]{0}}
%   \put(30,0){\circle{2}}
%   \put(32,0){\makebox(0,0)[bl]{2}}
%   ^^A polygon
%   \thinlines
%   ^^A \put(15,30){\vector(-1,-2){15}}
%   ^^A \put(0,0){\vector(1,0){30}}
%   ^^A dashed line from 0 to 2
%   ^^A \lbezier(15,30)(30,0)
%   ^^A \lbezier[50](15,30)(30,0)
%   ^^A \thinlines
%   ^^A \put(15,26){\makebox(0,0){$\beta$}}
%   \thicklines
%   \setweightscale{100}
%   \rqbezier[100](15,30)(0,0)(30,0)(4)
%   \rqbezier[100](15,30)(0,0)(30,0)(2)
%   \rqbezier(15,30)(0,0)(30,0)(1)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.75)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.5)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.25)
%   \rqbezier(15,30)(0,0)(30,0)(0)
%   \put(15,-5){\makebox(0,0)[b]{\texttt{weightscale = \theweightscale}}}
%   \end{picture}}
% \put(0,45){\begin{picture}(30,30)
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{1}}
%   \put(15,30){\circle{2}}
%   \put(13,30){\makebox(0,0)[br]{0}}
%   \put(30,0){\circle{2}}
%   \put(32,0){\makebox(0,0)[bl]{2}}
%   ^^A polygon
%   \thinlines
%   ^^A \put(15,30){\vector(-1,-2){15}}
%   ^^A \put(0,0){\vector(1,0){30}}
%   ^^A dashed line from 0 to 2
%   ^^A \lbezier(15,30)(30,0)
%   ^^A \lbezier[50](15,30)(30,0)
%   ^^A \thinlines
%   ^^A \put(15,26){\makebox(0,0){$\beta$}}
%   \thicklines
%   \setweightscale{1000}
%   \rqbezier[100](15,30)(0,0)(30,0)(4)
%   \rqbezier[100](15,30)(0,0)(30,0)(2)
%   \rqbezier(15,30)(0,0)(30,0)(1)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.75)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.5)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.25)
%   \rqbezier(15,30)(0,0)(30,0)(0)
%   \put(15,-5){\makebox(0,0)[b]{\texttt{weightscale = \theweightscale}}}
%   \end{picture}}
% \put(45,5){\begin{picture}(30,30)
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{1}}
%   \put(15,30){\circle{2}}
%   \put(13,30){\makebox(0,0)[br]{0}}
%   \put(30,0){\circle{2}}
%   \put(32,0){\makebox(0,0)[bl]{2}}
%   ^^A polygon
%   \thinlines
%   ^^A \put(15,30){\vector(-1,-2){15}}
%   ^^A \put(0,0){\vector(1,0){30}}
%   ^^A dashed line from 0 to 2
%   ^^A \lbezier(15,30)(30,0)
%   ^^A \lbezier[50](15,30)(30,0)
%   ^^A \thinlines
%   ^^A \put(15,26){\makebox(0,0){$\beta$}}
%   \thicklines
%   \resetweightscale
%   \rqbezier[100](15,30)(0,0)(30,0)(4)
%   \rqbezier[100](15,30)(0,0)(30,0)(2)
%   \rqbezier(15,30)(0,0)(30,0)(1)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.75)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.5)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.25)
%   \rqbezier(15,30)(0,0)(30,0)(0)
%   \put(15,-5){\makebox(0,0)[b]{\texttt{weightscale = \theweightscale}}}
%   \end{picture}}
% \put(45,45){\begin{picture}(30,30)
%   \put(0,0){\circle{2}}
%   \put(-2,0){\makebox(0,0)[br]{1}}
%   \put(15,30){\circle{2}}
%   \put(13,30){\makebox(0,0)[br]{0}}
%   \put(30,0){\circle{2}}
%   \put(32,0){\makebox(0,0)[bl]{2}}
%   ^^A polygon
%   \thinlines
%   ^^A \put(15,30){\vector(-1,-2){15}}
%   ^^A \put(0,0){\vector(1,0){30}}
%   ^^A dashed line from 0 to 2
%   ^^A \lbezier(15,30)(30,0)
%   ^^A \lbezier[50](15,30)(30,0)
%   ^^A \thinlines
%   ^^A \put(15,26){\makebox(0,0){$\beta$}}
%   \thicklines
%   \setweightscale{100000}
%   \rqbezier[100](15,30)(0,0)(30,0)(4)
%   \rqbezier[100](15,30)(0,0)(30,0)(2)
%   \rqbezier(15,30)(0,0)(30,0)(1)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.75)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.5)
%   \rqbezier[100](15,30)(0,0)(30,0)(0.25)
%   \rqbezier(15,30)(0,0)(30,0)(0)
%   \put(15,-5){\makebox(0,0)[b]{\texttt{weightscale = \theweightscale}}}
%   \end{picture}}
% \end{picture}
% \setlength{\unitlength}{1pt}
% \resetweightscale
% \caption{The effect of \texttt{weightscale} on the drawing
%          of rational quadratic Bezier curves}
% \label{fig:qrw}
% \end{figure}
%
%    Figure~\ref{fig:qrw} illustrates the effect on changing the 
% \texttt{weightscale} used for drawing the same curves as shown
% in figure~\ref{fig:qrb}. Note that the \texttt{weightscale}
%  has no effect when
% $W = 1$ or $W = 0$ as in these cases the curves are drawn using the
% algorithms for the |\qbezier| and |\lbezier| commands respectively.
%
%    It is obvious that some choices give very poorly formed curves. In
% other cases the curves may be poorly formed but do result in interesting
% cross-stitch like patterns.
%
%    Table~\ref{tab:rq} indicates that it is possible to draw circular
% arcs using a rational quadratic Bezier curves. The two legs of the
% control polygon define the tangents to the curve at the
% end points.
% Therefore, to draw a circular arc the two legs must be equal in length.
% That is, the convex hull is an isosceles triangle. In the special case
% when the convex hull forms an equilateral triangle, the required
% weight ($\cos \beta$, see figure~\ref{fig:beta}) for drawing a circular 
% arc is $\cos \beta = 0.5$. Further,
% for any given control polygon the the curves drawn with weights of
% $\pm W$ are complementary. That is, the curve with weight $-W$ is
% the `remainder' of the curve drawn with weight $W$. Thus, we have a
% simple means of drawing a complete circle, as shown in figure~\ref{fig:qrc}.
% The plotting commands of interest were:
% \begin{verbatim}
%     \lbezier[25](0,0)(15,26)
%     \lbezier[25](0,0)(30,0)
%     \setweightscale{50000}
%     \rqbezier[100](15,26)(0,0)(30,0)(0.5)
%     \rqbezier[200](15,26)(0,0)(30,0)(-0.5)
%     \resetweightscale
% \end{verbatim}
% where the |\lbezier| drawing commands were used to draw the dotted outline
% of the control polygon.
%
% \begin{figure}
% \centering
% \setlength{\unitlength}{1mm}
% \setweightscale{50000}
% \begin{picture}(60,62)
%   \put(0,5){\begin{picture}(30,30)
%     \thinlines
%     \put(0,0){\circle{2}}
%     \put(-2,0){\makebox(0,0)[br]{1}}
%     \put(15,26){\circle{2}}
%     \put(13,26){\makebox(0,0)[br]{0}}
%     \put(30,0){\circle{2}}
%     \put(32,0){\makebox(0,0)[bl]{2}}
%     ^^A polygon
%     \lbezier[25](0,0)(15,26)
%     \lbezier[25](0,0)(30,0)
%     ^^A RQBEZIER
%     \thicklines
%     \rqbezier[100](15,26)(0,0)(30,0)(0.5)
%     \rqbezier[200](15,26)(0,0)(30,0)(-0.5)
%     \end{picture}}
%   \end{picture}
% \setlength{\unitlength}{1pt}
% \caption{Rational quadratics with weights of $\pm 0.5$ and an equilateral
%          triangular convex hull
%          (\texttt{weightscale = \theweightscale}) } \label{fig:qrc}
% \resetweightscale
% \end{figure}
%
%    A more robust picture of the same circle is shown in 
% figure~\ref{fig:qr3c} where the complete circle is pieced together from
% three non-complementary circular arcs. The drawing commands of interest
% were
% \begin{verbatim}
%     \rqbezier[100](15,26)(0,0)(30,0)(0.5)
%     \rqbezier[100](30,0)(60,0)(45,26)(0.5)
%     \rqbezier[100](45,26)(30,52)(15,26)(0.5)
% \end{verbatim}
%
% \begin{figure}
% \centering
% \setlength{\unitlength}{1mm}
% \begin{picture}(60,62)
%   \put(0,5){\begin{picture}(60,52)
%     \thinlines
%     \put(0,0){\circle{2}}
% ^^A    \put(-2,0){\makebox(0,0)[br]{1}}
%     \put(15,26){\circle{2}}
% ^^A    \put(13,30){\makebox(0,0)[br]{0}}
%     \put(30,0){\circle{2}}
% ^^A    \put(32,0){\makebox(0,0)[bl]{2}}
%     \put(60,0){\circle{2}}
%     \put(45,26){\circle{2}}
%     \put(30,52){\circle{2}}
%     ^^A polygon
%     \lbezier[50](0,0)(30,52)
%     \lbezier[50](30,52)(60,0)
%     \lbezier[50](0,0)(60,0)
%     ^^A RQBEZIER
%     \thicklines
%     \rqbezier[100](15,26)(0,0)(30,0)(0.5)
%     \rqbezier[100](30,0)(60,0)(45,26)(0.5)
%     \rqbezier[100](45,26)(30,52)(15,26)(0.5)
%     \end{picture}}
%   \end{picture}
% \setlength{\unitlength}{1pt}
% \caption{Three rational quadratics with weights of $0.5$ 
%          (\texttt{weightscale = \theweightscale}) }
% \label{fig:qr3c}
% \end{figure}
%
%    The astute reader will have realised that the divisor in 
% \eqref{eq:rqfinal} can go to zero, and can even be negative.
% This has interesting consequences, both when trying to do computer
% arithmetic, and also on the the kind of curve that results. Essentially,
% the curve tends to $\infty$ as $W \rightarrow +0$. At $W = -0$ the curve
% is at $-\infty$ and then it tends to $-0$ as $W \rightarrow -\infty$.
% We will get a curve point at $\infty$ whenever $W = -1$ and a `negative'
% curve for $W < -1$. 
%
% \begin{figure}
% \centering
% \setlength{\unitlength}{1mm}
% \begin{picture}(60,60)
%   \put(30,20){\begin{picture}(30,20)
%     \thinlines
%     \put(0,10){\circle{2}}
%     \put(30,0){\circle{2}}
%     \put(30,20){\circle{2}}
%     ^^A polygon
%     \lbezier[25](30,20)(0,10)
%     \lbezier[25](0,10)(30,0)
%     ^^A RQBEZIER
%     \thicklines
%     \rqbezier[100](30,20)(0,10)(30,0)(2)
%     \rqbezier[100](30,20)(0,10)(30,0)(-2)
%     \end{picture}}
%   \end{picture}
% \setlength{\unitlength}{1pt}
% \caption{A rational quadratic that has gone negative; weights of $\pm 2$ 
%          (\texttt{weightscale = \theweightscale}) }
% \label{fig:neg}
% \end{figure}
%
% This effect is shown in figure~\ref{fig:neg} which draws the two branches
% of a hyperbola. The basic code for the illustration was
% \begin{verbatim}
%     \lbezier[25](30,20)(0,10)
%     \lbezier[25](0,10)(30,0)
%     \rqbezier[100](30,20)(0,10)(30,0)(2)
%     \rqbezier[100](30,20)(0,10)(30,0)(-2)
% \end{verbatim}
% where the control polygon was drawn using the |\lbezier| commands.
%
%
% \section{The \Lpack{bez123} package implementation} \label{sec:bez}
%
%    \LaTeX{} provides a facility for drawing quadratic Bezier curves.
% This package provides additional facilities for drawing linear, 
% rational quadratic, and cubic Bezier curves. 
%
%
% Announce the name and version of the package, which requires \LaTeXe.
%    \begin{macrocode}
%<*bez>
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{bez123}[1998/10/14 v1.1 Bezier curves]
%    \end{macrocode}
% \changes{v1.1}{1998/10/14}{Added call to include multiply package}
%    The package also requires the \Lpack{multiply} package.
%    \begin{macrocode}
\RequirePackage{multiply}[1998/10/14]
%</bez>
%    \end{macrocode}
%
%
% \subsection{Arithmetic in \TeX}
%
%    All arithmetic in \TeX{} is based on integer arithmetic, with a
% maximum integer value of $M = \number\maxdimen$. For example,
% \setcounter{weightscale}{8}\makeatletter\divide\c@weightscale by 3\makeatother
% $8/3 = \theweightscale$,
% \setcounter{weightscale}{9}\makeatletter\divide\c@weightscale by 3\makeatother
% $9/3 = \theweightscale$, and
% \setcounter{weightscale}{10}\makeatletter\divide\c@weightscale by 3\makeatother
% $10/3 = \theweightscale$.
% In other words, division always reduces the absolute value of the dividend,
% and also possibly truncates the value. One consequence of this is that the
% ordering of multiplication and division is important. For instance,
% \setcounter{weightscale}{8}\makeatletter\multiply\c@weightscale by 3 \divide\c@weightscale by 3\makeatother
% $(8 \times 3)/3 = \theweightscale$ but
% \setcounter{weightscale}{8}\makeatletter\divide\c@weightscale by 3 \multiply\c@weightscale by 3\makeatother
% $(8/3) \times 3 = \theweightscale$!
% Thus, in arithmetic calculations involving both multiplication and
% division, the dividend should be maximised and the divisor minimised,
% with multiplication preceeding division; also remembering that there
% is a limit on the size of an integer. To avoid multiplication overflow
% when calculating say, $a \times b$, we must ensure that 
% $\|a\| \leq \|M/b\|$.
%
%    When calculating polynomials, such as that in \eqref{eq:gen},
% we use a technique called Horner's schema, which is also known as nested
% multiplication. A general cubic equation, for example, can be written as:
% \begin{equation}
% p(t) - a_{0} = t(a_{1} + t(a_{2} + ta_{3})) \label{eq:horn}
% \end{equation}
% The following pseudo-code shows one way to implement Horner's schema for
% plotting $N$ points in the interval $0 \leq t \leq 1$
% of \eqref{eq:horn} using integer arithmetic.
% \begin{verbatim}
% procedure plot_cubic(a0, a1, a2, a3:vector; N:integer);
%   local p:vector; end_local;
%   a3 := a3/N;
%   repeat i := 0 to N by 1;
%     p := a3*i;
%     p := p + a2; p := p/N; p := p*i;
%     p := p + a1; p := p/N; p := p*i;
%     draw(p + a0);
%   end_repeat;
%   return;
% end_procedure;
% \end{verbatim}
% We use the above algorithm, with suitable modifications according to the
% degree of the polynomial, for plotting the points along Bezier curves.
%
% \subsection{Linear Bezier curves}
%
%    \begin{macrocode}
%<*bez>
%    \end{macrocode}
%
%    As a linear curve is simpler than a quadratic curve there is no
% need to declare extra variables from those used in the kernel by the
% |\qbezier| macro.
%
% \begin{macro}{\lbezier}
%    The user command to draw a linear Bezier curve represented by
% \eqref{eq:lin3}. The form of the command is:\\
% |\lbezier[|\meta{N}|]{(|\meta{p0}|)(|\meta{p1}|)| \\
% where \meta{pN} is the comma seperated X and Y coordinate values of
% point \textit{pN}.
%
%    \begin{macrocode}
\newcommand{\lbezier}[2][0]{\@lbez{#1}#2}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@lbez}
%    The drawing macro.
%    \begin{macrocode}
\gdef\@lbez#1(#2,#3)(#4,#5){%
%%%%\def\lbezier#1(#2,#3)(#4,#5){%
  \ifnum #1<\@ne
%    \end{macrocode}
%    When the number of plotting points are not given, then we calculate
% how many are needed. First determine the X distance between the end points.
%    \begin{macrocode}
    \@ovxx = #4\unitlength
      \advance\@ovxx by -#2\unitlength
      \ifdim \@ovxx < \z@
        \@ovxx = -\@ovxx
      \fi
%    \end{macrocode}
% Similarly calculate the Y distance.
%    \begin{macrocode}
    \@ovyy = #5\unitlength
      \advance\@ovyy by -#3\unitlength
      \ifdim \@ovyy < \z@
        \@ovyy = -\@ovyy
      \fi
%    \end{macrocode}
% Temporarily store the maximum distance in |\@multicnt|.
%    \begin{macrocode}
    \ifdim \@ovxx > \@ovyy
      \@multicnt = \@ovxx
    \else
      \@multicnt = \@ovyy
    \fi
%    \end{macrocode}
% We use a small square as the visual representation of a point.
% Calculate the number of points required to give 50\% overlap of adjacent
% squares, making
% sure that it doesn't exceed the limit. Store the result in |\@multicnt|.
%    \begin{macrocode}
    \@ovxx = 0.5\@halfwidth
    \divide\@multicnt by \@ovxx
    \ifnum \qbeziermax < \@multicnt
      \@multicnt = \qbeziermax\relax
    \fi
  \else
%    \end{macrocode}
% The number of points is given.
%    \begin{macrocode}
    \@multicnt = #1\relax
  \fi
%    \end{macrocode}
% 
%    Now we can prepare the constants for the plotting loop.
%    \begin{macrocode}
  \@tempcnta = \@multicnt
  \advance\@tempcnta by \@ne
  \@ovdx = #4\unitlength
    \advance\@ovdx by -#2\unitlength
    \divide\@ovdx by \@multicnt
  \@ovdy = #5\unitlength
    \advance\@ovdy by -#3\unitlength
    \divide\@ovdy by \@multicnt
%    \end{macrocode}
% The next bit of code defines the size of the square representing a point.
%    \begin{macrocode}
  \setbox\@tempboxa\hbox{\vrule \@height\@halfwidth
                                \@depth \@halfwidth
                                \@width \@wholewidth}%
%    \end{macrocode}
% Start the plot at the first point.
%    \begin{macrocode}
  \put(#2,#3){%
    \count@ = \z@
    \@whilenum{\count@ < \@tempcnta}\do
%    \end{macrocode}
% Evaluate the polynomial (simple in this case) using Horner's schema.
%    \begin{macrocode}
      {\@xdim = \count@\@ovdx
       \@ydim = \count@\@ovdy
%    \end{macrocode}
% Plot this point.
%    \begin{macrocode}
       \raise \@ydim
         \hb@xt@\z@{\kern\@xdim
                    \unhcopy\@tempboxa\hss}%
       \advance\count@\@ne}}%
%    \end{macrocode}
%  The end of the definition of |\@lbez|.
%    \begin{macrocode}
}
%    \end{macrocode}
% \end{macro}
%
% \subsection{Cubic Bezier curves}
%
%    As cubic curves are more complex than quadratic curves we need some
% extra variables.
% \begin{macro}{\@wxc}
% \begin{macro}{\@wyc}
%    Lengths.
%    \begin{macrocode}
\newlength{\@wxc}
\newlength{\@wyc}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\cbezier}
%    The user command for drawing a cubic Bezier curve as represented
% by \eqref{eq:cubic}. It is called as: \\
% |\cbezier[|\meta{N}|](|\meta{p0}|)(|\meta{p1}|)(|\meta{p2}|)(|\meta{p3}|)|.
%
%    \begin{macrocode}
\newcommand{\cbezier}[2][0]{\@cbez{#1}#2}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@cbez}
%    The drawing macro for cubic Bezier curves.
%    \begin{macrocode}
\gdef\@cbez#1(#2,#3)(#4,#5)(#6,#7)(#8,#9){%
  \ifnum #1<\@ne
%    \end{macrocode}
%    We have to calculate the number of plotting points required. We will
% use the maximum of the box enclosing the convex hull as a measure.
% First do the X value, using |\@ovxx| to store the maximum X coordinate
% and |\@ovdx| the minimum.
%    \begin{macrocode}
    \@ovxx = #2\unitlength
    \@ovdx = \@ovxx
    \@ovdy = #4\unitlength
    \ifdim \@ovdy > \@ovxx
      \@ovxx = \@ovdy
    \fi
    \ifdim \@ovdy < \@ovdx
      \@ovdx = \@ovdy
    \fi
    \@ovdy = #6\unitlength
    \ifdim \@ovdy > \@ovxx
      \@ovxx = \@ovdy
    \fi
    \ifdim \@ovdy < \@ovdx
      \@ovdx = \@ovdy
    \fi
    \@ovdy = #8\unitlength
    \ifdim \@ovdy > \@ovxx
      \@ovxx = \@ovdy
    \fi
    \ifdim \@ovdy < \@ovdx
      \@ovdx = \@ovdy
    \fi
%    \end{macrocode}
% Store the maximum X in |\@ovxx|.
%    \begin{macrocode}
    \advance\@ovxx by -\@ovdx
%    \end{macrocode}
% Repeat the process for the maximum Y value, finally storing 
% this in |\@ovyy|.
%    \begin{macrocode}
    \@ovyy = #3\unitlength
    \@ovdy = \@ovyy
    \@ovdx = #5\unitlength
    \ifdim \@ovdx > \@ovyy
      \@ovyy = \@ovdx
    \fi
    \ifdim \@ovdx < \@ovdy
      \@ovdy = \@ovdx
    \fi
    \@ovdx = #7\unitlength
    \ifdim \@ovdx > \@ovyy
      \@ovyy = \@ovdx
    \fi
    \ifdim \@ovdx < \@ovdy
      \@ovdy = \@ovdx
    \fi
    \@ovdx = #9\unitlength
    \ifdim \@ovdx > \@ovyy
      \@ovyy = \@ovdx
    \fi
    \ifdim \@ovdx < \@ovdy
      \@ovdy = \@ovdx
    \fi
    \advance\@ovyy by -\@ovdy
%    \end{macrocode}
% Temporarily store the max of X and Y in |\@multicnt|.
%    \begin{macrocode}
    \ifdim \@ovxx > \@ovyy
      \@multicnt = \@ovxx
    \else
      \@multicnt = \@ovyy
    \fi
%    \end{macrocode}
% Calculate the number of points required to give 50\% overlap, making
% sure that it doesn't exceed the limit. Store the number of points in
% |\@multicnt|.
%    \begin{macrocode}
    \@ovxx = 0.5\@halfwidth
    \divide\@multicnt by \@ovxx
    \ifnum \qbeziermax < \@multicnt
      \@multicnt = \qbeziermax\relax
    \fi
  \else
%    \end{macrocode}
% The number of points is given.
%    \begin{macrocode}
    \@multicnt = #1\relax
  \fi
%    \end{macrocode}
%
%    Now we can prepare the constants for the plotting loop. First the control
% counts.
%    \begin{macrocode}
  \@tempcnta = \@multicnt
  \advance\@tempcnta by \@ne
%    \end{macrocode}
% Then the cubic coefficients, firstly for X.
%    \begin{macrocode}
  \@ovdx = #4\unitlength  \advance\@ovdx by -#2\unitlength
  \@ovxx = #6\unitlength  \advance\@ovxx by -\@ovdx
  \multiply\@ovdx by \thr@@
  \advance\@ovxx by -#4\unitlength  \multiply\@ovxx by \thr@@
  \@wxc = #4\unitlength  \advance\@wxc by -#6\unitlength
  \multiply\@wxc by \thr@@  \advance\@wxc by #8\unitlength
  \advance\@wxc by -#2\unitlength \divide\@wxc by \@multicnt
%    \end{macrocode}
% And similarly for Y.
%    \begin{macrocode}
  \@ovdy = #5\unitlength  \advance\@ovdy by -#3\unitlength
  \@ovyy = #7\unitlength  \advance\@ovyy by -\@ovdy
  \multiply\@ovdy by \thr@@
  \advance\@ovyy by -#5\unitlength  \multiply\@ovyy by \thr@@
  \@wyc = #5\unitlength  \advance\@wyc by -#7\unitlength
  \multiply\@wyc by \thr@@  \advance\@wyc by #9\unitlength
  \advance\@wyc by -#3\unitlength \divide\@wyc by \@multicnt
%    \end{macrocode}
% Set up the plotting box.
%    \begin{macrocode}
  \setbox\@tempboxa\hbox{\vrule \@height\@halfwidth
                                \@depth \@halfwidth
                                \@width \@wholewidth}%
%    \end{macrocode}
% Start the plot at the first point.
%    \begin{macrocode}
  \put(#2,#3){%
    \count@ = \z@
    \@whilenum{\count@ < \@tempcnta}\do
      {\@xdim = \count@\@wxc
         \advance\@xdim by \@ovxx
         \divide\@xdim by \@multicnt
         \multiply\@xdim by \count@
         \advance\@xdim by \@ovdx
         \divide\@xdim by \@multicnt
         \multiply\@xdim by \count@
       \@ydim = \count@\@wyc
         \advance\@ydim by \@ovyy
         \divide\@ydim by \@multicnt
         \multiply\@ydim by \count@
         \advance\@ydim by \@ovdy
         \divide\@ydim by \@multicnt
         \multiply\@ydim by \count@
%    \end{macrocode}
% Plot the point.
%    \begin{macrocode}
       \raise \@ydim
         \hb@xt@\z@{\kern\@xdim
                    \unhcopy\@tempboxa\hss}%
       \advance\count@\@ne}}% 
%    \end{macrocode}
% The end of the definition of |\@cbez|.
%    \begin{macrocode}
}
%    \end{macrocode}
% \end{macro}
%
%
% \subsection{Quadratic rational Bezier curve}
%
%    This is the most complex of the Bezier curves that we deal with. 
% We need yet more variables.
%
% \begin{macro}{\@ww}
% \begin{macro}{\@wwa}
% \begin{macro}{\@wwb}
% \begin{macro}{\@wwo}
% \begin{macro}{\@wwi}
% Variables for the weight calculations.
%    \begin{macrocode}
\newlength{\@ww}
\newlength{\@wwa}
\newlength{\@wwb}
\newlength{\@wwo}
\newlength{\@wwi}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\c@@pntscale}
% Scale factor for points.
%    \begin{macrocode}
\newcounter{@pntscale}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\c@weightscale}
% Scale factor for divisor.
%    \begin{macrocode}
\newcounter{weightscale}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\botscale}
% Scale factor for bottom weights.
%    \begin{macrocode}
\newlength{\botscale}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\setweightscale}
% User level command |\setweightscale{|\meta{number}|}| for setting the
% divisor scaling.
%    \begin{macrocode}
\newcommand{\setweightscale}[1]{\setcounter{weightscale}{#1}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\resetweightscale}
% User level command for setting the divisor scaling to its default
% value ($10^{4}$). We also ensure that the scaling is set to this value.
%    \begin{macrocode}
\newcommand{\resetweightscale}{\setcounter{weightscale}{10000}}
\resetweightscale
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\rqbezier}
%    The user level command for drawing a rational quadratic Bezier curve
% as represented by \eqref{eq:rqfinal}. The form of the command is \\
% |\rqbezier[|\meta{N}|](|\meta{p0}|)(|\meta{p1}|)(|\meta{p2}|)(|\meta{W}|)| \\
% where the arguments are as per the other Bezier drawing commands, but with
% the final argument being the weight.
%
%    \begin{macrocode}
\newcommand{\rqbezier}[2][0]{\@rqbez{#1}#2}
%    \end{macrocode}
% \end{macro}
%
%
% \begin{macro}{\@rqbez}
%    The drawing macro for a rational quadratic Bezier curve. If the weight
% is such that the curve is either rational quadratic ($W = 1$) or 
% linear ($W = 0$), we use the simpler drawing macro.
%    \begin{macrocode}
\gdef\@rqbez#1(#2,#3)(#4,#5)(#6,#7)(#8){%
  \@ovxx = #8\unitlength
  \ifdim\@ovxx = \unitlength
    \PackageWarning{bez123}{Rational quadratic denerates to quadratic}
    \qbezier[#1](#2,#3)(#4,#5)(#6,#7)
  \else
    \ifdim\@ovxx = \z@
      \PackageWarning{bez123}{Rational quadratic degenerates to linear}
      \lbezier[#1](#2,#3)(#6,#7)
    \else
%    \end{macrocode}
% Calculate the maximum length of the control polygon's bounding box.
% Store the result in |\@wwi|.
%    \begin{macrocode}
    \@ovxx = #4\unitlength
      \advance\@ovxx by -#2\unitlength
      \ifdim \@ovxx < \z@
        \@ovxx = -\@ovxx
      \fi
    \@ovdx = #6\unitlength
      \advance\@ovdx by -#4\unitlength
      \ifdim \@ovdx < \z@
        \@ovdx = -\@ovdx
      \fi
    \ifdim \@ovxx < \@ovdx
      \@ovxx = \@ovdx
    \fi
    \@ovyy = #5\unitlength
      \advance\@ovyy by -#3\unitlength
      \ifdim \@ovyy < \z@
        \@ovyy = -\@ovyy
      \fi
    \@ovdy = #7\unitlength
      \advance\@ovdy by -#5\unitlength
      \ifdim \@ovdy < \z@
        \@ovdy = -\@ovdy
      \fi
    \ifdim \@ovyy < \@ovdy
      \@ovyy = \@ovdy
    \fi
    \ifdim \@ovxx > \@ovyy
      \@multicnt = \@ovxx
    \else
      \@multicnt = \@ovyy
    \fi
    \@wwi = \@multicnt sp
%    \end{macrocode}
% Now determine the number of points to be plotted.
%    \begin{macrocode}
  \ifnum #1<\@ne
    \@ovxx = 0.5\@halfwidth
    \divide\@multicnt by \@ovxx
    \ifnum\qbeziermax < \@multicnt
      \@multicnt = \qbeziermax\relax
    \fi
  \else
%    \end{macrocode}
% Number of points is a given.
%    \begin{macrocode}
    \@multicnt = #1\relax
  \fi
%    \end{macrocode}
% We are going to plot the curve in two halves in an attempt to reduce 
% roundoff problems. At
% a minimum this should at least make a symmetrical curve look symmetric
% about its mid point.
%    \begin{macrocode}
  \@tempcnta = \@multicnt
  \advance\@tempcnta by \@ne
  \divide\@tempcnta by \tw@
  \advance\@tempcnta by \@ne
%    \end{macrocode}
% We now have to deal with a possible multiplication overflow problem due
% to multiplication by the weight. In \eqref{eq:rqfinal} the potentially
% largest term is the coefficient of $t^{2}$ (i.e., $(l_{20}-2Wl_{10})$).
% The maximum length likely to be encountered is, say, 10 inches for a
% drawing on either A4 or US letterpaper. This is approximately 
% $4.8\times10^{8}$sp. Doing a little arithmetic, and remembering that the
% maximum length in \TeX{} is $M = \number\maxdimen$sp, it means that we must
% have $\|W\| \leq 1$ to prevent overflow. However, a typical range for
% $W$ is $-10 \leq W \leq 10$. Therefore we might have to do some scaling.
% Being pessimistic, we'll assume that $l_{20} = - l_{10}$ and that $l_{10}$
% is the largest dimension in the drawing. To prevent overflow we then have to
% meet the condition $\|W\| \leq (M - l_{20})/2l_{20}$, where all lengths are
% positive. We will use |\c@@pntscale| as a scale factor on $W$ to meet this
% condition. Earlier we set |\@wwi| to be the positive value of the largest
% dimension in the drawing.
%
% Set the distance scale factor. First evaluating the test condition.
%    \begin{macrocode}
  \@wwo = \maxdimen
    \advance\@wwo by -\@wwi
    \divide\@wwo by \tw@
    \divide\@wwo by \@wwi
%    \end{macrocode}
% Now perform the check and set the scale factor. We have to get a positive
%  integer value for $W$ as it may be a fraction. Actually, we only need to
% be concerned if $\|W\| > 1$.
%    \begin{macrocode}
  \@wwi = 10sp
  \@wwi = #8\@wwi
    \ifdim\@wwi < \z@
      \@wwi = -\@wwi
    \fi
    \divide\@wwi by 10\relax
  \ifdim\@wwi < \@wwo
    \c@@pntscale = \@ne
  \else
    \divide\@wwi by \tw@
    \ifdim\@wwi < \@wwo
      \c@@pntscale = \tw@
    \else
      \divide\@wwi by \tw@
      \ifdim\@wwi < \@wwo
        \c@@pntscale = 4\relax
      \else
        \divide\@wwi by \tw@
        \ifdim\@wwi < \@wwo
          \c@@pntscale = 8\relax
        \else
          \c@@pntscale = 16\relax
        \fi
      \fi
    \fi
  \fi
%    \end{macrocode}
% Calculate the constants for the top line of the function.
%    \begin{macrocode}
  \@ovxx = #4\unitlength \advance\@ovxx by -#2\unitlength
    \multiply\@ovxx by \tw@
    \divide\@ovxx by \c@@pntscale
    \@ovdx = #8\@ovxx
  \@ovxx = #6\unitlength  \advance\@ovxx by -#2\unitlength
    \divide\@ovxx by \c@@pntscale
    \advance\@ovxx by -\@ovdx
    \divide\@ovxx by \@multicnt
  \@ovyy = #5\unitlength \advance\@ovyy by -#3\unitlength
    \multiply\@ovyy by \tw@
    \divide\@ovyy by \c@@pntscale
    \@ovdy = #8\@ovyy
  \@ovyy = #7\unitlength  \advance\@ovyy by -#3\unitlength
    \divide\@ovyy by \c@@pntscale
    \advance\@ovyy by -\@ovdy
    \divide\@ovyy by \@multicnt
%    \end{macrocode}
% Now the constants for the bottom line. We also need to do some scaling
% here. This scaling can be set by the user.
%    \begin{macrocode}
  \setlength{\botscale}{\c@weightscale sp}
  \@wwo = \botscale
  \@wwi = #8\@wwo
  \@wwa = \@wwo  \advance\@wwa by -\@wwi
    \multiply\@wwa by \tw@
  \@wwb = \@wwa
    \divide\@wwb by \@multicnt
%    \end{macrocode}
% Prepare for the drawing.
%    \begin{macrocode}
  \@wwi = \botscale
  \setbox\@tempboxa\hbox{\vrule \@height\@halfwidth
                                \@depth \@halfwidth
                                \@width \@wholewidth}%
%    \end{macrocode}
% Draw the first half of the curve.
%    \begin{macrocode}
  \put(#2,#3){%
    \count@ = \z@
    \@whilenum{\count@ < \@tempcnta}\do
      {\@xdim = \count@\@ovxx
         \advance\@xdim by \@ovdx
         \divide\@xdim by \@multicnt
         \multiply\@xdim by \count@
       \@ydim = \count@\@ovyy
         \advance\@ydim by \@ovdy
         \divide\@ydim by \@multicnt
         \multiply\@ydim by \count@
       \@ww = \count@\@wwb
         \advance\@ww by -\@wwa
         \divide\@ww by \@multicnt
         \multiply\@ww by \count@
         \advance\@ww by \@wwo
         \divide\@ww by \c@@pntscale
         \ifdim\@ww = \z@
%    \end{macrocode}
% We are about to divide by |\@ww| which is zero. Treat |\@ww| as unity.
%    \begin{macrocode}
         \else
           \divide\@xdim by \@ww
           \divide\@ydim by \@ww
         \fi
%    \end{macrocode}
% For reasons I don't understand, the \% signs at the end of the next few
% lines are important!
%    \begin{macrocode}
         \multnooverflow{\@xdim}{\botscale}%
         \multnooverflow{\@ydim}{\botscale}%
       \raise \@ydim
         \hb@xt@\z@{\kern\@xdim
                    \unhcopy\@tempboxa\hss}%
       \advance\count@\@ne}}
%    \end{macrocode}
%
%    We now repeat the above process for plotting the second half of the
% curve, starting at the end point.
%
% Calculate the constants for the top line of the function.
%    \begin{macrocode}
  \@ovxx = #4\unitlength \advance\@ovxx by -#6\unitlength
    \multiply\@ovxx by \tw@
    \divide\@ovxx by \c@@pntscale
    \@ovdx = #8\@ovxx
  \@ovxx = #2\unitlength  \advance\@ovxx by -#6\unitlength
    \divide\@ovxx by \c@@pntscale
    \advance\@ovxx by -\@ovdx
    \divide\@ovxx by \@multicnt
  \@ovyy = #5\unitlength \advance\@ovyy by -#7\unitlength
    \multiply\@ovyy by \tw@
    \divide\@ovyy by \c@@pntscale
    \@ovdy = #8\@ovyy
  \@ovyy = #3\unitlength  \advance\@ovyy by -#7\unitlength
    \divide\@ovyy by \c@@pntscale
    \advance\@ovyy by -\@ovdy
    \divide\@ovyy by \@multicnt
%    \end{macrocode}
% The constants for the bottom line are the same as before as the function
% is symmetric. Similarly we don't need to recalculate the size of the
% rule box.
%
% Draw the second half of the curve.
%    \begin{macrocode}
  \put(#6,#7){%
    \count@ = \z@
    \@whilenum{\count@ < \@tempcnta}\do
      {\@xdim = \count@\@ovxx
         \advance\@xdim by \@ovdx
         \divide\@xdim by \@multicnt
         \multiply\@xdim by \count@
       \@ydim = \count@\@ovyy
         \advance\@ydim by \@ovdy
         \divide\@ydim by \@multicnt
         \multiply\@ydim by \count@
       \@ww = \count@\@wwb
         \advance\@ww by -\@wwa
         \divide\@ww by \@multicnt
         \multiply\@ww by \count@
         \advance\@ww by \@wwo
         \divide\@ww by \c@@pntscale
         \ifnum\@ww = \z@
%    \end{macrocode}
% We are about to divide by |\@ww| which is zero. Treat |\@ww| as unity.
%    \begin{macrocode}
         \else
           \divide\@xdim by \@ww
           \divide\@ydim by \@ww
         \fi
%    \end{macrocode}
% For reasons I don't understand, the \% signs at the end of the next few
% lines are important!
%    \begin{macrocode}
         \multnooverflow{\@xdim}{\botscale}%
         \multnooverflow{\@ydim}{\botscale}%
       \raise \@ydim
         \hb@xt@\z@{\kern\@xdim
                    \unhcopy\@tempboxa\hss}%
       \advance\count@\@ne}}
% End of definition of |\@rqbez|.
%    \begin{macrocode}
  \fi\fi}
%    \end{macrocode}
% \end{macro}
%
%    The end of this package.
%    \begin{macrocode}
%</bez>
%    \end{macrocode}
%
% \section{Multiplication without overflow: The \Lpack{multiply} package} \label{sec:mnoflow}
%
%    \TeX{} provides for integer arithmetic, subject to an upper limit
% given by |\maxdim|. For at least the \Lpack{bez123} package we need to 
% be able to multiply without overflow.
%
%    Announce the name of the package.
% \changes{v1.1}{1998/10/14}{Put multnooverflow into a seperate package}
%    \begin{macrocode}
%<*mult>
\ProvidesPackage{multiply}[1998/10/14 v1.1 Multiplication of lengths without overflow]
%    \end{macrocode}
%
% \begin{macro}{\n@fl@wa}
% \begin{macro}{\n@fl@wb}
% \begin{macro}{\n@fl@wc}
% \begin{macro}{\ifch@nge}
%    We need three length variables for this function. 
% We also need a boolean flag for dealing with negative numbers.
%    \begin{macrocode}
\newlength{\n@fl@wa}
\newlength{\n@fl@wb}
\newlength{\n@fl@wc}
\newif\ifch@nge
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\multnooverflow}
%
%    The routine |\multnooverflow{|\meta{a}|}{|\meta{b}|}| sets 
% $a$ to the minimum of $ab$ and |\maxdimen|, preserving signs. \meta{a}
% must be a length; it must not be a number literal.
%
%    \begin{macrocode}
\newcommand{\multnooverflow}[2]{%
  \n@fl@wa = #1\relax%
  \n@fl@wb = #2\relax%
  \ch@ngefalse%
%    \end{macrocode}
% Easy if $-1 \leq b \leq 1$.
%    \begin{macrocode}
  \ifnum\n@fl@wb = \@ne%
  \else%
    \ifnum\n@fl@wb = \z@%
      \n@fl@wa = \z@%
    \else%
      \ifnum\n@fl@wb = \m@ne%
        \ch@ngetrue%
      \else%
%    \end{macrocode}
% Also easy if $-1 \leq a \leq 1$.
%    \begin{macrocode}
        \ifnum\n@fl@wa = \z@%
        \else%
          \ifnum\n@fl@wa = \@ne%
            \n@fl@wa = \n@fl@wb%
          \else%
            \ifnum\n@fl@wa = \m@ne%
              \n@fl@wa = -\n@fl@wb%
            \else%
%    \end{macrocode}
% We have to check for potential overflow. First make sure that we deal
% only with positive values.
%    \begin{macrocode}
              \ifnum\n@fl@wa < \z@%
                \ch@ngetrue%
                \n@fl@wa = -\n@fl@wa%
              \fi%
              \ifnum\n@fl@wb < \z@%
                \n@fl@wb = -\n@fl@wb%
                \ifch@nge%
                  \ch@ngefalse%
                \else%
                  \ch@ngetrue%
                \fi%
              \fi%
%    \end{macrocode}
% Check for overflow.
%    \begin{macrocode}
              \n@fl@wc = \maxdimen%
              \divide\n@fl@wc by \n@fl@wb%
              \advance\n@fl@wc by -1sp% \m@ne
              \ifnum\n@fl@wa > \n@fl@wc%
%    \end{macrocode}
% We have overflow. Set the multiplication result to |\maxdimen|.
%    \begin{macrocode}
                \n@fl@wa = \maxdimen%
                \PackageWarning{multiply}{Multiplication overflow}%
              \else%
%    \end{macrocode}
% It is safe to do the multiplication.
%    \begin{macrocode}
                \multiply\n@fl@wa by \n@fl@wb%
              \fi%
            \fi%
          \fi%
        \fi%
      \fi%
    \fi%
  \fi%
%    \end{macrocode}
% The result of $ab$ is in |\n@fl@wa|. Adjust the sign if necessary.
%    \begin{macrocode}
  \ifch@nge%
    \n@fl@wa = -\n@fl@wa%
  \fi%
%    \end{macrocode}
% Return the result in the first argument variable.
%    \begin{macrocode}
  #1 = \n@fl@wa%
}
%    \end{macrocode}
% \end{macro}
%
%
%    The end of this package.
%    \begin{macrocode}
%</mult>
%    \end{macrocode}
%
%
% \bibliographystyle{alpha}
%
% \begin{thebibliography}{GMS94}
%
% \bibitem[Far90]{FARIN90}
% Gerald Farin.
% \newblock {\em Curves and Surfaces for Computer Aided Geometric Design --- A
%   Practical Guide}.
% \newblock Academic Press, Inc., second edition, 1990.
%
% \bibitem[FP81]{FandP}
% I.~D. Faux and M.~J. Pratt.
% \newblock {\em Computational Geometry for Design and Manufacture}.
% \newblock Ellis Horwood, 1981.
%
% \bibitem[GMS94]{GOOSSENS94}
% Michel Goossens, Frank Mittelbach, and Alexander Samarin.
% \newblock {\em The LaTeX Companion}.
% \newblock Addison-Wesley Publishing Company, 1994.
%
% \bibitem[Lam94]{LAMPORT94}
% Leslie Lamport.
% \newblock {\em LaTeX: A Document Preparation System}.
% \newblock Addison-Wesley Publishing Company, second edition, 1994.
%
% \bibitem[Mor85]{MORTENSON85}
% Michael~E. Mortenson.
% \newblock {\em Geometric Modeling}.
% \newblock John Wiley \& Sons, Inc., 1985.
%
% \bibitem[Wil96]{PRW96i}
% Peter~R. Wilson.
% \newblock {\em {LaTeX for standards: The LaTeX package files user manual}}.
% \newblock NIST Report NISTIR, June 1996.
%
% \end{thebibliography}
%
%
% \Finale
% \PrintIndex
%
\endinput

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