% \iffalse
%%
%% File `curve2e.dtx'.
%% Copyright (C)  2005--2006 Claudio Beccari  all rights reserved.
%%
% What follows is the usual trick that is not typeset in the documentation
% dvi file that is produced by LaTeX. It is used to define the date, the version
% and the short description that characterizes both this file and the package;
% the point is that |\ProvidesFile| is being read only by the driver, while
% |\ProvidePackage| goes to the stripped package file; It must be done before
% starting the documentation otherwise |\GetFileInfo|  can't get the necessary
% information.
% \fi
%<*package>
%    \begin{macrocode}
\NeedsTeXFormat{LaTeX2e}
%</package>
%<*driver>
\ProvidesFile{curve2e.dtx}%
%</driver>
%<+package>\ProvidesPackage{curve2e}%
        [2008/05/04 v.1.01 Extension package for pict2e]
%<*package>
%    \end{macrocode}
%</package>
% \iffalse
%<*driver>
\documentclass{ltxdoc}
\hfuzz 10pt
\usepackage{multicol}
\usepackage[ansinew]{inputenc}
\GetFileInfo{curve2e.dtx}
\title{The extension package \textsf{curve2e}\thanks{Version number
\fileversion; last revised \filedate.}}
\author{Claudio Beccari}
\begin{document}
 \maketitle
\begin{multicols}{2}
 \tableofcontents
 \end{multicols}
 \DocInput{curve2e.dtx}
\end{document}
%</driver>
% \fi
%
% \CheckSum{2222}
% \begin{abstract}
% This file documents the |curve2e| extension package to the recent
% implementation of the |pict2e| bundle that has been described by Lamport
% himself in the second edition of his \LaTeX\ handbook.
%
% This extension redefines a couple of commands and introduces some more drawing
% facilities that allow to draw circular arcs and arbitrary curves with the
% minimum of user intervention. This beta version is open to the contribution of
% other users as well as it may be incorporated in other people's packages.
% Please cite the original author and the chain of contributors.
% \end{abstract}
%
% \section{Package \texttt{pict2e} and this extension \texttt{curve2e}}
% Package \texttt{pict2e} was announced in issue 15 of \texttt{latexnews}
% around December 2003; it was declared that the new package would replace the
% dummy one that has been accompanying every release of \LaTeXe\ since its
% beginnings in 1994. The dummy package was just issuing an info message that
% simply announced the temporary unavailability of the real package.
%
% Eventually Gäßlein and Niepraschk implemented what Lamport himself had already
% documented in the second edition of his \LaTeX\ handbook, that is a \LaTeX\
% package that contained the macros capable of removing all the limitations
% contained in the standard commands of the original \texttt{picture}
% environment; specifically:
% \begin{enumerate}
% \item the line and vector slopes were limited to the ratios of relatively
% prime one digit integers of magnitude not exceeding 6 for lines and 4 for
% vectors;
% \item filled and unfilled full circles were limited by the necessarily
% bounded number of specific glyphs contained in the special \LaTeX\
% \texttt{picture} fonts;
% \item quarter circles were also limited in their radii for the same reason;
% \item ovals (rectangles with rounded corners) could not be too small because
% of the unavailability of small radius quarter circles, nor could be too
% large, in the sense that after a certain radius the rounded corners remained
% the same and would not increase proportionally to the oval size.
% \item vector arrows had only one possible shape besides matching the limited
% number of vector slopes;
% \item for circles and inclined lines and vectors there were available just
% two possible thicknesses.
% \end{enumerate}
%
% The package \texttt{pict2e} removes most if not all the above limitations:
% \begin{enumerate}
% \item line and vector slopes are virtually unlimited; the only remaining
% limitation is that the direction coefficients must be three-digit integer
% numbers; they need not be relatively prime;
% \item filled and unfilled circles can be of any size;
% \item ovals can be designed with  any specified corner curvature and there is
% virtually no limitation to such curvatures; of course corner radii should not
% exceed half the lower value between the base and the height of the oval;
% \item there are two shapes for the arrow tips; the triangular one traditional
% with \LaTeX\ vectors, or the arrow tip with PostScript style.
% \item the |\linethickness| command changes the thickness of all lines, straight,
% curved, vertical, horizontal, arrow tipped, et cetera.
% \end{enumerate}
%
%  This specific extension adds the following features
% \begin{enumerate}
% \item commands for setting the line terminations are introduced; the user can
% chose between square or rounded caps; the default is set to rounded caps;
% \item the |\line| macro is redefined so as to allow integer and fractional
% direction coefficients, but maintaining the same syntax as in the original
% \texttt{picture} environment;
% \item a new macro |\Line| is defined so as to avoid the need to specify the
% horizontal projection of inclined lines;
% \item a new macro |\LINE| joins two points specified with their coordinates;
% of course there is no need to use the |\put| command with this line
% specification;
% \item similar macros are redefined for vectors; |\vector| redefines the
% original macro but with the vector slope limitation removed; |\Vector| gets
% specified with its two horizontal and vertical components; |\VECTOR|
% joins two specified points (without using the |\put| command) with  the arrow
% pointing to the second point;
% \item a new macro |\polyline| for drawing polygonal lines is defined that
% accepts from two vertices up to an arbitrary (reasonably limited) number of
% them;
% \item a new macro |\Arc| is defined in order to draw an arc with arbitrary
% radius and arbitrary angle amplitude; this amplitude is specified in
% sexagesimal degrees, not in radians;
% \item two new macros are defined in order to draw circular arcs with one
% arrow at one or both ends;
% \item a new macro |\Curve| is defined so as to draw arbitrary curved lines
% by means of third order Bézier splines; the |\Curve| macro requires only the
% curve nodes and the direction of the tangents at each node.
% \end{enumerate}
%
% In order to make the necessary calculations many macros have been defined so
% as to use complex number to manipulate point coordinates, directions,
% rotations and the like. The trigonometric functions have also been defined in
% a way that the author believes to be more efficient that that implied by the
%  \texttt{trig} package; in any case the macro names are sufficiently
% different to accommodate both definitions in the same \LaTeX\ run.
%
% Many aspects of this extension could be fine tuned for better performance;
% many new commands could be defined in order to further extend this extension.
% If the new service macros are accepted by other \TeX\ and \LaTeX\ programmers,
% this beta version could become the start for a real extension of the
% \texttt{pict2e} package or even become a part of it.
%
% For this reason I suppose that every enhancement should be submitted to
% Gäßlein and Niepraschk  who are the prime maintainers of \texttt{pict2e};
% they only can decide whether or not to incorporate new macros in their package.
%
% \section{Acknowledgements}
% I wish to express my deepest thanks to Michel Goosens who spotted some errors
% and very kindly submitted them to me so that I was able to correct them.
%
% \StopEventually{%
% \begin{thebibliography}{9}
% \bibitem{pict2e} Gäßlein H.\ and Niepraschk R., \emph{The \texttt{pict2e}
% package}, PDF document attached to the ``new'' \texttt{pict2e} bundle; the
% bundle may be downloaded from any CTAN archive or one of their mirrors.
% \end{thebibliography}
% }
%
% \section{Source code}
% \subsection{Some preliminary extensions to the \texttt{pict2e} package}
% The necessary preliminary code has already been introduced. Here we require
% the \texttt{color} package and the \texttt{pict2e} one; for the latter one we
% make sure that a sufficiently recent version is used.
%    \begin{macrocode}
\RequirePackage{color}
\RequirePackageWithOptions{pict2e}[2004/06/01]
%    \end{macrocode}
%  Next we define the line terminators and joins; the following definitions work
% correctly  if the \texttt{dvips} or the \texttt{pdftex} driver are specified;
% probably other modes should be added so as to be consistent with
% \texttt{pict2e}.
%    \begin{macrocode}
\ifcase\pIIe@mode\relax
\or %Postscript
 \def\roundcap{\special{ps:: 1 setlinecap}}%
 \def\squarecap{\special{ps:: 0 setlinecap}}%
 \def\roundjoin{\special{ps:: 1 setlinejoin}}%
 \def\beveljoin{\special{ps:: 2 setlinejoin}}%
\or %pdf
 \def\roundcap{\pdfliteral{1 J}}%
 \def\squarecap{\pdfliteral{0 J}}%
 \def\roundjoin{\pdfliteral{1 j}}%
 \def\beveljoin{\pdfliteral{2 j}}%
\fi
%    \end{macrocode}
%
% The next macros are just for debugging. With the \texttt{tracing} package it
% would probably be better to define other macros, but this is not for the
% users, but for the developers.
%    \begin{macrocode}
\def\TRON{\tracingcommands\tw@ \tracingmacros\tw@}%
\def\TROF{\tracingcommands\z@ \tracingmacros\z@}%
%    \end{macrocode}
%
% Next we define some new dimension registers that will be used by the
% subsequent macros; should they be already defined, there will not be any
% redefinition; nevertheless the macros should be sufficiently protected so as
% to avoid overwriting register values loaded by other macro packages.
%    \begin{macrocode}
\ifx\undefined\@tdA \newdimen\@tdA \fi
\ifx\undefined\@tdB \newdimen\@tdB \fi
\ifx\undefined\@tdC \newdimen\@tdC \fi
\ifx\undefined\@tdD \newdimen\@tdD \fi
\ifx\undefined\@tdE \newdimen\@tdE \fi
\ifx\undefined\@tdF \newdimen\@tdF \fi
\ifx\undefined\defaultlinewidth \newdimen\defaultlinewidth \fi
%    \end{macrocode}
%
% It is better to define a macro for setting a different value for the line and
% curve thicknesses; the `|\defaultlinewidth| should contain the
% equivalent of |\@wholewidth|, that is the thickness of thick lines; thin lines
% are half as thick; so when the default line thickness is specified to, say,
% 1pt, thick lines will be 1pt thick and thin lines will be 0.5pt thick.  The
% default whole width of thick lines is 0,8pt, but this is specified in the
% kernel of \LaTeX\ and\slash or in \texttt{pict2e}. On the opposite it is
% necessary to redefine |\linethickness| because the \LaTeX\ kernel global definition
% does not hide the space after the closed brace when you enter something such as
% |\linethickness{1mm}| followed by a space or a new line.\footnote{Thanks to
% Daniele Degiorgi (\texttt{degiorgi@inf.ethz.ch}).}
%    \begin{macrocode}
\gdef\linethickness#1{\@wholewidth#1\@halfwidth.5\@wholewidth\ignorespaces}%
\newcommand\defaultlinethickness[1]{\defaultlinewidth=#1\relax
\def\thicklines{\linethickness{\defaultlinewidth}}%
\def\thinlines{\linethickness{.5\defaultlinewidth}}%
\thinlines\ignorespaces}
%    \end{macrocode}
% The |\ignorespaces| at the end of this and the subsequent macros is for
% avoiding spurious spaces to get into the picture that is being drawn, because
% these spaces introduce picture deformities often difficult to spot and
% eliminate.
%
% \subsubsection{Improved line and vector macros}
% The new macro |\Line| allows to draw an arbitrary inclination line as if it
% was a polygon with just two vertices. This line should be set by means of a
% |\put| command so that its starting point is always at a relative 0,0
% coordinate point. The two arguments define the horizontal and the
% vertical component respectively.
%    \begin{macrocode}
\def\Line(#1,#2){\pIIe@moveto\z@\z@
      \pIIe@lineto{#1\unitlength}{#2\unitlength}\pIIe@strokeGraph}%
%    \end{macrocode}
%
% A similar macro |\LINE| operates between two explicit points with absolute
% coordinates, instead of relative to the position specified by a |\put|
% command; it resorts to the |\polyline| macro that is to be defined in a while.
% The |\@killglue|command might be unnecessary, but it does not harm; it eliminates any
% explicit or implicit spacing that might precede this command.
%    \begin{macrocode}
\def\LINE(#1)(#2){\@killglue\polyline(#1)(#2)}%
%    \end{macrocode}
%
%  The |\line| macro is redefined by making use of a new division routine that
%  receives in input two dimensions and yields on output their fractional ratio.
% The beginning of the macro definition is the same as that of \texttt{pict2e}:
%    \begin{macrocode}
\def\line(#1)#2{\begingroup
  \@linelen #2\unitlength
  \ifdim\@linelen<\z@\@badlinearg\else
%    \end{macrocode}
% but as soon as it is verified that the line length is not negative, things
% change remarkably; in facts the machinery for complex numbers is invoked:
% |\DirOfVect| takes the only macro argument (that actually contains a comma
% separated pair of fractional numbers) and copies it to |\Dir@line| (an
% arbitrarily named control sequence) after re-normalizing to unit magnitude;
% this is passed to |GetCoord| that separates the two components into the
% control sequences |\d@mX| and|\d@mY|; these in turn are the values that are
% actually operated upon by the subsequent commands.
%    \begin{macrocode}
    \expandafter\DirOfVect#1to\Dir@line
    \GetCoord(\Dir@line)\d@mX\d@mY
%    \end{macrocode}
% The normalized vector direction is actually formed with the directing cosines
% of the line direction; since the line length is actually the horizontal
% component for non vertical lines, it is necessary to compute the actual line
% length for non vertical lines by dividing the given length by the
% magnitude of horizontal cosine |\d@mX|, and the line length is accordingly
% scaled:
%    \begin{macrocode}
    \ifdim\d@mX\p@=\z@\else
      \DividE\ifdim\d@mX\p@<\z@-\fi\p@ by\d@mX\p@ to\sc@lelen
      \@linelen=\sc@lelen\@linelen
    \fi
%    \end{macrocode}
%  Finally the \texttt{moveto}, \texttt{lineto} and \texttt{stroke} language
% keywords are invoked by means of the internal \texttt{pict2e} commands in
% order to draw the line. Notice that even vertical lines are drawn with the
% ``PostScript'' commands instead of resorting to the dvi low level language
% that was used both in \texttt{pict2e} and in the original \texttt{picture}
% commands; it had a meaning in the old times, but it certainly does not have
% any when lines are drawn by the driver that drives the output to a visible
% document form, not by \TeX\ the program.
%    \begin{macrocode}
    \pIIe@moveto\z@\z@
    \pIIe@lineto{\d@mX\@linelen}{\d@mY\@linelen}%
    \pIIe@strokeGraph
\fi
\endgroup\ignorespaces}%
%    \end{macrocode}
% The new macro |\GetCoord| splits a vector (or complex number) specification
% into its components:
%    \begin{macrocode}
\def\GetCoord(#1)#2#3{%
\expandafter\SplitNod@\expandafter(#1)#2#3\ignorespaces}
%    \end{macrocode}
% But the macro that does the real work is |\SplitNod@|:
%    \begin{macrocode}
\def\SplitNod@(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}%
%    \end{macrocode}
%
% The redefinitions and the new definitions for vectors are a little more
% complicated than with segments, because each vector is drawn as a filled
% contour; the original \texttt{pict2e} macro checks if the slopes are
% corresponding to the limitations specified by Lamport (integer three digit
% signed numbers) and sets up a transformation in order to make it possible to
% draw each vector as an horizontal left-to-right arrow and then to rotate it by
% its angle about its tail point; actually there are two macros for tracing the
% contours that are eventually filled by the principal macro; each contour
% macro draws the vector with a \LaTeX\ or a PostScript arrow whose parameters
% are specified by default or may be taken from the parameters taken from the
%\texttt{PSTricks} package if this one is loaded before \texttt{pict2e}; in any
% case we did not change the contour drawing macros because if they are
% modified the same modification is passed on to the arrows drawn with the
% \texttt{curve2e} package redefinitions.
%
% Because of these features the redefinitions and the new macros are different
% from those used for straight lines.
%
% We start with the redefinition of |\vector| and we use the machinery for
% vectors (as complex numbers) we used for |\line|.
%    \begin{macrocode}
  \def\vector(#1)#2{%
    \begingroup
      \GetCoord(#1)\d@mX\d@mY
      \@linelen#2\unitlength
%    \end{macrocode}
% As in \texttt{pict2e} we avoid tracing vectors if the slope parameters are
% both zero.
%    \begin{macrocode}
      \ifdim\d@mX\p@=\z@\ifdim\d@mY\p@=\z@\@badlinearg\fi\fi
%    \end{macrocode}
% But we check only for the positive nature of the $l_x$ component; if it is
% negative, we simply change sign instead of blocking the typesetting process.
% This is useful also for macros |\Vector| and |\VECTOR| to be defined in a
% while.
%    \begin{macrocode}
      \ifdim\@linelen<\z@ \@linelen=-\@linelen\fi
%    \end{macrocode}
% We now make a vector with the slope direction even if one or the other is
% zero and we determine its direction; the real and imaginary parts of the
% direction vector are also the values we need for the subsequent rotation.
%    \begin{macrocode}
      \MakeVectorFrom\d@mX\d@mY to\@Vect
      \DirOfVect\@Vect to\Dir@Vect
%    \end{macrocode}
% In order to be compatible with the original \texttt{pict2e} I need to
% transform the components of the vector direction in lengths with the specific
% names |\@xdim| and |\@ydim|
%    \begin{macrocode}
        \YpartOfVect\Dir@Vect to\@ynum \@ydim=\@ynum\p@
        \XpartOfVect\Dir@Vect to\@xnum \@xdim=\@xnum\p@
%    \end{macrocode}
% If the vector is really sloping we need to scale the $l_x$ component in order
% to get the vector total length; we have to divide by the cosine of the vector
% inclination which is the real part of the vector direction. I use my division
% macro; since it yields a ``factor'' I directly use it to scale the length of
% the vector. I finally memorize the true vector length in the internal
% dimension |@tdB|
%    \begin{macrocode}
        \ifdim\d@mX\p@=\z@
        \else\ifdim\d@mY\p@=\z@
          \else
            \DividE\ifdim\@xnum\p@<\z@-\fi\p@ by\@xnum\p@ to\sc@lelen
            \@linelen=\sc@lelen\@linelen
          \fi
        \fi
      \@tdB=\@linelen
%    \end{macrocode}
%  The remaining code is definitely similar to that of \texttt{pict2e}; the
% real difference consists in the fact that the arrow is designed by itself
% without the stem; but it is placed at the vector end; therefore the first
% statement is just the transformation matrix used by the output driver to
% rotate the arrow tip and to displace it the right amount. But in order
% to draw only the arrow tip I have to set the |\@linelen| length to zero.
%    \begin{macrocode}
\pIIe@concat\@xdim\@ydim{-\@ydim}\@xdim{\@xnum\@linelen}{\@ynum\@linelen}%
        \@linelen\z@
        \pIIe@vector
        \pIIe@fillGraph
%    \end{macrocode}
% Now we can restore the stem length that must be shortened by the dimension of
% the arrow; examining the documentation of \texttt{pict2e} we discover that
% we have to shorten it by an approximate amount of $AL$ (with the notations of
% \texttt{pict2e}, figs~10 and~11); the arrow tip parameters are stored in
% certain variables with which we can determine the amount of the stem
% shortening; if the stem was too short and the new length is negative, we
% refrain from designing such stem.
%    \begin{macrocode}
      \@linelen=\@tdB
      \@tdA=\pIIe@FAW\@wholewidth
      \@tdA=\pIIe@FAL\@tdA
      \advance\@linelen-\@tdA
      \ifdim\@linelen>\z@
        \pIIe@moveto\z@\z@
        \pIIe@lineto{\@xnum\@linelen}{\@ynum\@linelen}%
        \pIIe@strokeGraph\fi
    \endgroup}
%    \end{macrocode}
%
% Now we define the macro that does not require the specification of the length
% or the $l_x$ length component; the way the new |\vector| macro works does not
% actually  require this specification, because \TeX\ can compute the vector
% length, provided the two direction components are exactly the horizontal and
% vertical vector components. If the horizontal component is zero, the actual length
% must be specified as the vertical component.
%    \begin{macrocode}
\def\Vector(#1,#2){%
\ifdim#1\p@=\z@\vector(#1,#2){#2}
\else
\vector(#1,#2){#1}\fi}
%    \end{macrocode}
%
% On the opposite the next macro specifies a vector by means of the coordinates
% of its end points; the first point is where the vector starts, and the second
% point is  the arrow tip side. We need the difference as these two coordinates,because % it  represents the actual vector.
%    \begin{macrocode}
\def\VECTOR(#1)(#2){\begingroup
\SubVect#1from#2to\@tempa
\expandafter\put\expandafter(#1){\expandafter\Vector\expandafter(\@tempa)}%
\endgroup\ignorespaces}
%    \end{macrocode}
%
% The \texttt{pict2e} documentation says that if the vector length is zero the
% macro designs only the arrow tip; this may work with macro |\vector|,
% certainly not with |\Vector| and |\VECTOR|. This might be useful for adding
% an arrow tip to a circular arc
%
% \subsubsection{Polygonal lines}
% We now define the polygonal line macro; its syntax is very simple
% \begin{flushleft}\ttfamily
% \cs{polygonal}(\rmfamily{P}$_0$)(\rmfamily{P}$_1$)\rmfamily{P}$_2$)\dots
% (\rmfamily{P}$_n$)
% \end{flushleft}
% In order to write a recursive macro we need aliases for the parentheses;
% actually we need only the left parenthesis, but some editors complain about
% unmatched delimiters, so we define an alias also for the right parenthesis.
%    \begin{macrocode}
\let\lp@r( \let\rp@r)
%    \end{macrocode}
% The first call to |\polyline| examines the first point coordinates and moves
% the drawing position to this point; afterwards it looks for the second point
% coordinates; they start with a left parenthesis; if this is found the
% coordinates should be there, but if the left parenthesis is missing (possibly
% preceded by spaces that are ignored by the |\@ifnextchar| macro) then a
% warning message is output together with the line number where the missing
% parenthesis causes the warning: beware, this line number might point to
% several lines further on along the source file! In any case it's necessary to insert
% a |\@killglue| command, because |\polyline| refers to absolute coordinates
% not necessarily is put in position through a |\put| command that provides to
% eliminate any spurious spaces preceding this command.
%    \begin{macrocode}
\def\polyline(#1){\@killglue\beveljoin\GetCoord(#1)\d@mX\d@mY
    \pIIe@moveto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\p@lyline}{%
    \PackageWarning{curve2e}%
    {Polygonal lines require at least two vertices!\MessageBreak
     Control your polygonal line specification\MessageBreak}%
    \ignorespaces}}
%    \end{macrocode}
% But if there is a second or further point coordinate the recursive macro
% |\p@lyline| is called; it works on the next point and checks for a further
% point; if such a point exists it calls itself, otherwise it terminates the
% polygonal line by stroking it.
%    \begin{macrocode}
\def\p@lyline(#1){\GetCoord(#1)\d@mX\d@mY
    \pIIe@lineto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\p@lyline}{\pIIe@strokeGraph\ignorespaces}}
%    \end{macrocode}
%
% \subsubsection{The red service grid}
% The next command is very useful for debugging while editing one's drawings;
% it draws a red grid with square meshes that are ten drawing units apart;
% there is no graduation along the grid, since it is supposed to be a debugging
% aid and the user should know what he/she is doing; nevertheless it is
% advisable to displace the grid by means of a |\put| command so that its grid
% lines coincide with the graph coordinates multiples of 10. Missing to do so
% the readings become cumbersome. The |\RoundUp| macros provide to increase the
% grid dimensions to integer multiples of ten.
%    \begin{macrocode}
\def\GraphGrid(#1,#2){\begingroup\textcolor{red}{\linethickness{.1\p@}%
\RoundUp#1modulo10to\@GridWd \RoundUp#2modulo10to\@GridHt
\@tempcnta=\@GridWd \divide\@tempcnta10\relax \advance\@tempcnta\@ne
\multiput(0,0)(10,0){\@tempcnta}{\line(0,1){\@GridHt}}%
\@tempcnta=\@GridHt \divide\@tempcnta10\advance\@tempcnta\@ne
\multiput(0,0)(0,10){\@tempcnta}{\line(1,0){\@GridWd}}\thinlines}%
\endgroup\ignorespaces}
%    \end{macrocode}
%  Rounding up is useful because also the grid margins fall on coordinates
% multiples of 10. It resorts to the |\Integer| macro that will be described in
% a while.
%    \begin{macrocode}
\def\RoundUp#1modulo#2to#3{\expandafter\@tempcnta\Integer#1.??%
\count254\@tempcnta\divide\count254by#2\relax
\multiply\count254by#2\relax
\count252\@tempcnta\advance\count252-\count254
\ifnum\count252>0\advance\count252-#2\relax
\advance\@tempcnta-\count252\fi\edef#3{\number\@tempcnta}\ignorespaces}%
%    \end{macrocode}
% The |\Integer| macro takes a possibly fractional number whose decimal
% separator, if present, \textit{must} be the decimal point and uses the point
% as an argument delimiter  If one has the doubt that the number being passed
%  to |\Integer| might be an integer, he/she should call the macro with a
% further point;
% if the argument is truly integer this point works as the delimiter of the
% integer part; if the argument being passed is fractional this extra point
% gets discarded as well as the fractional part of the number.
%    \begin{macrocode}
\def\Integer#1.#2??{#1}%
%    \end{macrocode}
%
% \subsection{The new division macro}
% Now comes one of the most important macros in the whole package: the division
% macro; it takes two lengths as input values ant computes their fractional
% ratio.
% It must take care of the signs, so that it examines the operand signs and
% determines the result sign separately conserving this computed sign in the
% macro |\segno|; this done, we are sure that both operands are or are
% made positive; should the
% numerator be zero it directly issues the zero quotient; should the
% denominator be zero it outputs a signed ``infinity'', that is the maximum
% allowable length measured in points that \TeX\ can deal with.
% Since the result is assigned a value, the calling statement must pass as the
% third argument either a control sequence or an active character. Of course the
% first operand is the dividend, the second the divisor and the third the
% quotient.
%    \begin{macrocode}
\ifx\DividE\undefined
  \def\DividE#1by#2to#3{%
      \begingroup
      \dimendef\Numer=254\relax \dimendef\Denom=252\relax
      \countdef\Num 254\relax
      \countdef\Den 252\relax
      \countdef\I=250\relax
      \Numer #1\relax \Denom #2\relax
      \ifdim\Denom<\z@ \Denom -\Denom \Numer -\Numer\fi
      \def\segno{}\ifdim\Numer<\z@ \def\segno{-}\Numer -\Numer\fi
      \ifdim\Denom=\z@
        \ifdim\Numer>\z@\def\Q{16383.99999}\else\def\Q{-16383.99999}\fi
      \else
        \Num=\Numer \Den=\Denom \divide\Num\Den
        \edef\Q{\number\Num.}%
        \advance\Numer -\Q\Denom \I=6\relax
        \@whilenum \I>\z@ \do{\DividEDec\advance\I\m@ne}%
      \fi
      \xdef#3{\segno\Q}\endgroup
  }%
%    \end{macrocode}
%  The |\DividEDec| macro takes the remainder of the previous division,
% multiplies it by 10, computes a one digit quotient that postfixes to the
% previous overall quotient, and computes the next remainder; all operations
% are done on integer registers to whom the dimensional operands are assigned
%  so that the mentioned registers acquire the measures of the dimensions in
% scaled points; \TeX\ is called to perform integer arithmetics, but the long
% division takes care of the decimal separator and of the suitable number of
% fractional digits.
%    \begin{macrocode}
  \def\DividEDec{\Numer=10\Numer \Num=\Numer \divide\Num\Den
              \edef\q{\number\Num}\edef\Q{\Q\q}\advance\Numer -\q\Denom}%
\fi
%    \end{macrocode}
% In the above code the |\begingroup|\dots|\endgroup| maintain all registers
% local so that only the result must be globally defined. The |\ifx|\dots|\fi|
% construct assures the division machinery is not redefined; I use it in so
% many packages that its better not to mix up things even with slightly
% different definitions.
%
% The next two macros are one of the myriad variants of the dirty trick used by
% Knuth for separating a measure from its units that \textit{must} be points,
% ``\texttt{pt}''; One has to call |\Numero| with a control sequence and a
% dimension; the dimension value in points is assigned to the control sequence.
%    \begin{macrocode}
\ifx\undefined\@Numero%                         s
  {\let\cc\catcode \cc`p=12\cc`t=12\gdef\@Numero#1pt{#1}}%
\fi
\ifx\undefined\Numero
  \def\Numero#1#2{\dimen254
#2\edef#1{\expandafter\@Numero\the\dimen254}\ignorespaces}%
\fi
%    \end{macrocode}
% For both macros the |\ifx|\dots|\fi| constructs avoids messing up the
% definitions I have in several packages.
%
% \subsection{Trigonometric functions}
%  We now start with trigonometric functions. We define the macros |\SinOf|,
% |\CosOf| and |\TanOf| (we might define also |\CotOf|, but the cotangent does
% not appear so essential) by means of the parametric formulas that require the
% knowledge of the tangent of the half angle. We want to specify the angles
% in sexagesimal degrees, not in radians, so we can make accurate reductions to
% the main quadrants. we use the formulas
% \begin{eqnarray*}
% \sin\theta &=& \frac{2}{\cot x + \tan x}\\
% \cos\theta &=& \frac{\cot x - \tan x}{\cot x + \tan x}\\
% \tan\theta &=& \frac{2}{\cot x - \tan x}\\
% \noalign{\hbox{where}}
% x          &=& \theta/114.591559
% \end{eqnarray*}
% is the half angle in degrees converted to radians.
%
% We use this slightly modified set of parametric formulas because the cotangent
% of $x$ is a by product of the computation of the tangent of $x$; in this way
% we avoid computing the squares of numbers that might lead to overflows. For
% the same reason we avoid computing the value of the trigonometric functions
% in proximity of the value zero (and the other values that might involve high
% tangent or cotangent values) and in that case we prefer to approximate the
% small angle function value with its first or second order truncation of the
% McLaurin series; in facts for angles whose magnitude is smaller than $1^\circ$
% the magnitude of the independent variable $y=2x$ (the angle in degrees
% converted to radians) is so small (less than 0.017) that the sine and tangent
% can be freely approximated with $y$ itself (the error being smaller than
% approximately $10^{-6}$), while the cosine can be freely approximated with
% the formula $1-0.5y^2$ (the error being smaller than about $4\cdot10^{-9}$).
%
% We keep using grouping so that internal variables are local to these groups
% and do not mess up other things.
%
% The first macro is the service routine that computes the tangent and the
% cotangent of the half angle in radians; since we have to use always the
% reciprocal if this value, we call it |\X| but ins spite of the similarity it
% is the reciprocal of $x$. Notice that parameter \texttt{\#1} must be a length.
%    \begin{macrocode}
\def\g@tTanCotanFrom#1to#2and#3{%
\DividE 114.591559\p@ by#1to\X \@tdB=\X\p@
%    \end{macrocode}
% Computations are done with the help of counter |\I|, of the length |\@tdB|,
% and the auxiliary control sequences |\Tan| and |\Cot| whose meaning is
% transparent. The iterative process controlled by |\@whilenum| implements the
% (truncated) continued fraction expansion of the tangent function
% \[
% \tan x = \frac{1}{\displaystyle \frac{1\mathstrut}{\displaystyle x}
%     -\frac{1}{\displaystyle \frac{3\mathstrut}{\displaystyle x}
%     -\frac{1}{\displaystyle \frac{5\mathstrut}{\displaystyle x}
%     -\frac{1}{\displaystyle \frac{7\mathstrut}{\displaystyle x}
%     -\frac{1}{\displaystyle \frac{9\mathstrut}{\displaystyle x}
%     -\frac{1}{\displaystyle \frac{11\mathstrut}{\displaystyle x}
%     -\cdots}}}}}}
% \]
%    \begin{macrocode}
\countdef\I=254\def\Tan{0}\I=11\relax
\@whilenum\I>\z@\do{%
   \@tdC=\Tan\p@ \@tdD=\I\@tdB
   \advance\@tdD-\@tdC \DividE\p@ by\@tdD to\Tan
   \advance\I-2\relax}%
\def#2{\Tan}\DividE\p@ by\Tan\p@ to\Cot \def#3{\Cot}%
\ignorespaces}%
%    \end{macrocode}
%
%  Now that we have the macro for computing the tangent and cotangent of the
% half angle, we can compute the real trigonometric functions we are interested
% in. The sine value is computed after reducing the sine argument to the
% interval $0^\circ< \theta<180^\circ$; actually special values such as
% $0^\circ$,$90^\circ$, $180^\circ$, et cetera, are taken care separately, so
% that CPU time is saved for these special cases. The sine sign is taken care
% separately according to the quadrant of the sine argument.
%    \begin{macrocode}
\def\SinOf#1to#2{\begingroup%
\@tdA=#1\p@%
\ifdim\@tdA>\z@%
  \@whiledim\@tdA>180\p@\do{\advance\@tdA -360\p@}%
\else%
  \@whiledim\@tdA<-180\p@\do{\advance\@tdA 360\p@}%
\fi \ifdim\@tdA=\z@
  \gdef#2{0}%
\else
  \ifdim\@tdA>\z@
    \def\Segno{+}%
  \else
    \def\Segno{-}%
    \@tdA=-\@tdA
  \fi
  \ifdim\@tdA>90\p@
    \@tdA=-\@tdA \advance\@tdA 180\p@
  \fi
  \ifdim\@tdA=90\p@
    \xdef#2{\Segno1}%
  \else
    \ifdim\@tdA=180\p@
      \gdef#2{0}%
    \else
      \ifdim\@tdA<\p@
        \@tdA=\Segno0.0174533\@tdA
        \DividE\@tdA by\p@ to#2%
      \else
        \g@tTanCotanFrom\@tdA to\T and\Tp
        \@tdA=\T\p@ \advance\@tdA \Tp\p@
        \DividE \Segno2\p@ by\@tdA to#2%
      \fi
    \fi
  \fi
\fi
\endgroup\ignorespaces}%
%    \end{macrocode}
%
% For the computation of the cosine we behave in a similar way.
%    \begin{macrocode}
\def\CosOf#1to#2{\begingroup%
\@tdA=#1\p@%
\ifdim\@tdA>\z@%
  \@whiledim\@tdA>360\p@\do{\advance\@tdA -360\p@}%
\else%
  \@whiledim\@tdA<\z@\do{\advance\@tdA 360\p@}%
\fi
%
\ifdim\@tdA>180\p@
  \@tdA=-\@tdA \advance\@tdA 360\p@
\fi
%
\ifdim\@tdA<90\p@
  \def\Segno{+}%
\else
  \def\Segno{-}%
  \@tdA=-\@tdA \advance\@tdA 180\p@
\fi
\ifdim\@tdA=\z@
  \gdef#2{\Segno1}%
\else
  \ifdim\@tdA<\p@
    \@tdA=0.0174533\@tdA \Numero\@tempA\@tdA
    \@tdA=\@tempA\@tdA \@tdA=-.5\@tdA
    \advance\@tdA \p@
    \DividE\@tdA by\p@ to#2%
  \else
    \ifdim\@tdA=90\p@
      \gdef#2{0}%
    \else
      \g@tTanCotanFrom\@tdA to\T and\Tp
      \@tdA=\Tp\p@ \advance\@tdA-\T\p@
      \@tdB=\Tp\p@ \advance\@tdB\T\p@
      \DividE\Segno\@tdA by\@tdB to#2%
    \fi
  \fi
\fi
\endgroup\ignorespaces}%
%    \end{macrocode}
%
% For the tangent computation we behave in a similar way, except that we
% consider the fundamental interval as $0^\circ<\theta<90^\circ$; for the odd
% multiples of $90^\circ$ we assign the result a \TeX\ infinity value, that is
% the maximum number in points a dimension can be.
%    \begin{macrocode}
\def\TanOf#1to#2{\begingroup%
\@tdA=#1\p@%
\ifdim\@tdA>90\p@%
  \@whiledim\@tdA>90\p@\do{\advance\@tdA -180\p@}%
\else%
  \@whiledim\@tdA<-90\p@\do{\advance\@tdA 180\p@}%
\fi%
\ifdim\@tdA=\z@%
  \gdef#2{0}%
\else
  \ifdim\@tdA>\z@
    \def\Segno{+}%
  \else
    \def\Segno{-}%
    \@tdA=-\@tdA
  \fi
  \ifdim\@tdA=90\p@
    \xdef#2{\Segno16383.99999}%
  \else
    \ifdim\@tdA<\p@
        \@tdA=\Segno0.0174533\@tdA
        \DividE\@tdA by\p@ to#2%
    \else
      \g@tTanCotanFrom\@tdA to\T and\Tp
      \@tdA\Tp\p@ \advance\@tdA -\T\p@
      \DividE\Segno2\p@ by\@tdA to#2%
    \fi
  \fi
\fi
\endgroup\ignorespaces}%
%    \end{macrocode}
%
% \subsection{Arcs and curves preliminary information}
% We would like to define now a macro for drawing circular arcs of any radius
% and any angular aperture; the macro should require the arc center, the
% arc starting point and the angular aperture. The command should have the
% following syntax:
% \begin{flushleft}\ttfamily
% \cs{Arc}(\meta{{\rmfamily center}})(\meta{{\rmfamily starting
%                 point}}){\meta{{\rmfamily angle}}}
% \end{flushleft}
%  If the \meta{angle} is positive the arc runs counterclockwise from the
% starting point; clockwise if it's negative.
%
% It's necessary to determine the end point and the control points of the
% Bézier spline(s) that make up the circular arc.
%
% The end point is obtained from the rotation of the starting point around the
% center; but the \texttt{pict2e} command |\pIIe@rotate| is such that the
% pivoting point appears to be non relocatable.
% It is therefore necessary to resort to low level \TeX\ commands and the
% defined trigonometric functions and a set of macros that operate on complex
% numbers used as vector scale-rotate operators.
%
% \subsection{Complex number macros}
% We need therefore macros for summing, subtracting, multiplying, dividing
% complex numbers, for determining they directions (unit vectors); a unit vector
% is the complex number divided by its magnitude so that the result is the
% Cartesian form of the Euler's equation
% \[
% \mathrm{e}^{\mathrm{j}\phi} = \cos\phi+\mathrm{j}\sin\phi
% \]
%
% The magnitude of a vector id determined by taking a clever square root of a
% function of the real and the imaginary parts; see further on.
%
% It's better to represent each complex number with one control sequence; this
% implies frequent assembling and disassembling the pair of real numbers that
% make up a complex number. These real components are assembled into the
% defining control sequence as a couple of coordinates, i.e.\ two comma
% separated integer or fractional signed decimal numbers.
%
% For assembling two real numbers into a complex number we use the following
% elementary macro:
%    \begin{macrocode}
\def\MakeVectorFrom#1#2to#3{\edef#3{#1,#2}\ignorespaces}%
%    \end{macrocode}
% Another elementary macro copies a complex number into another one:
%    \begin{macrocode}
\def\CopyVect#1to#2{\edef#2{#1}\ignorespaces}%
%    \end{macrocode}
% The magnitude is determined with the macro |\ModOfVect| with delimited
% arguments; as usual it is assumed that the results are retrieved by means of
% control sequences, not used directly.
%
%  The magnitude $M$ is determined by taking the moduli of the real and
% imaginary parts, changing their signs if necessary; the larger component is
% then taken as the reference one so that, if $a$ is larger than $b$, the
% square root of the sum of their squares is computed as such:
% \[
% M = \sqrt{a^2+b^2} = a\sqrt{1+(b/a)^2}
% \]
% In this way the radicand never exceeds 2 and its is quite easy taking its
% square root by means of the Newton iterative process; due to the quadratic
% convergence, five iterations are more than sufficient. When one of the
% components is zero, the Newton iterative process is skipped. The overall
% macro is the following:
%    \begin{macrocode}
\def\ModOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
\@tempdima=\t@X\p@ \ifdim\@tempdima<\z@ \@tempdima=-\@tempdima\fi
\@tempdimb=\t@Y\p@ \ifdim\@tempdimb<\z@ \@tempdimb=-\@tempdimb\fi
\ifdim\@tempdima>\@tempdimb
   \DividE\@tempdimb by\@tempdima to\@T
   \@tempdimc=\@tempdima
\else
   \DividE\@tempdima by\@tempdimb to\@T
   \@tempdimc=\@tempdimb
\fi
\ifdim\@T\p@>\z@
   \@tempdima=\@T\p@ \@tempdima=\@T\@tempdima
   \advance\@tempdima\p@ %
   \@tempdimb=\p@%
   \@tempcnta=5\relax
   \@whilenum\@tempcnta>\z@\do{\DividE\@tempdima by\@tempdimb to\@T
   \advance\@tempdimb \@T\p@ \@tempdimb=.5\@tempdimb
   \advance\@tempcnta\m@ne}%
   \@tempdimc=\@T\@tempdimc
\fi
\Numero#2\@tempdimc
\ignorespaces}%
%    \end{macrocode}
% As a byproduct of the computation the control sequence |\@tempdimc| contains
% the vector or complex number magnitude multiplied by the length of one point.
%
% Since the macro for determining the magnitude of a vector is available, we
% can now normalize the vector to its magnitude, therefore getting the Cartesian
% form of the direction vector. If by any chance the direction of the null
% vector is requested, the output is again the null vector, without
% normalization.
%    \begin{macrocode}
\def\DirOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
\ModOfVect#1to\@tempa
\ifdim\@tempdimc=\z@\else
   \DividE\t@X\p@ by\@tempdimc to\t@X
   \DividE\t@Y\p@ by\@tempdimc to\t@Y
   \MakeVectorFrom\t@X\t@Y to#2\relax
\fi\ignorespaces}%
%    \end{macrocode}
%
% A cumulative macro uses the above ones for determining with one call both the
% magnitude and the direction of a complex number. The first argument is the
% input complex number, the second its magnitude, and the third is again a
% complex number normalized to unit magnitude (unless the input was the null
% complex number); remember always that output quantities must be specified
% with control sequences to be used at a later time.
%    \begin{macrocode}
\def\ModAndDirOfVect#1to#2and#3{%
\GetCoord(#1)\t@X\t@Y
\ModOfVect#1to#2%
\DividE\t@X\p@ by\@tempdimc to\t@X \DividE\t@Y\p@ by\@tempdimc to\t@Y
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
%    \end{macrocode}
% The next macro computes the magnitude and the direction of the difference of
% two complex numbers; the first input argument is the minuend, the second is
% the subtrahend; the output quantities are the third argument containing the
% magnitude of the difference and the fourth is the direction of the difference.
%  The service macro |\SubVect| executes the difference of two complex numbers
% and is described further on.
%    \begin{macrocode}
\def\DistanceAndDirOfVect#1minus#2to#3and#4{%
\SubVect#2from#1to\@tempa \ModAndDirOfVect\@tempa to#3and#4\relax
\ignorespaces}%
%    \end{macrocode}
% We now have two macros intended to fetch just the real or, respectively, the
% imaginary part of the input complex number.
%    \begin{macrocode}
\def\XpartOfVect#1to#2{%
\GetCoord(#1)#2\@tempa
\ignorespaces}%
%
\def\YpartOfVect#1to#2{%
\GetCoord(#1)\@tempa#2\relax
\ignorespaces}%
%    \end{macrocode}
% With the next macro we create a direction vector (second argument) from a
% given angle (first argument).
%    \begin{macrocode}
\def\DirFromAngle#1to#2{\CosOf#1to\t@X%
\SinOf#1to\t@Y\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
%    \end{macrocode}
%
% Sometimes it is necessary to scale a vector by an arbitrary real factor; this
% implies scaling both the real and imaginary part of the input given vector.
%    \begin{macrocode}
\def\ScaleVect#1by#2to#3{\GetCoord(#1)\t@X\t@Y
\@tempdima=\t@X\p@ \@tempdima=#2\@tempdima\Numero\t@X\@tempdima
\@tempdima=\t@Y\p@ \@tempdima=#2\@tempdima\Numero\t@Y\@tempdima
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
%    \end{macrocode}
% Again, sometimes it is necessary to reverse the direction of rotation; this
% implies changing the sign of the imaginary part of a given complex number;
% this operation produces the complex conjugate of the given number.
%    \begin{macrocode}
\def\ConjVect#1to#2{\GetCoord(#1)\t@X\t@Y
\@tempdima=-\t@Y\p@\Numero\t@Y\@tempdima
\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
%    \end{macrocode}
%
% With all the low level elementary operations we can now proceed to the
% definitions of the binary operations on complex numbers. We start with the
% addition:
%    \begin{macrocode}
\def\AddVect#1and#2to#3{\GetCoord(#1)\tu@X\tu@Y
\GetCoord(#2)\td@X\td@Y \@tempdima\tu@X\p@
\advance\@tempdima\td@X\p@ \Numero\t@X\@tempdima \@tempdima\tu@Y\p@
\advance\@tempdima\td@Y\p@ \Numero\t@Y\@tempdima
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
%    \end{macrocode}
% Then the subtraction:
%    \begin{macrocode}
\def\SubVect#1from#2to#3{\GetCoord(#1)\tu@X\tu@Y
\GetCoord(#2)\td@X\td@Y \@tempdima\td@X\p@
\advance\@tempdima-\tu@X\p@ \Numero\t@X\@tempdima \@tempdima\td@Y\p@
\advance\@tempdima-\tu@Y\p@ \Numero\t@Y\@tempdima
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
%    \end{macrocode}
%
% For the multiplication we need to split the operation according to the fact
% that we want to multiply by the second operand or by the complex conjugate of
% the second operand; it would be nice if we could use the usual
% postfixed asterisk notation for the complex conjugate, but I could not find
%  a simple means for doing so; therefore I use the prefixed notation, that is
% I put the asterisk before the second operand. The first part of the
% multiplication macro just takes care of the multiplicand and then checks for
% the asterisk; if there is no asterisk it calls a second service macro that
%  performs a regular complex multiplication, otherwise it calls a third
% service macro that executes the conjugate multiplication.
%    \begin{macrocode}
\def\MultVect#1by{\@ifstar{\@ConjMultVect#1by}{\@MultVect#1by}}%
%
\def\@MultVect#1by#2to#3{\GetCoord(#1)\tu@X\tu@Y
\GetCoord(#2)\td@X\td@Y \@tempdima\tu@X\p@
\@tempdimb\tu@Y\p@
\@tempdimc=\td@X\@tempdima\advance\@tempdimc-\td@Y\@tempdimb
\Numero\t@X\@tempdimc
\@tempdimc=\td@Y\@tempdima\advance\@tempdimc\td@X\@tempdimb
\Numero\t@Y\@tempdimc
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
%
\def\@ConjMultVect#1by#2to#3{\GetCoord(#1)\tu@X\tu@Y
\GetCoord(#2)\td@X\td@Y \@tempdima\tu@X\p@ \@tempdimb\tu@Y\p@
\@tempdimc=\td@X\@tempdima\advance\@tempdimc+\td@Y\@tempdimb
\Numero\t@X\@tempdimc
\@tempdimc=\td@X\@tempdimb\advance\@tempdimc-\td@Y\@tempdima
\Numero\t@Y\@tempdimc
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}
%    \end{macrocode}
%
% The division of two complex numbers implies scaling down the dividend by the
% magnitude of the divisor and by rotating the dividend scaled vector by the
% opposite direction of the divisor; therefore:
%    \begin{macrocode}
\def\DivVect#1by#2to#3{\ModAndDirOfVect#2to\@Mod and\@Dir
\DividE\p@ by\@Mod\p@ to\@Mod \ConjVect\@Dir to\@Dir
\ScaleVect#1by\@Mod to\@tempa
\MultVect\@tempa by\@Dir to#3\ignorespaces}%
%    \end{macrocode}
%
% \subsection{Arcs and curved vectors}
% We are now in the position of really doing graphic work.
% \subsubsection{Arcs}
% We start with tracing
% a circular arc of arbitrary center, arbitrary starting point and arbitrary
% aperture; The first macro checks the aperture; if this is not zero it
% actually proceeds with the necessary computations, otherwise it does
% nothing.
%    \begin{macrocode}
\def\Arc(#1)(#2)#3{\begingroup
\@tdA=#3\p@ \ifdim\@tdA=\z@\else
  \@Arc(#1)(#2)%
\fi
\endgroup\ignorespaces}%
%    \end{macrocode}
% The aperture is already memorized in |\@tdA|; the |\@Arc| macro receives
% the center coordinates in the first argument and the coordinates of the
% starting point in the second argument.
%    \begin{macrocode}
\def\@Arc(#1)(#2){%
\ifdim\@tdA>\z@
  \let\Segno+%
\else
  \@tdA=-\@tdA \let\Segno-%
\fi
%    \end{macrocode}
% The rotation angle sign is memorized in |\Segno| and |\@tdA| now contains the
% absolute value of the arc aperture.
% If the rotation angle is larger than $360^\circ$ a message is issued that
% informs the user that the angle will be reduced modulo $360^\circ$; this
% operation is performed by successive subtractions rather than with modular
%  arithmetics on the assumption that in general one subtraction suffices.
%    \begin{macrocode}
\Numero\@gradi\@tdA
\ifdim\@tdA>360\p@
 \PackageWarning{curve2e}{The arc aperture is \@gradi\space degrees
      and gets reduced\MessageBreak%
      to the range 0--360 taking the sign into consideration}%
 \@whiledim\@tdA>360\p@\do{\advance\@tdA-360\p@}%
\fi
%    \end{macrocode}
% Now the radius is determined and the drawing point is moved to the stating
% point.
%    \begin{macrocode}
\SubVect#2from#1to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
\CopyVect#1to\@Cent \GetCoord(\@pPun)\@pPunX\@pPunY
%    \end{macrocode}
% From now on it's better to define a new macro that will be used also in the
% subsequent macros that trace arcs; here we already have the starting point
% coordinates and the angle to draw the arc, therefore we just call the new
% macro, stroke the line and exit.
%    \begin{macrocode}
\@@Arc
\pIIe@strokeGraph\ignorespaces}%
%    \end{macrocode}
% And the new macro |\@@Arc| starts with moving the drawing point to the first
% point and does everything needed for tracing the requested arc, except
% stroking it; I leave the \texttt{stroke} command to the completion of the
% calling macro and nobody forbids to use the |\@@Arc| macro for other purposes.
%    \begin{macrocode}
\def\@@Arc{%
\pIIe@moveto{\@pPunX\unitlength}{\@pPunY\unitlength}%
%    \end{macrocode}
% If the aperture is larger than $180^\circ$ it traces a semicircle in the
% right direction and correspondingly reduces the overall aperture.
%    \begin{macrocode}
\ifdim\@tdA>180\p@
  \advance\@tdA-180\p@
  \Numero\@gradi\@tdA
  \SubVect\@pPun from\@Cent to\@V
  \AddVect\@V and\@Cent to\@sPun
  \MultVect\@V by0,-1.3333333to\@V \if\Segno-\ScaleVect\@V by-1to\@V\fi
  \AddVect\@pPun and\@V to\@pcPun
  \AddVect\@sPun and\@V to\@scPun
  \GetCoord(\@pcPun)\@pcPunX\@pcPunY
  \GetCoord(\@scPun)\@scPunX\@scPunY
  \GetCoord(\@sPun)\@sPunX\@sPunY
  \pIIe@curveto{\@pcPunX\unitlength}{\@pcPunY\unitlength}%
               {\@scPunX\unitlength}{\@scPunY\unitlength}%
               {\@sPunX\unitlength}{\@sPunY\unitlength}%
  \CopyVect\@sPun to\@pPun
\fi
%    \end{macrocode}
% If the remaining aperture is not zero it continues tracing the rest of the arc.
% Here we need the extrema of the arc and the coordinates of the control points
% of the Bézier cubic spline that traces the arc. The control points lay on the
% perpendicular to the vectors that join the arc center to the starting
% and end points respectively. Their distance $K$ from the adjacent nodes is
% determined with the formula
% \[
% K= \frac{4}{3}\,\frac{1-\cos\theta}{\sin\theta}R
% \]
% where $\theta$ is half the arc aperture and $R$ is its radius.
%    \begin{macrocode}
\ifdim\@tdA>\z@
  \DirFromAngle\@gradi to\@Dir \if\Segno-\ConjVect\@Dir to\@Dir \fi
  \SubVect\@Cent from\@pPun to\@V
  \MultVect\@V by\@Dir to\@V
  \AddVect\@Cent and\@V to\@sPun
  \@tdA=.5\@tdA \Numero\@gradi\@tdA
  \DirFromAngle\@gradi to\@Phimezzi
  \GetCoord(\@Phimezzi)\@cosphimezzi\@sinphimezzi
  \@tdB=1.3333333\p@ \@tdB=\@Raggio\@tdB
  \@tdC=\p@ \advance\@tdC -\@cosphimezzi\p@ \Numero\@tempa\@tdC
  \@tdB=\@tempa\@tdB
  \DividE\@tdB by\@sinphimezzi\p@ to\@cZ
  \ScaleVect\@Phimezzi by\@cZ to\@Phimezzi
  \ConjVect\@Phimezzi to\@mPhimezzi
  \if\Segno-%
    \let\@tempa\@Phimezzi
    \let\@Phimezzi\@mPhimezzi
    \let\@mPhimezzi\@tempa
  \fi
  \SubVect\@sPun from\@pPun to\@V
  \DirOfVect\@V to\@V
  \MultVect\@Phimezzi by\@V to\@Phimezzi
  \AddVect\@sPun and\@Phimezzi to\@scPun
  \ScaleVect\@V by-1to\@V
  \MultVect\@mPhimezzi by\@V to\@mPhimezzi
  \AddVect\@pPun and\@mPhimezzi to\@pcPun
  \GetCoord(\@pcPun)\@pcPunX\@pcPunY
  \GetCoord(\@scPun)\@scPunX\@scPunY
  \GetCoord(\@sPun)\@sPunX\@sPunY
  \pIIe@curveto{\@pcPunX\unitlength}{\@pcPunY\unitlength}%
               {\@scPunX\unitlength}{\@scPunY\unitlength}%
               {\@sPunX\unitlength}{\@sPunY\unitlength}%
\fi}
%    \end{macrocode}
%
% \subsubsection{Arc vectors}
% We exploit much of the above definitions for the |\Arc| macro for drawing
% circular arcs with an arrow at one or both ends; the first macro
% |\VerctorArc| draws an arrow at the ending point of the arc; the second macro
% |\VectorARC| draws arrows at both ends; the arrows have the same shape as
% those for vectors; actually they are drawn by putting a vector of zero
% length at the proper arc end(s), therefore they are styled as traditional or
% PostScript arrows according to the option of the \texttt{pict2e} package.
%
% But the specific drawing done here shortens the arc so as not to overlap on
% the arrow(s); the only or both arrows are also lightly tilted in order to
% avoid the impression of a corner where the arc enters the arrow tip.
%
%  All these operations require a lot of ``playing'' with vector directions,
% but even if the operations are numerous, they do not do anything else but:
% (a) determining the end point and its direction ; (b) determining the arrow
% length as an angular quantity, i.e. the arc amplitude that must be subtracted
% from the total arc to be drawn; (c) the direction of the arrow should be
% corresponding to the tangent to the arc at the point where the arrow tip is
% attached;(d) tilting the arrow tip by half its angular amplitude; (e)
% determining the resulting position and direction of the arrow tip so as to
% draw a zero length vector; (f) possibly repeating the same procedure for the
% other end of the arc; (g) shortening the total arc angular amplitude by the
% amount of the  arrow tip(s) already set, and (h) then drawing the final circular
% arc that joins the starting point to the final arrow or one arrow to the other
% one.
%
%  The calling macros are very similar to the |\Arc| macro initial one:
%    \begin{macrocode}
\def\VectorArc(#1)(#2)#3{\begingroup
\@tdA=#3\p@ \ifdim\@tdA=\z@\else
  \@VArc(#1)(#2)%
\fi
\endgroup\ignorespaces}%
%
\def\VectorARC(#1)(#2)#3{\begingroup
\@tdA=#3\p@
\ifdim\@tdA=\z@\else
  \@VARC(#1)(#2)%
\fi
\endgroup\ignorespaces}%
%    \end{macrocode}
%  The single arrowed arc is defined with the following long macro where all the
% described operations are performed more or less in the described succession;
% probably the macro requires a little cleaning, but since it works fine I did
% not try to optimize it for time or number of tokens. The final part of the
% macro is almost identical to that of the plain arc; the beginning also is
% quite similar. The central part is dedicated to the positioning of the arrow
% tip and to the necessary calculations for determining the tip tilt and the
% reduction of the total arc length;pay attention that the arrow length, stored in 
% |\@tdE| is a real length, while the radius stored in |\@Raggio| is just a multiple
% of the |\unitlength|, so that the division (that yields a good angular approximation
% to the arrow length as seen from the center of the arc) must be done with real
% lengths. The already defined |\@@Arc| macro actually draws the curved vector
% stem without stroking it.
%    \begin{macrocode}
\def\@VArc(#1)(#2){%
\ifdim\@tdA>\z@
  \let\Segno+%
\else
  \@tdA=-\@tdA \let\Segno-%
\fi \Numero\@gradi\@tdA
\ifdim\@tdA>360\p@
 \PackageWarning{curve2e}{The arc aperture is \@gradi\space degrees
      and gets reduced\MessageBreak%
      to the range 0--360 taking the sign into consideration}%
 \@whiledim\@tdA>360\p@\do{\advance\@tdA-360\p@}%
\fi
\SubVect#1from#2to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
\@tdE=\pIIe@FAW\@wholewidth \@tdE=\pIIe@FAL\@tdE
\DividE\@tdE by \@Raggio\unitlength to\DeltaGradi
\@tdD=\DeltaGradi\p@
\@tdD=57.29578\@tdD \Numero\DeltaGradi\@tdD
\@tdD=\ifx\Segno--\fi\@gradi\p@ \Numero\@tempa\@tdD
\DirFromAngle\@tempa to\@Dir
\MultVect\@V by\@Dir to\@sPun
\edef\@tempA{\ifx\Segno-\m@ne\else\@ne\fi}%
\MultVect\@sPun by 0,\@tempA to\@vPun
\DirOfVect\@vPun to\@Dir
\AddVect\@sPun and #1 to \@sPun
\GetCoord(\@sPun)\@tdX\@tdY
\@tdD\ifx\Segno--\fi\DeltaGradi\p@
\@tdD=.5\@tdD \Numero\DeltaGradi\@tdD
\DirFromAngle\DeltaGradi to\@Dird
\MultVect\@Dir by*\@Dird to\@Dir
\GetCoord(\@Dir)\@xnum\@ynum
\put(\@tdX,\@tdY){\vector(\@xnum,\@ynum){0}}%
\@tdE =\ifx\Segno--\fi\DeltaGradi\p@
\advance\@tdA -\@tdE \Numero\@gradi\@tdA
\CopyVect#1to\@Cent \GetCoord(\@pPun)\@pPunX\@pPunY
\@@Arc
\pIIe@strokeGraph\ignorespaces}%
%    \end{macrocode}
%
% The macro for the arc terminated with arrow tips at both ends is again very
% similar, except it is necessary to repeat the arrow tip positioning also at
% the starting point. The |\@@Arc| macro draws the curved stem.
%    \begin{macrocode}
\def\@VARC(#1)(#2){%
\ifdim\@tdA>\z@
  \let\Segno+%
\else
  \@tdA=-\@tdA \let\Segno-%
\fi \Numero\@gradi\@tdA
\ifdim\@tdA>360\p@
 \PackageWarning{curve2e}{The arc aperture is \@gradi\space degrees
      and gets reduced\MessageBreak%
      to the range 0--360 taking the sign into consideration}%
 \@whiledim\@tdA>360\p@\do{\advance\@tdA-360\p@}%
\fi
\SubVect#1from#2to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
\@tdE=\pIIe@FAW\@wholewidth \@tdE=0.8\@tdE
\DividE\@tdE by \@Raggio\unitlength to\DeltaGradi
\@tdD=\DeltaGradi\p@ \@tdD=57.29578\@tdD \Numero\DeltaGradi\@tdD
\@tdD=\ifx\Segno--\fi\@gradi\p@ \Numero\@tempa\@tdD
\DirFromAngle\@tempa to\@Dir
\MultVect\@V by\@Dir to\@sPun
\edef\@tempA{\ifx\Segno-\m@ne\else\@ne\fi}%
\MultVect\@sPun by 0,\@tempA to\@vPun
\DirOfVect\@vPun to\@Dir
\AddVect\@sPun and #1 to \@sPun
\GetCoord(\@sPun)\@tdX\@tdY
\@tdD\ifx\Segno--\fi\DeltaGradi\p@
\@tdD=.5\@tdD \Numero\@tempB\@tdD
\DirFromAngle\@tempB to\@Dird
\MultVect\@Dir by*\@Dird to\@Dir
\GetCoord(\@Dir)\@xnum\@ynum
\put(\@tdX,\@tdY){\vector(\@xnum,\@ynum){0}}%
\@tdE =\DeltaGradi\p@
\advance\@tdA -2\@tdE \Numero\@gradi\@tdA
\CopyVect#1to\@Cent \GetCoord(\@pPun)\@pPunX\@pPunY
\SubVect\@Cent from\@pPun to \@V
\edef\@tempa{\ifx\Segno-\else-\fi\@ne}%
\MultVect\@V by0,\@tempa to\@vPun
\@tdE\ifx\Segno--\fi\DeltaGradi\p@
\Numero\@tempB{0.5\@tdE}%
\DirFromAngle\@tempB to\@Dird
\MultVect\@vPun by\@Dird to\@vPun
\DirOfVect\@vPun to\@Dir\GetCoord(\@Dir)\@xnum\@ynum
\put(\@pPunX,\@pPunY){\vector(\@xnum,\@ynum){0}}
\edef\@tempa{\ifx\Segno--\fi\DeltaGradi}%
\DirFromAngle\@tempa to \@Dir
\SubVect\@Cent from\@pPun to\@V
\MultVect\@V by\@Dir to\@V
\AddVect\@Cent and\@V to\@pPun
\GetCoord(\@pPun)\@pPunX\@pPunY
\@@Arc
\pIIe@strokeGraph\ignorespaces}%
%    \end{macrocode}
%
% It must be understood that the curved vectors, the above circular arcs
% terminated with an arrow tip at one or both ends, have a nice appearance only
% if the arc radius is not too small, or, said in a different way, if the arrow
% tip  angular width does not exceed a maximum of a dozen degrees (and this is
% probably already too much); the tip does not get curved as the arc is,
% therefore there is not a smooth transition from the curved stem and the
% straight arrow tip if this one is large in comparison to the arc radius.
%
% \subsection{General curves}
% Now we define a macro for tracing a general, not necessarily circular arc.
% This macro resorts to a general triplet of macros with which it is possible
% to draw almost anything. It traces a single Bézier spline from a first point
% where the tangent direction is specified to a second point where again it is
% specified the tangent direction. Actually this is a special (possibly useless)
% case where the general |\Curve| macro could do the same or a better job. In
% any case\dots
%    \begin{macrocode}
\def\CurveBetween#1and#2WithDirs#3and#4{%
\StartCurveAt#1WithDir{#3}\relax
\CurveTo#2WithDir{#4}\CurveFinish}%
%    \end{macrocode}
%
%  Actually the above macro is a special case of concatenation of the triplet
% formed by macros |\StartCurve|, |\CurveTo| and|\CurveFinish|; the second of
% which can be repeated an arbitrary number of times.
%
% The first macro initializes the drawing and the third one strokes it; the
% real work is done by the second macro. The first macro initializes the
% drawing but also memorizes the starting direction; the second macro traces
% the current Bézier arc reaching the destination point with the specified
% direction, but memorizes this direction as the one with which to start the
% next arc. The overall curve is then always smooth because the various
% Bézier arcs join with continuous tangents. If a cusp is desired it is
% necessary to change the memorized direction at the end of the arc before the
% cusp and before the start of the next arc; this is better than stroking the
% curve before the cusp and then starting another curve, because the curve
% joining point at the cusp is not stroked with the same command, therefore we get
% two superimposed curve terminations. We therefore need another small macro
% |\ChangeDir| to perform this task.
%
% It is necessary to recall that the directions point to the control points,
% but they do not define the control points themselves; they are just
% directions, or, even better, they are simply vectors with the desired
% direction; the macros themselves provide to the normalization and
% memorization.
%
% The next desirable point would be to design a macro that accepts optional node
% directions and computes the missing ones according to a suitable strategy. I
% can think of many such strategies, but none seems to be generally applicable,
% in the sense that one strategy might give good results, say, with sinusoids
% and another one, say, with cardioids, but neither one is suitable for both
% cases.
%
% For the moment we refrain from automatic direction computation, but we design
% the general macro as if directions were optional.
%
%  Here we begin with the first initializing macro that receives in the  first
% argument the starting point and in the second argument the direction of the
% tangent (not necessarily normalized to a unit vector)
%    \begin{macrocode}
\def\StartCurveAt#1WithDir#2{%
\begingroup
\GetCoord(#1)\@tempa\@tempb
\CopyVect\@tempa,\@tempb to\@Pzero
\pIIe@moveto{\@tempa\unitlength}{\@tempb\unitlength}%
\GetCoord(#2)\@tempa\@tempb
\CopyVect\@tempa,\@tempb to\@Dzero
\DirOfVect\@Dzero to\@Dzero}
%    \end{macrocode}
% And this re-initializes the direction after a cusp
%    \begin{macrocode}
\def\ChangeDir<#1>{%
\GetCoord(#1)\@tempa\@tempb
\CopyVect\@tempa,\@tempb to\@Dzero
\DirOfVect\@Dzero to\@Dzero
\ignorespaces}
%    \end{macrocode}
%
% The next macro is the finishing one; it strokes the whole curve and closes the
% group that was opened with |\StartCurve|.
%    \begin{macrocode}
\def\CurveFinish{\pIIe@strokeGraph\endgroup\ignorespaces}%
%    \end{macrocode}
%
% The ``real'' curve macro comes next; it is supposed to determine the control
% points for joining the previous point (initial node) with the specified
% direction to the next point with another specified direction (final node).
% Since the control points are along the specified directions, it is necessary
% to determine the distances from the adjacent curve nodes. This must work
% correctly even if nodes and directions imply an inflection point somewhere
% along the arc.
%
% The strategy I devised consists in determining each control point as if it
% were the control point of a circular arc, precisely an arc of an
% osculating circle, a circle tangent to the curve at that node. The ambiguity
% of the stated problem may be solved by establishing that the chord of the
% osculating circle has the same direction as the chord of the arc being drawn,
% and that the curve chord is divided into two parts each of which should be
% interpreted as half the chord of the osculating circle; this curve chord
% division is made proportionally to the projection of the tangent directions
% on the chord itself. Excluding degenerate cases that may be dealt with
% directly, imagine the triangle built with the chord and the two tangents;
% this triangle is straightforward if there is no inflection point; otherwise it
% is necessary to change one of the two directions by reflecting it about the
% chord. This is much simpler to view if a general rotation of the whole
% construction is made so as to bring the curve chord on the $x$ axis, because
% the reflection about the chord amounts to taking the complex conjugate of one
% of the directions. In facts with a concave curve the ``left'' direction
% vector arrow and the ``right'' direction vector tail lay in the same half
% plane, while with an inflected curve, they lay in opposite half plains, so
% that taking the complex conjugate of one of directions re-establishes the
% correct situation for the triangle we are looking for.
%
% This done the perpendicular from the triangle vertex to the cord divides the
% chord in two parts (the foot of this perpendicular may lay outside the chord,
% but this is no problem since we are looking for positive solutions, so that
% if we get negative numbers we just negate them); these two parts are taken as
% the half chords of the osculating circles, therefore there is no problem
% determining the distances $K_{\mathrm{left}}$ and $K_{\mathrm{right}}$ from
% the left and right
% nodes by using the same formula we used with circular arcs. Well\dots\ the
% same formula means that we have to determine the radius from the half chord
% and its inclination with the node tangent; all things we can do with the
% complex number algebra and macros we already have at our disposal. If we look
% carefully at this computation done for the circular arc we discover that in
% practice we used the half chord length instead of the radius; so the coding
% is actually the same, may be just with different variable names.
%
% We therefore start with getting the points and directions and calculating the
% chord and its direction
%    \begin{macrocode}
\def\CurveTo#1WithDir#2{%
\def\@Puno{#1}\def\@Duno{#2}\DirOfVect\@Duno to\@Duno
\DistanceAndDirOfVect\@Puno minus\@Pzero to\@Chord and\@DirChord
%    \end{macrocode}
% Then we rotate everything about the starting point so as to bring the chord on
% the real axis
%    \begin{macrocode}
\MultVect\@Dzero by*\@DirChord to \@Dpzero
\MultVect\@Duno by*\@DirChord to \@Dpuno
\GetCoord(\@Dpzero)\@Xpzero\@Ypzero
\GetCoord(\@Dpuno)\@Xpuno\@Ypuno
%    \end{macrocode}
% The chord needs not be actually rotated because it suffices its length
% along the real axis; the chord length is memorized in |\@Chord|.
%
% We now examine the various degenerate cases, when either tangent is
% perpendicular to the chord, or when it is parallel pointing inward or outward,
% with or without inflection.
%
% We start with the $90^\circ$ case for the ``left'' direction
% separating the cases when the other direction is or is not $90^\circ$~\dots
%    \begin{macrocode}
\ifdim\@Xpzero\p@=\z@
  \ifdim\@Xpuno\p@=\z@
    \@tdA=0.666666\p@
    \Numero\@Mcpzero{\@Chord\@tdA}%
    \edef\@Mcpuno{\@Mcpzero}%
  \else
    \@tdA=0.666666\p@
    \Numero\@Mcpzero{\@Chord\@tdA}%
    \SetCPmodule\@Mcpuno from\@ne\@Chord\@Dpuno%
  \fi
%    \end{macrocode}
% \dots\ from when the ``left'' direction is not perpendicular to the chord; it
% might
% be  parallel and we must distinguish the cases for the other direction~\dots
%    \begin{macrocode}
\else
  \ifdim\@Xpuno\p@=\z@
    \@tdA=0.666666\p@
    \Numero\@Mcpuno{\@Chord\@tdA}%
    \SetCPmodule\@Mcpzero from\@ne\@Chord\@Dpzero%
  \else
     \ifdim\@Ypzero\p@=\z@
       \@tdA=0.333333\p@
       \Numero\@Mcpzero{\@Chord\@tdA}%
       \ifdim\@Ypuno\p@=\z@
            \edef\@Mcpuno{\@Mcpzero}%
       \fi
%    \end{macrocode}
% \dots\ from when the left direction is oblique and the other direction is
% either parallel to the chord~\dots
%    \begin{macrocode}
     \else
       \ifdim\@Ypuno\p@=\z@
         \@tdA=0.333333\p@
         \Numero\@Mcpuno{\@Chord\@tdA}%
         \SetCPmodule\@Mcpzero from\@ne\@Chord\@Dpzero
%    \end{macrocode}
% \dots\ and, finally, from when both directions are oblique with respect to
% the chord; we must see if there is an inflection point; if both direction
% point to the same half plane we have to take the complex conjugate of one
% direction so as to define the triangle we were speaking about above.
%    \begin{macrocode}
       \else
         \@tdA=\@Ypzero\p@ \@tdA=\@Ypuno\@tdA
         \ifdim\@tdA>\z@
           \ConjVect\@Dpuno to\@Dwpuno
         \else
           \edef\@Dwpuno{\@Dpuno}%
         \fi
%    \end{macrocode}
% The control sequence |\@Dwpuno| contains the right direction for forming the
% triangle; we cam make the weighed subdivision of the chord according to the
% horizontal components of the directions; we eventually turn negative values
% to positive ones since we are interested in the magnitudes of the control
% vectors.
%    \begin{macrocode}
         \GetCoord(\@Dwpuno)\@Xwpuno\@Ywpuno
         \@tdA=\@Xpzero\p@ \@tdA=\@Ywpuno\@tdA
         \@tdB=\@Xwpuno\p@ \@tdB=\@Ypzero\@tdB
         \DividE\@tdB by\@tdA to\@Fact
         \@tdC=\p@ \advance\@tdC-\@Fact\p@
         \ifdim\@tdC<\z@ \@tdC=-\@tdC\fi
         \DividE\p@ by \@Fact\p@ to\@Fact
         \@tdD=\p@ \advance\@tdD-\@Fact\p@
         \ifdim\@tdD<\z@ \@tdD=-\@tdD\fi
%    \end{macrocode}
%  before dividing by the denominator we have to check the directions, although
% oblique to the chord are not parallel to one another; in this case there is
% no question of a weighed subdivision of the chord
%    \begin{macrocode}
         \ifdim\@tdD<0.0001\p@
           \def\@factzero{1}%
           \def\@factuno{1}%
         \else
           \DividE\p@ by\@tdC to\@factzero
           \DividE\p@ by\@tdD to\@factuno
         \fi
%    \end{macrocode}
% We now have the subdivision factors and we call another macro for determining
% the required magnitudes
%    \begin{macrocode}
         \SetCPmodule\@Mcpzero from\@factzero\@Chord\@Dpzero
         \SetCPmodule\@Mcpuno from\@factuno\@Chord\@Dwpuno
      \fi
     \fi
  \fi
\fi
%    \end{macrocode}
% Now we have all data we need and we determine the positions of the control
% points; we do not work any more on the rotated diagram of the horizontal
% chord, but we operate on the original points and directions; all we had to
% compute, after all, were the distances of the control point along the
% specified directions; remember that the ``left'' control point is along the
% positive ``left'' direction, while the ``right'' control point precedes the
% curve node along the ``right'' direction, so that a vector subtraction must
% be done.
%    \begin{macrocode}
\ScaleVect\@Dzero by\@Mcpzero to\@CPzero
\AddVect\@Pzero and\@CPzero to\@CPzero
\ScaleVect\@Duno by\@Mcpuno to\@CPuno
\SubVect\@CPuno from\@Puno to\@CPuno
%    \end{macrocode}
% Now we have the four points and we can instruct the internal \texttt{pict2e}
% macros to do the path tracing.
%    \begin{macrocode}
\GetCoord(\@Puno)\@XPuno\@YPuno
\GetCoord(\@CPzero)\@XCPzero\@YCPzero
\GetCoord(\@CPuno)\@XCPuno\@YCPuno
\pIIe@curveto{\@XCPzero\unitlength}{\@YCPzero\unitlength}%
             {\@XCPuno\unitlength}{\@YCPuno\unitlength}%
             {\@XPuno\unitlength}{\@YPuno\unitlength}%
%    \end{macrocode}
%  It does not have to stroke the curve because other Bézier splines might still
%  be added to the path. On the opposite it memorizes the final point as the
% initial point of the next spline
%    \begin{macrocode}
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}%
%    \end{macrocode}
%
%  The next macro is used to determine the control vectors lengths when we have
% the chord fraction, the chord length and the direction along which to compute
% the vector; all the input data (arguments from \#2 to \#4) may be passed as
% control sequences so the calling statement needs not use any curly braces.
%    \begin{macrocode}
\def\SetCPmodule#1from#2#3#4{%
\GetCoord(#4)\t@X\t@Y
\@tdA=#3\p@
\@tdA=#2\@tdA
\@tdA=1.333333\@tdA
\@tdB=\p@ \advance\@tdB +\t@X\p@
\DividE\@tdA by\@tdB to#1\relax
\ignorespaces}%
%    \end{macrocode}
%
% We finally define the overall |\Curve| macro that recursively examines an
% arbitrary list of nodes and directions; node coordinates are grouped within
% regular parentheses while direction components are grouped within angle
% brackets. The first call of the macro initializes the drawing process and
% checks for the next node and direction; if a second node is missing, it issues
% a warning message and does not draw anything. The second macro defines the
% path to the next point and checks for another node; if the next list item is
% a square bracket delimited argument, it interprets it as a change of
% direction, while if it is another parenthesis delimited argument it interprets
% it as a new node-direction specification;
% if the node and direction list is terminated, it issues the stroking command
% and exits the recursive process. The |@ChangeDir| macro is just an interface
% for executing the regular |\ChangeDir| macro, but also for recursing again by
% recalling |\@Curve|.
%    \begin{macrocode}
\def\Curve(#1)<#2>{%
    \StartCurveAt#1WithDir{#2}%
    \@ifnextchar\lp@r\@Curve{%
    \PackageWarning{curve2e}{%
    Curve specifications must contain at least two nodes!\Messagebreak
    Please, control your Curve specifications\MessageBreak}}}
\def\@Curve(#1)<#2>{%
    \CurveTo#1WithDir{#2}%
    \@ifnextchar\lp@r\@Curve{%
    \@ifnextchar[\@ChangeDir\CurveFinish}}
\def\@ChangeDir[#1]{\ChangeDir<#1>\@Curve}
%    \end{macrocode}
%
% As a concluding remark, please notice the the |\Curve| macro is certainly the
% most comfortable to use, but it is sort of frozen in its possibilities. The
% user may certainly use the |\StartCurve|, |\CurveTo|, |\ChangeDir|, and
% |\CurveFinish| for a more versatile set of drawing macros; evidently nobody
% forbids to exploit the full power of the |\cbezier| original macro for cubic
% splines.
%
% I believe that the set of new macros can really help the user to draw his/her
% diagrams with more agility; it will be the accumulated experience to decide if
% this is true.
% \Finale
% \endinput