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\preamble

  Copyright (C)  2005--2019 Claudio Beccari  all rights reserved.
  License information appended
  
\endpreamble

\postamble


Distributable under the LaTeX Project Public License,
version 1.3c or higher (your choice). The latest version of
this license is at: http://www.latex-project.org/lppl.txt

This work is "author-maintained"

This work consists of file curve2e.dtx, and the derived files 
curve2e.sty and curve2e.pdf, plus the auxiliary derived files 
README.txt and curve2e-v161.sty.

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%<+package>\ProvidesPackage{curve2e}%
%<+readme>File README.txt for package curve2e
%<*package|readme>
        [2019-10-21 v.2.0.4 Extension package for pict2e]
%</package|readme>
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\usepackage{multicol,amsmath,fancyvrb,trace}
\usepackage{xcolor,curve2e}
\GetFileInfo{curve2e.dtx}
\title{The extension package \textsf{curve2e}}
\author{Claudio Beccari\thanks{E-mail: \texttt{claudio dot beccari at gmai dot com}}}
\date{Version \fileversion~--~Last revised \filedate.}
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\begin{document}
\maketitle
\columnseprule=0.4pt
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 \tableofcontents
 \end{multicols}
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% \CheckSum{5569}
% \begin{abstract}
% This file documents the |curve2e| extension package to  the |pict2e|
% bundle implementation that has been described by Lamport
% himself in the 1994 second edition of his \LaTeX\ handbook.
%
% Please take notice that in April 2011 a new updated version of the
% package |pict2e| has been released that incorporates some of the
% commands defined in early versions of this package; apparently there
% are no conflicts, but only the advanced features of |curve2e| remain
% available for extending the above package.
%
% 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 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}
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{The configuration file}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This package |curve2e| is distributed with a |ltxdoc.cfg| configuration
% file that contains, besides the preamble and the postamble comment lines,
% the following code line:
%\begin{verbatim}
%\AtBeginDocument{\OnlyDescription}
%\end{verbatim}
%
% If you want to type the whole documentation, comment out that code line
% in  the |ltxdoc.cfg| file. This is the only modification allowed by the
% LPPL licence that does not require to change the file name.
%
% For your information the initial part is about 20~pages long; the whole
% documentation is about 80~pages long.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Package \texttt{pict2e} and this extension \texttt{curve2e}}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% 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 what follows.
% \begin{enumerate}
% \item The line and vector slopes were limited to the ratios of relative
% prime one-digit integers of magnitude not exceeding 6 for lines and 4 for
% vectors.
%^^A
% \item Filled and unfilled full circles were limited by the necessarily
% limited number of specific glyphs contained in the special \LaTeX\
% \texttt{picture} fonts.
%^^A
% \item Quarter circles were also limited in their radii for the same
% reason.
%^^A
% \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.
%^^A
% \item Vector arrows had only one possible shape and matched the limited
% number of vector slopes.
%^^A
% \item For circles and inclined lines and vectors just two possible
% thicknesses  were available.
% \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; with the 2009 upgrade even
% this limitation was removed and now slope coefficients can be any
% fractional number whose magnitude does not exceed 16\,384, the maximum
% dimension in points that \TeX\ can handle.
%^^A
% \item Filled and unfilled circles can be of any size.
%^^A
% \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.
%^^A
% \item There are two shapes for the arrow tips; the triangular one
% traditional with \LaTeX\ vectors, or the arrow tip with PostScript style.
%^^A
% \item The |\linethickness| command changes the thickness of all lines,
% straight, curved, vertical, horizontal, arrow tipped, et cetera.
% \end{enumerate}
%
% This specific extension package |curve2e| adds the following features.
% \begin{enumerate}
% \item Point coordinates my be specified in both cartesian and polar form:
% internally they are handeld as cartesian coordinates, but the user can
% specify  his/her points also in polar form. In order to avoid confusion
% with other grapgic packages, |curve2e| uses the usual comma separated
% couple of integer or fractional numebrs for cartesian coordinates, and
% the couple \texttt{\meta{angle}:\meta{radius}} for polar coordinates.
% All graphic object commands accept polar or cartesian coordinates at the
% choice of the user who may use for each object the formalism he/she
% prefers. Also the |put| and |\multiput| commands have been redefined so
% as to accept cartesian or polar coordinates.
%^^A
%\item Most if not all cartesian coordinate pairs and slope pairs are
% treated as \emph{ordered pairs}, that is \emph{complex numbers}; in
% practice the user does not notice any difference from what he/she was
% used to, but all the mathematical treatment to be applied to these
% entities is coded as complex number operations, since complex numbers may
% be viewed non only as ordered pairs, but also as vectors or as
% roto-amplification operators.
%^^A
% \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 (now this original feature is directly available also with
% |pict2e|).
%^^A
% \item Commands for specifying the way two lines or curves join to one
% another.
% ^^A
% \item originally 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; now this functionality
% available directly with |pict2e|.
% ^^A
% \item A new macro |\Line| was originally defined so as to avoid the need
% to specify the horizontal projection of inclined lines; now this
% functionality id available directly with |pict2e|; but this macro name
% now conflicts with |pict2e| 2009 version; therefore its name is changed
% to |\LIne| and supposedly it will not be used very often, if ever,
% by the end user (but it is used within this package macros).
% ^^A
% \item A new macro |\LINE| was defined in order to join two points
% specified with their coordinates; this is now the normal behavior of the
% |\Line| macro of |pict2e| so that in this package |\LINE| is now renamed
% |\segment|; there is no need to use the |\put| command with this line
% specification.
% ^^A
% \item A new macro |\DLine| is defined in order to draw dashed lines
% joining any two given points; the dash length and gap (equal to one another) get specified through one of the macro arguments.
% ^^A
% \item A new macro |\Dotline| is defined in order to draw dotted straight
% lines as a sequence of equally spaced dots, where the gap can be
% specified by the user; such straight line may have any inclination, as
% well as the above dashed lines.
% ^^A
% \item Similar macros are redefined for vectors; |\vector| redefines the
% original macro but with the vector slope limitations removed; |\Vector|
% gets specified with its two horizontal and vertical components in analogy 
% with |\LIne|; |\VECTOR| joins two specified points (without using the
% |\put| command) with the arrow pointing to the second point.
%^^A
% \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 (available now also in |pict2e|); here it is redefined so as to
% allow an optional specification of the way segments for the polyline are
% joined to one another. Vertices may be specified with polar coordinates
%^^A
% \item The |pict2e| |polygon| macro to draw closed polylines, in practice
% general polygons, has been redefined in such a way that it can accept the
% various vertices specified with polar coordinates. The |polygon*| macro
% produces a color filled polygon; the default color is black, but a
% different color may be specified with the usual |\color| command given
% within the same group where |\polygon*| is enclosed.
%^^A
% \item A new macro |\Arc| is defined in order to draw an arc with
% arbitrary radius and arbitrary aperture (angle amplitude); this amplitude
% is specified in sexagesimal degrees, not in radians; a similar
% functionality is now achieved with the |\arc| macro of |pict2e|, which
% provides also the starred version |\arc*| that fills up the interior of
% the generated circular arc with the current color. It must be noticed
% that the syntax is slightly different, so that it's reasonable that these
% commands, in spite of producing identical arcs, might be more comfortable
% with this or that syntax.
%^^A
% \item Two new macros |\VectorArc| and |\VectorARC| are defined in order
% to draw circular arcs with an arrow at one or both ends.
%^^A
% \item A new macro |\Curve| is defined so as to draw arbitrary curved
% lines by means of cubic Bézier splines; the |\Curve| macro requires only
% the curve nodes and the directions of the tangents at each node.The
% starred version fills up the interior of the curve with the current
% color.
%^^A
% \item |\Curve| is a recursive macro that can draw an unlimited
% (reasonably limited) number of connected Bézier spline arcs with
% continuous tangents except for cusps; these arcs require only the
% specification of the tangent direction at the interpolation nodes. It is
% possible to use a lower level macro |\CbezierTo| that does the same but
% lets the user specify the control points of each arc; it is more
% difficult to use but it is more performant.
%^^A
% \item The basic macros used within the cumulative |\Curve| macro can be
% used individually in order to draw any curve, one cubic arc at the time;
% but they are intended for internal use, even if it is not prohibited to
% use them; by themselves such arcs are not different form those used by
% |Curve|, but the final command, |\FillCurve|, should be used in place of
% |\CurveFinish|, so as to fill up the closed path with the locally
% specified color; see figure~\ref{fig:colored-curve}.  It is much more
% convenient to use the starred version of the |\Curve| macro.
% \end{enumerate}
%
% The |pict2e| package already defines macros such as |\moveto|, |\lineto|,
% |\curveto|, |\closepath|, |\fillpath|, and |\strokepath|; of course these
% macros can be used by the end user, and sometimes they perform better
% than the macros defined in this package, because the user has a better
% control on the position of the Bézier splines  control points, while here
% the control points are sort of rigid. It would be very useful to resort
% to the |hobby| package, but its macros are conforming with those of the
% |tikz| and |pgf| packages, not with |curve2e|; an interface should be
% created in order to deal with the |hobby| package, but this has not been
% done yet.
%
% In order to make the necessary calculations many macros have been defined 
% so as to use complex number arithmetics to manipulate point coordinates,
% directions (unit vectors, also known as `versors'), rotations and the
% like. In the first versions of this package the trigonometric functions
% were also been defined in a way that the author believed to be more
% efficient than those defined by the \texttt{trig} package; in any case
% the macro names were sufficiently different to accommodate both
% definition sets in the same \LaTeX\ run. With the progress of the
% \LaTeX\,3 language, the |xfp| has recently become available, by which any
% sort of calculations can be done with floating point numbers; therefore
% the most common algebraic, irrational and transcendental functions can
% be computed in the background with the stable internal floating point
% facilities. We maintain some computation with complex number algebra,
% but use the |xfp| functionalities for other computations.
%
% 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 version could become the start for a
% real extension of the \texttt{pict2e} package or even become a part of
% it. Actually some macros have already been included in the
% \texttt{pict2e} package. The |\Curve| algorithm, as I said before, might
% be redefined so as to use the macros  introduced in the \texttt{hobby}
% package, that implements for the |tikz| and |pgf| packages the same
% functionalities that John Hobby implemented for the \MF\ and \MP\
% programs. 
%
% For these reasons I suppose that every enhancement should be submitted to
% Gäßlein, Niepraschk, and Tkadlec  who are the prime maintainers of
% \texttt{pict2e}; they are the only ones who can decide whether or not to
% incorporate new macros in their package.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Summary and examples of new commands}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% This package \texttt{curve2e} extends the power of \texttt{pict2e} with
% the following modifications and the following new commands.
% \begin{enumerate}
% \item This package |curve2e| calls directly the \LaTeX\ packages
% |color| and |pict2e| to which it passes any possible option that the
% latter can receive; actually the only options that make sense are
% those concerning the arrow tips, either \LaTeX\ or PostScript styled,
% because it is assumed that if you use this package you are not
% interested in using the original \LaTeX\ commands. See the |pict2e|
% documentation in order to see the correct options |pict2e| can receive.
%^^A
% \item The user is offered new commands in order to control the line
% terminators and the line joins; specifically:
% \begin{itemize}
% \item |\roundcap|: the line is terminated with a semicircle;
% \item |\squarecap|: the line is terminated with a half square;
% \item |\roundjoin|: two lines are joined with a rounded join;
% \item |\beveljoin|: two lines are joined with a bevel join;
% \item |\miterjoin|: two lines are joined with a miter join.
% \end{itemize}
% All the above commands should respect the intended range; but since they
% act at the PostScript or PDF level, not at \TeX\ level, it might be
% necessary to issue the necessary command in order to restore the previous
% terminator or join.
%^^A
% \item The commands |\linethickness|, |\thicklines|, |\thinlines| together
% with |\defaultlinethickness| always redefine the internal |\@wholewidth|
% and |\@halfwidth| so that the latter always refer to a full width and to
% a half of it in this way: if you issue the command
% |\defaultlinewidth{2pt}| all thin lines will be drawn with a thickness
% of 1\,pt while, if a drawing command directly refers to the internal
% value |\@wholewidth|, its line will be drawn with a thickness of 2\,pt.
% If one issues the declaration |\thinlines| all lines will be drawn with
% a  1\,pt width, but if a command refers to the internal value
% |\@halfwidth| the line will be drawn with a thickness of 0.5\,pt. The
% command |\linethickness| redefines the above internals but does not
% change the default width value; all these width specifications apply to
% all lines, straight ones, curved ones, circles, ovals, vectors, dashed,
% et cetera. It's better to recall that |\thinlines| and |\thicklines| are
% declarations that do not take arguments; on the opposite the other two
% commands follow the standard syntax:
% \begin{flushleft}
% |\linethickness|\marg{dimensioned value}\\
% |\defaultlinewidth|\marg{dimensioned value}
% \end{flushleft}
% where \meta{dimensioned value} means a length specification complete of
% its units or a dimensional expression.
%^^A
% \item Straight lines and vectors are redefined in such a way that
% fractional slope coefficients may be specified; the zero length line does
% not produce errors and is ignored; the zero length vectors draw only the
% arrow tips. 
%^^A
% \item New line and vector macros are defined that avoid the necessity of
% specifying the horizontal component; |\put(3,4){\LIne(25,15)}| specifies
% a segment that starts at point $(3,4)$ and goes to point $(3+25,4+15)$;
% the command |\segment(3,4)(28,19)| achieves the same result without the
% need of using command |\put|.
% The same applies to the vector commands |\Vector| and |\VECTOR| and
% |\VVECTOR|; the latter command behaves as |\VECTOR| but draws a vector
% with arrow tips at both ends; furthermore this command is available only with this new release of the |curve2e| package. 
% Experience has shown that the commands intended to join two specified
% points are particularly useful.
%	\begin{figure}
%	\begin{minipage}{.48\textwidth}
%	\begin{verbatim}
%	\unitlength=.5mm
%	\begin{picture}(60,20)
%	\put(0,0){\GraphGrid(80,20)}
%	\put(0,0){\vector(1.5,2.3){10}}
%	\put(20,0){\Vector(10,15.33333)}
%	\VECTOR(40,0)(50,15.33333)
%	\ifdefined\VVECTOR \VVECTOR(60,0)(80,10)\fi
%	\end{picture}
%	\end{verbatim}
%	\end{minipage}
%	\hfill
%	\begin{minipage}{.48\textwidth}\centering
%	\unitlength=.5mm
%	\begin{picture}(60,20)
%	\put(0,0){\GraphGrid(80,20)}
%	\put(0,0){\vector(1.5,2.3){10}}
%	\put(20,0){\Vector(10,15.33333)}
%	\VECTOR(40,0)(50,15.33333)
%	\ifdefined\VVECTOR \VVECTOR(60,0)(80,10)\fi
%	\end{picture}
%	\end{minipage}
%  \caption{Three (displaced) identical vectors obtained with the three
%           vector macros\ifdefined\VVECTOR; a double tipped vector is also shown\fi.}\label{fig:vectors}
%	\end{figure}
%^^A
% \item The |\polyline| command has been introduced: it accepts an
% unlimited list of  point coordinates enclosed within round parentheses;
% the command draws a sequence of connected segments that join in order the
% specified points; the syntax is:
%	\begin{flushleft}
%\cs{polyline}\oarg{optional join style}\parg{$P_1$}\parg{$P_2$}\texttt{...}\parg{$P_n$}
%	\end{flushleft}
% See figure~\ref{fig:polyline} where a regular pentagon is drawn; usage
% of polar coordinates is also shown.
%
%	\begin{figure}[!ht]
%	\begin{minipage}{.48\linewidth}
%	\begin{verbatim}
%	\unitlength=.5mm
%	\begin{picture}(40,32)(-20,-20)
%	\polyline(90:20)(162:20)(234:20)(306:20)(378:20)(90:20)
%	\end{picture}
%	\end{verbatim}
%	\end{minipage}
%   \hfill
%	\begin{minipage}{.48\linewidth}\raggedleft
%   \unitlength=.5mm
%	\begin{picture}(40,32)(-20,-20)
%	\polyline(90:20)(162:20)(234:20)(306:20)(378:20)(90:20)
%	\end{picture}\hspace*{2em}
%	\end{minipage}
%	\caption{Polygonal line obtained by means of the \texttt{\string\polyline}
%   command; vertex coordinates are in polar form.}
%	\label{fig:polyline}
%	\end{figure}
%
% Examples of using polar and cartesian coordinates are shown in 
% figure~\ref{fig:polar}.
%
%\begin{figure}[htb]\unitlength=0.01\textwidth
%\begin{minipage}{0.55\textwidth}
%\begin{verbatim}
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\put(40,0){\circle*{1.5}}
%   \put(41,0){\makebox(0,0)[bl]{40,0}}
%\put(90:30){\circle*{1.5}}
%   \put(90:31){\makebox(0,0)[bl]{90:30}}
%\put(60:30){\circle*{1.5}}
%   \put(60:31){\makebox(0,0)[bl]{60:30}}
%\put(30,30){\circle*{1.5}}
%   \put(30.7,30.7){\makebox(0,0)[bl]{30,30}}
%\multiput(0,0)(30:10){5}%
%   {\makebox(0,0){\rule{1.5mm}{1.5mm}}}
%\end{picture}
%\end{verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.4\textwidth}
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\put(40,0){\circle*{1.5}}\put(41,0){\makebox(0,0)[l]{40,0}}
%\put(90:30){\circle*{1.5}}\put(90:31){\makebox(0,0)[bc]{90:30}}
%\put(60:30){\circle*{1.5}}\put(60:3){\Zbox(60:27)[bc]{60{:}30}}
%\put(30,30){\circle*{1.5}}\put(30.7,30.7){\makebox(0,0)[bc]{30,30}}
%\multiput(0,0)(30:10){5}{\makebox(0,0){\rule{1.5mm}{1.5mm}}}
%\end{picture}
%\end{minipage}
%\caption{Use of cartesian and polar coordinates}
%\label{fig:polar}
%\end{figure}
%
% A similar example may be obtained with the |\polygon| macro that does
% not require to terminate the polyline at the starting point.
% Figure~\ref{fig:polygon} shows how to get a coloured filled pentagon.
%
%	\begin{figure}[!ht]
%	\begin{minipage}{.48\linewidth}
%	\begin{verbatim}
%	\unitlength=.5mm
%	\begin{picture}(40,32)(-20,-20)
%	\color{magenta}
%	\polygon*(90:20)(162:20)(234:20)(306:20)(378:20)
%	\end{picture}
%	\end{verbatim}
%	\end{minipage}
%   \hfill
%	\begin{minipage}{.48\linewidth}\raggedleft
%   \unitlength=.5mm
%	\begin{picture}(40,32)(-20,-20)
%	\color{magenta}
%	\polygon*(90:20)(162:20)(234:20)(306:20)(378:20)
%	\end{picture}\hspace*{2em}
%	\end{minipage}
%	\caption{A pentagon obtained by means of the \texttt{\string\polygon*}
%   command; vertex coordinates are in polar form.}
%	\label{fig:polygon}
%	\end{figure}
%
% \item The new command |\Dashline| (alias: |\Dline| for backwards
% compatibility)
% \begin{flushleft}
% |\Dashline|\parg{first point}\parg{second point}\marg{dash length}
% \end{flushleft}
% draws a dashed line containing as many dashes as possible, just as long
% as specified, and separated by a gap exactly the same size; actually,
% in order to make an even gap-dash sequence, the desired dash length is
% used to do some computations in order to find a suitable length, close
% to the one specified, such that the distance of the end points is evenly
% divided in equally sized dashes and gaps.
% The end points may be anywhere in the drawing area, without any
% constraint on the slope of the joining segment. The desired dash length
% is specified as a fractional multiple of |\unitlength|; see
% figure~\ref{fig:dashline}.
%	\begin{figure}[!ht]
%	\begin{minipage}{.48\textwidth}
%	\begin{verbatim}
%	\unitlength=1mm
%	\begin{picture}(40,40)
%	\put(0,0){\GraphGrid(40,40)}
%	\Dashline(0,0)(40,10){4}
%	\put(0,0){\circle*{2}}
%	\Dashline(40,10)(0,25){4}
%	\put(40,10){\circle*{2}}
%	\Dashline(0,25)(20,40){4}
%	\put(0,25){\circle*{2}}
%	\put(20,40){\circle*{2}}
%	\Dotline(0,0)(40,40){2}
% \put(40,40){\circle*{2}}
%	\end{picture}
%	\end{verbatim}
%	\end{minipage}
%	\hfill
%	\begin{minipage}{.48\textwidth}\centering
%	\unitlength=1mm
%	\begin{picture}(40,40)
%	\put(0,0){\GraphGrid(40,40)}
%	\Dashline(0,0)(40,10){4}
%	\put(0,0){\circle*{2}}
%	\Dashline(40,10)(0,25){4}
%	\put(40,10){\circle*{2}}
%	\Dashline(0,25)(20,40){4}
%	\put(0,25){\circle*{2}}
%	\put(20,40){\circle*{2}}
% \Dotline(0,0)(40,40){2}
% \put(40,40){\circle*{2}}
%	\end{picture}
%	\end{minipage}
%	\caption{Dashed lines and graph grid}\label{fig:dashline}
%	\end{figure}
% Another example of usage of cartesian and polar coordinates usage is 
% shown in figure~\ref{fig:polar} together with its code.
%
%\begin{figure}\unitlength=0.007\textwidth
%\begin{minipage}{0.55\textwidth}
%\begin{verbatim}
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\Dashline(0,0)(40,10){2}\Dashline(0,0)(40,20){2}
%\Dashline(0,0)(40,30){2}\Dashline(0,0)(30,30){2}
%\Dashline(0,0)(20,30){2}\Dashline(0,0)(10,30){2}
%{\color{blue}%
%\Dashline*(40,0)(108:30){2}
%\Dashline*(40,0)(126:30){2}
%\Dashline*(40,0)(144:30){2}
%\Dashline*(40,0)(162:30){2}}
%\end{picture}
%\end{verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.4\textwidth}\raggedleft
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\Dashline(0,0)(40,10){2}
%\Dashline(0,0)(40,20){2}
%\Dashline(0,0)(40,30){2}
%\Dashline(0,0)(30,30){2}
%\Dashline(0,0)(20,30){2}
%\Dashline(0,0)(10,30){2}
%{\color{blue}%
%\Dashline*(40,0)(108:30){2}
%\Dashline*(40,0)(126:30){2}
%\Dashline*(40,0)(144:30){2}
%\Dashline*(40,0)(162:30){2}}%
%\end{picture}
%\end{minipage}
%\caption{Different length dashed lines with the same nominal dash length}
%\label{fig:dashedlines}
%\end{figure}
%
%^^A
%\item Analogous to |\Dashline|, a new command |\Dotline| draws a dotted
% line with the syntax:
% \begin{flushleft}
% |\Dotline|\parg{first point}\parg{end point}\marg{dot gap}
% \end{flushleft}
% See figures~\ref{fig:dashline} and~\ref{fig:dottedlines} for examples.
%
%\begin{figure}[htb]\unitlength=0.007\textwidth
%\begin{minipage}{0.55\textwidth}
%\begin{verbatim}
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\Dotline(0,0)(40,10){1.5}\Dotline(0,0)(40,20){1.5}
%\Dotline(0,0)(40,30){1.5}\Dotline(0,0)(30,30){1.5}
%\Dotline(0,0)(20,30){1.5}\Dotline(0,0)(10,30){1.5}
%{\color{red}\Dotline*(40,0)(108:30){1.5}
%\Dotline*(40,0)(126:30){1.5}
%\Dotline*(40,0)(144:30){1.5}
%\Dotline*(40,0)(162:30){1.5}}%
%\end{picture}
%\end{verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.4\textwidth}\raggedleft
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\Dotline(0,0)(40,10){1.5}
%\Dotline(0,0)(40,20){1.5}
%\Dotline(0,0)(40,30){1.5}
%\Dotline(0,0)(30,30){1.5}
%\Dotline(0,0)(20,30){1.5}
%\Dotline(0,0)(10,30){1.5}
%{\color{red}%
%\Dotline*(40,0)(108:30){1.5}
%\Dotline*(40,0)(126:30){1.5}
%\Dotline*(40,0)(144:30){1.5}
%\Dotline*(40,0)(162:30){1.5}}%
%\end{picture}
%\end{minipage}
%\caption{Different length dotted lines with the same nominal dot gap}
%\label{fig:dottedlines}
%\end{figure}
%^^A
% \item |\GraphGrid| is a command that draws a red grid under the drawing 
% with lines separated |10\unitlength|s apart; it is described only with a
% comma separated couple of numbers, representing the base and the height
% of the grid, see figure~\ref{fig:dashline}; it's better to specify
% multiples of ten and the grid can be placed anywhere in the drawing
% canvas by means of |\put|, whose cartesian coordinates are multiples of
% 10; nevertheless the grid line distance is rounded to the nearest
% multiple of 10, while the point coordinates specified to |\put| are not
% rounded at all; therefore some care should be used to place the working
% grid on the drawing canvas. This grid is intended as an aid while
% drawing; even if you sketch your drawing on millimetre paper, the
% drawing grid turns out to be very useful; one must only delete or comment
% out the command when the drawing is finished. Several examples of usage of such grid are shown in several figures.
%^^A
% \item New trigonometric function macros have been  computed by means of
% the functionalities of the |xfp| package. The compared to the other
% existing macros is that angles are specified in sexagesimal degrees, so
% that the user needs not transform to radians. The computations are done
% taking into account that abnormal values can occasionally be avoided,
% for example $\tan90^\circ$ must be avoided and replaced with a suitably
% large number, because the TeX\ system does not handle “infinity”.
%
% These trigonometric functions are used within the complex number macros;
% but if the user wants to use them the syntax is the following:
%\begin{flushleft}
%      \cs{SinOf}\meta{angle}\texttt{to}\meta{control sequence}
%\\
%      \cs{CosOf}\meta{angle}\texttt{to}\meta{control sequence}
%\\
%      \cs{TanOf}\meta{angle}\texttt{to}\meta{control sequence}
%\end{flushleft}
% The \meta{control sequence} may then be used as a multiplying factor of a
% length.
% 
%^^A
% \item Arcs can be drawn as simple circular arcs, or with one or two
% arrows at their ends (curved vectors);  the syntax is:
%\begin{flushleft}
% \cs{Arc}\parg{center}\parg{starting point}\marg{angle}\\
% \cs{VectorArc}\parg{center}\parg{starting point}\marg{angle}\\
% \cs{VectorARC}\parg{center}\parg{starting point}\marg{angle}\\
%\end{flushleft}
% If the angle is specified numerically it must be enclosed in braces,
% while if it is specified with a control sequence the braces (curly
% brackets) are not necessary. The above macro |\Arc| draws a simple
% circular arc without arrows; |\VectorArc| draws an arc with an arrow
% tip at the ending point; |\VectorARC| draws an arc with arrow tips at
% both ends; see figure~\ref{fig:arcs}.
%	\begin{figure}
%	\begin{minipage}{.48\textwidth}
%	\begin{verbatim}
%	\unitlength=0.5mm
%	\begin{picture}(60,40)
%	\put(0,0){\GraphGrid(60,40)}
%	\Arc(0,20)(30,0){60}
%	\VECTOR(0,20)(30,0)\VECTOR(0,20)(32.5,36)
%	\VectorArc(0,20)(15,10){60}
%	\put(20,20){\makebox(0,0)[l]{$60^\circ$}}
%	\VectorARC(60,20)(60,0){-180}
%	\end{picture}
%	\end{verbatim}
%	\end{minipage}
%	\hfill
%	\begin{minipage}{.48\textwidth}\centering
%	\unitlength=0.5mm
%	\begin{picture}(60,40)
%	\put(0,0){\GraphGrid(60,40)}
%	\Arc(0,20)(30,0){60}
%	\VECTOR(0,20)(30,0)\VECTOR(0,20)(32.5,36)
%	\VectorArc(0,20)(15,10){60}
%	\put(20,20){\makebox(0,0)[l]{$60^\circ$}}
%	\VectorARC(60,20)(60,0){-180}
%	\end{picture}
%	\end{minipage}
%	\caption{Arcs and curved vectors}\label{fig:arcs}
%	\end{figure}
%^^A
% \item A multitude of commands have been defined in order to manage
% complex numbers; actually complex numbers are represented as a comma
% separated pair of fractional numbers (here we use only cartesian
% coordinates). They are used to address specific points in the drawing
% plane, but also as operators so as to scale and rotate other objects. In
% the following \meta{vector} means a comma separated pair of fractional
% numbers, \meta{vector macro} means a macro that contains a comma
% separated pair of fractional numbers; \meta{angle macro} means a macro
% that contains the angle of a vector in sexagesimal degrees;
% \meta{argument} means a brace delimited numeric value, even a macro;
% \textit{macro} is a valid macro name, i.e.~a backslash followed by
% letters, or anything else that can receive a definition. A
% \emph{direction} of a vector is its versor;% the angle of a vector is
% the angle between the vector and the positive $x$ axis in
% counterclockwise direction, as generally directly used in the
% Euler formula $ \vec{v} = Me^{\mathrm{j}\varphi}$.
%
% {\footnotesize\begin{itemize}
% \item |\MakeVectorFrom|\meta{two arguments}|to|\meta{vector macro}
% \item |\CopyVect|\meta{first vector}|to|\meta{second vector macro}
% \item |\ModOfVect|\meta{vector}|to|\meta{macro}
% \item |\DirOfvect|\meta{vector}|to|\meta{versor macro}
% \item |\ModAndDirOfVect|\meta{vector}|to|\meta{1st macro}|and|\meta{2nd macro}
% \item |\DistanceAndDirOfVect|\meta{1st vector}|minus|\meta{2nd vector}|to|\meta{1st macro}|and|\meta{2nd macro}
% \item |\XpartOfVect|\meta{vector}|to|\meta{macro}
% \item |\YpartOfVect|\meta{vector}|to|\meta{macro}
% \item |\DirFromAngle|\meta{angle}|to|\meta{versor macro}
% \item |\ArgOfVect|\meta{vector}|to|\meta{angle macro}
% \item |\ScaleVect|\meta{vector}|by|\meta{scaling factor}|to|\meta{vector macro}
% \item |\ConjVect|\meta{vector}|to|\meta{conjugate vector macro}
% \item |\SubVect|\meta{first vector}|from|\meta{second vector}|to|\meta{vector macro}
% \item |\AddVect|\meta{first vector}|and|\meta{second vector}|to|\meta{vector macro}
% \item |\MultVect|\meta{first vector}|by|\meta{second vector}|to|\meta{vector macro}
% \item |\MultVect|\meta{first vector}|by*|\meta{second vector}|to|\meta{vector macro}
% \item |\DivVect|\meta{first vector}|by|\meta{second vector}|to|\meta{vector macro}
% \end{itemize}}
%^^A
% \item General curves can be drawn with the |pict2e| macro |\curve| but it
% requires the specification of the third-order Bézier-spline control
% points; sometimes it's better to be very specific with the control points
% and there is no other means to do a decent graph; sometimes the curves to
% be drawn are not so tricky and a general set of macros can be defined so
% as to compute the control points, while letting the user specify only the
% nodes through which the curve must pass, and the tangent direction of the
% curve in such nodes. Such commands are the following:
%\begin{itemize}
%
%\item \cs{Curve} to draw a sequence of arcs as explained above, using
% third order (cubic) Bézier splines. The starred version of this command
% fills the internal part of the curve with the current color; if the last
% arc finishes where the fist arc starts, it is clear what is the interior;
% if it does not, the driver  (not the code of this package, but the
% driver between this code and the physical representation on paper or
% screen) assumes a straight line closure of the whole path.
%
%\item \cs{Qurve} similar to |\Curve|, but with second order (quadratic)
% Bézier splines. The starred version fills the interior with the current
% color
%
%\item \cs{CurveBetween} draws a single cubic Bézier spline between two
% given nodes and with two given directions vectors.
%
%\item \cs{CBezierBetween}  draws a single cubic Bézier spline between two
% given nodes, with two given directions versors along which the
% control node distances are specified. This is the most general macro
% (rather difficult to use) with which not only the arc end points are
% specified but also the control nodes coordinates are given.
%
%\end{itemize}
%
% The main macro is |\Curve| and must be followed by an
% ``unlimited'' sequence of node-direction coordinates as a quadruple
% defined as
%\[
% \parg{node coordinates}\aarg{direction vector}
%\]
% Possibly if a sudden change of direction has to be performed (cusp)
% another item can be inserted after one of those quadruples in the form
%\[
% \mbox{\dots\parg{...}\aarg{...}\oarg{new direction vector}\parg{...}\aarg{...}\dots}
%\]
%
% Possibly it is necessary to specifiy the “tension” or the “looseness”
% of a specific Bézier arc; such tension parameters range from 0 (zero)
% to~4; the zero value implies a very stiff arc, as if it was a string
% subject to a high tension (i.e. with zero looseness); a value of~4
% implies a very low tension (very high looseness), almost as if the string
% was not subject to any tension. In \MF\ or \MP\ language such a concept
% is used very often; in this package, where the Hobby  algorithms are
% not used, the parameter value appears to mean the opposite of tension.
% A couple of comma separated tension values may be optionally used, they
% are separated with a semicolon form the direction vector,
% and they apply to the arc terminating with the last node; their
% specification  must
% precede any possible change of tangent according to this
% syntax\footnote{The tension may be specified only for cubic splines,
% because the quadratic ones do not use enough parameters to control the
% tension; not all commands for drawing cubic splines accept this
% optional tension specification.}:
%\[
%\mbox{\dots\parg{...}\aarg{\upshape{\em direction vector}\texttt{;}{\em start tension}|,|{\em end tension}}\parg{...}\aarg{...}\dots}
%\]
%
% The |\Curve| macro does not (still) have facilities for cycling the path,
% that is to close the path from the last specified node-direction to the
% first specified node-direction; but, as already mentioned, if the ending
% node of the last arc does not coincide with the stating node of the
% first arc, a straight line is assumed to join such nodes; this line does
% not get drawn, but with starred commands no lines are drawn because only
% the interior is coloured.
% The tangent direction need not be specified with a unit vector, although
% only its direction is relevant; the scaling of the specified direction
% vector to a unit vector is performed by the macro itself.
% Therefore one cannot specify the fine tuning of the curve convexity as it
% can be done with other programs or commands, as for example with \MF\
% or the |pgf/tikz| package and environment. See figure~\ref{fig:curve} for
% an example.
%	\begin{figure}[htb]
%	\begin{minipage}{.48\textwidth}
%   \begin{verbatim}
%	\unitlength=8mm\relax
%	\begin{picture}(5,5)
%	\put(0,0){\framebox(5,5){}}\thicklines\roundcap
%	\Curve(2.5,0)<1,1>(5,3.5)<0,1>%
%	  (4,5)<-1,0>(2.5,3.5)<-.5,-1.2>[-.5,1.2]%
%	  (1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<1,-1>
%	\end{picture}
%   \end{verbatim}
%	\end{minipage}
%	\hfill
%	\begin{minipage}{.48\textwidth}\raggedleft\relax
%	\unitlength=8mm\relax
%	\begin{picture}(5,5)
%	\put(0,0.5){\put(0,0){\framebox(5,5){}}\thicklines\roundcap
%	\Curve(2.5,0)<1,1>(5,3.5)<0,1>(4,5)<-1,0>(2.5,3.5)<-0.5,-1.2>[-0.5,1.2](1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<1,-1>}
%	\end{picture}
%	\end{minipage}
%	\caption{A heart shaped curve with cusps drawn with \texttt{\string\Curve}}
%	\label{fig:curve}

%\vspace*{2\baselineskip}
%
%	\begin{minipage}{.48\textwidth}
%   \begin{verbatim}
%	\unitlength=8mm\relax
%	\begin{picture}(5,5)
%	\put(0,0){\framebox(5,5){}}\thicklines\roundcap
%   \color{green}\relax
%	\Curve*(2.5,0)<1,1>(5,3.5)<0,1>%
%	  (4,5)<-1,0>(2.5,3.5)<-.5,-1.2>[-.5,1.2]%
%	  (1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<1,-1>
%	\end{picture}
%   \end{verbatim}
%	\end{minipage}
%	\hfill
%	\begin{minipage}{.48\textwidth}\raggedleft\relax
%	\unitlength=8mm\relax
%	\begin{picture}(5,5)
%	\put(0,0.5){\put(0,0){\framebox(5,5){}}\thicklines\roundcap
%   \color{green}\relax
%	\Curve*(2.5,0)<1,1>(5,3.5)<0,1>(4,5)<-1,0>(2.5,3.5)<-0.5,-1.2>[-0.5,1.2](1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<1,-1>}
%	\end{picture}
%	\end{minipage}
%\caption{Coloring the inside of a closed path drawn with \texttt{\string\Curve*}}
%\label{fig:colored-curve}

%	\end{figure}
%
% With the starred version of |\Curve|, instead of stroking the contour,
% the macro fills up the contour with the selected current color,
% figure~\ref{fig:colored-curve}.
%
% Figure~\ref{fig:arcspline}  shows a geometric construction that contains the geometric elements and symbols used to determine the parameters of a cubic spline required to draw a quarter circle. This construction containa many of the commands described so far.
%
%\begin{figure}[p]
%\begin{minipage}{\linewidth}\small
%\begin{verbatim}
%\unitlength=0.007\textwidth
%\begin{picture}(100,90)(-50,-50)
%\put(-50,0){\vector(1,0){100}}\put(50,1){\makebox(0,0)[br]{$x$}}%
%\put(20,-1){\makebox(0,0)[t]{$s$}}%
%\put(0,0){\circle*{2}}\put(-1,-1){\makebox(0,0)[tr]{$M$}}%
%\legenda(12,-45){s=\overline{MP_2}=R\sin\theta}%
%\put(0,-50){\vector(0,1){90}}%
%\put(1,40){\makebox(0,0)[tl]{$y$}}%
%\put(0,-40){\circle*{2}}\put(1,-41){\makebox(0,0)[lt]{$C$}}%
%\segment(0,-40)(-40,0)\segment(0,-40)(40,0)%
%\put(-41,1){\makebox(0,0)[br]{$P_1$}}\put(-40,0){\circle*{2}}%
%\put(41,1){\makebox(0,0)[bl]{$P_2$}}\put(40,0){\circle*{2}}%
%\put(0,0){\linethickness{1pt}\Arc(0,-40)(40,0){90}}%
%\segment(-40,0)(-20,20)\put(-20,20){\circle*{2}}%
%\put(-20,21.5){\makebox(0,0)[b]{$C_1$}}%
%\segment(40,0)(20,20)\put(20,20){\circle*{2}}%
%\put(20,21.5){\makebox(0,0)[b]{$C_2$}}%
%\put(0,-40){\put(0,56.5685){\circle*{2}}%
%\put(1,58){\makebox(0,0)[bl]{$P$}}}%
%\VectorARC(0,-40)(15,-25){45}\put(10,-18){\makebox(0,0)[c]{$\theta$}}%
%\VectorARC(40,0)(20,0){-45}\put(19,5){\makebox(0,0)[r]{$\theta$}}%
%\VectorARC(-40,0)(-20,0){45}\put(-19,5){\makebox(0,0)[l]{$\theta$}}%
%\put(-20,-18){\makebox(0,0)[bl]{$R$}}%
%\put(-32,13){\makebox(0,0)[bl]{$K$}}%
%\put(32,13){\makebox(0,0)[br]{$K$}}%
%\end{picture}
%\end{verbatim}
%\end{minipage}\vspace{\stretch{1}}
%
%\begin{minipage}{\linewidth}\centering
%\unitlength=0.007\textwidth
%\begin{picture}(100,90)(-50,-50)
%\put(-50,0){\vector(1,0){100}}\put(50,1){\makebox(0,0)[br]{$x$}}%
%\put(20,-1){\makebox(0,0)[t]{$s$}}%
%\put(0,0){\circle*{2}}\put(-1,-1){\makebox(0,0)[tr]{$M$}}%
%\legenda(12,-45){s=\overline{MP_2}=R\sin\theta}%
%\put(0,-50){\vector(0,1){90}}%
%\put(1,40){\makebox(0,0)[tl]{$y$}}%
%\put(0,-40){\circle*{2}}\put(1,-41){\makebox(0,0)[lt]{$C$}}%
%\segment(0,-40)(-40,0)\segment(0,-40)(40,0)%
%\put(-41,1){\makebox(0,0)[br]{$P_1$}}\put(-40,0){\circle*{2}}%
%\put(41,1){\makebox(0,0)[bl]{$P_2$}}\put(40,0){\circle*{2}}%
%\put(0,0){\linethickness{1pt}\Arc(0,-40)(40,0){90}}%
%\segment(-40,0)(-20,20)\put(-20,20){\circle*{2}}%
%\put(-20,21.5){\makebox(0,0)[b]{$C_1$}}%
%\segment(40,0)(20,20)\put(20,20){\circle*{2}}%
%\put(20,21.5){\makebox(0,0)[b]{$C_2$}}%
%\put(0,-40){\put(0,56.5685){\circle*{2}}%
%\put(1,58){\makebox(0,0)[bl]{$P$}}}%
%\VectorARC(0,-40)(15,-25){45}\put(10,-18){\makebox(0,0)[c]{$\theta$}}%
%\VectorARC(40,0)(20,0){-45}\put(19,5){\makebox(0,0)[r]{$\theta$}}%
%\VectorARC(-40,0)(-20,0){45}\put(-19,5){\makebox(0,0)[l]{$\theta$}}%
%\put(-20,-18){\makebox(0,0)[bl]{$R$}}%
%\put(-32,13){\makebox(0,0)[bl]{$K$}}%
%\put(32,13){\makebox(0,0)[br]{$K$}}%
%\end{picture}
%\end{minipage}
%\caption{The code to display the Nodes and control points for an arc to
% be approximated with a cubic Bézier spline}
%\label{fig:arcspline}
%\end{figure}
%
%
% To show what you can do with |\CurveBetween| see the code and result
% shown in figure~\ref{fig:curva-due-punti}. Notice the effect of changing the directions at both or a the end nodes os a single cubic spline. The directions are conveniently expressed with unit vectors described by polar coordinates.

%\begin{figure}\centering\unitlength=0.004\textwidth
%\begin{picture}(220,120)(-50,-20)
%\put(0,60){\Line(-50,0)(50,0)
%\CurveBetween-50,0and50,0WithDirs15:1and{-15:1}
%\CurveBetween-50,0and50,0WithDirs30:1and{-30:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-45:1}
%\CurveBetween-50,0and50,0WithDirs60:1and{-60:1}
%\CurveBetween-50,0and50,0WithDirs75:1and{-75:1}
%\CurveBetween-50,0and50,0WithDirs90:1and{-90:1}}
%\put(120,60){%
%\Line(-50,0)(50,0)
%\CurveBetween-50,0and50,0WithDirs15:1and{15:1}
%\CurveBetween-50,0and50,0WithDirs30:1and{30:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{45:1}
%\CurveBetween-50,0and50,0WithDirs60:1and{60:1}
%\CurveBetween-50,0and50,0WithDirs75:1and{75:1}
%\CurveBetween-50,0and50,0WithDirs90:1and{90:1}}
%\put(0,0){%
%\Line(-50,0)(50,0)
%\CurveBetween-50,0and50,0WithDirs45:1and{-15:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-30:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-45:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-60:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-75:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-90:1}}
%\put(120,0){%
%\Line(-50,0)(50,0)
%\CurveBetween-50,0and50,0WithDirs45:1and{15:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{30:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{45:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{60:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{75:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{90:1}}
%\end{picture}
%\caption{Curves between two points with different start and end slopes}\label{fig:curva-due-punti}
%\end{figure}
%
% A little more complicated is the use of the |\CBezierBetween| macro,
% figure~\ref{fig:Cbezier}. The directions are specified with unit
% vectors in polar form; the control points are specified by adding their
% distances from their neighbouring nodes; actually the right distance
% is maintained to the value~1, while the left one increases from~4 to~10.
% The black line corresponds to the standard |\CurveBetween| where the
% default distance is computed by default to trace an arc of a circle and
% is approximately~3.5. 
%
%\begin{figure}[!tb]
%\begin{minipage}[t]{0.52\textwidth}
%\begin{verbatim}
%\unitlength=0.1\textwidth
%\begin{picture}(10,3)
%\CurveBetween0,0and10,0WithDirs1,1and{1,-1}
%\color{red}%
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists4And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists6And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists8And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists10And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists12And{1}
%\end{picture}
%\end{verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.40\textwidth}\raggedleft
%\unitlength=0.1\textwidth
%\begin{picture}(10,3)(0,1.25)
%\CurveBetween0,0and10,0WithDirs1,1and{1,-1}
%\color{red}%
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists4And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists6And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists8And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists10And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists12And{1}
%\end{picture}
%\end{minipage}
%\caption{Comparison between similar arcs drawn with \cs{CurveBetween} (black)
% and \cs{CbezierTo} (red)}
%\label{fig:Cbezier}
%\end{figure}
%
% In figure~\ref{fig:tensions} the effect of tension specification is
% shown. The red line corresponds to the default tension, since the
% tension values are not specified. The black lines correspond to the
% various values used in the various commands to the |\Curve| macro.
% With a tension of zero, the spline is almost coincident wit the
% horizontal base line of the frame. Increasing the tension value
% to~4.5, the curved becomes taller and taller, until it wraps itself
% displaying an evident loop. We would say that the value of ~2 is a
% reasonable  maximum  and  increasing that tension value is just to
% obtain special effects.
%
%\begin{figure}[!htb]\centering
%\begin{minipage}{0.48\textwidth}\small
%\begin{verbatim}
%\raggedleft\unitlength=0.01\textwidth
%\begin{picture}(70,70)
%\put(0,0){\color{blue}\frame(70,70){}}
%\put(0,0){\color{red}\Curve(0,0)<1,1>(70,0)<1,-1>}
%\Curve(0,0)<1,1>(70,0)<1,-1;0,0>
%\Curve(0,0)<1,1>(70,0)<1,-1;0.2,0.2>
%\Curve(0,0)<1,1>(70,0)<1,-1;2,2>
%\Curve(0,0)<1,1>(70,0)<1,-1;4.5,4.5>
%\Curve(0,0)<1,1>(70,0)<1,-1;0,3>
%\Curve(0,0)<1,1>(70,0)<1,-1;3,0>
%\end{picture}
%\end{verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.46\textwidth}
%\raggedleft\unitlength=0.01\textwidth
%\begin{picture}(70,70)
%\put(0,0){\color{blue}\framebox(70,70){}}
%\put(0,0){\color{red}\Curve(0,0)<1,1>(70,0)<1,-1>}
%\Curve(0,0)<1,1>(70,0)<1,-1;0,0>
%\Curve(0,0)<1,1>(70,0)<1,-1;0.2,0.2>
%\Curve(0,0)<1,1>(70,0)<1,-1;2,2>
%\Curve(0,0)<1,1>(70,0)<1,-1;4.5,4.5>
%\Curve(0,0)<1,1>(70,0)<1,-1;0,3>
%\Curve(0,0)<1,1>(70,0)<1,-1;3,0>
%\end{picture}
%\end{minipage}
%\caption{The effects of tension factors}\label{fig:tensions}
%\end{figure}
%
% Figure~\ref{fig:sinewave} displays two approximations of a sine wave;
% Bézier splines can approximate transcendental curves, but the
% approximation may be a poor one, depending on the approximated curve,
% if few arcs are used to draw it. With arcs specified with more
% complicated macros the approximation is better even with a lower number
% of arcs.  With many arcs it is possible to approximate almost anything.
% On the left side a modest approximation is obtained with just three
% standard arcs obtained with |\Curve| and four node specifications;
% on the right we have just two arcs created with |CBezierBetween| 
% with tension specification and control point distances; this drawing 
%is almost undistinguishable from a real sinusoid.
%
%\begin{figure}[!htb]
%\begin{minipage}{\linewidth}
%\begin{verbatim}
%\unitlength=0.01\textwidth
%\begin{picture}(100,50)(0,-25)
%\put(0,0){\VECTOR(0,0)(45,0)\VECTOR(0,-25)(0,25)
%\Zbox(45,0)[br]{x}\Zbox(0,26)[tl]{y}
%\Curve(0,0)<77:1>(10,20)<1,0;2,0.4>(30,-20)<1,0;0.4,0.4>(40,0)<77:1;0.4,2>
%}
%\put(55,0){\VECTOR(0,0)(45,0)\VECTOR(0,-25)(0,25)
%\Zbox(45,0)[br]{x}\Zbox(0,26)[tl]{y}
%\CbezierBetween0,0And20,0WithDirs77:1And-77:1UsingDists28And{28}
%\CbezierBetween20,0And40,0WithDirs-77:1And77:1UsingDists28And{28}}
%\end{picture}
%\end{verbatim}
%\end{minipage}\vspace{\baselineskip}
%
%\begin{minipage}{\linewidth}
%\unitlength=0.01\textwidth
%\begin{picture}(100,50)(0,-25)
%\put(0,0){\VECTOR(0,0)(45,0)\VECTOR(0,-25)(0,25)
%\Zbox(45,0)[br]{x}\Zbox(0,26)[tl]{y}
%\Curve(0,0)<77:1>(10,20)<1,0;2,0.4>(30,-20)<1,0;0.4,0.4>(40,0)<77:1;0.4,2>
%}
%\put(55,0){\VECTOR(0,0)(45,0)\VECTOR(0,-25)(0,25)
%\Zbox(45,0)[br]{x}\Zbox(0,26)[tl]{y}
%\CbezierBetween0,0And20,0WithDirs77:1And-77:1UsingDists28And{28}
%\CbezierBetween20,0And40,0WithDirs-77:1And77:1UsingDists28And{28}}
%\end{picture}
%\end{minipage}
%\caption{A sequence of arcs; the left figure has been drawn with the
% \cs{Curve} command with a sequence of four couples of point-direction
% arguments; the right figure has been drawn with two commands
% \cs{CbezierBetween} that include also the specification of the control
% points}
%\label{fig:sinewave}
%\end{figure}
%
% In figure~\ref{fig:quadratic-arcs} some lines drawn with quadratic
% splines by means of the |\Qurve| macro are shown. In the left there are
% some open and closed curves inscribed within a square. On the right a
% “real" circle is compared to a quadratic spline circle; the word “real”
% is emphasised because it actually is an approximation with four
% quarter-circle cubic splines that, in spite of being drawn with third
% degree parametric polynomials, approximate very well a real circle; on
% the opposite the quadratic spline circle is clearly a poor
% approximation even if the maximum radial error amounts just to about
% 6\% of the radius.
%
%\begin{figure}[p]
%\begin{minipage}{\linewidth}
%\begin{Verbatim}[fontsize=\setfontsize{7.75}]
%\unitlength=0.0045\textwidth
%\begin{picture}(100,100)
%\put(0,0){\framebox(100,100){}}
%\put(50,50){%
%  \Qurve(0,-50)<1,0>(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>(0,-50)<1,0>
%\color{green}
%  \Qurve*(0,-50)<0,1>(50,0)<1,0>[-1,0](0,50)<0,1>[0,-1](-50,0)<-1,0>[1,0](0,-50)<0,-1> 
%}
%\Qurve(0,0)<1,4>(50,50)<1,0>(100,100)<1,4>
%\put(5,50){\Qurve(0,0)<1,1.5>(22.5,20)<1,0>(45,0)<1,-1.5>%
%(67.5,-20)<1,0>(90,0)<1,1.5>}
%\Zbox(0,0)[tc]{0,0}\Zbox(100,0)[tc]{100,0}
%\Zbox(100,100)[bc]{100,100}\Zbox(0,100)[bc]{0,100}
%\Pall[2](0,0)\Pall[2](100,0)\Pall[2](100,100)\Pall[2](0,100)
%\end{picture}
%\hfill
%\begin{picture}(100,100)
%\put(0,0){\framebox(100,100){}}
%\put(50,50){%
%\Qurve(0,-50)<1,0>(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>(0,-50)<1,0>
%\Curve(0,-50)<1,0>(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>(0,-50)<1,0>}
%\Zbox(50,50)[t]{O}\Pall[2](50,50)\put(50,50){\Vector(45:50)}\Zbox(67,70)[tl]{R}
%\end{picture}
%\end{Verbatim}
%\end{minipage}\vspace{2\baselineskip}
%
%\begin{minipage}{\linewidth}
%\unitlength=0.0045\textwidth
%\begin{picture}(100,100)
%\put(0,0){\framebox(100,100){}}
%\put(50,50){\Qurve(0,-50)<1,0>(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>(0,-50)<1,0>}
%\put(50,50){\color{green}%
%\Qurve*(0,-50)<0,1>(50,0)<1,0>[-1,0](0,50)<0,1>[0,-1](-50,0)<-1,0>[1,0](0,-50)<0,-1>}
%\Qurve(0,0)<1,4>(50,50)<1,0>(100,100)<1,4>
%\put(5,50){\Qurve(0,0)<1,1.5>(22.5,20)<1,0>(45,0)<1,-1.5>(67.5,-20)<1,0>(90,0)<1,1.5>}
%\Zbox(0,0)[tc]{0,0}\Zbox(100,0)[tc]{100,0}
%\Zbox(100,100)[bc]{100,100}\Zbox(0,100)[bc]{0,100}
%\Pall[2](0,0)\Pall[2](100,0)\Pall[2](100,100)\Pall[2](0,100)
%\end{picture}
%\hfill
%\begin{picture}(100,100)
%\put(0,0){\framebox(100,100){}}
%\put(50,50){\Qurve(0,-50)<1,0>(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>(0,-50)<1,0>
%\Curve(0,-50)<1,0>(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>(0,-50)<1,0>}
%\Zbox(50,50)[t]{O}\Pall[2](50,50)\put(50,50){\Vector(45:50)}\Zbox(67,70)[tl]{R}
%\end{picture}
%\end{minipage}
%
%\caption{\rule{0pt}{4ex}Several graphs drawn with quadratic Bézier
% splines. On the right a quadratic spline circle is compared with a
% cubic line circle.}
%\label{fig:quadratic-arcs}
%\end{figure}
%
% Notice that the previous version of |curve2e| contained an error and
% would color the outside of the green four-pointed star. 
% The |curve2e-v161|, attached to this bundle, has been corrected;
% therefore it is not actually identical to the previous version,
% although the latter one performed correctly for everything else except
% for color-filled quadratic paths.
% ^^A
% \item The new version of |\mulpiput| is backwards compatibile with
% the original version contained in the \LaTeX\ kernel. The new part
% consists into the handling of the coordinate increments from one
% position to the next for the \meta{object} to include in the drawing.
% On page~\pageref{pag:multiput} we show the code for the figure shown
% there. The red grid is nothing new, except that id demonstrate the the
% traditional |\multiput| used in tis code, shown in a previous example,
% produces exactly the same result. But the for “graphs” on the grid,
% display an alignment of black dots along the diagonal of the grid
% (again traditional |\multiput| rendered with the new version);
% a number of blue dots along a parabola; another number of magenta
% dots alined along a half sine wave; a number of little green squares
% aligned along a $-15~\circ$ line starting from the center of the grid.
%
%\noindent
%\begin{minipage}{0.45\linewidth}
%\begin{Verbatim}[fontsize=\setfontsize{8.25}]
%\unitlength=0.01\linewidth
%\begin{picture}(100,100)
%\put(0,0){\GraphGrid(100,100)}
%\multiput(0,0)(10,10){11}{\circle*{2}}
%\color{blue!70!white}
%\multiput(0,0)(10,0){11}{%
%  \circle*{2}}%
%  [\edef\X{\fpeval{\X+10}}%
%  \edef\Y{\fpeval{((\X/10)**2)}}]
%\color{magenta}
%\multiput(0,0)(10,1){11}{%
%\circle*{2}}%
%  [\edef\X{\fpeval{\X+10}}%
%  \edef\Y{\fpeval{sind(\X*1.8)*100}}]
%\color{green!60!black}
%\multiput(50,50)(-15:5){11}}{%
%\polygon*(-1,-1)(1,-1)(1,1)(-1,1)}
%\end{picture}
%\end{Verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.45\linewidth}
%\unitlength=0.01\linewidth
%\begin{picture}(100,100)
%\put(0,0){\GraphGrid(100,100)}
%\multiput(0,0)(10,10){11}{\circle*{2}}
%\color{blue!70!white}
%\multiput(0,0)(10,0){11}{\circle*{2}}%
%[\edef\X{\fpeval{\X+10}}\edef\Y{\fpeval{((\X/10)**2)}}]
%\color{magenta}
%\multiput(0,0)(10,1){11}{\circle*{2}}%
%[\edef\X{\fpeval{\X+10}}\edef\Y{\fpeval{sind(\X*1.8)*100}}]
%\color{green!60!black}
%\multiput(50,50)(-15:5){11}{\polygon*(-1,-1)(1,-1)(1,1)(-1,1)}
%\end{picture}\label{pag:multiput}
%\end{minipage}
%
% \end{enumerate}
%
%
% In spite of the relative simplicity of the macros contained in this
% package, the described macros, as well as the original ones included in
% the |pict2e| package, allow to produce fine drawings that were
% unconceivable with the original \LaTeX\ picture environment. Leslie
% Lamport himself announced an extension to his environment when \LaTeXe\
% was first released in 1994; in the |latexnews| news-letter of December
% 2003; the first implementation was announced; the first version of this
% package was issued in 2006. It was time to have a  better drawing
% environment; this package is a simple attempt to follow the initial
% path while extending the drawing facilities; but Till Tantau's |pgf|
% package has gone much farther.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Remark}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% There are other packages in the \textsc{ctan} archives that deal with
% tracing curves of various kinds. |PSTricks| and |tikz/pgf| are the most
% powerful ones.
% But there is also the package |curves| that is intended to draw almost
% anything by using little dots or other symbols partially superimposed to
% one another. It uses only quadratic Bézier curves and the curve tracing
% is eased by specifying only the curve nodes, without specifying the
% control nodes; with a suitable option to the package call it is possible
% to reduce the memory usage by using short straight segments drawn with
% the PostScript facilities offered by the |dvips| driver.
%
% Another package |ebezier| performs about the same as |curve2e| but draws
% its Bézier curves by using little dots partially superimposed to one
% another. The documentation is quite interesting but since it explains
% very clearly what exactly are the Bézier splines. Apparently |ebezier|
% should be used only for dvi output without recourse to PostScript machinery.
%
% The |picture| package extends the performance of the |picture|
% environment (extended with \texttt{pict2e}) by accepting coordinates
% and lengths in real absolute dimensions, not only as multiples of
% |\unitlength|; it provides commands to extend that functionality to
% other packages. In certain circumstances it is very useful.
%
% Package \texttt{xpicture} builds over the |picture| \LaTeX\ environment
% so as to allow to draw the usual curves that are part of an introductory
% analytic geometry course; lines, circles, parabolas, ellipses,
% hyperbolas, and polynomials; the syntax is very comfortable; for all
% these curves it uses the quadratic Bézier splines.
% 
% Package |hobby| extends the cubic Bézier spline handling with the
% algorithms John Hobby created for \MF\ and \MP. But by now this package
% interfaces very well with |tikz|; it has not (yet) been adapted to the
% common |picture| environment, even extended with |pict2e|, and, why not,
% with |curve2e|.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Acknowledgements}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% 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.
%
% Josef Tkadlec and the author collaborated extensively in order to make a
% better real long division so as to get correctly the quotient fractional
% part and to avoid as much as possible any numeric overflow; many Josef's
% ideas are incorporated in the macro that was implemented in the previous
% version of this package, although the macro used by Josef was slightly 
% different. Both versions aim/aimed at a better accuracy and at widening
% the operand ranges. In this version we abandoned the long division macro,
% and substituted it with the floating point division provided by the
% |xfp| package.
%
% Daniele Degiorgi spotted a fault in the kernel definition of
% |\linethickness| that heavily influenced also |curve2e|; see below in
% the code documentation part.
%
% Thanks also to Jin-Hwan Cho and Juho Lee who suggested a small but
% crucial modification in order to have \texttt{curve2e} work smoothly also
% with XeTeX (XeLaTeX). Actually if version 0.2x or later, dated 2009/08/05 
% or later, of |pict2e| is being used, such modification is not necessary,
% but it's true that it becomes imperative if older versions are used.
%
% \StopEventually{%
% \begin{thebibliography}{9}
% \bibitem{pict2e} Gäßlein H.,  Niepraschk R., and Tkadlec J.
% \emph{The \texttt{pict2e} package}, 2014, PDF documentation of
% \texttt{pict2e}; this package is part of any modern complete distribution
% of the \TeX\ system. In case of a basic or partial system installation,
% the package may be installed by means of the specific facilities of the
% distribution. It may be read by means of the line command \texttt{texdoc
% pict2e}.
% \end{thebibliography}
% }
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Source code}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Some preliminary extensions to the \texttt{pict2e} package}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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.
% If you want to use package \texttt{xcolor}, load it after
% \texttt{curve2e}.
%
% Here we load also the |xparse| and |xfp| packages because we use their
% functionalities; but we do load them only if they are not already loaded
% with or without options; nevertheless we warn the user who wants to load
% them explicitly, to do this action before loading \texttt{curve2e}. 
% The |xfp| package is absolutely required; if this package can not found
% it in the \TeX\ system installation, the loading of this package is
% aborted, and the previous version 1.61 of |curve2e| is loaded in its
% place; the overall functionalities should non change much, nevertheless
% the functionalities of |xfp| are not available.
%\iffalse
%<*package>
%\fi
%    \begin{macrocode}
\IfFileExists{xfp.sty}{%
    \RequirePackage{color}
    \RequirePackageWithOptions{pict2e}[2014/01/01]
    \@ifl@aded{sty}{xparse}{}{\RequirePackage{xparse}}
    \@ifl@aded{sty}{xfp}{}{\RequirePackage{xfp}}%
}{%
    \RequirePackage{curve2e-v161}%
    \PackageWarningNoLine{curve2e}{%
      Package xfp is required, but apparently\MessageBreak%
      such package cannot be found in this \MessageBreak%
      TeX system installation\MessageBreak%
      Either your installation is not complete \MessageBreak%
      or it is older than 2018-10-17.\MessageBreak%
      \MessageBreak%
      ***************************************\MessageBreak%
      Version 1.61 of curve2e has been loaded\MessageBreak%
      instead of the current version\MessageBreak%
      ***************************************\MessageBreak}%
      \endinput
}
%    \end{macrocode}
%
% The next macros are just for debugging. With the \texttt{trace} 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}
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Line thickness macros}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 these 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.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Improved line and vector macros}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The macro |\LIne| allows to draw a line with arbitrary inclination 
% as if it was a polygonal with just two vertices; actually it joins the
% canvas coordinate origin with the specified relative coordinate;
% therefore this object must be set in place by means of a |\put| command.
% Since its starting point is always at a relative 0,0 coordinate point
% inside the box created with |\put|, the two arguments define the
% horizontal and the vertical component respectively.
%    \begin{macrocode}
\def\LIne(#1){{\GetCoord(#1)\@tX\@tY
  \moveto(0,0)
  \pIIe@lineto{\@tX\unitlength}{\@tY\unitlength}\strokepath}\ignorespaces
}%
%    \end{macrocode}
%
% A similar macro |\segment| 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 shall 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\segment(#1)(#2){\@killglue\polyline(#1)(#2)}%
%    \end{macrocode}
% By passing its ending points coordinates to the |\polyline| macro, both
% macro arguments are a pair of coordinates, not their components; in other
% words, if $P_1=(x_1, y_2)$ and $P_2=(x_2, y_2)$, then the first argument
% is the couple $x_1, y_1$ and likewise the second argument is $x_2, y_2$.
% Notice that since |\polyline| accepts also the corner coordinates in
% polar form, also|\segment| accepts the polar form. Please remember that
% the decimal separator is the decimal \emph{point}, while the \emph{comma}
% acts as coordinate separator. This recommendation is particularly
% important for non-English speaking users, since in all other languages
% the comma must be used as the decimal separator.
%
% The |\line| macro is redefined by making use of a division routine
% performed in floating point arithmetics; for this reason the \LaTeX\
% kernel and the overall \TeX\ system installation must be as recent as
% the release date of the \texttt{xfp} package, i.e. 2018-10-17.
% The floating point division macro receives in input two fractional
% numbers and yields on output their fractional ratio.
% Notice that this command |\line| should follow the same syntax as the
% original pre~1994 \LaTeX\ version; but the new definition  accepts the
% direction coefficients in polar mode; that is, instead of specifying a
% slope of $30^\circ$ with the actual sine and cosine (or values 
% proportional to such functions), for example |(0.5,0.866025)|, you may
% specify it as |(30:1)|, i.e. as a unit vector with the required slope of
% $30^\circ$.
%
% 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. This makes the code much simpler, not necessarily more
% efficient; nevertheless |\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 normalised 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 the horizontal cosine |\d@mX|, and the line
% length is accordingly scaled:
%    \begin{macrocode}
    \ifdim\d@mX\p@=\z@\else
      \edef\sc@lelen{\fpeval{1 / abs(\d@mX)}}\relax
      \@linelen=\sc@lelen\@linelen
    \fi
%    \end{macrocode}
% Of course, if the line is vertical this division must not take place.
% 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 PDF language commands instead of resorting to the DVI low level
% language that was used in both \texttt{pict2e} and the original
% (pre 1994) \texttt{picture} commands; it had a meaning in the old times,
% but it certainly does not have any, since lines are drawn by the driver
% that produces the output in a human visible document form, not by \TeX\
% the program.
%    \begin{macrocode}
    \moveto(0,0)
    \pIIe@lineto{\d@mX\@linelen}{\d@mY\@linelen}%
    \strokepath
  \fi
\endgroup\ignorespaces}%
%    \end{macrocode}
% The new definition of the command |\line|, besides the ease with which is
% readable, does not do different things from the definition of |pict2e|
% 2009, but it did perform in a better way compared to the 2004 version
% that was limited to integer direction coefficients up to 999 in
% magnitude. In any case this |curve2e| version accepts polar coordinates
% as slope couples, making it much simpler to draw lines with specific
% slopes.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Dashed and dotted lines}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dashed and dotted lines are very useful in technical drawings; here we
% introduce four macros that help drawing them in the proper way; besides
% the obvious difference between the use of dashes or dots, they may refer
% in a different way to the end points that must be specified to the
% various macros. 
%
% The coordinates of the first point $P_1$, where le line starts, are
% always referred to the origin of the coordinate axes; the end point $P_2$
% coordinates with the first macro type are referred to the origin of the
% axes, while with the second macro type they are referred to $P_1$; both
% macro types have their usefulness and figures~\ref{fig:dashedlines} on
% page~\pageref{fig:dashedlines} and~\ref{fig:dottedlines} on
% page~\pageref{fig:dottedlines} show how to use these macros.
%
% We distinguish these macros with an asterisk; the unstarred version is
% the first macro type, while the starred one refers to the second macro
% type.
%
% The above mentioned macros create dashed lines between two given
% points, with a dash length that must be specified, or dotted lines, with
% a dot gap that must be specified; actually the specified dash length or
% dot gap is a desired one; the actual length or gap is computed by integer
% division between the distance of the given points and the desired dash
% length or dot gap; when dashes are involved, this integer is tested in
% order to see if it is an odd number; if it's not, it is increased by
% unity. Then the actual dash length or dot gap is obtained by dividing the
% above distance by this number.
%
% Another vector $P_2-P_1$ is created by dividing it by this number;
% then, when dashes are involved, it is multiplied by two in order to
% have the increment from one dash to the next; finally the number of
% patterns is obtained by integer division of this number by 2 and
% increasing it by~1.
% A simple |\multiput| completes the job, but in order to use the various
% vectors and numbers within a group and to throw the result outside the
% group while restoring all the intermediate counters and registers, a
% service macro is created with an expanded definition and then this
% service macro is executed.
% Figure~\ref{fig:dashedlines} on page~\pageref{fig:dashedlines} shows
% the effect of the slight changing of the dash length in order to
% maintain approximately the same dash-space pattern along the line,
% irrespective of the line length. The syntax is the following:
% \begin{flushleft}
% \cs{Dashline}\meta{\texttt{*}}\parg{first point}\parg{second point}\marg{dash length}
% \end{flushleft}
% where \meta{first point} contains the coordinates of the starting point
% and \meta{second point} those of the ending point; of course the
% \meta{dash length}, which equals the dash gap, is mandatory. The
% asterisk plays a specific role; in facts, if coordinates are specified
% in polar form, without the optional asterisk the dashed line is
% misplaced, while if the asterisk is specified, the whole object is put
% in the proper position. On the opposite, if the coordinates are in
% cartesian form the \meta{first point} coordinates play the role they
% are supposed to do even without the asterisk.
%    \begin{macrocode}
\ifx\Dashline\undefined
  \def\Dashline{\@ifstar{\Dashline@@}{\Dashline@}}

  \def\Dashline@(#1)(#2)#3{%
  \bgroup
     \countdef\NumA3254\countdef\NumB3252\relax
     \GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttA
     \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttB
     \SubVect\V@ttA from\V@ttB to\V@ttC
     \ModOfVect\V@ttC to\DlineMod
     \DivideFN\DlineMod by#3 to\NumD
     \NumA=\fpeval{trunc(\NumD,0)}\relax
     \unless\ifodd\NumA\advance\NumA\@ne\fi
     \NumB=\NumA \divide\NumB\tw@
     \DividE\DlineMod\p@ by\NumA\p@ to\D@shMod
     \DividE\p@ by\NumA\p@ to \@tempa
     \MultVect\V@ttC by\@tempa,0 to\V@ttB
     \MultVect\V@ttB by 2,0 to\V@ttC
     \advance\NumB\@ne
     \edef\@mpt{\noexpand\egroup
     \noexpand\multiput(\V@ttA)(\V@ttC){\number\NumB}%
        {\noexpand\LIne(\V@ttB)}}%
     \@mpt\ignorespaces}%
  \let\Dline\Dashline
  
  \def\Dashline@@(#1)(#2)#3{\put(#1){\Dashline@(0,0)(#2){#3}}}
\fi
%    \end{macrocode}
%
% A simpler |\Dotline| macro can draw a dotted line between two given
% points; the dots are rather small, therefore the inter dot distance is
% computed in such a way as to have the first and the last dot at the
% exact position of the dotted-line end-points; again the specified dot
% distance is nominal in the sense that it is recalculated in such a way
% that the first and last dots coincide with the line end points. The
% syntax is as follows:
%\begin{flushleft}
%\cs{Dotline}\meta{\texttt{*}}\parg{start point}\parg{end point}\marg{dot distance}
%\end{flushleft}
%    \begin{macrocode}
\ifx\Dotline\undefined
  \def\Dotline{\@ifstar{\Dotline@@}{\Dotline@}}
  \def\Dotline@(#1)(#2)#3{%
  \bgroup
     \countdef\NumA 3254\relax \countdef\NumB 3255\relax
     \GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttA
     \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttB
     \SubVect\V@ttA from\V@ttB to\V@ttC
     \ModOfVect\V@ttC to\DotlineMod
     \DivideFN\DotlineMod by#3 to\NumD 
     \NumA=\fpeval{trunc(\NumD,0)}\relax
     \DivVect\V@ttC by\NumA,0 to\V@ttB                     
     \advance\NumA\@ne 
     \edef\@mpt{\noexpand\egroup
     \noexpand\multiput(\V@ttA)(\V@ttB){\number\NumA}%
        {\noexpand\makebox(0,0){\noexpand\circle*{0.5}}}}%
     \@mpt\ignorespaces}%
  
  \def\Dotline@@(#1)(#2)#3{\put(#1){\Dotline@(0,0)(#2){#3}}}%
\fi
%    \end{macrocode}
%
% Notice that vectors as complex numbers in their cartesian and polar forms
% always represent a point position referred to the origin of the axes;
% this is why in figures~\ref{fig:dashedlines} on
% page~\pageref {fig:dashedlines} and~\ref{fig:dottedlines} on
% page~\pageref{fig:dottedlines} the dashed and dotted line that depart
% from the lower right corner of the graph grid, and that use polar
% coordinates, have to be put at the proper position with the starred
% version of the commands that take care of the relative specification
% made with the polar coordinates.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Coordinate handling}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The new macro |\GetCoord| splits a vector (or complex number)
% specification into its components; in particular it distinguishes the
% polar from the cartesian form of the coordinates. The latter have the
% usual syntax \meta{x\texttt{,}y}, while the former have the syntax
% \meta{angle\texttt{:}radius}. The |\put| and |\multiput| commands are
% redefined to accept the same syntax; the whole work is done by
% |\SplitNod@| and its subsidiaries.
%
% Notice that package |eso-pic| uses |picture| macros in its definitions,
% but its original macro |\LenToUnit| is incompatible with  this
% |\GetCoord| macro; its function is to translate real lengths into
% coefficients to be used as multipliers of the current |\unitlength|; in
% case that the |eso-pic| had been loaded, at the |\begin{document}|
% execution the |eso-pic| macro is redefined using the e-\TeX\ commands
% so as to make it compatible with these local macros.\footnote{Thanks to
% Franz-Joseph Berthold who was so kind to spot the bug.}
%    \begin{macrocode}
\AtBeginDocument{\@ifpackageloaded{eso-pic}{%
\renewcommand\LenToUnit[1]{\strip@pt\dimexpr#1*\p@/\unitlength}}{}}%
%    \end{macrocode}
% The above redefinition is delayed at |\AtBeginDocument| in order to
% have the possibility to check the the |eso-pic| package had actually
% been loaded. Nevertheless the code is defined here just because the
% original |eso-pic| macro was interfering with the algorithms of
% coordinate handling.
%
% But let us come to the real subject of this section. We define a
% |\GettCoord| macro that passes control to the service macro with the
% expanded arguments; expanding arguments allows to use macros to name
% points, instead of explicit coordinates; with this version of |curve2e|
% this facility is not fully exploited, but a creative user can use this
% feature.  
%    \begin{macrocode}
\def\GetCoord(#1)#2#3{%
\expandafter\SplitNod@\expandafter(#1)#2#3\ignorespaces}
\def\SplitNod@(#1)#2#3{\isnot@polar#1:!!(#1)#2#3}%
%    \end{macrocode}
% The macro that detects the form of the coordinates is |\isnot@polar|;
% it examines the parameter syntax in order to see if it contains a
% colon; if it does, the coordinates are in polar form, otherwise they
% are in cartesian form. this macro uses delimited arguments, therefore
% low level definition syntax must be used.
%    \begin{macrocode}
\def\isnot@polar#1:#2!!{\def\@tempOne{#2}\ifx\@tempOne\empty
\expandafter\@firstoftwo\else
\expandafter\@secondoftwo\fi
{\SplitNod@@}{\SplitPolar@@}}

\def\SplitNod@@(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}%
\def\SplitPolar@@(#1:#2)#3#4{\DirFromAngle#1to\@DirA
\ScaleVect\@DirA by#2to\@DirA
\expandafter\SplitNod@@\expandafter(\@DirA)#3#4}
%    \end{macrocoe}
% In order to accept polar coordinates with |\put| and |\multiput| 
% we  resort to using  |\GetCoord|; therefore the redefinition of
% |\put| is very simple because it suffices to save the original
% meaning of that macro and redefine the new one in terms of the
% old one. 
%    \begin{macrocode}
\let\originalput\put
\def\put(#1){\bgroup\GetCoord(#1)\@tX\@tY
\edef\x{\noexpand\egroup\noexpand\originalput(\@tX,\@tY)}\x}
%    \end{macrocode}
% For |\multiput| it is mor complicated, because the increments from one
% position to the next cannot be done efficiently because the increments
% in the original definition are executed within boxes, therefore any 
% macro instruction inside these boxes is lost. It is a good occasion to
% modify the |\multiput|  definition by means of the advanced macro
% definitions provided by package |xparse|; we can add also some error
% messages for avoiding doing anything when son mandatory parameters are
% missing ore are empty, or do not contain anything different from an
% ordered pair or a polar form. We ad also an optional argument to
% handle the increments outside the boxes.
% The new macro has the following syntax:\\[2ex]
% \mbox{\cs{multiput}\texttt{(\meta{initial})}\texttt{(\meta{increment})}\marg{number}\marg{objext}\oarg{handler}}\\[2ex]
% where \meta{initial} contains the cartesian or polar coordinates
% of the initial point; \meta{increment} contains the cartesian or
% polar increment for the coordinates to be used from the second
% argument to the last; \meta{number} il the total number of points
% to be drawn; \meta{object} is the object to be put in position at
% each cycle repetition; the optional \meta{handler} may be used to
% control the current values of the horizontal and vertical increments.
% The new definition contains two |\put| commands where the second is
% nested within a while loop which in turn is within the argument of
% the first |\put| command. Basically it is the same idea that the
% original macros, but now the increment are computed within the While
% loop, bit outside the argument of the inner |\put| command. If the
% optional \meta{handler} is specified the increments are computed
% from the macros specified by the user. 
%
% The two increments components inside the optional argument may be set
% by means of mathematical expressions operated upon by the |\fpeval|
% function given by the |\xfp| package already loaded by |curve2e|. Of
% course it the user responsibility to pay attention to the scales of
% the two axes and to write meaningful expressions; the figure and code
% shown in the first part of this documentation show some examples:
% see page~\pageref{pag:multiput}.
%    \begin{macrocode}
\RenewDocumentCommand{\multiput}{ d() d() m m o }{%
\IfNoValueTF{#1}{\PackageError{curve2e}{%
  \string\multiput\space initial point coordinates missing}%
  {Nothing done}}%
  {\IfNoValueTF{#2}{\PackageError{curve2e}{%
    \string\multiput\space Increment components missing}%
    {Nothing done}%
  }%
  {\GetCoord(#2)\dX\dY
    \put(#1){\def\X{0}\def\Y{0}\@multicnt=#3\relax
      \@whilenum \@multicnt > \z@\do{%
        \put(\X,\Y){#4}\IfValueTF{#5}{#5}{%
          \edef\X{\fpeval{\X+\dX}}\edef\Y{\fpeval{\Y+\dY}}}%
          \advance\@multicnt\m@ne
       }%
    }}
  }%
}
%    \end{macrocode}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Vectors}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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} 2004  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; with |pict2e| 2009,
% possibly this redefinition of |\vector| is not necessary, but we do it
% as well and for the same reasons we had for redefining |\line|; 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 styled arrow tip whose parameters are specified
% by default or may be taken from the parameters taken from the|PSTricks|
% package if this one is loaded before |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 |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|, |\VECTOR|, and
% |\VVECTOR| to be defined in a while.
%    \begin{macrocode}
      \ifdim\@linelen<\z@ \@linelen=-\@linelen\fi
%    \end{macrocode}
% We now make a vector with the slope coefficients 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} we need to
% transform the components of the vector direction in lengths with the
% specific names |\@xdim| and |\@ydim|^^A--------! Necessario?
%    \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.
% We use the floating point division function; since it yields a ``factor''
% We directly use it to scale the length of the vector. I finally memorise
% the true vector length in the internal dimension |@tdB|
%    \begin{macrocode}
        \ifdim\d@mX\p@=\z@
        \else\ifdim\d@mY\p@=\z@
          \else
            \edef\sc@lelen{\fpeval{1 / abs(\@xnum)}}\relax
            \@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
        \fillpath
%    \end{macrocode}
% Now we can restore the stem length that must be shortened by the
% dimension of the arrow; by 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 avoid designing such a stem.
%    \begin{macrocode}
      \@linelen=\@tdB
      \@tdA=\pIIe@FAW\@wholewidth
      \@tdA=\pIIe@FAL\@tdA
      \advance\@linelen-\@tdA
      \ifdim\@linelen>\z@
        \moveto(0,0)
        \pIIe@lineto{\@xnum\@linelen}{\@ynum\@linelen}%
        \strokepath\fi
    \endgroup}
%    \end{macrocode}
%
% 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. The object defined with |\Vector|, as well as |\vector|,
% must be put in place by means of a |\put| command.
%    \begin{macrocode}
\def\Vector(#1){{%
\GetCoord(#1)\@tX\@tY
\ifdim\@tX\p@=\z@
  \vector(\@tX,\@tY){\@tY}%
\else
  \vector(\@tX,\@tY){\@tX}%
\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 of 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 double tipped vector is built on  the |\VECTOR| macro by simply
% drawing two vectors from the middle point of the double tipped vector.
%    \begin{macrocode}
\def\VVECTOR(#1)(#2){{\SubVect#1from#2to\@tempb
\ScaleVect\@tempb by0.5to\@tempb
\AddVect\@tempb and#1to\@tempb
\VECTOR(\@tempb)(#2)\VECTOR(\@tempb)(#1)\ignorespaces}}
%    \end{macrocode}
%
% The \texttt{pict2e} documentation says that if the vector length is zero
% the macro draws 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. See examples in
% figure~\ref{fig:vectors} on page~\pageref{fig:vectors}.
%
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Polylines and polygons}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% We now define the polygonal line macro; its syntax is very simple:
% \begin{flushleft}
% \cs{polygonal}\oarg{join}\texttt{(}$P_0$\texttt{)(}$P_1$\texttt{)(}$P_2$)%
%  \texttt{\dots(}$P_n$\texttt{)}
% \end{flushleft}
% Remember: |\polyline| has been incorporated into |pict2e| 2009, but we
% redefine it so as to allow an optional argument to specify the line join
% type.
%
% 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|, besides setting the line joints, 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,
% and not necessarily is put in position through a |\put| command that
% provides to eliminate any spurious spaces preceding this command.
%
%  \begin{figure}[!hb]
%  \begin{minipage}{0.55\textwidth}
%\begin{verbatim}
%  \unitlength=0.07\hsize
%  \begin{picture}(8,8)(-4,-4)\color{red}
%  \polygon*(45:4)(135:4)(-135:4)(-45:4)
%  \end{picture}
%\end{verbatim}
%  \end{minipage}
%  \hfill
%  \begin{minipage}{0.4\textwidth}\centering
%  \unitlength=0.07\hsize
%  \begin{picture}(8,8)(-4,-4)\color{red}
%  \polygon*(45:4)(135:4)(-135:4)(-45:4)
%  \end{picture}
%  \end{minipage}
%  \caption{The code and the result of defining a polygon with its vertex
%  polar coordinates}\label{fig:filled-polygon}
%  \end{figure}
%  
% In order to allow a specification for the joints of the various segments 
% of a polyline it is necessary to allow for an optional parameter;
% the default is the bevel join.
%    \begin{macrocode}
\renewcommand*\polyline[1][\beveljoin]{\p@lylin@[#1]}

\def\p@lylin@[#1](#2){\@killglue#1\GetCoord(#2)\d@mX\d@mY
    \pIIe@moveto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\p@lyline}{%
    \PackageWarning{curve2e}%
    {Polylines require at least two vertices!\MessageBreak
     Control your polyline 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}{\strokepath\ignorespaces}}
%    \end{macrocode}
%
% The same treatment must be done for the \cs{polygon} macros; we use the
% defining commands of package |xparse|, in order to use an optional
% asterisk; as it is usual with |picture| convex lines, the command with
% asterisk does not trace the contour, but fills the contour with the
% current color.The asterisk is tested at the beginning and, depending on
% its presence, a temporary switch is set to \texttt{true}; this being the
% case the contour is filled, otherwise it is simply stroked.
%    \begin{macrocode}
\providecommand\polygon{} 
\RenewDocumentCommand\polygon{s O{\beveljoin} }{\@killglue\begingroup
\IfBooleanTF{#1}{\@tempswatrue}{\@tempswafalse}%
\@polygon[#2]} 
 
\def\@polygon[#1](#2){\@killglue#1\GetCoord(#2)\d@mX\d@mY
    \pIIe@moveto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\@@polygon}{%
    \PackageWarning{curve2e}%
    {Polygons require at least two vertices!\MessageBreak
     Control your polygon specification\MessageBreak}%
    \ignorespaces}}
      
  \def\@@polygon(#1){\GetCoord(#1)\d@mX\d@mY
    \pIIe@lineto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\@@polygon}{\pIIe@closepath
      \if@tempswa\pIIe@fillGraph\else\pIIe@strokeGraph\fi
      \endgroup
      \ignorespaces}}
%    \end{macrocode}
% Now, for example, a filled polygon can be drawn using polar coordinates
% for its vertices; see figure~\ref{fig:filled-polygon} on
% page~\pageref{fig:filled-polygon}.
%
% Remember; the polygon polar coordinates are relative to the origin of
% the local axes; therefore in order to position a polygon in a different
% position, it is necessary to do it through a |\put| command.

%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{The red service grid}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The next command is very useful for debugging while editing one's
% drawing;  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 that
% are multiples of 10. Missing to do so the readings become cumbersome.
% The |\RoundUp| macro provides to increase the
% grid dimensions to integer multiples of ten.
%    \begin{macrocode}
\def\GraphGrid(#1,#2){\bgroup\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}%
\egroup\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}
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Math operations on fractional operands}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This is not the place to complain about the fact that all programs of the
% \TeX\ system use only integer arithmetics; luckily enough, in 218 the
% package |xfp| was released: that package resorts in the background
% language \LaTeX\,3; this language now can compute fractional number
% operations coded in decimal digits and accepts also numbers written in
% the usual way in computer science, that is as a fractional, possibly
% signed, number followed by an expression that contains the  exponent to
% 10 necessary to (ideally) move the fractional separator  in one or the
% other direction according to the sign of the exponent of 10; in other
% words  the L3 library for floating point calculations accepts such
% expressions as  \texttt{123.456}, \texttt{0.12345e3}, and
% \texttt{12345e-3}, and any other equivalent expression. If the first
% number is integer, it assumes that the decimal separator is to the
% right of the rightmost digits of the digit string.
%
% Floating pint calculations may be done through the |\fpeval| L3
% function with a very simple syntax:
%  \begin{flushleft}
%  \cs{fpeval}\marg{mathematical expression}
%  \end{flushleft}
% where \meta{mathematica exression} can contain the usual algebraic
% operation sings, |= - * / ** ^| and the function names of the most
% common algebraic, trigonometric, and transcendental functions; for direct
% and inverse trigonometric functions it accepts arguments in radians and
% in sexagesimal degrees; it accepts the group of rounding/truncating
% operators; it can perform several kinds of comparisons;  as to this date
% the todo list includes the direct and inverse hyperbolic functions. The
% mantissa length of the floating point operands amounts to 16 decimal
% digits. Further details may be read in the documentations of the |xfp|
% and |interface3|  documents, just by typing into a command line window
% the command \texttt{texdoc \meta{document}}, where \meta{document} is
% just the name of the above named files without extension.
%
% Before the availability of the |xfp| package, it was necessary to fake
% fractional number computations by means of the native e-\TeX\ commands
% |\dimexpr|, i.e. to multiply each fractional number by the unit |\p@|
% (1\,pt) so as to get a length; operate un such lengths, and then
% stripping of the `pt' component from the result; very error prone and
% with less precision as the one that the modern decimal floating point
% calculations can do. Of course it is not so important to use fractional
% numbers with more that 5 or 6 fractional digits, because the other
% \TeX\ and \LaTeX\ macros cannot handle them, but it is very convenient
% to have simpler and more readable code. We therefore switched to the
% new floating point functionality, even if this maintains the |curve2e|
% functionality, but renders this package unusable with older \LaTeX\ 
% kernel installations. It has already been explained that the input of
% |curve2e| gets aborted if the |xfp| package is not available.
%
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{The division macro}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% The most important macro is the division of two fractional numbers; we
% seek a macro that gets dividend and divisor as fractional numbers and
% saves their ratio in a macro; this is done in a simple way with the
% following code.
%    \begin{macrocode}
\def\DividE#1by#2to#3{%
  \edef#3{\fpeval{#1 / #2}}\relax
}
%    \end{macrocode}
% In order to avoid problems with divisions by zero, or with numbers that
% yield results to large to be used as multipliers of lengths, it would
% be preferable that the above code be preceded or followed by some tests
% and possible messages. Actually we decided to avoid such tests and
% messages, because the internal L3 functions already provide some. This
% is what it was done in the previous versions of this package, when the
% |\fpeval| L3 function was not available.
%
% Notice that operands |#1| and |#2| may be integer numbers or
% fractional, or mixed numbers. They may be also dimensions, but while
% dimensions in printer points (72.27pt=1in) are handled as assumed, when
% different units are used, the length  must be enclosed in parentheses: 
%\begin{verbatim}
%\DividE(1mm)by(3mm) to\result
%\end{verbatim}
% yields correctly |\result=0.33333333|. Without parentheses the result
% is unpredictable.
%
% For backward compatibility we need an alias.
%    \begin{macrocode}
\let\DivideFN\DividE
%    \end{macrocode}
%
% We do the same in order to multiply two integer o fractional numbers
% held in the first two arguments and the third argument is a definable
% token that will hold the result of multiplication in the form of a
% fractional number, possibly with a non null fractional part; a null
% fractional part is stripped away
%    \begin{macrocode} 
\def\MultiplY#1by#2to#3{\edef#3{\fpeval{#1 * #2}}}\relax
\let\MultiplyFN\MultiplY
%    \end{macrocode}
% but with multiplication it is better to avoid computations with
% lengths.
%
% The next macro uses the \verb|\strip@pt| \LaTeX\ kernel macro to get
% the numerical value of a measure in points. 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}
\unless\ifdefined\Numero
  \def\Numero#1#2{\bgroup\dimen3254=#2\relax
    \edef\x{\noexpand\egroup\noexpand\edef\noexpand#1{%
      \strip@pt\dimen3254}}\x\ignorespaces}%
\fi
%    \end{macrocode}
% The \verb|\ifdefined| primitive command is provided by the e-\TeX\
% extension of the typesetting engine; the test does not create any hash
% table entry; it is a different way than the
% \verb|\ifx\csname ...\endcsname| test,
% because the latter first possibly creates a macro meaning \verb|\relax|
% then executes the test; therefore an undefined macro name is always
% defined to mean |\relax|.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Trigonometric functions}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  We now start with trigonometric functions. In previous versions of this
% package we defined the macros |\SinOf|, |\CosOf| and |\TanOf| (|\CotOf|
% does not appear so essential) by means of the parametric
% formulas that require the knowledge of the tangent of the half angle.
% We wanted, and still want, to specify the angles in sexagesimal degrees,
% not in radians, so that accurate reductions to the main quadrants are
% possible. The formulas are
% \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.
%
% But now, in this new version, the availability of the floating point
% computations with the specific L3 library makes al the above superfluous;
% actually the above approach gave good results but it was cumbersome and
% limited by the fixed radix computations of the \TeX\ system programs.
% Matter of facts, we compared the results (with 6 fractional digits) the
% computations executed with the \texttt{sind} function name, in order to
% use the angles in degrees, and a table of trigonometric functions with
% the same number of fractional digits, and we di not find and difference,
% not even one unit on the sixth decimal digit. Probably the |\fpeval|
% computations, without rounding before the sixteenth significant digit,
% are much more accurate, but it is useless to have a better accuracy when
% the other \TeX\ and \LaTeX\ macros would not be able to exploit them.
%
% Having available such powerful instrument, even the tangent appears to
% be of little use for the kind of computations that are supposed to be
% required in this package.
%
% The codes for the computation of |\SinOf| and |\CosOf| of the angle in
% degrees is therefore the following
%    \begin{macrocode}
\def\SinOf#1to#2{\edef#2{\fpeval{round(sind#1,6)}}}\relax
\def\CosOf#1to#2{\edef#2{\fpeval{round(cosd#1,6)}}}\relax
%    \end{macrocode}
%
%
% As of today the anomaly (angle) of a complex number may not be necessary,
% but it might become useful in the future; therefore with macro
% \verb|\ArgOfVect| we calculate the four quadrant arctangent (in degrees)
% of the given vector taking into account the sings of the vector
% components. The |\ArgOfVect| macro receives on input a vector;
% from the signs of the horizontal and vertical components it determines
% the ratio and from this ratio the arctangent; but before doing this it
% tests the components in order to determine the quadrant of the vector
% tip; depending on signs it possibly adds what is necessary to determine
% the angle in the range $-180^\circ < \varphi \le  +180^\circ$. If both
% components are zero, the angle is undefined, but for what concerns
% |curve2e| it is assigned the angle $0^circ$.
%    \begin{macrocode}
\def\ArgOfVect#1to#2{\bgroup\GetCoord(#1){\t@X}{\t@Y}%
\def\s@gno{}%
\ifdim\t@X\p@=\z@
  \ifdim\t@Y\p@=\z@
    \def\ArcTan{0}% vettore nullo
  \else
    \def\ArcTan{90}% vettore verticale
    \ifdim\t@Y\p@<\z@\def\ArcTan{-90}\fi
  \fi
\else
  \ifdim\t@Y\p@=\z@% vettore orizzontale
    \ifdim\t@X\p@<\z@
      \def\ArcTan{180}%
    \else
      \def\ArcTan{0}%
    \fi
  \else % vettore qualsiasi
    \edef\ArcTan{\fpeval{atand(\t@Y / \t@X)}}\relax
    \ifdim\t@X\p@<\z@% vettore nei quadranti di sinistra
      \ifdim\t@Y\p@<\z@
        \edef\ArcTan{\fpeval{\ArcTan - 180}}\relax
      \else
        \edef\ArcTan{\fpeval{\ArcTan + 180}}\relax
      \fi
    \fi
  \fi
\fi
\edef\x{\noexpand\egroup\noexpand\edef\noexpand#2{\ArcTan}}%
\x\ignorespaces}
%    \end{macrocode}
%
% It is worth examining the following table, where the angles of nine
% vectors $45^\circ$ degrees apart from one another are computed from
% this macro.
% \begin{center}
% \begin{tabular}{l*9r}
% Vector &0,0 &1,0 &1,1 & 0,1 & -1,1& -1,0&-1,-1&0,-1&1,-1\\
% Angle  & 0  & 0  & 45 & 90  & 135 & 180 &-135 & -90& -45
% \end{tabular}
% \end{center}
% Real computations with the |\ArgOfVect| macro produce those very numbers 
% without the need of rounding; |\fpeval| produces all trimming of lagging
% zeros and rounding by itself.
%
%
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Arcs and curves preliminary information}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 arc has its
% reference point in its center, therefore it does not need to be put in
% place by the command |\put|; nevertheless if |\put| is used, it may
% displace the arc into another position.
%
% The command should have the following syntax:
% \begin{flushleft}\ttfamily
% \cs{Arc}(\meta{center})(\meta{starting point})\marg{angle}
% \end{flushleft}
% which is totally equivalent to:
% \begin{flushleft}\ttfamily
% \cs{put}(\meta{center})\marg{\upshape\cs{Arc}(0,0)(\meta{starting point})\marg{angle}}
% \end{flushleft}
% If the \meta{angle}, i.e. the arc angular aperture, is positive the arc
% runs counterclockwise from the starting point; clockwise if it's
% negative.
% Notice that since the \meta{starting point} is relative to the
% \meta{center} point, its polar coordinates are very convenient, since
% they become \parg{\meta{start angle}:\meta{radius}}, where the
% \meta{start angle} is relative to the arc center. Therefore you can think
% about a syntax such as this one:
%\begin{flushleft}
%\cs{Arc}\parg{\meta{center}}\parg{\normalfont{\itshape start angle}\texttt{:}{\itshape radius}}\marg{angle}
%\end{flushleft}
%
% The difference between the |pict2e| |\arc| definition  consists in a very
% different syntax:
%\begin{flushleft}
%\cs{arc}\texttt{[}\meta{start angle}\texttt{,}\meta{end angle}\texttt{]}\marg{radius}
%\end{flushleft}
% and the center is assumed to be at the coordinate established with a
% required |\put| command; moreover the difference in specifying angles
% is that \meta{end angle} equals the sum of \meta{start angle} and
% \meta{angle}. With the definition of this |curve2e| package
% use of a |\put| command is not prohibited, but it may be used for fine
% tuning the arc position by means of a simple displacement; moreover the
% \meta{starting point} may be specified with polar coordinates (that are
% relative to the arc center). 
%
% 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 roto-amplification operators.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Complex number macros}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% In this package \emph{complex number} is a vague phrase; it may be used
% in the mathematical sense of an ordered pair of real numbers; it can be
% viewed as a vector joining the origin of the coordinate axes to the
% coordinates indicated by the ordered pair; it can be interpreted as a
% roto-amplification operator that scales its operand and rotates it
% about a pivot point; besides the usual conventional representation used
% by the mathematicians where the ordered pair is enclosed in round
% parentheses (which is in perfect agreement with the standard code used
% by the |picture| environment) there is the other conventional
% representation used by the engineers that stresses the
% roto-amplification nature of a complex number:
%\[
%(x, y) = x + \mathrm{j}y =M \mathrm{e}^{\mathrm{j}\theta}
%\]
% Even the imaginary unit is indicated with $\mathrm{i}$ by the
% mathematicians and with $\mathrm{j}$ by the engineers. In spite of
% these differences, these objects, the \emph{complex numbers}, are used
% without any problem by both mathematicians and engineers.
% 
% The important point is that these objects can be summed, subtracted,
% multiplied, divided, raised to any power (integer, fractional, positive
% or negative), be the argument of transcendental functions according to
% rules that are agreed upon by everybody. We do not need all these
% properties, but we need some and we must create the suitable macros for
% doing some of these operations.
%
% In facts we need macros for summing, subtracting, multiplying, dividing
% complex numbers, for determining their directions (unit vectors); a
% unit vector is the complex number divided by its magnitude so that the
% result is the cartesian or polar form of the Euler's formula
% \[
% \mathrm{e}^{\mathrm{j}\phi} = \cos\phi+\mathrm{j}\sin\phi
% \]
%
% The magnitude of a vector is 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.
%
% In the preceding version of package |curve2e| the magnitude $M$ was
% determined by taking the moduli of the real and imaginary parts, by
% changing their signs if necessary; the larger component was
% 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:
% \[
% M = \sqrt{a^2+b^2} = \vert a\vert\sqrt{1+(b/a)^2}
% \]
% In this way the radicand never exceeds 2 and it was quite easy to get
% its square root by means of the Newton iterative process; due to the
% quadratic convergence, five iterations were more than sufficient. When
% one of the components was zero, the Newton iterative process was
% skipped.
% With the availability of the |xfp| package and its floating point
% algorithms it is much easier to compute the magnitude of a complex
% number; since these algorithms allow to use very large numbers, it is
% not necessary to normalise the complex number components to the largest
% one; therefore the code is much simpler than the one used for
% implementing the Newton method in the previous versions of this
% package.
%    \begin{macrocode}
\def\ModOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
\edef#2{\fpeval{sqrt(\t@X*\t@X + \t@Y*\t@Y)}}\relax
\ignorespaces}%
%    \end{macrocode}
%
% Since the macro for determining the magnitude of a vector is available,
% we can now normalise 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 normalisation.
%    \begin{macrocode}
\def\DirOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
\ModOfVect#1to\@tempa
\unless\ifdim\@tempa\p@=\z@
   \DividE\t@X by\@tempa to\t@X
   \DividE\t@Y by\@tempa to\t@Y
\fi
\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
%    \end{macrocode}
%
% A cumulative macro uses the above ones to determine 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 normalised 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%
\DirOfVect#1to#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\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\ignorespaces}%
%    \end{macrocode}
% With the next macro we create a direction vector (second argument) from 
% a given angle (first argument, in degrees).
%    \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
\edef\t@X{\fpeval{#2 * \t@X}}\relax
\edef\t@Y{\fpeval{#2 * \t@Y}}\relax
\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
\edef\t@Y{-\t@Y}%
\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 
\edef\t@X{\fpeval{\tu@X + \td@X}}\relax
\edef\t@Y{\fpeval{\tu@Y + \td@Y}}\relax
\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 
\edef\t@X{\fpeval{\td@X - \tu@X}}\relax
\edef\t@Y{\fpeval{\td@Y - \tu@Y}}\relax
\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 we
% could not find a simple means for doing so\footnote{Actually the simple
% means do exist now through the use of the \texttt{xparse} package, but
% they became available only when the syntax of this
% \texttt{\string\MultVect} macro had been already in use for a long
% time; changing the position of the asterisk at this moment, would mean
% to render the new definition incompatible with the existing source
% document files.}; therefore we use the prefixed notation, that is we
% put the asterisk \emph{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 
\edef\t@X{\fpeval{\tu@X * \td@X - \tu@Y * \td@Y}}\relax
\edef\t@Y{\fpeval{\tu@Y * \td@X + \tu@X * \td@Y}}\relax
\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 
\edef\t@X{\fpeval{\tu@X * \td@X + \tu@Y * \td@Y}}\relax
\edef\t@Y{\fpeval{\tu@Y * \td@X - \tu@X * \td@Y}}\relax
\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 conjugate direction of the divisor;
%\[
% \frac{\vec{N}}{\vec{D}}= \frac{\vec{N}}{M\vec{u}}=
% \frac{\vec{N}}{M}\vec{u}^{\mkern2mu\star}
%\] 
%therefore:
%    \begin{macrocode}
\def\DivVect#1by#2to#3{\ModAndDirOfVect#2to\@Mod and\@Dir
\edef\@Mod{\fpeval{1 / \@Mod}}\relax
\ConjVect\@Dir to\@Dir
\ScaleVect#1by\@Mod to\@tempa
\MultVect\@tempa by\@Dir to#3\ignorespaces}%
%    \end{macrocode}
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Arcs and curved vectors}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% We are now in the position of really doing graphic work.
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsubsection{Arcs}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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@ 
\unless\ifdim\@tdA=\z@
  \@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 memorised 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
% starting 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 draw 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\strokepath\ignorespaces}%
%    \end{macrocode}
% And the new macro |\@@Arc| starts with moving the drawing point to the
% first% point and does everything needed for drawing 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. 
%
% With reference to figure~\ref{fig:arcspline} on
% page~\pageref{fig:arcspline}, the points $P_1$ and $P_2$
% are the arc end-points; $C_1$ and $C_2$ are the Bézier-spline
% control-points; $P$ is the arc mid-point, that should be distant from
% the center of the arc the same as $P_1$ and $P_2$.
% Choosing a convenient orientation of the arc relative to the coordinate
% axes, the coordinates of these five points are:
%\begin{align*}
%P_1 &= (-R\sin\theta, 0)\\
%P_2 &= (R\sin\theta, 0)\\
%C_1 &= (-R\sin\theta+K\cos\theta, K\sin\theta)\\
%C_2 &= (R\sin\theta-K\cos\theta, K\sin\theta)\\
%P   &= (0, R(1-\cos\theta))
%\end{align*}
% The Bézier cubic spline interpolating the end and mid points is given by
% the parametric equation:
%\begin{equation*}
%P= P_1(1-t)^3 + C_1 3(1-t)^2t + C_2 3(1-t)t^2 + P_2t^3
%\end{equation*}
% where the mid point is obtained for $t=0.5$; the four coefficients then
% become $1/8, 3/8, 3/8, 1/8$ and the only unknown remains $K$. Solving
% for $K$ we obtain the formula
% \begin{equation}\label{equ:corda}
% K= \frac{4}{3}\,\frac{1-\cos\theta}{\sin\theta}R
%= \frac{4}{3}\,\frac{1-\cos\theta}{\sin^2\theta}s
% \end{equation}
% where $\theta$ is half the arc aperture, $R$ is its radius, and $s$ is
% half the arc chord.
%    \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}
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsubsection{Arc vectors}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 \LaTeX\ or PostScript arrows according to the specific
% option to the \texttt{pict2e} package.
%
% But the arc drawing done here shortens it so as not to overlap on the
% arrow(s); the only arrow (or both ones) 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 finally (h) drawing the
% 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 we did not try to optimise 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
\strokepath\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=\if\Segno--\fi\@gradi\p@ \Numero\@tempa\@tdD
\DirFromAngle\@tempa to\@Dir
\MultVect\@V by\@Dir to\@sPun% corrects the end point
\edef\@tempA{\if\Segno--\fi1}%
\MultVect\@sPun by 0,\@tempA to\@vPun
\DirOfVect\@vPun to\@Dir
\AddVect\@sPun and #1 to \@sPun
\GetCoord(\@sPun)\@tdX\@tdY
\@tdD\if\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}}% end point arrowt ip
\@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{\if\Segno-\else-\fi\@ne}%
\MultVect\@V by0,\@tempa to\@vPun
\@tdE\if\Segno--\fi\DeltaGradi\p@
\Numero\@tempB{0.5\@tdE}%
\DirFromAngle\@tempB to\@Dird
\MultVect\@vPun by\@Dird to\@vPun% corrects the starting point
\DirOfVect\@vPun to\@Dir\GetCoord(\@Dir)\@xnum\@ynum
\put(\@pPunX,\@pPunY){\vector(\@xnum,\@ynum){0}}% starting point arrow tip
\edef\@tempa{\if\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
\strokepath\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.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{General curves}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The most used method to draw curved lines with computer programs is to
% connect several simple curved lines, general ``arcs'', one to another
% generally maintaining the same tangent at the junction. If the direction
% changes we are dealing with a cusp.
%
% The simple general arcs that are directly implemented in every program
% that display typeset documents, are those drawn with the parametric
% curves called \emph{Béźier splines}; given a sequence of points in the
% $x,y$ plane, say $P_0, P_1, P_2, p_3, \dots$ (represented as coordinate
% pairs, i.e. by complex numbers), the most common Bézier splines are the
% following ones:
% \begin{align}
% \mathcal{B}_1 &= P_0(1-t) + P_1t               \label{equ:B-1}      \\
% \mathcal{B}_2 &= P_0(1-t)^2 + P_1 2(1-t)t + P_2t^2 \label{equ:B-2}  \\
% \mathcal{B}_3 &= P_0(1-t)^3 + P_1 3(1-t)^2t +P_2 3(1-t)t^2 +P_3t^3
% \label{equ:B-3}
% \end{align}
%
% All these splines depend on parameter $t$; they have the property that
% for $t=0$ each line starts at the first point, while for $t=1$ they
% reach the last point; in each case the generic point $P$ on each curve
% takes off with a direction that points to the next point, while it
% lands on the destination point with a direction coming from the 
% penultimate point; moreover, when $t$ varies from 0 to 1, the curve arc
% is completely contained within the convex hull formed by the polygon that
% has the spline points as vertices.
%
% Last but not least first order splines implement just straight lines
% and they are out of question for what concerns maxima, minima,
% inflection points and the like. Quadratic splines draw just
% parabolas, therefore they draw arcs that have the concavity just on one
% side of the path; therefore no inflection points. Cubic splines are
% extremely versatile and can draw lines with maxima, minima and inflection
% points. Virtually a multi-arc curve may be drawn by a set of cubic 
% splines as well as a set of quadratic splines (fonts are a good example:
% Adobe Type~1 fonts have their contours described by cubic splines, while
% TrueType fonts have their contours described with quadratic splines;
% with a naked eye it is impossible to notice the difference).
%
% Each program that processes the file to be displayed is capable of
% drawing first order Bézier splines (segments) and third order Bézier
% splines, for no other reason, at least, because they have to draw vector
% fonts whose contours are described by Bézier splines; sometimes they
% have also the program commands to draw second order Bézier splines, but
% not always these machine code routines are available to the user for
% general use. For what concerns |pdftex|, |xetex| and |luatex|, they have
% the user commands for straight lines and cubic arcs. At least with
% |pdftex|, quadratic arcs must be simulated with a clever use of third
% order Bézier splines.
%
% Notice that \LaTeXe\ environment |picture| by itself is capable of
% drawing both cubic and quadratic Bézier splines as single arcs; but it
% resorts to ``poor man'' solutions. The |pict2e| package removes all the
% old limitations and implements the interface macros for sending the
% driver the necessary drawing information, including the transformation
% from typographical points (72.27\,pt/inch) to PostScript big points
% (72\,bp/inch). But for what concerns the quadratic spline it resorts to
% the clever use of a cubic spline.
%
% Therefore here we treat first the drawings that can be made with cubic
% splines; then we describe the approach to quadratic splines.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Cubic splines}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 of
% |pict2e| 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\ignorespaces
}%
%    \end{macrocode}
%
% Actually the above macro is a special case of concatenation of the
% triplet formed by macros |\StartCurve|, |\CurveTo| and|\CurveFinish|;
% the second macro can be repeated an arbitrary number of times.
% In any case the directions specified with the direction arguments, both
% here and with the more general macro|\Curve|, the angle between the
% indicated tangent and the arc chord may give raise to some little
% problems when they are very close to 90° in absolute value. Some control
% is exercised on these values, but some tests might fail if the angle
% derives from computations; this is a good place to use polar forms for
% the direction vectors.
%
% 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 memorises the starting direction; the second macro
% traces the current Bézier arc reaching the destination point with the
% specified direction, but memorises 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 memorised 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 direction vectors 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 normalisation
% and memorization.
%
% The next desirable feature would be to design a macro that accepts
% optional node directions and computes the missing ones according to a
% suitable strategy. We 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 initialising macro that receives with the
%  first argument the starting point and with the second argument the
% direction of the tangent (not necessarily normalised 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
\ignorespaces}
%    \end{macrocode}
% And this re-initializes the direction to create  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 macros are the finishing ones; the first strokes the whole
% curve, while the second fills the (closed) curve with the default color;
% both close the group that was opened with |\StartCurve|. The third macro
% is explained in a while; we anticipate it is functional to chose between
% the first two macros when a star is possibly used to switch between
% stroking and filling.
%    \begin{macrocode}
\def\CurveFinish{\strokepath\endgroup\ignorespaces}% 
\def\FillCurve{\fillpath\endgroup\ignorespaces}
\def\CurveEnd{\fillstroke\endgroup\ignorespaces}
%    \end{macrocode}
%
% In order to draw the internal arcs it would be desirable to have a single
% macro that, given the destination point, computes the control points that
% produce a cubic Bézier spline that joins the starting point with the
% destination point in the best possible way. The problem is strongly ill
% defined and has an infinity of solutions; here we give two solutions:
% $(a)$ a supposedly smart one that resorts to osculating circles and
% requires only the direction at the destination point; and $(b)$ a less
% smart solution that requires the control points to be specified in a
% certain format.
%
% We start with solution $(b)$, |\CbezierTo|, the code of which is simpler
% than that of solution $(a)$; then we will produce the solution $(a)$,
% |\CurveTo|,  that will become the main building block for a general path
% construction macro, |\Curve|. 
%
% The “naïve” macro |\CBezierTo| simply uses the previous point direction
% saved in |\@Dzero| as a unit vector by the starting macro; specifies
% a destination point, the distance of the first control point from the
% starting point, the destination point direction that will save also for
% the next arc-drawing macro as a unit vector, and the distance of the
% second control point from the destination point along this last
% direction. Both distances must be  positive possibly fractional numbers.
% The syntax therefore is the follwing:
%\begin{flushleft}
%\cs{CbezierTo}\meta{end
% point}|WithDir|\meta{direction}|AndDists|\meta{$K_0$}|And|\meta{$K_1$}
%\end{flushleft}
% where \meta{end point} is a vector macro or a comma separated pair of
% values; again \meta{direction} is another vector macro or a comma
% separated pair of values, that not necessarily indicate a unit vector,
% since the macro provides to normalise it to unity; \meta{$K_0$} and
% \meta{$K_1$} are the distances of the control points from their
% respective node points; they must be positive integers or fractional
% numbers. If \meta{$K_1$} is a number must be enclosed in curly braces,
% while if it is a macro name (containing the desired fractional or
% integer value) there is no need for braces.
%
% This macro uses the input information to use the internal |pict2e| macro
% |\pIIe@curveto| with the proper arguments, and to save the final
% direction into the same |\@Dzero| macro for successive use of other
% arc-drawing macros.
%    \begin{macrocode}
\def\CbezierTo#1WithDir#2AndDists#3And#4{%
\GetCoord(#1)\@tX\@tY \MakeVectorFrom\@tX\@tY to\@Puno
\GetCoord(#2)\@tX\@tY \MakeVectorFrom\@tX\@tY to \@Duno
\DirOfVect\@Duno to\@Duno
\ScaleVect\@Dzero by#3to\@Czero \AddVect\@Pzero and\@Czero to\@Czero
\ScaleVect\@Duno by-#4to \@Cuno \AddVect\@Puno and\@Cuno to \@Cuno
\GetCoord(\@Czero)\@XCzero\@YCzero
\GetCoord(\@Cuno)\@XCuno\@YCuno
\GetCoord(\@Puno)\@XPuno\@YPuno
\pIIe@curveto{\@XCzero\unitlength}{\@YCzero\unitlength}%
             {\@XCuno\unitlength}{\@YCuno\unitlength}%
             {\@XPuno\unitlength}{\@YPuno\unitlength}%
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}%
%    \end{macrocode}
%
% With this building block it is not difficult to set up a macro that draws
% a Bézier arc between two given points, similarly to the other macro
% |\CurveBetween| previously described and defined here:
%
%    \begin{macrocode}
\def\CbezierBetween#1And#2WithDirs#3And#4UsingDists#5And#6{%
\StartCurveAt#1WithDir{#3}\relax
\CbezierTo#2WithDir#4AndDists#5And{#6}\CurveFinish}
%    \end{macrocode}

%
% An example of use is shown in figure~\ref{fig:Cbezier} on
% page~\pageref{fig:Cbezier}; notice that the tangents at the end points
% are the same for the black curve drawn with |\CurveBetween| and the five
% red curves drawn with |\CbezierBetween|; the five red curves differ only
% for the distance of their control point $C_0$ from the starting  point;
% the differences are remarkable and the topmost curve even presents a
% slight inflection close to the end point. These effects cannot be
% obtained with the ``smarter'' macro |\CurveBetween|. But certainly this
% simpler macro is more difficult to use because the distances of the
% control points are difficult to estimate and require a number of
% cut-and-try experiments.
%
%
% The ``smarter'' 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 we 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, i.e. 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 equal parts
% each of which should be interpreted as half the chord of the osculating
% circle. 
%
% This makes the algorithm a little rigid; sometimes the path drawn is very
% pleasant, while in other circumstances the determined curvatures are too
% large or too small. We therefore add some optional information that lets
% us have some control over the curvatures; the idea is based on the
% concept of \emph{tension}, similar but not identical to the one used in
% the drawing programs \MF\ and \MP. We add to the direction information,
% with which the control nodes of the osculating circle arcs are
% determined, a scaling factor that should be intuitively related to the
% tension of the arc (actually, since the tension of the ‘rope’ is high
% when this parameter is low, probably a name such as ‘looseness’ would be
% better suited): the smaller this number, the closer the arc resembles
% a straight line as a rope subjected to a high tension; value zero is
% allowed, while a value of 4 is close to ``infinity'' and turns a quarter
% circle into a line with an unusual loop; a value of 2 turns a quarter
% circle almost into a polygonal line with rounded corner. Therefore these
% tension factors should be used only for fine tuning the arcs, not when
% a path is drawn for the first time.
%
% We devised a syntax for specifying direction and tensions:
%\begin{flushleft}
% \meta{direction\texttt{\upshape;}tension factors}
%\end{flushleft}
% where \emph{direction} contains a pair of fractional number that not
% necessarily refer to the components of a unit vector direction, but
% simply to a vector with the desired orientation (polar form is OK); the
% information contained from the semicolon (included) to the rest of the
% specification is optional; if it is present, the \emph{tension factors}
% is simply a comma separated pair of fractional or integer numbers that
% represent respectively the tension at the starting or the ending node of
% a path arc.
%
% We therefore need a macro to extract the mandatory and optional parts:
%    \begin{macrocode}
\def\@isTension#1;#2!!{\def\@tempA{#1}%
\def\@tempB{#2}\unless\ifx\@tempB\empty\strip@semicolon#2\fi}

\def\strip@semicolon#1;{\def\@tempB{#1}}
%    \end{macrocode}
% By changing the tension values we can achieve different results: see
% figure~\ref{fig:tensions} on page~\pageref{fig:tensions}.
%
% We use the formula we got for arcs~\eqref{equ:corda}, where the half
% chord is indicated with $s$,  and we derive the necessary distances:
%\begin{subequations}\label{equ:Kzero-Kuno}
%\begin{align}
%K_0 &= \frac{4}{3} s\frac{1-\cos\theta_0}{\sin^2\theta_0}\\
%K_1 &=\frac{4}{3}s\frac{1-\cos\theta_1}{\sin^2\theta_1}
%\end{align}
%\end{subequations}
%
% We therefore start with getting the points and directions and calculating
% the chord and its direction:
%    \begin{macrocode}
\def\CurveTo#1WithDir#2{%
\def\@Tuno{1}\def\@Tzero{1}\relax
\edef\@Puno{#1}\@isTension#2;!!%
\expandafter\DirOfVect\@tempA to\@Duno
\bgroup\unless\ifx\@tempB\empty\GetCoord(\@tempB)\@Tzero\@Tuno\fi
\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)\@DXpzero\@DYpzero
\GetCoord(\@Dpuno)\@DXpuno\@DYpuno
\DivideFN\@Chord by2 to\@semichord
%    \end{macrocode}
% The chord needs not be actually rotated because it suffices its length
% along the real axis; the chord length is memorised in |\@Chord| and
% its half is saved in |\@semichord|.
%
% We now examine the various degenerate cases, when either tangent is
% perpendicular or parallel to the chord. Notice that we are calculating
% the distances of the control points from the adjacent nodes using the
% half chord length, not the full length. We also distinguish between the
% computations relative to the arc starting point and those relative to
% the end point.
%
%    \begin{macrocode}
\ifdim\@DXpzero\p@=\z@
   \@tdA=1.333333\p@
   \Numero\@KCzero{\@semichord\@tdA}%
\fi
\ifdim\@DYpzero\p@=\z@
  \@tdA=1.333333\p@
  \Numero\@Kpzero{\@semichord\@tdA}%
\fi
%    \end{macrocode}
% The distances we are looking for are positive generally fractional
% numbers; so if the components are negative, we take the absolute values.
% Eventually we determine the absolute control point coordinates. 
%    \begin{macrocode}
\unless\ifdim\@DXpzero\p@=\z@
  \unless\ifdim\@DYpzero\p@=\z@
    \edef\@CosDzero{\ifdim\@DXpzero\p@<\z@ -\fi\@DXpzero}%
    \edef\@SinDzero{\ifdim\@DYpzero\p@<\z@ -\fi\@DYpzero}%
    \@tdA=\@semichord\p@ \@tdA=1.333333\@tdA
    \DividE\@tdA by\@SinDzero\p@ to \@KCzero
    \@tdA=\dimexpr(\p@-\@CosDzero\p@)\relax
    \DividE\@KCzero\@tdA by\@SinDzero\p@ to \@KCzero
  \fi
\fi
\MultiplyFN\@KCzero by \@Tzero to \@KCzero
\ScaleVect\@Dzero by\@KCzero to\@CPzero
\AddVect\@Pzero and\@CPzero to\@CPzero
%    \end{macrocode}
% We now repeat the calculations for the arc end point, taking into
% consideration that the end point direction points outwards, so that in
% computing the end point control point we have to take this fact into
% consideration by using a negative sign for the distance; in this way
% the displacement of the control point from the end point takes place
% in a backwards direction.
%    \begin{macrocode}
\ifdim\@DXpuno\p@=\z@
   \@tdA=-1.333333\p@
   \Numero\@KCuno{\@semichord\@tdA}%
\fi
\ifdim\@DYpuno\p@=\z@
  \@tdA=-1.333333\p@
  \Numero\@KCuno{\@semichord\@tdA}%
\fi
\unless\ifdim\@DXpuno\p@=\z@
    \unless\ifdim\@DYpuno\p@=\z@
    \edef\@CosDuno{\ifdim\@DXpuno\p@<\z@ -\fi\@DXpuno}%
    \edef\@SinDuno{\ifdim\@DYpuno\p@<\z@ -\fi\@DYpuno}%
    \@tdA=\@semichord\p@ \@tdA=-1.333333\@tdA
    \DividE\@tdA by \@SinDuno\p@ to \@KCuno
    \@tdA=\dimexpr(\p@-\@CosDuno\p@)\relax
    \DividE\@KCuno\@tdA by\@SinDuno\p@ to \@KCuno
  \fi
\fi
\MultiplyFN\@KCuno by \@Tuno to \@KCuno
\ScaleVect\@Duno by\@KCuno to\@CPuno
\AddVect\@Puno and\@CPuno to\@CPuno
%    \end{macrocode}
% Now we have the four points and we can instruct the internal
% \texttt{pict2e} macros to do the path drawing.
%    \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}\egroup
%    \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 memorises the final point
% as the initial point of the next spline
%    \begin{macrocode}
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}%
%    \end{macrocode}
%
%
% We finally define the overall |\Curve| macro that has two flavours:
% starred and unstarred; the former fills the curve path with the locally
% selected color, while the latter just strokes the path. Both recursively
% examine 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 initialises 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. It does
% not check for a change in direction, because it would be meaningless at
% the beginning of a curve. 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 or filling command through
% |\CurveEnd|, and exits the recursive process. The |\CurveEnd| control
% sequence has a different meaning depending on the fact that the main
% macro was starred or unstarred. The |@ChangeDir| macro is just an
% interface to execute the regular |\ChangeDir| macro, but also for
% recursing again by recalling |\@Curve|.
%    \begin{macrocode}
\def\Curve{\@ifstar{\let\fillstroke\fillpath\Curve@}%
{\let\fillstroke\strokepath\Curve@}}

\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 \string\Curve\space specifications\MessageBreak}}}
\def\@Curve(#1)<#2>{%
    \CurveTo#1WithDir{#2}%
    \@ifnextchar\lp@r\@Curve{%
    \@ifnextchar[\@ChangeDir\CurveEnd}}
\def\@ChangeDir[#1]{\ChangeDir<#1>\@Curve}
%    \end{macrocode}
%
% As a concluding remark, please notice that 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| or  |\FillCurve| for a more versatile
% set of drawing macros; evidently nobody forbids to exploit the full power
% of the |\cbezier| original macro for cubic splines; we made available
% macros |\CbezierTo| and the isolated arc macro |\CbezierBetween| in order
% to use the general internal cubic Bézier splines in a more comfortable
% way.
%
% As it can be seen in figure~\ref{fig:sinewave} on
% page~\pageref{fig:sinewave} the two diagrams should approximately
% represent a sine wave. With Bézier curves, that resort on polynomials,
% it is impossible to represent a transcendental function, but it is only
% possible to approximate it. It is evident that the approximation obtained 
% with full control on the control points requires less arcs and it is more
% accurate than the approximation obtained with the recursive |\Curve|
% macro; this macro requires almost two times as many pieces of information
% in order to minimise the effects of the lack of control on the control
% points, and even with this added information the macro approaches the
% sine wave with less accuracy. At the same time for many applications the
% |\Curve| recursive macro proves to be much easier to use than with single 
% arcs drawn with the |\CbezierBetween| macro.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Quadratic splines}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% We want to create a recursive macro with the same properties as the above
% described |\Curve| macro, but that uses quadratic splines; we call it
% |\Qurve| so that the initial macro name letter reminds us of the nature
% of the splines being used. For the rest they have an almost identical
% syntax; with quadratic spline it is not possible to specify the distance
% of the control points from the extrema, since quadratic spline have just
% one control point that must lay at the intersection of the two tangent
% directions therefore with quadratic splines the tangents at each point
% cannot have the optional part that starts with a semicolon. The syntax,
% therefore, is just:
%\begin{flushleft}
%\cs{Qurve}\parg{first point}\aarg{direction}...\parg{any point}\aarg{direction}...\parg{last point}\aarg{direction}
%\end{flushleft}
% As with |\Curve|, also with |\Qurve| there is no limitation on the number
% of points, except for the computer memory size; it is advisable not to
% use many arcs otherwise it might become very difficult to find errors.
% 
% The first macros that set up the recursion are very similar to those we
% wrote for |\Curve|:
%    \begin{macrocode}
\def\Qurve{\@ifstar{\let\fillstroke\fillpath\Qurve@}%
{\let\fillstroke\strokepath\Qurve@}}

\def\Qurve@(#1)<#2>{%
    \StartCurveAt#1WithDir{#2}%
    \@ifnextchar\lp@r\@Qurve{%
    \PackageWarning{curve2e}{%
      Quadratic curve specifications must contain at least 
      two nodes!\Messagebreak
      Please, control your Qurve specifications\MessageBreak}}}%
      
\def\@Qurve(#1)<#2>{\QurveTo#1WithDir{#2}%
    \@ifnextchar\lp@r\@Qurve{%
    \@ifnextchar[\@ChangeQDir\CurveEnd}}%
    
\def\@ChangeQDir[#1]{\ChangeDir<#1>\@Qurve}%
%    \end{macrocode}
%
% Notice that in case of long paths it might be better to use the single
% macros |\StartCurveAt|, |\QurveTo|, |\ChangeDir| and |\CurveFinish|
% (or |\FillCurve|), with their respective syntax, in such a way that a
% long list % of node-direction specifications passed to |\Qurve| may be
% split into shorter input lines in order to edit the input data in a more
% comfortable way.
%
%
% The macro that does everything is |\QurveTo|. it starts with reading its
% arguments received through the calling macro |\@Qurve|
%    \begin{macrocode}
\def\QurveTo#1WithDir#2{%
\edef\@Puno{#1}\DirOfVect#2to\@Duno\bgroup
\DistanceAndDirOfVect\@Puno minus\@Pzero to\@Chord and\@DirChord
%    \end{macrocode}
% It verifies if |\@Dpzero| and |\@Dpuno|, the directions at the two
% extrema of the arc, are parallel or anti-parallel by taking their
% ``scalar'' product (|\@Dpzero| times |\@Dpuno*|); if  the imaginary
% component  of the scalar product vanishes the two directions are
% parallel; in this case we produce an error message, but we continue by
% skipping this arc destination point; evidently the drawing will not be
% the desired one, but the job should not abort.
%    \begin{macrocode}
\MultVect\@Dzero by*\@Duno to \@Scalar
\YpartOfVect\@Scalar to \@YScalar
\ifdim\@YScalar\p@=\z@
\PackageWarning{curve2e}%
   {Quadratic Bezier arcs cannot have their starting\MessageBreak
   and ending directions parallel or antiparallel with\MessageBreak
   each other. This arc is skipped and replaced with 
   a dotted line.\MessageBreak}%
   \Dotline(\@Pzero)(\@Puno){2}\relax
\else
%    \end{macrocode}
% Otherwise we rotate everything about the starting point so as to bring
% the chord on the real axis; we get also the components of the two
% directions that, we should remember, are unit vectors, not generic
% vectors, although the user can use the vector specifications that are
% more understandable to him/her:
%    \begin{macrocode}
\MultVect\@Dzero by*\@DirChord to \@Dpzero
\MultVect\@Duno by*\@DirChord to \@Dpuno
\GetCoord(\@Dpzero)\@DXpzero\@DYpzero
\GetCoord(\@Dpuno)\@DXpuno\@DYpuno
%    \end{macrocode}
% We check if the two directions point to the same half plane; this implies
% that these rotated directions point to different sides of the chord
% vector; all this is equivalent that the two direction Y components have
% opposite signs, so that their product is strictly negative, while the two
% X components product is not negative.
%    \begin{macrocode}
\MultiplyFN\@DXpzero by\@DXpuno to\@XXD
\MultiplyFN\@DYpzero by\@DYpuno to\@YYD
\unless\ifdim\@YYD\p@<\z@\ifdim\@XXD\p@<\z@
\PackageWarning{curve2e}%
   {Quadratic Bezier arcs cannot have inflection points\MessageBreak
   Therefore the tangents to the starting and ending arc\MessageBreak
   points cannot be directed to the same half plane.\MessageBreak
   This arc is skipped and replaced by a dotted line\MessageBreak}%
   \Dotline(\@Pzero)(\@Puno){2}\fi
\else
%    \end{macrocode}
%
% After these tests we should be in a ``normal'' situation. We first copy
% the expanded input information into new macros that have more explicit
% names: macros stating wit `S' denote the sine of the direction angle,
% while those starting with `C' denote the cosine of that angle. We will
% use these expanded definitions as we know we are working with the actual
% values. These directions are those relative to the arc chord.
%    \begin{macrocode}
\edef\@CDzero{\@DXpzero}\relax
\edef\@SDzero{\@DYpzero}\relax
\edef\@CDuno{\@DXpuno}\relax
\edef\@SDuno{\@DYpuno}\relax
%    \end{macrocode}
% Suppose we write the parametric equations of a straight line that departs
% from the beginning of the chord with direction angle $\phi_0$ and the
% corresponding equation of the straight line departing from the end of the
% chord (of length $c$) with direction angle $\phi_1$. We have to find the
% coordinates of the intersection point of these two straight lines.
%\begin{subequations}
%\begin{align}
% t \cos\phi_0 - s \cos\phi_1 &=  c\\
% t \sin\phi_0 - s \sin\phi_1 &= 0
%\end{align}
%\end{subequations}
% The parameters $t$ and $s$ are just the running parameters; we have
% to solve those simultaneous equations in the unknown variables $t$ and
% $s$; these values let us compute the coordinates of the intersection
% point:
%\begin{subequations}\begin{align}
% X_C &=\dfrac{c\cos\phi_0\sin\phi_1}{\sin\phi_0\cos\phi_1 - \cos\phi_0\sin\phi_1} \\
% Y_C &=\dfrac{c\sin\phi_0\sin\phi_1}{\sin\phi_0\cos\phi_1 - \cos\phi_0\sin\phi_1}
%\end{align}\end{subequations}
%
% Having performed the previous tests we are sure that the denominator is
% not vanishing (direction are not parallel or anti-parallel) and that it
% lays at the same side as the direction with angle $\phi_0$ with respect
% to the chord.
%
% The coding then goes on like this:
%    \begin{macrocode}
\MultiplyFN\@SDzero by\@CDuno to\@tempA 
\MultiplyFN\@SDuno by\@CDzero to\@tempB 
\edef\@tempA{\strip@pt\dimexpr\@tempA\p@-\@tempB\p@}\relax
\@tdA=\@SDuno\p@ \@tdB=\@Chord\p@ \@tdC=\@tempA\p@
\edef\@tempC{\strip@pt\dimexpr \@tdA*\@tdB/\@tdC}\relax
\MultiplyFN\@tempC by\@CDzero to \@XC 
\MultiplyFN\@tempC by\@SDzero to \@YC 
\ModOfVect\@XC,\@YC to\@KC            
%    \end{macrocode}
% Now we have the coordinates and the module  of the intersection point
% vector taking into account the rotation of the real axis; getting back
% to the original coordinates before rotation, we get:
%    \begin{macrocode}
\ScaleVect\@Dzero by\@KC to\@CP       
\AddVect\@Pzero and\@CP to\@CP       
\GetCoord(\@Pzero)\@XPzero\@YPzero    
\GetCoord(\@Puno)\@XPuno\@YPuno      
\GetCoord(\@CP)\@XCP\@YCP             
%    \end{macrocode}
% We have now the coordinates of the two extrema point of the quadratic arc
% and of the control point. Keeping in mind that the symbols $P_0$, $P_1$
% and $C$ denote geometrical points but also their coordinates as ordered
% pairs of real numbers (i.e. they are complex numbers) we have to
% determine the parameters of a cubic spline that with suitable  values
% get simplifications in its parametric equation so that it becomes a
% second degree function instead of a third degree one. It is possible,
% in spite of the fact the it appears impossible that e cubic form becomes
% a quadratic one; we should determine the values of $P_a$ and $P_b$ such
% that:
%\[
% P_0(1-t)^3 +3P_a(1-t)^2t +3P_b(1-t)t^2 +P_1t^3
%\]
% is equivalent to
%\[
% P_0(1-t)^2 + 2C(1-t)t + P_1t^2
%\]
% It turns out that the solution is given by
%\begin{equation}
%    P_a= C+(P_0-C)/3 \qquad \text{and}\qquad P_b = C+(P_1- C)/3
%\label{equ:spline3}
%\end{equation}
%
% The transformations implied by equations~\eqref{equ:spline3} are
% performed by the following macros already available from the |pict2e|
% package; we use them here with the actual arguments used for this task:
%    \begin{macrocode}
\@ovxx=\@XPzero\unitlength \@ovyy=\@YPzero\unitlength
\@ovdx=\@XCP\unitlength \@ovdy=\@YCP\unitlength
\@xdim=\@XPuno\unitlength \@ydim=\@YPuno\unitlength
      \pIIe@bezier@QtoC\@ovxx\@ovdx\@ovro
      \pIIe@bezier@QtoC\@ovyy\@ovdy\@ovri
      \pIIe@bezier@QtoC\@xdim\@ovdx\@clnwd
      \pIIe@bezier@QtoC\@ydim\@ovdy\@clnht
%    \end{macrocode}
%
% We call the basic |pict2e| macro to draw a cubic spline and we finish
% the conditional statements with which we started these calculations;
% eventually we close the group we opened at the beginning and we copy
% the terminal node information (position and direction) into the
% 0-labelled macros that indicate the starting point of the next arc.
%    \begin{macrocode}
      \pIIe@curveto\@ovro\@ovri\@clnwd\@clnht\@xdim\@ydim
\fi\fi\egroup
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}
%    \end{macrocode}
%
% An example of usage is shown at the left in
% figure~\ref{fig:quadratic-arcs}\footnote{The commands \cs{legenda},
% \cs{Pall} and \cs{Zbox} are specifically defined in the preamble of this
% document; they must be used within a \texttt{picture} environment.
% \cs{legenda} draws a framed legend made up of a single (short) math
% formula; \cs{Pall} is just a shorthand to put a filled small circle at a
% specified position' \cs{Zbox} puts a symbol in math mode a little
% displaced in the proper direction relative to a specified position.
% They are just handy to label certain objects in a \texttt{picture}
% diagram, but they are not part of the \texttt{curve2e} package.} on
% page~\pageref{fig:quadratic-arcs} created with the code shown in the
% same page.
%
% Notice also that the inflexed line is made with two arcs that meet at
% the inflection point; the same is true for the line that resembles
% a sine wave. The cusps of the inner border of the green area are
% obtained with the usual optional star already used also with the
% |\Curve| recursive macro.
%
% The ``circle'' inside the square frame is visibly different from a real
% circle, in spite of the fact that the maximum deviation from the true
% circle is just about 6\% relative to the radius; a quarter circle
% obtained with a single parabola is definitely a poor approximation of a
% real quarter circle; possibly by splitting each quarter circle in three
% or four partial arcs the approximation of a real quarter circle would be
% much better. On the right of figure~\ref{fig:quadratic-arcs} on
% page~\pageref{fig:quadratic-arcs} it is possible to compare a “circle”
% obtained with quadratic arcs with the the internal circle obtained with
% cubic arcs; the difference is easily seen even without using measuring
% instruments.
%
% With quadratic arcs we decided to avoid defining specific macros similar
% to |\CurveBetween| and |\CbezierBetween|; the first macro would not save
% any typing to the operator; furthermore it may be questionable if it was
% really useful even with cubic splines; the second macro with quadratic
% arcs is meaningless, since with quadratic arcs there is just one control
% point and there is no choice on its position.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Conclusion}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% I believe that the set of new macros provided by this package  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.
%
% As a personal experience we found very comfortable to draw ellipses and
% to define macros to draw  not only such shapes or filled elliptical
% areas, but also to create “legends” with coloured backgrounds and
% borders. But this is an application of the functionality implemented in
% this package.
%
%\iffalse
%</package>
%\fi
%
%
%\iffalse
%<*readme>
%\fi
% \section{The \texttt{README.txt} file}
%The following it the text that forms the contents of the |README.txt|
% file that accompanies the package. We fount it handy to have it in
% the documented source, because in this way certain informations
% don't need to be repeated again and again in different files. 
%    \begin{macrocode}
The package bundle curve2e is composed of the following files

curve2e.dtx
curve2e.pdf
README.txt
curve2e-v161.sty
ltxdoc.cfg

curve2e.dtx is the documented TeX source file of file curve2e.sty; you 
get both curve2e.sty and curve2e.pdf by running pdflatex on curve2e.dtx. 
The ltxdoc.cfg file customises the way the documentation file is typeset. 
This .cfg file is not subject to the LPPL licence.

README.txt, this file, contains general information.

Curve2e-v161.sty contains the previous version of the package; see below 
why the older version might become necessary for the end user.

Curve2e.sty is an extension of the package pict2e.sty which extends the 
standard picture LaTeX environment according to what Leslie Lamport 
specified in the second edition of his LaTeX manual.

This further extension allows to draw lines and vectors with any non 
integer slope parameters, to draw dashed lines of any slope, to draw arcs 
and curved vectors, to draw curves where just the interpolating nodes are 
specified together with the slopes at the nodes; closed paths of any 
shape can be filled with color; all coordinates are treated as ordered 
pairs, i.e. 'complex numbers'; coordinates may be expressed also in 
polar form. 
Some of these features have been incorporated in the 2011 version of 
pict2e; therefore this package avoids any modification to the original 
pict2e commands.


Curve2e now accepts polar coordinates in addition to the usual cartesian 
ones; several macros have been upgraded and a new macro for tracing cubic 
Bezier splines with their control nodes specified in polar form is 
available.  The same applies to quadratic Bezier splines. The multiput 
command has been completely modified in a backwards compatible way, as
to manipulate the increment components.

This version solves a conflict with package eso-pic.

This version of curve2e is almost fully compatible with pict2e dated 
2014/01/12 version 0.2z.

If you specify

\usepackage[<pict2e options>]{curve2e}

the package pict2e is automatically invoked with the specified options.

The -almost compatible- frase is necessary to explain that this version 
of curve2e uses some `functions' of the LaTeX3 language that were made
available to the LaTeX developer by mid October 2018. Should the user
have an older or a basic/incomplete installation of the TeX system, 
such L3 functions might not be available. This is why this 
package checks the presence of the developer interface; in case 
such interface is not available it falls back to the previous version 
renamed curve2e-v161.sty, which is part of this bundle, and that must 
not be renamed in any way. The compatibility mentioned above implies 
that the user macros remain the same, but their implementation requires 
the L3 interface.

The package has the LPPL status of author maintained.

According to the LPPL licence, you are entitled to modify this package, 
as long as you fulfil the few conditions set forth by the Licence.

Nevertheless this package is an extension to the standard LaTeX package 
pict2e (2014). Therefore  any change must be controlled on the 
parent package pict2e, so as to avoid redefining what has already been 
incorporated in the official package. 

If you prefer sending me your modifications, as long as I will maintain 
this package, I will possibly include every (documented) suggestion or 
modification into this package and, of course, I will acknowledge your 
contribution.

Claudio Beccari

claudio dot beccari at gmail dot com
%    \end{macrocode}
%\iffalse
%</readme>
%\fi
%\iffalse
%<*v161>
%\fi
% \section{The fall-back package version \texttt{curve2e-v161}}
% this is the fall-back version of |curve2e-v161.sty| to which the main
% file |curve2e.sty| falls back in case the interface package |xfp| is not
% available.
%    \begin{macrocode}
\NeedsTeXFormat{LaTeX2e}[2016/01/01]
\ProvidesPackage{curve2e-v161}%
        [2019/02/07 v.1.61 Extension package for pict2e]

\RequirePackage{color}
\RequirePackageWithOptions{pict2e}[2014/01/01]
\RequirePackage{xparse}
\def\TRON{\tracingcommands\tw@ \tracingmacros\tw@}%
\def\TROF{\tracingcommands\z@ \tracingmacros\z@}%
\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
\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}
\def\LIne(#1){{\GetCoord(#1)\@tX\@tY
      \moveto(0,0)
      \pIIe@lineto{\@tX\unitlength}{\@tY\unitlength}\strokepath}\ignorespaces}%
\def\segment(#1)(#2){\@killglue\polyline(#1)(#2)}%
\def\line(#1)#2{\begingroup
  \@linelen #2\unitlength
  \ifdim\@linelen<\z@\@badlinearg\else
    \expandafter\DirOfVect#1to\Dir@line
    \GetCoord(\Dir@line)\d@mX\d@mY
    \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
    \moveto(0,0)
    \pIIe@lineto{\d@mX\@linelen}{\d@mY\@linelen}%
    \strokepath
  \fi
\endgroup\ignorespaces}%
\ifx\Dashline\undefined
\def\Dashline{\@ifstar{\Dashline@@}{\Dashline@}}
\def\Dashline@(#1)(#2)#3{%
\bgroup
   \countdef\NumA3254\countdef\NumB3252\relax
   \GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttA
   \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttB
   \SubVect\V@ttA from\V@ttB to\V@ttC  
   \ModOfVect\V@ttC to\DlineMod  
   \DivideFN\DlineMod by#3 to\NumD  
   \NumA\expandafter\Integer\NumD.??  
   \ifodd\NumA\else\advance\NumA\@ne\fi  
   \NumB=\NumA \divide\NumB\tw@  
   \DividE\DlineMod\p@ by\NumA\p@  to\D@shMod  
   \DividE\p@ by\NumA\p@ to \@tempa  
   \MultVect\V@ttC by\@tempa,0 to\V@ttB 
   \MultVect\V@ttB by 2,0 to\V@ttC
   \advance\NumB\@ne 
   \edef\@mpt{\noexpand\egroup
   \noexpand\multiput(\V@ttA)(\V@ttC){\number\NumB}%
      {\noexpand\LIne(\V@ttB)}}%
   \@mpt\ignorespaces}%
\let\Dline\Dashline

\def\Dashline@@(#1)(#2)#3{\put(#1){\Dashline@(0,0)(#2){#3}}}
\fi
\ifx\Dotline\undefined
\def\Dotline{\@ifstar{\Dotline@@}{\Dotline@}}
\def\Dotline@(#1)(#2)#3{%
\bgroup
   \countdef\NumA 3254\relax \countdef\NumB 3255\relax
   \GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttA
   \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttB
   \SubVect\V@ttA from\V@ttB to\V@ttC
   \ModOfVect\V@ttC to\DotlineMod
   \DivideFN\DotlineMod by#3 to\NumD
   \NumA=\expandafter\Integer\NumD.??
   \DivVect\V@ttC by\NumA,0 to\V@ttB
   \advance\NumA\@ne
   \edef\@mpt{\noexpand\egroup
   \noexpand\multiput(\V@ttA)(\V@ttB){\number\NumA}%
      {\noexpand\makebox(0,0){\noexpand\circle*{0.5}}}}%
   \@mpt\ignorespaces}%

\def\Dotline@@(#1)(#2)#3{\put(#1){\Dotline@(0,0)(#2){#3}}}
\fi
\AtBeginDocument{\@ifpackageloaded{eso-pic}{%
\renewcommand\LenToUnit[1]{\strip@pt\dimexpr#1*\p@/\unitlength}}{}}

\def\GetCoord(#1)#2#3{%
\expandafter\SplitNod@\expandafter(#1)#2#3\ignorespaces}
\def\isnot@polar#1:#2!!{\def\@tempOne{#2}\ifx\@tempOne\empty
\expandafter\@firstoftwo\else
\expandafter\@secondoftwo\fi
{\SplitNod@@}{\SplitPolar@@}}

\def\SplitNod@(#1)#2#3{\isnot@polar#1:!!(#1)#2#3}%
\def\SplitNod@@(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}%
\def\SplitPolar@@(#1:#2)#3#4{\DirFromAngle#1to\@DirA
\ScaleVect\@DirA by#2to\@DirA
\expandafter\SplitNod@@\expandafter(\@DirA)#3#4}

\let\originalput\put
\def\put(#1){\bgroup\GetCoord(#1)\@tX\@tY
\edef\x{\noexpand\egroup\noexpand\originalput(\@tX,\@tY)}\x}

\let\originalmultiput\multiput
\let\original@multiput\@multiput

\long\def\@multiput(#1)#2#3{\bgroup\GetCoord(#1)\@mptX\@mptY
\edef\x{\noexpand\egroup\noexpand\original@multiput(\@mptX,\@mptY)}%
\x{#2}{#3}\ignorespaces}

\gdef\multiput(#1)#2{\bgroup\GetCoord(#1)\@mptX\@mptY
\edef\x{\noexpand\egroup\noexpand\originalmultiput(\@mptX,\@mptY)}\x(}%)
  \def\vector(#1)#2{%
    \begingroup
      \GetCoord(#1)\d@mX\d@mY
      \@linelen#2\unitlength
      \ifdim\d@mX\p@=\z@\ifdim\d@mY\p@=\z@\@badlinearg\fi\fi
      \ifdim\@linelen<\z@ \@linelen=-\@linelen\fi
      \MakeVectorFrom\d@mX\d@mY to\@Vect
      \DirOfVect\@Vect to\Dir@Vect
        \YpartOfVect\Dir@Vect to\@ynum \@ydim=\@ynum\p@
        \XpartOfVect\Dir@Vect to\@xnum \@xdim=\@xnum\p@
        \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
\pIIe@concat\@xdim\@ydim{-\@ydim}\@xdim{\@xnum\@linelen}{\@ynum\@linelen}%
        \@linelen\z@
        \pIIe@vector
        \fillpath
      \@linelen=\@tdB
      \@tdA=\pIIe@FAW\@wholewidth
      \@tdA=\pIIe@FAL\@tdA
      \advance\@linelen-\@tdA
      \ifdim\@linelen>\z@
        \moveto(0,0)
        \pIIe@lineto{\@xnum\@linelen}{\@ynum\@linelen}%
        \strokepath\fi
    \endgroup}
\def\Vector(#1){{%
\GetCoord(#1)\@tX\@tY
\ifdim\@tX\p@=\z@\vector(\@tX,\@tY){\@tY}
\else
\vector(\@tX,\@tY){\@tX}\fi}}
\def\VECTOR(#1)(#2){\begingroup
\SubVect#1from#2to\@tempa
\expandafter\put\expandafter(#1){\expandafter\Vector\expandafter(\@tempa)}%
\endgroup\ignorespaces}
\let\lp@r( \let\rp@r)
\renewcommand*\polyline[1][\beveljoin]{\p@lylin@[#1]}

\def\p@lylin@[#1](#2){\@killglue#1\GetCoord(#2)\d@mX\d@mY
    \pIIe@moveto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\p@lyline}{%
    \PackageWarning{curve2e}%
    {Polylines require at least two vertices!\MessageBreak
     Control your polyline specification\MessageBreak}%
    \ignorespaces}}

\def\p@lyline(#1){\GetCoord(#1)\d@mX\d@mY
    \pIIe@lineto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\p@lyline}{\strokepath\ignorespaces}}
\providecommand\polygon{}
\RenewDocumentCommand\polygon{s O{\beveljoin} }{\@killglue\begingroup
\IfBooleanTF{#1}{\@tempswatrue}{\@tempswafalse}%
\@polygon[#2]}

\def\@polygon[#1](#2){\@killglue#1\GetCoord(#2)\d@mX\d@mY
    \pIIe@moveto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\@@polygon}{%
    \PackageWarning{curve2e}%
    {Polygons require at least two vertices!\MessageBreak
     Control your polygon specification\MessageBreak}%
    \ignorespaces}}

  \def\@@polygon(#1){\GetCoord(#1)\d@mX\d@mY
    \pIIe@lineto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\@@polygon}{\pIIe@closepath
      \if@tempswa\pIIe@fillGraph\else\pIIe@strokeGraph\fi
      \endgroup
      \ignorespaces}}
\def\GraphGrid(#1,#2){\bgroup\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}%
\egroup\ignorespaces}
\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}%
\def\Integer#1.#2??{#1}%
\ifdefined\dimexpr
    \unless\ifdefined\DividE
\def\DividE#1by#2to#3{\bgroup
\dimendef\Num2254\relax \dimendef\Den2252\relax
\dimendef\@DimA 2250
\Num=\p@ \Den=#2\relax
\ifdim\Den=\z@
  \edef\x{\noexpand\endgroup\noexpand\def\noexpand#3{\strip@pt\maxdimen}}%
\else
  \@DimA=#1\relax
  \edef\x{%
    \noexpand\egroup\noexpand\def\noexpand#3{%
       \strip@pt\dimexpr\@DimA*\Num/\Den\relax}}%
\fi
\x\ignorespaces}%
\fi
    \unless\ifdefined\DivideFN
         \def\DivideFN#1by#2to#3{\DividE#1\p@ by#2\p@ to{#3}}%
    \fi
    \unless\ifdefined\MultiplY
        \def\MultiplY#1by#2to#3{\bgroup
        \dimendef\@DimA 2254 \dimendef\@DimB2255
        \@DimA=#1\p@\relax \@DimB=#2\p@\relax
        \edef\x{%
           \noexpand\egroup\noexpand\def\noexpand#3{%
             \strip@pt\dimexpr\@DimA*\@DimB/\p@\relax}}%
        \x\ignorespaces}%
        \let\MultiplyFN\MultiplY
    \fi
\fi

\unless\ifdefined\Numero
  \def\Numero#1#2{\bgroup\dimen3254=#2\relax
    \edef\x{\noexpand\egroup\noexpand\edef\noexpand#1{%
      \strip@pt\dimen3254}}\x\ignorespaces}%
\fi
\def\g@tTanCotanFrom#1to#2and#3{%
\DividE 114.591559\p@ by#1to\X@ \@tdB=\X@\p@
\countdef\I=2546\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}%
\def\SinOf#1to#2{\bgroup%
\@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@
  \def\@tempA{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@
    \def\@tempA{\Segno1}%
  \else
    \ifdim\@tdA=180\p@
      \def\@tempA{0}%
    \else
      \ifdim\@tdA<\p@
        \@tdA=\Segno0.0174533\@tdA
        \DividE\@tdA by\p@ to \@tempA%
      \else
        \g@tTanCotanFrom\@tdA to\T and\Tp
        \@tdA=\T\p@ \advance\@tdA \Tp\p@
        \DividE \Segno2\p@ by\@tdA to \@tempA%
      \fi
    \fi
  \fi
\fi
\edef\endSinOf{\noexpand\egroup
  \noexpand\def\noexpand#2{\@tempA}\noexpand\ignorespaces}%
\endSinOf}%
\def\CosOf#1to#2{\bgroup%
\@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@
  \def\@tempA{\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\@tempA%
  \else
    \ifdim\@tdA=90\p@
      \def\@tempA{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\@tempA%
    \fi
  \fi
\fi
\edef\endCosOf{\noexpand\egroup
    \noexpand\def\noexpand#2{\@tempA}\noexpand\ignorespaces}%
\endCosOf}%
\def\TanOf#1to#2{\bgroup%
\@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@%
  \def\@tempA{0}%
\else
  \ifdim\@tdA>\z@
    \def\Segno{+}%
  \else
    \def\Segno{-}%
    \@tdA=-\@tdA
  \fi
  \ifdim\@tdA=90\p@
    \def\@tempA{\Segno16383.99999}%
  \else
    \ifdim\@tdA<\p@
        \@tdA=\Segno0.0174533\@tdA
        \DividE\@tdA by\p@ to\@tempA%
    \else
      \g@tTanCotanFrom\@tdA to\T and\Tp
      \@tdA\Tp\p@ \advance\@tdA -\T\p@
      \DividE\Segno2\p@ by\@tdA to\@tempA%
    \fi
  \fi
\fi
\edef\endTanOf{\noexpand\egroup
   \noexpand\def\noexpand#2{\@tempA}\noexpand\ignorespaces}%
\endTanOf}%
\def\ArcTanOf#1to#2{\bgroup
\countdef\Inverti 4444\Inverti=0
\def\Segno{}
\edef\@tF{#1}\@tdF=\@tF\p@ \@tdE=57.295778\p@
\@tdD=\ifdim\@tdF<\z@ -\@tdF\def\Segno{-}\else\@tdF\fi
\ifdim\@tdD>\p@
\Inverti=\@ne
\@tdD=\dimexpr\p@*\p@/\@tdD\relax
\fi
\unless\ifdim\@tdD>0.02\p@
    \def\@tX{\strip@pt\dimexpr57.295778\@tdD\relax}%
\else
    \edef\@tX{45}\relax
    \countdef\I 2523 \I=9\relax
    \@whilenum\I>0\do{\TanOf\@tX to\@tG
    \edef\@tG{\strip@pt\dimexpr\@tG\p@-\@tdD\relax}\relax
    \MultiplY\@tG by57.295778to\@tG
    \CosOf\@tX to\@tH
    \MultiplY\@tH by\@tH to\@tH
    \MultiplY\@tH by\@tG to \@tH
    \edef\@tX{\strip@pt\dimexpr\@tX\p@ - \@tH\p@\relax}\relax
    \advance\I\m@ne}%
\fi
\ifnum\Inverti=\@ne
\edef\@tX{\strip@pt\dimexpr90\p@-\@tX\p@\relax}
\fi
\edef\x{\egroup\noexpand\edef\noexpand#2{\Segno\@tX}}\x\ignorespaces}%
\def\MakeVectorFrom#1#2to#3{\edef#3{#1,#2}\ignorespaces}%
\def\CopyVect#1to#2{\edef#2{#1}\ignorespaces}%
\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=\z@
    \ifdim\@tempdimb=\z@
        \def\@T{0}\@tempdimc=\z@
    \else
        \def\@T{0}\@tempdimc=\@tempdimb
    \fi
\else
    \ifdim\@tempdima>\@tempdimb
       \DividE\@tempdimb by\@tempdima to\@T
       \@tempdimc=\@tempdima
    \else
          \DividE\@tempdima by\@tempdimb to\@T
          \@tempdimc=\@tempdimb
    \fi
\fi
\unless\ifdim\@tempdimc=\z@
    \unless\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
\fi
\Numero#2\@tempdimc
\ignorespaces}%
\def\DirOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
\ModOfVect#1to\@tempa
\unless\ifdim\@tempdimc=\z@
   \DividE\t@X\p@ by\@tempdimc to\t@X
   \DividE\t@Y\p@ by\@tempdimc to\t@Y
\fi
\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
\def\ModAndDirOfVect#1to#2and#3{%
\GetCoord(#1)\t@X\t@Y
\ModOfVect#1to#2%
\ifdim\@tempdimc=\z@\else
  \DividE\t@X\p@ by\@tempdimc to\t@X
  \DividE\t@Y\p@ by\@tempdimc to\t@Y
\fi
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
\def\DistanceAndDirOfVect#1minus#2to#3and#4{%
\SubVect#2from#1to\@tempa
\ModAndDirOfVect\@tempa to#3and#4\ignorespaces}%
\def\XpartOfVect#1to#2{%
\GetCoord(#1)#2\@tempa\ignorespaces}%
\def\YpartOfVect#1to#2{%
\GetCoord(#1)\@tempa#2\ignorespaces}%
\def\DirFromAngle#1to#2{%
\CosOf#1to\t@X
\SinOf#1to\t@Y
\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
\def\ArgOfVect#1to#2{\bgroup\GetCoord(#1){\t@X}{\t@Y}%
\def\s@gno{}\def\addflatt@ngle{0}
\ifdim\t@X\p@=\z@
  \ifdim\t@Y\p@=\z@
    \def\ArcTan{0}%
  \else
    \def\ArcTan{90}%
    \ifdim\t@Y\p@<\z@\def\s@gno{-}\fi
  \fi
\else
  \ifdim\t@Y\p@=\z@
    \ifdim\t@X\p@<\z@
      \def\ArcTan{180}%
    \else
      \def\ArcTan{0}%
    \fi
  \else
    \ifdim\t@X\p@<\z@%
      \def\addflatt@ngle{180}%
      \edef\t@X{\strip@pt\dimexpr-\t@X\p@}%
      \edef\t@Y{\strip@pt\dimexpr-\t@Y\p@}%
      \ifdim\t@Y\p@<\z@
        \def\s@gno{-}%
        \edef\t@Y{-\t@Y}%
      \fi
    \fi
  \DivideFN\t@Y by\t@X to \t@A
  \ArcTanOf\t@A to\ArcTan
  \fi
\fi
\edef\ArcTan{\unless\ifx\s@gno\empty\s@gno\fi\ArcTan}%
\unless\ifnum\addflatt@ngle=0\relax
   \edef\ArcTan{%
   \strip@pt\dimexpr\ArcTan\p@\ifx\s@gno\empty-\else+\fi
   \addflatt@ngle\p@\relax}%
\fi
\edef\x{\noexpand\egroup\noexpand\edef\noexpand#2{\ArcTan}}%
\x\ignorespaces}
\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}%
\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}%
\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}%
\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}%
\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}
\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}%
\def\Arc(#1)(#2)#3{\begingroup
\@tdA=#3\p@
\unless\ifdim\@tdA=\z@
  \@Arc(#1)(#2)%
\fi
\endgroup\ignorespaces}%
\def\@Arc(#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#2from#1to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
\CopyVect#1to\@Cent \GetCoord(\@pPun)\@pPunX\@pPunY
\@@Arc
\strokepath\ignorespaces}%
\def\@@Arc{%
\pIIe@moveto{\@pPunX\unitlength}{\@pPunY\unitlength}%
\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
\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}
\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}%
\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
\strokepath\ignorespaces}%
\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=\if\Segno--\fi\@gradi\p@ \Numero\@tempa\@tdD
\DirFromAngle\@tempa to\@Dir
\MultVect\@V by\@Dir to\@sPun% corrects the end point
\edef\@tempA{\if\Segno--\fi1}%
\MultVect\@sPun by 0,\@tempA to\@vPun
\DirOfVect\@vPun to\@Dir
\AddVect\@sPun and #1 to \@sPun
\GetCoord(\@sPun)\@tdX\@tdY
\@tdD\if\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}}% end point arrowt ip
\@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{\if\Segno-\else-\fi\@ne}%
\MultVect\@V by0,\@tempa to\@vPun
\@tdE\if\Segno--\fi\DeltaGradi\p@
\Numero\@tempB{0.5\@tdE}%
\DirFromAngle\@tempB to\@Dird
\MultVect\@vPun by\@Dird to\@vPun% corrects the starting point
\DirOfVect\@vPun to\@Dir\GetCoord(\@Dir)\@xnum\@ynum
\put(\@pPunX,\@pPunY){\vector(\@xnum,\@ynum){0}}% starting point arrow tip
\edef\@tempa{\if\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
\strokepath\ignorespaces}%
\def\CurveBetween#1and#2WithDirs#3and#4{%
\StartCurveAt#1WithDir{#3}\relax
\CurveTo#2WithDir{#4}\CurveFinish\ignorespaces}%
\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
\ignorespaces}
\def\ChangeDir<#1>{%
\GetCoord(#1)\@tempa\@tempb
\CopyVect\@tempa,\@tempb to\@Dzero
\DirOfVect\@Dzero to\@Dzero
\ignorespaces}
\def\CurveFinish{\strokepath\endgroup\ignorespaces}%
\def\FillCurve{\fillpath\endgroup\ignorespaces}
\def\CurveEnd{\fillstroke\endgroup\ignorespaces}
\def\CbezierTo#1WithDir#2AndDists#3And#4{%
\GetCoord(#1)\@tX\@tY \MakeVectorFrom\@tX\@tY to\@Puno
\GetCoord(#2)\@tX\@tY \MakeVectorFrom\@tX\@tY to \@Duno
\DirOfVect\@Duno to\@Duno
\ScaleVect\@Dzero by#3to\@Czero \AddVect\@Pzero and\@Czero to\@Czero
\ScaleVect\@Duno by-#4to \@Cuno \AddVect\@Puno and\@Cuno to \@Cuno
\GetCoord(\@Czero)\@XCzero\@YCzero
\GetCoord(\@Cuno)\@XCuno\@YCuno
\GetCoord(\@Puno)\@XPuno\@YPuno
\pIIe@curveto{\@XCzero\unitlength}{\@YCzero\unitlength}%
             {\@XCuno\unitlength}{\@YCuno\unitlength}%
             {\@XPuno\unitlength}{\@YPuno\unitlength}%
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}%
\def\CbezierBetween#1And#2WithDirs#3And#4UsingDists#5And#6{%
\StartCurveAt#1WithDir{#3}\relax
\CbezierTo#2WithDir#4AndDists#5And{#6}\CurveFinish}

\def\@isTension#1;#2!!{\def\@tempA{#1}%
\def\@tempB{#2}\unless\ifx\@tempB\empty\strip@semicolon#2\fi}
\def\strip@semicolon#1;{\def\@tempB{#1}}
\def\CurveTo#1WithDir#2{%
\def\@Tuno{1}\def\@Tzero{1}\relax
\edef\@Puno{#1}\@isTension#2;!!%
\expandafter\DirOfVect\@tempA to\@Duno
\bgroup\unless\ifx\@tempB\empty\GetCoord(\@tempB)\@Tzero\@Tuno\fi
\DistanceAndDirOfVect\@Puno minus\@Pzero to\@Chord and\@DirChord
\MultVect\@Dzero by*\@DirChord to \@Dpzero
\MultVect\@Duno by*\@DirChord to \@Dpuno
\GetCoord(\@Dpzero)\@DXpzero\@DYpzero
\GetCoord(\@Dpuno)\@DXpuno\@DYpuno
\DivideFN\@Chord by2 to\@semichord
\ifdim\@DXpzero\p@=\z@
   \@tdA=1.333333\p@
   \Numero\@KCzero{\@semichord\@tdA}%
\fi
\ifdim\@DYpzero\p@=\z@
  \@tdA=1.333333\p@
  \Numero\@Kpzero{\@semichord\@tdA}%
\fi
\unless\ifdim\@DXpzero\p@=\z@
  \unless\ifdim\@DYpzero\p@=\z@
    \edef\@CosDzero{\ifdim\@DXpzero\p@<\z@ -\fi\@DXpzero}%
    \edef\@SinDzero{\ifdim\@DYpzero\p@<\z@ -\fi\@DYpzero}%
    \@tdA=\@semichord\p@ \@tdA=1.333333\@tdA
    \DividE\@tdA by\@SinDzero\p@ to \@KCzero
    \@tdA=\dimexpr(\p@-\@CosDzero\p@)\relax
    \DividE\@KCzero\@tdA by\@SinDzero\p@ to \@KCzero
  \fi
\fi
\MultiplyFN\@KCzero by \@Tzero to \@KCzero
\ScaleVect\@Dzero by\@KCzero to\@CPzero
\AddVect\@Pzero and\@CPzero to\@CPzero
\ifdim\@DXpuno\p@=\z@
   \@tdA=-1.333333\p@
   \Numero\@KCuno{\@semichord\@tdA}%
\fi
\ifdim\@DYpuno\p@=\z@
  \@tdA=-1.333333\p@
  \Numero\@KCuno{\@semichord\@tdA}%
\fi
\unless\ifdim\@DXpuno\p@=\z@
    \unless\ifdim\@DYpuno\p@=\z@
    \edef\@CosDuno{\ifdim\@DXpuno\p@<\z@ -\fi\@DXpuno}%
    \edef\@SinDuno{\ifdim\@DYpuno\p@<\z@ -\fi\@DYpuno}%
    \@tdA=\@semichord\p@ \@tdA=-1.333333\@tdA
    \DividE\@tdA by \@SinDuno\p@ to \@KCuno
    \@tdA=\dimexpr(\p@-\@CosDuno\p@)\relax
    \DividE\@KCuno\@tdA by\@SinDuno\p@ to \@KCuno
  \fi
\fi
\MultiplyFN\@KCuno by \@Tuno to \@KCuno
\ScaleVect\@Duno by\@KCuno to\@CPuno
\AddVect\@Puno and\@CPuno to\@CPuno
\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}\egroup
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}%
\def\Curve{\@ifstar{\let\fillstroke\fillpath\Curve@}%
{\let\fillstroke\strokepath\Curve@}}
\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\CurveEnd}}
\def\@ChangeDir[#1]{\ChangeDir<#1>\@Curve}
\def\Qurve{\@ifstar{\let\fillstroke\fillpath\Qurve@}%
{\let\fillstroke\strokepath\Qurve@}}

\def\Qurve@(#1)<#2>{%
    \StartCurveAt#1WithDir{#2}%
    \@ifnextchar\lp@r\@Qurve{%
    \PackageWarning{curve2e}{%
      Quadratic curve specifications must contain at least
      two nodes!\Messagebreak
      Please, control your Qurve specifications\MessageBreak}}}%
\def\@Qurve(#1)<#2>{\QurveTo#1WithDir{#2}%
    \@ifnextchar\lp@r\@Qurve{%
    \@ifnextchar[\@ChangeQDir\CurveEnd}}%
\def\@ChangeQDir[#1]{\ChangeDir<#1>\@Qurve}%
\def\QurveTo#1WithDir#2{%
\edef\@Puno{#1}\DirOfVect#2to\@Duno\bgroup
\DistanceAndDirOfVect\@Puno minus\@Pzero to\@Chord and\@DirChord
\MultVect\@Dzero by*\@Duno to \@Scalar
\YpartOfVect\@Scalar to \@YScalar
\ifdim\@YScalar\p@=\z@
\PackageWarning{curve2e}%
   {Quadratic Bezier arcs cannot have their starting\MessageBreak
   and ending directions parallel or antiparallel with\MessageBreak
   each other. This arc is skipped and replaced with
   a dotted line.\MessageBreak}%
   \Dotline(\@Pzero)(\@Puno){2}\relax
\else
\MultVect\@Dzero by*\@DirChord to \@Dpzero
\MultVect\@Duno by*\@DirChord to \@Dpuno
\GetCoord(\@Dpzero)\@DXpzero\@DYpzero
\GetCoord(\@Dpuno)\@DXpuno\@DYpuno
\MultiplyFN\@DXpzero by\@DXpuno to\@XXD
\MultiplyFN\@DYpzero by\@DYpuno to\@YYD
\unless\ifdim\@YYD\p@<\z@\ifdim\@XXD\p@<\z@
\PackageWarning{curve2e}%
   {Quadratic Bezier arcs cannot have inflection points\MessageBreak
   Therefore the tangents to the starting and ending arc\MessageBreak
   points cannot be directed to the same half plane.\MessageBreak
   This arc is skipped and replaced by a dotted line\MessageBreak}%
   \Dotline(\@Pzero)(\@Puno){2}\fi
\else
\edef\@CDzero{\@DXpzero}\relax
\edef\@SDzero{\@DYpzero}\relax
\edef\@CDuno{\@DXpuno}\relax
\edef\@SDuno{\@DYpuno}\relax
\MultiplyFN\@SDzero by\@CDuno to\@tempA
\MultiplyFN\@SDuno by\@CDzero to\@tempB
\edef\@tempA{\strip@pt\dimexpr\@tempA\p@-\@tempB\p@}\relax
\@tdA=\@SDuno\p@ \@tdB=\@Chord\p@ \@tdC=\@tempA\p@
\edef\@tempC{\strip@pt\dimexpr \@tdA*\@tdB/\@tdC}\relax
\MultiplyFN\@tempC by\@CDzero to \@XC
\MultiplyFN\@tempC by\@SDzero to \@YC
\ModOfVect\@XC,\@YC to\@KC
\ScaleVect\@Dzero by\@KC to\@CP
\AddVect\@Pzero and\@CP to\@CP
\GetCoord(\@Pzero)\@XPzero\@YPzero
\GetCoord(\@Puno)\@XPuno\@YPuno
\GetCoord(\@CP)\@XCP\@YCP
\@ovxx=\@XPzero\unitlength \@ovyy=\@YPzero\unitlength
\@ovdx=\@XCP\unitlength \@ovdy=\@YCP\unitlength
\@xdim=\@XPuno\unitlength \@ydim=\@YPuno\unitlength
      \pIIe@bezier@QtoC\@ovxx\@ovdx\@ovro
      \pIIe@bezier@QtoC\@ovyy\@ovdy\@ovri
      \pIIe@bezier@QtoC\@xdim\@ovdx\@clnwd
      \pIIe@bezier@QtoC\@ydim\@ovdy\@clnht
      \pIIe@moveto\@ovxx\@ovyy
      \pIIe@curveto\@ovro\@ovri\@clnwd\@clnht\@xdim\@ydim
\fi\fi\egroup
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}

%    \end{macrocode}
%\iffalse
%</v161>
%\fi
% \Finale
% \endinput

%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
