texlive[50663] Master/texmfdist: curve2e (30mar19)
commits+karl at tug.org
commits+karl at tug.org
Sat Mar 30 22:26:00 CET 2019
Revision: 50663
http://tug.org/svn/texlive?view=revision&revision=50663
Author: karl
Date: 20190330 22:26:00 +0100 (Sat, 30 Mar 2019)
Log Message:

curve2e (30mar19)
Modified Paths:

trunk/Master/texmfdist/doc/latex/curve2e/README.txt
trunk/Master/texmfdist/doc/latex/curve2e/curve2e.pdf
trunk/Master/texmfdist/source/latex/curve2e/curve2e.dtx
trunk/Master/texmfdist/tex/latex/curve2e/curve2e.sty
Added Paths:

trunk/Master/texmfdist/source/latex/curve2e/ltxdoc.cfg
trunk/Master/texmfdist/tex/latex/curve2e/curve2ev161.sty
Removed Paths:

trunk/Master/texmfdist/doc/latex/curve2e/manifest.txt
Modified: trunk/Master/texmfdist/doc/latex/curve2e/README.txt
===================================================================
 trunk/Master/texmfdist/doc/latex/curve2e/README.txt 20190330 21:25:18 UTC (rev 50662)
+++ trunk/Master/texmfdist/doc/latex/curve2e/README.txt 20190330 21:26:00 UTC (rev 50663)
@@ 10,29 +10,50 @@
%% License information appended
%%
File README.txt for package curve2e
 [2019/02/07 v.1.61 Extension package for pict2e]
+ [20190329 v.2.0.1 Extension package for pict2e]
This file 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 lined 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'.
Some of these features have been incorporated in the 2011 version of pict2e;
therefore this package avoids any modification to the original pict2e commands.
+The package bundle curve2e is composed of the following files
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.
+curve2e.dtx
+curve2e.pdf
+README.txt
+curve2ev161.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.
+
+Curve2ev161.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 lined 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.
+
This version solves a conflict with package esopic.
This version of curve2e is fully compatible with pict2e dated 2014/01/12
version 0.2z.
+This version of curve2e is almost fully compatible with pict2e dated
+2014/01/12 version 0.2z.
If you specify
@@ 40,6 +61,37 @@
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 curve2ev161.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
+
%%
%%
%% Distributable under the LaTeX Project Public License,
Modified: trunk/Master/texmfdist/doc/latex/curve2e/curve2e.pdf
===================================================================
(Binary files differ)
Deleted: trunk/Master/texmfdist/doc/latex/curve2e/manifest.txt
===================================================================
 trunk/Master/texmfdist/doc/latex/curve2e/manifest.txt 20190330 21:25:18 UTC (rev 50662)
+++ trunk/Master/texmfdist/doc/latex/curve2e/manifest.txt 20190330 21:26:00 UTC (rev 50663)
@@ 1,61 +0,0 @@
%%
%% This is file `manifest.txt',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% curve2e.dtx (with options: `manifest')
%%
%% Copyright (C) 20052019 Claudio Beccari all rights reserved.
%% License information appended
%%
File manifest.txt for package curve2e
 [2019/02/07 v.1.61 Extension package for pict2e]


The package bundle curve2e is composed of the following files

curve2e.dtx
curve2e.pdf
mainfest.txt
README.txt

Manifest.txt is this file.

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.

README.txt contains general information.

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 against 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.

Claudio Beccari

claudio.beccari at gmail.com

%%
%%
%% Distributable under the LaTeX Project Public License,
%% version 1.3c or higher (your choice). The latest version of
%% this license is at: http://www.latexproject.org/lppl.txt
%%
%% This work is "authormaintained"
%%
%% This work consists of file curve2e.dtx, and the derived files
%% curve2e.sty and curve2e.pdf, plus the auxiliary derived files
%% README.txt and manifest.txt.
%%
%%
%% End of file `manifest.txt'.
Modified: trunk/Master/texmfdist/source/latex/curve2e/curve2e.dtx
===================================================================
 trunk/Master/texmfdist/source/latex/curve2e/curve2e.dtx 20190330 21:25:18 UTC (rev 50662)
+++ trunk/Master/texmfdist/source/latex/curve2e/curve2e.dtx 20190330 21:26:00 UTC (rev 50663)
@@ 29,7 +29,7 @@
\generate{\file{curve2e.sty}{\from{curve2e.dtx}{package}}}
\generate{\file{README.txt}{\from{curve2e.dtx}{readme}}}
\generate{\file{manifest.txt}{\from{curve2e.dtx}{manifest}}}
+\generate{\file{curve2ev161.sty}{\from{curve2e.dtx}{v161}}}
\def\tmpa{plain}
\ifx\tmpa\fmtname\endgroup\expandafter\bye\fi
@@ 46,10 +46,9 @@
%</driver>
%<+package>\ProvidesPackage{curve2e}%
%<+readme>File README.txt for package curve2e
%<+manifest>File manifest.txt for package curve2e
%<*packagereadmemanifest>
 [2019/02/07 v.1.61 Extension package for pict2e]
%</packagereadmemanifest>
+%<*packagereadme>
+ [20190329 v.2.0.1 Extension package for pict2e]
+%</packagereadme>
%<*driver>
\documentclass{ltxdoc}\errorcontextlines=9
\hfuzz 10pt
@@ 56,15 +55,15 @@
\usepackage[utf8]{inputenc}
\usepackage{lmodern,textcomp}
\usepackage{mflogo}
\usepackage{multicol,amsmath,trace}
+\usepackage{multicol,amsmath,fancyvrb,trace}
\usepackage{curve2e}
\GetFileInfo{curve2e.dtx}
\title{The extension package \textsf{curve2e}}
\author{Claudio Beccari}
\date{Version number \fileversion; last revised \filedate.}
+\author{Claudio Beccari\thanks{Email: \texttt{claudio dot beccari at gmai dot com}}}
+\date{Version \fileversion~~Last revised \filedate.}
\providecommand*\diff{\mathop{}\!\mathrm{d}}
\renewcommand\meta[1]{{\normalfont\textlangle\textit{#1}\textrangle}}
\renewcommand\marg[1]{\texttt{\{\meta{#1}\}}}
+\renewcommand\marg[1]{\texttt{\char123\meta{#1}\char125}}
\providecommand\oarg{}
\renewcommand\oarg[1]{\texttt{[\meta{#1}]}}
\providecommand\aarg{}
@@ 83,13 +82,18 @@
\def\Zbox(#1){\bgroup\edef\@tempA{#1}\@Zbox}
\newcommand*\@Zbox[2][]{\fboxrule\z@\fboxsep=0.75ex\relax
+\newcommand*\@Zbox[2][]{\fboxrule\z@\fboxsep=0.75ex\def\@tempB{#1}%
\setbox2575\hbox{\fbox{$\relax\rule[0.5ex]{0pt}{2.5ex}#2\relax$}}\relax
\ifx\@tempB\empty
\put(\@tempA){\makebox(0,0){\box2575}}\else
\put(\@tempA){\makebox(0,0)[#1]{\box2575}}\fi\egroup\ignorespaces}
\begin{document}
+\providecommand\setfontsize{}
+\DeclareRobustCommand\setfontsize[2][1.2]{%
+\linespread{#1}\fontsize{#2}{#2}\selectfont}
+
+
+\begin{document}%\OnlyDescription
\maketitle
\columnseprule=0.4pt
\begin{multicols}{2}
@@ 100,37 +104,59 @@
%</driver>
% \fi
%
% \CheckSum{3138}
+% \CheckSum{5560}
% \begin{abstract}
% This file documents the curve2e extension package to the recent
% implementation of the pict2e bundle that has been described by Lamport
% himself in the second edition of his \LaTeX\ handbook.
+% This 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
% this package; apparently there are no conflicts, but only the advanced features
% of curve2e remain available for extending the above package.
+% 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 beta version is open to the contribution of
% other users as well as it may be incorporated in other people's packages.
% Please cite the original author and the chain of contributors.
+% 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.
+% 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.
+% 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 onedigit integers of magnitude not exceeding 6 for lines and 4 for
@@ 140,160 +166,194 @@
% 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.
+% \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.
+% \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.
+% \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.
+% 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 threedigit 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.
+% 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.
+% \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.
+% \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.
+% \item The \linethickness command changes the thickness of all lines,
+% straight, curved, vertical, horizontal, arrow tipped, et cetera.
% \end{enumerate}
%
% This specific extension adds the following features.
+% This specific extension package curve2e adds the following features.
% \begin{enumerate}
%\item Most if not all 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 rotoamplification operators.
+% \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 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
% available also with pict2e).
+%\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
+% rotoamplification operators.
%^^A
% \item Commands for specifying the way two lines or curves join to one another.
+% \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 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 available also with pict2e).
+% \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 defined so as to avoid the need to specify the
% horizontal projection of inclined lines (now available also with pict2e);
% 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,
+% \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 \LINE is now renamed \segment; there is no need
% to use the \put command with this line specification.
+% \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.
+% \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.
+% 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.
+% 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.
+% 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 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. 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
+% \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.
+% \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 currently specified color.
+% \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
% low) number of connected Bézier spline arcs with continuos tangents except
% for cusps; these arcs require only the specification of te 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.
+% \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 Last but not least, all these commands accept polar coordinates or
% cartesian ones 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 the cartesian or the polar coordinates.
%^^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
+% 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:coloredcurve}. It is much more
% convenient to use the starred version of \Curve macro.
+% 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 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 yet been done.
+% 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 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.
% The trigonometric functions have also been defined in a way that the author
% believes to be more efficient than those defined by the \texttt{trig} package;
% in any case the macro names are sufficiently different to accommodate both
% definition sets in the same \LaTeX\ run.
+% 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.
+% 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
@@ 300,19 +360,23 @@
% \texttt{pict2e}; they are the only ones who can decide whether or not to
% incorporate new macros in their package.
%
% \section{Summary of modifications and new commands}
% This package \texttt{curve2e} extends the power of \texttt{pict2e} with the
% following modifications and the following new commands.
+%^^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 use the correct options pict2e can receive.
+% \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:
+% \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;
@@ 320,54 +384,86 @@
% \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.
+% 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:
+% \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.
+% 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.
+% \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. Experience has
% shown that the commands intended to join two specified coordinates are
% particularly useful.
+% 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 joins in order the specified
% points; the syntax is:
+% \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.
+% 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}
@@ 386,29 +482,87 @@
% \end{picture}\hspace*{2em}
% \end{minipage}
% \caption{Polygonal line obtained by means of the \texttt{\string\polyline}
% command; coordinates are in polar form.}
+% command; vertex coordinates are in polar form.}
% \label{fig:polyline}
% \end{figure}
%
% Although you can draw polygons with \polyline, as it was done in
% figure~\ref{fig:polyline}, do not confuse this command with the command
% \polygon defined in pict2e 2009; the latter automatically joins the
% last specified coordinate to the first one with a straight line, therefore
% closing the path. pict2e defines also the starred command that fills up
% the inside of the generated polygon.
%^^A
% \item The new command \Dashline (alias: \Dline for backwards compatibility)
+% 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, long as specified,
% and separated by a gap exactly the same size; actually, in order to make an
% even gapdash 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
+% 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 gapdash 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}
@@ 424,6 +578,7 @@
% \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}
@@ 439,50 +594,120 @@
% \Dashline(0,25)(20,40){4}
% \put(0,25){\circle*{2}}
% \put(20,40){\circle*{2}}
% \Dotline(0,0)(40,40){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:
+%\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.
+% 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\unitlengths 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 in 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.
+% with lines separated 10\unitlengths 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 implemented; possibly they
% are not better than the corresponding macros of the trig package, but they
% are supposed to be more accurate at least they were intended to be so. The
% other difference is that angles are specified in sexagesimal degrees
% (360° to one revolution), so that reduction to the fundamental quadrant
% is supposed to be more accurate; the tangent of odd multiples of 90°
% are approximated with a ``\TeX\ infinity'', that is the signed value
% 16383.99999. This will possibly produce computational errors in the
% subsequent calculations, but at least it does not stop the tangent
% computation. In order to avoid overflows or underflows in the computation
% of small angles (reduced to the first quadrant), the sine and the tangent
% of angles smaller than 1° are approximated by the first term of the
% McLaurin series, while for the cosine the approximation is given by the first
% two terms of the McLaurin series. In both cases theoretical errors are smaller
% than what \TeX\ arithmetics can handle.
+% \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:
+% 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}
%\\
@@ 492,19 +717,21 @@
%\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:
+% \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}.
+% 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}
@@ 534,20 +761,22 @@
% \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. They are used to address to 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 the 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, that
% is a backslash followed by letters, or anything else that can receive a
% definition. A `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, generally directly used in the Euler formula $ \vec{v} =
% Me^{\mathrm{j}\varphi}$.
+% \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}
@@ 570,31 +799,85 @@
% \end{itemize}}
%^^A
% \item General curves can be drawn with the pict2e macro \curve but it
% requires the specification of the thirdorder Bézierspline 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. This macro is \Curve and must be followed by an ``unlimited''
% sequence of nodedirection coordinates as a quadruple defined as
+% requires the specification of the thirdorder Bézierspline 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 nodedirection 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
+% 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 nodedirection to the first
% specified nodedirection. 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.
+% that is to close the path from the last specified nodedirection to the
+% first specified nodedirection; 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, as for example with \MF\ or the pgf/tikz
% package and environment. See figure~\ref{fig:curve} for an example.
% \end{enumerate}
+% 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}
@@ 650,96 +933,467 @@
% the macro fills up the contour with the selected current color,
% figure~\ref{fig:coloredcurve}.
%
% 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 issued in 1994; in the
% latexnews news letter of December 2003; the first implementation 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
+% 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:curvaduepunti}. 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)
+%\CurveBetween50,0and50,0WithDirs15:1and{15:1}
+%\CurveBetween50,0and50,0WithDirs30:1and{30:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{45:1}
+%\CurveBetween50,0and50,0WithDirs60:1and{60:1}
+%\CurveBetween50,0and50,0WithDirs75:1and{75:1}
+%\CurveBetween50,0and50,0WithDirs90:1and{90:1}}
+%\put(120,60){%
+%\Line(50,0)(50,0)
+%\CurveBetween50,0and50,0WithDirs15:1and{15:1}
+%\CurveBetween50,0and50,0WithDirs30:1and{30:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{45:1}
+%\CurveBetween50,0and50,0WithDirs60:1and{60:1}
+%\CurveBetween50,0and50,0WithDirs75:1and{75:1}
+%\CurveBetween50,0and50,0WithDirs90:1and{90:1}}
+%\put(0,0){%
+%\Line(50,0)(50,0)
+%\CurveBetween50,0and50,0WithDirs45:1and{15:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{30:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{45:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{60:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{75:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{90:1}}
+%\put(120,0){%
+%\Line(50,0)(50,0)
+%\CurveBetween50,0and50,0WithDirs45:1and{15:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{30:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{45:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{60:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{75:1}
+%\CurveBetween50,0and50,0WithDirs45:1and{90:1}}
+%\end{picture}
+%\caption{Curves between two points with different start and end slopes}\label{fig:curvaduepunti}
+%\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 WithDirs45:1And45:1UsingDists4And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists6And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists8And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists10And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists12And{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 WithDirs45:1And45:1UsingDists4And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists6And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists8And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists10And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists12And{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}
+%\CbezierBetween0,0And20,0WithDirs77:1And77:1UsingDists28And{28}
+%\CbezierBetween20,0And40,0WithDirs77:1And77:1UsingDists28And{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}
+%\CbezierBetween0,0And20,0WithDirs77:1And77:1UsingDists28And{28}
+%\CbezierBetween20,0And40,0WithDirs77:1And77:1UsingDists28And{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 pointdirection
+% 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:quadraticarcs} 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
+% quartercircle 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:quadraticarcs}
+%\end{figure}
+%
+% Notice that the previous version of curve2e contained an error and
+% would color the outside of the green fourpointed star.
+% The curve2ev161, 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 colorfilled quadratic paths.
+%
+% \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 newsletter 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}
% 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.
+%^^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.
+% 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, it appears that ebezier should be used only
% for dvi output without recourse to PostScript machinery.
+% 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.
+% 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 \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.
+% 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}
% 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.
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Josef Tkadlec and the author collaborated extensively in order to make a better
% real long division so as to get the fractional part and to avoid as much as
% possible any numeric overflow; many Josef's ideas are incorporated in the macro
% that is implemented in this package, although the macro used by Josef is
% slightly different from this one. Both versions aim at a better accuracy and
% at widening the operand ranges.
+% 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.
%
% Daniele Degiorgi spotted a fault in the kernel definition of \linethickness
% that heavily influenced also curve2e; see below.
+% 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.
%
% Thanks also to JinHwan 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.
+% 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 JinHwan 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} 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}.
+% \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}
% 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.
+%^^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}
\RequirePackage{color}
\RequirePackageWithOptions{pict2e}[2014/01/01]
\RequirePackage{xparse}
+\IfFileExists{xfp.sty}{%
+ \RequirePackage{color}
+ \RequirePackageWithOptions{pict2e}[2014/01/01]
+ \@ifl at aded{sty}{xparse}{}{\RequirePackage{xparse}}
+ \@ifl at aded{sty}{xfp}{}{\RequirePackage{xfp}}%
+}{%
+ \RequirePackage{curve2ev161}%
+ \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 20181017.\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.
+% 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@}%
@@ 747,8 +1401,9 @@
%
% 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.
+% 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
@@ 759,61 +1414,84 @@
\ifx\undefined\defaultlinewidth \newdimen\defaultlinewidth \fi
% \end{macrocode}
%
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Line thickness macros}
% 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 at inf.ethz.ch}).}
+%^^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 at inf.ethz.ch}).}
% \begin{macrocode}
\gdef\linethickness#1{\@wholewidth#1\@halfwidth.5\@wholewidth\ignorespaces}%
+\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\thinlines{\linethickness{.5\defaultlinewidth}}\thinlines \ignorespaces}%
% \end{macrocode}
% The \ignorespaces at the end of this and the subsequent macros is for
% avoiding spurious spaces to get into the picture that is being drawn, because
+% 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}
% The new macro \LIne allows to draw an arbitrary inclination line as if it
% was a polygonal with just two vertices. This line should be set by means of a
% \put command so that its starting point is always at a relative 0,0
% coordinate point inside the box created with \put. The two arguments
% define the horizontal and the vertical component respectively.
+%^^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 at lineto{\@tX\unitlength}{\@tY\unitlength}\strokepath}\ignorespaces}%
+ \moveto(0,0)
+ \pIIe at 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.
+% 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 $x_2, y_2$. 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
% nonEnglish speaking users, since in all other languages the comma
% must be used as the decimal separator.
+% 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 nonEnglish 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 new division routine that
% receives in input two dimensions and yields on output their fractional ratio.
+% 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. 20181017.
+% 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
@@ 820,28 +1498,29 @@
\@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 at line (an
% arbitrarily named control sequence) after renormalizing to unit magnitude;
% this is passed to GetCoord that separates the two components into the
% control sequences \d at mX and\d at mY; these in turn are the values that are
% actually operated upon by the subsequent commands.
+% 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 at line (an arbitrarily named control sequence) after
+% renormalizing to unit magnitude; this is passed to GetCoord that
+% separates the two components into the control sequences \d at mX and
+% \d at mY; these in turn are the values that are actually operated upon by
+% the subsequent commands.
% \begin{macrocode}
\expandafter\DirOfVect#1to\Dir at line
\GetCoord(\Dir at line)\d at mX\d at mY
% \end{macrocode}
% The normalized vector direction is actually formed with the directing cosines
% of the line direction; since the line length is actually the horizontal
% component for non vertical lines, it is necessary to compute the actual line
% length for non vertical lines by dividing the given length by the
% magnitude of horizontal cosine \d at mX, and the line length is accordingly
% scaled:
+% 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 at mX, and the line
+% length is accordingly scaled:
% \begin{macrocode}
\ifdim\d at mX\p@=\z@\else
 \DividE\ifdim\d at mX\p@<\z at \fi\p@ by\d at mX\p@ to\sc at lelen
+ \edef\sc at lelen{\fpeval{1 / abs(\d at mX)}}\relax
\@linelen=\sc at lelen\@linelen
\fi
% \end{macrocode}
@@ 848,12 +1527,13 @@
% 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
% ``PostScript'' commands instead of resorting to the dvi low level language
% that was used both in \texttt{pict2e} and in the original \texttt{picture}
% commands; it had a meaning in the old times, but it certainly does not have
% any when lines are drawn by the driver that drives the output to a visible
% document form, not by \TeX\ the program.
+% 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 at lineto{\d at mX\@linelen}{\d at mY\@linelen}%
@@ 862,215 +1542,185 @@
\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 preform in a better way with the 2004 version that was limited to
% integer direction coefficients up to 999 in magnitude.
+% 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.
+% 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 $P_2$
+% 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}
% and~\ref{fig:dottedlines} show how to use these macro types.
+% and~\ref{fig:dottedlines} show how to use these macros.
%
% We distinguish these macro types with an asterisk; the unstarred version is
% the first macro type, while the starred one refers to the second macro type.
+% 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 can 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 one. Then the actual
% dash length or dot gap is obtained by dividing the above distance by this
% number.
+% 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 $P_2P_1$ 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.
+% 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.
+% 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} shows the effect of the slight changing
% of the dash length in order to maintain approximately the same dashspace
% pattern along the line, irrespective o the line length.
+% pattern along the line, irrespective of the line length.
+% The syntac 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 pout 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 at ttA
 \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
 \SubVect\V at ttA from\V at ttB to\V at ttC
 \ModOfVect\V at 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 at shMod
 \DividE\p@ by\NumA\p@ to \@tempa
 \MultVect\V at ttC by\@tempa,0 to\V at ttB
 \MultVect\V at ttB by 2,0 to\V at ttC
 \advance\NumB\@ne
 \edef\@mpt{\noexpand\egroup
 \noexpand\multiput(\V at ttA)(\V at ttC){\number\NumB}%
 {\noexpand\LIne(\V at ttB)}}%
 \@mpt\ignorespaces}%
\let\Dline\Dashline
+ \def\Dashline{\@ifstar{\Dashline@@}{\Dashline@}}
\def\Dashline@@(#1)(#2)#3{\put(#1){\Dashline@(0,0)(#2){#3}}}
+ \def\Dashline@(#1)(#2)#3{%
+ \bgroup
+ \countdef\NumA3254\countdef\NumB3252\relax
+ \GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttA
+ \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
+ \SubVect\V at ttA from\V at ttB to\V at ttC
+ \ModOfVect\V at 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 at shMod
+ \DividE\p@ by\NumA\p@ to \@tempa
+ \MultVect\V at ttC by\@tempa,0 to\V at ttB
+ \MultVect\V at ttB by 2,0 to\V at ttC
+ \advance\NumB\@ne
+ \edef\@mpt{\noexpand\egroup
+ \noexpand\multiput(\V at ttA)(\V at ttC){\number\NumB}%
+ {\noexpand\LIne(\V at ttB)}}%
+ \@mpt\ignorespaces}%
+ \let\Dline\Dashline
+
+ \def\Dashline@@(#1)(#2)#3{\put(#1){\Dashline@(0,0)(#2){#3}}}
\fi
% \end{macrocode}
%
%\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{red}\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{red}\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 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 dottedline endpoints; 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:
+% 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 dottedline endpoints; 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}\texttt{(}\meta{start point}\texttt{)(}\meta{end point}\texttt{)\{}\meta{dot distance}\texttt{\}}
+%\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 at ttA
 \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
 \SubVect\V at ttA from\V at ttB to\V at ttC
 \ModOfVect\V at ttC to\DotlineMod
 \DivideFN\DotlineMod by#3 to\NumD
 \NumA=\expandafter\Integer\NumD.??
 \DivVect\V at ttC by\NumA,0 to\V at ttB
 \advance\NumA\@ne
 \edef\@mpt{\noexpand\egroup
 \noexpand\multiput(\V at ttA)(\V at 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}}}
+ \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 at ttA
+ \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
+ \SubVect\V at ttA from\V at ttB to\V at ttC
+ \ModOfVect\V at ttC to\DotlineMod
+ \DivideFN\DotlineMod by#3 to\NumD
+ \NumA=\fpeval{trunc(\NumD,0)}\relax
+ \DivVect\V at ttC by\NumA,0 to\V at ttB
+ \advance\NumA\@ne
+ \edef\@mpt{\noexpand\egroup
+ \noexpand\multiput(\V at ttA)(\V at 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}
%
%\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}
%
% 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} and~\ref{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.
+% always represent a point position referred to the origin of the axes;
+% this is why in figures~\ref{fig:dashedlines} and~\ref{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}
% 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 command is redefined to accept
% the same syntax; the whole work is done by \SplitNod@
% and its subsidiaries.
% Notice that package esopic uses 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 esopic
% had been loaded at the \begin{document} execution, the esopic
% macro is redefined using the e\TeX\ commands so as to make it compatible
% with these local macros. Thanks to FranzJoseph Berthold who was so kind
% to spot the bug.
+%^^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 esopic 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 esopic had been loaded, at the \begin{document}
+% execution the esopic macro is redefined using the e\TeX\ commands
+% so as to make it compatible with these local macros.\footnote{Thanks to
+% FranzJoseph Berthold who was so kind to spot the bug.}
% \begin{macrocode}
\AtBeginDocument{\@ifpackageloaded{esopic}{%
\renewcommand\LenToUnit[1]{\strip at pt\dimexpr#1*\p@/\unitlength}}{}}

+\renewcommand\LenToUnit[1]{\strip at pt\dimexpr#1*\p@/\unitlength}}{}}%
+% \end{macrocode}
+% The above redefinition is delayed at \AtBeginDocument in order to have
+% the possibility to check the the esopic package had actually been
+% loaded. Nevertheless the code is defined here just because the origina
+%l esopic 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 at polar#1:!!(#1)#2#3}%
% \end{macrocode}
% But the macro that detects the form of the coordinates is \isnot at polar,
% that 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:
+% The macro that detects the form of the coordinates is \isnot at 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 at polar#1:#2!!{\def\@tempOne{#2}\ifx\@tempOne\empty
\expandafter\@firstoftwo\else
@@ 1077,7 +1727,6 @@
\expandafter\@secondoftwo\fi
{\SplitNod@@}{\SplitPolar@@}}
\def\SplitNod@(#1)#2#3{\isnot at 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
@@ 1097,62 +1746,29 @@
\gdef\multiput(#1)#2{\bgroup\GetCoord(#1)\@mptX\@mptY
\edef\x{\noexpand\egroup\noexpand\originalmultiput(\@mptX,\@mptY)}\x(}%)
% \end{macrocode}
% 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)[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{minipage}
%\caption{Use of cartesian and polar coordinates}
%\label{fig:polar}
%\end{figure}
%
+%^^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 lefttoright 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 arrow 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.
+% 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 lefttoright 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 thePSTricks
+% 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.
+% 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.
@@ 1162,43 +1778,44 @@
\GetCoord(#1)\d at mX\d at mY
\@linelen#2\unitlength
% \end{macrocode}
% As in \texttt{pict2e} we avoid tracing vectors if the slope parameters are
% both zero.
+% As in \texttt{pict2e} we avoid tracing vectors if the slope parameters
+% are both zero.
% \begin{macrocode}
\ifdim\d at mX\p@=\z@\ifdim\d at mY\p@=\z@\@badlinearg\fi\fi
% \end{macrocode}
% But we check only for the positive nature of the $l_x$ component; if it is
% negative, we simply change sign instead of blocking the typesetting process.
% This is useful also for macros \Vector and \VECTOR to be defined in a
% while.
+% 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.
+% 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 at mX\d at mY to\@Vect
\DirOfVect\@Vect to\Dir at 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
+% transform the components of the vector direction in lengths with the
+% specific names \@xdim and \@ydim^^A! Necessario?
% \begin{macrocode}
\YpartOfVect\Dir at Vect to\@ynum \@ydim=\@ynum\p@
\XpartOfVect\Dir at Vect to\@xnum \@xdim=\@xnum\p@
% \end{macrocode}
% If the vector is really sloping we need to scale the $l_x$ component in order
% to get the vector total length; we have to divide by the cosine of the vector
% inclination which is the real part of the vector direction. I use my division
% macro; since it yields a ``factor'' I directly use it to scale the length of
% the vector. I finally memorize the true vector length in the internal
% dimension @tdB
+% 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 at mX\p@=\z@
\else\ifdim\d at mY\p@=\z@
\else
 \DividE\ifdim\@xnum\p@<\z at \fi\p@ by\@xnum\p@ to\sc at lelen
+ \edef\sc at lelen{\fpeval{1 / abs(\@xnum)}}\relax
\@linelen=\sc at lelen\@linelen
\fi
\fi
@@ 1216,13 +1833,13 @@
\pIIe at vector
\fillpath
% \end{macrocode}
% Now we can restore the stem length that must be shortened by the dimension of
% the arrow; examining the documentation of \texttt{pict2e} we discover that
% we have to shorten it by an approximate amount of $AL$ (with the notations of
% \texttt{pict2e}, figs~10 and~11); the arrow tip parameters are stored in
% certain variables with which we can determine the amount of the stem
% shortening; if the stem was too short and the new length is negative, we
% refrain from designing such stem.
+% 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 at FAW\@wholewidth
@@ 1235,24 +1852,29 @@
\endgroup}
% \end{macrocode}
%
% Now we define the macro that does not require the specification of the length
% or the $l_x$ length component; the way the new \vector macro works does not
% actually require this specification, because \TeX\ can compute the vector
% length, provided the two direction components are exactly the horizontal and
% vertical vector components. If the horizontal component is zero, the actual
% length% must be specified as the vertical component.
+% 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}
+\ifdim\@tX\p@=\z@
+ \vector(\@tX,\@tY){\@tY}%
\else
\vector(\@tX,\@tY){\@tX}\fi}}
+ \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.
+% 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
@@ 1260,49 +1882,37 @@
\endgroup\ignorespaces}
% \end{macrocode}
%
% The \texttt{pict2e} documentation says that if the vector length is zero the
% macro designs only the arrow tip; this may work with macro \vector,
% certainly not with \Vector and \VECTOR. This might be useful for adding
% an arrow tip to a circular arc. See examples in figure~\ref{fig:vectors}.
+% 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}
%
% \begin{figure}
% \begin{minipage}{.48\textwidth}
% \begin{verbatim}
% \unitlength=.5mm
% \begin{picture}(60,20)
% \put(0,0){\GraphGrid(60,20)}
% \put(0,0){\vector(1.5,2.3){10}}
% \put(20,0){\Vector(10,15.33333)}
% \VECTOR(40,0)(50,15.33333)
% \end{picture}
% \end{verbatim}
% \end{minipage}
% \hfill
% \begin{minipage}{.48\textwidth}\centering
% \unitlength=.5mm
% \begin{picture}(60,20)
% \put(0,0){\GraphGrid(60,20)}
% \put(0,0){\vector(1.5,2.3){10}}
% \put(20,0){\Vector(10,15.33333)}
% \VECTOR(40,0)(50,15.33333)
% \end{picture}
% \end{minipage}
% \caption{Three (displaced) identical vectors obtained with the three vector
% macros.}\label{fig:vectors}
% \end{figure}
+% 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}.
%
% \subsection{Polylines}
% We now define the polygonal line macro; its syntax is very simple
+%
+%^^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.
+% 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.
+% 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 at r( \let\rp at r)
% \end{macrocode}
@@ 1314,14 +1924,34 @@
% ignored by the \@ifnextchar macro) then a warning message is output
% together with the line number where the missing parenthesis causes the
% warning: beware, this line number might point to several lines further on
% along the source file! In any case it's necessary to insert a \@killglue
% command, because \polyline refers to absolute coordinates not necessarily
% is put in position through a \put command that provides to eliminate any
% spurious spaces preceding this command.
+% along the source file! In any case it's necessary to insert a
+%\@killgluecommand, 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.
%
% In order to allow a specification for the joints of the various segments of
% a polygonal line it is necessary to allow for an optional parameter; the default
% join is the bevel join.
+% \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:filledpolygon}
+% \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 at lylin@[#1]}
@@ 1335,9 +1965,9 @@
% \end{macrocode}
% But if there is a second or further point coordinate, the recursive macro
% \p at 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.
+% \p at 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 at lyline(#1){\GetCoord(#1)\d at mX\d at mY
\pIIe at lineto{\d at mX\unitlength}{\d at mY\unitlength}%
@@ 1344,7 +1974,13 @@
\@ifnextchar\lp at r{\p at lyline}{\strokepath\ignorespaces}}
% \end{macrocode}
%
% The same treatment must be done for the \cs{polygon} macros
+% 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
@@ 1369,35 +2005,21 @@
% Now, for example, a filled polygon can be drawn using polar coordinates
% for its vertices; see figure~\ref{fig:filledpolygon}
%
% \begin{figure}
% \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:filledpolygon}
% \end{figure}
%
%
+% 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}
% The next command is very useful for debugging while editing one's drawings;
% it draws a red grid with square meshes that are ten drawing units apart;
% there is no graduation along the grid, since it is supposed to be a debugging
% aid and the user should know what he/she is doing; nevertheless it is
% advisable to displace the grid by means of a \put command so that its grid
% lines coincide with the graph coordinates multiples of 10. Missing to do so
% the readings become cumbersome. The \RoundUp macro provides to increase the
+%^^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@}%
@@ 1408,9 +2030,9 @@
\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.
+% 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
@@ 1420,184 +2042,116 @@
\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.
+% 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}
%
% \section{Math operations on fixed radix operands}
+%^^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; LuaTeX can do floating point
% arithmetics through the Lua language that it partially incorporates. But
% this curve2e package is supposed to work also with pdfTeX and XeTeX.
% Therefore the Lua language should not be used.
+% \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{12345e3}, 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.
%
% The only possibility to fake fractional arithmetics is to use fractional
% numbers as multipliers of the unit length \p@ that is 1\,pt long;
% calculations are performed on lengths, and eventually their value,
% extracted from the length registers with the \the command is stripped
% off the ``pt'' component. The \LaTeX\ kernel macro does this in one step.
% At the same time the dimensional expressions made available by the eTeX
% extension to all the \TeX\ system engines, allows to perform all operations
% directly on suitable length registers.
+% 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.
%
% The drawback of working with \TeX\ arithmetics for dimensions is that they
% are saved in binary form in computer words of 32 bits; the sixteen less
% significant bits are reserved for the fractional part; the two more
% significant bits are reserved for the sign and the type of dimension.
% There remain in total 30 bits available for the entire number; just to
% simplify this representation the \TeX\book explains that the computer
% 32bit word contains the dimension in \emph{scaled points}, where 1\,pt
% equals $2^{16}$\,sp.
+% 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.
%
% Since the number of bits of the fractional part is constant (16) it is said
% that the number representation is in \emph{fixed radix}. This is much
% different form the scientific approach to fractional numbers where
% a 32bit word reserves 24 bits to the significant digits, one bit for the sign,
% and a signed exponent of 2 that has 7 significant bits and represents the
% number of binary digits that is necessary to move the binary fractional
% sign to the right or to the left in order to remain with a number greater
% or equal to 1, but lower than 2; this way of coding numbers is called
% \emph{floating point} representation (of course special numbers, such as
% zero, require special codes); \TeX\ fixed radix representation may code
% numbers with absolute value not exceeding ($2^{30}1$)\,sp =1073741823\,sp
% =16383.99998\,pt; a floating point 32bit number cannot exceed in magnitude
% the value of approximately $1.8446744\times 10^{19}$; with fixed radix
% numbers it is possible to evaluate the absolute value of the imprecision
% of the results by summing the absolute imprecision of the terms of
% summation and subtraction; with floating point numbers it is possible to
% estimate the relative imprecision by summing the relative imprecisions of
% the terms of multiplication and division.
%
% Working with fixed radix numbers one must keep in mind that 16 fractional
% binary digits are more or less equivalent to 5 decimal fractional
% digits; and that 16383,99998\,pt are a little less than six meters (5,75832\,m).
% These limits appear completely sufficient to do most computations necessary
% for typography, but when we pretend to make computations of mathematical
% functions with such a poor ``calculator'', we must expect poorly approximated
% results. Nevertheless using the proper iterative algorithms the results are
% not too bad, but certainly it is necessary to accept the situation.
%
% Then why not using the fp package that allows to do computations in \TeX\
% with the floating point representation of numbers? Simply because the results
% would require a lot of time for their execution; this is a serious problem
% with package pgfplots with which it is possible to draw beautiful 2D and
% 3D color diagrams, but at the expense of even dozens of seconds of computation
% time instead of milliseconds.
%
% \subsection{The new division macro}
% The most important macro in the whole package is the division
% macro; it takes two lengths as input values and computes their fractional
% ratio into a control sequence.
%
% It must take care of the signs, so that it examines the operand signs and
% determines the result sign separately conserving this computed sign in the
% macro \segno; this done, we are sure that both operands are or are
% made positive; should the numerator be zero it directly issues the zero
% quotient; should the denominator be zero it outputs
% ``infinity'' (\maxdimen in points), that is the maximum allowable length
% measured in points that \TeX\ can deal with.
%
% Since the result is assigned a value, the calling statement must pass as the
% third argument either a control sequence or an active character. Of course the
% first operand is the dividend, the second the divisor and the third the
% quotient.
%
% Since curve2e is supposed to be an extension of pict2e and this macro
% package already contains a division macro, we might not define any other
% division macro; nevertheless, since the macro in pict2e may not be so
% efficient as it might be if the etex extensions of the interpreter program
% were available, here we check and eventually provide a more efficient macro.
% The latter exploits the scaling mechanism embedded in pdftex since 2007,
% when the extended mode is enabled; it is used to scale a dimension by a
% fraction: $L\times N/D$, where $L$ is a dimension, and $N$ and $D$ are the
% numerator an denominator of the scaling factor; these might be integers, but
% it's better they are both represented by dimension registers, that contain
% two lengths expressed in the same units, possibly the fractional scaling
% factor numerator and denominator that `scale'' the unit length \p@.
%
% Therefore first we test if the extended mode exists and/or is enabled:
+%^^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}
\ifdefined\dimexpr
+\def\DividE#1by#2to#3{%
+ \edef#3{\fpeval{#1 / #2}}\relax
+}
% \end{macrocode}
% then we test if the macro is already defined:
% \begin{macrocode}
 \unless\ifdefined\DividE
% \end{macrocode}
% Notice that \dimexpr is the specific extended mode control sequence we are
% going to use in order to perform our task; if the interpreter program is too
% old and/or it is a recent version, but it was compiled without activating the
% extended mode, the macro \dimexpr is undefined.
+% 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.
%
% The macro \DividE, creates a group where the names of two counters and a
% dimensional register are defined; the numbers of these integer and dimension
% registers are expressly above the value 255, because one of the extensions is
% the possibility of using a virtually unlimited number of registers; moreover
% even if these registers were used within other macros, their use within a group
% does not damage the other macros; we just have to use a Knuthian dirty trick
% to throw the result beyond the endgroup command.
+% 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.
%
% The efficiency of this macro is contained in the extended command \dimexpr;
% both the \@DimA and \Num registers are program words of 32\,bits; the result
% is stored into an internal register of 64\,bits; the final division by a factor
% stored into a register of 32 bits, so that in terms of scaled points a division
% by 1\,pt = $1\times 2^{16}$, scales down the result by 16 bits, and if the total
% length of the result is smaller than $2^{30}$, the result can be correctly
% assigned to a dimension register. In any other case the extended features imply
% suitable error messages but not the termination of the program. During the
% division and a scaling down by 16 bits, the result is not simply truncated, but
% it is rounded to the nearest integer (in scaled points). The first two operands
% are lengths and the third is a macro.
%
+% For backward compatibility we need an alias.
% \begin{macrocode}
 \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 at pt\maxdimen}}%
 \else
 \@DimA=#1\relax
 \edef\x{%
 \noexpand\egroup\noexpand\def\noexpand#3{%
 \strip at pt\dimexpr\@DimA*\Num/\Den\relax}}%
 \fi
 \x\ignorespaces}%
 \fi
+\let\DivideFN\DividE
% \end{macrocode}
%
% We need a similar macro to divide two fractional or integer numbers,
% not dimensions, and produce a macro that contains the fractional result.
% \begin{macrocode}
 \unless\ifdefined\DivideFN
 \def\DivideFN#1by#2to#3{\DividE#1\p@ by#2\p@ to{#3}}%
 \fi
+% 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.
%
% 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
% eliminated by \verb\strip at pt.
% \begin{macrocode}
 \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 at pt\dimexpr\@DimA*\@DimB/\p@\relax}}%
 \x\ignorespaces}%
 \let\MultiplyFN\MultiplY
 \fi
\fi
% \end{macrocode}

% The next macro uses the \verb\strip at 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
@@ 1609,20 +2163,24 @@
\strip at 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,
+% 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.
+% then executes the test; therefore an undefined macro name is always
+% defined to mean \relax.
%
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Trigonometric functions}
% We now start with trigonometric functions. We define the macros \SinOf,
% \CosOf and \TanOf (we might define also \CotOf, but the cotangent does
% not appear so essential) by means of the parametric formulas that require the
% knowledge of the tangent of the half angle. We want to specify the angles
% in sexagesimal degrees, not in radians, so we can make accurate reductions to
% the main quadrants. We use the formulas
+%^^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}\\
@@ 1632,333 +2190,99 @@
% \end{eqnarray*}
% is the half angle in degrees converted to radians.
%
% We use this slightly modified set of parametric formulas because the cotangent
% of $x$ is a by product of the computation of the tangent of $x$; in this way
% we avoid computing the squares of numbers that might lead to overflows. For
% the same reason we avoid computing the value of the trigonometric functions
% in proximity of the value zero (and the other values that might involve high
% tangent or cotangent values) and in that case we prefer to approximate the
% small angle function value with its first or second order truncation of the
% McLaurin series; in facts for angles whose magnitude is smaller than 1°
% the magnitude of the independent variable $y=2x$ (the angle in degrees
% converted to radians) is so small (about 0.017) that the sine and tangent
% can be freely approximated with $y$ itself (the error being smaller than
% approximately $10^{6}$), while the cosine can be freely approximated with
% the formula $10.5y^2$ (the error being smaller than about $10^{6}$).
+% 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.
%
% We keep using grouping so that internal variables are local to these groups
% and do not mess up other things.
+% 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 first macro is the service routine that computes the tangent and the
% cotangent of the half angle in radians; since we have to use always the
% reciprocal of this value, we call it \X@ but in spite of the similarity it
% is the reciprocal of $x$. Notice that parameter \texttt{\#1} must be a length.
+% The codes for the computation of \SinOf and \CosOf of the angle in
+% degrees is therefore the following
% \begin{macrocode}
\def\g at tTanCotanFrom#1to#2and#3{%
\DividE 114.591559\p@ by#1to\X@ \@tdB=\X@\p@
+\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}
%
% Computations are done with the help of counter \I, of the length \@tdB,
% and the auxiliary control sequences \Tan and \Cot whose meaning is
% transparent. The iterative process controlled by \@whilenum implements the
% (truncated) continued fraction expansion of the tangent function:
% \[
% \tan x = \frac{1}{\displaystyle \frac{1\mathstrut}{\displaystyle x}
% \frac{1}{\displaystyle \frac{3\mathstrut}{\displaystyle x}
% \frac{1}{\displaystyle \frac{5\mathstrut}{\displaystyle x}
% \frac{1}{\displaystyle \frac{7\mathstrut}{\displaystyle x}
% \frac{1}{\displaystyle \frac{9\mathstrut}{\displaystyle x}
% \frac{1}{\displaystyle \frac{11\mathstrut}{\displaystyle x}
% \cdots}}}}}}
% \]
% \begin{macrocode}
\countdef\I=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\I2\relax}%
\def#2{\Tan}\DividE\p@ by\Tan\p@ to\Cot \def#3{\Cot}\ignorespaces}%
% \end{macrocode}
%
% Now that we have the macro for computing the tangent and cotangent of the
% half angle, we can compute the real trigonometric functions we are interested
% in. The sine value is computed after reducing the sine argument to the
% interval $0^\circ< \theta<180^\circ$; actually special values such as
% 0°, 90°, 180°, et cetera, are taken care separately, so
% that CPU time is saved for these special cases. The sine sign is taken care
% separately according to the quadrant of the sine argument.
%
% Since all computations are done within a group, a trick is necessary in order to
% extract the sine value from the group; this is done by defining within the group
% a macro (in this case \endSinOf) with the expanded definition of the result,
% but in charge of of closing the group, so that when the group is closed the
% auxiliary function is not defined any more, although its expansion keeps getting
% executed so that the expanded result is thrown beyond the group end.
+% 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\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{+}%
+\def\ArgOfVect#1to#2{\bgroup\GetCoord(#1){\t at X}{\t at Y}%
+\def\s at gno{}%
+\ifdim\t at X\p@=\z@
+ \ifdim\t at Y\p@=\z@
+ \def\ArcTan{0}% vettore nullo
\else
 \def\Segno{}%
 \@tdA=\@tdA
+ \def\ArcTan{90}% vettore verticale
+ \ifdim\t at Y\p@<\z@\def\ArcTan{90}\fi
\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\t at Y\p@=\z@% vettore orizzontale
+ \ifdim\t at X\p@<\z@
+ \def\ArcTan{180}%
\else
 \ifdim\@tdA<\p@
 \@tdA=\Segno0.0174533\@tdA
 \DividE\@tdA by\p@ to \@tempA%
+ \def\ArcTan{0}%
+ \fi
+ \else % vettore qualsiasi
+ \edef\ArcTan{\fpeval{atand(\t at Y / \t at X)}}\relax
+ \ifdim\t at X\p@<\z@% vettore nei quadranti di sinistra
+ \ifdim\t at Y\p@<\z@
+ \edef\ArcTan{\fpeval{\ArcTan  180}}\relax
\else
 \g at tTanCotanFrom\@tdA to\T and\Tp
 \@tdA=\T\p@ \advance\@tdA \Tp\p@
 \DividE \Segno2\p@ by\@tdA to \@tempA%
+ \edef\ArcTan{\fpeval{\ArcTan + 180}}\relax
\fi
\fi
\fi
\fi
\edef\endSinOf{\noexpand\egroup
 \noexpand\def\noexpand#2{\@tempA}\noexpand\ignorespaces}%
\endSinOf}%
+\edef\x{\noexpand\egroup\noexpand\edef\noexpand#2{\ArcTan}}%
+\x\ignorespaces}
% \end{macrocode}
%
% For the computation of the cosine we behave in a similar way using also the
% identical trick for throwing the result beyond the group end.
% \begin{macrocode}
\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
+% It is worth examining the following table, where the angles of nine
+% vectors $45^circ$ degrees 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.
%
\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 at 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}%
% \end{macrocode}
%
% For the tangent computation we behave in a similar way, except that we
% consider the fundamental interval as $0^\circ<\theta<90^\circ$; for the odd
% multiples of 90° we assign the result a \TeX\ infinity value, i.e. \maxdimen,
% the maximum dimension \TeX\ can handle.
% \begin{macrocode}
\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 at 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}%
% \end{macrocode}
+%^^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.
%
% 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. For the principal
% value of the arctangent we would like to use the continued fraction:
%\begin{equation}
%\arctan x = \cfrac{x}{1+ \cfrac{x^2}{3x^2 + \cfrac{(3x)^2}{53x^2 +
% \cfrac{(5x)^2}{75x^2 + \cfrac{(7x)^2}{97x^2 + \ddots}}}}}
%\label{equ:arctanfrazcont}
%\end{equation}
% but after some testing we had to give up due to the slow convergence of
% continued fraction~\eqref{equ:arctanfrazcont}, strictly connected with
% the slow convergence of the McLaurin series from which it is derived.
%
% Waiting for a faster convergence continued fraction, we examined the
% parametric formula and its inverse:
%\begin{subequations}
%\begin{align}
%\tan\theta &= \frac{2\tan(\theta/2))}{1  \tan^2(\theta/2)}\\
%\tan(\theta/2) &= \frac{\sqrt{\tan^2\theta +1}1}{\tan\theta}
%\label{equ:tanfimezzi}
%\end{align}
%\end{subequations}
% If we count the times we use the above formula we can arrive at a point
% where we have to compute the arctangent of a very small value, where the
% arctangent and its argument are approximately equal, so that the angle value
% in radians is equal to its tangent; at that point we multiply by $2^n$,
% where $n$ is the number of bisections, and transform the radians in degrees.
% The procedure is pretty good, even if is is very rudimental and based on an
% approximation; the fixed radix computation of the typesetting engine does
% not help, but we get pretty decent results, although we loose some accuracy
% that hopefully would not harm further computations.
%
% The results obtainable with equation~\eqref{equ:tanfimezzi} are possibly
% acceptable, but the square that must be computed in it tends to go in
% underflow if too many iterations are performed and the algorithm crashes;
% therefore it's virtually impossibile to get an absolute error lower than 0.0005.
%
% It is probably better to refer to the Newton iterations for solving the
% equation:
%\begin{equation}
% \tan\theta \tan\theta_\infty= 0
%\end{equation}
% in the unknown $\theta$ given the value $t=\tan\theta_\infty$; see
% figure~\ref{fig:tangenti}%
%\begin{figure}[!htb]\centering\unitlength=0.01\textwidth
%\begin{picture}(100,80)
%\legenda(15,73){y=\tan\theta}
%\legenda(37,73){t=\tan\theta_\infty}
%\put(0,0){\vector(1,0){100}}\Zbox(100,0)[br]{\theta}
%\put(0,0){\vector(0,1){80}}\Zbox(0,80)[tl]{y}
%\Dashline(90,0)(90,80){2.5}
%\Zbox(90,0)[br]{\pi/2}
%\Pall[2](28.8,11)\Line(0,11)(80,11)%\Dashline(28.8,0)(28.8,11){2.5}
%\Zbox(0,11)[bl]{t}\Zbox(28.8,0)[br]{\theta_\infty}
%
%\Dashline(34,0)(34,11){1.5}\Zbox(34,0)[bl]{\theta_{i+1}}
%
%\Pall[2](47.5,21.83)\Dashline(47.5,0)(47.5,21.83){2.5}
%\Dashline(0,21.83)(47.5,21.83){2.5}
%\Zbox(0,20.83)[bl]{y_i}\Zbox(47.5,0)[bl]{\theta_{i}}
%
%\Pall[2](61.2,36.38)\Dashline(61.2,0)(61.2,36.38){2.5}
%\Dashline(0,36.38)(61.2,36.38){2.5}
%\Zbox(0,36.38)[bl]{y_{i1}}\Zbox(61.2,0)[bl]{\theta_{i1}}
%
%\Pall[2](72,61.55367)\Dashline(72,0)(72,61.55367){2.5}
%\Dashline(0,61.55367)(72,61.55367){2.5}
%\Zbox(0,61.55367)[bl]{y_{i2}}\Zbox(72,0)[bl]{\theta_{i2}}
%
%\put(0,0){\linethickness{1pt}
%\Curve%(0,0)<1,0.34907>(28.8,11)<1,0.45456>(47.5,21.83)<1,0.76479>%
%(61.2,36.38)<1,1.91499>(72,61.55367)<1,4.65428>(74.4,71.6)<1,6.14571>
%}%
%
%\Line(34,11)(47.5,21.83)
%\Line(61.2,36.38)(47.5,11)
%\Line(72,61.55367)(61.2,11)
%\end{picture}
%\caption{Newton's method of tangents}\label{fig:tangenti}
%\end{figure}
%
% The iterative algorithm with Newton method implies the recurrence
%\begin{subequations}\begin{align}
%y'_{i1} &= \frac{\diff\tan(\theta_{i1})}{\diff\theta}
% = \frac{1}{\cos^2\theta_{i1}}\\
%\theta_i &= \theta_{i1}  \frac{\tan \theta_{i1}  t}{y'_{i1}}
% =\theta_{i1}  \cos^2 \theta_{i1}(\tan \theta_{i1}  t)
%\label{equ:iterazione}
%\end{align}
%\end{subequations}
%
% The algorithm starts with an initial value $\theta_0$; at each iteration
% for $i=1, 2, 3,\dots$ a new value of $\theta_i$ is computed from the data
% of the previous iteration $i1$. When for a certain $i$, $\tan\theta_i$
% is sufficiently close to $t$, the iterations may be stopped; since we
% already have the algorithms for computing both the tangent and the cosine;
% such Newton iterative method does not set forth any problem, especially if we
% use the properties of the trigonometric functions and we confine the
% computations to the first quadrant; possibly we limit the computations
% to the first octant and we resort to the cotangent when the tangent exceeds
% one; in this case we use the same algorithm, but we have to get the
% complementary angle; in order to make computations with positive numbers,
% we save the initial tangent sign and we restore it to the result.
% \begin{macrocode}
\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 at 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 at pt\dimexpr\@tG\p at \@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 at pt\dimexpr\@tX\p@  \@tH\p@\relax}\relax
 \advance\I\m at ne}%
\fi
\ifnum\Inverti=\@ne
\edef\@tX{\strip at pt\dimexpr90\p at \@tX\p@\relax}
\fi
\edef\x{\egroup\noexpand\edef\noexpand#2{\Segno\@tX}}\x\ignorespaces}%
% \end{macrocode}
%
% \subsection{Arcs and curves preliminary information}
% We would like to define now a macro for drawing circular arcs of any radius
% and any angular aperture; the macro should require the arc center, the
% arc starting point and the angular aperture. The 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}
@@ 1968,14 +2292,15 @@
% \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
+% 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{\meta{start angle}:\meta{radius}}\marg{angle}
+%\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
@@ 1995,14 +2320,16 @@
% 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 at rotate is such that the
% pivoting point appears to be non relocatable.
+% The end point is obtained from the rotation of the starting point around
+% the center; but the \texttt{pict2e} command \pIIe at 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 rotoamplification operators.
+% defined trigonometric functions and a set of macros that operate on
+% complex numbers used as vector rotoamplification 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
@@ 2010,42 +2337,44 @@
% rotoamplification 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 stress the rotoamplification nature of a complex number:
+% (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 rotoamplification 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.
+% 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,
+% 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.
+% 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 wee 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
+% 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.
+% 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.
+% 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:
+% 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}
@@ 2054,102 +2383,74 @@
\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.
+% arguments; as usual it is assumed that the results are retrieved by means
+% of control sequences, not used directly.
%
% The magnitude $M$ is determined by taking the moduli of the real and
% imaginary parts, changing their signs if necessary; the larger component is
+% 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 is quite easy to get its
% square root by means of the Newton iterative process; due to the quadratic
% convergence, five iterations are more than sufficient. When one of the
% components is zero, the Newton iterative process is skipped. The overall
% macro is the following:
+% 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 ne used for implementing
+% the Newton method in the previous versions of this package.
% \begin{macrocode}
\def\ModOfVect#1to#2{\GetCoord(#1)\t at X\t at Y
\@tempdima=\t at X\p@ \ifdim\@tempdima<\z@ \@tempdima=\@tempdima\fi
\@tempdimb=\t at 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 at ne}%
 \@tempdimc=\@T\@tempdimc
 \fi
\fi
\Numero#2\@tempdimc
+\edef#2{\fpeval{sqrt(\t at X*\t at X + \t at Y*\t at Y)}}\relax
\ignorespaces}%
% \end{macrocode}
% As a byproduct of the computation the control sequence \@tempdimc contains
% a length the value in points of which is the computed root.
%
% Since the macro for determining the magnitude of a vector is available, we
% can now normalize the vector to its magnitude, therefore getting the Cartesian
% form of the direction vector. If by any chance the direction of the null
% vector is requested, the output is again the null vector, without
% normalization.
+% Since the macro for determining the magnitude of a vector is available,
+% we can now normalize the vector to its magnitude, therefore getting the
+% Cartesian form of the direction vector. If by any chance the direction
+% of the null vector is requested, the output is again the null vector,
+% without normalization.
% \begin{macrocode}
\def\DirOfVect#1to#2{\GetCoord(#1)\t at X\t at Y
\ModOfVect#1to\@tempa
\unless\ifdim\@tempdimc=\z@
 \DividE\t at X\p@ by\@tempdimc to\t at X
 \DividE\t at Y\p@ by\@tempdimc to\t at Y
+\unless\ifdim\@tempa\p@=\z@
+ \DividE\t at X\p@ by\@tempa to\t at X
+ \DividE\t at Y\p@ by\@tempa to\t at Y
\fi
\MakeVectorFrom\t at X\t at 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 normalized to unit magnitude (unless the input was the null
% complex number); remember always that output quantities must be specified
% with control sequences to be used at a later time.
+% 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 at X\t at Y
\ModOfVect#1to#2%
\ifdim\@tempdimc=\z@\else
 \DividE\t at X\p@ by\@tempdimc to\t at X
 \DividE\t at Y\p@ by\@tempdimc to\t at Y
\fi
\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
+\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.
+% 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.
+% 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}%
@@ 2158,7 +2459,7 @@
\GetCoord(#1)\@tempa#2\ignorespaces}%
% \end{macrocode}
% With the next macro we create a direction vector (second argument) from a
% given angle (first argument).
+% given angle (first argument, in degrees).
% \begin{macrocode}
\def\DirFromAngle#1to#2{%
\CosOf#1to\t at X
@@ 2166,92 +2467,32 @@
\MakeVectorFrom\t at X\t at Y to#2\ignorespaces}%
% \end{macrocode}
%
% Since we have the algorithm to compute the arctangent of a number,
% it should be relatively easy to compute the angle of a complex number.
% We just have to pay attention that the algorithm to compute the arctangent
% does not care about the quadrant where the complex number lays in, and
% it yields the principal value of the arctan in the domain $\pi/2 <
% \theta \leq \pi/2$. With complex numbers we have just a sign change in
% their angle when they lay in the first or the fourth quadrants; while
% for the third and second quadrants we have to reflect the complex number
% to its opposite and in the result we have to add a ``flat angle'', that
% is 180° since we are working in degrees. Even if mathematically it
% is undefined we decided to assign a null angle to a null complex number;
% possibly a warning message would be helpful, but for drawing purposes
% we think that the problem is irrelevant.
%
+% 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\ArgOfVect#1to#2{\bgroup\GetCoord(#1){\t at X}{\t at Y}%
\def\s at gno{}\def\addflatt at ngle{0}
\ifdim\t at X\p@=\z@
 \ifdim\t at Y\p@=\z@
 \def\ArcTan{0}%
 \else
 \def\ArcTan{90}%
 \ifdim\t at Y\p@<\z@\def\s at gno{}\fi
 \fi
\else
 \ifdim\t at Y\p@=\z@
 \ifdim\t at X\p@<\z@
 \def\ArcTan{180}%
 \else
 \def\ArcTan{0}%
 \fi
 \else
 \ifdim\t at X\p@<\z@%
 \def\addflatt at ngle{180}%
 \edef\t at X{\strip at pt\dimexpr\t at X\p@}%
 \edef\t at Y{\strip at pt\dimexpr\t at Y\p@}%
 \ifdim\t at Y\p@<\z@
 \def\s at gno{}%
 \edef\t at Y{\t at Y}%
 \fi
 \fi
 \DivideFN\t at Y by\t at X to \t at A
 \ArcTanOf\t at A to\ArcTan
 \fi
\fi
\edef\ArcTan{\unless\ifx\s at gno\empty\s at gno\fi\ArcTan}%
\unless\ifnum\addflatt at ngle=0\relax
 \edef\ArcTan{%
 \strip at pt\dimexpr\ArcTan\p@\ifx\s at gno\empty\else+\fi
 \addflatt at ngle\p@\relax}%
\fi
\edef\x{\noexpand\egroup\noexpand\edef\noexpand#2{\ArcTan}}%
\x\ignorespaces}
% \end{macrocode}
%
% It is worth noting that the absolute average error in these computations is
% much lower than 0.0001°; pretty satisfactory since the typesetting engines work
% in fixed radix notation with 16 fractional binary digits, and an error on
% the fourth or fifth fractional decimal digit is almost the best it can be
% expected from this kind of arithmetics.
%
% 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 at X\t at Y
\@tempdima=\t at X\p@ \@tempdima=#2\@tempdima\Numero\t at X\@tempdima
\@tempdima=\t at Y\p@ \@tempdima=#2\@tempdima\Numero\t at Y\@tempdima
+\edef\t at X{\fpeval{#2 * \t at X}}\relax
+\edef\t at Y{\fpeval{#2 * \t at Y}}\relax
\MakeVectorFrom\t at X\t at 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.
+% 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 at X\t at Y
\@tempdima=\t at Y\p@\Numero\t at Y\@tempdima
+\edef\t at Y{\t at Y}%
\MakeVectorFrom\t at X\t at 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:
+% definitions of the binary operations on complex numbers. We start with
+% the addition:
% \begin{macrocode}
\def\AddVect#1and#2to#3{\GetCoord(#1)\tu at X\tu at Y
\GetCoord(#2)\td at X\td at Y
\@tempdima\tu at X\p@\advance\@tempdima\td at X\p@ \Numero\t at X\@tempdima
\@tempdima\tu at Y\p@\advance\@tempdima\td at Y\p@ \Numero\t at Y\@tempdima
+\edef\t at X{\fpeval{\tu at X + \td at X}}\relax
+\edef\t at Y{\fpeval{\tu at Y + \td at Y}}\relax
\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
% \end{macrocode}
% Then the subtraction:
@@ 2258,60 +2499,60 @@
% \begin{macrocode}
\def\SubVect#1from#2to#3{\GetCoord(#1)\tu at X\tu at Y
\GetCoord(#2)\td at X\td at Y
\@tempdima\td at X\p@\advance\@tempdima\tu at X\p@ \Numero\t at X\@tempdima
\@tempdima\td at Y\p@\advance\@tempdima\tu at Y\p@ \Numero\t at Y\@tempdima
+\edef\t at X{\fpeval{\td at X  \tu at X}}\relax
+\edef\t at Y{\fpeval{\td at Y  \tu at Y}}\relax
\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
% \end{macrocode}
%
% For the multiplication we need to split the operation according to the fact
% that we want to multiply by the second operand or by the complex conjugate of
% the second operand; it would be nice if we could use the usual
% postfixed asterisk notation for the complex conjugate, but I could not find
% a simple means for doing so; therefore I use the prefixed notation, that is
% I put the asterisk before the second operand. The first part of the
% multiplication macro just takes care of the multiplicand and then checks for
% the asterisk; if there is no asterisk it calls a second service macro that
% performs a regular complex multiplication, otherwise it calls a third
% service macro that executes the conjugate multiplication.
+% 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; therefore we use the prefixed
+% notation, that is I put the asterisk before the second operand. The first
+% part of the multiplication macro just takes care of the multiplicand and
+% then checks for the asterisk; if there is no asterisk it calls a second
+% service macro that performs a regular complex multiplication, otherwise
+% it calls a third service macro that executes the conjugate
+% multiplication.
% \begin{macrocode}
\def\MultVect#1by{\@ifstar{\@ConjMultVect#1by}{\@MultVect#1by}}%
%
\def\@MultVect#1by#2to#3{\GetCoord(#1)\tu at X\tu at Y
\GetCoord(#2)\td at X\td at Y
\@tempdima\tu at X\p@ \@tempdimb\tu at Y\p@
\@tempdimc=\td at X\@tempdima\advance\@tempdimc\td at Y\@tempdimb
\Numero\t at X\@tempdimc
\@tempdimc=\td at Y\@tempdima\advance\@tempdimc\td at X\@tempdimb
\Numero\t at Y\@tempdimc
+\edef\t at X{\fpeval{\tu at X * \td at X  \tu at Y * \td at Y}}\relax
+\edef\t at Y{\fpeval{\tu at Y * \td at X + \tu at X * \td at Y}}\relax
\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
%
\def\@ConjMultVect#1by#2to#3{\GetCoord(#1)\tu at X\tu at Y
\GetCoord(#2)\td at X\td at Y \@tempdima\tu at X\p@ \@tempdimb\tu at Y\p@
\@tempdimc=\td at X\@tempdima\advance\@tempdimc+\td at Y\@tempdimb
\Numero\t at X\@tempdimc
\@tempdimc=\td at X\@tempdimb\advance\@tempdimc\td at Y\@tempdima
\Numero\t at Y\@tempdimc
+\GetCoord(#2)\td at X\td at Y
+\edef\t at X{\fpeval{\tu at X * \td at X + \tu at Y * \td at Y}}\relax
+\edef\t at Y{\fpeval{\tu at Y * \td at X  \tu at X * \td at Y}}\relax
\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}
% \end{macrocode}
%
% The division of two complex numbers implies scaling down the dividend by the
% magnitude of the divisor and by rotating the dividend scaled vector by the
% opposite direction of the divisor; therefore:
+% The division of two complex numbers implies scaling down the dividend
+% by the magnitude of the divisor and by rotating the dividend scaled
+% vector by the opposite direction of the divisor; therefore:
% \begin{macrocode}
\def\DivVect#1by#2to#3{\ModAndDirOfVect#2to\@Mod and\@Dir
\DividE\p@ by\@Mod\p@ to\@Mod \ConjVect\@Dir to\@Dir
+\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}
% 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.
+%^^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@
@@ 2331,12 +2572,14 @@
\@tdA=\@tdA \let\Segno%
\fi
% \end{macrocode}
% The rotation angle sign is memorized in \Segno and \@tdA now contains the
% absolute value of the arc aperture.
+% 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.
+% 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@
@@ 2346,24 +2589,25 @@
\@whiledim\@tdA>360\p@\do{\advance\@tdA360\p@}%
\fi
% \end{macrocode}
% Now the radius is determined and the drawing point is moved to the stating
% point.
+% 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
+\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.
+% 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}%
+\@@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.
+% 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 at moveto{\@pPunX\unitlength}{\@pPunY\unitlength}%
@@ 2376,7 +2620,8 @@
\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 by1to\@V\fi
+ \MultVect\@V by0,1.3333333to\@V
+ \if\Segno\ScaleVect\@V by1to\@V\fi
\AddVect\@pPun and\@V to\@pcPun
\AddVect\@sPun and\@V to\@scPun
\GetCoord(\@pcPun)\@pcPunX\@pcPunY
@@ 2388,47 +2633,18 @@
\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.
+% 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.
%
%\begin{figure}\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]{$C$}}
%\Line(0,40)(40,0)\Line(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}}
%\Line(40,0)(20,20)\put(20,20){\circle*{2}}
%\put(20,21.5){\makebox(0,0)[b]{$C_1$}}
%\Line(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}
%\caption{Nodes and control points for an arc to be approximated with a cubic Bézier spline}
%\label{fig:arcspline}
%\end{figure}
%
% With reference to figure~\ref{fig:arcspline},
% the points $P_1$ and $P_2$ are the arc endpoints; $C_1$ and $C_2$ are the
% Bézierspline controlpoints; $P$ is the arc midpoint, 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:
+% With reference to figure~\ref{fig:arcspline}, the points $P_1$ and $P_2$
+% are the arc endpoints; $C_1$ and $C_2$ are the Bézierspline
+% controlpoints; $P$ is the arc midpoint, 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)\\
@@ 2441,7 +2657,9 @@
%\begin{equation*}
%P= P_1(1t)^3 + C_1 3(1t)^2t + C_2 3(1t)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
+% 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
@@ 2484,32 +2702,36 @@
\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.
+% \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.
+% 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
+% 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:
@@ 2528,19 +2750,20 @@
\endgroup\ignorespaces}%
% \end{macrocode}
%
% The single arrowed arc is defined with the following long macro where all the
% described operations are performed more or less in the described succession;
% probably the macro requires a little cleaning, but since it works fine I did
% not try to optimize it for time or number of tokens. The final part of the
% macro is almost identical to that of the plain arc; the beginning also is
% quite similar. The central part is dedicated to the positioning of the arrow
% tip and to the necessary calculations for determining the tip tilt and the
% reduction of the total arc length; pay attention that the arrow length, stored
% in \@tdE is a real length, while the radius stored in \@Raggio is just
% a multiple of the \unitlength, so that the division (that yields a good
% angular approximation to the arrow length as seen from the center of the arc)
% must be done with real lengths. The already defined \@@Arc macro actually
% draws the curved vector stem without stroking it.
+% 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@
@@ 2580,9 +2803,9 @@
\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.
+% 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@
@@ 2637,24 +2860,27 @@
% \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.
+% 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. I the direction
+% 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 parametri 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:
+% 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(1t) + P_1t \label{equ:B1} \\
% \mathcal{B}_2 &= P_0(1t)^2 + P_1 2(1t)t + P_2t^2 \label{equ:B2} \\
@@ 2662,141 +2888,111 @@
% \label{equ:B3}
% \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 reaches 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
+% 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 multiarc 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).
+% points. Virtually a multiarc 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.
+% 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.
+% 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}
% 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
+%^^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}%
+ \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 of
% which 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.
+% 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.
%
%\begin{figure}\centering\unitlength=0.004\textwidth
%\begin{picture}(220,120)(50,20)
%\put(0,60){\Line(50,0)(50,0)
%\CurveBetween50,0and50,0WithDirs15:1and{15:1}
%\CurveBetween50,0and50,0WithDirs30:1and{30:1}
%\CurveBetween50,0and50,0WithDirs45:1and{45:1}
%\CurveBetween50,0and50,0WithDirs60:1and{60:1}
%\CurveBetween50,0and50,0WithDirs75:1and{75:1}
%\CurveBetween50,0and50,0WithDirs90:1and{90:1}}
%\put(120,60){%
%\Line(50,0)(50,0)
%\CurveBetween50,0and50,0WithDirs15:1and{15:1}
%\CurveBetween50,0and50,0WithDirs30:1and{30:1}
%\CurveBetween50,0and50,0WithDirs45:1and{45:1}
%\CurveBetween50,0and50,0WithDirs60:1and{60:1}
%\CurveBetween50,0and50,0WithDirs75:1and{75:1}
%\CurveBetween50,0and50,0WithDirs90:1and{90:1}}
%\put(0,0){%
%\Line(50,0)(50,0)
%\CurveBetween50,0and50,0WithDirs45:1and{15:1}
%\CurveBetween50,0and50,0WithDirs45:1and{30:1}
%\CurveBetween50,0and50,0WithDirs45:1and{45:1}
%\CurveBetween50,0and50,0WithDirs45:1and{60:1}
%\CurveBetween50,0and50,0WithDirs45:1and{75:1}
%\CurveBetween50,0and50,0WithDirs45:1and{90:1}}
%\put(120,0){%
%\Line(50,0)(50,0)
%\CurveBetween50,0and50,0WithDirs45:1and{15:1}
%\CurveBetween50,0and50,0WithDirs45:1and{30:1}
%\CurveBetween50,0and50,0WithDirs45:1and{45:1}
%\CurveBetween50,0and50,0WithDirs45:1and{60:1}
%\CurveBetween50,0and50,0WithDirs45:1and{75:1}
%\CurveBetween50,0and50,0WithDirs45:1and{90:1}}
%\end{picture}
%\caption{Curves between two points}\label{fig:curvaduepunti}
%\end{figure}
%
% The first macro initializes the drawing and the third one strokes it; the
% real work is done by the second macro. The first macro initializes the
% drawing but also memorizes the starting direction; the second macro traces
% the current Bézier arc reaching the destination point with the specified
% direction, but memorizes this direction as the one with which to start the
% next arc. The overall curve is then always smooth because the various
% Bézier arcs join with continuous tangents. If a cusp is desired it is
% necessary to change the memorized direction at the end of the arc before the
% cusp and before the start of the next arc; this is better than stroking the
% curve before the cusp and then starting another curve, because the curve
% joining point at the cusp is not stroked with the same command, therefore we get
% two superimposed curve terminations. We therefore need another small macro
% \ChangeDir to perform this task.
+% 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 directions point to the control points,
% but they do not define the control points themselves; they are just
% directions, or, even better, they are simply vectors with the desired
% direction; the macros themselves provide to the normalization and
% memorization.
+% 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 point would be to design a macro that accepts optional node
% directions and computes the missing ones according to a suitable strategy. I
% can think of many such strategies, but none seems to be generally applicable,
% in the sense that one strategy might give good results, say, with sinusoids
% and another one, say, with cardioids, but neither one is suitable for both
% cases.
+% 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.
+% For the moment we refrain from automatic direction computation, but we
+% design the general macro as if directions were optional.
%
% Here we begin with the first initializing macro that receives in the first
% argument the starting point and in the second argument the direction of the
% tangent (not necessarily normalized to a unit vector)
+% 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
@@ 2817,11 +3013,12 @@
\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.
+% 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}
@@ 2843,30 +3040,32 @@
% \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
+% 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 will
% be therefore:
+% starting point, the destination point direction that will save also for
+% the next arcdrawing 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 point 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.
+% 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 at curveto with the proper arguments, and to save the final direction
% into the same \@Dzero macro for successive use of other macros.
+% \pIIe at curveto with the proper arguments, and to save the final
+% direction into the same \@Dzero macro for successive use of other
+% arcdrawing macros.
% \begin{macrocode}
\def\CbezierTo#1WithDir#2AndDists#3And#4{%
\GetCoord(#1)\@tX\@tY \MakeVectorFrom\@tX\@tY to\@Puno
@@ 2886,8 +3085,8 @@
% \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 as the other macro
% \CurveBetween described previously.
+% 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{%
@@ 2902,74 +3101,47 @@
% 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 point are difficult to estimate and require a
+% 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 cutandtry experiments.
%
%\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 WithDirs45:1And45:1UsingDists4And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists6And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists8And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists10And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists12And{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 WithDirs45:1And45:1UsingDists4And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists6And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists8And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists10And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And45:1UsingDists12And{1}
%\end{picture}
%\end{minipage}
%\caption{Comparison between similar arcs drawn with \cs{CurveBetween} (black)
% and \cs{CbezierTo} (red)}
%\label{fig:Cbezier}
%\end{figure}
%
+% 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 ``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.
%
% The strategy I devised consists in determining each control point as if it
% were the control point of a circular arc, precisely an arc of an osculating
% circle, 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: 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 as the first time a path is drawn.
+% 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}
@@ 2976,57 +3148,26 @@
% \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; 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.
+% 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 therefor need a macro to extract the mandatory and optional parts:
+% 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 at semicolon#2\fi}
+
\def\strip at semicolon#1;{\def\@tempB{#1}}
% \end{macrocode}
% By changing the tension values we can achieve different results: see
% figure~\ref{fig:tensions}.
%\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}
%
%
% 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:
+% 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:KzeroKuno}
%\begin{align}
%K_0 &= \frac{4}{3} s\frac{1\cos\theta_0}{\sin^2\theta_0}\\
@@ 3034,8 +3175,8 @@
%\end{align}
%\end{subequations}
%
% We therefore start with getting the points and directions and calculating the
% chord and its direction:
+% 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
@@ 3044,8 +3185,8 @@
\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
+% 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
@@ 3074,9 +3215,9 @@
\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.
+% 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@
@@ 3121,8 +3262,8 @@
\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.
+% 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
@@ 3131,9 +3272,9 @@
{\@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
+% 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
@@ 3141,34 +3282,35 @@
% \end{macrocode}
%
%
% We finally define the overall \Curve macro that has two flavors: 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 nodedirection specification; if the node and direction
% list is terminated, it issues the stroking or filling command through
+% 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 nodedirection 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.
+% 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 at r\@Curve{%
\PackageWarning{curve2e}{%
Curve specifications must contain at least two nodes!\Messagebreak
 Please, control your Curve specifications\MessageBreak}}}
+ Please, control your \string\Curve\space specifications\MessageBreak}}}
\def\@Curve(#1)<#2>{%
\CurveTo#1WithDir{#2}%
\@ifnextchar\lp at r\@Curve{%
@@ 3176,49 +3318,33 @@
\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 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.
%
%\begin{figure}[!htb]
%\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}
%\CbezierBetween0,0And20,0WithDirs77:1And77:1UsingDists28And{28}
%\CbezierBetween20,0And40,0WithDirs77:1And77:1UsingDists28And{28}}
%\end{picture}
%\caption{A sequence of arcs; the left figure has been drawn with the
% \cs{Curve} command with a sequence of four couples of pointdirection
% 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}
%
% As it can be seen in figure~\ref{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.
+% 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
@@ 3227,13 +3353,14 @@
% 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:
+% 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.
+% 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:
@@ 3248,19 +3375,23 @@
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 at 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 nodedirection specifications passed to \Qurve may be split into
% shorter input lines in order to edit the input data in a more comfortable way.
+% (or \FillCurve), with their respective syntax, in such a way that a
+% long list % of nodedirection 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 start with reading its
+% 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{%
@@ 3267,13 +3398,13 @@
\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 antiparallel by taking their
+% It verifies if \@Dpzero and \@Dpuno, the directions at the two
+% extrema of the arc, are parallel or antiparallel 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 skipping this arc
% destination point; evidently the drawing will not be the desired one, but
% the job should not abort.
+% 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
@@ 3286,10 +3417,11 @@
\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:
+% 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
@@ 3297,9 +3429,9 @@
\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, and therefore their product is strictly negative, and that the two
+% 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
@@ 3314,7 +3446,7 @@
\else
% \end{macrocode}
%
% After these tests we should be in a ``normal'' situation.We first copy
+% 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
@@ 3338,16 +3470,19 @@
%\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 comupte the coordinates of the intersection point:
+% 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 antiparallel) and that it lays at
% the same side as the direction with angle $\phi_0$ with respect to the chord.
+% Having performed the previous tests we are sure that the denominator is
+% not vanishing (direction are not parallel or antiparallel) 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
@@ 3359,9 +3494,9 @@
\MultiplyFN\@tempC by\@SDzero to \@YC
\ModOfVect\@XC,\@YC to\@KC
% \end{macrocode}
% We eventually computed 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:
+% 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
@@ 3372,9 +3507,13 @@
% 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 smart cubic arc nodes and control points; we should determine the
% values of $P_a$ and $P_b$ such that
+% 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:
%\[
% P_0(1t)^3 +3P_a(1t)^2t +3P_b(1t)t^2 +P_1t^3
%\]
@@ 3384,13 +3523,13 @@
%\]
% It turns out that the solution is given by
%\begin{equation}
%P_a= C+(P_0C)/3 \qquad \text{and}\qquad P_b = C+(P_1 C)/3
+% P_a= C+(P_0C)/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:
+% 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
@@ 3399,7 +3538,6 @@
\pIIe at bezier@QtoC\@ovyy\@ovdy\@ovri
\pIIe at bezier@QtoC\@xdim\@ovdx\@clnwd
\pIIe at bezier@QtoC\@ydim\@ovdy\@clnht
 \pIIe at moveto\@ovxx\@ovyy
% \end{macrocode}
%
% We call the basic pict2e macro to draw a cubic spline and we finish
@@ 3415,83 +3553,34 @@
\ignorespaces}
% \end{macrocode}
%
% An example of usage is shown at the left in figure~\ref{fig:quadraticarcs}\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.}.
% created with the following code:
%\begin{verbatim}
%\begin{figure}[!htp]
%\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)[tl]{0,0}\Zbox(100,0)[tr]{100,0}
%\Zbox(100,100)[br]{100,100}\Zbox(0,100)[bl]{0,100}
%\end{picture}
%\end{figure}
%\end{verbatim}
+% An example of usage is shown at the left in
+% figure~\ref{fig:quadraticarcs}\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.}.
+% created with the code shown within figure~\ref{fig:quadraticarcs}.
%
%\begin{figure}[!tb]
%\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)[tl]{0,0}\Zbox(100,0)[tr]{100,0}
%\Zbox(100,100)[br]{100,100}\Zbox(0,100)[bl]{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}
+% 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 argument already used also with the
+% \Curve recursive macro.
%
%\caption{\rule{0pt}{4ex}Several graphs drawn with quadratic Bézier splines}
%\label{fig:quadraticarcs}
%\end{figure}
%
% Notice the green filled path: that result is not expected, but the filling
% operation is controlled by the inner workings of the typesetting program,
% where it fills what \emph{it} considers the interior of a path, not what
% \emph{we} think it is the interior of a path.Knowing this feature it is not
% difficult to fill the external lozenge with green, and then fill the internal
% path with white; the result would be to cover with white the external part of
% the interior path. Sort of odd way of getting the result, but this is not due
% to the quadratic splines but to the internal workings of pdftex and its
% companion typesetting engines, that consider ``interior'' the concave side of
% the closed path, not the convex one.
%
% 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 argument 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:quadraticarcs} 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 with a naked eye.
+% 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:quadraticarcs} 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
@@ 3500,89 +3589,1003 @@
% 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
%<*manifest>
+%<*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
mainfest.txt
README.txt
+curve2ev161.sty
+ltxdoc.cfg
Manifest.txt is this file.

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.
+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 contains general information.
+README.txt, this file, contains general information.
+Curve2ev161.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 lined 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.
+
+This version solves a conflict with package esopic.
+
+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 curve2ev161.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 against the parent package
pict2e so as to avoid redefining what has already been incorporated in the
official package.
+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.
+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.beccari at gmail.com
+claudio dot beccari at gmail dot com
% \end{macrocode}
%\iffalse
%</manifest>
+%</readme>
%\fi
%
%\iffalse
%<*readme>
+%<*v161>
%\fi
+% \section{The fallback package version \texttt{curve2ev161}}
+% this is the fallback version of curve2ev161.sty to which the main
+% file curve2e.sty falls back in case the interface package xfp is not
+% available.
% \begin{macrocode}
This file 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.
+\NeedsTeXFormat{LaTeX2e}[2016/01/01]
+\ProvidesPackage{curve2ev161}%
+ [2019/02/07 v.1.61 Extension package for pict2e]
This further extension allows to draw lines and vectors with any non integer
slope parameters, to draw dashed lined 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'.
Some of these features have been incorporated in the 2011 version of pict2e;
therefore this package avoids any modification to the original pict2e commands.
+\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 at 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 at line
+ \GetCoord(\Dir at line)\d at mX\d at mY
+ \ifdim\d at mX\p@=\z@\else
+ \DividE\ifdim\d at mX\p@<\z at \fi\p@ by\d at mX\p@ to\sc at lelen
+ \@linelen=\sc at lelen\@linelen
+ \fi
+ \moveto(0,0)
+ \pIIe at lineto{\d at mX\@linelen}{\d at 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 at ttA
+ \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
+ \SubVect\V at ttA from\V at ttB to\V at ttC
+ \ModOfVect\V at 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 at shMod
+ \DividE\p@ by\NumA\p@ to \@tempa
+ \MultVect\V at ttC by\@tempa,0 to\V at ttB
+ \MultVect\V at ttB by 2,0 to\V at ttC
+ \advance\NumB\@ne
+ \edef\@mpt{\noexpand\egroup
+ \noexpand\multiput(\V at ttA)(\V at ttC){\number\NumB}%
+ {\noexpand\LIne(\V at 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 at ttA
+ \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
+ \SubVect\V at ttA from\V at ttB to\V at ttC
+ \ModOfVect\V at ttC to\DotlineMod
+ \DivideFN\DotlineMod by#3 to\NumD
+ \NumA=\expandafter\Integer\NumD.??
+ \DivVect\V at ttC by\NumA,0 to\V at ttB
+ \advance\NumA\@ne
+ \edef\@mpt{\noexpand\egroup
+ \noexpand\multiput(\V at ttA)(\V at ttB){\number\NumA}%
+ {\noexpand\makebox(0,0){\noexpand\circle*{0.5}}}}%
+ \@mpt\ignorespaces}%
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.
+\def\Dotline@@(#1)(#2)#3{\put(#1){\Dotline@(0,0)(#2){#3}}}
+\fi
+\AtBeginDocument{\@ifpackageloaded{esopic}{%
+\renewcommand\LenToUnit[1]{\strip at pt\dimexpr#1*\p@/\unitlength}}{}}
This version solves a conflict with package esopic.
+\def\GetCoord(#1)#2#3{%
+\expandafter\SplitNod@\expandafter(#1)#2#3\ignorespaces}
+\def\isnot at polar#1:#2!!{\def\@tempOne{#2}\ifx\@tempOne\empty
+\expandafter\@firstoftwo\else
+\expandafter\@secondoftwo\fi
+{\SplitNod@@}{\SplitPolar@@}}
This version of curve2e is fully compatible with pict2e dated 2014/01/12
version 0.2z.
+\def\SplitNod@(#1)#2#3{\isnot at 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}
If you specify
+\let\originalput\put
+\def\put(#1){\bgroup\GetCoord(#1)\@tX\@tY
+\edef\x{\noexpand\egroup\noexpand\originalput(\@tX,\@tY)}\x}
\usepackage[<pict2e options>]{curve2e}
+\let\originalmultiput\multiput
+\let\original at multiput\@multiput
the package pict2e is automatically invoked with the specified options.
+\long\def\@multiput(#1)#2#3{\bgroup\GetCoord(#1)\@mptX\@mptY
+\edef\x{\noexpand\egroup\noexpand\original at 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 at mX\d at mY
+ \@linelen#2\unitlength
+ \ifdim\d at mX\p@=\z@\ifdim\d at mY\p@=\z@\@badlinearg\fi\fi
+ \ifdim\@linelen<\z@ \@linelen=\@linelen\fi
+ \MakeVectorFrom\d at mX\d at mY to\@Vect
+ \DirOfVect\@Vect to\Dir at Vect
+ \YpartOfVect\Dir at Vect to\@ynum \@ydim=\@ynum\p@
+ \XpartOfVect\Dir at Vect to\@xnum \@xdim=\@xnum\p@
+ \ifdim\d at mX\p@=\z@
+ \else\ifdim\d at mY\p@=\z@
+ \else
+ \DividE\ifdim\@xnum\p@<\z at \fi\p@ by\@xnum\p@ to\sc at lelen
+ \@linelen=\sc at lelen\@linelen
+ \fi
+ \fi
+ \@tdB=\@linelen
+\pIIe at concat\@xdim\@ydim{\@ydim}\@xdim{\@xnum\@linelen}{\@ynum\@linelen}%
+ \@linelen\z@
+ \pIIe at vector
+ \fillpath
+ \@linelen=\@tdB
+ \@tdA=\pIIe at FAW\@wholewidth
+ \@tdA=\pIIe at FAL\@tdA
+ \advance\@linelen\@tdA
+ \ifdim\@linelen>\z@
+ \moveto(0,0)
+ \pIIe at 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 at r( \let\rp at r)
+\renewcommand*\polyline[1][\beveljoin]{\p at lylin@[#1]}
+
+\def\p at lylin@[#1](#2){\@killglue#1\GetCoord(#2)\d at mX\d at mY
+ \pIIe at moveto{\d at mX\unitlength}{\d at mY\unitlength}%
+ \@ifnextchar\lp at r{\p at lyline}{%
+ \PackageWarning{curve2e}%
+ {Polylines require at least two vertices!\MessageBreak
+ Control your polyline specification\MessageBreak}%
+ \ignorespaces}}
+
+\def\p at lyline(#1){\GetCoord(#1)\d at mX\d at mY
+ \pIIe at lineto{\d at mX\unitlength}{\d at mY\unitlength}%
+ \@ifnextchar\lp at r{\p at 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 at mX\d at mY
+ \pIIe at moveto{\d at mX\unitlength}{\d at mY\unitlength}%
+ \@ifnextchar\lp at r{\@@polygon}{%
+ \PackageWarning{curve2e}%
+ {Polygons require at least two vertices!\MessageBreak
+ Control your polygon specification\MessageBreak}%
+ \ignorespaces}}
+
+ \def\@@polygon(#1){\GetCoord(#1)\d at mX\d at mY
+ \pIIe at lineto{\d at mX\unitlength}{\d at mY\unitlength}%
+ \@ifnextchar\lp at r{\@@polygon}{\pIIe at closepath
+ \if at tempswa\pIIe at fillGraph\else\pIIe at 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 at pt\maxdimen}}%
+\else
+ \@DimA=#1\relax
+ \edef\x{%
+ \noexpand\egroup\noexpand\def\noexpand#3{%
+ \strip at 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 at 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 at pt\dimen3254}}\x\ignorespaces}%
+\fi
+\def\g at 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\I2\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 at 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 at 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 at 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 at 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 at pt\dimexpr\@tG\p at \@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 at pt\dimexpr\@tX\p@  \@tH\p@\relax}\relax
+ \advance\I\m at ne}%
+\fi
+\ifnum\Inverti=\@ne
+\edef\@tX{\strip at pt\dimexpr90\p at \@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 at X\t at Y
+\@tempdima=\t at X\p@ \ifdim\@tempdima<\z@ \@tempdima=\@tempdima\fi
+\@tempdimb=\t at 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 at ne}%
+ \@tempdimc=\@T\@tempdimc
+ \fi
+\fi
+\Numero#2\@tempdimc
+\ignorespaces}%
+\def\DirOfVect#1to#2{\GetCoord(#1)\t at X\t at Y
+\ModOfVect#1to\@tempa
+\unless\ifdim\@tempdimc=\z@
+ \DividE\t at X\p@ by\@tempdimc to\t at X
+ \DividE\t at Y\p@ by\@tempdimc to\t at Y
+\fi
+\MakeVectorFrom\t at X\t at Y to#2\ignorespaces}%
+\def\ModAndDirOfVect#1to#2and#3{%
+\GetCoord(#1)\t at X\t at Y
+\ModOfVect#1to#2%
+\ifdim\@tempdimc=\z@\else
+ \DividE\t at X\p@ by\@tempdimc to\t at X
+ \DividE\t at Y\p@ by\@tempdimc to\t at Y
+\fi
+\MakeVectorFrom\t at X\t at 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 at X
+\SinOf#1to\t at Y
+\MakeVectorFrom\t at X\t at Y to#2\ignorespaces}%
+\def\ArgOfVect#1to#2{\bgroup\GetCoord(#1){\t at X}{\t at Y}%
+\def\s at gno{}\def\addflatt at ngle{0}
+\ifdim\t at X\p@=\z@
+ \ifdim\t at Y\p@=\z@
+ \def\ArcTan{0}%
+ \else
+ \def\ArcTan{90}%
+ \ifdim\t at Y\p@<\z@\def\s at gno{}\fi
+ \fi
+\else
+ \ifdim\t at Y\p@=\z@
+ \ifdim\t at X\p@<\z@
+ \def\ArcTan{180}%
+ \else
+ \def\ArcTan{0}%
+ \fi
+ \else
+ \ifdim\t at X\p@<\z@%
+ \def\addflatt at ngle{180}%
+ \edef\t at X{\strip at pt\dimexpr\t at X\p@}%
+ \edef\t at Y{\strip at pt\dimexpr\t at Y\p@}%
+ \ifdim\t at Y\p@<\z@
+ \def\s at gno{}%
+ \edef\t at Y{\t at Y}%
+ \fi
+ \fi
+ \DivideFN\t at Y by\t at X to \t at A
+ \ArcTanOf\t at A to\ArcTan
+ \fi
+\fi
+\edef\ArcTan{\unless\ifx\s at gno\empty\s at gno\fi\ArcTan}%
+\unless\ifnum\addflatt at ngle=0\relax
+ \edef\ArcTan{%
+ \strip at pt\dimexpr\ArcTan\p@\ifx\s at gno\empty\else+\fi
+ \addflatt at ngle\p@\relax}%
+\fi
+\edef\x{\noexpand\egroup\noexpand\edef\noexpand#2{\ArcTan}}%
+\x\ignorespaces}
+\def\ScaleVect#1by#2to#3{\GetCoord(#1)\t at X\t at Y
+\@tempdima=\t at X\p@ \@tempdima=#2\@tempdima\Numero\t at X\@tempdima
+\@tempdima=\t at Y\p@ \@tempdima=#2\@tempdima\Numero\t at Y\@tempdima
+\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
+\def\ConjVect#1to#2{\GetCoord(#1)\t at X\t at Y
+\@tempdima=\t at Y\p@\Numero\t at Y\@tempdima
+\MakeVectorFrom\t at X\t at Y to#2\ignorespaces}%
+\def\AddVect#1and#2to#3{\GetCoord(#1)\tu at X\tu at Y
+\GetCoord(#2)\td at X\td at Y
+\@tempdima\tu at X\p@\advance\@tempdima\td at X\p@ \Numero\t at X\@tempdima
+\@tempdima\tu at Y\p@\advance\@tempdima\td at Y\p@ \Numero\t at Y\@tempdima
+\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
+\def\SubVect#1from#2to#3{\GetCoord(#1)\tu at X\tu at Y
+\GetCoord(#2)\td at X\td at Y
+\@tempdima\td at X\p@\advance\@tempdima\tu at X\p@ \Numero\t at X\@tempdima
+\@tempdima\td at Y\p@\advance\@tempdima\tu at Y\p@ \Numero\t at Y\@tempdima
+\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
+\def\MultVect#1by{\@ifstar{\@ConjMultVect#1by}{\@MultVect#1by}}%
+\def\@MultVect#1by#2to#3{\GetCoord(#1)\tu at X\tu at Y
+\GetCoord(#2)\td at X\td at Y
+\@tempdima\tu at X\p@ \@tempdimb\tu at Y\p@
+\@tempdimc=\td at X\@tempdima\advance\@tempdimc\td at Y\@tempdimb
+\Numero\t at X\@tempdimc
+\@tempdimc=\td at Y\@tempdima\advance\@tempdimc\td at X\@tempdimb
+\Numero\t at Y\@tempdimc
+\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
+\def\@ConjMultVect#1by#2to#3{\GetCoord(#1)\tu at X\tu at Y
+\GetCoord(#2)\td at X\td at Y \@tempdima\tu at X\p@ \@tempdimb\tu at Y\p@
+\@tempdimc=\td at X\@tempdima\advance\@tempdimc+\td at Y\@tempdimb
+\Numero\t at X\@tempdimc
+\@tempdimc=\td at X\@tempdimb\advance\@tempdimc\td at Y\@tempdima
+\Numero\t at Y\@tempdimc
+\MakeVectorFrom\t at X\t at 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 0360 taking the sign into consideration}%
+ \@whiledim\@tdA>360\p@\do{\advance\@tdA360\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 at moveto{\@pPunX\unitlength}{\@pPunY\unitlength}%
+\ifdim\@tdA>180\p@
+ \advance\@tdA180\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 by1to\@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 at 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 by1to\@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 at 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 0360 taking the sign into consideration}%
+ \@whiledim\@tdA>360\p@\do{\advance\@tdA360\p@}%
+\fi
+\SubVect#1from#2to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
+\@tdE=\pIIe at FAW\@wholewidth \@tdE=\pIIe at 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 at 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 0360 taking the sign into consideration}%
+ \@whiledim\@tdA>360\p@\do{\advance\@tdA360\p@}%
+\fi
+\SubVect#1from#2to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
+\@tdE=\pIIe at 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 at 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 at 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 at semicolon#2\fi}
+\def\strip at 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 at \@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 at \@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 at 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 at 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 at 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 at 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 at 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 at pt\dimexpr\@tempA\p at \@tempB\p@}\relax
+\@tdA=\@SDuno\p@ \@tdB=\@Chord\p@ \@tdC=\@tempA\p@
+\edef\@tempC{\strip at 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 at bezier@QtoC\@ovxx\@ovdx\@ovro
+ \pIIe at bezier@QtoC\@ovyy\@ovdy\@ovri
+ \pIIe at bezier@QtoC\@xdim\@ovdx\@clnwd
+ \pIIe at bezier@QtoC\@ydim\@ovdy\@clnht
+ \pIIe at moveto\@ovxx\@ovyy
+ \pIIe at curveto\@ovro\@ovri\@clnwd\@clnht\@xdim\@ydim
+\fi\fi\egroup
+\CopyVect\@Puno to\@Pzero
+\CopyVect\@Duno to\@Dzero
+\ignorespaces}
+
% \end{macrocode}
%\iffalse
%</readme>
+%</v161>
%\fi
% \Finale
% \endinput
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Added: trunk/Master/texmfdist/source/latex/curve2e/ltxdoc.cfg
===================================================================
 trunk/Master/texmfdist/source/latex/curve2e/ltxdoc.cfg (rev 0)
+++ trunk/Master/texmfdist/source/latex/curve2e/ltxdoc.cfg 20190330 21:26:00 UTC (rev 50663)
@@ 0,0 +1,7 @@
+%%
+%% This is file `ltxdoc.cfg',
+
+\AtBeginDocument{\OnlyDescription}
+
+%%
+%% End of file `ltxdoc.cfg'.
Property changes on: trunk/Master/texmfdist/source/latex/curve2e/ltxdoc.cfg
___________________________________________________________________
Added: svn:eolstyle
## 0,0 +1 ##
+native
\ No newline at end of property
Added: trunk/Master/texmfdist/tex/latex/curve2e/curve2ev161.sty
===================================================================
 trunk/Master/texmfdist/tex/latex/curve2e/curve2ev161.sty (rev 0)
+++ trunk/Master/texmfdist/tex/latex/curve2e/curve2ev161.sty 20190330 21:26:00 UTC (rev 50663)
@@ 0,0 +1,904 @@
+%%
+%% This is file `curve2ev161.sty',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% curve2e.dtx (with options: `v161')
+%%
+%% Copyright (C) 20052019 Claudio Beccari all rights reserved.
+%% License information appended
+%%
+
+
+
+\NeedsTeXFormat{LaTeX2e}[2016/01/01]
+\ProvidesPackage{curve2ev161}%
+ [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 at 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 at line
+ \GetCoord(\Dir at line)\d at mX\d at mY
+ \ifdim\d at mX\p@=\z@\else
+ \DividE\ifdim\d at mX\p@<\z at \fi\p@ by\d at mX\p@ to\sc at lelen
+ \@linelen=\sc at lelen\@linelen
+ \fi
+ \moveto(0,0)
+ \pIIe at lineto{\d at mX\@linelen}{\d at 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 at ttA
+ \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
+ \SubVect\V at ttA from\V at ttB to\V at ttC
+ \ModOfVect\V at 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 at shMod
+ \DividE\p@ by\NumA\p@ to \@tempa
+ \MultVect\V at ttC by\@tempa,0 to\V at ttB
+ \MultVect\V at ttB by 2,0 to\V at ttC
+ \advance\NumB\@ne
+ \edef\@mpt{\noexpand\egroup
+ \noexpand\multiput(\V at ttA)(\V at ttC){\number\NumB}%
+ {\noexpand\LIne(\V at 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 at ttA
+ \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
+ \SubVect\V at ttA from\V at ttB to\V at ttC
+ \ModOfVect\V at ttC to\DotlineMod
+ \DivideFN\DotlineMod by#3 to\NumD
+ \NumA=\expandafter\Integer\NumD.??
+ \DivVect\V at ttC by\NumA,0 to\V at ttB
+ \advance\NumA\@ne
+ \edef\@mpt{\noexpand\egroup
+ \noexpand\multiput(\V at ttA)(\V at 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{esopic}{%
+\renewcommand\LenToUnit[1]{\strip at pt\dimexpr#1*\p@/\unitlength}}{}}
+
+\def\GetCoord(#1)#2#3{%
+\expandafter\SplitNod@\expandafter(#1)#2#3\ignorespaces}
+\def\isnot at polar#1:#2!!{\def\@tempOne{#2}\ifx\@tempOne\empty
+\expandafter\@firstoftwo\else
+\expandafter\@secondoftwo\fi
+{\SplitNod@@}{\SplitPolar@@}}
+
+\def\SplitNod@(#1)#2#3{\isnot at 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 at multiput\@multiput
+
+\long\def\@multiput(#1)#2#3{\bgroup\GetCoord(#1)\@mptX\@mptY
+\edef\x{\noexpand\egroup\noexpand\original at 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 at mX\d at mY
+ \@linelen#2\unitlength
+ \ifdim\d at mX\p@=\z@\ifdim\d at mY\p@=\z@\@badlinearg\fi\fi
+ \ifdim\@linelen<\z@ \@linelen=\@linelen\fi
+ \MakeVectorFrom\d at mX\d at mY to\@Vect
+ \DirOfVect\@Vect to\Dir at Vect
+ \YpartOfVect\Dir at Vect to\@ynum \@ydim=\@ynum\p@
+ \XpartOfVect\Dir at Vect to\@xnum \@xdim=\@xnum\p@
+ \ifdim\d at mX\p@=\z@
+ \else\ifdim\d at mY\p@=\z@
+ \else
+ \DividE\ifdim\@xnum\p@<\z at \fi\p@ by\@xnum\p@ to\sc at lelen
+ \@linelen=\sc at lelen\@linelen
+ \fi
+ \fi
+ \@tdB=\@linelen
+\pIIe at concat\@xdim\@ydim{\@ydim}\@xdim{\@xnum\@linelen}{\@ynum\@linelen}%
+ \@linelen\z@
+ \pIIe at vector
+ \fillpath
+ \@linelen=\@tdB
+ \@tdA=\pIIe at FAW\@wholewidth
+ \@tdA=\pIIe at FAL\@tdA
+ \advance\@linelen\@tdA
+ \ifdim\@linelen>\z@
+ \moveto(0,0)
+ \pIIe at 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 at r( \let\rp at r)
+\renewcommand*\polyline[1][\beveljoin]{\p at lylin@[#1]}
+
+\def\p at lylin@[#1](#2){\@killglue#1\GetCoord(#2)\d at mX\d at mY
+ \pIIe at moveto{\d at mX\unitlength}{\d at mY\unitlength}%
+ \@ifnextchar\lp at r{\p at lyline}{%
+ \PackageWarning{curve2e}%
+ {Polylines require at least two vertices!\MessageBreak
+ Control your polyline specification\MessageBreak}%
+ \ignorespaces}}
+
+\def\p at lyline(#1){\GetCoord(#1)\d at mX\d at mY
+ \pIIe at lineto{\d at mX\unitlength}{\d at mY\unitlength}%
+ \@ifnextchar\lp at r{\p at 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 at mX\d at mY
+ \pIIe at moveto{\d at mX\unitlength}{\d at mY\unitlength}%
+ \@ifnextchar\lp at r{\@@polygon}{%
+ \PackageWarning{curve2e}%
+ {Polygons require at least two vertices!\MessageBreak
+ Control your polygon specification\MessageBreak}%
+ \ignorespaces}}
+
+ \def\@@polygon(#1){\GetCoord(#1)\d at mX\d at mY
+ \pIIe at lineto{\d at mX\unitlength}{\d at mY\unitlength}%
+ \@ifnextchar\lp at r{\@@polygon}{\pIIe at closepath
+ \if at tempswa\pIIe at fillGraph\else\pIIe at 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 at pt\maxdimen}}%
+\else
+ \@DimA=#1\relax
+ \edef\x{%
+ \noexpand\egroup\noexpand\def\noexpand#3{%
+ \strip at 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 at 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 at pt\dimen3254}}\x\ignorespaces}%
+\fi
+\def\g at 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\I2\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 at 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 at 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 at 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 at 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 at pt\dimexpr\@tG\p at \@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 at pt\dimexpr\@tX\p@  \@tH\p@\relax}\relax
+ \advance\I\m at ne}%
+\fi
+\ifnum\Inverti=\@ne
+\edef\@tX{\strip at pt\dimexpr90\p at \@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 at X\t at Y
+\@tempdima=\t at X\p@ \ifdim\@tempdima<\z@ \@tempdima=\@tempdima\fi
+\@tempdimb=\t at 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 at ne}%
+ \@tempdimc=\@T\@tempdimc
+ \fi
+\fi
+\Numero#2\@tempdimc
+\ignorespaces}%
+\def\DirOfVect#1to#2{\GetCoord(#1)\t at X\t at Y
+\ModOfVect#1to\@tempa
+\unless\ifdim\@tempdimc=\z@
+ \DividE\t at X\p@ by\@tempdimc to\t at X
+ \DividE\t at Y\p@ by\@tempdimc to\t at Y
+\fi
+\MakeVectorFrom\t at X\t at Y to#2\ignorespaces}%
+\def\ModAndDirOfVect#1to#2and#3{%
+\GetCoord(#1)\t at X\t at Y
+\ModOfVect#1to#2%
+\ifdim\@tempdimc=\z@\else
+ \DividE\t at X\p@ by\@tempdimc to\t at X
+ \DividE\t at Y\p@ by\@tempdimc to\t at Y
+\fi
+\MakeVectorFrom\t at X\t at 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 at X
+\SinOf#1to\t at Y
+\MakeVectorFrom\t at X\t at Y to#2\ignorespaces}%
+\def\ArgOfVect#1to#2{\bgroup\GetCoord(#1){\t at X}{\t at Y}%
+\def\s at gno{}\def\addflatt at ngle{0}
+\ifdim\t at X\p@=\z@
+ \ifdim\t at Y\p@=\z@
+ \def\ArcTan{0}%
+ \else
+ \def\ArcTan{90}%
+ \ifdim\t at Y\p@<\z@\def\s at gno{}\fi
+ \fi
+\else
+ \ifdim\t at Y\p@=\z@
+ \ifdim\t at X\p@<\z@
+ \def\ArcTan{180}%
+ \else
+ \def\ArcTan{0}%
+ \fi
+ \else
+ \ifdim\t at X\p@<\z@%
+ \def\addflatt at ngle{180}%
+ \edef\t at X{\strip at pt\dimexpr\t at X\p@}%
+ \edef\t at Y{\strip at pt\dimexpr\t at Y\p@}%
+ \ifdim\t at Y\p@<\z@
+ \def\s at gno{}%
+ \edef\t at Y{\t at Y}%
+ \fi
+ \fi
+ \DivideFN\t at Y by\t at X to \t at A
+ \ArcTanOf\t at A to\ArcTan
+ \fi
+\fi
+\edef\ArcTan{\unless\ifx\s at gno\empty\s at gno\fi\ArcTan}%
+\unless\ifnum\addflatt at ngle=0\relax
+ \edef\ArcTan{%
+ \strip at pt\dimexpr\ArcTan\p@\ifx\s at gno\empty\else+\fi
+ \addflatt at ngle\p@\relax}%
+\fi
+\edef\x{\noexpand\egroup\noexpand\edef\noexpand#2{\ArcTan}}%
+\x\ignorespaces}
+\def\ScaleVect#1by#2to#3{\GetCoord(#1)\t at X\t at Y
+\@tempdima=\t at X\p@ \@tempdima=#2\@tempdima\Numero\t at X\@tempdima
+\@tempdima=\t at Y\p@ \@tempdima=#2\@tempdima\Numero\t at Y\@tempdima
+\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
+\def\ConjVect#1to#2{\GetCoord(#1)\t at X\t at Y
+\@tempdima=\t at Y\p@\Numero\t at Y\@tempdima
+\MakeVectorFrom\t at X\t at Y to#2\ignorespaces}%
+\def\AddVect#1and#2to#3{\GetCoord(#1)\tu at X\tu at Y
+\GetCoord(#2)\td at X\td at Y
+\@tempdima\tu at X\p@\advance\@tempdima\td at X\p@ \Numero\t at X\@tempdima
+\@tempdima\tu at Y\p@\advance\@tempdima\td at Y\p@ \Numero\t at Y\@tempdima
+\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
+\def\SubVect#1from#2to#3{\GetCoord(#1)\tu at X\tu at Y
+\GetCoord(#2)\td at X\td at Y
+\@tempdima\td at X\p@\advance\@tempdima\tu at X\p@ \Numero\t at X\@tempdima
+\@tempdima\td at Y\p@\advance\@tempdima\tu at Y\p@ \Numero\t at Y\@tempdima
+\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
+\def\MultVect#1by{\@ifstar{\@ConjMultVect#1by}{\@MultVect#1by}}%
+\def\@MultVect#1by#2to#3{\GetCoord(#1)\tu at X\tu at Y
+\GetCoord(#2)\td at X\td at Y
+\@tempdima\tu at X\p@ \@tempdimb\tu at Y\p@
+\@tempdimc=\td at X\@tempdima\advance\@tempdimc\td at Y\@tempdimb
+\Numero\t at X\@tempdimc
+\@tempdimc=\td at Y\@tempdima\advance\@tempdimc\td at X\@tempdimb
+\Numero\t at Y\@tempdimc
+\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
+\def\@ConjMultVect#1by#2to#3{\GetCoord(#1)\tu at X\tu at Y
+\GetCoord(#2)\td at X\td at Y \@tempdima\tu at X\p@ \@tempdimb\tu at Y\p@
+\@tempdimc=\td at X\@tempdima\advance\@tempdimc+\td at Y\@tempdimb
+\Numero\t at X\@tempdimc
+\@tempdimc=\td at X\@tempdimb\advance\@tempdimc\td at Y\@tempdima
+\Numero\t at Y\@tempdimc
+\MakeVectorFrom\t at X\t at 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 0360 taking the sign into consideration}%
+ \@whiledim\@tdA>360\p@\do{\advance\@tdA360\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 at moveto{\@pPunX\unitlength}{\@pPunY\unitlength}%
+\ifdim\@tdA>180\p@
+ \advance\@tdA180\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 by1to\@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 at 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 by1to\@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 at 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 0360 taking the sign into consideration}%
+ \@whiledim\@tdA>360\p@\do{\advance\@tdA360\p@}%
+\fi
+\SubVect#1from#2to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
+\@tdE=\pIIe at FAW\@wholewidth \@tdE=\pIIe at 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 at 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 0360 taking the sign into consideration}%
+ \@whiledim\@tdA>360\p@\do{\advance\@tdA360\p@}%
+\fi
+\SubVect#1from#2to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
+\@tdE=\pIIe at 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 at 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 at 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 at semicolon#2\fi}
+\def\strip at 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 at \@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 at \@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 at 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 at 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 at 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 at 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 at 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 at pt\dimexpr\@tempA\p at \@tempB\p@}\relax
+\@tdA=\@SDuno\p@ \@tdB=\@Chord\p@ \@tdC=\@tempA\p@
+\edef\@tempC{\strip at 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 at bezier@QtoC\@ovxx\@ovdx\@ovro
+ \pIIe at bezier@QtoC\@ovyy\@ovdy\@ovri
+ \pIIe at bezier@QtoC\@xdim\@ovdx\@clnwd
+ \pIIe at bezier@QtoC\@ydim\@ovdy\@clnht
+ \pIIe at moveto\@ovxx\@ovyy
+ \pIIe at curveto\@ovro\@ovri\@clnwd\@clnht\@xdim\@ydim
+\fi\fi\egroup
+\CopyVect\@Puno to\@Pzero
+\CopyVect\@Duno to\@Dzero
+\ignorespaces}
+
+
+%%
+%%
+%% Distributable under the LaTeX Project Public License,
+%% version 1.3c or higher (your choice). The latest version of
+%% this license is at: http://www.latexproject.org/lppl.txt
+%%
+%% This work is "authormaintained"
+%%
+%% This work consists of file curve2e.dtx, and the derived files
+%% curve2e.sty and curve2e.pdf, plus the auxiliary derived files
+%% README.txt and manifest.txt.
+%%
+%%
+%% End of file `curve2ev161.sty'.
Property changes on: trunk/Master/texmfdist/tex/latex/curve2e/curve2ev161.sty
___________________________________________________________________
Added: svn:eolstyle
## 0,0 +1 ##
+native
\ No newline at end of property
Modified: trunk/Master/texmfdist/tex/latex/curve2e/curve2e.sty
===================================================================
 trunk/Master/texmfdist/tex/latex/curve2e/curve2e.sty 20190330 21:25:18 UTC (rev 50662)
+++ trunk/Master/texmfdist/tex/latex/curve2e/curve2e.sty 20190330 21:26:00 UTC (rev 50663)
@@ 11,12 +11,30 @@
%%
\NeedsTeXFormat{LaTeX2e}[2016/01/01]
\ProvidesPackage{curve2e}%
 [2019/02/07 v.1.61 Extension package for pict2e]
+ [20190329 v.2.0.1 Extension package for pict2e]
\RequirePackage{color}
\RequirePackageWithOptions{pict2e}[2014/01/01]
\RequirePackage{xparse}
+
+\IfFileExists{xfp.sty}{%
+ \RequirePackage{color}
+ \RequirePackageWithOptions{pict2e}[2014/01/01]
+ \@ifl at aded{sty}{xparse}{}{\RequirePackage{xparse}}
+ \@ifl at aded{sty}{xfp}{}{\RequirePackage{xfp}}%
+}{%
+ \RequirePackage{curve2ev161}%
+ \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 20181017.\MessageBreak%
+ \MessageBreak%
+ ***************************************\MessageBreak%
+ Version 1.61 of curve2e has been loaded\MessageBreak%
+ instead of the current version\MessageBreak%
+ ***************************************\MessageBreak}%
+ \endinput
+}
\def\TRON{\tracingcommands\tw@ \tracingmacros\tw@}%
\def\TROF{\tracingcommands\z@ \tracingmacros\z@}%
\ifx\undefined\@tdA \newdimen\@tdA \fi
@@ 26,14 +44,15 @@
\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}%
+\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\thinlines{\linethickness{.5\defaultlinewidth}}\thinlines \ignorespaces}%
\def\LIne(#1){{\GetCoord(#1)\@tX\@tY
 \moveto(0,0)
 \pIIe at lineto{\@tX\unitlength}{\@tY\unitlength}\strokepath}\ignorespaces}%
+ \moveto(0,0)
+ \pIIe at lineto{\@tX\unitlength}{\@tY\unitlength}\strokepath}\ignorespaces
+}%
\def\segment(#1)(#2){\@killglue\polyline(#1)(#2)}%
\def\line(#1)#2{\begingroup
\@linelen #2\unitlength
@@ 41,7 +60,7 @@
\expandafter\DirOfVect#1to\Dir at line
\GetCoord(\Dir at line)\d at mX\d at mY
\ifdim\d at mX\p@=\z@\else
 \DividE\ifdim\d at mX\p@<\z at \fi\p@ by\d at mX\p@ to\sc at lelen
+ \edef\sc at lelen{\fpeval{1 / abs(\d at mX)}}\relax
\@linelen=\sc at lelen\@linelen
\fi
\moveto(0,0)
@@ 50,62 +69,62 @@
\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 at ttA
 \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
 \SubVect\V at ttA from\V at ttB to\V at ttC
 \ModOfVect\V at 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 at shMod
 \DividE\p@ by\NumA\p@ to \@tempa
 \MultVect\V at ttC by\@tempa,0 to\V at ttB
 \MultVect\V at ttB by 2,0 to\V at ttC
 \advance\NumB\@ne
 \edef\@mpt{\noexpand\egroup
 \noexpand\multiput(\V at ttA)(\V at ttC){\number\NumB}%
 {\noexpand\LIne(\V at ttB)}}%
 \@mpt\ignorespaces}%
\let\Dline\Dashline
+ \def\Dashline{\@ifstar{\Dashline@@}{\Dashline@}}
\def\Dashline@@(#1)(#2)#3{\put(#1){\Dashline@(0,0)(#2){#3}}}
+ \def\Dashline@(#1)(#2)#3{%
+ \bgroup
+ \countdef\NumA3254\countdef\NumB3252\relax
+ \GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttA
+ \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
+ \SubVect\V at ttA from\V at ttB to\V at ttC
+ \ModOfVect\V at 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 at shMod
+ \DividE\p@ by\NumA\p@ to \@tempa
+ \MultVect\V at ttC by\@tempa,0 to\V at ttB
+ \MultVect\V at ttB by 2,0 to\V at ttC
+ \advance\NumB\@ne
+ \edef\@mpt{\noexpand\egroup
+ \noexpand\multiput(\V at ttA)(\V at ttC){\number\NumB}%
+ {\noexpand\LIne(\V at 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 at ttA
 \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
 \SubVect\V at ttA from\V at ttB to\V at ttC
 \ModOfVect\V at ttC to\DotlineMod
 \DivideFN\DotlineMod by#3 to\NumD
 \NumA=\expandafter\Integer\NumD.??
 \DivVect\V at ttC by\NumA,0 to\V at ttB
 \advance\NumA\@ne
 \edef\@mpt{\noexpand\egroup
 \noexpand\multiput(\V at ttA)(\V at ttB){\number\NumA}%
 {\noexpand\makebox(0,0){\noexpand\circle*{0.5}}}}%
 \@mpt\ignorespaces}%
+ \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 at ttA
+ \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V at ttB
+ \SubVect\V at ttA from\V at ttB to\V at ttC
+ \ModOfVect\V at ttC to\DotlineMod
+ \DivideFN\DotlineMod by#3 to\NumD
+ \NumA=\fpeval{trunc(\NumD,0)}\relax
+ \DivVect\V at ttC by\NumA,0 to\V at ttB
+ \advance\NumA\@ne
+ \edef\@mpt{\noexpand\egroup
+ \noexpand\multiput(\V at ttA)(\V at 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}}}
+ \def\Dotline@@(#1)(#2)#3{\put(#1){\Dotline@(0,0)(#2){#3}}}%
\fi
\AtBeginDocument{\@ifpackageloaded{esopic}{%
\renewcommand\LenToUnit[1]{\strip at pt\dimexpr#1*\p@/\unitlength}}{}}

+\renewcommand\LenToUnit[1]{\strip at pt\dimexpr#1*\p@/\unitlength}}{}}%
\def\GetCoord(#1)#2#3{%
\expandafter\SplitNod@\expandafter(#1)#2#3\ignorespaces}
+\def\SplitNod@(#1)#2#3{\isnot at polar#1:!!(#1)#2#3}%
\def\isnot at polar#1:#2!!{\def\@tempOne{#2}\ifx\@tempOne\empty
\expandafter\@firstoftwo\else
\expandafter\@secondoftwo\fi
{\SplitNod@@}{\SplitPolar@@}}
\def\SplitNod@(#1)#2#3{\isnot at 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
@@ 137,7 +156,7 @@
\ifdim\d at mX\p@=\z@
\else\ifdim\d at mY\p@=\z@
\else
 \DividE\ifdim\@xnum\p@<\z at \fi\p@ by\@xnum\p@ to\sc at lelen
+ \edef\sc at lelen{\fpeval{1 / abs(\@xnum)}}\relax
\@linelen=\sc at lelen\@linelen
\fi
\fi
@@ 157,13 +176,19 @@
\endgroup}
\def\Vector(#1){{%
\GetCoord(#1)\@tX\@tY
\ifdim\@tX\p@=\z@\vector(\@tX,\@tY){\@tY}
+\ifdim\@tX\p@=\z@
+ \vector(\@tX,\@tY){\@tY}%
\else
\vector(\@tX,\@tY){\@tX}\fi}}
+ \vector(\@tX,\@tY){\@tX}%
+\fi}}
\def\VECTOR(#1)(#2){\begingroup
\SubVect#1from#2to\@tempa
\expandafter\put\expandafter(#1){\expandafter\Vector\expandafter(\@tempa)}%
\endgroup\ignorespaces}
+\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}}
\let\lp at r( \let\rp at r)
\renewcommand*\polyline[1][\beveljoin]{\p at lylin@[#1]}
@@ 197,6 +222,7 @@
\if at tempswa\pIIe at fillGraph\else\pIIe at 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
@@ 211,234 +237,65 @@
\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 at pt\maxdimen}}%
\else
 \@DimA=#1\relax
 \edef\x{%
 \noexpand\egroup\noexpand\def\noexpand#3{%
 \strip at 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 at pt\dimexpr\@DimA*\@DimB/\p@\relax}}%
 \x\ignorespaces}%
 \let\MultiplyFN\MultiplY
 \fi
\fi
+\def\DividE#1by#2to#3{%
+ \edef#3{\fpeval{#1 / #2}}\relax
+}
+\let\DivideFN\DividE
+\def\MultiplY#1by#2to#3{\edef#3{\fpeval{#1 * #2}}}\relax
+\let\MultiplyFN\MultiplY
\unless\ifdefined\Numero
\def\Numero#1#2{\bgroup\dimen3254=#2\relax
\edef\x{\noexpand\egroup\noexpand\edef\noexpand#1{%
\strip at pt\dimen3254}}\x\ignorespaces}%
\fi
\def\g at 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\I2\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{+}%
+\def\SinOf#1to#2{\edef#2{\fpeval{round(sind#1,6)}}}\relax
+\def\CosOf#1to#2{\edef#2{\fpeval{round(cosd#1,6)}}}\relax
+\def\ArgOfVect#1to#2{\bgroup\GetCoord(#1){\t at X}{\t at Y}%
+\def\s at gno{}%
+\ifdim\t at X\p@=\z@
+ \ifdim\t at Y\p@=\z@
+ \def\ArcTan{0}% vettore nullo
\else
 \def\Segno{}%
 \@tdA=\@tdA
+ \def\ArcTan{90}% vettore verticale
+ \ifdim\t at Y\p@<\z@\def\ArcTan{90}\fi
\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\t at Y\p@=\z@% vettore orizzontale
+ \ifdim\t at X\p@<\z@
+ \def\ArcTan{180}%
\else
 \ifdim\@tdA<\p@
 \@tdA=\Segno0.0174533\@tdA
 \DividE\@tdA by\p@ to \@tempA%
+ \def\ArcTan{0}%
+ \fi
+ \else % vettore qualsiasi
+ \edef\ArcTan{\fpeval{atand(\t at Y / \t at X)}}\relax
+ \ifdim\t at X\p@<\z@% vettore nei quadranti di sinistra
+ \ifdim\t at Y\p@<\z@
+ \edef\ArcTan{\fpeval{\ArcTan  180}}\relax
\else
 \g at tTanCotanFrom\@tdA to\T and\Tp
 \@tdA=\T\p@ \advance\@tdA \Tp\p@
 \DividE \Segno2\p@ by\@tdA to \@tempA%
+ \edef\ArcTan{\fpeval{\ArcTan + 180}}\relax
\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 at 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 at 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 at 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 at pt\dimexpr\@tG\p at \@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 at pt\dimexpr\@tX\p@  \@tH\p@\relax}\relax
 \advance\I\m at ne}%
\fi
\ifnum\Inverti=\@ne
\edef\@tX{\strip at pt\dimexpr90\p at \@tX\p@\relax}
\fi
\edef\x{\egroup\noexpand\edef\noexpand#2{\Segno\@tX}}\x\ignorespaces}%
+\edef\x{\noexpand\egroup\noexpand\edef\noexpand#2{\ArcTan}}%
+\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 at X\t at Y
\@tempdima=\t at X\p@ \ifdim\@tempdima<\z@ \@tempdima=\@tempdima\fi
\@tempdimb=\t at 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 at ne}%
 \@tempdimc=\@T\@tempdimc
 \fi
\fi
\Numero#2\@tempdimc
+\edef#2{\fpeval{sqrt(\t at X*\t at X + \t at Y*\t at Y)}}\relax
\ignorespaces}%
\def\DirOfVect#1to#2{\GetCoord(#1)\t at X\t at Y
\ModOfVect#1to\@tempa
\unless\ifdim\@tempdimc=\z@
 \DividE\t at X\p@ by\@tempdimc to\t at X
 \DividE\t at Y\p@ by\@tempdimc to\t at Y
+\unless\ifdim\@tempa\p@=\z@
+ \DividE\t at X\p@ by\@tempa to\t at X
+ \DividE\t at Y\p@ by\@tempa to\t at Y
\fi
\MakeVectorFrom\t at X\t at Y to#2\ignorespaces}%
\def\ModAndDirOfVect#1to#2and#3{%
\GetCoord(#1)\t at X\t at Y
\ModOfVect#1to#2%
\ifdim\@tempdimc=\z@\else
 \DividE\t at X\p@ by\@tempdimc to\t at X
 \DividE\t at Y\p@ by\@tempdimc to\t at Y
\fi
\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
+\DirOfVect#1to#3\ignorespaces}%
\def\DistanceAndDirOfVect#1minus#2to#3and#4{%
\SubVect#2from#1to\@tempa
\ModAndDirOfVect\@tempa to#3and#4\ignorespaces}%
@@ 450,79 +307,37 @@
\CosOf#1to\t at X
\SinOf#1to\t at Y
\MakeVectorFrom\t at X\t at Y to#2\ignorespaces}%
\def\ArgOfVect#1to#2{\bgroup\GetCoord(#1){\t at X}{\t at Y}%
\def\s at gno{}\def\addflatt at ngle{0}
\ifdim\t at X\p@=\z@
 \ifdim\t at Y\p@=\z@
 \def\ArcTan{0}%
 \else
 \def\ArcTan{90}%
 \ifdim\t at Y\p@<\z@\def\s at gno{}\fi
 \fi
\else
 \ifdim\t at Y\p@=\z@
 \ifdim\t at X\p@<\z@
 \def\ArcTan{180}%
 \else
 \def\ArcTan{0}%
 \fi
 \else
 \ifdim\t at X\p@<\z@%
 \def\addflatt at ngle{180}%
 \edef\t at X{\strip at pt\dimexpr\t at X\p@}%
 \edef\t at Y{\strip at pt\dimexpr\t at Y\p@}%
 \ifdim\t at Y\p@<\z@
 \def\s at gno{}%
 \edef\t at Y{\t at Y}%
 \fi
 \fi
 \DivideFN\t at Y by\t at X to \t at A
 \ArcTanOf\t at A to\ArcTan
 \fi
\fi
\edef\ArcTan{\unless\ifx\s at gno\empty\s at gno\fi\ArcTan}%
\unless\ifnum\addflatt at ngle=0\relax
 \edef\ArcTan{%
 \strip at pt\dimexpr\ArcTan\p@\ifx\s at gno\empty\else+\fi
 \addflatt at ngle\p@\relax}%
\fi
\edef\x{\noexpand\egroup\noexpand\edef\noexpand#2{\ArcTan}}%
\x\ignorespaces}
\def\ScaleVect#1by#2to#3{\GetCoord(#1)\t at X\t at Y
\@tempdima=\t at X\p@ \@tempdima=#2\@tempdima\Numero\t at X\@tempdima
\@tempdima=\t at Y\p@ \@tempdima=#2\@tempdima\Numero\t at Y\@tempdima
+\edef\t at X{\fpeval{#2 * \t at X}}\relax
+\edef\t at Y{\fpeval{#2 * \t at Y}}\relax
\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
\def\ConjVect#1to#2{\GetCoord(#1)\t at X\t at Y
\@tempdima=\t at Y\p@\Numero\t at Y\@tempdima
+\edef\t at Y{\t at Y}%
\MakeVectorFrom\t at X\t at Y to#2\ignorespaces}%
\def\AddVect#1and#2to#3{\GetCoord(#1)\tu at X\tu at Y
\GetCoord(#2)\td at X\td at Y
\@tempdima\tu at X\p@\advance\@tempdima\td at X\p@ \Numero\t at X\@tempdima
\@tempdima\tu at Y\p@\advance\@tempdima\td at Y\p@ \Numero\t at Y\@tempdima
+\edef\t at X{\fpeval{\tu at X + \td at X}}\relax
+\edef\t at Y{\fpeval{\tu at Y + \td at Y}}\relax
\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
\def\SubVect#1from#2to#3{\GetCoord(#1)\tu at X\tu at Y
\GetCoord(#2)\td at X\td at Y
\@tempdima\td at X\p@\advance\@tempdima\tu at X\p@ \Numero\t at X\@tempdima
\@tempdima\td at Y\p@\advance\@tempdima\tu at Y\p@ \Numero\t at Y\@tempdima
+\edef\t at X{\fpeval{\td at X  \tu at X}}\relax
+\edef\t at Y{\fpeval{\td at Y  \tu at Y}}\relax
\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
\def\MultVect#1by{\@ifstar{\@ConjMultVect#1by}{\@MultVect#1by}}%
\def\@MultVect#1by#2to#3{\GetCoord(#1)\tu at X\tu at Y
\GetCoord(#2)\td at X\td at Y
\@tempdima\tu at X\p@ \@tempdimb\tu at Y\p@
\@tempdimc=\td at X\@tempdima\advance\@tempdimc\td at Y\@tempdimb
\Numero\t at X\@tempdimc
\@tempdimc=\td at Y\@tempdima\advance\@tempdimc\td at X\@tempdimb
\Numero\t at Y\@tempdimc
+\edef\t at X{\fpeval{\tu at X * \td at X  \tu at Y * \td at Y}}\relax
+\edef\t at Y{\fpeval{\tu at Y * \td at X + \tu at X * \td at Y}}\relax
\MakeVectorFrom\t at X\t at Y to#3\ignorespaces}%
\def\@ConjMultVect#1by#2to#3{\GetCoord(#1)\tu at X\tu at Y
\GetCoord(#2)\td at X\td at Y \@tempdima\tu at X\p@ \@tempdimb\tu at Y\p@
\@tempdimc=\td at X\@tempdima\advance\@tempdimc+\td at Y\@tempdimb
\Numero\t at X\@tempdimc
\@tempdimc=\td at X\@tempdimb\advance\@tempdimc\td at Y\@tempdima
\Numero\t at Y\@tempdimc
+\GetCoord(#2)\td at X\td at Y
+\edef\t at X{\fpeval{\tu at X * \td at X + \tu at Y * \td at Y}}\relax
+\edef\t at Y{\fpeval{\tu at Y * \td at X  \tu at X * \td at Y}}\relax
\MakeVectorFrom\t at X\t at 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
+\edef\@Mod{\fpeval{1 / \@Mod}}\relax
+\ConjVect\@Dir to\@Dir
\ScaleVect#1by\@Mod to\@tempa
\MultVect\@tempa by\@Dir to#3\ignorespaces}%
\def\Arc(#1)(#2)#3{\begingroup
@@ 544,10 +359,10 @@
to the range 0360 taking the sign into consideration}%
\@whiledim\@tdA>360\p@\do{\advance\@tdA360\p@}%
\fi
\SubVect#2from#1to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
+\SubVect#2from#1to\@V \ModOfVect\@V to\@Raggio
+\CopyVect#2to\@pPun
\CopyVect#1to\@Cent \GetCoord(\@pPun)\@pPunX\@pPunY
\@@Arc
\strokepath\ignorespaces}%
+\@@Arc\strokepath\ignorespaces}%
\def\@@Arc{%
\pIIe at moveto{\@pPunX\unitlength}{\@pPunY\unitlength}%
\ifdim\@tdA>180\p@
@@ 555,7 +370,8 @@
\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 by1to\@V\fi
+ \MultVect\@V by0,1.3333333to\@V
+ \if\Segno\ScaleVect\@V by1to\@V\fi
\AddVect\@pPun and\@V to\@pcPun
\AddVect\@sPun and\@V to\@scPun
\GetCoord(\@pcPun)\@pcPunX\@pcPunY
@@ 697,8 +513,9 @@
\@@Arc
\strokepath\ignorespaces}%
\def\CurveBetween#1and#2WithDirs#3and#4{%
\StartCurveAt#1WithDir{#3}\relax
\CurveTo#2WithDir{#4}\CurveFinish\ignorespaces}%
+ \StartCurveAt#1WithDir{#3}\relax
+ \CurveTo#2WithDir{#4}\CurveFinish\ignorespaces
+}%
\def\StartCurveAt#1WithDir#2{%
\begingroup
\GetCoord(#1)\@tempa\@tempb
@@ 737,6 +554,7 @@
\def\@isTension#1;#2!!{\def\@tempA{#1}%
\def\@tempB{#2}\unless\ifx\@tempB\empty\strip at semicolon#2\fi}
+
\def\strip at semicolon#1;{\def\@tempB{#1}}
\def\CurveTo#1WithDir#2{%
\def\@Tuno{1}\def\@Tzero{1}\relax
@@ 802,12 +620,13 @@
\ignorespaces}%
\def\Curve{\@ifstar{\let\fillstroke\fillpath\Curve@}%
{\let\fillstroke\strokepath\Curve@}}
+
\def\Curve@(#1)<#2>{%
\StartCurveAt#1WithDir{#2}%
\@ifnextchar\lp at r\@Curve{%
\PackageWarning{curve2e}{%
Curve specifications must contain at least two nodes!\Messagebreak
 Please, control your Curve specifications\MessageBreak}}}
+ Please, control your \string\Curve\space specifications\MessageBreak}}}
\def\@Curve(#1)<#2>{%
\CurveTo#1WithDir{#2}%
\@ifnextchar\lp at r\@Curve{%
@@ 823,9 +642,11 @@
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 at r\@Qurve{%
\@ifnextchar[\@ChangeQDir\CurveEnd}}%
+
\def\@ChangeQDir[#1]{\ChangeDir<#1>\@Qurve}%
\def\QurveTo#1WithDir#2{%
\edef\@Puno{#1}\DirOfVect#2to\@Duno\bgroup
@@ 878,7 +699,6 @@
\pIIe at bezier@QtoC\@ovyy\@ovdy\@ovri
\pIIe at bezier@QtoC\@xdim\@ovdx\@clnwd
\pIIe at bezier@QtoC\@ydim\@ovdy\@clnht
 \pIIe at moveto\@ovxx\@ovyy
\pIIe at curveto\@ovro\@ovri\@clnwd\@clnht\@xdim\@ydim
\fi\fi\egroup
\CopyVect\@Puno to\@Pzero
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