curve2e (19jun15)

git-svn-id: svn://tug.org/texlive/trunk@37619 c570f23f-e606-0410-a88d-b1316a301751

5 files changed, 1497 insertions, 742 deletions

diff --git a/Master/texmf-dist/doc/latex/curve2e/README b/Master/texmf-dist/doc/latex/curve2e/README
index 8230ad92e95..a1390fa48ec 100644
--- a/Master/texmf-dist/doc/latex/curve2e/README
+++ b/Master/texmf-dist/doc/latex/curve2e/README
@@ -1,15 +1,18 @@
 Curve2e.sty
-version 1.32
-filedate 06 June 2015
+version: 1.50
+filedate: 19 June 2015
 
-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 the LaTeX manual.
+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. Some of these features, implemented in this package previous versions, have been incorporated in the 2011 version of pict2e;  therefore  this package  avoids redefining 
-the original commands.
+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. Òcomplex numbersÓ. 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 BŽzier splines with their control nodes specified in polar form is available.
 
-This version id fully compatible with pict2e version 0.2x dated 2011/05/01.
+This version is fully compatible with pict2e version 0.2x dated 2011/05/01.
 
-I you specify
+If you specify
 
 \usepackage[<pict2e options>]{curve2e}
 
diff --git a/Master/texmf-dist/doc/latex/curve2e/curve2e.pdf b/Master/texmf-dist/doc/latex/curve2e/curve2e.pdf
index 78ba5350c21..aa8c5dd79ff 100644
--- a/Master/texmf-dist/doc/latex/curve2e/curve2e.pdf
+++ b/Master/texmf-dist/doc/latex/curve2e/curve2e.pdf
diff --git a/Master/texmf-dist/doc/latex/curve2e/manifest.txt b/Master/texmf-dist/doc/latex/curve2e/manifest.txt
index 7938ea3786c..04b8fefba5d 100644
--- a/Master/texmf-dist/doc/latex/curve2e/manifest.txt
+++ b/Master/texmf-dist/doc/latex/curve2e/manifest.txt
@@ -5,24 +5,30 @@ curve2e.pdf
 mainfest.txt
 README
 
-Maninfest.txt is this file.
+Manifest.txt is this file.
 
-curve2e.dtx is the documented source file of package curve2e.sty; you get both
+curve2e.dtx is the documented TeX source file of package curve2e.sty; you get both
 curve2e.sty and curve2e.pdf by running pdflatex on curve2e.dtx.
 
 README contains general information
 
-The package has the lppl status of author maintained.
+The package has the LPPL status of author maintained.
+
+This package is subject to the LaTeX Project Public Licence, LPPL, version 1.3 or any
+successive version.
+
+Therefore, according to the 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 
 (2011). 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.  
+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.
+package, I will possibly include every (documented) suggestion or modification 
+into this package.
 
 Claudio Beccari
 
-claudio.beccari@gmail.com
+claudio.beccari@gmail.com
\ No newline at end of file
diff --git a/Master/texmf-dist/source/latex/curve2e/curve2e.dtx b/Master/texmf-dist/source/latex/curve2e/curve2e.dtx
index 3af8caafb3e..7e5e7bae80f 100644
--- a/Master/texmf-dist/source/latex/curve2e/curve2e.dtx
+++ b/Master/texmf-dist/source/latex/curve2e/curve2e.dtx
@@ -9,7 +9,6 @@
   The curve2e package for LaTeX and XeLATeX
   Copyright (C) 2010 Claudio Beccari 
   All rights reserved
-
   License information appended
 
 \endpreamble
@@ -58,23 +57,38 @@ and the derived files curve2e.sty and curve2e.pdf.
 %</driver>
 %<+package>\ProvidesPackage{curve2e}%
 %<*package>
-        [2015/06/06 v.1.42 Extension package for pict2e]
+        [2015/06/19 v.1.50 Extension package for pict2e]
 %</package>
 %<*driver>
 \documentclass{ltxdoc}\errorcontextlines=9
 \hfuzz 10pt
-\usepackage{multicol,amsmath}
 \usepackage[utf8]{inputenc}
 \usepackage{lmodern,textcomp}
 \usepackage{mflogo}
+\usepackage{multicol,amsmath,trace}
 \usepackage{curve2e}
 \GetFileInfo{curve2e.dtx}
 \title{The extension package \textsf{curve2e}}
 \author{Claudio Beccari}
 \date{Version number \fileversion; last revised \filedate.}
 \providecommand*\diff{\mathop{}\!\mathrm{d}}
+\renewcommand\meta[1]{{\normalfont\textlangle\textit{#1}\textrangle}}
+\renewcommand\marg[1]{\texttt{\{\meta{#1}\}}}
+\providecommand\oarg{}
+\renewcommand\oarg[1]{\texttt{[\meta{#1}]}}
+\providecommand\parg{}
+\renewcommand\parg[1]{\texttt{(#1)}}
+\makeatletter
+\newcommand*\Pall[1][1.5]{\def\circdiam{#1}\@Pall}
+   \def\@Pall(#1){\put(#1){\circle*{\circdiam}}}
+\def\legenda(#1,#2)#3{\put(#1,#2){\setbox3333\hbox{$#3$}%
+   \dimen3333\dimexpr\wd3333*\p@/\unitlength +3\p@\relax
+   \edef\@tempA{\strip@pt\dimen3333}%
+   \framebox(\@tempA,7){\box3333}}}
+\def\Zbox(#1)[#2]#3{\put(#1){\makebox(0,0)[#2]{$#3$}}}
 \begin{document}
- \maketitle
+\maketitle
+\columnseprule=0.4pt
 \begin{multicols}{2}
  \tableofcontents
  \end{multicols}
@@ -83,7 +97,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 %</driver>
 % \fi
 %
-% \CheckSum{2484}
+% \CheckSum{2756}
 % \begin{abstract}
 % This file documents the |curve2e| extension package to the recent
 % implementation of the |pict2e| bundle that has been described by Lamport
@@ -92,11 +106,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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. Moreover
-% the |xetex.def| driver was introduced so that certain commands previously 
-% defined in this extension not only become unnecessary, but also would produce
-% errors when the program is used under XeLaTeX. Therefore these commands were
-% either eliminated or corrected.
+% 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
@@ -105,6 +115,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 % Please cite the original author and the chain of contributors.
 % \end{abstract}
 %
+%
 % \section{Package \texttt{pict2e} and this extension \texttt{curve2e}}
 % Package \texttt{pict2e} was announced in issue 15 of \texttt{latexnews}
 % around December 2003; it was declared that the new package would replace the
@@ -116,125 +127,187 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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:
+% environment; specifically what follows.
 % \begin{enumerate}
-% \item the line and vector slopes were limited to the ratios of relatively
+% \item The line and vector slopes were limited to the ratios of relative
 % prime one-digit integers of magnitude not exceeding 6 for lines and 4 for
-% vectors;
-% \item filled and unfilled full circles were limited by the necessarily
+% vectors.
+%^^A
+% \item Filled and unfilled full circles were limited by the necessarily
 % limited number of specific glyphs contained in the special \LaTeX\
-% \texttt{picture} fonts;
-% \item quarter circles were also limited in their radii for the same reason;
-% \item ovals (rectangles with rounded corners) could not be too small because
+% \texttt{picture} fonts.
+%^^A
+% \item Quarter circles were also limited in their radii for the same reason.
+%^^A
+% \item Ovals (rectangles with rounded corners) could not be too small because
 % of the unavailability of small radius quarter circles, nor could be too
 % large, in the sense that after a certain radius the rounded corners remained
 % the same and would not increase proportionally to the oval size.
-% \item vector arrows had only one possible shape and matched the limited
-% number of vector slopes;
-% \item for circles and inclined lines and vectors just two possible thicknesses
+%^^A
+% \item Vector arrows had only one possible shape and matched the limited
+% number of vector slopes.
+%^^A
+% \item For circles and inclined lines and vectors just two possible thicknesses
 %  were available.
 % \end{enumerate}
 %
-% The package \texttt{pict2e} removes most if not all the above limitations:
+% 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
+% \item Line and vector slopes are virtually unlimited; the only remaining
 % limitation is that the direction coefficients must be three-digit integer
 % numbers; they need not be relatively prime; with the 2009 upgrade even this
 % limitation was removed and now slope coefficients can be any fractional number
 % whose magnitude does not exceed 16\,384, the maximum dimension in points that
-% \TeX\ can handle;
-% \item filled and unfilled circles can be of any size;
-% \item ovals can be designed with  any specified corner curvature and there is
+% \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 there are two shapes for the arrow tips; the triangular one traditional
+% 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 the |\linethickness| command changes the thickness of all lines, straight,
+%^^A
+% \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 adds the following features.
 % \begin{enumerate}
-% \item commands for setting the line terminations are introduced; the user can
+%\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 roto-amplification operators.
+%^^A
+% \item Commands for setting the line terminations are introduced; the user can
 % chose between square or rounded caps; the default is set to rounded caps (now
-% available also with |pict2e|); 
-% \item commands for specifying the way two lines or curves join to one nanother;
+% 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
+% \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|);
+% \texttt{picture} environment (now available also with |pict2e|).
 % ^^A
-% \item a new macro |\Line| was defined so as to avoid the need to specify the
+% \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 conflicts with |pict2e| 2009 version; therefore its name is changed to
-% |\LIne| and supposedly it will not be used very often, if ever used; 
+% 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
+% \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|; of course there is no need
-% to use the |\put| command with this line specification;
+% |pict2e| so that |\LINE| is now renamed |\segment|; there is no need
+% to use the |\put| command with this line specification.
+% ^^A
+% \item A new macro |\DLine| is defined in order to draw dashed lines joining any
+% two given points; the dash length and gap (equal to one another) get 
+% specified through one of the macro arguments.
 % ^^A
-% \item a new macro |\DLine| is defined in order to draw dashed lines joining any
-% two given points; the dash length and gap (equal to one another) must be
-% specified;
+% \item A new macro |\Dotline| is defined in order to draw dotted straight
+% lines as a sequence of equally spaced dots, where the gap can be specified
+% by the user; such straight line may have any inclination, as well as the
+% above dashed lines.
 % ^^A
-% \item similar macros are redefined for vectors; |\vector| redefines the
+% \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; |\VECTOR|
-% joins two specified points (without using the |\put| command) with  the arrow
-% pointing to the second point;
-% \item a new macro |\polyline| for drawing polygonal lines is defined that
+% 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 if is redefined so as to allow
-% an optional specification of the way segments fo the polyline are join to
-% one another.;
-% \item a new macro |\Arc| is defined in order to draw an arc with arbitrary
-% radius and arbitrary angle amplitude; this amplitude is specified in
-% sexagesimal degrees, not in radians; the same functionality is now achieved with
-% the |\arc| macro of |pict2e|, which provides also the strar version |\arc*| that
-% fills up the interior of the generated circular arc. It must be noticed that
-% the syntax is slighltly different, so that it's reasonable that both commands,
-% in spite  of producing identical arcs, might be more comfortable with this or
-% that syntax.
-% \item two new macros are defined in order to draw circular arcs with one
-% arrow at one or both ends;
-% \item a new macro |\Curve| is defined so as to draw arbitrary curved lines
-% by means of third order Bézier splines; the |\Curve| macro requires only the
-% curve nodes and the direction of the tangents at each node.
+% 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.
+%^^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
+% with this or that syntax.
+%^^A
+% \item Two new macros |\VectorArc| and |\VectorARC| are defined in order to
+% draw circular arcs with an
+% arrow at one or both ends.
+%^^A
+% \item A new macro |\Curve| is defined so as to draw arbitrary curved lines
+% by means of cubic Bézier splines; the |\Curve| macro requires only the
+% curve nodes and the directions of the tangents at each node.The starred
+% version fills up the interior of the curve with the currently specified color.
+%^^A
+% \item |\Curve| is a recursive macro that can draw an unlimited (reasonably
+% low) number of connecter 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.
+%^^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
+% |\CurveFinish|, so as to fill up the closed path with the locally
+% specified color; see figure~\ref{fig:colored-curve}.  It is much more
+% convenient to use the starred version of  |\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.
+%
 % In order to make the necessary calculations many macros have been defined so
 % as to use complex number arithmetics to manipulate point coordinates,
-% directions (directional versors), rotations and the like. The trigonometric
-% functions have also been defined in a way that the author believes to be
-% more efficient than that implied by the \texttt{trig} package; in any case
-% the macro names are sufficiently different to accommodate both definitions
-% in the same \LaTeX\ run.
+% 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.
 %
 % Many aspects of this extension could be fine tuned for better performance;
 % many new commands could be defined in order to further extend this extension.
 % If the new service macros are accepted by other \TeX\ and \LaTeX\ programmers,
-% this beta version could become the start for a real extension of the
+% 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. Actually the
-% \verb|\Curve| algorithm might be redefined so as to use the macros introduced
-% in the \texttt{hobby} package, that implements for the typesetting engines
-% the same functionalities that John Hobby wrote for \MF\ and \MP\ programs. 
+% 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 this reason I suppose that every enhancement should be submitted to
-% Gäßlein and Niepraschk  who are the prime maintainers of \texttt{pict2e};
-% they only can decide whether or not to incorporate new macros in their package.
+% For these reasons I suppose that every enhancement should be submitted to
+% Gäßlein, Niepraschk, and Tkadlec  who are the prime maintainers of
+% \texttt{pict2e}; they are the only ones who can decide whether or not to
+% incorporate new macros in their package.
 %
 % \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.
 % \begin{enumerate}
 % \item This package |curve2e| calls directly the \LaTeX\ packages |color| and 
-% |pict2e| to whom it passes any possible option that the latter can receive;
+% |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.
+%^^A
 % \item The user is offered new commands in order to control the line terminators
 % and the line joins; specifically:
 % \begin{itemize}
@@ -247,11 +320,12 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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 thickeness of 1\,pt while if a drawing command
+% 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
@@ -268,47 +342,50 @@ and the derived files curve2e.sty and curve2e.pdf.
 % \end{flushleft}
 % where \meta{dimensioned value} means a length specification complete of its
 % units or a dimensional expression.
+%^^A
 % \item Straight lines and vectors are redefined in such a way that fractional
 % slope coefficients may be specified; the zero length line does not produce
 % errors and is ignored; the zero length vectors draw only the arrow tips.
+%^^A
 % \item New line and vector macros are defined that avoid the necessity of
-% specifying the horizontal component |\put(3,4){\LIne(25,15)}| specifies a
+% 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
-% the using command |\put|.
+% using command |\put|.
 % The same applies to the vector commands |\Vector| and |\VECTOR|. Experience has
 % shown that the commands intended to joint two specified coordinates are
 % particularly useful.
+%^^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 sequence the specified
+% draws a sequence of connected segments that joins in order the specified
 % points; the syntax is:
 %	\begin{flushleft}
-%	\cs{polyline[}\marg{optional join style}\texttt{](}\meta{$P_1$}\texttt{)(}%
-%	\meta{$P_2$}\texttt{)...(}\meta{$P_n$}\texttt{)}
+%\cs{polyline}\texttt{[}\marg{optional join style}\texttt{]%
+%(}\meta{$P_1$}\texttt{)(}\meta{$P_2$}\texttt{)...(}\meta{$P_n$}\texttt{)}
 %	\end{flushleft}
-% See figure~\ref{fig:polyline} where a pentagon is designed..
+% See figure~\ref{fig:polyline} where a regular pentagon is drawn; usage of polar
+% coordinates is also shown.
 %
 %	\begin{figure}[!ht]
 %	\begin{minipage}{.48\linewidth}
 %	\begin{verbatim}
 %	\unitlength=.5mm
-%	\begin{picture}(40,32)(-20,0)
-%	\polyline(0,0)(19.0211,13,8197)(11.7557,36.1803)%
-%	(-11.7557,36.1803)(-19.0211,13,8197)(0,0)
+%	\begin{picture}(40,32)(-20,-20)
+%	\polyline(90:20)(162:20)(234:20)(306:20)(378:20)(90:20)
 %	\end{picture}
 %	\end{verbatim}
 %	\end{minipage}
 %   \hfill
 %	\begin{minipage}{.48\linewidth}\raggedleft
 %   \unitlength=.5mm
-%	\begin{picture}(40,32)(-20,0)
-%	\polyline(0,0)(19.0211,13,8197)(11.7557,36.1803)%
-%	(-11.7557,36.1803)(-19.0211,13,8197)(0,0)
+%	\begin{picture}(40,32)(-20,-20)
+%	\polyline(90:20)(162:20)(234:20)(306:20)(378:20)(90:20)
 %	\end{picture}\hspace*{2em}
 %	\end{minipage}
 %	\caption{Polygonal line obtained by means of the \texttt{\string\polyline}
-%   command}	\label{fig:polyline}
+%   command; coordinates are in polar form.}
+%	\label{fig:polyline}
 %	\end{figure}
 %
 % Although you can draw polygons with |\polyline|, as it was done in 
@@ -317,9 +394,10 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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.
-% \item The new command
+%^^A
+% \item The new command |\Dashline| (alias: |\Dline| for backwards compatibility)
 % \begin{flushleft}
-% |\Dline(|\textit{first point}|)(|\textit{second point}|)(|\textit{dash length}|)|
+% |\Dashline(|\meta{first point}|)(|\meta{second point}|){|\meta{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
@@ -333,59 +411,70 @@ and the derived files curve2e.sty and curve2e.pdf.
 %	\begin{figure}[!ht]
 %	\begin{minipage}{.48\textwidth}
 %	\begin{verbatim}
-%	\unitlength.5mm
+%	\unitlength=1mm
 %	\begin{picture}(40,40)
 %	\put(0,0){\GraphGrid(40,40)}
-%	\Dline(0,0)(40,10){4}
+%	\Dashline(0,0)(40,10){4}
 %	\put(0,0){\circle*{2}}
-%	\Dline(40,10)(0,25){4}
+%	\Dashline(40,10)(0,25){4}
 %	\put(40,10){\circle*{2}}
-%	\Dline(0,25)(20,40){4}
+%	\Dashline(0,25)(20,40){4}
 %	\put(0,25){\circle*{2}}
 %	\put(20,40){\circle*{2}}
+%	\Dotline(0,0)(40,40){2}
 %	\end{picture}
 %	\end{verbatim}
 %	\end{minipage}
 %	\hfill
 %	\begin{minipage}{.48\textwidth}\centering
-%	\unitlength.5mm
+%	\unitlength=1mm
 %	\begin{picture}(40,40)
 %	\put(0,0){\GraphGrid(40,40)}
-%	\Dline(0,0)(40,10){4}
+%	\Dashline(0,0)(40,10){4}
 %	\put(0,0){\circle*{2}}
-%	\Dline(40,10)(0,25){4}
+%	\Dashline(40,10)(0,25){4}
 %	\put(40,10){\circle*{2}}
-%	\Dline(0,25)(20,40){4}
+%	\Dashline(0,25)(20,40){4}
 %	\put(0,25){\circle*{2}}
 %	\put(20,40){\circle*{2}}
+%   \Dotline(0,0)(40,40){2}
 %	\end{picture}
 %	\end{minipage}
 %	\caption{Dashed lines and graph grid}\label{fig:dashline}
 %	\end{figure}
-% \item |\GraphGrid| is a command that draws a red grid over the drawing area
-% with lines separated |10\unitlength|s; it is described only with a comma
+%^^A
+%\item Analogous to |\Dashline|, a new command |\Dotline| draws a dotted line with
+% the syntax:
+% \begin{flushleft}
+% |\Dotline(|\meta{first point}|)(|\meta{end point}|){|\meta{dot gap}|}|
+% \end{flushleft}
+% See figures~\ref{fig:dashline} and~\ref{fig:dottedlines} for  examples.
+%^^A
+% \item |\GraphGrid| is a command that draws a red grid under the drawing 
+% with lines separated |10\unitlength|s apart; it is described only with a comma
 % separated couple of numbers, representing the base and the height of the grid,
 % see figure~\ref{fig:dashline}; it's better to specify multiples of ten and
-% the grid can be placed anywhere in the drawing plane by means of |\put|,
-% whose 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 plane. This grid is intended as an aid in
-% 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.
+% 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.
+%^^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
-% ($360^\circ$ to one revolution), so that reduction to the fundamental quadrant
-% is supposed to be more accurate; the tangent of odd multiples of $90^\circ$
+% (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 $1^\circ$ are approximated by the first term of the
+% 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.
@@ -399,13 +488,18 @@ and the derived files curve2e.sty and curve2e.pdf.
 %\\
 %      \texttt{\char92TanOf}\meta{angle}\texttt{to}\meta{control sequence}
 %\end{flushleft}
-% The \meta{control sequence} may then be used as a multiplying factor of a length.
+% The \meta{control sequence} may then be used as a multiplying factor of a
+% length.
+%^^A
 % \item Arcs can be drawn as simple circular arcs, or with one or two arrows at
 % their ends (curved vectors);  the syntax is:
 %\begin{flushleft}
-% \texttt{\char92Arc(}\meta{center}\texttt{)(}\meta{starting point}\texttt{)}\marg{angle}\\
-% \texttt{\char92VectorArc(}\meta{center}\texttt{)(}\meta{starting point}\texttt{)}\marg{angle}\\
-% \texttt{\char92VectorARC(}\meta{center}\texttt{)(}\meta{starting point}\texttt{)}\marg{angle}\\
+% \texttt{\char92Arc(}\meta{center}\texttt{)(}\meta{starting point}\texttt{)}%
+%\marg{angle}\\
+% \texttt{\char92VectorArc(}\meta{center}\texttt{)(}\meta{starting point}%
+%\texttt{)}\marg{angle}\\
+% \texttt{\char92VectorARC(}\meta{center}\texttt{)(}\meta{starting point}%
+%\texttt{)}\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
@@ -440,6 +534,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 %	\end{minipage}
 %	\caption{Arcs and curved vectors}\label{fig:arcs}
 %	\end{figure}
+%^^A
 % \item A multitude of commands have been defined in order to manage complex
 % numbers; actually complex numbers are represented as a comma separated pair of
 % fractional numbers. They are used to address to specific points in the drawing
@@ -460,7 +555,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 % \item |\ModOfVect|\meta{vector}|to|\meta{macro}
 % \item |\DirOfvect|\meta{vector}|to|\meta{versor macro}
 % \item |\ModAndDirOfVect|\meta{vector}|to|\meta{1st macro}|and|\meta{2nd macro}
-% \item |\DistanceAndDirOfVect|\meta{first vector}|minus|\meta{second vector}|to|\meta{1st macro}|and|\meta{2nd macro}
+% \item |\DistanceAndDirOfVect|\meta{1st vector}|minus|\meta{2nd vector}|to|\meta{1st macro}|and|\meta{2nd macro}
 % \item |\XpartOfVect|\meta{vector}|to|\meta{macro}
 % \item |\YpartOfVect|\meta{vector}|to|\meta{macro}
 % \item |\DirFromAngle|\meta{angle}|to|\meta{versor macro}
@@ -473,9 +568,9 @@ and the derived files curve2e.sty and curve2e.pdf.
 % \item |\MultVect|\meta{first vector}|by*|\meta{second vector}|to|\meta{vector macro}
 % \item |\DivVect|\meta{first vector}|by|\meta{second vector}|to|\meta{vector macro}
 % \end{itemize}}
-%
+%^^A
 % \item General curves can be drawn with the |pict2e| macro |\curve| but it
-% requires the specification of the Bézier third order spline control points;
+% requires the specification of the third-order Bézier-spline control points;
 % sometimes it's better to be very specific with the control points and there
 % is no other means to do a decent graph; sometimes the curves to be drawn
 % are not so tricky and a general set of macros can be defined so as to compute
@@ -500,15 +595,15 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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}
-%	\begin{figure}
+%	\begin{figure}[htb]
 %	\begin{minipage}{.48\textwidth}
 %   \begin{verbatim}
-%	\unitlength=8mm
+%	\unitlength=8mm\relax
 %	\begin{picture}(5,5)
 %	\put(0,0){\framebox(5,5){}}\thicklines\roundcap
 %	\Curve(2.5,0)<1,1>(5,3.5)<0,1>%
-%	      (2.5,3.5)<-.5,-1.2>[-.5,1.2]%
-%	      (0,3.5)<0,-1>(2.5,0)<1,-1>
+%	  (4,5)<-1,0>(2.5,3.5)<-.5,-1.2>[-.5,1.2]%
+%	  (1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<1,-1>
 %	\end{picture}
 %   \end{verbatim}
 %	\end{minipage}
@@ -517,29 +612,61 @@ and the derived files curve2e.sty and curve2e.pdf.
 %	\unitlength=8mm\relax
 %	\begin{picture}(5,5)
 %	\put(0,0.5){\put(0,0){\framebox(5,5){}}\thicklines\roundcap
-%	\Curve(2.5,0)<1,1>(5,3.5)<0,1>(2.5,3.5)<-0.5,-1.2>[-0.5,1.2](0,3.5)<0,-1>(2.5,0)<1,-1>}
+%	\Curve(2.5,0)<1,1>(5,3.5)<0,1>(4,5)<-1,0>(2.5,3.5)<-0.5,-1.2>[-0.5,1.2](1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<1,-1>}
 %	\end{picture}
 %	\end{minipage}
 %	\caption{A heart shaped curve with cusps drawn with \texttt{\string\Curve}}
 %	\label{fig:curve}
+
+%\vspace*{2\baselineskip}
+%
+%	\begin{minipage}{.48\textwidth}
+%   \begin{verbatim}
+%	\unitlength=8mm\relax
+%	\begin{picture}(5,5)
+%	\put(0,0){\framebox(5,5){}}\thicklines\roundcap
+%   \color{green}\relax
+%	\Curve*(2.5,0)<1,1>(5,3.5)<0,1>%
+%	  (4,5)<-1,0>(2.5,3.5)<-.5,-1.2>[-.5,1.2]%
+%	  (1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<1,-1>
+%	\end{picture}
+%   \end{verbatim}
+%	\end{minipage}
+%	\hfill
+%	\begin{minipage}{.48\textwidth}\raggedleft\relax
+%	\unitlength=8mm\relax
+%	\begin{picture}(5,5)
+%	\put(0,0.5){\put(0,0){\framebox(5,5){}}\thicklines\roundcap
+%   \color{green}\relax
+%	\Curve*(2.5,0)<1,1>(5,3.5)<0,1>(4,5)<-1,0>(2.5,3.5)<-0.5,-1.2>[-0.5,1.2](1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<1,-1>}
+%	\end{picture}
+%	\end{minipage}
+%\caption{Coloring the inside of a closeded path drawn with \texttt{\string\Curve*}}
+%\label{fig:colored-curve}
+
 %	\end{figure}
 %
-% In spite of the relative simplicity of the macros contained in this package, the
-% described macros, as well as the original macros included in the |pict2e| package,
-% allow to produce fine drawings that were inconceivable of 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 appeared; 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.
+% With the starred version of |\Curve|, instead of stroking the contour,
+% the macro fills up the contour with the selected current color,
+% figure~\ref{fig:colored-curve}.
+%
+% 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|
+% package has gone much farther.
 %
 % \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.
-% But there are also the package |curves| that is intended to draw almost
+% 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 used only quadratic Bézier curves and the curve tracing is eased
+% 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
@@ -551,6 +678,23 @@ and the derived files curve2e.sty and curve2e.pdf.
 % exactly are the Bézier splines, it appears that |ebezier| should be used only
 % for dvi output without recourse to PostScript machinery.
 %
+% The |picture| package extends the performance of the |picture| environment
+% (extended with \texttt{pict2e}) by accepting coordinates and lengths in real
+% absolute dimensions, not only as multiples of |\unitlength|; it provides
+% commands to extend that functionality to other packages. In certain
+% circumstances it is very useful.
+%
+% Package \texttt{xpicture} builds over the |picture| \LaTeX\ environment so
+% as to allow to draw the usual curves that are part of an introductory
+% analytic geometry course; lines, circles, parabolas, ellipses, hyperbolas, and
+% polynomials; the syntax is very comfortable; for all these curves it uses
+% the quadratic Bézier splines.
+% 
+% Package |hobby| extends the cubic Bézier spline handling with the algorithms
+% John Hobby created for \MF\ and \MP. But by now this package interfaces very
+% well with |tikz|; it has not (yet) been adapted to the common |picture|
+% environment, even extended with |pict2e|, and, why not, with |curve2e|.
+%
 % \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.
@@ -585,12 +729,15 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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.
+%\iffalse
+%<*package>
+%\fi
 %    \begin{macrocode}
 \RequirePackage{color}
 \RequirePackageWithOptions{pict2e}[2014/01/01]
 %    \end{macrocode}
 %
-% The next macros are just for debugging. With the \texttt{tracing} package it
+% 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}
@@ -612,6 +759,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 \ifx\undefined\defaultlinewidth \newdimen\defaultlinewidth \fi
 %    \end{macrocode}
 %
+% \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
@@ -635,20 +783,21 @@ and the derived files curve2e.sty and curve2e.pdf.
 % these spaces introduce picture deformities often difficult to spot and
 % eliminate.
 %
-% \subsubsection{Improved line and vector macros}
+% \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. The two arguments define the horizontal and the
-% vertical component respectively.
+% coordinate point inside the box created with |\put|. The two arguments
+% define the horizontal and the vertical component respectively.
 %    \begin{macrocode}
-\def\LIne(#1,#2){\moveto(0,0)
-      \pIIe@lineto{#1\unitlength}{#2\unitlength}\strokepath}%
+\def\LIne(#1){{\GetCoord(#1)\@tX\@tY
+      \moveto(0,0)
+      \pIIe@lineto{\@tX\unitlength}{\@tY\unitlength}\strokepath}\ignorespaces}%
 %    \end{macrocode}
 %
 % A similar macro |\segment| operates between two explicit points with absolute
 % coordinates, instead of relative to the position specified by a |\put|
-% command; it resorts to the |\polyline| macro that is to be defined in a while.
+% 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}
@@ -660,8 +809,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 % $x_1, y_1$ 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
-% non-English speaking users, since the ISO regulations allow the decimal point
-% only for English speaking countries, while in all other countries the comma
+% non-English speaking users, since  in all other languages the comma
 % must be used as the decimal separator.
 %
 % The |\line| macro is redefined by making use of a new division routine that
@@ -715,35 +863,59 @@ and the derived files curve2e.sty and curve2e.pdf.
 %    \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 whith the 2004 version that was limited to
+% it did preform in a better way with the 2004 version that was limited to
 % integer direction coefficients up to 999 in magnitude.
 %
-% Another useful line-type macro creates a dashed line between two given points
-% with a dash length that must be specified; actually the specified dash length
-% is a desired dash length; the actual length is computed by integer division
-% between the distance of the given points and the desired dash length; this
-% integer is tested in order to see if it's odd; if it's not, it is increased by
-% one. Then the actual dash length is obtained by dividing the above distance by
-% this odd number.
-% Another vector is created from $P_1-P_0$ by dividing it by the magic odd number;
-% then it is multiplied by two in order to have the increment from one dash to the
-% next, and finally the number of patterns is obtained by integer dividing the
-% magic odd 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.
-%    \begin{macrocode}
-\ifx\Dline\undefined
-\def\Dline(#1,#2)(#3,#4)#5{%
-\begingroup
-   \countdef\NumA254\countdef\NumB252\relax
-   \MakeVectorFrom{#1}{#2}to\V@ttA 
-   \MakeVectorFrom{#3}{#4}to\V@ttB 				
+% \subsection{Dashed and dotted lines}
+% Dashed and dotted lines are very useful in technical drawings; here we
+% introduce four macros that help drawing them in the proper way; besides
+% the obvious difference between the use of dashes or dots, they may refer
+% in a different way to the end points that must be specified to the various
+% macros. 
+%
+% The coordinates of the first point $P_1$, where le line starts, are always
+% referred to the origin of the coordinate axes; the end point $P_2$
+% coordinates with the first macro type are referred to the origin of the
+% axes, while with the second macro type they are referred to $P_1$; both
+% macro types have their usefulness and figures~\ref{fig:dashedlines}
+% and~\ref{fig:dottedlines} show how to use these macro types.
+%
+% 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.
+%
+% 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.
+%
+% Another vector $P_2-P_1$ is created by dividing it by this number;
+% then, when dashes are involved, it is multiplied by two in order to have
+% the increment from one dash to the next; finally the number of patterns
+% is obtained by integer division of this number by 2 and increasing it by 1.
+% A simple |\multiput| completes the job, but in order to use the various
+% vectors and numbers within a group and to throw the result outside the group
+% while restoring all the intermediate counters and registers, a service macro
+% is created with an expanded definition and then this service macro is executed.
+% Figure~\ref{fig:dashedlines} shows the effect of the slight changing
+% of the dash length in order to maintain approximately the same dash-space
+% pattern along the line, irrespective o the line length.
+%    \begin{macrocode}
+\ifx\Dashline\undefined
+\def\Dashline{\@ifstar{\Dashline@@}{\Dashline@}}
+\def\Dashline@(#1)(#2)#3{%
+\bgroup
+   \countdef\NumA3254\countdef\NumB3252\relax
+   \GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttA
+   \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttB
    \SubVect\V@ttA from\V@ttB to\V@ttC 			
    \ModOfVect\V@ttC to\DlineMod 				
-   \DividE\DlineMod\p@ by#5\p@ to\NumD 			
-   \NumA\expandafter\Integer\NumD?? 	
+   \DivideFN\DlineMod by#3 to\NumD 			
+   \NumA\expandafter\Integer\NumD.?? 	
    \ifodd\NumA\else\advance\NumA\@ne\fi 		
    \NumB=\NumA \divide\NumB\tw@ 				
    \DividE\DlineMod\p@ by\NumA\p@  to\D@shMod 	
@@ -751,23 +923,206 @@ and the derived files curve2e.sty and curve2e.pdf.
    \MultVect\V@ttC by\@tempa,0 to\V@ttB	
    \MultVect\V@ttB by 2,0 to\V@ttC
    \advance\NumB\@ne			
-   \edef\@mpt{\noexpand\endgroup
-   \noexpand\multiput(\V@ttA)(\V@ttC){\number\NumB}{\noexpand\LIne(\V@ttB)}}%
+   \edef\@mpt{\noexpand\egroup
+   \noexpand\multiput(\V@ttA)(\V@ttC){\number\NumB}%
+      {\noexpand\LIne(\V@ttB)}}%
    \@mpt\ignorespaces}%
+\let\Dline\Dashline
+
+\def\Dashline@@(#1)(#2)#3{\put(#1){\Dashline@(0,0)(#2){#3}}}
 \fi
 %    \end{macrocode}
 %
+%\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 to given points; 
+% the dots are rather small, therefore the inter dot distance is computed in
+% such a way as to have the first and the last dot at the exact position of
+% the dotted-line end-points; again the specified dot distance is nominal in
+% the sense that it is recalculated in such a way that the first and last
+% dots coincide with the line end points. The syntax is as follows:
+%\begin{flushleft}
+%\cs{Dotline}\texttt{(}\meta{start point}\texttt{)(}\meta{end point}\texttt{)\{}\meta{dot distance}\texttt{\}}
+%\end{flushleft}
+%    \begin{macrocode}
+\ifx\Dotline\undefined
+\def\Dotline{\@ifstar{\Dotline@@}{\Dotline@}}
+\def\Dotline@(#1)(#2)#3{%
+\bgroup
+   \countdef\NumA 3254\relax \countdef\NumB 3255\relax
+   \GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttA
+   \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttB
+   \SubVect\V@ttA from\V@ttB to\V@ttC                    
+   \ModOfVect\V@ttC to\DotlineMod                        
+   \DivideFN\DotlineMod by#3 to\NumD                     
+   \NumA=\expandafter\Integer\NumD.??                    
+   \DivVect\V@ttC by\NumA,0 to\V@ttB                     
+   \advance\NumA\@ne 
+   \edef\@mpt{\noexpand\egroup
+   \noexpand\multiput(\V@ttA)(\V@ttB){\number\NumA}%
+      {\noexpand\makebox(0,0){\noexpand\circle*{0.5}}}}%
+   \@mpt\ignorespaces}%
+
+\def\Dotline@@(#1)(#2)#3{\put(#1){\Dotline@(0,0)(#2){#3}}}
+\fi
+%    \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.
+%
+% \subsection{Coordinate handling}
 % The new macro |\GetCoord| splits a vector (or complex number) specification
-% into its components:
+% 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.
 %    \begin{macrocode}
 \def\GetCoord(#1)#2#3{%
 \expandafter\SplitNod@\expandafter(#1)#2#3\ignorespaces}
 %    \end{macrocode}
-% But the macro that does the real work is |\SplitNod@|:
+% But the macro that detects the form of the coordinates is |\isnot@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:
 %    \begin{macrocode}
-\def\SplitNod@(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}%
-%    \end{macrocode}
+\def\isnot@polar#1:#2!!{\def\@tempOne{#2}\ifx\@tempOne\empty
+\expandafter\@firstoftwo\else
+\expandafter\@secondoftwo\fi
+{\SplitNod@@}{\SplitPolar@@}}
+
+\def\SplitNod@(#1)#2#3{\isnot@polar#1:!!(#1)#2#3}%
+\def\SplitNod@@(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}%
+\def\SplitPolar@@(#1:#2)#3#4{\DirFromAngle#1to\@DirA
+\ScaleVect\@DirA by#2to\@DirA
+\expandafter\SplitNod@@\expandafter(\@DirA)#3#4}
+
+\let\originalput\put
+\def\put(#1){\bgroup\GetCoord(#1)\@tX\@tY
+\edef\x{\noexpand\egroup\noexpand\originalput(\@tX,\@tY)}\x}
+
+\let\originalmultiput\multiput
+\let\original@multiput\@multiput
+
+\long\def\@multiput(#1)#2#3{\bgroup\GetCoord(#1)\@mptX\@mptY
+\edef\x{\noexpand\egroup\noexpand\original@multiput(\@mptX,\@mptY)}%
+\x{#2}{#3}\ignorespaces}
+
+\gdef\multiput(#1)#2{\bgroup\GetCoord(#1)\@mptX\@mptY
+\edef\x{\noexpand\egroup\noexpand\originalmultiput(\@mptX,\@mptY)}\x(}%)
+%    \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}
 %
+% \subsection{Vectors}
 % 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
@@ -780,10 +1135,10 @@ and the derived files curve2e.sty and curve2e.pdf.
 % contours that are eventually filled by the principal macro; each contour
 % macro draws the vector with a \LaTeX\ or a PostScript arrow whose parameters
 % are specified by default or may be taken from the parameters taken from the
-%\texttt{PSTricks} package if this one is loaded before \texttt{pict2e}; in any
+%|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
-% \texttt{curve2e} package redefinitions.
+% |curve2e| package redefinitions.
 %
 % Because of these features the redefinitions and the new macros are different
 % from those used for straight lines.
@@ -876,15 +1231,17 @@ and the derived files curve2e.sty and curve2e.pdf.
 % vertical vector components. If the horizontal component is zero, the actual
 % length% must be specified as the vertical component.
 %    \begin{macrocode}
-\def\Vector(#1,#2){%
-\ifdim#1\p@=\z@\vector(#1,#2){#2}
+\def\Vector(#1){{%
+\GetCoord(#1)\@tX\@tY
+\ifdim\@tX\p@=\z@\vector(\@tX,\@tY){\@tY}
 \else
-\vector(#1,#2){#1}\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.
+% 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
@@ -923,7 +1280,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 %            macros.}\label{fig:vectors}
 %	\end{figure}
 %
-% \subsubsection{Polylines}
+% \subsection{Polylines}
 % We now define the polygonal line macro; its syntax is very simple
 % \begin{flushleft}
 % \cs{polygonal}\texttt{(}$P_0$\texttt{)(}$P_1$\texttt{)(}$P_2$)%
@@ -943,13 +1300,12 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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.
+% 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.
 %
 % Remember: |\polyline| has been incorporated into |pict2e| 2009, but we
-% redefine it so as to allow an optional argument to allow the line join
-% specification.
+% redefine it so as to allow an optional argument to specify the line join type.
 %
 % 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
@@ -966,7 +1322,7 @@ and the derived files curve2e.sty and curve2e.pdf.
     \ignorespaces}}
 
 %    \end{macrocode}
-% But if there is a second or further point coordinate the recursive macro
+% But if there is a second or further point coordinate, the recursive macro
 % |\p@lyline| is called; it works on the next point and checks for a further
 % point; if such a point exists it calls itself, otherwise it terminates the
 % polygonal line by stroking it.
@@ -976,7 +1332,7 @@ and the derived files curve2e.sty and curve2e.pdf.
     \@ifnextchar\lp@r{\p@lyline}{\strokepath\ignorespaces}}
 %    \end{macrocode}
 %
-% \subsubsection{The red service grid}
+% \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
@@ -986,13 +1342,13 @@ and the derived files curve2e.sty and curve2e.pdf.
 % the readings become cumbersome. The |\RoundUp| macro provides to increase the
 % grid dimensions to integer multiples of ten.
 %    \begin{macrocode}
-\def\GraphGrid(#1,#2){\begingroup\textcolor{red}{\linethickness{.1\p@}%
+\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}%
-\endgroup\ignorespaces}
+\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
@@ -1007,20 +1363,80 @@ and the derived files curve2e.sty and curve2e.pdf.
 %    \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
+% 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.
+% 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}
+% 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.
+%
+% 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 |e-TeX|
+% extension to all the \TeX\ system engines, allows to perform all operations
+% directly on suitable length registers.
+%
+% 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
+% 32-bit word contains the dimension in \emph{scaled points}, where 1\,pt
+% equals $2^{16}$\,sp.
+%
+% Since the number of digits 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 32-bit 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 32-bit 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 microseconds.
+%
 % \subsection{The new division macro}
-% Now comes one of the most important macros in the whole package: the division
+% 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
@@ -1028,29 +1444,30 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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 do not define any other division
-% macro; nevertheless, since the macro in |pict2e| may not be so efficient as it
-% might be if the |e-tex| 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, if the extended mode is
-% enabled, that 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 represent the
-% numbers of scaled points another two dimensions correspond to, in the philosophy
-% that floating point numbers are represented by the measures of lengths in
-% points.
-%
-% Therefore first we test if the macro is already defined:
+% 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 |e-tex| 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:
 %    \begin{macrocode}
 \ifdefined\dimexpr
 %    \end{macrocode}
-% then we test if the extended mode exists and/or is enabled:
+% then we test if the macro is already defined:
 %    \begin{macrocode}
     \unless\ifdefined\DividE
 %    \end{macrocode}
@@ -1059,13 +1476,13 @@ and the derived files curve2e.sty and curve2e.pdf.
 % old and/or it is a recent version, but it was compiled without activating the
 % extended mode, the macro |\dimexpr| is undefined.
 %
-% The macro, creates a group where the names of two counters and a
+% 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 dirty trick to throw
-% the result beyond the end-group command.
+% does not damage the other macros; we just have to use a Knuthian dirty trick
+% to throw the result beyond the end-group command.
 %
 % 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
@@ -1074,22 +1491,23 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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 end the termination of the program. During the division
-% 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.
+% suitable error messages and 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.
 %
 %    \begin{macrocode}
 	\def\DividE#1by#2to#3{\bgroup
-		\countdef\Num2254\relax \countdef\Den2252\relax
-		\dimendef\@DimA 2254
-		\Num=\p@ \@DimA=#2\relax \Den=\@DimA
-		\ifnum\Den=\z@
+		\dimendef\Num2254\relax \dimendef\Den2252\relax
+		\dimendef\@DimA 2250
+		\Num=\p@ \Den=#2\relax 
+		\ifdim\Den=\z@
 		  \edef\x{\noexpand\endgroup\noexpand\def\noexpand#3{\strip@pt\maxdimen}}%
 		\else
 		  \@DimA=#1\relax
-		  \@DimA=\dimexpr\@DimA*\Num/\Den\relax
-		  \edef\x{\noexpand\egroup\noexpand\def\noexpand#3{\strip@pt\@DimA}}%
+		  \edef\x{%
+		    \noexpand\egroup\noexpand\def\noexpand#3{%
+		       \strip@pt\dimexpr\@DimA*\Num/\Den\relax}}%
 		\fi
 	\x\ignorespaces}%
 	\fi
@@ -1099,22 +1517,23 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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}%
+         \def\DivideFN#1by#2to#3{\DividE#1\p@ by#2\p@ to{#3}}%
     \fi
 %    \end{macrocode}
 %
 % We do the same in order to multiply two integer o fractional numbers held
 % in the first two arguments and the third argument is a definable token that
 % will hold the result of multiplication in the form of a fractional number,
-% possibly with  a non null fractional part; a null fractional part is
-% eliminated by \verb|strip@pt|.
+% possibly with a non null fractional part; a null fractional part is
+% eliminated by \verb|\strip@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
-        \@DimA=\dimexpr\@DimA*\@DimB/\p@\relax
-        \edef\x{\noexpand\egroup\noexpand\def\noexpand#3{\strip@pt\@DimA}}%
+        \edef\x{%
+           \noexpand\egroup\noexpand\def\noexpand#3{%
+             \strip@pt\dimexpr\@DimA*\@DimB/\p@\relax}}%
         \x\ignorespaces}%
     \fi
 \fi
@@ -1126,16 +1545,17 @@ and the derived files curve2e.sty and curve2e.pdf.
 % assigned to the control sequence.
 %    \begin{macrocode}
 \unless\ifdefined\Numero
-  \def\Numero#1#2{\dimen3254#2\relax
-  \edef#1{\strip@pt\dimen3254}\ignorespaces}%
+  \def\Numero#1#2{\bgroup\dimen3254=#2\relax
+    \edef\x{\noexpand\egroup\noexpand\edef\noexpand#1{%
+      \strip@pt\dimen3254}}\x\ignorespaces}%
 \fi
 %    \end{macrocode}
 % The \verb|\ifdefined| primitive command is provided by the e-\TeX\ extension
 % of the typesetting engine; the test does not create any hash table entry;
 % it is a different way than the \verb|\ifx\csname ....\endcsname| test,
-% because the latter first possibly creates a macro with meaning \verb|relax|
+% 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 \verb|\relax|.
+% to mean |\relax|.
 %
 % \subsection{Trigonometric functions}
 %  We now start with trigonometric functions. We define the macros |\SinOf|,
@@ -1160,12 +1580,12 @@ and the derived files curve2e.sty and curve2e.pdf.
 % in proximity of the value zero (and the other values that might involve high
 % tangent or cotangent values) and in that case we prefer to approximate the
 % small angle function value with its first or second order truncation of the
-% McLaurin series; in facts for angles whose magnitude is smaller than $1^\circ$
+% 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 (less than 0.017) that the sine and tangent
+% 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 $1-0.5y^2$ (the error being smaller than about $4\cdot10^{-9}$).
+% the formula $1-0.5y^2$ (the error being smaller than about $\cdot10^{-6}$).
 %
 % We keep using grouping so that internal variables are local to these groups
 % and do not mess up other things.
@@ -1178,6 +1598,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 \def\g@tTanCotanFrom#1to#2and#3{%
 \DividE 114.591559\p@ by#1to\X@ \@tdB=\X@\p@
 %    \end{macrocode}
+%
 % Computations are done with the help of counter |\I|, of the length |\@tdB|,
 % and the auxiliary control sequences |\Tan| and |\Cot| whose meaning is
 % transparent. The iterative process controlled by |\@whilenum| implements the
@@ -1197,15 +1618,14 @@ and the derived files curve2e.sty and curve2e.pdf.
    \@tdC=\Tan\p@ \@tdD=\I\@tdB
    \advance\@tdD-\@tdC \DividE\p@ by\@tdD to\Tan
    \advance\I-2\relax}%
-\def#2{\Tan}\DividE\p@ by\Tan\p@ to\Cot \def#3{\Cot}%
-\ignorespaces}%
+\def#2{\Tan}\DividE\p@ by\Tan\p@ to\Cot \def#3{\Cot}\ignorespaces}%
 %    \end{macrocode}
 %
 %  Now that we have the macro for computing the tangent and cotangent of the
 % half angle, we can compute the real trigonometric functions we are interested
 % in. The sine value is computed after reducing the sine argument to the
 % interval $0^\circ< \theta<180^\circ$; actually special values such as
-% $0^\circ$, $90^\circ$, $180^\circ$, et cetera, are taken care separately, so
+% 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.
 %
@@ -1303,8 +1723,8 @@ and the derived files curve2e.sty and curve2e.pdf.
 %
 % For the tangent computation we behave in a similar way, except that we
 % consider the fundamental interval as $0^\circ<\theta<90^\circ$; for the odd
-% multiples of $90^\circ$ we assign the result a \TeX\ infinity value, that is
-% the maximum a dimension can be.
+% 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@%
@@ -1340,6 +1760,127 @@ and the derived files curve2e.sty and curve2e.pdf.
 \endTanOf}%
 %    \end{macrocode}
 %
+% As of today the anomaly (angle) of a complex number may not be necessary, but
+% it might become useful in the future; therefore with macro \verb|\ArgOfVect|
+% we calculate the four quadrant arctangent (in degrees) of the given vector
+% taking into account the sings of the vector components. 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}{3-x^2 + \cfrac{(3x)^2}{5-3x^2 + 
+%            \cfrac{(5x)^2}{7-5x^2 + \cfrac{(7x)^2}{9-7x^2 + \ddots}}}}}
+%\label{equ:arctan-fraz-cont}
+%\end{equation}
+% but after some testing we had to give up due to the slow convergence of 
+% continued fraction~\eqref{equ:arctan-fraz-cont}, 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 algorythim crashes;
+% therefore it's virtually impossibile to get more than three correct digits
+% after the decimal separator.
+% 
+% 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}\centering\unitlength=0.007\textwidth
+%\begin{picture}(100,60)
+%\legenda(15,73){y=\tan\theta}
+%\legenda(35,73){t=\tan\theta_\infty}
+%\put(0,0){\vector(1,0){100}}\Zbox(100,1)[br]{\theta}
+%\put(0,0){\vector(0,1){80}}\Zbox(1,80)[tl]{y}
+%\Dashline(75,0)(75,80){2.5}
+%\put(76,1){\makebox(0,0)[bl]{$\pi/2$}}
+%\put(0,0){\linethickness{1pt}
+%\Curve(0,0)<1,0.8>(24,20)<1,0.90>(51,49.5)<17,29,5>(60,70)<1,5>(62,80)<1,8>}
+%\put(51,49.5){\circle*{2}}
+%\Dashline(51,0)(51,49.5){2.5}
+%\put(52,1){\makebox(0,0)[bl]{$\theta_{i-1}$}}
+%\Dashline(0,49.5)(51,49.5){2.5}
+%\put(1,51){\makebox(0,0)[bl]{$y_{i-1}$}}
+%\put(0,20){\line(1,0){70}}\put(1,21){\makebox(0,0)[bl]{$t$}}
+%\Line(34,20)(51,49.25)
+%\Line(60.15,70)(51,20)
+%\put(51,20){\circle*{2}}\put(60,70){\circle*{2}}
+%\Dashline(60,0)(60,70){2.5}
+%\put(61,1){\makebox(0,0)[bl]{$\theta_{i-2}$}}
+%\Dashline(0,70)(60,70){2.5}
+%\put(1,71){\makebox(0,0)[bl]{$y_{i-2}$}}
+%\put(34,20){\circle*{2}}\put(34,29.5){\circle*{2}}
+%\Dashline(34,0)(34,29.5){2.5}
+%\Dashline(0,29.5)(34,29.5){2.5}
+%\put(1,30.5){\makebox(0,0)[bl]{$y_i$}}
+%\put(35,1){\makebox(0,0)[bl]{$\theta_{i}$}}
+%\put(24,20){\circle*{2}}
+%\Dashline(24,0)(24,20){2.5}
+%\put(25,1){\makebox(0,0)[bl]{$\theta_\infty$}}
+%\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'_{i-1} &= \frac{\diff\tan(\theta_{i-1})}{\diff\theta}
+%         = \frac{1}{\cos^2\theta_{i-1}}\\
+%\theta_i &= \theta_{i-1} - \frac{\tan \theta_{i-1} - t}{y'_{i-1}}
+%         =\theta_{i-1} - \cos^2 \theta_{i-1}(\tan \theta_{i-1} - 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 $i-1$. 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.
+%    \begin{macrocode}
+\def\ArcTanOf#1to#2{\bgroup
+\edef\@tF{#1}\@tdF=\@tF\p@ \@tdE=57.295778\p@
+\@tdD=\ifdim\@tdF>\z@ \@tdF\else -\@tdF\fi
+\unless\ifdim\@tdD>0.02\p@
+    \def\@tX{\strip@pt\dimexpr57.295778\@tdF\relax}%
+\else
+    \edef\@tX{45}\relax
+    \countdef\I 2523 \I=8\relax
+    \@whilenum\I>0\do{\TanOf\@tX to\@tG
+    \edef\@tG{\strip@pt\dimexpr\@tG\p@-\@tdF\relax}\relax
+    \MultiplY\@tG by57.295778to\@tG
+    \CosOf\@tX to\@tH
+    \MultiplY\@tH by\@tH to\@tH
+    \MultiplY\@tH by\@tG to \@tH
+    \edef\@tX{\strip@pt\dimexpr\@tX\p@ - \@tH\p@\relax}\relax
+    \advance\I\m@ne}%
+\fi
+\edef\x{\egroup\noexpand\edef\noexpand#2{\@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
@@ -1348,16 +1889,36 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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{{\rmfamily center}})(\meta{{\rmfamily starting
-%                 point}}){\marg{{\rmfamily angle}}}
+% \cs{Arc}(\meta{center})(\meta{starting point})\marg{angle}
 % \end{flushleft}
 % which is totally equivalent to:
 % \begin{flushleft}\ttfamily
-% \string\put(\meta{\rmfamily center})\string{\string\Arc(0,0)(\meta{\rmfamily starting
-% point})\marg{\rmfamily angle}\string}
+% \cs{put}(\meta{center})\marg{\upshape\cs{Arc}(0,0)(\meta{starting point})\marg{angle}}
 % \end{flushleft}
-%  If the \meta{angle} is positive the arc runs counterclockwise from the
-% starting point; clockwise if it's negative.
+% If the \meta{angle}, i.e. the arc angular aperture, is positive the arc
+% runs counterclockwise from the starting point; clockwise if it's negative.
+% Notice that since the \meta{starting point} is relative to the \meta{center}
+% point, its polar coordinates are very convenient, since they become
+% \parg{\meta{start angle}:\meta{radius}}, where the
+% \meta{start angle} is relative to the arc center. Therefore you can think
+% about a syntax such as this one:
+%\begin{flushleft}
+%\cs{Arc}\parg{\meta{center}}\parg{\meta{start angle}:\meta{radius}}\marg{angle}
+%\end{flushleft}
+%
+% The difference between the |pict2e| |\arc| definition  consists in a very
+% different syntax:
+%\begin{flushleft}
+%\cs{arc}\texttt{[}\meta{start angle}\texttt{,}\meta{end angle}\texttt{]}\marg{radius}
+%\end{flushleft}
+% and the center is assumed to be at the coordinate established with a
+% required |\put| command; moreover the difference in specifying angles
+% is that \meta{end angle} equals the sum of \meta{start angle} and
+% \meta{angle}. With the definition of this |curve2e| package
+% use of a |\put| command is not prohibited, but it may be used for fine
+% tuning the arc position by means of a simple displacement; moreover the
+% \meta{starting point} may be specified with polar coordinates (that are
+% relative to the arc center). 
 %
 % It's necessary to determine the end point and the control points of the
 % Bézier spline(s) that make up the circular arc.
@@ -1367,13 +1928,37 @@ and the derived files curve2e.sty and curve2e.pdf.
 % pivoting point appears to be non relocatable.
 % It is therefore necessary to resort to low level \TeX\ commands and the
 % defined trigonometric functions and a set of macros that operate on complex
-% numbers used as vector scale-rotate operators.
+% numbers used as vector roto-amplification operators.
 %
 % \subsection{Complex number macros}
-% We need therefore macros for summing, subtracting, multiplying, dividing
-% complex numbers, for determining they directions (unit vectors); a unit vector
+% In this package \emph{complex number} is a vague phrase; it may be used
+% in the mathematical sense of an ordered pair of real numbers; it can be
+% viewed as a vector joining the origin of the coordinate axes to the
+% coordinates indicated by the ordered pair; it can be interpreted as a
+% roto-amplification operator that scales its operand and rotates it about
+% a pivot point; besides the usual conventional representation used by the
+% mathematicians where the ordered pair is enclosed in round parentheses
+% (which is in perfect agreement with the standard code use by the |picture|
+% environment) there is the other conventional representation used by the
+% engineers that stress the roto-amplification nature of a complex number:
+%\[
+%(x, y) = x + \mathrm{j}y =M \mathrm{e}^{\mathrm{j}\theta}
+%\]
+% Even the imaginary unit is indicated with $\mathrm{i}$ by the mathematicians
+% and with $\mathrm{j}$ by the engineers. In spite of these differences,
+% these objects, the \emph{complex numbers}, are used without any problem by
+% both mathematicians and engineers.
+% 
+%The important point is that these objects can be summed, subtracted,
+% multiplied, divided, raised to any power (integer, fractional, positive
+% or negative), be the argument of transcendental functions according to
+% rules that are agreed upon by everybody. We do not need all these properties, but we need some and we must create the suitable macros for doing some of
+% these operations.
+%
+% In facts 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 form of the Euler's formula
+% cartesian or polar form of the Euler's formula
 % \[
 % \mathrm{e}^{\mathrm{j}\phi} = \cos\phi+\mathrm{j}\sin\phi
 % \]
@@ -1416,29 +2001,38 @@ and the derived files curve2e.sty and curve2e.pdf.
 \def\ModOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
 \@tempdima=\t@X\p@ \ifdim\@tempdima<\z@ \@tempdima=-\@tempdima\fi
 \@tempdimb=\t@Y\p@ \ifdim\@tempdimb<\z@ \@tempdimb=-\@tempdimb\fi
-\ifdim\@tempdima>\@tempdimb
-   \DividE\@tempdimb by\@tempdima to\@T
-   \@tempdimc=\@tempdima
+\ifdim\@tempdima=\z@
+    \ifdim\@tempdimb=\z@
+        \def\@T{0}\@tempdimc=\z@
+    \else
+        \def\@T{0}\@tempdimc=\@tempdimb
+    \fi
 \else
-   \DividE\@tempdima by\@tempdimb to\@T
-   \@tempdimc=\@tempdimb
+    \ifdim\@tempdima>\@tempdimb
+       \DividE\@tempdimb by\@tempdima to\@T
+       \@tempdimc=\@tempdima
+    \else
+          \DividE\@tempdima by\@tempdimb to\@T
+          \@tempdimc=\@tempdimb
+    \fi
 \fi
-\ifdim\@T\p@=\z@
-\else
-   \@tempdima=\@T\p@ \@tempdima=\@T\@tempdima
-   \advance\@tempdima\p@%
-   \@tempdimb=\p@%
-   \@tempcnta=5\relax
-   \@whilenum\@tempcnta>\z@\do{\DividE\@tempdima by\@tempdimb to\@T
-   \advance\@tempdimb \@T\p@ \@tempdimb=.5\@tempdimb
-   \advance\@tempcnta\m@ne}%
-   \@tempdimc=\@T\@tempdimc
+\unless\ifdim\@tempdimc=\z@
+    \unless\ifdim\@T\p@=\z@
+       \@tempdima=\@T\p@ \@tempdima=\@T\@tempdima
+       \advance\@tempdima\p@%
+       \@tempdimb=\p@%
+       \@tempcnta=5\relax
+       \@whilenum\@tempcnta>\z@\do{\DividE\@tempdima by\@tempdimb to\@T
+       \advance\@tempdimb \@T\p@ \@tempdimb=.5\@tempdimb
+       \advance\@tempcnta\m@ne}%
+       \@tempdimc=\@T\@tempdimc
+    \fi
 \fi
 \Numero#2\@tempdimc
 \ignorespaces}%
 %    \end{macrocode}
 % As a byproduct of the computation the control sequence |\@tempdimc| contains
-% the vector or complex number magnitude multiplied by the length of one point.
+% 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
@@ -1500,124 +2094,16 @@ and the derived files curve2e.sty and curve2e.pdf.
 \MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
 %    \end{macrocode}
 %
-% As of today the anomaly (angle) of a complex number may not be necessary, but
-% it might become useful in the future; therefore with macro \verb|\ArgOfVect|
-% we calculate the four quadrant arctangent (in degrees) of the given vector
-% taking into account the sings of the vector components. 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}{3-x^2 + \cfrac{(3x)^2}{5-3x^2 + 
-%            \cfrac{(5x)^2}{7-5x^2 + \cfrac{(7x)^2}{9-7x^2 + \ddots}}}}}
-%\label{equ:arctan-fraz-cont}
-%\end{equation}
-% but after some testing we had to give up due to the slow convergence of 
-% continued fraction~\eqref{equ:arctan-fraz-cont}, 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{equation}
-%\begin{subequations}
-%\begin{aligned}
-%\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{aligned}
-%\end{subequations}
-%\end{equation}
-% 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 it 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 algorthim crashes;
-% therefore it's virtually impossibile to get more than three correct digits
-% after the decimal separator.
-% 
-% 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}\centering\unitlength=0.007\textwidth
-%\begin{picture}(100,70)
-%\put(10,63){\framebox(18,7){$y=\tan\theta$}}
-%\put(30,63){\framebox(20,7){$t=\tan\theta_\infty$}}
-%\put(0,0){\vector(1,0){100}}\put(100,3){\makebox(0,0)[br]{$\theta$}}
-%\put(0,0){\vector(0,1){70}}\put(3,70){\makebox(0,0)[tl]{$y$}}
-%\multiput(75,0)(0,5){14}{\line(0,1){2.5}}\put(77,2){\makebox(0,0)[bl]{$\pi/2$}}
-%{\linethickness{1pt}\cbezier(0,0)(5,5)(55,40)(60,70)}
-%\put(51,50){\circle*{2}}
-%\multiput(51,0)(0,5){10}{\line(0,1){2.5}}\put(54,3){\makebox(0,0)[bl]{$\theta_{i-1}$}}
-%\multiput(0,50)(5,0){10}{\line(1,0){2.5}}\put(3,53){\makebox(0,0)[bl]{$y_{i-1}$}}
-%\put(0,20){\line(1,0){70}}\put(3,23){\makebox(0,0)[bl]{$t$}}
-%\Line(34,20)(51,50)
-%\put(34,20){\circle*{2}}
-%\multiput(34,0)(0,5){4}{%
-%  \line(0,1){2.5}}\put(36,3){\makebox(0,0)[bl]{$\theta_{i}$}}
-%\put(24,20){\circle*{2}}
-%\multiput(24,0)(0,5){4}{\line(0,1){2.5}}\put(21,3){\makebox(0,0)[br]{$\theta_\infty$}}
-%\end{picture}
-%\caption{Newton method}\label{fig:tangenti}
-%\end{figure}
-%
-% The iterative algorithm with Newton method implies the recurrence
-%\begin{equation}\begin{subequations}\begin{aligned}
-%y'_{i-1} &= \frac{\diff\tan(\theta_{i-1})}{\diff\theta} 
-%         = \frac{1}{\cos^2\theta_{i-1}}\\
-%\theta_i &= \theta_{i-1} - y'_{i-1}(\tan \theta_{i-1} - t)
-%         =\theta_{i-1} - \cos^2 \theta_{i-1}(\tan \theta_{i-1} - t)
-%\end{aligned}
-%\label{equ:iterazione}
-%\end{subequations}\end{equation}
-%
-% 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 $i-1$. 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 dos not pose any problems, especially if we
-% use the properties of the trigonometric functions and we confine the
-% computations to the first quadrant.
-%    \begin{macrocode}
-\def\ArcTanOf#1to#2{\bgroup
-\edef\@tF{#1}\@tdF=\@tF\p@
-\@tdE=57.295779\p@
-\ifdim\@tdF=\z@\def\@tX{0}\else
-\edef\@tXX{1}%
-\MultiplY57.295779by\@tXX to \@tX
-\countdef\I 2323 \I=7\relax
-\@whilenum\I>0\do{\TanOf\@tX to\@tG 
-\CosOf\@tX to \@tH 
-\edef\@tG{\strip@pt\dimexpr\@tG\p@-\@tdF\relax}%
-\MultiplY\@tH by\@tH to\@tH
-\MultiplY\@tH by\@tG to \@tH
-\edef\@tXX{\strip@pt\dimexpr\@tXX\p@ - \@tH\p@\relax}%
-\MultiplY57.295779by\@tXX to\@tX
-\advance\I\m@ne}\fi
-\edef\x{\egroup\noexpand\edef\noexpand#2{\@tX}}\x}%
-%    \end{macrocode} 
-%
-% Now we have the algorithm to compute the arctangent of a number; and
+% 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 have to pay attention that the algorithm to compute the arctangent
+% 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 the lay in the first or the fourth quadrants; while
+% 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 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.
@@ -1662,16 +2148,27 @@ and the derived files curve2e.sty and curve2e.pdf.
 \edef\x{\noexpand\egroup\noexpand\edef\noexpand#2{\ArcTan}}%
 \x\ignorespaces}
 %    \end{macrocode}
-%^^A \begin{tabular}{ll}
-%^^A 0		& \ArcTanOf 0 to\Res \Res\\
-%^^A 1		& \ArcTanOf 1 to\Res \Res\\
-%^^A 2		& \ArcTanOf 2 to\Res \Res\\
-%^^A 0.5	& \ArcTanOf 0.5 to\Res \Res\\
-%^^A 0.707	& \ArcTanOf 0.707 to\Res \Res\\
-%^^A \end{tabular}
-%^^A
-%^^A\bigskip
-%^^A
+%^^A  \begin{tabular}{ll}
+%^^A  0		& \ArcTanOf 0 to\Res \Res\\
+%^^A  0.01		& \ArcTanOf 0.01 to\Res \Res\\
+%^^A  0.02		& \ArcTanOf 0.02 to\Res \Res\\
+%^^A  0.04		& \ArcTanOf 0.04 to\Res \Res\\
+%^^A  0.05		& \ArcTanOf 0.05 to\Res \Res\\
+%^^A  0.06		& \ArcTanOf 0.06 to\Res \Res\\
+%^^A  0.09		& \ArcTanOf 0.09 to\Res \Res\\
+%^^A  0.1		& \ArcTanOf 0.1 to\Res \Res\\
+%^^A  0.2		& \ArcTanOf 0.2 to\Res \Res\\
+%^^A  0.4		& \ArcTanOf 0.4 to\Res \Res\\
+%^^A  0.5		& \ArcTanOf 0.5 to\Res \Res\\
+%^^A  0.6		& \ArcTanOf 0.6 to\Res \Res\\
+%^^A  0.8		& \ArcTanOf 0.8 to\Res \Res\\
+%^^A  0.707	& \ArcTanOf 0.707 to\Res \Res\\
+%^^A  1		& \ArcTanOf 1 to\Res \Res\\
+%^^A  2		& \ArcTanOf 2 to\Res \Res\\
+%^^A  \end{tabular}
+% 
+% \bigskip
+%
 %^^A \begin{tabular}{rl}
 %^^A 0,0		& \ArgOfVect0,0to\Res \Res\\
 %^^A 1,0		& \ArgOfVect1,0to\Res \Res\\
@@ -1684,9 +2181,9 @@ and the derived files curve2e.sty and curve2e.pdf.
 % ^^A-1,-1		& \ArgOfVect-1,-1to\Res \Res\\
 %^^A \end{tabular}
 % It is worth noting that the absolute error in these computations is lower
-% than 0.001°, that is 0.000017\,rad; pretty satisfactory since the typesetting
-% engines work in fixed radix notation with 16 fractional binary digits, and
-% an error on the fifth fractional digit is almost the best it can be expected
+% than 0.0001°; pretty satisfactory since the typesetting engines work in
+% fixed radix notation with 16 fractional binary digits, and an error on
+% the 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
@@ -1777,7 +2274,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 %    \begin{macrocode}
 \def\Arc(#1)(#2)#3{\begingroup
 \@tdA=#3\p@ 
-\ifdim\@tdA=\z@\else
+\unless\ifdim\@tdA=\z@
   \@Arc(#1)(#2)%
 \fi
 \endgroup\ignorespaces}%
@@ -1854,12 +2351,62 @@ and the derived files curve2e.sty and curve2e.pdf.
 % Here we need the extrema of the arc and the coordinates of the control points
 % of the Bézier cubic spline that traces the arc. The control points lay on the
 % perpendicular to the vectors that join the arc center to the starting
-% and end points respectively. Their distance $K$ from the adjacent nodes is
-% determined with the formula
-% \[
+% 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 end-points; $C_1$ and $C_2$ are the
+% Bézier-spline control-points; $P$ is the arc mid-point, that should be
+% distant from the center of the arc the same as $P_1$ and $P_2$. Choosing a
+% convenient orientation of the arc relative to the coordinate axes, the
+% coordinates of these five points are:
+%\begin{align*}
+%P_1 &= (-R\sin\theta, 0)\\
+%P_2 &= (R\sin\theta, 0)\\
+%C_1 &= (-R\sin\theta+K\cos\theta, K\sin\theta)\\
+%C_2 &= (R\sin\theta-K\cos\theta, K\sin\theta)\\
+%P   &= (0, R(1-\cos\theta))
+%\end{align*}
+% The Bézier cubic spline interpolating the end and mid points is given by
+% the parametric equation:
+%\begin{equation*}
+%P= P_1(1-t)^3 + C_1 3(1-t)^2t + C_2 3(1-t)t^2 + P_2t^3
+%\end{equation*}
+% where the mid point is obtained for $t=0.5$; the four coefficients then become $1/8, 3/8, 3/8, 1/8$ and the only unknown remains $K$. Solving for $K$ we obtain the formula
+% \begin{equation}\label{equ:corda}
 % K= \frac{4}{3}\,\frac{1-\cos\theta}{\sin\theta}R
-% \]
-% where $\theta$ is half the arc aperture and $R$ is its radius.
+%= \frac{4}{3}\,\frac{1-\cos\theta}{\sin^2\theta}s
+% \end{equation}
+% where $\theta$ is half the arc aperture, $R$ is its radius, and $s$ is
+% half the arc chord.
 %    \begin{macrocode}
 \ifdim\@tdA>\z@
   \DirFromAngle\@gradi to\@Dir \if\Segno-\ConjVect\@Dir to\@Dir \fi
@@ -1902,10 +2449,11 @@ and the derived files curve2e.sty and curve2e.pdf.
 % |\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 option of the \texttt{pict2e} package.
+% 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 specific drawing done here shortens the arc so as not to overlap on
+% 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.
 %
@@ -1915,11 +2463,11 @@ and the derived files curve2e.sty and curve2e.pdf.
 % 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)
+% 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
+% draw a zero length vector; (f\/) possibly repeating the same procedure for the
 % other end of the arc; (g) shortening the total arc angular amplitude by the
-% amount of the  arrow tip(s) already set, and (h) then drawing the final circular
+% 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.
 %
@@ -2011,33 +2559,33 @@ and the derived files curve2e.sty and curve2e.pdf.
 \@tdE=\pIIe@FAW\@wholewidth \@tdE=0.8\@tdE
 \DividE\@tdE by \@Raggio\unitlength to\DeltaGradi
 \@tdD=\DeltaGradi\p@ \@tdD=57.29578\@tdD \Numero\DeltaGradi\@tdD
-\@tdD=\ifx\Segno--\fi\@gradi\p@ \Numero\@tempa\@tdD
+\@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{\ifx\Segno-\m@ne\else\@ne\fi}%
+\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\ifx\Segno--\fi\DeltaGradi\p@
+\@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}}% arrow tip at the end point
+\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{\ifx\Segno-\else-\fi\@ne}%
+\edef\@tempa{\if\Segno-\else-\fi\@ne}%
 \MultVect\@V by0,\@tempa to\@vPun
-\@tdE\ifx\Segno--\fi\DeltaGradi\p@
+\@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}}% arrow tip at the starting point
-\edef\@tempa{\ifx\Segno--\fi\DeltaGradi}%
+\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
@@ -2056,7 +2604,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 % straight arrow tip if this one is large in comparison to the arc radius.
 %
 % \subsection{General curves}
-% Now we define a macro for tracing a general, not necessarily circular arc.
+% 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
@@ -2066,12 +2614,58 @@ and the derived files curve2e.sty and curve2e.pdf.
 %    \begin{macrocode}
 \def\CurveBetween#1and#2WithDirs#3and#4{%
 \StartCurveAt#1WithDir{#3}\relax
-\CurveTo#2WithDir{#4}\CurveFinish}%
+\CurveTo#2WithDir{#4}\CurveFinish\ignorespaces}%
 %    \end{macrocode}
+% For backwards compatibility the old command with lower case |and| is made
+% to do the same as this macro |\CurveBetween| with capitalised |And|.
 %
 % 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  should never exceed 90° in absolute value;
+% strange error messages may be issued by the interpreter. 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)
+%\CurveBetween-50,0and50,0WithDirs15:1and{-15:1}
+%\CurveBetween-50,0and50,0WithDirs30:1and{-30:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{-45:1}
+%\CurveBetween-50,0and50,0WithDirs60:1and{-60:1}
+%\CurveBetween-50,0and50,0WithDirs75:1and{-75:1}
+%\CurveBetween-50,0and50,0WithDirs90:1and{-90:1}}
+%\put(120,60){%
+%\Line(-50,0)(50,0)
+%\CurveBetween-50,0and50,0WithDirs15:1and{15:1}
+%\CurveBetween-50,0and50,0WithDirs30:1and{30:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{45:1}
+%\CurveBetween-50,0and50,0WithDirs60:1and{60:1}
+%\CurveBetween-50,0and50,0WithDirs75:1and{75:1}
+%\CurveBetween-50,0and50,0WithDirs90:1and{90:1}}
+%\put(0,0){%
+%\Line(-50,0)(50,0)
+%\CurveBetween-50,0and50,0WithDirs45:1and{-15:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{-30:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{-45:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{-60:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{-75:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{-90:1}}
+%\put(120,0){%
+%\Line(-50,0)(50,0)
+%\CurveBetween-50,0and50,0WithDirs45:1and{15:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{30:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{45:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{60:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{75:1}
+%\CurveBetween-50,0and50,0WithDirs45:1and{90:1}}
+%\end{picture}
+%\caption{Curves between two points}\label{fig:curva-due-punti}
+%\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
@@ -2116,7 +2710,7 @@ and the derived files curve2e.sty and curve2e.pdf.
 \CopyVect\@tempa,\@tempb to\@Dzero
 \DirOfVect\@Dzero to\@Dzero}
 %    \end{macrocode}
-% And this re-initializes the direction after a cusp
+% And this re-initializes the direction to create  a cusp:
 %    \begin{macrocode}
 \def\ChangeDir<#1>{%
 \GetCoord(#1)\@tempa\@tempb
@@ -2125,13 +2719,128 @@ and the derived files curve2e.sty and curve2e.pdf.
 \ignorespaces}
 %    \end{macrocode}
 %
-% The next macro is the finishing one; it strokes the whole curve and closes the
-% group that was opened with |\StartCurve|.
+% The next macros are the finishing ones; the first strokes the whole curve,
+% while the second fills the (closed) curve with the default color; both close
+% the group that was opened with |\StartCurve|. The third macro is explained
+% in a while; we anticipate it is functional to chose between the first two
+% macros when a star is possibly used to switch between stroking and filling.
+%    \begin{macrocode}
+\def\CurveFinish{\strokepath\endgroup\ignorespaces}% 
+\def\FillCurve{\fillpath\endgroup\ignorespaces}
+\def\CurveEnd{\fillstroke\endgroup\ignorespaces}
+%    \end{macrocode}
+%
+% In order to draw the internal arcs it would be desirable to have a single
+% macro that, given the destination point, computes the control points that
+% produce a cubic Bézier spline that joins the starting point with the
+% destination point in the best possible way. The problem is strongly ill
+% defined and has an infinity of solutions; here we give two solutions:
+% $(a)$ a supposedly smart one that resorts to osculating circles and
+% requires only the direction at the destination point; and $(b)$ a less
+% smart solution that requires the control points to be specified in a
+% certain format.
+%
+% We start with solution $(b)$, |\CbezierTo|, the code of which is simpler
+% than that of solution $(a)$; then we will produce the solution $(a)$,
+% |\CurveTo|,  that will become the main building block for a general path
+% construction macro, |\Curve|. 
+%
+% The ``naïve'' macro |\CBezierTo| simply uses the previous point direction saved in |\@Dzero| as a unit vector by the starting macro; specifies
+% a destination point, the distance of the first control point from the
+% starting point, the destination point direction that will save also for the
+% next arc drawing macro as a unit vector, and the distance of the second
+% control point from the destination point along this last direction. Both
+% distances must be  positive possibly fractional numbers. The syntax will
+% be therefore:
+%\begin{flushleft}
+%\cs{CbezierTo}\meta{end
+% point}|WithDir|\meta{direction}|AndDist|\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.
+%
+% This macro uses the input information to use the internal |pict2e| macro
+% |\pIIe@curveto| with the proper arguments, and to save the final direction
+% into the same |\@Dzero| macro for successive use of other macros.
+%    \begin{macrocode}
+\def\CbezierTo#1WithDir#2AndDists#3And#4{%
+\GetCoord(#1)\@tX\@tY \MakeVectorFrom\@tX\@tY to\@Puno
+\GetCoord(#2)\@tX\@tY \MakeVectorFrom\@tX\@tY to \@Duno
+\DirOfVect\@Duno to\@Duno
+\ScaleVect\@Dzero by#3to\@Czero \AddVect\@Pzero and\@Czero to\@Czero
+\ScaleVect\@Duno by-#4to \@Cuno \AddVect\@Puno and\@Cuno to \@Cuno
+\GetCoord(\@Czero)\@XCzero\@YCzero
+\GetCoord(\@Cuno)\@XCuno\@YCuno
+\GetCoord(\@Puno)\@XPuno\@YPuno
+\pIIe@curveto{\@XCzero\unitlength}{\@YCzero\unitlength}%
+             {\@XCuno\unitlength}{\@YCuno\unitlength}%
+             {\@XPuno\unitlength}{\@YPuno\unitlength}%
+\CopyVect\@Puno to\@Pzero
+\CopyVect\@Duno to\@Dzero
+\ignorespaces}%
+%    \end{macrocode}
+%
+% With this building block it is not difficult to set up a macro that draws
+% a Bézier arc between two given points, similarly as the other macro
+% |\CurveBetween| described previously.
+%
 %    \begin{macrocode}
-\def\CurveFinish{\strokepath\endgroup\ignorespaces}%
+\def\CbezierBetween#1And#2WithDirs#3And#4UsingDists#5And#6{%
+\StartCurveAt#1WithDir{#3}\relax
+\CbezierTo#2WithDir#4AndDists#5And{#6}\CurveFinish}
 %    \end{macrocode}
+
+%
+% An example of use is shown in figure~\ref{fig:Cbezier}; notice that the
+% tangents at the end points are the same for the black curve drawn with
+% |\CurveBetween| and the four red curves drawn with |\CbezierBetween|; the
+% five red curves differ only for the distance of their control point $C_0$
+% from the starting  point; the differences are remarkable and the topmost
+% curve even presents a slight inflection close to the end point. These
+% effects cannot be obtained with the ``smarter'' macro |\CurveBetween|. But
+% certainly this simpler macro is more difficult to use because of the
+% distances of the control point are sort of unpredictable and require a
+% number of cut-and-try 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 WithDirs45:1And-45:1UsingDists4And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists6And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists8And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists10And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists12And{1}
+%\end{picture}
+%\end{verbatim}
+%\end{minipage}
+%\hfill
+%\begin{minipage}{0.40\textwidth}\raggedleft
+%\unitlength=0.1\textwidth
+%\begin{picture}(10,3)(0,1.25)
+%\CurveBetween0,0and10,0WithDirs1,1and{1,-1}
+%\color{red}%
+%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists4And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists6And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists8And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists10And{1}
+%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists12And{1}
+%\end{picture}
+%\end{minipage}
+%\caption{Comparison between similar arcs drawn with \cs{CurveBetween} (black)
+% and \cs{CbezierTo} (red)}
+%\label{fig:Cbezier}
+%\end{figure}
 %
-% The ``real'' curve macro comes next; it is supposed to determine the control
+%
+% 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
@@ -2144,39 +2853,20 @@ and the derived files curve2e.sty and curve2e.pdf.
 % osculating circle, a circle tangent to the curve at that node. The ambiguity
 % of the stated problem may be solved by establishing that the chord of the
 % osculating circle has the same direction as the chord of the arc being drawn,
-% and that the curve chord is divided into two parts each of which should be
-% interpreted as half the chord of the osculating circle; this curve chord
-% division is made proportionally to the projection of the tangent directions
-% on the chord itself. Excluding degenerate cases that may be dealt with
-% directly, imagine the triangle built with the chord and the two tangents;
-% this triangle is straightforward if there is no inflection point; otherwise it
-% is necessary to change one of the two directions by reflecting it about the
-% chord. This is much simpler to view if a general rotation of the whole
-% construction is made so as to bring the curve chord on the $x$ axis, because
-% the reflection about the chord amounts to taking the complex conjugate of one
-% of the directions. In facts with a concave curve the ``left'' direction
-% vector arrow and the ``right'' direction vector tail lay in the same half
-% plane, while with an inflected curve, they lay in opposite half plains, so
-% that taking the complex conjugate of one of directions re-establishes the
-% correct situation for the triangle we are looking for.
-%
-% This done the perpendicular from the triangle vertex to the cord divides the
-% chord in two parts (the foot of this perpendicular may lay outside the chord,
-% but this is no problem since we are looking for positive solutions, so that
-% if we get negative numbers we just negate them); these two parts are taken as
-% the half chords of the osculating circles, therefore there is no problem
-% determining the distances $K_{\mathrm{left}}$ and $K_{\mathrm{right}}$ from
-% the left and right
-% nodes by using the same formula we used with circular arcs. Well\dots\ the
-% same formula means that we have to determine the radius from the half chord
-% and its inclination with the node tangent; all things we can do with the
-% complex number algebra and macros we already have at our disposal. If we look
-% carefully at this computation done for the circular arc we discover that in
-% practice we used the half chord length instead of the radius; so the coding
-% is actually the same, may be just with different variable names.
+% 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. 
+%
+% We use the formula we got for arcs~\eqref{equ:corda}, where the half chord is
+% indicated with $s$,  and we derive the necessary distances:
+%\begin{subequations}\label{equ:Kzero-Kuno}
+%\begin{align}
+%K_0 &= \frac{4}{3} s\frac{1-\cos\theta_0}{\sin^2\theta_0}\\
+%K_1 &=\frac{4}{3}s\frac{1-\cos\theta_1}{\sin^2\theta_1}
+%\end{align}
+%\end{subequations}
 %
 % We therefore start with getting the points and directions and calculating the
-% chord and its direction
+% chord and its direction:
 %    \begin{macrocode}
 \def\CurveTo#1WithDir#2{%
 \def\@Puno{#1}\def\@Duno{#2}\DirOfVect\@Duno to\@Duno
@@ -2187,117 +2877,75 @@ and the derived files curve2e.sty and curve2e.pdf.
 %    \begin{macrocode}
 \MultVect\@Dzero by*\@DirChord to \@Dpzero
 \MultVect\@Duno by*\@DirChord to \@Dpuno
-\GetCoord(\@Dpzero)\@Xpzero\@Ypzero
-\GetCoord(\@Dpuno)\@Xpuno\@Ypuno
+\GetCoord(\@Dpzero)\@DXpzero\@DYpzero
+\GetCoord(\@Dpuno)\@DXpuno\@DYpuno
+\DivideFN\@Chord by2 to\@semichord
 %    \end{macrocode}
 % The chord needs not be actually rotated because it suffices its length
-% along the real axis; the chord length is memorized in |\@Chord|.
+% along the real axis; the chord length is memorised in |\@Chord| and
+% its half is saved in |\@semichord|.
 %
 % We now examine the various degenerate cases, when either tangent is
-% perpendicular to the chord, or when it is parallel pointing inward or outward,
-% with or without inflection.
+% perpendicular or parallel to the chord. Notice that we are calculating
+% the distances of the control points from the adjacent nodes using the
+% half chord length, not the full length. We also distinguish between the
+% computations relative to the arc starting point and those relative to
+% the end point.
 %
-% We start with the $90^\circ$ case for the ``left'' direction
-% separating the cases when the other direction is or is not $90^\circ$~\dots
 %    \begin{macrocode}
-\ifdim\@Xpzero\p@=\z@
-  \ifdim\@Xpuno\p@=\z@
-    \@tdA=0.666666\p@
-    \Numero\@Mcpzero{\@Chord\@tdA}%
-    \edef\@Mcpuno{\@Mcpzero}%
-  \else
-    \@tdA=0.666666\p@
-    \Numero\@Mcpzero{\@Chord\@tdA}%
-    \SetCPmodule\@Mcpuno from\@ne\@Chord\@Dpuno%
-  \fi
+\ifdim\@DXpzero\p@=\z@
+   \@tdA=1.333333\p@
+   \Numero\@KCzero{\@semichord\@tdA}%
+\fi
+\ifdim\@DYpzero\p@=\z@
+  \@tdA=1.333333\p@
+  \Numero\@Kpzero{\@semichord\@tdA}%
+\fi
 %    \end{macrocode}
-% \dots\ from when the ``left'' direction is not perpendicular to the chord; it
-% might be  parallel and we must distinguish the cases for the other direction~\dots
-%    \begin{macrocode}
-\else
-  \ifdim\@Xpuno\p@=\z@
-    \@tdA=0.666666\p@
-    \Numero\@Mcpuno{\@Chord\@tdA}%
-    \SetCPmodule\@Mcpzero from\@ne\@Chord\@Dpzero%
-  \else
-     \ifdim\@Ypzero\p@=\z@
-       \@tdA=0.333333\p@
-       \Numero\@Mcpzero{\@Chord\@tdA}%
-       \edef\@Mcpuno{\@Mcpzero}% 
-%    \end{macrocode}
-% \dots\ from when the left direction is oblique and the other direction is
-% either parallel to the chord~\dots
-%    \begin{macrocode}
-     \else
-       \ifdim\@Ypuno\p@=\z@
-         \@tdA=0.333333\p@
-         \Numero\@Mcpuno{\@Chord\@tdA}%
-         \SetCPmodule\@Mcpzero from\@ne\@Chord\@Dpzero
-%    \end{macrocode}
-% \dots\ and, finally, from when both directions are oblique with respect to
-% the chord; we must see if there is an inflection point; if both direction
-% point to the same half plane we have to take the complex conjugate of one
-% direction so as to define the triangle we were speaking about above.
-%    \begin{macrocode}
-       \else
-         \@tdA=\@Ypzero\p@ \@tdA=\@Ypuno\@tdA
-         \ifdim\@tdA>\z@
-           \ConjVect\@Dpuno to\@Dwpuno
-         \else
-           \edef\@Dwpuno{\@Dpuno}%
-         \fi
-%    \end{macrocode}
-% The control sequence |\@Dwpuno| contains the right direction for forming the
-% triangle; we can make the weighed subdivision of the chord according to the
-% horizontal components of the directions; we eventually turn negative values
-% to positive ones since we are interested in the magnitudes of the control
-% vectors.
-%    \begin{macrocode}
-         \GetCoord(\@Dwpuno)\@Xwpuno\@Ywpuno
-         \@tdA=\@Xpzero\p@ \@tdA=\@Ywpuno\@tdA
-         \@tdB=\@Xwpuno\p@ \@tdB=\@Ypzero\@tdB
-         \DividE\@tdB by\@tdA to\@Fact
-         \@tdC=\p@ \advance\@tdC-\@Fact\p@
-         \ifdim\@tdC<\z@ \@tdC=-\@tdC\fi
-         \DividE\p@ by \@Fact\p@ to\@Fact
-         \@tdD=\p@ \advance\@tdD-\@Fact\p@
-         \ifdim\@tdD<\z@ \@tdD=-\@tdD\fi
-%    \end{macrocode}
-% Before dividing by the denominator we have to check the directions, although
-% oblique to the chord are not parallel to one another; in this case there is
-% no question of a weighed subdivision of the chord
-%    \begin{macrocode}
-         \ifdim\@tdD<0.0001\p@
-           \def\@factzero{1}%
-           \def\@factuno{1}%
-         \else
-           \DividE\p@ by\@tdC to\@factzero
-           \DividE\p@ by\@tdD to\@factuno
-         \fi
-%    \end{macrocode}
-% We now have the subdivision factors and we call another macro for determining
-% the required magnitudes
-%    \begin{macrocode}
-         \SetCPmodule\@Mcpzero from\@factzero\@Chord\@Dpzero
-         \SetCPmodule\@Mcpuno from\@factuno\@Chord\@Dwpuno
-      \fi
-     \fi
+% The distances we are looking for are positive generally fractional numbers;
+% so if the components are negative, we take the absolute values. Eventually
+% we determine the absolute control point coordinates. 
+%    \begin{macrocode}
+\unless\ifdim\@DXpzero\p@=\z@
+  \unless\ifdim\@DYpzero\p@=\z@
+    \edef\@CosDzero{\ifdim\@DXpzero\p@<\z@ -\fi\@DXpzero}%
+    \edef\@SinDzero{\ifdim\@DYpzero\p@<\z@ -\fi\@DYpzero}%
+    \@tdA=\@semichord\p@ \@tdA=1.333333\@tdA
+    \DividE\@tdA by\@SinDzero\p@ to \@KCzero
+    \@tdA=\dimexpr(\p@-\@CosDzero\p@)
+    \DividE\@KCzero\@tdA by\@SinDzero\p@ to \@KCzero
   \fi
 \fi
+\ScaleVect\@Dzero by\@KCzero to\@CPzero
+\AddVect\@Pzero and\@CPzero to\@CPzero
 %    \end{macrocode}
-% Now we have all data we need and we determine the positions of the control
-% points; we do not work any more on the rotated diagram of the horizontal
-% chord, but we operate on the original points and directions; all we had to
-% compute, after all, were the distances of the control points along the
-% specified directions; remember that the ``left'' control point is along the
-% positive ``left'' direction, while the ``right'' control point precedes the
-% curve node along the ``right'' direction, so that a vector subtraction must
-% be done.
+% We now repeat the calculations for the arc end point, taking into
+% consideration that the end point direction points outwards, so that in
+% computing the end point control point we have to take this fact into
+% consideration by using a negative sign for the distance; in this way
+% the displacement of the control point from the end point takes place
+% in a backwards direction.
 %    \begin{macrocode}
-\ScaleVect\@Dzero by\@Mcpzero to\@CPzero
-\AddVect\@Pzero and\@CPzero to\@CPzero
-\ScaleVect\@Duno by\@Mcpuno to\@CPuno
-\SubVect\@CPuno from\@Puno to\@CPuno
+\ifdim\@DXpuno\p@=\z@
+   \@tdA=-1.333333\p@
+   \Numero\@KCuno{\@semichord\@tdA}%
+\fi
+\ifdim\@DYpuno\p@=\z@
+  \@tdA=-1.333333\p@
+  \Numero\@KCuno{\@semichord\@tdA}%
+\fi
+\unless\ifdim\@DXpuno\p@=\z@
+    \unless\ifdim\@DYpuno\p@=\z@
+    \edef\@CosDuno{\ifdim\@DXpuno\p@<\z@ -\fi\@DXpuno}%
+    \edef\@SinDuno{\ifdim\@DYpuno\p@<\z@ -\fi\@DYpuno}%
+    \@tdA=\@semichord\p@ \@tdA=-1.333333\@tdA
+    \DividE\@tdA by \@SinDuno\p@ to \@KCuno
+    \@tdA=\dimexpr(\p@-\@CosDuno\p@)
+    \DividE\@KCuno\@tdA by\@SinDuno\p@ to \@KCuno
+  \fi
+\fi
+\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 tracing.
@@ -2310,7 +2958,7 @@ and the derived files curve2e.sty and curve2e.pdf.
              {\@XPuno\unitlength}{\@YPuno\unitlength}%
 %    \end{macrocode}
 %  It does not have to stroke the curve because other Bézier splines might still
-%  be added to the path. On the opposite it memorizes the final point as the
+%  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
@@ -2318,37 +2966,30 @@ and the derived files curve2e.sty and curve2e.pdf.
 \ignorespaces}%
 %    \end{macrocode}
 %
-%  The next macro is used to determine the control vectors lengths when we have
-% the chord fraction, the chord length and the direction along which to compute
-% the vector; all the input data (arguments from \#2 to \#4) may be passed as
-% control sequences so the calling statement needs not use any curly braces.
-%    \begin{macrocode}
-\def\SetCPmodule#1from#2#3#4{%
-\GetCoord(#4)\t@X\t@Y
-\@tdA=#3\p@
-\@tdA=#2\@tdA
-\@tdA=1.333333\@tdA
-\@tdB=\p@ \advance\@tdB +\t@X\p@
-\DividE\@tdA by\@tdB to#1\relax
-\ignorespaces}%
-%    \end{macrocode}
 %
-% We finally define the overall |\Curve| macro that recursively examines an
+% 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 initializes the drawing process and
+% 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
+% direction, because it would be meaningless at the beginning of a curve. The
+% second macro defines the path to the next point and checks for another node; if
+% the next list item is a square bracket delimited argument, it interprets it as
 % a change of direction, while if it is another parenthesis delimited argument it
 % interprets it as a new node-direction specification; if the node and direction
-% list is terminated, it issues the stroking command and exits the recursive
-% process. The |@ChangeDir| macro is just an interface for executing the regular
-% |\ChangeDir| macro, but also for recursing again by recalling |\@Curve|.
-%    \begin{macrocode}
-\def\Curve(#1)<#2>{%
+% list is terminated, it issues the stroking or filling command through
+% |\CurveEnd|, and exits the recursive process. The |\CurveEnd| control
+% sequence has a different meaning depending on the fact that the main macro
+% was starred or unstarred. The |@ChangeDir| macro is just an interface to
+% execute the regular |\ChangeDir| macro, but also for recursing again by
+% recalling |\@Curve|.
+%    \begin{macrocode}
+\def\Curve{\@ifstar{\let\fillstroke\fillpath\Curve@}%
+{\let\fillstroke\strokepath\Curve@}}
+\def\Curve@(#1)<#2>{%
     \StartCurveAt#1WithDir{#2}%
     \@ifnextchar\lp@r\@Curve{%
     \PackageWarning{curve2e}{%
@@ -2357,20 +2998,63 @@ and the derived files curve2e.sty and curve2e.pdf.
 \def\@Curve(#1)<#2>{%
     \CurveTo#1WithDir{#2}%
     \@ifnextchar\lp@r\@Curve{%
-    \@ifnextchar[\@ChangeDir\CurveFinish}}
+    \@ifnextchar[\@ChangeDir\CurveEnd}}
 \def\@ChangeDir[#1]{\ChangeDir<#1>\@Curve}
 %    \end{macrocode}
 %
 % As a concluding remark, please notice that the |\Curve| macro is certainly the
 % most comfortable to use, but it is sort of frozen in its possibilities. The
 % user may certainly use the |\StartCurve|, |\CurveTo|, |\ChangeDir|, and
-% |\CurveFinish| for a more versatile set of drawing macros; evidently nobody
-% forbids to exploit the full power of the |\cbezier| original macro for cubic
-% splines.
+% |\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(45,1)[br]{x}\Zbox(1,25)[tl]{y}
+%\Curve(0,0)<1,3.927>%
+%(5,14.14)<1,2.776>%
+%(10,20)<1,0>%
+%(15,14.14)<1,-2.776>%
+%(20,0)<1,-3.927>%
+%(25,-14.14)<1,-2.776>%
+%(30,-20)<1,0>%
+%(35,-14.14)<1,2.776>%
+%(40,0)<1,3.927>%
+%}
+%\put(50,0){\VECTOR(0,0)(45,0)\VECTOR(0,-25)(0,25)
+%\Zbox(45,1)[br]{x}\Zbox(1,25)[tl]{y}
+%\CbezierBetween0,0And20,0WithDirs77:1And-77:1UsingDists28And{28}
+%\CbezierBetween20,0And40,0WithDirs-77:1And77:1UsingDists28And{28}}
+%\end{picture}
+%\caption{A sequence of arcs; the left figure has been drawn with the \cs{Curve} command with a sequence of nine couples of point-direction arguments; the right figure has been drawn with two commands \cs{CbezierBetween} that include also the specification of the control points}
+%\label{fig:sinewawe}
+%\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 three 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 far much easier to use than with
+% single arcs drawn with the |\CbezierBetween| macro.
+%
+% I believe that the set of new macrosprovided 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.
+%\iffalse
+%</package>
+%\fi
 %
-% I believe that the set of new macros can really help the user to draw his/her
-% diagrams with more agility; it will be the accumulated experience to decide if
-% this is true.
 % \Finale
 % \endinput
 
diff --git a/Master/texmf-dist/tex/latex/curve2e/curve2e.sty b/Master/texmf-dist/tex/latex/curve2e/curve2e.sty
index e97c8f5f16b..db002fd2a51 100644
--- a/Master/texmf-dist/tex/latex/curve2e/curve2e.sty
+++ b/Master/texmf-dist/tex/latex/curve2e/curve2e.sty
@@ -9,7 +9,6 @@
 %%   The curve2e package for LaTeX and XeLATeX
 %%   Copyright (C) 2010 Claudio Beccari
 %%   All rights reserved
-%% 
 %%   License information appended
 %% 
 %%
@@ -18,7 +17,9 @@
 %%
 \NeedsTeXFormat{LaTeX2e}[2014/05/01]
 \ProvidesPackage{curve2e}%
-        [2015/06/06 v.1.42 Extension package for pict2e]
+        [2015/06/19 v.1.50 Extension package for pict2e]
+
+
 \RequirePackage{color}
 \RequirePackageWithOptions{pict2e}[2014/01/01]
 \def\TRON{\tracingcommands\tw@ \tracingmacros\tw@}%
@@ -35,8 +36,9 @@
 \def\thicklines{\linethickness{\defaultlinewidth}}%
 \def\thinlines{\linethickness{.5\defaultlinewidth}}%
 \thinlines\ignorespaces}
-\def\LIne(#1,#2){\moveto(0,0)
-      \pIIe@lineto{#1\unitlength}{#2\unitlength}\strokepath}%
+\def\LIne(#1){{\GetCoord(#1)\@tX\@tY
+      \moveto(0,0)
+      \pIIe@lineto{\@tX\unitlength}{\@tY\unitlength}\strokepath}\ignorespaces}%
 \def\segment(#1)(#2){\@killglue\polyline(#1)(#2)}%
 \def\line(#1)#2{\begingroup
   \@linelen #2\unitlength
@@ -52,16 +54,17 @@
     \strokepath
   \fi
 \endgroup\ignorespaces}%
-\ifx\Dline\undefined
-\def\Dline(#1,#2)(#3,#4)#5{%
-\begingroup
-   \countdef\NumA254\countdef\NumB252\relax
-   \MakeVectorFrom{#1}{#2}to\V@ttA
-   \MakeVectorFrom{#3}{#4}to\V@ttB
+\ifx\Dashline\undefined
+\def\Dashline{\@ifstar{\Dashline@@}{\Dashline@}}
+\def\Dashline@(#1)(#2)#3{%
+\bgroup
+   \countdef\NumA3254\countdef\NumB3252\relax
+   \GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttA
+   \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttB
    \SubVect\V@ttA from\V@ttB to\V@ttC
    \ModOfVect\V@ttC to\DlineMod
-   \DividE\DlineMod\p@ by#5\p@ to\NumD
-   \NumA\expandafter\Integer\NumD??
+   \DivideFN\DlineMod by#3 to\NumD
+   \NumA\expandafter\Integer\NumD.??
    \ifodd\NumA\else\advance\NumA\@ne\fi
    \NumB=\NumA \divide\NumB\tw@
    \DividE\DlineMod\p@ by\NumA\p@  to\D@shMod
@@ -69,13 +72,60 @@
    \MultVect\V@ttC by\@tempa,0 to\V@ttB
    \MultVect\V@ttB by 2,0 to\V@ttC
    \advance\NumB\@ne
-   \edef\@mpt{\noexpand\endgroup
-   \noexpand\multiput(\V@ttA)(\V@ttC){\number\NumB}{\noexpand\LIne(\V@ttB)}}%
+   \edef\@mpt{\noexpand\egroup
+   \noexpand\multiput(\V@ttA)(\V@ttC){\number\NumB}%
+      {\noexpand\LIne(\V@ttB)}}%
    \@mpt\ignorespaces}%
+\let\Dline\Dashline
+
+\def\Dashline@@(#1)(#2)#3{\put(#1){\Dashline@(0,0)(#2){#3}}}
+\fi
+\ifx\Dotline\undefined
+\def\Dotline{\@ifstar{\Dotline@@}{\Dotline@}}
+\def\Dotline@(#1)(#2)#3{%
+\bgroup
+   \countdef\NumA 3254\relax \countdef\NumB 3255\relax
+   \GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttA
+   \GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttB
+   \SubVect\V@ttA from\V@ttB to\V@ttC
+   \ModOfVect\V@ttC to\DotlineMod
+   \DivideFN\DotlineMod by#3 to\NumD
+   \NumA=\expandafter\Integer\NumD.??
+   \DivVect\V@ttC by\NumA,0 to\V@ttB
+   \advance\NumA\@ne
+   \edef\@mpt{\noexpand\egroup
+   \noexpand\multiput(\V@ttA)(\V@ttB){\number\NumA}%
+      {\noexpand\makebox(0,0){\noexpand\circle*{0.5}}}}%
+   \@mpt\ignorespaces}%
+
+\def\Dotline@@(#1)(#2)#3{\put(#1){\Dotline@(0,0)(#2){#3}}}
 \fi
 \def\GetCoord(#1)#2#3{%
 \expandafter\SplitNod@\expandafter(#1)#2#3\ignorespaces}
-\def\SplitNod@(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}%
+\def\isnot@polar#1:#2!!{\def\@tempOne{#2}\ifx\@tempOne\empty
+\expandafter\@firstoftwo\else
+\expandafter\@secondoftwo\fi
+{\SplitNod@@}{\SplitPolar@@}}
+
+\def\SplitNod@(#1)#2#3{\isnot@polar#1:!!(#1)#2#3}%
+\def\SplitNod@@(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}%
+\def\SplitPolar@@(#1:#2)#3#4{\DirFromAngle#1to\@DirA
+\ScaleVect\@DirA by#2to\@DirA
+\expandafter\SplitNod@@\expandafter(\@DirA)#3#4}
+
+\let\originalput\put
+\def\put(#1){\bgroup\GetCoord(#1)\@tX\@tY
+\edef\x{\noexpand\egroup\noexpand\originalput(\@tX,\@tY)}\x}
+
+\let\originalmultiput\multiput
+\let\original@multiput\@multiput
+
+\long\def\@multiput(#1)#2#3{\bgroup\GetCoord(#1)\@mptX\@mptY
+\edef\x{\noexpand\egroup\noexpand\original@multiput(\@mptX,\@mptY)}%
+\x{#2}{#3}\ignorespaces}
+
+\gdef\multiput(#1)#2{\bgroup\GetCoord(#1)\@mptX\@mptY
+\edef\x{\noexpand\egroup\noexpand\originalmultiput(\@mptX,\@mptY)}\x(}%)
   \def\vector(#1)#2{%
     \begingroup
       \GetCoord(#1)\d@mX\d@mY
@@ -107,10 +157,11 @@
         \pIIe@lineto{\@xnum\@linelen}{\@ynum\@linelen}%
         \strokepath\fi
     \endgroup}
-\def\Vector(#1,#2){%
-\ifdim#1\p@=\z@\vector(#1,#2){#2}
+\def\Vector(#1){{%
+\GetCoord(#1)\@tX\@tY
+\ifdim\@tX\p@=\z@\vector(\@tX,\@tY){\@tY}
 \else
-\vector(#1,#2){#1}\fi}
+\vector(\@tX,\@tY){\@tX}\fi}}
 \def\VECTOR(#1)(#2){\begingroup
 \SubVect#1from#2to\@tempa
 \expandafter\put\expandafter(#1){\expandafter\Vector\expandafter(\@tempa)}%
@@ -129,13 +180,13 @@
 \def\p@lyline(#1){\GetCoord(#1)\d@mX\d@mY
     \pIIe@lineto{\d@mX\unitlength}{\d@mY\unitlength}%
     \@ifnextchar\lp@r{\p@lyline}{\strokepath\ignorespaces}}
-\def\GraphGrid(#1,#2){\begingroup\textcolor{red}{\linethickness{.1\p@}%
+\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}%
-\endgroup\ignorespaces}
+\egroup\ignorespaces}
 \def\RoundUp#1modulo#2to#3{\expandafter\@tempcnta\Integer#1.??%
 \count254\@tempcnta\divide\count254by#2\relax
 \multiply\count254by#2\relax
@@ -146,34 +197,37 @@
 \ifdefined\dimexpr
     \unless\ifdefined\DividE
 \def\DividE#1by#2to#3{\bgroup
-\countdef\Num2254\relax \countdef\Den2252\relax
-\dimendef\@DimA 2254
-\Num=\p@ \@DimA=#2\relax \Den=\@DimA
-\ifnum\Den=\z@
+\dimendef\Num2254\relax \dimendef\Den2252\relax
+\dimendef\@DimA 2250
+\Num=\p@ \Den=#2\relax
+\ifdim\Den=\z@
   \edef\x{\noexpand\endgroup\noexpand\def\noexpand#3{\strip@pt\maxdimen}}%
 \else
   \@DimA=#1\relax
-  \@DimA=\dimexpr\@DimA*\Num/\Den\relax
-  \edef\x{\noexpand\egroup\noexpand\def\noexpand#3{\strip@pt\@DimA}}%
+  \edef\x{%
+    \noexpand\egroup\noexpand\def\noexpand#3{%
+       \strip@pt\dimexpr\@DimA*\Num/\Den\relax}}%
 \fi
 \x\ignorespaces}%
 \fi
     \unless\ifdefined\DivideFN
-         \def\DivideFN#1by#2to#3{\DividE#1\p@ by#2\p@ to#3}%
+         \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
-        \@DimA=\dimexpr\@DimA*\@DimB/\p@\relax
-        \edef\x{\noexpand\egroup\noexpand\def\noexpand#3{\strip@pt\@DimA}}%
+        \edef\x{%
+           \noexpand\egroup\noexpand\def\noexpand#3{%
+             \strip@pt\dimexpr\@DimA*\@DimB/\p@\relax}}%
         \x\ignorespaces}%
     \fi
 \fi
 
 \unless\ifdefined\Numero
-  \def\Numero#1#2{\dimen3254#2\relax
-  \edef#1{\strip@pt\dimen3254}\ignorespaces}%
+  \def\Numero#1#2{\bgroup\dimen3254=#2\relax
+    \edef\x{\noexpand\egroup\noexpand\edef\noexpand#1{%
+      \strip@pt\dimen3254}}\x\ignorespaces}%
 \fi
 \def\g@tTanCotanFrom#1to#2and#3{%
 \DividE 114.591559\p@ by#1to\X@ \@tdB=\X@\p@
@@ -182,8 +236,7 @@
    \@tdC=\Tan\p@ \@tdD=\I\@tdB
    \advance\@tdD-\@tdC \DividE\p@ by\@tdD to\Tan
    \advance\I-2\relax}%
-\def#2{\Tan}\DividE\p@ by\Tan\p@ to\Cot \def#3{\Cot}%
-\ignorespaces}%
+\def#2{\Tan}\DividE\p@ by\Tan\p@ to\Cot \def#3{\Cot}\ignorespaces}%
 \def\SinOf#1to#2{\bgroup%
 \@tdA=#1\p@%
 \ifdim\@tdA>\z@%
@@ -292,28 +345,55 @@
 \edef\endTanOf{\noexpand\egroup
    \noexpand\def\noexpand#2{\@tempA}\noexpand\ignorespaces}%
 \endTanOf}%
+\def\ArcTanOf#1to#2{\bgroup
+\edef\@tF{#1}\@tdF=\@tF\p@ \@tdE=57.295778\p@
+\@tdD=\ifdim\@tdF>\z@ \@tdF\else -\@tdF\fi
+\unless\ifdim\@tdD>0.02\p@
+    \def\@tX{\strip@pt\dimexpr57.295778\@tdF\relax}%
+\else
+    \edef\@tX{45}\relax
+    \countdef\I 2523 \I=8\relax
+    \@whilenum\I>0\do{\TanOf\@tX to\@tG
+    \edef\@tG{\strip@pt\dimexpr\@tG\p@-\@tdF\relax}\relax
+    \MultiplY\@tG by57.295778to\@tG
+    \CosOf\@tX to\@tH
+    \MultiplY\@tH by\@tH to\@tH
+    \MultiplY\@tH by\@tG to \@tH
+    \edef\@tX{\strip@pt\dimexpr\@tX\p@ - \@tH\p@\relax}\relax
+    \advance\I\m@ne}%
+\fi
+\edef\x{\egroup\noexpand\edef\noexpand#2{\@tX}}\x\ignorespaces}%
 \def\MakeVectorFrom#1#2to#3{\edef#3{#1,#2}\ignorespaces}%
 \def\CopyVect#1to#2{\edef#2{#1}\ignorespaces}%
 \def\ModOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
 \@tempdima=\t@X\p@ \ifdim\@tempdima<\z@ \@tempdima=-\@tempdima\fi
 \@tempdimb=\t@Y\p@ \ifdim\@tempdimb<\z@ \@tempdimb=-\@tempdimb\fi
-\ifdim\@tempdima>\@tempdimb
-   \DividE\@tempdimb by\@tempdima to\@T
-   \@tempdimc=\@tempdima
+\ifdim\@tempdima=\z@
+    \ifdim\@tempdimb=\z@
+        \def\@T{0}\@tempdimc=\z@
+    \else
+        \def\@T{0}\@tempdimc=\@tempdimb
+    \fi
 \else
-   \DividE\@tempdima by\@tempdimb to\@T
-   \@tempdimc=\@tempdimb
+    \ifdim\@tempdima>\@tempdimb
+       \DividE\@tempdimb by\@tempdima to\@T
+       \@tempdimc=\@tempdima
+    \else
+          \DividE\@tempdima by\@tempdimb to\@T
+          \@tempdimc=\@tempdimb
+    \fi
 \fi
-\ifdim\@T\p@=\z@
-\else
-   \@tempdima=\@T\p@ \@tempdima=\@T\@tempdima
-   \advance\@tempdima\p@%
-   \@tempdimb=\p@%
-   \@tempcnta=5\relax
-   \@whilenum\@tempcnta>\z@\do{\DividE\@tempdima by\@tempdimb to\@T
-   \advance\@tempdimb \@T\p@ \@tempdimb=.5\@tempdimb
-   \advance\@tempcnta\m@ne}%
-   \@tempdimc=\@T\@tempdimc
+\unless\ifdim\@tempdimc=\z@
+    \unless\ifdim\@T\p@=\z@
+       \@tempdima=\@T\p@ \@tempdima=\@T\@tempdima
+       \advance\@tempdima\p@%
+       \@tempdimb=\p@%
+       \@tempcnta=5\relax
+       \@whilenum\@tempcnta>\z@\do{\DividE\@tempdima by\@tempdimb to\@T
+       \advance\@tempdimb \@T\p@ \@tempdimb=.5\@tempdimb
+       \advance\@tempcnta\m@ne}%
+       \@tempdimc=\@T\@tempdimc
+    \fi
 \fi
 \Numero#2\@tempdimc
 \ignorespaces}%
@@ -343,22 +423,6 @@
 \CosOf#1to\t@X
 \SinOf#1to\t@Y
 \MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
-\def\ArcTanOf#1to#2{\bgroup
-\edef\@tF{#1}\@tdF=\@tF\p@
-\@tdE=57.295779\p@
-\ifdim\@tdF=\z@\def\@tX{0}\else
-\edef\@tXX{1}%
-\MultiplY57.295779by\@tXX to \@tX
-\countdef\I 2323 \I=7\relax
-\@whilenum\I>0\do{\TanOf\@tX to\@tG
-\CosOf\@tX to \@tH
-\edef\@tG{\strip@pt\dimexpr\@tG\p@-\@tdF\relax}%
-\MultiplY\@tH by\@tH to\@tH
-\MultiplY\@tH by\@tG to \@tH
-\edef\@tXX{\strip@pt\dimexpr\@tXX\p@ - \@tH\p@\relax}%
-\MultiplY57.295779by\@tXX to\@tX
-\advance\I\m@ne}\fi
-\edef\x{\egroup\noexpand\edef\noexpand#2{\@tX}}\x}%
 \def\ArgOfVect#1to#2{\bgroup\GetCoord(#1){\t@X}{\t@Y}%
 \def\s@gno{}\def\addflatt@ngle{0}
 \ifdim\t@X\p@=\z@
@@ -436,7 +500,7 @@
 \MultVect\@tempa by\@Dir to#3\ignorespaces}%
 \def\Arc(#1)(#2)#3{\begingroup
 \@tdA=#3\p@
-\ifdim\@tdA=\z@\else
+\unless\ifdim\@tdA=\z@
   \@Arc(#1)(#2)%
 \fi
 \endgroup\ignorespaces}%
@@ -571,33 +635,33 @@
 \@tdE=\pIIe@FAW\@wholewidth \@tdE=0.8\@tdE
 \DividE\@tdE by \@Raggio\unitlength to\DeltaGradi
 \@tdD=\DeltaGradi\p@ \@tdD=57.29578\@tdD \Numero\DeltaGradi\@tdD
-\@tdD=\ifx\Segno--\fi\@gradi\p@ \Numero\@tempa\@tdD
+\@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{\ifx\Segno-\m@ne\else\@ne\fi}%
+\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\ifx\Segno--\fi\DeltaGradi\p@
+\@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}}% arrow tip at the end point
+\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{\ifx\Segno-\else-\fi\@ne}%
+\edef\@tempa{\if\Segno-\else-\fi\@ne}%
 \MultVect\@V by0,\@tempa to\@vPun
-\@tdE\ifx\Segno--\fi\DeltaGradi\p@
+\@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}}% arrow tip at the starting point
-\edef\@tempa{\ifx\Segno--\fi\DeltaGradi}%
+\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
@@ -607,7 +671,7 @@
 \strokepath\ignorespaces}%
 \def\CurveBetween#1and#2WithDirs#3and#4{%
 \StartCurveAt#1WithDir{#3}\relax
-\CurveTo#2WithDir{#4}\CurveFinish}%
+\CurveTo#2WithDir{#4}\CurveFinish\ignorespaces}%
 \def\StartCurveAt#1WithDir#2{%
 \begingroup
 \GetCoord(#1)\@tempa\@tempb
@@ -622,71 +686,75 @@
 \DirOfVect\@Dzero to\@Dzero
 \ignorespaces}
 \def\CurveFinish{\strokepath\endgroup\ignorespaces}%
+\def\FillCurve{\fillpath\endgroup\ignorespaces}
+\def\CurveEnd{\fillstroke\endgroup\ignorespaces}
+\def\CbezierTo#1WithDir#2AndDists#3And#4{%
+\GetCoord(#1)\@tX\@tY \MakeVectorFrom\@tX\@tY to\@Puno
+\GetCoord(#2)\@tX\@tY \MakeVectorFrom\@tX\@tY to \@Duno
+\DirOfVect\@Duno to\@Duno
+\ScaleVect\@Dzero by#3to\@Czero \AddVect\@Pzero and\@Czero to\@Czero
+\ScaleVect\@Duno by-#4to \@Cuno \AddVect\@Puno and\@Cuno to \@Cuno
+\GetCoord(\@Czero)\@XCzero\@YCzero
+\GetCoord(\@Cuno)\@XCuno\@YCuno
+\GetCoord(\@Puno)\@XPuno\@YPuno
+\pIIe@curveto{\@XCzero\unitlength}{\@YCzero\unitlength}%
+             {\@XCuno\unitlength}{\@YCuno\unitlength}%
+             {\@XPuno\unitlength}{\@YPuno\unitlength}%
+\CopyVect\@Puno to\@Pzero
+\CopyVect\@Duno to\@Dzero
+\ignorespaces}%
+\def\CbezierBetween#1And#2WithDirs#3And#4UsingDists#5And#6{%
+\StartCurveAt#1WithDir{#3}\relax
+\CbezierTo#2WithDir#4AndDists#5And{#6}\CurveFinish}
+
 \def\CurveTo#1WithDir#2{%
 \def\@Puno{#1}\def\@Duno{#2}\DirOfVect\@Duno to\@Duno
 \DistanceAndDirOfVect\@Puno minus\@Pzero to\@Chord and\@DirChord
 \MultVect\@Dzero by*\@DirChord to \@Dpzero
 \MultVect\@Duno by*\@DirChord to \@Dpuno
-\GetCoord(\@Dpzero)\@Xpzero\@Ypzero
-\GetCoord(\@Dpuno)\@Xpuno\@Ypuno
-\ifdim\@Xpzero\p@=\z@
-  \ifdim\@Xpuno\p@=\z@
-    \@tdA=0.666666\p@
-    \Numero\@Mcpzero{\@Chord\@tdA}%
-    \edef\@Mcpuno{\@Mcpzero}%
-  \else
-    \@tdA=0.666666\p@
-    \Numero\@Mcpzero{\@Chord\@tdA}%
-    \SetCPmodule\@Mcpuno from\@ne\@Chord\@Dpuno%
-  \fi
-\else
-  \ifdim\@Xpuno\p@=\z@
-    \@tdA=0.666666\p@
-    \Numero\@Mcpuno{\@Chord\@tdA}%
-    \SetCPmodule\@Mcpzero from\@ne\@Chord\@Dpzero%
-  \else
-     \ifdim\@Ypzero\p@=\z@
-       \@tdA=0.333333\p@
-       \Numero\@Mcpzero{\@Chord\@tdA}%
-       \edef\@Mcpuno{\@Mcpzero}%
-     \else
-       \ifdim\@Ypuno\p@=\z@
-         \@tdA=0.333333\p@
-         \Numero\@Mcpuno{\@Chord\@tdA}%
-         \SetCPmodule\@Mcpzero from\@ne\@Chord\@Dpzero
-       \else
-         \@tdA=\@Ypzero\p@ \@tdA=\@Ypuno\@tdA
-         \ifdim\@tdA>\z@
-           \ConjVect\@Dpuno to\@Dwpuno
-         \else
-           \edef\@Dwpuno{\@Dpuno}%
-         \fi
-         \GetCoord(\@Dwpuno)\@Xwpuno\@Ywpuno
-         \@tdA=\@Xpzero\p@ \@tdA=\@Ywpuno\@tdA
-         \@tdB=\@Xwpuno\p@ \@tdB=\@Ypzero\@tdB
-         \DividE\@tdB by\@tdA to\@Fact
-         \@tdC=\p@ \advance\@tdC-\@Fact\p@
-         \ifdim\@tdC<\z@ \@tdC=-\@tdC\fi
-         \DividE\p@ by \@Fact\p@ to\@Fact
-         \@tdD=\p@ \advance\@tdD-\@Fact\p@
-         \ifdim\@tdD<\z@ \@tdD=-\@tdD\fi
-         \ifdim\@tdD<0.0001\p@
-           \def\@factzero{1}%
-           \def\@factuno{1}%
-         \else
-           \DividE\p@ by\@tdC to\@factzero
-           \DividE\p@ by\@tdD to\@factuno
-         \fi
-         \SetCPmodule\@Mcpzero from\@factzero\@Chord\@Dpzero
-         \SetCPmodule\@Mcpuno from\@factuno\@Chord\@Dwpuno
-      \fi
-     \fi
+\GetCoord(\@Dpzero)\@DXpzero\@DYpzero
+\GetCoord(\@Dpuno)\@DXpuno\@DYpuno
+\DivideFN\@Chord by2 to\@semichord
+\ifdim\@DXpzero\p@=\z@
+   \@tdA=1.333333\p@
+   \Numero\@KCzero{\@semichord\@tdA}%
+\fi
+\ifdim\@DYpzero\p@=\z@
+  \@tdA=1.333333\p@
+  \Numero\@Kpzero{\@semichord\@tdA}%
+\fi
+\unless\ifdim\@DXpzero\p@=\z@
+  \unless\ifdim\@DYpzero\p@=\z@
+    \edef\@CosDzero{\ifdim\@DXpzero\p@<\z@ -\fi\@DXpzero}%
+    \edef\@SinDzero{\ifdim\@DYpzero\p@<\z@ -\fi\@DYpzero}%
+    \@tdA=\@semichord\p@ \@tdA=1.333333\@tdA
+    \DividE\@tdA by\@SinDzero\p@ to \@KCzero
+    \@tdA=\dimexpr(\p@-\@CosDzero\p@)
+    \DividE\@KCzero\@tdA by\@SinDzero\p@ to \@KCzero
   \fi
 \fi
-\ScaleVect\@Dzero by\@Mcpzero to\@CPzero
+\ScaleVect\@Dzero by\@KCzero to\@CPzero
 \AddVect\@Pzero and\@CPzero to\@CPzero
-\ScaleVect\@Duno by\@Mcpuno to\@CPuno
-\SubVect\@CPuno from\@Puno to\@CPuno
+\ifdim\@DXpuno\p@=\z@
+   \@tdA=-1.333333\p@
+   \Numero\@KCuno{\@semichord\@tdA}%
+\fi
+\ifdim\@DYpuno\p@=\z@
+  \@tdA=-1.333333\p@
+  \Numero\@KCuno{\@semichord\@tdA}%
+\fi
+\unless\ifdim\@DXpuno\p@=\z@
+    \unless\ifdim\@DYpuno\p@=\z@
+    \edef\@CosDuno{\ifdim\@DXpuno\p@<\z@ -\fi\@DXpuno}%
+    \edef\@SinDuno{\ifdim\@DYpuno\p@<\z@ -\fi\@DYpuno}%
+    \@tdA=\@semichord\p@ \@tdA=-1.333333\@tdA
+    \DividE\@tdA by \@SinDuno\p@ to \@KCuno
+    \@tdA=\dimexpr(\p@-\@CosDuno\p@)
+    \DividE\@KCuno\@tdA by\@SinDuno\p@ to \@KCuno
+  \fi
+\fi
+\ScaleVect\@Duno by\@KCuno to\@CPuno
+\AddVect\@Puno and\@CPuno to\@CPuno
 \GetCoord(\@Puno)\@XPuno\@YPuno
 \GetCoord(\@CPzero)\@XCPzero\@YCPzero
 \GetCoord(\@CPuno)\@XCPuno\@YCPuno
@@ -696,15 +764,9 @@
 \CopyVect\@Puno to\@Pzero
 \CopyVect\@Duno to\@Dzero
 \ignorespaces}%
-\def\SetCPmodule#1from#2#3#4{%
-\GetCoord(#4)\t@X\t@Y
-\@tdA=#3\p@
-\@tdA=#2\@tdA
-\@tdA=1.333333\@tdA
-\@tdB=\p@ \advance\@tdB +\t@X\p@
-\DividE\@tdA by\@tdB to#1\relax
-\ignorespaces}%
-\def\Curve(#1)<#2>{%
+\def\Curve{\@ifstar{\let\fillstroke\fillpath\Curve@}%
+{\let\fillstroke\strokepath\Curve@}}
+\def\Curve@(#1)<#2>{%
     \StartCurveAt#1WithDir{#2}%
     \@ifnextchar\lp@r\@Curve{%
     \PackageWarning{curve2e}{%
@@ -713,7 +775,7 @@
 \def\@Curve(#1)<#2>{%
     \CurveTo#1WithDir{#2}%
     \@ifnextchar\lp@r\@Curve{%
-    \@ifnextchar[\@ChangeDir\CurveFinish}}
+    \@ifnextchar[\@ChangeDir\CurveEnd}}
 \def\@ChangeDir[#1]{\ChangeDir<#1>\@Curve}
 
 %%