euclideangeometry (6feb20)

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+%%
+%% This is file `README.txt',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% euclideangeometry.dtx  (with options: `readme')
+%% 
+%%   Copyright (C)  2020 Claudio Beccari all rights reserved.
+%%   License information appended
+%% 
+File README.txt for package euclideangeometry
+        [2020-02-02 v.0.1.2 Extension package for curve2e]
+
+The package bundle euclideangeometry is composed of the following files
+
+euclideangeometry.dtx
+euclideangeometry.pdf
+euclideangeometry-man.tex
+euclideangeometry-man.pdf
+README.txt
+
+euclideangeometry.dtx is the documented TeX source file of package
+euclideangeometry.sty; you get euclideangeometry.sty,
+euclideangeometry.pdf by running pdflatex on euclideangeometry.dtx.
+
+README.txt, this file, contains general information.
+
+euclideangeometry-man.tex and euclideangeometry-man.pdf are
+the source file and the readable document containing the end
+user manual.
+In other words euclideangeometry.pdf is oriented towards the
+developers and euclideangeometry-man.pdf to the end users.
+
+Claudio Beccari
+
+claudio dot beccari at gmail dot com
+%% 
+%% Distributable under the LaTeX Project Public License,
+%% version 1.3c or higher (your choice). The latest version of
+%% this license is at: http://www.latex-project.org/lppl.txt
+%% 
+%% This work is "maintained"
+%% 
+%% This work consists of files:
+%% 1) euclideangeometry.dtx, and the derived files euclideangeometry.sty
+%%                           and euclideangeometry.pdf,
+%% 2) euclideangeometry-man.tex and the derived file euclideangeometry-man.pdf,
+%% 3) the auxiliary derived file README.txt
+%% 
+%%
+%% End of file `README.txt'.
diff --git a/Master/texmf-dist/doc/latex/euclideangeometry/euclideangeometry-man.pdf b/Master/texmf-dist/doc/latex/euclideangeometry/euclideangeometry-man.pdf
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+% !TEX encoding = UTF-8 Unicode
+% !TEX TS-program = pdflatex
+
+\documentclass[11pt,titlepage]{article}\errorcontextlines=100
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage[english]{babel}
+\usepackage{lmodern,textcomp,mflogo}
+\usepackage{amsmath,fancyvrb,graphicx,verbatim,afterpage,trace}
+\usepackage{etoolbox,enumitem,xspace,xcolor,euclideangeometry}
+\usepackage{url}
+
+\AfterEndPreamble{%
+\apptocmd{\TeX}{{}\xspace}{}{}
+\apptocmd{\LaTeX}{{}\xspace}{}{}}
+
+\newcommand\heart[3]{%
+\edef\X{\fpeval{#1*16*(sind(#2)^3)}}
+\edef\Y{\fpeval{#1*(13*cosd(#2) - 5*cosd(2*#2) 
+  - 2*cosd(3*#2) -cosd(4*#2)+2.4)}}
+\CopyVect\X,\Y to#3}
+\let\cuore\heart
+
+\begin{document}
+
+\begin{titlepage}
+\thispagestyle{empty}
+\centering
+\textsc{\large Claudio Beccari}\\[2ex]
+\texttt{claudio dot beccari at gmail dot com}
+
+\vspace{\stretch{2}}
+
+{\LARGE\scshape The \pack{euclideangeometry} package\\
+{\large user manual}\par}
+
+\vspace{\baselineskip}
+
+
+\GetFileInfo{euclideangeometry.sty}
+Version \fileversion\ of \filedate
+
+\vspace*{\stretch{5}}
+\end{titlepage}
+
+ \begin{abstract}
+ This file further extends the functionalities of the \pack{curve2e} 
+ package, which, on turn, is an extension of the \pack{pict2e} package
+ to the standard \env{picture} environment as defined in the \LaTeX\ 
+ kernel source file.
+
+ The \pack{curve2e} package was upgraded a the beginning of 2020; the 
+ material
+ of this new package, might have been included in the former one, but is
+ is sospecific, that we preferred defining a standalone one; this package 
+ takes care of requesting the packages it depends from.
+
+ The purpose is to provide the tools to draw most of the geometrical
+ constructions that a high school instructor or bachelor degree professor
+ might need to teach geometry. The connection to Euclide depends on the
+ fact that in its times calculations were made with ruler, compass, and,
+ apparently, also with ellipsograph,
+
+ The user of this package has available all the machinery provided by
+ the \pack{pict2e} and \pack{curve2e} packages, in order to define new functionalities
+ and build macros that draw the necessary lines, circles, and other such
+ objects, as they would have done in the ancient times. Actually just one 
+ macro is programmed to solve a linear system of equations
+ \end{abstract}
+ 
+ \tableofcontents
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section*{Warning}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+The \pack{euclideangeometry} package requires the advanced functionalities of the \LaTeX\!3 (L3) language; if such functionalities are not available for any reason (incomplete/basic installation of the \TeX system; legacy installation of the \TeX system; the \TeX system has not been updated; \dots) input of this package is stopped, the whole job is aborted, and a visible message is issued. 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section{Introduction}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ The \pack{picture} environment has been available since the very beginning
+ of \LaTeX in 1985. At that time it was a very simple environment
+ that allowed to draw very simple line graphics with many limitations.
+ When \LaTeX was upgraded from \LaTeX\!2.09 to \LaTeXe in 1994, Leslie
+ Lamport announced an upgrade that eventually became available in 2003
+ with package \pack{pict2e}; in 2006 I wrote the \pack{curve2e} package that added
+ many more functionalities; both packages were upgraded during these
+ years; and now line graphics with the \env{picture} environment can perform
+ pretty well. The package \pack{euclideangeometry} adds even more specific
+ functionalities in order to produce geometric drawings as they were
+ possible in the old times, when calculus and analytic geometry were
+ not available.
+ 
+ In these years other drawing programs were made available to the \TeX 
+ community; \pack{PSTricks} and \pack{TikZ} are the most known ones, but there are 
+ other less known packages, that perform very well; among the latter 
+ I would like to mention \pack{xpicture}, that relies on \pack{pict2e} and 
+ \pack{curve2e}, but extends the functionalities with a very smart handling 
+ of coordinate systems, that allow to draw many line drawings suitable 
+ for teaching geometry in high schools and introductory courses in the 
+ university bachelor degree programs.
+ 
+ This package \pack{euclideangeomery} in a certain way follows the same
+ path of \pack{xpicture} but it avoids defining a new user language
+ interface; rather it builds new macros by using the same philosophy of
+ the recent \pack{curve2e} package.
+ 
+ It is worth mentioning that now \pack{curve2e} accepts coordinates in both 
+ cartesian and  polar form; it allows to identify specific points of the 
+ drawing with macros, so the same macro can be used over and over again to 
+ address the same points.The package can draw lines, vectors, arcs 
+ with no arrow tips, or with one arrow tip, or with arrow tips at both ends,
+ arcs included. The macros for drawing poly lines, polygons, circles,
+ generic curves (by means of Bézier cubic or quadratic splines) are
+ already available; such facilities are well documented and exemplified
+ in the user manual of \pack{curve2e} package. 
+
+ In what follows there will be several figures drawn with this package;
+ in the background there is a red grid where the meshes are
+ 10\,\cs{unitlenth} apart in both directions; they should help to
+ understand the position of the various drawings on the picture canvas.
+ This grid is useful also to the end user, while s/he is working on a 
+ particular drawing, but when the drawing is finished, the user can 
+ delete the grid command or comment ot that line of code.
+ For what regards the commands used to render the images, their codes can
+ be found in the documented code file \pack{euclideangeometry.pdf}.
+ 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section{Installing \pack{euclideangeometry}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ You are not supposed to manually install package \pack{euclideangeometry}.
+ In facts you have to work with a complete and updated/upgraded \TeX 
+ installation, otherwise this package won't work; this means that
+ you have done your updating after 2020-01-18. And this package is already
+ present in any modern updated complete installation of the \TeX system.
+ Nevertheless the package will load \pack{curve2e} with the wrong version
+ and file date, but this package will abort its own loading.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section{Loading \pack{euclideangeometry}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ If you want to use the \pack{euclideangeometry} package, we suggest you load it with the following command:
+\begin{flushleft}\obeylines
+\cs{usepackage}\oarg{options}\Marg{euclideangeomery}
+\end{flushleft}
+ The package will take care of managing the possible \meta{options}
+ and to call \pack{curve2e} with the specified options; on turn
+ \pack{curve2e} calls \pack{pict2e} passing on the \meta{options}; such
+ \meta{options} are only those usable by \pack{pict2e} because neither
+ \pack{curve2e} nor \pack{euclideangeometry} use any option. If the
+ user is invoking \pack{euclideangeometry}, it is certain s/he does
+ not want to use the native picture environment, but the modern
+ extended one; therefore the only meaningful possible options are
+ \opt{latex} and\opt{pstricks}; such options influence only the shape
+ of the arrow tips; with option \opt{latex} they are triangular,
+ while with \opt{pstricks} they have the shape of a stealth
+ aircraft. The difference is very small; therefore we imagine that
+ even if these options are available, they might never be used.
+
+ Nothing happens if the user forgets this mechanism; therefore if s/he
+ loads \pack{curve2e} and/or \pack{pict2e}, before
+ \pack{euclideangeomentry} the only problem that might arise is an
+ “Option clash” error message; if two of these packages are selected
+ with different arrow tips; not impossible, of course, by we deem
+ it very unlikely.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section{Available commands}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ The commands available with the first extension \pack{pict2e} to the native
+ \amb{picture} environment, maintain their names but do not maintain the
+ same restrictions; in particular there are the following improvements.
+\begin{enumerate}[noitemsep]
+ \item Lines and vectors are drawn as usual by \cs{put}ting in place
+ their forms, but their inclinations are not limited to a limited
+ number of slope parameters, originally specified with reciprocally
+ prime single digit values not exceeding~6 for lines, and~4 for vectors;
+ the length of these sloped objects is still their horizontal component;
+ now, the slopes may be described with any signed fractional number not
+ exceeding $2^{30}-1$ in absolute value; it still is a limited number of
+ slopes, but their combinations are practically countless.
+%
+\item There is no restriction on the minimum length of lines and vectors.
+%
+\item Circles and dots can be drawn at any size, not at that dozen
+ or so of finite sizes that were accepted with the original environment.
+%
+\item Ovals may be specified the corner curvature; the default
+ size of the quarter circles that make up the oval corners may be
+ specified; if no specification is given the radius of such corners
+ is the maximum that can be fitted in the oval; in practice it is
+ half the shortest value between the oval height and width.
+
+\item The quadratic Bézier splines do not require the specification
+ of the number of dots that were used by the native environment to
+ draw “arbitrary” curves; now they are drawn with continuous curved
+ lines.
+\end{enumerate}
+ Some new commands were added by \pack{pict2e}
+\begin{enumerate}[noitemsep]
+
+\item The third degree (cubic) Bézier splines are sort of new; certainly
+ now they are traced with continuous lines; if it is desired, it is
+ possible to replace the continuous line with a number of dots so as
+ to have a (unevenly) dotted curve. It suffices to specify the number 
+ of dots the curve should be made with.
+
+\item\cs{arc} and \cs{arc*} draw an arc or a filled circular sector,
+ with their centers at the axes origin; therefore  they need to be put
+ in place somewhere else by means of the usual \cs{put} command.
+
+\item The new command \cs{Line} traces a segment from one given point
+ to another point; it is very convenient to specify the end points
+ instead of the slope the line must have to go form the starting
+ point to the ending one. The command does not require the \cs{put}
+ command to put the segment in place; nevertheless it can be shifted
+ somewhere else with \cs{put} if it becomes necessary.
+
+\item the new command \cs{polyline} draws a sequence of connected
+ segments that form a piecewise linear “curve”; the way segments are
+ joined to one another depend from the “join” specifiers that
+ \pack{pict2e} has introduced; they will be described further on.
+
+\item \cs{polygon} and \cs{polygon*} produce closed paths as it would
+ be possible when using \cs{polyline} and specifying the last point
+ coincident with the first point of that curve. The closed path is
+ filled with the default color if the asterisk is used.
+\end{enumerate}
+
+ There were also the low level commands user interfaces to the various 
+ drivers; these drivers really exist, but \pack{pict2e}
+ knows how to detect the correct language of the necessary drive;
+ the user is therefore allowed to pretend to ignore the existence
+ of such drivers; s/he can simply use these commands; their names are
+ almost self explanatory.
+\begin{enumerate}[noitemsep]
+\item \cs{moveto} Sets the start of a line tracing to an initial point.
+
+\item \cs{lineto} traces a segment up to a specified point.
+
+\item \cs{curveto} traces a third degree Bézier up to the third specified
+ point, while using the other two ones as control points.\footnote{If
+ these terms are unfamiliar, please read the \pack{pict2e} documentation.}
+
+ \item \cs{circlearc} traces a circumference arc from the last line
+ point to a specified destination; its center, its angle amplitude, its
+ initial point are among the specified arguments, but the reader should
+ check on the \pack{pict2e} documentation for the details.
+
+\item[]\mbox{\bfseries Attention!} Notice that these commands produce
+ just information to trace lines, but by themselves they do not trace 
+ anything; in order to actually trace the curve or do other operations 
+ with what has been done after the user finished describing the line to 
+ trace, the following low level commands must be used.
+
+\item A \cs{closepath} is necessary if it is desired to
+ join the last position to the initial one. But if the last point
+ specified coincides with the very first one, a closed loop is
+ effectively already completed.
+
+\item If a \cs{strokepath} command is used the line is drawn.
+
+\item If a \cs{fillpath} command is used, the line loop is filled by
+ the current color. Notice, if the described line is not a closed loop,
+ this filling command acts as if the line first point and last point were
+ joined by a straight line.
+
+\end{enumerate}
+
+ While describing a line with the above low level commands, or with
+ the previous high level commands, lines and segments join and finish
+ as described hereafter; the following commands must be used, possibly
+ within a group, before actually tracing a specific line made up with
+ several joined lines or curves. Notice that their effect is just
+ visible with lines as thin as 1\,pt, and very visible with thicker
+ lines.
+\begin{enumerate}[noitemsep]
+\item \cs{buttcap} truncates each line with a sharp cut perpendicular
+ to the line axis exactly through the line end point (default).
+
+\item \cs{roundcap} adds a semicircle to the very end of each line.
+
+\item \cs{squarecap} adds the half square to the very end of each line.
+
+\item \cs{miterjoin} joins two (generally straight) lines with a miter
+ (or mitre) joint; this means that the borders of the line are prolonged
+ until they meet; it is very nice when the junction angle is not far away
+ from, or is larger than $90^\circ$. Apparently for \pack{pict2e} this
+ type of joint is the default.
+
+\item \cs{roundjoin} joins each (generally straight) line with
+ a \cs{roundcap}; it is good in most circumstances.
+
+\item \cs{beveljoin} joins two (generally straight) lines with a miter
+ joint truncated with a sharp cut perpendicular to the bisector of the
+ lines axes; with acute angles it is better than the miter joint, but
+ when angles are very small, even this joint is not adequate.
+\end{enumerate}
+
+ Notice that \cs{buttcap} is the default, but in general it might be
+ better to declare the \cs{roundcap} for the whole document.
+
+ We do not go further in the description of the new \pack{pict2e}
+ modified and new new commands; the reader unfamiliar with programmable
+ drawing and the \pack{pic2e} extensions can consult that package
+ documentation. Actually all commands have been redefined or modified
+ by \pack{curve2e} in order to render them at least compatible with
+ both the cartesian and polar coordinates. In oder to have a better understanding of these details, see figure~\ref{fig:joins}\footnote{The \cs{polyline} macro has the default join of type bevel; remember to specify a different join type if you want a different one.}.
+ 
+ \begin{figure}[!htb] \centering
+ \makebox[\textwidth]{\unitlength=0.009\textwidth
+ \begin{picture}(130,50)
+ \GraphGrid(130,50)
+ \linethickness{4mm}
+ \buttcap\miterjoin
+ \put(10,10){\polyline[\miterjoin](0,0)(30,0)(30,30)
+ \put(0,-10){\makebox(30,10)[b]{\cs{buttcap},\cs{miterjoin}}}}
+ \roundcap\roundjoin
+ \put(50,10){\polyline[\roundjoin](0,0)(30,0)(30,30)
+ \put(0,-10){\makebox(30,10)[b]{\cs{roundcap},\cs{roundjoin}}}}
+ \squarecap\beveljoin
+ \put(90,10){\polyline(0,0)(30,0)(30,30)
+ \put(0,-10){\makebox(30,10)[b]{\cs{squarecap},\cs{beveljoin}}}}
+ \end{picture}}
+ \caption{Different caps and joins}\label{fig:joins}
+ \end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%>>>
+\section{\pack{curve2e} extensions}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ Again we do not enter into the details, because the user can read the
+ new user manual \file{curve2e-manual.pdf} simply by entering and
+ executing the \verb|texdoc curve2e-manual| command into a terminal or 
+ command prompt window; this new manual is available with version~2.2.0 
+ (or higher) of \pack{curve2e} and it contains the extensions and sample 
+ codes for (simple) sample drawings; some examples are not so simple, but 
+ show the power of this package upgrade.
+
+ The most important two changes are $(a)$ the choice of different
+ coordinates for addressing points on the drawing canvas, and $(b)$
+ the possibility of using macros to identify specific points.
+ As already mentioned, such changes have been applied also to most,
+ if not all\footnote{I assume I have upgraded all such commands; if not,
+ please, send me a bug notice; I will acknowledge your contribution.}
+ commands defined by \pack{pict2e}.
+ 
+ \pack{curve2e} defines a lot of operations the user can do with
+ the point coordinates; this is done by assuming they are complex
+ numbers, or vectors, or roto\-amplification operators, and making
+ with such entities a lot of actions compatible with their “incarnation".
+ For example multiplying a vector by a roto\`amplification operator,
+ in spite the fact that internally they are both represented by ordered
+ pairs of (generally) fractional numbers, means simply obtaining a new
+ vector rotated and scaled with respect to the original one; the point
+ addressed by the first vector, becomes another point in a different
+ precise position.
+
+ Below you see several examples of usage of such commands; but here space
+ will be saved if a short list is made concerning these “complex number”
+ operations.
+ 
+ Remember the double nature of such complex numbers:
+\[
+z = x + \iu y = m\eu^{\iu \phi}
+\]
+ therefore addition and subtraction are simply done with
+\[
+z_1 \pm z_2 = x_1 \pm x_2 +\iu (y_1 \pm y_2)
+\]
+ Multiplications and divisions are simply done with
+\begin{align*}
+z_1 z_2 &=( m_1 m_2) \eu^{\iu(\phi_1 + \phi_2)} \\
+z_1/z_2 &=( m_1/m_2) \eu^{\iu(\phi_1 - \phi_2)}
+\end{align*}
+ Squares and square roots\footnote{The square root of a complex number 
+ has two values; here we do not go into the details on how \pack{curve2e} 
+ choses one or the other value. In practice, the \pack{curve2e} macros 
+ that use square roots, work mostly on scalars to find magnitudes that 
+ are always positive.} are simply done with:
+\begin{align*}
+z^2      &= m^2\eu^{\iu 2\phi}\\
+\sqrt{z} &= \sqrt{m}\eu^{\iu\phi/2}
+\end{align*}
+ The complex conjugate of a complex number is shown with a superscript
+ asterisk:
+\[\text{if } z= x+\iu y \text{ then } z^\star = x- \iu y\]
+ and from these simple formal rules many results can be obtained;
+ and therefore several macros must be defined.
+
+ But let us summarise. Here is a short list with a minimum of
+ explanation of the commands functionalities introduced by
+ \pack{curve2e}. The user notices that many commands rely on
+ a delimited argument command syntax; the first arguments can
+ generally be introduced with point macros, as well as numerical
+ coordinates (no matter if cartesian and polar ones) while
+ the output(s) should always be in form of point macro(s). Parentheses
+ for delimiting the ordered couples or the point macros are seldom
+ required. On the other side, the variety of multiple optional
+ arguments, sometimes requires the use of different delimiters,
+ most often than not the signs~\texttt{<~>}, in addition to the
+ usual brackets. These syntax functionalities are available with the
+ \pack{xparse} and \pack{xfp} packages, that render the language~L3
+ very useful and effective.
+
+ Handling of complex numbers is done with the following commands.
+ New commands to draw special objects, are also described.
+\begin{enumerate}[noitemsep]
+%
+\item Cartesian and polar coordinates; they are distinguished by
+ their separator; cartesian coordinates are the usual comma separated
+ couple \meta{$x,y$}; polar coordinates are specified with a colon
+ separated couple \meta{$\theta{:}\,\rho$}. In general they are
+ specified within parentheses, but some commands require them without
+ any parentheses. In what follows a generic math symbol, such as for
+ example $P_1$, is used to indicate a complex number that addresses
+ a particular point, irrespective of the chosen coordinate type,
+ or a macro  defined to contain those coordinates.
+
+\item The complex number/vector operations already available with
+ \pack{curve2e} are the following; we specify “macro” because in general
+ macros are used, instead of explicit numerical values, but for input
+ vector macros it is possible to use the comma or colon separated ordered
+ couple; “versor” means “unit vector”; angles are always expressed in
+ degrees; output quantities are everything follows the key word
+ \texttt{to}; output quantities are alway supposed to be in the form
+ of control sequences.
+\begin{itemize}\small
+\item \cs{MakeVectorFrom}\meta{number,number}\meta{numeric macro} to\meta{vector macro}
+\item \cs{CopyVect}\meta{vector macro} to\meta{vector macro}
+\item \cs{ModOfVect}\meta{vector macro} to\meta{modulus macro}
+\item \cs{DirOfVect}\meta{vector macro} to\meta{versor macro}
+\item \cs{ModAndDirOfVect}\meta{vector macro} to\meta{modulus macro}
+    and\meta{versor macro}
+\item \cs{ModAndAngleOfVect}\meta{vector macro} to \meta{modulus macro}
+ and\meta{angle macro}
+\item \cs{DistanceAndDirOfVect}\meta{1st vector macro}
+    minus\meta{2nd vector macro} to\meta{distance macro}
+    and\meta{versor macro}
+\item \cs{XpartOfVect}\meta{vector macro} to\meta{numerical macro}
+\item \cs{YpartOfVect}\meta{vector macro} to\meta{numerical macro}
+\item \cs{DirFromAngle}\meta{angle macro} to\meta{versor macro}
+\item \cs{ArgOfVect}\meta{vector macro} to\meta{angle macro}
+\item \cs{ScaleVect}\meta{vector macro} by\meta{scale factor}
+    to\meta{vector macro}
+\item \cs{ConjVect}\meta{vector macro} to\meta{conjugate vector macro}
+\item \cs{SubVect}\meta{subtrahend vector} from\meta{minuend vector}
+    to\meta{vector macro}
+\item \cs{AddVect}\meta{1st vector} and\meta{2nd vector}
+    to\meta{vector macro}
+\item \cs{Multvect}\marg{1st vector}\meta{$\star$}\marg{2nd vector}\meta{$
+    \star$}\meta{output vector macro}\newline the asterisks are optional;
+    either one changes the \meta{2nd vector} into its complex conjugate
+\item \cs{MultVect}\meta{1st vector}\meta{$\star$}\meta{2nd vector}
+    to\meta{vector macro}\newline discouraged; maintained for backward
+    compatibility; the only optional asterisk changes the \meta{2nd vector}
+    into its complex conjugate
+\item \cs{Divvect}\marg{dividend vector}\marg{divisor vector}\marg{output
+    vector macro}
+\item \cs{DivVect}\meta{dividend vector}\meta{divisor vector}
+    to\meta{vector macro}\newline maintained for backwards
+    compatibility
+\end{itemize}
+
+\item A new command \cs{segment}\parg{$P_1$}\parg{$P_2$} draws a line that
+ joins the specified points.
+
+\item Command \cs{Dashline}\parg{$P_1$}\parg{$P_2$}\marg{dash length}
+ draws a dashed line between the specified points; the
+ \meta{dash length} is specified as a coefficient of
+ \cs{unitlenth} so they are proportioned  to the diagram scale. The gap
+ between dashes is just as wide as the dashes; they are recomputed by
+ the command in order to slightly adjust the \meta{dash length} so
+ that the line starts at point $P_1$ with a dash, and ends at $P_2$
+ again with a dash.
+
+\item Command \cs{Dotline}\parg{$P_1$}\parg{$P_2$}\marg{gap}\oarg{diameter}
+ traces a dotted line between the specified points with dots \meta{gap}
+ units apart, starting and ending with a dot at the specified points.
+ Optionally the absolute diameter of the dots may be specified: a
+ diameter of 1\,pt (default) is visible, but it might be too small;
+ a diameter of 1\,mm is really very black, and may be too large; if
+ the diameter is specified without dimensions they are assumed by
+ default to be typographic points.
+
+\item Command \cs{polyline}, \cs{polygon} and \cs{polygon*} are redefined
+ to accept both coordinate kinds.
+
+\item Commands \cs{VECTOR}\parg{$P_1$}\parg{$P_2$} (and \cs{VVECTOR}, with the same syntax) draw vectors with one arrow tip at the end, or arrow tips at both ends respectively.
+
+\item New commands \cs{Arc}\parg{center}\parg{start}\marg{angle} and,
+ with the same syntax, \cs{VectorArc} and \cs{VectorARC} draw 
+ arcs without or with arrow tip(s), with the specified \meta{center},
+ starting at point \meta{start}, with an aperture of \meta{angle}
+ degrees (not radians). \cs{Arc} draws the arc without arrow tips;
+ \cs{VectorArc} draws the arc with one arrow tip at the end point;
+ \cs{VectorARC} draws an arc with arrow tips at both ends.
+
+\item Command \cs{multiput} has been redefined to accept optional
+ arguments, besides the use of coordinates of both kinds. The new syntax
+ is the following:
+\begin{flushleft}\obeylines
+\cs{multiput}\oarg{shift}\parg{origin}\parg{step}\marg{number}\marg{object}\oarg{handler}
+\end{flushleft}
+ where, if you neglect the first and the last (optional) arguments,
+ you have the original syntax; the \meta{origin} point is where the first
+ \meta{object} is placed; \meta{step} is the displacement of a new
+ \meta{object} relative to the previous one; \meta{number} is the
+ total number of \meta{object}s put in place by the command; possibly
+ the number may be an integer expression computed with the \cs{inteval}
+ function of the L3~language, accessed through the \pack{xfp} package
+ already loaded by \pack{curve2e}. The new features are \meta{shift},
+ that is used to displace the whole drawing somewhere else (in case
+ some fine tuning is required), and \meta{handler}; the latter is a
+ powerful means to control both the object to be set in place and its
+ position; further on there will be examples that show that the object
+ can be put not only on straight paths, but also un other curves,
+ including parabolas, circles, and other shapes.
+
+\item Another version of repetitive commands \cs{xmultiput} is very
+ similar to \cs{multiput} but the iterations are controlled in a
+ different way so that it is possible also to draw continuous
+ curves describing analytical functions even with parametric
+ equations. Further on there will be some examples.
+
+\item The preloaded \pack{xfp} package provides two important 
+ functionalities, i.e. two L3 “functions”, \cs{fpeval} and \cs{inteval};
+ the latter executes expressions on integer numbers containing the usual
+ operators \texttt{+, -, *, /}; the division quotient is rounded to the
+ nearest (positive or negative) integer. The former operates with real
+ fractional numbers and, in addition to the usual arithmetical operators as
+ \cs{inteval}, it can use many mathematical functions, from square roots,
+ to exponentials, logarithms, trigonometric and hyperbolic direct and
+ inverse functions\footnote{The implementation of inverse hyperbolic
+ function is on the L3~Team “to do” list.}, plus other ones. Normally
+ fractional numbers are operated on decimal strings, with 16 fractional
+ places, and 14 integer places but the L3~functions accept also scientific
+ notation. The user can specify truncation or rounding to a specified
+ number of digits. Such integer and fractional mathematical operations
+ are already integrated in most computations performed by \pack{curve2e}.
+
+\item \pack{curve2e} provides two more L3 functions: \cs{fptest} and
+ \cs{fpdowhile} with the following syntax:
+\begin{flushleft}\obeylines
+\cs{fptest}\marg{test}\marg{true}\marg{false}
+\cs{fpdowhile}\marg{test}\marg{actions}
+\end{flushleft}
+ For both macros the \meta{test} is a logical F3~expression; its operands
+ are logical constants, logical values, logical numeric comparisons; its
+ operators are the typical \texttt{||}, \texttt{\&\&}, and \texttt{!},
+ respectively for OR, AND, and NOT. The logical numerical comparisons
+ are mathematical constants or expressions connected with relation
+ operators, such as \texttt{>, =, <}; such operators may be negated with
+ the NOT operator; therefore, for example, \texttt{!>} means “not greater 
+ than”, therefore “lower or equal to”. 
+
+\item The above tests are very useful to control both \cs{fptest} and
+ \cs{fpdowhile}. The logical \meta{test} result lets \cs{fptest} execute
+ only the \meta{true} or the \meta{false} code. Before using \cs{fpdowhile}
+ the \meta{test} expression must be initialised to be \texttt{true};
+ the \meta{actions} should contain some code to be iteratively executed,
+ but they must contain some assignments, typically a change in an
+ iteration counter, such that eventually the \meta{test} logical
+ expression becomes \texttt{false}. Lacking this assignments, the
+ loop continues to infinity, or better, until a fatal error message
+ is issued that informs that the program working memory is exhausted.
+
+\item Such new commands are already used to code the \cs{multiput} and
+ \cs{xmultiput} commands, but they are available also to the user who can
+ operate in a very advanced way; further on, some examples will show
+ some advanced drawings.
+
+\item General curves can be drawn by \pack{pic2e} command \cs{curve}
+ that is sort of difficult to use, because the user has to specify
+ also the control points of the third order Bézier splines. Some other
+ new commands are available with \pack{curve2e}, that are supposed
+ to be easier to use; they are described in the following items.
+
+\item The new command \cs{Curve} joins a sequence of third order
+ splines by simply specifying the node-direction coordinates; i.e. at the
+ junction of two consecutive splines, in a certain interpolation node the
+ final previous spline tangent has the same direction of the tangent
+ at the second spline first node; if a change of direction is required, an
+ optional new direction can be specified. Therefore this triplet of
+ information has the following syntax:
+\begin{flushleft}
+ \parg{node}\aarg{direction}\oarg{new direction}
+\end{flushleft}
+ Evidently the \meta{new direction} is specified only for the nodes
+ that correspond to a cusp. A variation of the command arguments is
+ available by optionally specifying the “looseness” of the curve:
+\begin{flushleft}
+\parg{node}\aarg{direction\texttt{;}start\texttt{,}end}\oarg{\dots}
+\end{flushleft}
+ where \meta{start} is the spline starting “looseness” and \meta{end}
+ is the spline ending one. These (generally different) values
+ are an index of how far is the control point from the adjacent node.
+ With this functionality the user has a very good control on the curve
+ shape and curvature.
+
+\item A similar command \cs{Qurve} works almost the same way, but it
+ traces a quadratic Bézier spline; this one is specified only with two
+ nodes an a single control point, therefore is less configurable than
+ cubic splines; the same final line requires several quadratic splines
+ when just a single cubic spline might do the same job. Notice also that
+ quadratic splines are just parabolic arcs, therefore without inflections,
+ while a cubic spline can have one inflexion. 
+
+\item A further advanced variation is obtained with the new
+ \cs{CurveBetween} command that creates a single cubic spline between two
+ given points with the following syntax:
+\begin{flushleft}\ttfamily
+\cs{CurveBetween}\meta{node1} And\meta{node2} WithDirs \meta{dir1}
+    And\meta{dir2}
+\end{flushleft}
+
+\item A similar variant command is defined with the following syntax:
+\begin{flushleft}\ttfamily
+ \cs{CbezierBetween}\meta{node1} And\meta{node2} WithDirs\meta{dir1}
+    And\meta{dir2}\\ 
+\qquad UsingDists\meta{dist1} And\meta{dist2}
+\end{flushleft}
+ Usage examples are shown in section~\ref{sec:examples}
+\end{enumerate}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{ Euclidean geometry commands}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ With the already large power of \pack{curve2e} there was a push towards
+ specialised applications; the first of which was, evidently, geometry;
+ that kind of geometry that was used in the ancient times when
+ mathematicians did not have available the sophisticated means they
+ have today; they did not even have a positional numerical notation, that
+ arrived in the “west” of the world we are familiar with, just by
+ the XI-XII century; before replacing the roman numbering system another
+ couple of centuries passed by; real numbers with the notation we use
+ today with a decimal separator, had to wait till the XVI century (at
+ least); many things that naw are taught in elementary school were
+ still a sort of magic until the end of XVIII century.
+
+ Even a simple algebraic second degree equation was a problem.
+ In facts the Renaissance was the artistic period when the classical
+ proportions were brought back to the artists who could not solve the
+ simple equation where a segment of unit length is divided in two
+ unequal parts $x$  and $1-x$ such that the following proportion exists
+ among the various parts and the whole segment:
+\[
+\frac{x}{1} = \frac{1-x}{x} \Longrightarrow x = \frac{1}{x} - 1
+\]
+ today we cam solve the problem by manipulating that simple proportion
+ to get
+\[
+ x^2 + x -1 = 0
+\]
+ and we know that the equation has two solution of opposite signs, and that
+ their magnitudes are the reciprocal of one another. Since we are
+ interested in their magnitudes, we adapt the solutions in the form
+\begin{equation}
+ x_{1,2} = \frac{\sqrt{5} \pm 1}{2}= \sqrt{1+0.5^2}\pm 0.5 \Longrightarrow
+\begin{cases}x_1=1.618\ldots\\x_2=0.618\ldots \end{cases}\label{equ:aureo}
+\end{equation}
+ The larger number is called the \emph{golden number} and the smaller
+ one the \emph{golden section}. 
+
+ Luca Pacioli, by the turn of centuries XV–XVI, was the tutor of
+ Guidubaldo, the heir of Federico di Montefeltro, Duke of
+ Urbino\footnote{If you never visited this Renaissance city and its Ducal
+ Palace, consider visiting it; it is one of the many UNESCO Heritage
+ places.}; he wrote the famous book \emph{De Diuina Proportione} that
+ contained also the theory of the golden section accompanied by beautiful
+ drawings of many Platonic solids and other non convex ones, drawn by
+ Leonardo da Vinci. Everything was executed with perfect etchings, even the
+ construction of the golden section; in its basic form\footnote{The third
+ formula in equation~\eqref{equ:aureo} is written in such a way as to
+ explain the graphical construction in figure~\ref{fig:aureo}.} it is
+ replicated in figure~\ref{fig:aureo}. 
+\begin{figure}
+\begin{Esempio}[\setfontsize{7}](0.40)
+\unitlength=0.005\linewidth
+\begin{picture}(170,140)(0,-70)
+\GraphGrid(0,-70)(170,140)
+\VECTOR(0,0)(170,0)
+\Pbox(170,0)[t]{x}[0]
+\Pbox(100,0)[t]{\mathrm{1}}[2]
+\Pbox(0,0)[r]{O}[2]
+\Arc(100,0)(50,0){-90}
+\segment(100,0)(100,70)
+\segment(0,0)(100,50)
+\Pbox(50,0)[tr]{\mathrm{0.5}}[2]
+\ModAndAngleOfVect100,50 to\M and\A 
+\Arc(0,0)(\M,0){\A}\Pbox(\M,0)[bl]{C}[2]
+\Arc(\M,0)(\M,-50){90}
+\Arc(\M,0)(\M,-50){-90}
+\Pbox(\fpeval{\M-50},0)[b]{\mathit{x_2}}[3]
+\Pbox(\fpeval{\M+50},0)[b]{\mathit{x_1}}[3]
+\put(\M,0){\Vector(-70:50)}
+\Pbox(120,-25)[bl]{\mathit{r}=\mathrm{0.5}}[0]
+\thicklines
+\segment(0,0)(100,0)
+\end{picture}
+\end{Esempio}
+\caption{The golden section $x_2$ and the golden number $x_1$}\label{fig:aureo}
+\end{figure}
+ By the way figure~\ref{fig:aureo} shows also the code that is used
+ for the drawing done completely with the facilities available just with \pack{curv2e}. It is also a usage example of several commands. 
+
+ Illiteracy was very widespread; books were expensive and were
+ common just in the wealthy people mansions.
+
+ Mathematicians in the classical times B.C. up to the artists in the
+ Renaissance, had no other means but to use geometrical constructions with
+ ruler and compass. Even today in schools where calculus is not yet
+ taught as a normal subject, possibly not  in certainly high school degree
+ courses, but certainly not in elementary and junior high schools, the
+ instructors have to recourse to geometrical constructions. Sometimes, as
+ in Italy, access to public universities is open with no restrictions to
+ all students with a high school diploma for degree courses that are 
+ more vocational than cultural. Therefore such students in some university 
+ degree courses have to frequent upgrading courses in order to master some 
+ more mathematics compared to what they studied during their basic 
+ education.
+
+ The instructors nowadays very often prepare some booklets with their
+ lessons; such documents, especially in electronic form, are a nice help
+ for many students. And \LaTeX is used to write such documents.
+ Therefore this extension module is mostly dedicated to such instructors.
+
+ The contents of this module is not exhaustive; it just shows a way to
+ use the \pack{curve2e} facilities to extend it to be suited for the
+ kind of geometry they teach.
+ 
+ Here we describe the new commands provided by this package; then in
+ section~\ref{sec:examples} we show their usage by means  examples.
+
+\begin{enumerate}[noitemsep]
+\item Command \cs{IntersectionOfLines} is a fundamental one; its syntax is
+ the following:
+\begin{ttsintassi}
+\cs{IntersecionOfLines}\parg{point1}\parg{dir1} and\parg{point2}\parg{dir2} to\meta{vector}
+\end{ttsintassi}
+ were each line is identified with its \meta{point} and its direction
+ \meta{dir}; the intersection coordinates go to the output \meta{vector}.
+
+\item A second command\cs{IntersectionOfSegments} does almost the same
+ work, but the coordinates of a segment define also its direction,
+ which is the argument of the difference of the terminal nodes of each
+ segment; the syntax therefore is the following:
+\begin{ttsintassi}
+\cs{IntersectionOfSegments}\parg{point11}\parg{point12} 
+ and\parg{point21}\parg{point22}to\meta{vector}
+\end{ttsintassi}
+ Again the intersection point coordinates go to the output \meta{vector}.
+ The first segment is between points 11 and 12, and, similarly, the second
+ segment is between points 21 and 22.
+
+\item Command \cs{ThreePointCircle} draws a circle that goes through three
+ given points; the syntax is the following:
+\begin{ttsintassi}
+ \cs{ThreePointCircle}\meta{$\star$}\parg{point1}\parg{point2}\parg{point3}
+\end{ttsintassi}
+ A sub product of this macro is formed by the vector \cs{C} that contains
+ the coordinates of the center of the circle, that might be useful even
+ if the circle is not drawn; the optional asterisk, if present, does not
+ draw the circle, but the center is available.
+
+\item Alternatively
+\begin{ttsintassi}
+ \cs{ThreePointCircleCenter}\parg{point1}\parg{point2}\parg{point3}to\meta{vector}
+\end{ttsintassi}
+ computes the three point circle center assigning its coordinates to
+ \meta{vector}.
+
+\item Command \cs{CircleWithCenter} draws a circle given its center and it
+ radius; in facts the syntax is the following:
+\begin{ttsintassi}
+\cs{CircleWithCenter}\meta{center} Radius\meta{Radius}
+\end{ttsintassi}
+ This macro does not require the \cs{put} command to put the circle
+ in place.
+
+\item A similar macro \cs{Circlewithcenter} does almost the same; its
+ syntax is the following:
+\begin{ttsintassi}
+\cs{Circlewithcenter}\meta{center} radius\meta{radius}
+\end{ttsintassi}
+ Apparently these two commands do the same, but, no, they behave
+ differently: in the former command the \meta{Radius} is a vector the
+ modulus of which si computed and used as the radius; in the latter
+ command the \meta{radius} is a scalar and (its magnitude) is directly
+ used.
+
+\item Command with syntax:
+\begin{ttsintassi}
+\cs{AxisOf}\meta{point1} and\meta{point2} to \meta{point3} and\meta{point4}
+\end{ttsintassi}
+ is used to determine the axis of a segment; the given
+ segment is specified with its end points \meta{point1} and \meta{point2}
+ and the axis is determined by point \meta{point3} and \meta{point4};
+ actually \meta{point3} is the middle point of the given segment.
+
+\item These two commands with syntax:
+\begin{ttsintassi}
+\cs{SegmentCenter}\parg{point1}\parg{point2}to\meta{center}
+\cs{MiddlePointOf}\parg{point1}\parg{point2}to\meta{center}
+\end{ttsintassi}
+ determine just the middle point between two given points. They are
+ totally equivalent, aliases to one another; sometimes it is more
+ convenient to use a name, sometimes the other; it helps reading the
+ code and maintaining it.
+
+\item Given a triangle  and a specific vertex, it is possible to determine
+ the middle point of the opposite side; it is not very difficult, but it
+ is very handy to have all the necessary elements to draw the median line.
+ The simple syntax is the following:
+\begin{ttsintassi}
+\cs{TriangleMedianBase}\meta{vertex} on\meta{base1} and\meta{base2}
+\qquad to\meta{base middle point}
+\end{ttsintassi}
+
+\item A similar command \cs{TriangleHeightBase} is used to determine the
+ intersection of the height segment from one vertex to the opposite base;
+ with triangles that have an obtuse angle, the height base might lay
+ externally to one of the bases adjacent to such an angle. The syntax is
+ the following
+\begin{ttsintassi}
+\cs{TriangleHeigthtBase}\meta{vertex} on\meta{base1} and\meta{base2} to\meta{height base}
+\end{ttsintassi}
+
+\item Similarly there is the \cs{TriangleBisectorBase} macro with
+ a similar syntax:
+\begin{ttsintassi}
+\cs{TriangleBisectorBase}\meta{vertex} on\meta{base1} and\meta{base2}
+\qquad to\meta{bisector base}
+\end{ttsintassi}
+
+\item A triangle \emph{barycenter} is the point where its median lines
+ intersect; command \cs{TriangleBarycenter} determines its coordinates
+ with the following syntax.
+\begin{ttsintassi}
+\cs{TriangleBarycenter}\parg{vertex1}\parg{vertex2}\parg{vertex3} to\meta{barycenter}
+\end{ttsintassi}
+
+\item A triangle \emph{orthocenter} is the point where its height lines
+ intersect; command \cs{TriangleOrthocenter} determines its coordinates
+ with the following syntax:
+\begin{ttsintassi}
+\cs{TriangleOrthocenter}\parg{vertex1}\parg{vertex2}\parg{vertex3} to\meta{orthocenter}
+\end{ttsintassi}
+
+\item A triangle \emph{incenter} is the point where its bisector lines
+ intersect; command \cs{TriangleIncenter} determines its coordinates
+ with the following syntax:
+\begin{ttsintassi}
+\cs{TriangleIncenter}\parg{vertex1}\parg{vertex2}\parg{vertex3} to\meta{incenter}
+\end{ttsintassi}
+
+\item The distance of a specified point from a given segment or line is
+ computed with the following command
+\begin{ttsintassi}
+\cs{DistanceOfPoint}\meta{point} from\parg{point1}\parg{point2} to\meta{distance}
+\end{ttsintassi}
+ where \meta{point} specifies the point and \meta{point1} and \meta{point2}
+ identify two points on a segment or a line; \meta{distance} is a scalar
+ value.
+
+\item In a construction that will be examined in section~\ref{sec:examples}
+ we need to determine an ellipse axis if the other axis and the focal
+ distance are know; actually it solves the relation
+\begin{equation}a^2= b^2 +c^2\label{equ:axes-foci}\end{equation}
+ that connects such three quantities; $a$ is always the largest of the
+ three quantities; therefore the macro tests if the first entry is larger
+ than the second one: if is is, it computes a Pitagorean difference,
+ otherwise the user should pay attention to use as the first entry the
+ smaller among $b$ and $c$, so as to compute a Pitagorean sum. The command
+ is the following:
+\begin{ttsintassi}
+\cs{AxisFromAxisAndFocus}\meta{axis or focus} and\meta{focus or axis} 
+\qquad to\meta{other axis or focus}
+\end{ttsintassi}
+ The word “axis” stands for “semi axis length”; the word “focus" stands
+ for “focal semi distance”; actually the macro works equally well with
+ full lengths, instead of half lengths; its is important not to mix
+ full and half lengths. Such lengths are expressed as factors of
+ \cs{unitlength}, not as absolute values. This command is described 
+ again when dealing with the specific problem referred to at the 
+ beginning of this list item;  the description is going to be more 
+ detailed and another macro is added to avoid possible errors.
+
+\item Given a segment, i.e. the coordinates of its end points, it is
+ useful to have a macro that computes its length; at the same time it is
+ useful to to compute its direction; this operation is not the same as to
+ compute modulus and argument of a vector, but consists in computing such
+ quantities from the difference of the vectors pointing to the segment
+ end points. These two macros are the following:
+\begin{ttsintassi}
+\cs{SegmentLength}\parg{point1}\parg{point2} to\meta{length}
+\cs{SegmentArg}\parg{point1}\parg{point2} to\meta{argument}
+\end{ttsintassi}
+ The \meta{argument} is computed in the interval $-180^\circ < \phi \leq
+ +180^\circ$; it represents the argument of the vector that goes from
+ \meta{point1} to \meta{point2}, therefore the user must pay attention to
+ the order s/he enters the end points coordinates.
+
+\item The next command \cs{SymmetricalPointOf} is used to find the
+ reflection of a specified point with respect to a fixed point; of course
+ the latter is the middle point of the couple, but the unknown to be
+ determined is not the center of a segment, but one of its end points.
+ The syntax is the following:
+\begin{ttsintassi}
+\cs{SymmetricalPointOf}\meta{point1} respect\meta{fixed} to\meta{point2}
+\end{ttsintassi}
+
+\item  Command \cs{RegPolygon} draws a regular polygon inscribed within
+ a circle of given radius and center, with a specified number of sides;
+ optional arguments allow to  specify color and thickness of the sides,
+ or the polygon interior color; this macro operates differently from the 
+ one for drawing ellipses, that draws simultaneously an ellipse with
+ the border of a color and the interior of another one; with this macro the
+ user who wants to achieve this effect must superimpose to polygons with
+ different settings; but it would not be too difficult to arrange a new
+ macro or to modify this one in order to get “bicolor” polygons.
+ It is not necessary for the purpose of this package, therefore we
+ let the user express his/her phantasy with other macros. The actual
+ syntax is the following:
+\begin{ttsintassi}
+\cs{RegPolygon}\meta{$\star$}\parg{center}\marg{radius}\marg{sides}\oarg{angle}\aarg{settings}
+\end{ttsintassi}
+ The initial optional asterisk specifies if the interior has to be
+ coloured; if yes, the \meta{settings} refer to the color of the
+ interior; if not, the \meta{settings} refer to the thickness and
+ color of the sides; no \meta{settings} imply sides drawn with the
+ default line thickness, generally the one corresponding to \cs{thinlines},
+ and the default color (generally black) for the sides or the interior.
+ By default the first vertex is set to an angle of $0^\circ$ with
+ respect to the \meta{center}; the optional \meta{angle} modifies
+ this value to what is necessary for a particular polygon. The
+ \meta{center} itself is optional, in the sense that if it is not
+ specified the center lays in the origin of the \env{picture} axes; if this 
+ argument is specified, the polygon center is displaced accordingly.
+ The number of sides in theory may be very high, but it is not wise
+ to exceed a couple of dozen sides; if the number of sides is too
+ high, the polygon becomes undistinguishable from a circumference.
+
+\item Several macros are dedicated to ellipses; their names are spelled
+ in Italian, “ellisse”, because the name “ellipse” is already taken by
+ other packages; with Italian user command names there should be no
+ interference with other packages, or the risk is reduced to
+ a minimum. The various macros are \cs{ellisse}, \cs{Sellisse},
+ \cs{Xellisse}, \cs{XSellisse}, \cs{EllisseConFuoco} \cs{EllisseSteiner};
+ the last two control sequence names are aliased with the corresponding
+ English ones \cs{EllipseWithFocus} and \cs{SteinerEllipse}. For the
+ other four ones it is wise to avoid English names for the reasons
+ explained above. After all the Italian  and the English names are very
+ similar and are pronounced almost identically.
+
+ {\tolerance=3000 Actually \cs{ellisse} is practically a shorthand for 
+ \cs{Sellisse} because
+ some optional arguments are already fixed, but the meaning of
+ \cs{fillstroke} depends on the presence or absence of an initial
+ asterisk; similarly \cs{Xellisse} is a sort of a shorthand for
+ \cs{XSellisse}; in facts those commands, that contain
+ an ‘S’ in their names, can optionally perform also the affine
+ \emph{shear} transformation, while those without the ‘S’ do not execute
+ such transformation. Figure~\ref{fig:shear} displays a normal ellipse
+ with its bounding rectangle, and the same ellipse to which the shear
+ affine transformation is applied; the labeled points represent the
+ third order Bézier spline nodes and control points.\par}
+\begin{figure}[!htb]
+\dimendef\Wmp=2000 \Wmp=\dimexpr(\textwidth-\columnsep)/2\relax
+\begin{minipage}{\Wmp}\centering
+\unitlength=0.007\linewidth
+\begin{picture}(100,70)(-50,-35)
+\XSellisse(0,0){50}{25}*
+\end{picture}
+\end{minipage}
+\hfill
+\begin{minipage}{\Wmp}\centering
+\unitlength=0.007\linewidth
+\begin{picture}(100,70)(-50,-35)
+\XSellisse(0,0)<20>{50}{25}*
+\end{picture}
+\end{minipage}
+\caption{The effect of shearing an ellipse with its bounding rectangle}\label{fig:shear}
+\end{figure}
+
+\item The syntax of those six commands are the following:
+\begin{ttsintassi}
+\cs{Sellisse}\meta{$\star$}\marg{semiaxis-h}\marg{semiaxis-v}\oarg{shear}
+\cs{ellisse}\meta{$\star$}\marg{semiaxis-h}\marg{semiaxis-v}
+\cs{XSellisse}\meta{$\star$}\parg{center}\oarg{angle}\aarg{shear}\marg{semiaxis-h}\%
+\qquad \marg{semiaxis-v}\meta{$\star$}\oarg{settings1}\oarg{settings2}
+\cs{Xellisse}\meta{$\star$}\parg{center}\oarg{angle}\marg{semiaxis-h}\%
+\qquad\marg{semiaxis-v}\oarg{settings1}\marg{settings2}
+\cs{EllipseWithFocus}\meta{$\star$}\parg{vertex1}\parg{vertex2}\parg{vertex3}\parg{focus}
+\cs{SteinerEllipse}\meta{$\star$}\parg{vertex1}\parg{vertex2}\parg{vertex3}\oarg{diameter}
+\end{ttsintassi}
+ All require the semi axis lengths; the \meta{semiaxis-h} and
+ \meta{semiaxis-v} refer to the semi axes before possible rotation by
+ \meta{angle} degrees, and do not make assumptions on which axis is the
+ larger one. The optional parameter \meta{shear} is the angle in degrees
+ by which the vertical coordinate lines are rotated by effect of shearing. 
+ If \meta{shear}, that by default equals zero, is not set
+ to another value, the asterisks of command \cs{Sellisse} and
+ \cs{XSellisse} do not have any effect. Otherwise the asterisk of
+ \cs{Sellisse} forces to draw the ellipse bounding box (rectangle before
+ shearing, parallelogram after shearing) as shown together with some
+ marked special points (the vertices, spline nodes and control points
+ of the quarter circles or quarter ellipses) in figure~\ref{fig:shear}.
+ For \cs{ellipse} the asterisk implies filling, instead of stroking the
+ ellipse contour.
+ The \meta{setting}~1 and~2 refer to the color filling and/or border
+ color, and contour thickness, as already explained. For the
+ \cs{EllipseWithFocus}, the \meta{focus} contains the coordinates
+ of one of the two ellipse foci; such coordinates should point to
+ some position \emph{inside} the triangle. The \cs{SteinerEllipse}
+ requires less data, in the sense that such ellipse is unique; it
+ is the ellipse internally tangent to the triangle at its side middle
+ points.
+
+\end{enumerate}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Examples}\label{sec:examples}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ Here we can show some examples of the advanced \pack{curve2e} commands
+ and of what can be done with this \pack{euclideangeometry} extension.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Straight and curved vectors}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ 
+ Figure~\ref{fig:vectors} shows some vectors and vector arcs with the
+ code used to draw them; as usual some points are described with cartesian
+ coordinates and some with polar ones.
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{8.5}](0.45)
+\unitlength=0.01\linewidth
+\begin{picture}(100,60)
+\GraphGrid(100,60)
+\put(0,30){\vector(1,2){10}}
+\put(20,30){\Vector(10,20)}
+\VECTOR(40,30)(50,50)
+\VVECTOR(60,30)(70,60)
+\Arc(100,60)(80,60){90}
+\VectorArc(0,0)(20,0){90}
+\VectorARC(100,0)(80,0){-90}
+\polyvector(30,0)(35,10)(55,20)(60,0)
+\end{picture}
+\end{Esempio}
+\caption{Some vectors and vector arcs}\label{fig:vectors}
+\end{figure}
+
+
+\begin{figure*}[!tb]
+\begin{Esempio}*[\setfontsize{10}]\centering
+\unitlength=0.006\linewidth\begin{picture}(120,90)
+%
+\RegPolygon(9,20){20}{6}<\linethickness{3pt}\color{red}> 
+\RegPolygon(55,20){20}{7}[90]
+\RegPolygon(100,20){20}{8}[22.5]<\linethickness{0.5ex}\color{blue}>
+%
+\put(0,50){%
+  \RegPolygon(9,20){20}{3}\RegPolygon(9,20){20}{3}[30]
+  \RegPolygon(9,20){20}{3}[60]\RegPolygon(9,20){20}{3}[90]
+%
+  \RegPolygon*(55,20){20}{4}<\color{green}>
+  \RegPolygon(55,20){20}{4}<\linethickness{1ex}>
+%
+  \RegPolygon*(100,20){20}{4}[45]<\color{orange}>
+  \RegPolygon(100,20){20}{4}[45]<\linethickness{1ex}\color{blue}>
+}
+\end{picture}
+\end{Esempio}
+\caption{A variety of polygons and their codes}\label{fig:variety-of-polygons}
+\end{figure*}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Polygons}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ Figures~\ref{fig:pentagon} and~\ref{fig:filled-pentagon} display
+ a normal and a color filled pentagon with their codes.
+ Figure~\ref{fig:variety-of-polygons} shows a variety of polygons
+ with their codes.
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{10}](0.8)
+\unitlength=0.5mm
+\begin{picture}(40,32)(-20,-17)
+\polyline(90:20)(162:20)(234:20)(306:20)(378:20)(90:20)
+\end{picture}
+\end{Esempio}
+\caption{A normal polygon drawn with \cs{polyline}}
+\label{fig:pentagon}
+\end{figure}
+
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{10}](0.8)
+\unitlength=.5mm
+\begin{picture}(40,32)(-20,-20)
+\color{magenta}
+\polygon*(90:20)(162:20)(234:20)(306:20)(378:20)
+\end{picture}
+\end{Esempio}
+\caption{A filled polygon drawn with \cs{polygon}}
+\label{fig:filled-pentagon}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \subsection{Dashed and dotted lines}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ For dotted lines there is a possibility of specifying the dot size;
+ it can be specified with an explicit unit of measure, or, if no unit is
+ specified, it is assumed to be “points”. The \cs{Dotline} takes care
+ of transforming the implied or the explicit dimension in multiples of
+ \cs{unitlength}. Figure~\ref{fig:DashDot} shows some examples with their
+ codes.
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{10}](0.65)
+\unitlength=1mm
+\begin{picture}(40,40)
+\GraphGrid(40,40)
+\Dashline(0,0)(40,10){4}
+\put(0,0){\circle*{2}}
+\Dashline(40,10)(0,25){4}
+\put(40,10){\circle*{2}}
+\Dashline(0,25)(20,40){4}
+\put(0,25){\circle*{2}}
+\put(20,40){\circle*{2}}
+\Dotline(0,0)(40,40){2}[0.75mm]
+\put(40,40){\circle*{2}}
+\end{picture}
+\end{Esempio}
+\caption{Dashed and dotted lines}\label{fig:DashDot}
+\end{figure}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Generic curves}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ With the \cs{Curve} macro it is possible to make line art or filled shapes. Figures~\ref{fig:hearts} show the same shape, the first just stroked and the second color filled. 
+
+\begin{figure}[!htp]
+\begin{Esempio}[\setfontsize{9}](0.65)
+\unitlength=8mm\relax
+\begin{picture}(5,5)
+\put(0,0){\framebox(5,5){}}\thicklines\roundcap
+\Curve(2.5,0)<0.1,1>(5,3.5)<0,1>%
+  (4,5)<-1,0>(2.5,3.5)<-0.1,-1.2>[-0.1,1.2]%
+  (1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<0.1,-1>
+\end{picture}
+\end{Esempio}
+
+\vspace*{2\baselineskip}
+
+\begin{Esempio}[\setfontsize{9}](0.65)
+\unitlength=8mm\relax
+\begin{picture}(5,5)
+\put(0,0){\framebox(5,5){}}\thicklines\roundcap
+\color{orange}\relax
+\Curve*(2.5,0)<0.1,1>(5,3.5)<0,1>%
+(4,5)<-1,0>(2.5,3.5)<-0.1,-1.2>[-0.1,1.2]%
+(1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<0.1,-1>
+\end{picture}
+\end{Esempio}
+\caption{A stroked and a filled heart shaped contour}
+\label{fig:hearts}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{The \cs{multiput} command}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ The new \cs{multiput} and \cs{xmultiput} commands are extensions of the 
+ original \cs{multiput} macro; both are used to put a number of objects 
+ according to a discrete law; but they can produce surprising effects. 
+ Figure~\ref{fig:multiput} displays several examples. As it possible 
+ to see, the black dots are evenly distributed along the canvas diagonal; 
+ the green filled squares are along a sloping down line inclined by 
+ $15^\circ$ as specified by the polar coordinates of the \meta{increment}; 
+ the blue filled triangles are distributed along a parabola; the red 
+ stroked diamonds are distributed along a half sine wave.
+
+\noindent\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{8}](0.55)
+\unitlength=0.01\linewidth
+\begin{picture}(100,100)
+\GraphGrid(100,100)
+\multiput(0,0)(10,10){11}{\circle*{2}}
+\color{blue!70!white}
+\multiput(0,0)(10,0){11}{%
+\RegPolygon*{2}{3}<\color{blue!70!white}>}%
+  [\GetCoord(\R)\X\Y
+  \edef\X{\fpeval{\X+10}}
+  \edef\Y{\fpeval{(\X/10)**2}}
+  \CopyVect\X,\Y to\R]
+\multiput(0,0)(10,1){11}{%
+\RegPolygon{2}{4}<\color{magenta}>}%
+  [\GetCoord(\R)\X\Y
+   \edef\X{\fpeval{\X+10}}
+   \edef\Y{\fpeval{sind(\X*1.8)*100}}
+   \CopyVect\X,\Y to\R]
+\multiput(50,50)(-15:5){11}{%
+\RegPolygon*{2}{4}[45]<\color{green!60!black}>}
+\end{picture}
+\end{Esempio}
+\caption{Some examples of the \meta{handler} optional argument}
+\label{fig:multiput}
+\end{figure}
+
+ Another interesting construction is a clock quadrant; this is shown
+ in figure~\ref{fig:orologio}
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{9.5}](0.525)
+\unitlength=0.0095\linewidth
+\begin{picture}(100,100)
+\GraphGrid(100,100)
+\put(50,50){\thicklines\circle{100}}
+\xmultiput[50,50](60:35)(-30:1){12}%
+  {\makebox(0,0){\circle*{2}}}%
+    [\MultVect\R by\D to\R]% 
+\xmultiput[50,50](60:40)(-30:1){12}%
+  {\ArgOfVect\R to\Ang
+    \rotatebox{\fpeval{\Ang-90}}%
+    {\makebox(0,0)[b]{%
+       \Roman{multicnt}}}}%
+      [\Multvect{\R}{\D}\R]
+\thicklines\put(50,50){\circle*{4}}
+\put(50,50){\Vector(37.5:30)}
+\put(50,50){\Vector(180:33)}
+\end{picture}
+\end{Esempio}
+\caption{Usage example of the \texttt{\string\xmultiput} command}
+\label{fig:orologio}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \subsection{Drawing mathematical functions}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ Figure~\ref{fig:iperbole} shows an equilateral hyperbola; since it
+ has asymptotes, the drawing must be carefully done avoiding overflows,
+ parts of drawing out of the \env{picture} area. Nevertheless
+ the possibility of describing mathematical functions in terms of L3
+ functions (in spite of the same name, they are completely different
+ things) makes it possible to exploit the \meta{settings} argument to
+ do the job with \cs{xmultiput}.
+
+\begin{figure*}[!tp]
+\begin{Esempio}[\setfontsize{9}](0.5)%
+\unitlength=0.008\linewidth
+\begin{picture}(100,100)
+\GraphGrid(100,100)
+\VECTOR(0,0)(100,0)\Pbox(100,0)[tr]{x}[0]
+\VECTOR(0,0)(0,100)\Pbox(0,100)[tr]{y}[0]
+\Pbox(0,0)[r]{O}[3pt]
+\thicklines
+\moveto(10,100)\countdef\I=2560 \I=11
+\xmultiput(0,0)(1,0){101}%
+  {\lineto(\I,\fpeval{1000/\I})}%
+  [\advance\I by1 \value{multicnt}=\I]
+\strokepath
+\end{picture}
+\end{Esempio}
+\caption{An equilateral hyperbola drawn with a thinly sampled piecewise
+ continuous line}\label{fig:iperbole}
+\end{figure*}
+
+ A more complicated drawing can be done by expressing the function to draw
+ with parametric equations; the idea is to code the math formulas
+\[\begin{cases}
+x(t) = f_1(t)\\
+y(t) = f_2(t)
+\end{cases}\]
+because it is easy to code the $x$ and the $y$ component and use the
+ \cs{fpdowhile} command to trace the curve with a piecewise continuous
+ line; actually a continuous line with a piecewise continuous derivative;
+ it is important to sample the curve in a sufficient dense way.
+ A heart shaped mathematical function taken from the internet
+ \footnote{\url{http://mathworld.wolfram.com/HeartCurve.html} reports 
+ several formulas, including the cardioid, but the one we use here is 
+ a different function} is the following
+\begin{align*}
+x(t) &= \sin^3(t)\\
+y(t) &= \dfrac{13\cos(t)-5\cos(2t)-2\cos(3t)-\cos(4t)}{16}
+\end{align*}
+ Figure~\ref{fig:math-heart} displays the graph, and its code, and, most
+ important, the L3 definition of the parametric equations. Compared to
+ the previous equations we applied a scale factor and added the final
+ term \texttt(2.4) in order to shift a little bit the drawing so as to
+ vertically center it . 
+
+\begin{figure*}[!tbp]
+\begin{minipage}{\textwidth}
+\def\codefraction{0.5}
+\dimendef\Wuno=2000 \Wuno=\codefraction\textwidth
+\dimendef\Wdue=2002 \Wdue=\dimexpr\textwidth-\columnsep-\Wuno\relax
+\begin{minipage}{\Wuno}\setfontsize{6.9}%
+\begin{verbatim}
+%              Parametric equations of the "heart"
+\providecommand\heart[3]{%
+\edef\X{\fpeval{#1*16*(sind(#2)^3)}}
+\edef\Y{\fpeval{#1*(13*cosd(#2) - 5*cosd(2*#2) 
+  - 2*cosd(3*#2) -cosd(4*#2)+2.4)}}
+\CopyVect\X,\Y to#3}
+%
+%                                         Drawing
+\unitlength=0,005\linewidth
+\begin{picture}(200,200)(-100,-100)
+\GraphGrid(-100,-100)(200,200)
+\VECTOR(-100,0)(100,0)\Pbox(100,0)[tr]{x}[0]
+\VECTOR(0,-100)(0,100)\Pbox(0,100)[tl]{y}[0]
+\Pbox(0,0)[tr]{O}\linethickness{1pt}\bgroup
+\edef\scala{\fpeval{100/16}}
+\countdef\I=2560 \I=0\roundjoin
+\fpdowhile{\I !>360}{\heart\scala\I\Punto
+\ifnum\I=0 \moveto(\Punto)\else \lineto(\Punto)\fi
+\advance\I by 3}\strokepath\egroup
+\end{picture}\end{verbatim}
+\end{minipage}
+\hfill
+\begin{minipage}{\Wdue}\raggedleft
+\setfontsize{8}%
+%
+\providecommand\heart[3]{%
+\edef\X{\fpeval{#1*16*(sind(#2)^3)}}
+\edef\Y{\fpeval{#1*(13*cosd(#2) - 5*cosd(2*#2) 
+  - 2*cosd(3*#2) -cosd(4*#2)+2.4)}}
+\CopyVect\X,\Y to#3}
+%
+\unitlength=0,005\linewidth
+\begin{picture}(200,200)(-100,-100)
+\GraphGrid(-100,-100)(200,200)
+\VECTOR(-100,0)(100,0)\Pbox(100,0)[tr]{x}[0]
+\VECTOR(0,-100)(0,100)\Pbox(0,100)[tl]{y}[0]
+\Pbox(0,0)[tr]{O}\linethickness{1pt}\bgroup
+\edef\scala{\fpeval{100/16}}
+\countdef\I=2560 \I=0\roundjoin
+\fpdowhile{\I !>360}{\heart\scala\I\Punto
+\ifnum\I=0 \moveto(\Punto)\else \lineto(\Punto)\fi
+\advance\I by 3}\strokepath\egroup
+\end{picture}
+\end{minipage}
+\end{minipage}
+\caption{A heart shaped mathematical function drawn with a thinly sampled
+ piecewise continuous line}\label{fig:math-heart}
+\end{figure*}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Triangles and their special lines}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ Triangles have special lines; they are the median, the height, and the
+ bisector lines. They join each vertex with a specific point of the
+ apposite side, respectively with the middle point, the intersection 
+ with the side perpendicular line, and the intersection with the bisector 
+ line. Figure~\ref{fig:MHBlines}  displays the construction of the three 
+ special lines relative to a specific vertex. Thanks to the macros 
+ described earlier in this list, this drawing is particularly simple; most
+ of the code is dedicated to labelling the various points and to
+ assign coordinate values to the macros that are going to use them
+ in a symbolic way. The generic triangle (not a regular polygon) requires 
+ one line, and the determination of the intersections of the lines with
+ the suitable triangle side, and their tracing requires two code lines
+ each.
+
+\begin{figure}[!tb]\centering
+\begin{Esempio}[\setfontsize{6.9}](0.4)
+\unitlength=0.008\linewidth
+\begin{picture}(100,100)(0,-10)
+\GraphGrid(0,-10)(100,100)
+\def\Puno{0,0} \def\Pdue{0,80} \def\Ptre{100,60}
+ {\thicklines\polygon(\Puno)(\Pdue)(\Ptre)}%
+ \Pbox(\Puno)[tc]{P_1}[1]
+\Pbox(\Pdue)[bc]{P_2}[1]\Pbox(\Ptre)[bc]{P_3}[1]
+ % Median
+ \TriangleMedianBase\Puno on \Pdue and \Ptre to\M
+ \Pbox(\M)[bc]{M}[1.5]\segment(\Puno)(\M)
+% Height
+ \TriangleHeightBase\Puno on \Pdue and\Ptre to\H
+ \Dotline(\Puno)(\H){2}[1.5]\Pbox(\H)[bc]{H}[1.5]
+% Bisector
+ \TriangleBisectorBase\Puno on\Pdue and\Ptre to\B
+ \Dashline(\Puno)(\B){3}\Pbox(\B)[b]{B}[1.5]
+\end{picture}
+\end{Esempio}
+\caption{A triangle with the median, the height, and the bisector lines
+ from a specific vertex}\label{fig:MHBlines}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \subsection{Special triangle centers}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ Each triplet of a triangle special lines of the same kind intersect each
+ other in a special point; the median lines intersect in the
+ \emph{barycenter}, the height lines in the \emph{orthocenter}, the 
+ bisectors lines in the \emph{incenter}; these centers may be those of 
+ special circles: Figures~\ref{fig:barycenter} to~\ref{fig:circumcenter}; 
+ the \emph{incircle}, centered in the incenter,
+ has a special name, because it has the property of being tangent to all
+ the three triangle sides; there is also the circumcircle that passes
+ through the three vertices, its center is the intersection of the
+ three side axes. There is also the \emph{nine point circle}.
+ Figures~\ref{fig:barycenter}, \ref{fig:orthocenter}, \ref{fig:incenter},
+ and~\ref{fig:circumcenter} display the necessary constructions and,
+ possibly, also the special circles they are centers of.
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{6.5}](0.45)
+\unitlength=0.008\linewidth
+\begin{picture}(100,100)(0,-10)
+\GraphGrid(0,-10)(100,100)
+\def\Puno{0,0}\def\Pdue{0,80}\def\Ptre{100,60}
+{\linethickness{0.6pt}\polygon(\Puno)(\Pdue)(\Ptre)}%
+\Pbox(\Puno)[tl]{P_1}[1.5]%
+\Pbox(\Pdue)[bl]{P_2}[1.5]\Pbox(\Ptre)[bc]{P_3}[1.5]
+ \TriangleMedianBase\Puno on\Pdue and \Ptre to\Mu
+ \TriangleMedianBase\Pdue on\Ptre and \Puno to\Md
+ \TriangleMedianBase\Ptre on\Puno and \Pdue to\Mt
+\Dotline(\Puno)(\Mu){3}[1.5]
+\Dotline(\Pdue)(\Md){3}[1.5]
+\Dotline(\Ptre)(\Mt){3}[1.5]
+\IntersectionOfSegments(\Puno)(\Mu)and(\Pdue)(\Md)to\C
+\Pbox(\C)[t]{B}[2]
+\end{picture}
+\end{Esempio}
+\caption{Determination of the barycenter}\label{fig:barycenter}
+\end{figure}
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{6.5}](0.45)
+\unitlength=0.008\linewidth
+\begin{picture}(100,100)(0,-10)
+\GraphGrid(0,-10)(100,100)
+\def\Puno{0,0}\def\Pdue{0,80}\def\Ptre{100,60}
+{\linethickness{0.6pt}\polygon(\Puno)(\Pdue)(\Ptre)}%
+\Pbox(\Puno)[tl]{P_1}[1.5]%
+\Pbox(\Pdue)[bl]{P_2}[1.5]\Pbox(\Ptre)[bc]{P_3}[1.5]
+ \TriangleHeightBase\Puno on\Pdue and \Ptre to\Hu
+ \TriangleHeightBase\Pdue on\Ptre and \Puno to\Hd
+ \TriangleHeightBase\Ptre on\Puno and \Pdue to\Ht
+\Dotline(\Puno)(\Hu){3}[1.5]
+\Dotline(\Pdue)(\Hd){3}[1.5]
+\Dotline(\Ptre)(\Ht){3}[1.5]
+\IntersectionOfSegments(\Puno)(\Hu)and(\Pdue)(\Hd)to\C
+\Pbox(\C)[t]{H}[2]
+\end{picture}
+\end{Esempio}
+\caption{Determination of the orthocenter}\label{fig:orthocenter}
+\end{figure}
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{6}](0.45)
+\unitlength=0.008\linewidth
+\begin{picture}(100,100)(0,-10)
+\GraphGrid(0,-10)(100,100)
+\def\Puno{0,0}\def\Pdue{0,80}\def\Ptre{100,60}
+{\linethickness{0.6pt}%
+  \polygon(\Puno)(\Pdue)(\Ptre)}%
+\Pbox(\Puno)[tl]{P_1}[1.5]%
+\Pbox(\Pdue)[bl]{P_2}[1.5]
+\Pbox(\Ptre)[bc]{P_3}[1.5]
+ \TriangleBisectorBase\Puno on\Pdue and \Ptre to\Iu
+ \TriangleBisectorBase\Pdue on\Ptre and \Puno to\Id
+ \TriangleBisectorBase\Ptre on\Puno and \Pdue to\It
+\Dotline(\Puno)(\Iu){3}[1.5]
+\Dotline(\Pdue)(\Id){3}[1.5]
+\Dotline(\Ptre)(\It){3}[1.5]
+\IntersectionOfSegments(\Puno)(\Iu)%
+  and(\Pdue)(\Id)to\C
+\Pbox(\C)[t]{I}[2]
+\DistanceOfPoint\C from(\Puno)(\Pdue)to\R
+\Circlewithcenter\C radius\R
+\end{picture}
+\end{Esempio}
+\caption{Determination of the incenter and of the incircle}\label{fig:incenter}
+\end{figure}
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{6.25}](0.45)
+\unitlength=0.01\linewidth
+\begin{picture}(100,110)
+\GraphGrid(100,110)
+\CopyVect20,10to\Pu \Pbox(\Pu)[t]{P_1}
+\CopyVect10,90to\Pd \Pbox(\Pd)[br]{P_2}
+\CopyVect100,70to\Pt \Pbox(\Pt)[l]{P_3}
+{\linethickness{0.6pt}\polygon(\Pu)(\Pd)(\Pt)}%
+\AxisOf\Pd and\Pu to\Mu\Du 
+\AxisOf\Pu and\Pt to\Md\Dd
+\AxisOf\Pt and\Pd to\Mt\Dt
+\IntersectionOfLines(\Mu)(\Du)and(\Md)(\Dd)to\C
+\AddVect\Mu and\Du to\Du\Dotline(\Mu)(\Du){3}[2]
+\AddVect\Md and\Dd to\Dd\Dotline(\Md)(\Dd){3}[2]
+\AddVect\Mt and\Dt to\Dt\Dotline(\Mt)(\Dt){3}[2]
+\Pbox(\C)[t]{C}[2.5]
+\ThreePointCircle*(\Pu)(\Pd)(\Pt)
+\end{picture}\end{Esempio}
+\caption{Determination of the circumcenter and of the circumcircle}\label{fig:circumcenter}
+\end{figure}
+
+ Although these examples require some new simple macros, described
+ in the previous sections; some more more examples can be made that require 
+ more complex macros. Even these macros are just examples. For other
+ applications it is probably necessary to add more macros.
+
+ Let us proceed with the construction of the Steiner ellipse: given a
+ triangle, there exists only one ellipse that is internally tangent to
+ the side middle points.
+
+ The geometrical construction goes on this way; suppose you have to draw
+ the Steiner ellipse of triangle $T$; finding the side middle points
+ has already been shown, but the process to build the ellipse is still
+ to be found. So let us chose a side to work as the base of triangle $T$,
+ and perform an affine shear transformation parallel to the base 
+ so as to move the vertex of triangle $T$, opposite to the base, to the
+ base axis, we get another triangle $T_1$ that is isosceles; if it is
+ not yet so, let us make another compression/expansion affine
+ transformation, so as to get an equilateral triangle $T_2$; this last
+ triangle is particularly simple to handle, because its Steiner ellipse
+ reduces to its incircle. If we apply in reverse order the above
+ transformations we get the Steiner ellipse we were looking for.
+ The only difficult part is the affine shear transformation. 
+
+ The L3 functions we already created take care of all such
+ transformations, but with an optional asterisk we can draw the
+ intermediate passages where triangles $T_2$ and $T_1$ have their base
+ shifted and rotated to be horizontal, so that some translations and
+ rotations are also necessary. Figure~\ref{fig:Steiner-finale} displays
+ the final result and the code necessary to build it.
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{8}](0.5)
+\unitlength=0.01\linewidth
+\begin{picture}(100,110)
+\GraphGrid(100,110)%
+\SteinerEllipse(10,10)(90,20)(60,105)[2]
+\end{picture}\end{Esempio}
+\caption{The Steiner ellipse of a given triangle}
+\label{fig:Steiner-finale}
+\end{figure}
+
+ With just the addition of an asterisk we can draw the whole geometrical
+ construction; see figure~\ref{fig:Steiner-intermedio}
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{8}](0.5)
+\unitlength=0.01\linewidth
+\begin{picture}(100,110)(0,-10)
+\GraphGrid(0,-10)(100,110)%
+\SteinerEllipse*(10,10)(90,20)(60,105)[2]
+\end{picture}\end{Esempio}
+\caption{The construction of the Steiner ellipse of a given triangle}
+\label{fig:Steiner-intermedio}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{A triangle internally tangent ellipse given one of its foci}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ It is possible to draw an ellipse that is internally tangent to
+ a triangle if one of its foci is specified; without this specification
+ the problem is not definite, and the number o such ellipses is countless.
+ But with the focus specification, just one ellipse exists with that
+ tangency constraint. It suffices to find the other focus and at least
+ one point of tangency, because the focal distance and the sum of
+ distances of that tangency point from the foci, is sufficient to
+ determine all the parameters required to draw the ellipse.
+
+\begin{figure}[!tb]\centering
+\begin{Esempio}[\setfontsize{7}](0.375)
+\unitlength=0.0065\linewidth
+\begin{picture}(150,150)(-30,-20)
+\GraphGrid(-30,-20)(150,150)
+\EllipseWithFocus%
+  (10,40)(110,10)(0,110)(20,60)
+\end{picture}\end{Esempio}
+\caption{An ellipse internally tangent to a triangle, given a focus}
+\label{fig:ellisse-interna-finale}
+\end{figure}
+
+\begin{figure}[!tb]
+\begin{Esempio}[\setfontsize{7}](0.375)
+\unitlength0.0065\linewidth
+\begin{picture}(150,150)(-30,-20)
+\GraphGrid(-30,-20)(150,150)
+\EllipseWithFocus*%
+  (10,40)(110,10)(0,110)(20,60)
+\end{picture}\end{Esempio}
+\caption{Construction of the ellipse of figure~\ref{fig:ellisse-interna-finale}}\label{fig:ellisse-interna}
+\end{figure}
+
+
+ The geometrical construction is rather complicated; the steps to follow
+ are the following:
+\begin{itemize}[noitemsep]
+
+\item draw the triangle and the given focus $\mathsf{F}$;
+
+\item Find the symmetrical points $\mathsf G_i$ of this focus with 
+ respect to the sides of the triangle;
+
+\item use these three points $\mathsf G_i$ as the vertices of a triangle
+ with which to draw its circumcircle, actually only its center is of
+ interest, because it represents the second focus $\mathsf F'$; the
+ inter focal distance  $2c$; is just le length of vector $F' -F$;
+
+\item join with segments each symmetrical point $\mathsf G_i$ with the
+ second focus $\mathsf F'$ and find their intersections $\mathsf T_i$
+ with the triangle sides; they represent the tangency points of the
+ ellipse to be drawn;
+
+\item use one of these tangency points to find its distances from the
+ foci; their sum gives the ellipse larger axis length $2a$;
+
+\item equation~\eqref{equ:axes-foci} allows to find the second axis
+ length; the segment that joins the foci has the required inclination
+ of the main axis; therefore all necessary pieces of information to
+ draw the ellipse are known.
+
+\end{itemize}
+ Figures~\ref{fig:ellisse-interna-finale} and~\ref{fig:ellisse-interna}
+ display the construction steps and the final result.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section{Conclusion}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ We have shown that the \amb{picture} environment, extended with
+ this package \pack{euclideangeometry} (that takes care of loading
+ \pack{curve2e} and \pack{pict2e}) can make important diagrams that
+ certainly were not foreseen by Leslie Lamport when he first wrote the
+ code for the initial \env{picture} environment.
+ 
+ The reader can easily understand that this package is far from being
+ exhaustive for all geometrical problema]s to be solved with ruler and
+ compass; it shows a way to add more commands to approach further problems;
+ if any author, who creates new commands, would like to  contribute more
+ macros to this package, I will be happy to integrate his/her contribution
+ to a new version of this package; depending on the contribution, I would
+ be very happy to add its author name to this  package author list; for
+ simpler contributions each contributor will be duly acknowledged.
+
+ Creating new macros to solve more problems is pleasant and more
+ difficult is the problem, greater is the satisfaction in solving it.
+
+
+\begin{center}
+ Have fun with \LaTeX and its potential  applications!
+\end{center}
+
+\end{document}
\ No newline at end of file
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