% \iffalse
% !TEX encoding = UTF-8 Unicode
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\preamble
Copyright (C) 2020-2023 Claudio Beccari all rights reserved.
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
\endpreamble
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%<package>\NeedsTeXFormat{LaTeX2e}[2019/01/01]
%<package>\ProvidesPackage{euclideangeometry}%
%<readme>File README.txt for package euclideangeometry
%<*package|readme>
[2023-07-11 v.0.2.2 Extension package for curve2e]
%</package|readme>
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\begin{document}\errorcontextlines=100
\GetFileInfo{euclideangeometry.dtx}
\title{The \textsf{euclideangeometry} package}
\author{Claudio Beccari\\[1ex]\texttt{claudio dot beccari at gmail dot com}}
\date{Version \fileversion~--~Last revised \filedate.}
\maketitle
\columnseprule=0.4pt
\begin{multicols}{2}
\tableofcontents
\end{multicols}
\DocInput{euclideangeometry.dtx}
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%</driver>
% \fi
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section*{Preface}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% This file contains the documented code of \pack{euclideangeometry}.
% The user manual source file \file{euclideangeometry-man.tex} and the
% readable document is \file{euclideangeometry.pdf}; it should already be
% installed with your updated complete \TeX system installation.
% Please refer to the user manual before using this package.
% \CheckSum{1315}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\StopEventually{}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\iffalse
%<*package>
%\fi
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{The code}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Test the date of a sufficiently recent \texttt{curve2e}
% package}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This package has been already identified by the commands extracted
% by the |docstrip| package, during the |.dtx| file compilation.
% In any case, if the test checks that the |curve2e| file date is too old;
% it warns the user with an emphasised error message on the console,
% loading this |euclideangeometry| package is stopped and the whole
% job aborts. The emphasised error message appears like this:\\~
%\begin{flushleft}\ttfamily\obeylines
%*************************************************************
%Package curve2e too old
%Be sure that your TeX installation is complete and up to date
%*************************************************************
%Input of euclideangeometry is stopped and job aborted
%*************************************************************
%\end{flushleft}
% This message should be sufficiently strong in order to avoid using
% this package with a vintage version of \TeXLive or \MikTeX.
% \begin{macrocode}
\RequirePackage{curve2e}
\@ifpackagelater{curve2e}{2020/01/18}{}%
{%
\typeout{*************************************************************}
\typeout{Package curve2e too old}
\typeout{Be sure that your TeX installation is complete and up to date}
\typeout{*************************************************************}
\typeout{Input of euclideangeometry stopped and job aborted}
\typeout{*************************************************************}
\@@end
}%
% \end{macrocode}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Labelling}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% While doing any graphical geometrical drawing it is necessary to label
% points, lines, angles and other such items. Non measurable labels
% should be in upright sans serif font, according to the ISO regulations,
% but here we are dealing with point identified by macros the contain
% their (cartesian or polar) coordinates that very often are both labels
% and math variables.
%
% Here we provide a versatile macro that can do several things. Its name
% is |\Pbox| and it produces a box containing the label in math format. By
% default the point label is typeset with the math font variant produced
% by command |\mathsf|, but the macro is sufficiently versatile to allow
% other settings; it accepts several optional arguments, therefore it
% syntax is particular:
%\begin{ttsyntax}
%\cs{Pbox}\parg{coordinates}\oarg{alignment}\marg{label}\oarg{diameter}\meta{$\star$}\aarg{angle}
%\end{ttsyntax}
% where \meta{coordinates} are the coordinates where to possibly set a
% black dot with the specified \meta{diameter}; in any case it is the
% reference point of the \meta{label}; the \meta{alignment} is formed by
% the usual letters \texttt{t, b, c, l, r} that can be paired in a
% coherent way (for example the couple \texttt{tb} is evidently
% incoherent, as well as \texttt{lr}), but in absence of this optional
% specification, the couple \texttt{cc} is assumed; most often than
% not, the label position becomes such that when the user reviews the
% document drafts, s/he understands immediately that s/he forgot to
% specify some reasonable \meta{alignment} codes; in any case the
% \texttt{cc} code works fine to just put the dot of a specified
% diameter but with an empty label. Think of the \meta{alignment}
% letters as the position of the reference point with respect to the
% the \meta{label} optical center. The optional \meta{angle} argument
% produces a rotation of the whole label by that angle; it may be used
% in several circumstances, especially when the label is just text, to
% produce, for example, a sideways legend. It is useful also when the
% labels are produced within a rotated box, in order to counterrotate
% them.
%
% The optional asterisk draws a frame around the \emph{label}. Notice
% that the separator between the visible or the invisible frame and the
% box contents varies according the the fact that the \meta{alignment}
% specification contains just one or two letter codes; this is useful,
% because the diagonal position of the label should be optically equal
% to the gap that exists between the reference point and the \meta{label}
% box.
%
% If the \meta{diameter} is zero, no dot is drawn, the whole \meta{label}
% is typeset with the |\mathit| math font; otherwise only the first symbol
% of a math expression is typeset in sans serif. The presence of subscripts
% makes the labels appear more distant from their reference point; the same
% is true when math symbols, even without subscripts, are used, because
% of the oblique nature of the math ‘letters’ alphabet.
%
% If some text has to be printed as a label, it suffices to surround it
% with dollar signs, that switch back to text mode when the default mode
% is the math one. With this kind of textual labels it might be convenient
% to use the optional asterisk to frame the text.
% The final optional argument \meta{angle} (to be delimited with the
% \texttt{<~>} signs) specifies the inclination of the label with respect
% to the horizontal line; it is useful, for example to set a label along
% a sloping line.
% \begin{macrocode}
\providecommand\Pbox{}
\RenewDocumentCommand\Pbox{D(){0,0} O{cc} m O{0.5ex} s D<>{0}}{%
\put(#1){\rotatebox{#6}{\makebox(0,0){%
\settowidth\PbDim{#2}%
\edef\Rapp{\fpeval{\PbDim/{1ex}}}%
\fptest{\Rapp > 1.5}{\fboxsep=0.5ex}{\fboxsep=0.75ex}%
\IfBooleanTF{#5}{\fboxrule=0.4pt}{\fboxrule=0pt}%
\fptest{#4 = 0sp}%
{\makebox(0,0)[#2]{\fbox{$\relax#3\relax$}}}%
{\edef\Diam{\fpeval{(#4)/\unitlength}}%
\makebox(0,0){\circle*{\Diam}}%
\makebox(0,0)[#2]{\fbox{$\relax\mathsf#3\relax$}}%
}}}%
}\ignorespaces}
% \end{macrocode}
% The following command, to be used always within a group, or a
% environment or inside a box, works only with piecewise continuously
% scalable font collection, such as, for example, the Latin Modern
% fonts, or with continuously scalable fonts, such as, for example,
% the Times ones. They let the operator select, for the scope of the
% command ,any size, even fractional so as to fine adjust the text
% width in the space allowed for it; it is particularly useful with
% the monospaced fonts, that forbid hyphenation, and therefore cannot
% be adjusted to the current line width.
% \begin{macrocode}
\DeclareRobustCommand\setfontsize[2][1.2]{%
\linespread{#1}\fontsize{#2}{#2}\selectfont}
% \end{macrocode}
% With OpenType fonts there should not be any problems even with math
% fonts; with Type~1 fonts the only scalable fonts I know of, are the
% LibertinusMath fonts, usable through the LibertinusT1math package, are
% also the only ones that have 8~bit encoded math fonts (256
% glyph fonts), while the standard default math fonts are just
% 7~bit encoded (128 glyphs fonts).
%
% Another useful labelling command is |Zbox|; this command is an
% evolution of a command that I have been using for years in several
% documents of mine. It uses some general text, not necessarily
% connected to a particular point of the |picture| environment,
% as a legend; It can draw short text as a simple horizontal box,
% and longer texts as a vertical box of specified width and height
%
% Is syntax is the following:
%\begin{ttsyntax}
%\cs{Zbox}\parg{position}\parg{dimensions}\oarg{alignment}\marg{text}
%\end{ttsyntax}
% where \meta{position} is where the box reference point has
% to be put in the picture; \meta{dimensions} are optional; if not
% specified, the box is a horizontal one, and it is as wide as its
% contents; if it is specified, it must be a comma separated list
% of two integer or fractional numbers that are the width and the
% height of the box; if the height is specified as zero, the width
% specifies a horizontal box of that width; \meta{alignment} is optional
% and is formed by one or two coherent letter codes from the usual set
% \texttt{t, b, c, l, r}; if the \meta{alignment} is absent, the
% default alignment letters are \texttt{bl}, i.e. the box reference
% point is the bottom left corner; \meta{text} contains general text,
% even containing some math.
% \begin{macrocode}
\def\EUGsplitArgs(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}
\newlength\EUGZbox
\providecommand\Zbox{}
\RenewDocumentCommand\Zbox{r() D(){0,0} O{bl} m}{%
\EUGsplitArgs(#2)\ZboxX\ZboxY % splits box dimensions
\fboxsep=2\unitlength
\ifnum\ZboxX=\z@
\def\ZTesto{\fbox{#4}}%
\else
\ifnum\ZboxY=\z@
\def\ZTesto{\fbox{\parbox{\ZboxX\unitlength}{#4}}}%
\else
\def\ZTesto{%
\setbox\EUGZbox=\hbox{\fbox{%
\parbox[c][\ZboxY\unitlength][c]{\ZboxX\unitlength}{#4}}}%
\dimen\EUGZbox=\dimexpr(\ht\EUGZbox +\dp\EUGZbox)/2\relax
\ht\EUGZbox=\dimen\EUGZbox\relax
\dp\EUGZbox=\dimen\EUGZbox\relax
\box\EUGZbox%
}%
\fi
\fi
\put(#1){\makebox(0,0)[#3]{\ZTesto}}\ignorespaces}
% \end{macrocode}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Service macros for ellipses}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The |\ellisse| has a control sequence name in Italian; it differs for
% just one letter from the name |ellipse| English name, but we cannot use
% the latter one because it may conflict with other packages loaded by
% the user; actually this command and the next one are just shortcuts
% for executing more general commands with specific sets of arguments.
% For details and syntax, please refer yourself to
% section~\ref{ssec:ellissi}
% \begin{macrocode}
\NewDocumentCommand\ellisse{ s m m}{%
\IfBooleanTF{#1}%
{\let\fillstroke\fillpath}%
{\let\fillstroke\strokepath}%
\Sellisse{#2}{#3}%
}
\NewDocumentCommand\Xellisse{ s D(){0,0} O{0} m m O{} o}{%
\IfBooleanTF{#1}%
{\XSellisse*(#2)[#3]{#4}{#5}[#6][#7]}%
{\XSellisse(#2)[#3]{#4}{#5}[#6][#7]}%
}
% \end{macrocode}
%
% We do not know if the following macro |\polyvector| may be useful for
% euclidean geometry constructions, but it may be useful in block
% diagrams; it is simply a polyline where the last segment is a geometrical
% vector. As in polyline the number of recursions is done until the last
% specified coordinate pair; recognising that it is the last one, instead
% of drawing a segment, the macro draws a vector.
%
% \begin{macrocode}
\def\polyvector(#1){\roundcap\def\EUGpreviouspoint{#1}\EUGpolyvector}
\def\EUGpolyvector(#1){%
\@ifnextchar({%
\segment(\EUGpreviouspoint)(#1)\def\EUGpreviouspoint{#1}\EUGpolyvector}%
{\VECTOR(\EUGpreviouspoint)(#1)}%
}
% \end{macrocode}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Processing lines and segments}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% The next macros are functional for the geometric constructions we are
% going to make: finding the intersection of lines or segments,
% finding the lengths and arguments of segments, directions, distances,
% distance of a point from a line or a segment, the symmetrical
% point of a another one specified with respect to a given center of
% symmetry; the axes of segments, the solutions of the relationship
% between the semi axes of an ellipse and the semi focal distance,
% and so on.
%
% Most of these commands have delimited arguments; the delimiters
% may be the usual parentheses, but they may be keywords; many
% commands contain the keyword \texttt{to}, not necessarily the
% last one; the arguments before such keyword may be entered as
% ordered comma separated numerical couples, or comma separated
% macros the containing scalar values; or they may be macros that
% contain the ordered couples representing vectors or directions;
% they all may be in cartesian or polar form. Remember that such
% ordered couples are complex numbers, representable by
% vectors applied to the origin of the axes; therefore sometimes it is
% necessary that the underlying commands execute some vector
% differences so as to work with generic vectors.
%
% On the opposite the output values, i.e. the argument after that
% \texttt{to} keyword, should be tokens that can receive a definition,
% in general macros, to which the user should assign a mnemonic name;
% s/he should use such macros for further computations or for drawing
% commands.
%
% The first and principal command is |\IntersectionOfLines| and
% it has the following syntax:
%\begin{ttsyntax}
%\cs{IntersectionOfLines}\parg{point1}\parg{dir1}and\parg{point2}\parg{dir2}to\meta{crossing point}
%\end{ttsyntax}
% where \meta{point1} and \meta{dir1} are respectively a point of the
% first line and its \emph{direction}, not a second point, but the
% \emph{direction} — it is important to stress this point; similarly
% for the second line; the output is stored in
% the macro that identifies the \meta{crossing point}. The directions
% do not need to be expressed with unit vectors, but the lines must not
% be parallel or anti parallel (equal directions or differing by
% $180^\circ$); the macro contains a test that checks this anomalous
% situation because an intersection at infinity or too far away
% ($2^{14}-1$ typographical points, approximately 5,758\,m) is of no
% interest; in case, no warning message is issued, the result is
% put to \texttt{0,0}, and the remaining computations become nonsense.
% It is a very unusual situation and I never encountered~it;
% nevertheless\dots
% \begin{macrocode}
\def\IntersectionOfLines(#1)(#2)and(#3)(#4)to#5{\bgroup
\def\IntPu{#1}\def\Uu{#2}\def\IntPd{#3}\def\Ud{#4}%
\DirOfVect\Uu to\Du
\DirOfVect\Ud to\Dd
\XpartOfVect\Du to \a \YpartOfVect\Du to \b
\XpartOfVect\Dd to \c \YpartOfVect\Dd to \d
\XpartOfVect\IntPu to \xu \YpartOfVect\IntPu to \yu
\XpartOfVect\IntPd to \xd \YpartOfVect\IntPd to \yd
\edef\Den{\fpeval{-(\a*\d-\b*\c)}}%
\fptest{abs(\Den)<1e-5}{% Almost vanishing determinant
\def#5{0,0}%
}{% Determinant OK
\edef\Numx{\fpeval{(\c*(\b*\xu-\a*\yu)-\a*(\d*\xd-\c*\yd))/\Den}}%
\edef\Numy{\fpeval{(\d*(\b*\xu-\a*\yu)-\b*(\d*\xd-\c*\yd))/\Den}}%
\CopyVect\Numx,\Numy to\Paux
\edef\x{\egroup\noexpand\edef\noexpand#5{\Paux}}\x\ignorespaces}}
% \end{macrocode}
%
% The |IntersectionOfSegments| macro is similar but in input it
% contains the end points of two segments: internally it uses
% |\IntersectionOfLines| and to do so it has to determine the
% directions of both segments. The syntax is the following:
%\begin{ttsyntax}
%\cs{IntersectionOfSegments}\parg{point11}\parg{point12}and\parg{point21}\parg{point22}
%\qquad to\meta{crossing}
%\end{ttsyntax}
% The \meta{crossing} point might fall outside one or both segments.
% It is up to the users to find out if the result is meaningful
% or nonsense. Two non parallel lines are infinitely long in both
% directions and any \meta{crossing} point is acceptable; with
% segments the situation might become nonsense.
% \begin{macrocode}
\def\IntersectionOfSegments(#1)(#2)and(#3)(#4)to#5{%
\SubVect#1from#2to\IoSvectu \DirOfVect\IoSvectu to\DirIoSVecu
\SubVect#3from#4to\IoSvectd \DirOfVect\IoSvectd to\DirIoSVecd
\IntersectionOfLines(#1)(\DirIoSVecu)and(#3)(\DirIoSVecd)to#5\ignorespaces}
% \end{macrocode}
%
% An application of the above intersections is formed by the next
% two macros; they find the axes of a couple of sides of a triangle
% and use their base point and direction to identify two lines the
% intersection of which is the circumcenter; the distance of one
% base point from the circumcenter is the radius of the circumcircle
% that can be drawn with the usual macros. We have to describe
% the macros |\AxisOf| and |CircleWithCenter| and we will do it in
% a little while. Meanwhile the syntax of the whole macro is the
% following:
%\begin{ttsyntax}
%\cs{ThreePointCircle}\meta{$\star$}\parg{vertex1}\parg{vertex2}\parg{vertex3}
%\end{ttsyntax}
% where the three vertices are the three points where the circle
% must pass, but they identify also a triangle. Its side axes
% intersect in one point that by construction is at the same
% distance from the three vertices, therefore it is the center of
% the circle that passes through the three vertices. A sub product
% of the computations is the macro |\C| that contains the center
% coordinates. If the optional asterisk is used the whole drawing
% is executed, while if it is missing, only the |\C| macro remains
% available but the user is responsible to save/copy its value
% into another macro; for this reason another macro should be more
% easy to use; its syntax is the following:
%\begin{ttsyntax}
%\cs{ThreePointCircleCenter}\parg{vertex1}\parg{vertex2}\parg{vertex3}
%\qquad to\meta{center}
%\end{ttsyntax}
% where the vertices have the same meaning, but\meta{center} is
% the user chosen macro that contains the center coordinates.
%
% \begin{macrocode}
\NewDocumentCommand\ThreePointCircle{s r() r() r()}{%
\AxisOf#2and#3to\Mu\Du \AxisOf#2and#4to\Md\Dd
\IntersectionOfSegments(\Mu)(\Du)and(\Md)(\Dd)to\C
\SubVect#2from\C to\R
\IfBooleanTF{#1}{\CircleWithCenter\C Radius\R}{}\ignorespaces}
\NewDocumentCommand\ThreePointCircleCenter{r() r() r() m}{%
\ThreePointCircle(#1)(#2)(#3)\CopyVect\C to#4}
% \end{macrocode}
%
% There are some useful commands that help creating |picture|
% diagrams in an easier way; for example one of the above described
% commands internally uses |\CircleWithCenter|. It is well known
% that the native |picture| command |\circle| requires the
% specification of the diameter but many |euclideangeometry|
% commands already get the distance of two points, or the magnitude
% of a segment, or similar objects that may be used as a radius, rather
% than the diameter; why should we not have macros that simultaneously
% compute the require diameter and draw the circle. Here there are two
% such macros; they are similar to one another but their names differ in
% capitalisation, but also in the way they use the available input
% information. The syntax is the following:
%\begin{ttsyntax}
%\cs{CircleWithCenter}\meta{center} Radius\meta{Radius}
%\cs{Circlewithcenter}\meta{center} radius\meta{radius}
%\end{ttsyntax}
% where in both cases \meta{center} is a vector/ordered couple
% that points to the circle center. On the contrary \meta{Radius}
% is a vector obtained through previous calculations, while
% \meta{radius} is a scalar containing a previously calculated length.
% \begin{macrocode}
\def\CircleWithCenter#1Radius#2{\put(#1){\ModOfVect#2to\CWR
\circle{\fpeval{2*\CWR}}}\ignorespaces}
%
\def\Circlewithcenter#1radius#2{\put(#1){\circle{\fpeval{2*abs(#2)}}}%
\ignorespaces}
% \end{macrocode}
%
% As announced, here we have a macro to compute the axis of a segment;
% given two points $P_1$ and $P_2$, for example the end points of a
% segment, or better the end points of the vector that goes from
% $P_1$ to $P_2$, the macro determines the segment middle point and
% a second point the lays on the perpendicular at a distance equal to half
% the first two points distance; this second point lays at the left of
% vector $P_2-P_1$, therefore it is important to select the right initial
% vector, in order to have the second axis point on the desired side.
% This macro did not work well because sometimes its internal variables
% interfered with other commands internal ones; enclosing everything
% within an auto deleting group by means of the auto deleting |\x| macro
% usual trick, such interferences should be avoided.
%\begin{ttsyntax}
%\cs{AxisOf}\meta{P1} and\meta{P2} to\meta{Axis1}\meta{Axis2}
%\end{ttsyntax}
% Macros |\SegmentCenter| and |\MiddlePointOf| are alias of one another;
% their syntax is:
%\begin{ttsyntax}
%\cs{SegmentCenter}\parg{P1}\parg{P2}to\meta{center}
%\cs{MiddlePointOf}\parg{P1}\parg{P2}to\meta{center}
%\end{ttsyntax}
% \meta{P1}, \meta{P2}, \meta{Axis1}, \meta{Axis2} and \meta{center} are
% all ordered pairs (vectors).
% \begin{macrocode}
\def\AxisOf#1and#2to#3#4{\bgroup
\SubVect#1from#2to\Base \ScaleVect\Base by0.5to\Base
\AddVect\Base and#1to\Mout \MultVect\Base by0,1to\Base
\AddVect\Mout and\Base to\Dout
\edef\x{\egroup\noexpand\CopyVect\Mout to\noexpand#3\relax
\noexpand\CopyVect\Dout to\noexpand#4}\x\ignorespaces}
\def\SegmentCenter(#1)(#2)to#3{\AddVect#1and#2to\Segm
\ScaleVect\Segm by0.5to#3\ignorespaces}
\let\MiddlePointOf\SegmentCenter
% \end{macrocode}
%
% Some other macros are needed to solve certain triangle problems;
% one of such macros is the one allows to determine the length of one
% leg of a right triangle by knowing the lengths of the hypothenuse
% and the other leg. The syntax is the following:
%\begin{ttsyntax}
%\cs{LegFromHypotenuse}\meta{hypothenuse} AndOtherLeg\meta{leg1} to\meta{leg2}
%\end{ttsyntax}
% where the three parameters may be macros, especially the last one;
% all of them contina scalar values.
%
% \begin{macrocode}
\def\LegFromHypotenuse#1AndOtherLeg#2to#3{%
\edef#3{\fpeval{sqrt(#1**2-#2**2)}}}
% \end{macrocode}
%
% Another useful macro determines the two intersections of a line with
% a circumference if they exist; otherwise it issues a warning and sets
% both output values to vector \texttt{0,0}, which, of course, is wrong,
% but it allows to go on with typesetting, although with non sense results.
% Warnings do not stop the compilation program, therefore their message
% goes to the \file{.log} file and the user might not notice it; but
% since the results are probably absurd, s/he certainly notices this
% fact and looks for messages; the user, therefore, who has carefully
% read this user manual, immediately looks onto the \file{.log} file
% and realises the reason of the wrong results. A similar approach
% is used for the macro that determines the intersection of two circles;
% see below
% The syntax of this macro is the following:
%\begin{ttsyntax}
%\cs{IntersectionsOfLine}\parg{point}\parg{direction} WithCircle\parg{center}\marg{radius} to\meta{int1} and\meta{int2}
%\end{ttsyntax}
% where \meta{point} and \meta{direction} are the line parameters
% that can be explicit complex values or macros; \meta{center} is
% the circumference explicit complex value, or a macro, containing
% the center coordinates; \meta{radius} is the scalar explicit or
% macro radius length; The intersection points \meta{int1} and
% \meta{ind2} are supposed to be macros that get defined with the
% intersection point coordinates; \meta{int1} is the first intersection
% that is determined along the line \emph{direction}. Please notice the
% different first part of the macro name \texttt{IntersectionsOfLine}
% compared to the macro that determines the intersection of two lines
% \texttt{IntersectionOfLines}: two intersections and one line vs. one
% intersection with two lines.
% \begin{macrocode}
\def\IntersectionsOfLine(#1)(#2)WithCircle(#3)#4to#5and#6{%
\CopyVect#3 to\C \edef\R{#4}
\CopyVect#1to\Pu \CopyVect#2to\Pd
\Circlewithcenter\C radius\R
\segment(\Pu)(\Pd)\SegmentArg(\Pu)(\Pd)to\Diru
\edef\Dird{\fpeval{\Diru+90}}\Pbox(\C)[b]{C}[2]
\IntersectionOfLines(\Pu)(\Diru:1)and(\C)(\Dird:1)to\Int
\SegmentLength(\C)(\Int)to\A
\fptest{\A > \R}{\PackageError{euclideangeometry}%
{Distance of line \A\space larger than radius \R. No intersections}%
{Check your data; correct and retry}}{%
\LegFromHypotenuse\R AndOtherLeg\A to\B
\AddVect\Int and\Diru:-\B to\Pt \edef#5{\Pt}
\SymmetricalPointOf\Pt respect\Int to\Pq \edef#6{\Pq}
}}
% \end{macrocode}
%
% Another useful macro determines the point \meta{p2} symmetric
% to a given point \meta{p1} with respect to a given segment the
% end points of which are \meta{Segm1} and \meta{Segm2}:
%\begin{ttsyntax}
%\cs{Segment}\parg{Segm1}\parg{Segm2}SymmetricPointOf\marg{p1} to\meta{p2}
%\end{ttsyntax}
% where, as usual, the input data may be explicit or macro defined
% coordinates, while the output result should be a macro name.
% \begin{macrocode}
\def\Segment(#1)(#2)SymmetricPointOf#3to#4{%
\SegmentArg(#1)(#2)to\Sanguno\edef\Sangdue{\fpeval{\Sanguno+90}}
\IntersectionOfLines(#1)(\Sanguno:1)and(#3)(\Sangdue:1)to\Smed
\SymmetricalPointOf#3respect\Smed to#4\ignorespaces}
% \end{macrocode}
%
% This useful macro draws a circle given its \meta{center} and the
% coordinates of the \meta{point} which the circumference should pass
% through. The syntax is:
%\begin{ttsyntax}
%\cs{CircleThrough}\parg{[point}WithCenter\marg{center}
%\end{ttsyntax}
% As usual, the parameters are all explicit or macro defined complex
% numbers.
% \begin{macrocode}
\def\CircleThrough#1WithCenter#2{%
\SegmentLength(#1)(#2)to\Radius
\Circlewithcenter#2radius\Radius}
% \end{macrocode}
% The above macro is the building block for a simple macro that draws
% two circles that cross at a given point; but it is so simple that
% it is not worth defining a macro: if the user wants to try his/her
% ability, s/he may define:
%\begin{flushleft}\obeylines
%| \NewDocumentCommand{r() r() r()}{%|
%| \CircleThrough#3 WithCenter{#1}|
%| \CircleThrough#3 WithCenter{#2}\ignorespaces}|
%\end{flushleft}
% where \cs{ignorespaces} may be superfluous, but is always a safety action
% when defining commands to be used within the \amb{picture} environment.
% In any case see example~15 in \file{euclideangeometry-man.pdf}
%^^A\ref{fig:two-intersecting-circles}.
%
% If it is necessary to find the intersections of two circles that
% do not share a previously known point; we can use the following
% macro. Analytically given the equations of two circumferences,
% it is necessary to solve a system of two second degree equations
% the processing of which ends up with a second degree polynomial
% that might have real roots (the coordinates of the intersection
% points), or two coincident roots (the circles are tangent), or
% complex roots (the circles do not intersect), or they may be
% indefinite (the two circles heve the same center and the same
% radii). Let us exclude the last case, although it would be trivial
% to create the macro with a test that controls such situation. But
% even the analysis of the discriminant of the second degree equation
% requires a complicated code.
%
% On the opposite a simple drawing of the two circles, with centers
% $C_1$ and $C_2$ and radii $R_1$ and $R_2$, with the centers distance
% of $ a= |C_1-C_2|$, allows to understand that in order to have
% intersections: $(\alpha)$ if $a\leq \max(R_1,R_2)$ (the center
% of a circle is contained within the other one) then it must be
% $ a \geq |R_1 - R_2|$, where the ‘equals’ sign applies when the
% circles are internally tangent; $(\beta)$ otherwise
% $a \geq \max(R_1,R_2)$ and it must be $a \leq R_1+R_2$ , where
% the ‘equals’ sign applies when the circles are externally tangent.
% In conclusion in any case il the range $|R_1-R_2|\leq a\leq R_1+R_2$
% is where the two circles intersect, while outside this distance
% range the circles do not intersect.
%
% For simplicity let us assume that $R_1 \geq R_2$; the macro can
% receive the circle data in any order, but the macro very easily
% switches their data so that circle number~1 is the one with larger
% radius. If the distance $a$ is outside the allowed range, there
% are no intersections, therefore a warning message is output and the
% intersection point coordinates are both set to \texttt{0,0}, so that
% processing continues with non sense data; the remaining geometric
% construction based on such intersection points might continue with
% other error messages or to absurd results; a string message to the
% user who, having read the documentation, understand the problem and
% provides for.
%
% The new macro has the following syntax:
%\begin{ttsyntax}
%\cs{TwoCirclesIntersections}\parg{C1}\parg{C2}withradii\marg{R1} and\marg{R2} to\meta{P1} and\meta{P2}
%\end{ttsyntax}
% where the symbols in input may be macros or explicit numerical
% values; the output point coordinates \meta{P1} and \meta{P2}
% should be definable single tokens, therefore the surrounding
% braces are not necessary.
% \begin{macrocode}
\def\TwoCirclesIntersections(#1)(#2)withradii#3and#4to#5and#6{%
\fptest{#3 >=#4}{%
\edef\Cuno{#1}\edef\Cdue{#2}%
\edef\Runo{#3}\edef\Rdue{#4}%
}{%
\edef\Cdue{#2}\edef\Cuno{#2}%
\edef\Rdue{#3}\edef\Runo{#4}%
}
% \end{macrocode}
% Above we switched the circle data so as to be sure that symbols relating
% to circle ‘one’ refer to the circle with larger (or equal) radius.
% Now we define the centers distance in macro \cs{A}; the test if \cs{A}
% lays in the correct range, otherwise we output a warning message.
% \begin{macrocode}
\SegmentLength(\Cuno)(\Cdue)to\A
\edef\TCIdiffR{\fpeval{\Runo-\Rdue}}\edef\TCIsumR{\fpeval{\Runo+\Rdue}}
\fptest{\TCIdiffR > \A || \A > \TCIsumR}{%
\edef#5{0,0}\edef#6{0,0}% Valori assurdi se i cerchi non si intersecano
\PackageWarning{euclideangeometry}{%
***********************************\MessageBreak
Circles do not intersect \MessageBreak
Check centers and radii and retry \MessageBreak
Both intersection point are set to \MessageBreak
(0,0) therefore expect errors \MessageBreak
***********************************\MessageBreak}%
}{%
% \end{macrocode}
% Here we are within the correct range and we proceed with the
% calculations. We take as a temporary reference the segment
% that joins the centers. The common chord that joins the
% intersection points is perpendicular to such a segment crossing
% it by a distance $c$ form $C_1$, and, therefore by a distance
% $a-c$ from $C_2$; this chord forms two isosceles triangles with the
% centers; the above segment bisects such triangles, forming four right
% triangles; their hypotenuses equal the radii of the respective
% circles; their bases $h$ are all equal to half the chord; Pythagoras'
% theorem allows us to write:
%\[
%\left\{
% \begin{aligned}
% h^2 &= R_1^2 - c^2\\
% h^2 &= R_2^2 - (a-c)^2
% \end{aligned}
%\right.
%\]
% Solving for $c$, we get:
%\[\left\{
%\begin{aligned}
%c &= \frac{R_1^2 - R_2^2 + a^2 }{2a}\\
%h &= \sqrt{R_1^2 -c^2}
%\end{aligned}
%\right.
%\]
% \begin{macrocode}
\SegmentArg(\Cuno)(\Cdue)to\Acompl
\SubVect\Cuno from\Cdue to \Cdue
\edef\CI{\fpeval{(\Runo^2 - \Rdue^2 +\A^2)/(2*\A)}}
\edef\H{\fpeval{sqrt(\Runo^2 - \CI^2)}}
\CopyVect\CI,-\H to\Puno
\CopyVect\CI,\H to\Pdue
% \end{macrocode}
% Now we do not need anymore the chord intersection distance \cs{CI}
% any more, so we can use for other tasks, and we create a vector with
% absolute coordinates; We then add the rotated vector corresponding
% to the base \cs{H} so as to get the absolute chord extrema \cs{PPuno}
% and \cs{PPdue}.
% \begin{macrocode}
\MultVect\CI,0 by\Acompl:1 to\CI
\AddVect\Cuno and\CI to\CI
\MultVect\Puno by\Acompl:1 to\PPunorot
\AddVect\PPunorot and \Cuno to \PPuno
\MultVect\Pdue by\Acompl:1 to\PPduerot
\AddVect\PPduerot and \Cuno to \PPdue
\edef#5{\PPuno}\edef#6{\PPdue}%
}%
}
% \end{macrocode}
% It may be noticed that the first intersection point, assigned to
% parameter |#5| is the one found along the orthogonal direction
% to the vector form $C_1$ to $C_2$, obtained by a rotation of
% $90^\circ$ counterclockwise.
% The whole construction of the geometry described above is shown
% in figure~16 in the user manual \file{euclideangeometry-man.pdf}.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Triangle special points}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Here we have the macros to find the special points on a triangle
% side that are the “foot” of special lines from one vertex to the
% opposite side. We already described the circumcircle and the
% circumcenter, but that is a separate case, because the circumcenter
% is not the intersection of special lines from one vertex to the
% opposite base. The special lines we are interested in here are
% the height, the median, and the bisector
% The macros have the same aspect |\Triangle...Base|, where the dots
% are replaced with each of the (capitalised) special line names.
% Their syntaxes are therefore very similar:
%\begin{ttsyntax}
%\cs{TriangleMedianBase}\meta{vertex} on\meta{base1} and\meta{base2} to\meta{M}
%\cs{TriangleHeightBase}\meta{vertex} on\meta{base1} and\meta{base2} to\meta{H}
%\cs{TrinagleBisectorBase}\meta{vertex} on\meta{base1} and\meta{base2} to\meta{B}
%\end{ttsyntax}
% where \meta{vertex} contains one of the vertices coordinates, and
% \meta{base1} and \meta{base2} are the end points of the side
% opposite to that triangle vertex; \meta{M}, meta{H}, and \meta{B}
% are the intersections of these special lines from the \meta{vertex}
% to the opposite side; in order, they are the foot of the median,
% the foot of the height; the foot of the bisector. The construction
% of the median foot \meta{M} is trivial because this foot is the base
% center; the construction of the height foot is a little more
% complicated, because it is necessary to find the exact direction
% of the perpendicular from the vertex to the base in order to
% find the intersection \meta{H}; the construction of the bisector
% base implies finding the exact direction of the two sides starting at the
% \meta{vertex}, and taking the mean direction, which is trivial if
% polar coordinates are used; at this point the bisector line is completely
% determined and the intersection with the base line \meta{B} is
% easily obtained.
% \begin{macrocode}
\def\TriangleMedianBase#1on#2and#3to#4{%
\SubVect#1from#2to\TMBu \SubVect#1from#3to\TMBd
\SubVect\TMBu from\TMBd to\Base
\ScaleVect\Base by0.5to\TMBm\AddVect#2and\TMBm to#4\ignorespaces}
%
\def\TriangleHeightBase#1on#2and#3to#4{%
\SubVect#2from#3to\Base
\ArgOfVect\Base to\Ang \CopyVect\fpeval{\Ang+90}:1 to\Perp
\IntersectionOfLines(#1)(\Perp)and(#2)(\Base)to#4\ignorespaces}
%
\def\TriangleBisectorBase#1on#2and#3to#4{%
\SubVect#2from#1to\Luno \SubVect#3from#1to\Ldue
\SubVect#2from#3to\Base
\ArgOfVect\Luno to\Arguno \ArgOfVect\Ldue to\Argdue
\edef\ArgBis{\fpeval{(\Arguno+\Argdue)/2}}%
\CopyVect \ArgBis:1to \Bisect
\IntersectionOfLines(#2)(\Base)and(#1)(\Bisect)to#4\ignorespaces}
% \end{macrocode}
% Having defined the previous macros, it becomes very easy to create
% the macros to find the\emph{barycenter}, the \emph{orthocenter},
% the\emph{incenter}; for the \emph{circumcenter} and the
% \emph{circumcircle} we have already solved the question with the
% |\ThreePointCircleCenter| and the |ThreePointCircle| macros; for
% homogeneity, we create here their aliases with the same form as
% the new “center” macros. Actually, for the “circle” macros,
% once the center is known, there is no problem with the circumcircle,
% while for the incircle it suffices a macro to determine the distance
% of the incenter from one of the triangle sides; such a macro is going to
% be defined in a little while; it is more general than simply to
% determine the radius of the incircle.
% \begin{macrocode}
\let\TriangleCircumcenter\ThreePointCircleCenter
\let\TriangleCircummcircle\ThreePointCircle
% \end{macrocode}
%
% The other “center” macros are the following; they all consist
% in finding two of the specific triangle lines, and finding their
% intersection. Therefore for the barycenter we intersect two
% median lines; for the orthocenter we intersect two height lines;
% for the incenter we intersect two bisector lines;
% \begin{macrocode}
\def\TriangleBarycenter(#1)(#2)(#3)to#4{%
\TriangleMedianBase#1on#2and#3to\Pa
\TriangleMedianBase#2on#3and#1to\Pb
\DistanceAndDirOfVect#1minus\Pa to\ModPa and\AngPa
\DistanceAndDirOfVect#2minus\Pb to\ModPb and\AngPb
\IntersectionOfLines(#1)(\AngPa)and(#2)(\AngPb)to#4}
\def\TriangleOrthocenter(#1)(#2)(#3)to#4{%
\TriangleHeightBase#1on#2and#3to\Pa
\TriangleHeightBase#2on#3and#1to\Pb
\DistanceAndDirOfVect#1minus\Pa to\ModPa and\AngPa
\DistanceAndDirOfVect#2minus\Pb to\ModPb and\AngPb
\IntersectionOfLines(#1)(\AngPa)and(#2)(\AngPb)to#4}
\def\TriangleIncenter(#1)(#2)(#3)to#4{%
\TriangleBisectorBase#1on#2and#3to\Pa
\TriangleBisectorBase#2on#3and#1to\Pb
\DistanceAndDirOfVect#1minus\Pa to\ModPa and\AngPa
\DistanceAndDirOfVect#2minus\Pb to\ModPb and\AngPb
\IntersectionOfLines(#1)(\AngPa)and(#2)(\AngPb)to#4}
% \end{macrocode}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Other specific service macros}\label{ssec:ellissi}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% And here it comes the general macro to determine the distance
% of a point from a segment or from a line that contains that
% segment; it may be used for determining the radius of the
% incenter, but it is going to be used also for other purposes.
% Its syntax is the following:
%\begin{ttsyntax}
%\cs{DistanceOfPoint}\meta{point} from\parg{P1}\parg{P2}to\meta{distance}
%\end{ttsyntax}
% where \meta{point} is a generic point; \meta{P1} and \meta{P2}
% are a segment end points, or two generic points on a line;
% \meta{distance} is the macro that receives the computed scalar
% distance value.
% \begin{macrocode}
\def\DistanceOfPoint#1from(#2)(#3)to#4{%
\SubVect#2from#3to\Base \MultVect\Base by0,1to\AB
\IntersectionOfLines(#1)(\AB)and(#2)(\Base)to\D
\SubVect#1from\D to\D
\ModOfVect\D to#4}
% \end{macrocode}
% The following macros are specific to solve other little geometrical
% problems that arise when creating more complicated constructions.
%
% The |\AxisFromAxisAndFocus| is an unhappy name that describes
% the solution of an ellipse relationship between the ellipse axes
% and the focal distance
%\begin{equation} a^2 = b^2 + c^2\label{equ:axes-foci}\end{equation}
% This relation exists between the “semi” values, but it works equally
% well with the full values. Evidently $a$ is the largest quantity and
% refers to the main ellipse axis, the one that passes through the two
% foci; $b$ refers to the other shorter ellipse axis and $c$ refers to
% the foci; $b$ and $c$ are smaller than $a$, but there is no specific
% relationship among these two quantities It goes by itself that
% these statements apply to a veritable ellipse, not to a circle,
% that is the special case where $b=a$ and $c=0$.
%
% Since to solve the above equation we have one unknown and two
% known data, but we do not know what they represent, we have to
% assume some relationship exist between the known data; therefore
% if $a$ is known it must be entered as the first macro argument;
% otherwise $a$ is the unknown and the first Argument has to be the
% smaller one among $b$ and $c$. Since $b$ and$c$ may come from other
% computation the user has a dilemma: which is the smaller one?
% But this is a wrong approach; of course if the user knows which is the
% smaller, s/he can use the macro by entering the data in the proper
% order; but the user is determining the main axis, therefore it better
% that s/he uses directly the second macro |\MainAxisFromAxisAndFocus|
% that directly computes $a$ disregarding the order with which $b$
% and $c$ are entered; the macro name suggests to enter $b$ first
% and $c$ second, but it is irrelevant thanks to the sum properties.
% Summarising:
%\begin{itemize}[noitemsep]
%\item if the main axis is known use |\AxisFromAxisAndFocus| by
% entering the main axis as the first argument; otherwise
%\item ~
% \begin{itemize}
% \item if it is known which is smaller among $b$ and $c$, it is
% possible to use |\AxisFromAxisAndFocus| by entering the smaller
% one as the first argument; otherwise
% \item determine the main axis by using |\MainAxisFromAxisAndFocus|
% \end{itemize}
%\end{itemize}
% Their syntaxes of these two commands are basically the following:
%\begin{ttsyntax}
%\cs{AxisFromAxisAndFocus}\meta{main axis} and\meta{axis or focus} to\meta{focus or axis}
%\cs{MainAxisFromAxisAndFocus}\meta{axis or focus} and\meta{focus or axis} to\meta{main axis}
%\end{ttsyntax}
% but it is possible to enter the data in a different way with the
% first command; the described syntax is the suggested one.
% Evidently \meta{axis or focus} and \meta{focus or axis} imply
% that if you specify the focus in one of the two, you have to
% specify the axis in the other one.
% \begin{macrocode}
\def\AxisFromAxisAndFocus#1and#2to#3{%
\fptest{abs(#1)>abs(#2)}%
{\edef#3{\fpeval{sqrt(#1**2-#2**2)}}}%
{\edef#3{\fpeval{sqrt(#2**2+#1**2)}}}}
\def\MainAxisFromAxisAndFocus#1and#2to#3{%
\edef#3{\fpeval{sqrt(#2**2+#1**2)}}}
% \end{macrocode}
% The following macros allow to determine some scalar values relative
% to segments; in the second one the order of the segment end points is
% important, because the computed argument refers to the vector $P_2 - P_1$.
% Their syntaxes are the following:
%\begin{ttsyntax}
%\cs{SegmentLength}\parg{P1}\parg{P2}to\meta{length}
%\cs{SegmentArg}\parg{P1}\parg{P2}to\meta{argument}
%\end{ttsyntax}
% Both \meta{length} and \meta{argument} are macros that contain
% scalar quantities; the argument is in the range
% $-180^\circ <\Phi \leq +180^\circ$.
% \begin{macrocode}
\def\SegmentLength(#1)(#2)to#3{\SubVect#1from#2to\Segm
\ModOfVect\Segm to#3}
\def\SegmentArg(#1)(#2)to#3{\SubVect#1from#2to\Segm
\GetCoord(\Segm)\SegmX\SegmY\edef#3{\fpeval{atand(\SegmY,\SegmX)}}%
\ignorespaces}
% \end{macrocode}
%
% In the following sections we need some transformations, in
% particular the affine shear one. The macros we define here are
% not for general use, but are specific for the purpose of this package.
%
% The fist macro shears a segment, or better a vector that goes
% from point $P_1$ to point $P_2$ with a horizontal shear
% factor/angle $\alpha$; the origin of the vector does not vary
% and remains $P_1$ but the arrow tip of the vector is moved
% according to the shear factor; in practice this shearing macro
% is valid only for vectors that start from any point laying on
% the $x$ axis. The shear factor $\alpha$ is the angle of the
% \emph{clock wise} rotation vector operator by which the vertical
% coordinate lines get rotated with respect to their original position.
% The syntax is the following:
%\begin{ttsyntax}
%\cs{ShearVect}\parg{P1}\parg{P2}by\meta{shear} to\meta{vector}
%\end{ttsyntax}
% where \meta{P1} and \meta{P2} are the initial and final points of the
% vector to be sheared with the \meta{shear} angle, and the result is put
% in the output \meta{vector}
% \begin{macrocode}
\def\ShearVect(#1)(#2)by#3to#4{%
\SubVect#1from#2to\AUX
\GetCoord(\AUX)\Aux\Auy
\edef\Aux{\fpeval{\Aux + #3*\Auy}}%
\edef\Auy{\fpeval{\Auy}}%
\AddVect\Aux,\Auy and#1to#4\ignorespaces}
% \end{macrocode}
% Again we have another different |\ScaleVector| macro that takes in input
% the starting and ending points of a vector, and scales the vector
% independently of the initial point.
% \begin{macrocode}
\def\ScaleVector(#1)(#2)by#3to#4{%
% Scala per il fattore #3 il vettore da #1 a #2
\SubVect#1from#2to\AUX
\ScaleVect\AUX by#3to\AUX
\AddVect\AUX and#1to#4\ignorespaces}
% \end{macrocode}
%
% The following macro to draw a possibly sheared ellipse appears
% complicated; but in reality it is not much different from a
% “normal” ellipse drawing command. In oder to do the whole work
% the ellipse center is set in the origin of the axes, therefore
% it is not altered by the shearing process; everything else is
% horizontally sheared by the shear angle $\alpha$. In particular the
% 12~nodes and control point that are required by the Bézier
% splines that draw the four ellipse quarters. It is this multitude
% of shearing commands that makes the macro mach longer and apparently
% complicated.
% The syntax is the following:
%\begin{ttsyntax}
%\cs{Sellisse}\meta{$\star$}\marg{h-axis}\marg{v-axis}\oarg{shear}
% \end{ttsyntax}
% where the optional asterisk is used to mark and label the Bézier
% spline nodes and the control points of the possibly sheared ellipse;
% without the asterisk the ellipse is drawn without any “decoration”;
% the optional \meta{shear} is as usual the angle of the sheared
% vertical coordinate lines; its default value is zero.
% \begin{macrocode}
%
\NewDocumentCommand\Sellisse{s m m O{0}}{\bgroup
\CopyVect#2,#3to\Ptr \ScaleVect\Ptr by-1to\Pbl
\CopyVect#2,-#3to\Pbr \ScaleVect\Pbr by-1to\Ptl
\edef\Ys{\fpeval{tand{#4}}}%
\edef\K{\fpeval{4*(sqrt(2)-1)/3}}%
%
\ShearVect(0,0)(0,#3)by\Ys to\Pmt
\ShearVect(0,0)(0,-#3)by\Ys to\Pmb
\ShearVect(0,0)(#2,0)by\Ys to\Pmr
\ShearVect(0,0)(-#2,0)by\Ys to\Pml
%
\ShearVect(\Pmr)(\Ptr)by\Ys to\Ptr
\ShearVect(\Pml)(\Ptl)by\Ys to\Ptl
\ShearVect(\Pmr)(\Pbr)by\Ys to\Pbr
\ShearVect(\Pml)(\Pbl)by\Ys to\Pbl
%
\IfBooleanTF{#1}{\Pbox(\Ptr)[bl]{P_{tr}}\Pbox(\Pbl)[tr]{P_{bl}}%
\Pbox(\Pbr)[tl]{P_{br}}\Pbox(\Ptl)[br]{P_{tl}}%
\polygon(\Pbr)(\Ptr)(\Ptl)(\Pbl)}{}%
%
\ScaleVector(\Pmr)(\Ptr)by\K to\Crt
\ScaleVector(\Pmr)(\Pbr)by\K to\Crb
\ScaleVector(\Pml)(\Ptl)by\K to\Clt
\ScaleVector(\Pml)(\Pbl)by\K to\Clb
\ScaleVector(\Pmt)(\Ptr)by\K to\Ctr
\ScaleVector(\Pmt)(\Ptl)by\K to\Ctl
\ScaleVector(\Pmb)(\Pbr)by\K to\Cbr
\ScaleVector(\Pmb)(\Pbl)by\K to\Cbl
%
\IfBooleanTF{#1}{%
\Pbox(\Crt)[l]{C_{rt}}\Pbox(\Crb)[l]{C_{rb}}
\Pbox(\Clt)[r]{C_{lt}}\Pbox(\Clb)[r]{C_{lb}}
\Pbox(\Ctr)[b]{C_{tr}}\Pbox(\Ctl)[b]{C_{tl}}
\Pbox(\Cbr)[t]{C_{br}}\Pbox(\Cbl)[t]{C_{bl}}
%
\Pbox(\Pmr)[l]{P_{mr}}\Pbox(\Pmt)[b]{P_{mt}}%
\Pbox(\Pml)[r]{P_{ml}}\Pbox(\Pmb)[t]{P_{mb}}%
%
\polygon(\Pbr)(\Ptr)(\Ptl)(\Pbl)\thicklines}{}%
%
\moveto(\Pmr)
\curveto(\Crt)(\Ctr)(\Pmt)
\curveto(\Ctl)(\Clt)(\Pml)
\curveto(\Clb)(\Cbl)(\Pmb)
\curveto(\Cbr)(\Crb)(\Pmr)
\fillstroke
\egroup}
% \end{macrocode}
% This user macro is used to call the |\Sellisse| macro with
% the desired parameters, but also to act with it on order
% to fill or stroke the ellipse contour, and to select some
% settings such as the contour line thickness, or the color
% of the ellipse contour or interior.
% the syntax is the following:
%\begin{ttsyntax}
%\cs{XSellisse}\meta{$\star$1}\parg{center}\oarg{angle}\aarg{shear}\marg{h-axis}\marg{v axis}\meta{$\star$2}\oarg{settings1}\oarg{settings2}
%\end{ttsyntax}
% where there are two optional asterisks, \meta{$\star$1} and
% \meta{$\star$2}; the first one controls the coloring of the ellipse:
% if present the interior is filled, if absent the contour is stroked; the
% second one controls the way a possibly sheared ellipse appears:
% if present, the construction is shown, if absent only the final result
% is shown; \meta{center} is optional: if present, the ellipse center is
% specified; if absent, its center is at the origin of the picture axes;
% \meta{angle} is optional with default value zero: if absent,
% the ellipse is not rotated and the \meta{h-axis} remains horizontal,
% while the \meta{v-axis} remains vertical, while if present and with a
% non vanishing value, the ellipse is rotated counterclockwise
% the amount specified, and, of course, if the value is negative,
% the rotation is clockwise. The optional parameter \meta{shear},
% if present, shears the ellipse paralle the \meta{h-axis} direction;
% the \meta{settings1} and \meta{settings2} operate as described
% for command \cs{Xellisse}.
% \begin{macrocode}
\NewDocumentCommand\XSellisse{ s D(){0,0} O{0} D<>{0} m m s O{} o }%
{\IfBooleanTF#1{\let\fillstroke\fillpath}%
{\let\fillstroke\strokepath}%
\put(#2){\rotatebox{#3}{#8\relax
\IfBooleanTF{#7}{\Sellisse*{#5}{#6}[#4]}%
{\Sellisse{#5}{#6}[#4]}%
\IfValueTF{#9}{\let\fillstroke\strokepath
#9\Sellisse{#5}{#7}[#4]}{}}}%
\ignorespaces}
% \end{macrocode}
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Regular polygons and special ellipses}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% We finally arrive to more complex macros used to create special
% polygons and special ellipses.
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsubsection{Regular polygons}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Regular polygons are not that special; it is possible to draw them
% by using the |\multiput| or |\xmultiput| commands, but a single
% command that does everything by itself with more built in
% functionalities is much handier. The new command |\RegPolygon|
% has the following syntax:
%\begin{ttsyntax}
%\cs{RegPoligon}\meta{$\star$}\parg{center}\marg{radius}\marg{number}\oarg{angle}\aarg{settings}
%\end{ttsyntax}
% where \meta{$\star$} is an optional asterisk; its presence means
% that the polygon interior is filled, instead of the polygon
% contour being stroked; the \meta{center} specification of the
% polygon is optional; if it is omitted, the polygon center goes
% to the origin of the |picture| coordinates; \meta{radius} is the
% mandatory radius of the circumscribed circle, or, in other words,
% the distance of each polygon vertex form the \meta{center}; the
% mandatory \meta{number} is an integer that specifies the number of
% polygon sides; the first vertex that is being drawn by this
% command, has an angle of zero degrees with respect to the
% \meta{center}; if a different initial \meta{angle} different from
% zero is desired, it is specified through this optional argument;
% possibly the angle bracketed optional \meta{setting} parameter
% may be used to specify, for example, the line thickness for the
% contour, and/or the color for the polygon contour or interior.
% See the documentation \file{euclideangeometry-man.pdf} for more
% information and usage examples.
% \begin{macrocode}
\newcount\RPI
\NewDocumentCommand\RegPolygon{s D(){0,0} m m O{0} D<>{\relax} }{{%
%\countdef\RPI=258
\RPI=0
\CopyVect#5:#3to\P
\CopyVect\fpeval{360/#4}:1to\R
\put(#2){#6\relax
\moveto(\P)\fpdowhile{\RPI < #4}%
{\MultVect\P by\R to\P
\lineto(\P)\advance\RPI by 1}%
\IfBooleanTF{#1}%
{\fillpath}{#6\strokepath}}}\ignorespaces}
%%%%%%%%%
\def\DirOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
\ModOfVect#1to\@tempa
\unless\ifdim\@tempa\p@=\z@
\DividE\t@X by\@tempa to\t@X
\DividE\t@Y by\@tempa to\t@Y
\fi\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
% \end{macrocode}
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsubsection{The Steiner ellipse}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The construction of the Steiner ellipse is very peculiar; it is
% almost intuitive that any triangle has infinitely many internal
% tangent ellipses; therefore it is necessary to state some other
% constraints to find one specific ellipse out from this unlimited set.
%
% One such ellipse is the Steiner one, obtained by adding the
% constraint that the ellipse be tangent to the median points
% of the triangle sides. But one thing is the definition, and
% another totally different one is to find the parameters of
% such an ellipse; and working with ruler and compass, it is
% necessary to find a procedure to draw such an ellipse.
%
% The construction described here and implemented with the
% |SteinerEllipse| macro is based on the following steps, each
% one requiring the use of some of the commands and/or
% transformations described in the previous sections.
%\begin{enumerate}[noitemsep]
%^^A
%\item Given a generic triangle (the coordinates of its three
% vertices) it is not necessary, but it is clearer to explain,
% if the triangle is shifted and rotated so as to have one of
% its sides horizontal, and the third vertex in the upper part
% of the |picture| drawing. So we first perform the initial
% shift and rotation and memorise the parameters of this
% transformation so that, at the end of the procedure, we can
% put back the triangle (and its Steiner ellipse) in its
% original position. Let us call this shifted and rotated triangle
% with the symbol $T_0$.
%^^A
%\item We transform $T_0$ with an affine shear transformation into an
% isosceles triangle $T_1$ that has the same base and the same
% height as $T_0$. We memorise the shear “angle” so as to proceed
% to an inverse transformation when the following steps are completed:
% let be $\alpha$ this shear angle; geometrically it represents the
% angle of the sheared vertical coordinate lines with respect
% to the original vertical position.\label{enum:shear}
%^^A
%\item With another affine vertical scaling transformation we transform $T_1$ into an equilateral triangle $T_2$; the ratio of the vertical
% transformation equals the ratio between the $T_2$ to the
% $T_1$ heights; we memorise this ratio for the reverse transformation
% at the end of the procedure.\label{enum:ratio}
%^^A
%\item The Steiner ellipse of the equilateral triangle $T_2$ is its
% incircle. We are almost done; we just have to proceed to the
% inverse transformations; getting back from $T_2$ to $T_1$ first implies
% transforming the incircle of $T_2$ into an ellipse with its
% vertical axis scaled by the inverse ratio memorised in
% step~\ref{enum:ratio}.
%^^A
%\item The second inverse transformation by the shear angle is easy
% with the passage from $T_1$ to $T_0$, but it would be more difficult
% for transforming the ellipse into the sheared ellipse. We have already
% defined the |\Sellipse| and the |\XSellipse| macros that may take
% care of the ellipse shear transformation; we already memorised the
% shear angle in step~\ref{enum:shear}, therefore the whole procedure,
% except for putting back the triangle, is almost done.
%^^A
%\item Eventually we perform the last shifting and rotating transformation
% and the whole construction is completed.
%^^A
%\end{enumerate}
%
% The new macro Steiner ellipse has therefore the following syntax:
%\begin{ttsyntax}
%\cs{SteinerEllipse}\meta{$\star$}\parg{P1}\parg{P2}\parg{P3}\oarg{diameter}
%\end{ttsyntax}
% where \meta{P1}, \meta{P2}, \meta{P3} are the vertices of the
% triangle; \meta{$\star$} is an optional asterisk; without it
% the maro draws only the final result, that contains only the
% given triangle and its Steiner ellipse; on the opposite, if the asterisk
% is used the whole construction from $T_0$ to its Steiner ellipse
% is drawn; the labelling of points is done with little dots of
% the default \meta{diameter} or a specified value; by default it is a 1\,pt
% diameter, but sometimes it would be better to use a slightly
% larger value (remembering that 1\,mm — about three points —
% is already too much).
% Please refer to the documentation file \file{euclideangeometry-man.pdf}
% for usage examples and suggestions.
% \begin{macrocode}
%
\NewDocumentCommand\SteinerEllipse{s d() d() d() O{1}}{\bgroup
%
\IfBooleanTF{#1}{}{\put(#2)}{%
\CopyVect0,0to\Pu
\SubVect#2from#3to\Pd
\SubVect#2from#4to\Pt
\ModAndAngleOfVect\Pd to\M and\Rot
\MultVect\Pd by-\Rot:1 to\Pd \MultVect\Pt by-\Rot:1 to\Pt
\IfBooleanTF{#1}{}{\rotatebox{\Rot}}{\makebox(0,0)[bl]{%
\Pbox(\Pu)[r]{P_1}[#5]<-\Rot>\Pbox(\Pd)[t]{P_2}[#5]<-\Rot>
\Pbox(\Pt)[b]{P_3}[#5]<-\Rot>%
\polygon(\Pu)(\Pd)(\Pt)%
\edef\B{\fpeval{\M/2}}\edef\H{\fpeval{\B*tand(60)}}
\IfBooleanTF{#1}{\Pbox(\B,\H)[b]{H}[#5]
\polygon(\Pu)(\B,\H)(\Pd)}{}%
\edef\R{\fpeval{\B*tand(30)}}
\IfBooleanTF{#1}{\Pbox(\B,\R)[bl]{C}[#5]
\Circlewithcenter\B,\R radius{\R}}{}%
\GetCoord(\Pt)\Xt\Yt\edef\VScale{\fpeval{\Yt/\H}}
\IfBooleanTF{#1}{\polyline(\Pu)(\B,\Yt)(\Pd)
\Pbox(\B,\Yt)[b]{V}[#5]}{}%
\edef\Ce{\fpeval{\R*\VScale}}
\IfBooleanTF{#1}{\Xellisse(\B,\Ce){\R}{\Ce}
\Pbox(\B,\Ce)[r]{C_e}[#5]\Pbox(\B,0)[t]{B}[#5]}{}%
\SubVect\B,0 from\Pt to\SlMedian
\IfBooleanTF{#1}{\Dotline(\B,0)(\Pt){2}[1.5]}{}%
\ModAndAngleOfVect\SlMedian to\Med and\Alfa
\edef\Alfa{\fpeval{90-\Alfa}}
\IfBooleanTF{#1}{\Dotline(\B,\Yt)(\B,0){2}[1.5]
\Pbox(\fpeval{\B+\Ce*tand{\Alfa}},\Ce)[l]{C_i}[#5]
\VectorArc(\B,0)(\B,15){-\Alfa}
\Pbox(\fpeval{\B+2.5},14)[t]{\alpha}[0]}{}%
\edef\a{\R}\edef\b{\Ce}%
\CopyVect\fpeval{\B+\Ce*tand{\Alfa}},\Ce to\CI
\XSellisse(\CI)<\Alfa>{\R}{\Ce}
}}}%
\egroup\ignorespaces}
\let\EllisseSteiner\SteinerEllipse
% \end{macrocode}
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsubsection{The ellipse that is internally tangent to a triangle while one of its foci is prescribed}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% We now are going to tackle another problem. As we said before, any
% triangle has an infinite set of internally tangent circles, unless
% some further constraint is specified.
%
% Another problem of this kind is the determination and geometrical
% construction of an internally tangent ellipse when one focus is
% specified; of course since the whole ellipse is totally internal
% to the triangle, we assume that the user has already verified
% that the coordinates of the focus fall inside the triangle.
% We are not going to check this feature in place of the user;
% after all, if the user draws the triangle within a |picture| image,
% together with the chosen focus, is suffices a glance to verify that
% such focus lays within the triangle perimeter.
%
% The geometrical construction is quite complicated, but it is
% described in a paper by Estevão V.~Candia on \TB~2019 \textbf{40}(3);
% it consists of the following steps.
%\begin{enumerate}[noitemsep]
%^^A
%\item Suppose you have specified a triangle by means of its three
% vertices, and a point inside it to play the role of a focus; it
% is necessary to find the other focus and the main axis length in
% order to have a full description of the ellipse.
%^^A
%\item To do so, it is necessary to find the focus three symmetrical
% points with respect to the three sides.
%^^A
%\item The center of the three point circle through these symmetrical
% points is the second focus.
%^^A
%\item The lines that join the second focus to the three symmetrical
% points of the first focus, intersect the triangle sides in three
% points that result to be the tangency points of the ellipse to the
% triangle.
%^^A
%\item Chosen one of these tangency points and computing the sum of its
% distances from both foci, the total length of the ellipsis main axis
% is found.
%^^A
%\item Knowing both foci, the total inter focal distance is found,
% therefore equation~\eqref{equ:axes-foci} allows to find the other
% axis length.
%^^A
%\item The inclination of the focal segment gives us the the rotation
% to which the ellipse is subject, and the middle point of such
% segment gives the ellipse center.
%^^A
%\item At this point we have all the necessary elements to draw
% the ellipse.
%^^A
%\end{enumerate}
%
% We need another little macro to find the symmetrical points;
% if the focus $F$ and its symmetrical point $P$ with respect
% to a side/segment, the intersection of such segment $F-P$ with
% the side is the segment middle point $M$; from this property
% we derive the formula $P= 2M -F$. Now $M$ is also the
% intersection of the line passing through $F$ and perpendicular
% to the side. Therefore it is particularly simple to compute,
% but its better to have available a macro that does the whole
% work; here it is, but it assumes the the center of symmetry is
% already known:
% \begin{macrocode}
\def\SymmetricalPointOf#1respect#2to#3{\ScaleVect#2by2to\Segm
\SubVect#1from\Segm to#3\ignorespaces}
% \end{macrocode}
% And its syntax is the following:
%\begin{ttsyntax}
%\cs{SymmetricalPointOf}\meta{focus} respect\meta{symmetry center}
%\qquad to\meta{symmetrical point}
%\end{ttsyntax}
% where the argument names are self explanatory.
%
% The overall macro that executes all the passages described in
% the above enumeration follows; the reader can easily recognise
% the various steps, since the names of the macros are self
% explanatory; the $S_i$ point names are the symmetrical ones
% to the first focus $F$; the $M_i$ points are the centers of
% symmetry; the $F'$ point is the second focus; the $T_i$ points
% are the tangency points. The macro |\EllipseWithFOcus| has the
% following syntax:
%\begin{ttsyntax}
%\cs{EllipseWithFocus}\meta{$\star$}\parg{P1}\parg{P2}\parg{P3}\parg{focus}
%\end{ttsyntax}
% where \meta{P1}, \meta{P2}, \meta{P3} are the triangle vertices
% and \meta{focus} contains the first focus coordinates; the
% optional asterisk, as usual, selects the construction steps
% versus the final result: no asterisk, no construction steps.
% \begin{macrocode}
\NewDocumentCommand\EllipseWithFocus{s d() d() d() d()}{\bgroup%
\CopyVect#2to\Pu
\CopyVect#3to\Pd
\CopyVect#4to\Pt
\CopyVect#5to\F
\polygon(\Pu)(\Pd)(\Pt)
\Pbox(\Pu)[r]{P_1}[1.5pt]\Pbox(\Pd)[t]{P_2}[1.5pt]
\Pbox(\Pt)[b]{P_3}[1.5pt]\Pbox(\F)[b]{F}[1.5pt]
\SegmentArg(\Pu)(\Pt)to\At
\SegmentArg(\Pu)(\Pd)to\Ad
\SegmentArg(\Pd)(\Pt)to\Au
\IntersectionOfLines(\Pu)(\At:1)and(\F)(\fpeval{\At+90}:1)to\Mt
\IntersectionOfLines(\Pd)(\Ad:1)and(\F)(\fpeval{\Ad+90}:1)to\Md
\IntersectionOfLines(\Pd)(\Au:1)and(\F)(\fpeval{\Au+90}:1)to\Mu
\IfBooleanTF{#1}{\Pbox(\Mt)[br]{M_3}[1.5pt]\Pbox(\Md)[t]{M_2}[1.5pt]
\Pbox(\Mu)[b]{M_1}[1.5pt]}{}
\SymmetricalPointOf\F respect\Mu to\Su
\IfBooleanTF{#1}{\Pbox(\Su)[l]{S_1}[1.5pt]}{}
\SymmetricalPointOf\F respect \Md to\Sd
\IfBooleanTF{#1}{\Pbox(\Sd)[t]{S_2}[1.5pt]}{}
\SymmetricalPointOf\F respect \Mt to\St
\IfBooleanTF{#1}{\Pbox(\St)[r]{S_3}[1.5pt]}{}
\IfBooleanTF{#1}{\ThreePointCircle*(\Su)(\Sd)(\St)}%
{\ThreePointCircle(\Su)(\Sd)(\St)}
\CopyVect\C to\Fp \Pbox(\Fp)[l]{F'}[1.5pt]
\IfBooleanTF{#1}{%
\Dotline(\F)(\St){2}[1.5pt]
\Dotline(\F)(\Sd){2}[1.5pt]
\Dotline(\F)(\Su){2}[1.5pt]}{}
\IntersectionOfSegments(\Pu)(\Pt)and(\Fp)(\St)to\Tt
\IntersectionOfSegments(\Pu)(\Pd)and(\Fp)(\Sd)to\Td
\IntersectionOfSegments(\Pd)(\Pt)and(\Fp)(\Su)to\Tu
\IfBooleanTF{#1}{\Pbox(\Tu)[l]{T_1}[1.5pt]
\Pbox(\Td)[b]{T_2}[1.5pt]
\Pbox(\Tt)[tl]{T_3}[1.5pt]
\Dashline(\Fp)(\Su){1}\Dashline(\Fp)(\Sd){1}\Dashline(\Fp)(\St){1}}{}
\DistanceAndDirOfVect\Fp minus\Tt to\DFp and\AFu
\DistanceAndDirOfVect\F minus\Tt to\DF and\AF
\SegmentCenter(\F)(\Fp)to\CE \Pbox(\CE)[b]{C}[1.5pt]
\edef\a{\fpeval{(\DFp+\DF)/2}}
\SegmentArg(\F)(\Fp)to\AngFocalAxis
\SegmentLength(\F)(\CE)to\c
\AxisFromAxisAndFocus\a and\c to\b
\Xellisse(\CE)[\AngFocalAxis]{\a}{\b}[\thicklines]
\VECTOR(-30,0)(120,0)\Pbox(120,0)[t]{x}[0]
\VECTOR(0,-20)(0,130)\Pbox(0,130)[r]{y}[0]\Pbox(0,0)[tr]{O}[1.5pt]
\egroup\ignorespaces}
\let\EllisseConFuoco\EllipseWithFocus
% \end{macrocode}
%
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Comments on this package}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% In general we found very comfortable to draw ellipses and
% to define macros to draw not only such shapes or filled elliptical
% areas, but also to create “legends” with coloured backgrounds and
% borders; such applications found their way in other works.
% But here we dealt with other geometrical problems. The accompanying
% document |euclideangeometry-man.pdf| describes much clearly with
% examples what you can do with the macros described in this package.
% In facts, this file just describes the package macros, and it gives
% some ideas on how to extend the ability of |curve2e| to draw
% geometrical diagrams.
% The users who would like to modify or to add some functionalities
% are invited to do so; I will certainly acknowledge their contributions
% and even add their names to the list of authors.
%
% As long as I can, I enjoy playing with \LaTeX\ and its wonderful
% facilities; but, taking into consideration my age, I would invite
% the users to consider the possibility of assuming the maintenance
% of this package.
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Acknowledgements}
%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% I am very grateful to Enrico Gregorio who let me know the several
% glitches I made in my first version; besides being a real \TeX wizard,
% he is a wise person and suggested me several things that was important
% to change, because they could offer risks of confusion with other
% packages.
%
%\iffalse
%</package>
%\fi
% \Finale
%