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diff --git a/Master/texmf-dist/doc/latex/euclide/euclide_english.tex b/Master/texmf-dist/doc/latex/euclide/euclide_english.tex deleted file mode 100644 index 4c0f2a98b99..00000000000 --- a/Master/texmf-dist/doc/latex/euclide/euclide_english.tex +++ /dev/null @@ -1,1177 +0,0 @@ -\documentclass[12pt, draft]{article} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\usepackage[T1]{fontenc} -\usepackage[latin1]{inputenc} -\usepackage[a4paper]{geometry} -\usepackage[usenames]{pstcol} -\usepackage{pst-eucl} -\usepackage{pst-plot} -\usepackage{multido} -\usepackage{ifthen} -\usepackage{calc} -\usepackage{array} -\usepackage{moreverb} -\usepackage{multicol} -\usepackage{mathrsfs} -\usepackage[dvips]{changebar} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% QQ DEFINITIONS -\newcommand{\Vecteur}[1]{\ensuremath{\overrightarrow{#1}}}% -\newcommand{\Angle}[1]{\ensuremath{\widehat{#1}}}% -\makeatletter -\newcommand{\Arg}[1]{{\normalfont$\{$\@Arg{#1}$\}$}}% -\newcommand{\Argsans}[1]{\@Arg{#1}}% -\newcommand{\OptArg}[1]{{\normalfont[{\@Arg{#1}}]}}% -\newcommand{\@Arg}[1]% - {$\protect\langle${\itshape\mdseries\rmfamily#1}$\protect\rangle$}% -\makeatother -%% le backslash \ -\newcommand{\bs}{\symbol{'134}}% -\newcommand{\defcom}[2]% - {\begin{trivlist}\item\fbox{\texttt{\upshape\bs#1}#2}\end{trivlist}}% -\newcommand{\defcomdeux}[4]% - {% - \begin{center}% - \begin{minipage}[t]{.45\linewidth}% - \begin{trivlist}\item\fbox{\texttt{\upshape\bs#1}#2}\end{trivlist}% - \end{minipage}% - \hfill% - \begin{minipage}[t]{.45\linewidth}% - \begin{trivlist}\item\fbox{\texttt{\upshape\bs#3}#4}\end{trivlist}% - \end{minipage} - \end{center}}% -\newenvironment{tabexemple}[1]% - [@{}m{.3\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.677\linewidth}@{}]% - {\vspace{1em}\noindent\small\begin{center}% - \noindent\begin{tabular}{#1}}% - {\mbox{}\vspace{-1.5em}\end{tabular}\end{center}}% -\newcommand{\tabex}[2]% - [@{}m{.3\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.677\linewidth}@{}]% -{% - \begin{tabexemple}[#1]% - \input{Exemples/#2}&\verbatiminput{Exemples/#2_in}% - \end{tabexemple}% -}% -\newcommand{\param}[1]{\texttt{#1}} -\newcommand{\com}[1]{\texttt{\bs #1}} -\newcommand{\DefaultVal}[1]{(\texttt{#1} by default)} -\newcommand{\PStricks}{\texttt{PStricks}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\title{The \texttt{pst-euclide.sty} Package} -\author{Version $0.\beta.5$\\Dominique Rodriguez} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\geometry{bottom=1cm, left=1cm, right=1cm} -\psset{subgriddiv=0,griddots=5} -%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%% -\newcounter{i} -%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%% -\begin{document} -\renewcommand{\abstractname}{WARNING} -\maketitle -\cbstart -\begin{abstract} - This is a $\beta$-version, it is still in development, the macros - name can and will change without upward-compatibility. -\end{abstract} -\cbend -\begin{multicols}{2} -\tableofcontents{} -\end{multicols} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Special specifications} - -\cbstart - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{\PStricks\ Options} - -The package activates the \com{SpecialCoor} mode. This mode extend the -coordinates specification. Furthermore the plotting type is set to -\texttt{dimen=middle}, which indicates that the position of the -drawing is done according to the middle of the line. Please look at -the user manual for more information about these setting. - -At last, the working axes are supposed to be (ortho)normed. - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Conventions} - -For this manual, I used the geometric French conventions for naming -the points: - -\begin{itemize} -\item $O$ is a centre (circle, axes, symmetry, homothety, rotation); -\item $I$ defined the unity of the abscissa axe, or a midpoint; -\item $J$ defined the unity of the ordinate axe; -\item $A$, $B$, $C$, $D$ are points ; -\item $M'$ is the image of $M$ by a transformation ; -\end{itemize} - -At last, although these are nodes in \PStricks, I treat them -intentionally as points. -\cbend - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Basic Objects} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Points} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{default axes} - -\defcom{pstGeonode}{\OptArg{par}\Arg{$A$}} - -This command defines a geometrical point associated with a node. This -point has a node name \Argsans{$A$} which defines the default label put on -the picture. This label is managed by default in mathematical mode, -the boolean parameter \param{PtNameMath} can modify this behavior and -let manage the label in normal mode. It is put at a distance of -\texttt{1em} of the center of the node with a angle of -\param{PosAngle}\DefaultVal{0}. It is possible to -specify another label using the parameter \param{PointName}, and an -empty label can be specified by selecting the value \texttt{none}, in -that case the point will have no name on the picture. - -The point symbol is given by the parameter -\param{PointSymbol} \DefaultVal{*}. The symbol is the same as used by -the macro \com{pstdot}. This parameter can be set to \texttt{none}, -which means that neither the point nor its label will be drawn on the -picture. - -Here are the possible values for this parameter: - -\begin{multicols}{3} - \begin{itemize}\psset{dotscale=2} - \item \param{*}: \psdots(.5ex,.5ex) - \item \param{o}: \psdots[dotstyle=o](.5ex,.5ex) - \item \param{+}: \psdots[dotstyle=+](.5ex,.5ex) - \item \param{x}: \psdots[dotstyle=x](.5ex,.5ex) - \item \param{asterisk} : \psdots[dotstyle=asterisk](.5ex,.5ex) - \item \param{oplus} : \psdots[dotstyle=oplus](.5ex,.5ex) - \item \param{otimes} : \psdots[dotstyle=otimes](.5ex,.5ex) - \item \param{triangle}: \psdots[dotstyle=triangle](.5ex,.5ex) - \item \param{triangle*}: \psdots[dotstyle=triangle*](.5ex,.5ex) - \item \param{square}: \psdots[dotstyle=square](.5ex,.5ex) - \item \param{square*}: \psdots[dotstyle=square*](.5ex,.5ex) - \item \param{diamond} : \psdots[dotstyle=diamond](.5ex,.5ex) - \item \param{diamond*} : \psdots[dotstyle=diamond*](.5ex,.5ex) - \item \param{pentagon}: \psdots[dotstyle=pentagon](.5ex,.5ex) - \item \param{pentagon*} \psdots[dotstyle=pentagon*](.5ex,.5ex) - \item \param{|}: \psdots[dotstyle=|](.5ex,.5ex) - \end{itemize} -\end{multicols} - -% EXEMPLE GEONODE -\tabex{geonode} - -Obviously, the nodes appearing in the picture can be used as normal -pstricks nodes. Thus, it is possible to reference a point from -\rnode{ici}{here}. -\nccurve{->}{ici}{B_1} - -% There is a bug linked to the definition of a point, so a mandatory -% \verb$%$ symbol must be put at the end of each such command. If it is -% forgotten, a shift appear on the picture as it can be seen on the -% following diagram: - -% \begin{center} -% \begin{pspicture}(-2,-2)(2,2)\psgrid -% \pstGeonode(0,0){A}{A} -% \pstGeonode[PosAngle=-90](1,2){B1}{B_1} -% \pstGeonode[PointSymbol=pstSmallCircle, -% linecolor=red](-2,1){B2}{B_2}% -% \end{pspicture} -% \end{center} - -\cbstart - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{User defined axes} - -\defcom{pstOIJGeonode}{\OptArg{par}$(x,y)$\Arg{$A$}\Arg{$O$}\Arg{$I$}\Arg{$J$}} - -This command allows the placement of points in any landmark(?) defined -by the three points $(O;I;J)$. - -%% EXAMPLE -\tabex{oij} -\cbend - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Segment mark} - -A segment can be drawn using the \texttt{\bs ncline} command. However, -for marking a segment there is the following command: - -\defcom{pstMarkSegment}{\OptArg{par}\Arg{$A$}\Arg{$B$}} - -The symbol drawn on the segment is given by the parameter -\param{SegmentSymbol}. Its value can be any valid command which can be -used in math mode. Its default value is \texttt{pstSlashsSlash}, -which produced two slashes on the segment. The segment is drawn. - -%% EXAMPLE -\tabex{segmentmark} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Triangles} - -The more classical figure, it has its own macro for a quick definition: - -\defcom{pstTriangle}{% - \OptArg{par} - $(x_A;y_A)$\Arg{$A$}$(x_B;y_B)$\Arg{$B$}$(x_C;y_C)$\Arg{$C$}} - -In order to accurately put the name of the points, there are three -parameters \param{PosAngleA}, \param{PosAngleB} and \param{PosAngleC}, -which are associated respectively to the nodes \Argsans{$A$}, -\Argsans{$B$} et \Argsans{$C$}. Obviously they have the same meaning -as the parameter \param{PosAngle}. If one or more of such parameters is -omitted, the value of \param{PosAngle} is taken. - -In the same way there are parameters for controlling the symbol used -for each points: \param{PointSymbolA}, \param{PointSymbolB} and -\param{PointSymbolC}. They are equivalent to the parameter -\param{PointSymbol}. The management of the default value followed the -same rule. - -\tabex{triangle} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Angles} - -Each angle is defined with three points. The vertex is the second -point. Their order is important because it is assumed that the angle is -specified in the direct order. The first command is the marking of a -right angle: - -\defcom{pstRightAngle}% - {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$C$}} - -The only parameter controlling this command, excepting the ones which -controlled the line, is \param{RightAngleSize} which defines the size -of the symbol\DefaultVal{0.28 unit}. - -For other angles, there is the command: - -\defcom{pstMarkAngle}% - {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$C$}} - -The \param{label} can be any valid \TeX\ box, it is put at -\param{LabelSep} \DefaultVal{1 unit} of the node in the direction of -\cbstart the bisector of the angle modified by -\param{LabelAngleOffset}\DefaultVal{0} and positioned using -\param{LabelRefPt} \DefaultVal{c}. \cbend Furthermore the arc used for -marking has a radius of \param{MarkAngleRadius} \DefaultVal{.4~unit}. -At least, it is possible to place an arrow using the parameter -\param{arrows}. - -\tabex{angle} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Lines, half-lines and segments} - -The classical line! - -\defcom{pstLineAB}{\OptArg{par}\Arg{$A$}\Arg{$B$}} - -In order to control its length\footnote{which is the comble for a -line!}, the two parameters \param{nodesepA} et \param{nodesepB} -specify the abscissa of the extremity of the drawing part of the line. -A negative abscissa specify an outside point, while a positive -abscissa specify an internal point. If these parameters have to be -equal, \param{nodesep} can be used instead. The default value of these -parameters is equal to 0. - -\tabex{droite} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Circles} - -A circle can be defined either with its center and a point of its -circumference, or with two diameterly opposed points. There is two -commands : - -\defcomdeux{pstCircleOA}{\OptArg{par}\Arg{$O$}\Arg{$A$}}% - {pstCircleAB}{\OptArg{par}\Arg{$A$}\Arg{$B$}} - -For the first macro, it is possible to omit the second point and then -to specify a radius or a diameter using the parameters \param{Radius} -and \param{Diameter}. The values of these parameters must be specified -with one of the two following macros : - -\defcomdeux{pstDistAB}{\OptArg{par}\Arg{$A$}\Arg{$B$}}% - {pstDistVal}{\OptArg{par}\Arg{x}} - -The first specifies a distance between two points. The parameter -\param{DistCoef} can be used to specify a coefficient to reduce or -enlarge this distance. To be taken into account this last parameter -must be specified before the distance. The second macro can be used to -specify an explicit numeric value. - -We will see later how to draw the circle crossing three points. - -\vspace{1.1\baselineskip} -\begin{minipage}[m]{.45\linewidth} - With this package, it becomes possible to draw: - - \begin{itemize} - \item {\color{red} the circle of center $A$ crossing $B$ ;} - \item {\color{green} the circle of center $A$ whose radius is $AC$ ;} - \item {\color{blue} the circle of center $A$ whose radius is $BC$ ;} - \item {\color{Sepia} the circle of center $B$ whose radius is $AC$ ;} - \item {\color{Aquamarine} the circle of center $B$ of diameter $AC$ ;} - \item {\color{RoyalBlue} the circle whose diameter is $BC$ ;} - \end{itemize} -\end{minipage} -% -\input{Exemples/cercle} - -\verbatiminput{Exemples/cercle_in} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Circle arcs} - -\defcomdeux{pstArcOAB}{\OptArg{par}\Arg{$O$}\Arg{$A$}\Arg{$B$}}% - {pstArcnOAB}{\OptArg{par}\Arg{$O$}\Arg{$A$}\Arg{$B$}} - -These two macros draw circle arcs, $O$ is the center, the radius -defined by $OA$, the beginning angle given by $A$ and the final angle -by $B$. Finally, the first macro draws the arc in the direct way, -whereas the second in the indirect way. It is not necessary that the -two points are at the same distance of $O$. - -\tabex{arc} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Curved abscissa} - -A point can be positioned on a circle using its curved abscissa. - -\defcom{pstCurvAbsNode}{\OptArg{par}\Arg{$O$}\Arg{$A$}\Arg{$B$}\Arg{Abs}} - -The point \Argsans{$B$} is positioned on the circle of center -\Argsans{$O$} crossing \Argsans{$A$}, with the curved abscissa -\Argsans{Abs}. The origin is \Argsans{$A$} and the direction is -anti-clockwise by default. The parameter \param{CurvAbsNeg} -\DefaultVal{false} can change this behavior. - -\tabex{abscur} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Généric curve} - -It is possible to generate a set of point using a loop, and to give -them a generic name defined by a radical and a number. The following -command can draw a interpolated curve crossing all such kind of -points. - -\defcom{pstGenericCurve}{\OptArg{par}\Arg{Radical}\Arg{$n_1$}\Arg{$n_2$}} - -The curve is drawn on the points whose name is defined using the -radical \Argsans{Radical} followed by a number from \Argsans{$n_1$} to -\Argsans{$n_2$}. In order to manage side effect, the parameters -\param{GenCurvFirst} et \param{GenCurvLast} can be used to specified -special first or last point. The parameter \param{GenCurvInc} can be -used to modify the increment from a point to the next one -\DefaultVal{1}. - -\tabex{gencur} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Geometric Transformations} - -The geometric transformation are the ideal tools to construct -geometric figures. All the classical transformation are accessible -with the following macros. - -They share the parameter \param{CodeFig} which draws the specific -constructions lines. Its default value is \param{false}, and a -\param{true} value activates this optional drawing. - -The drawing is done using the line style \param{CodeFigStyle} -\DefaultVal{dashed}, with the color \param{CodeFigColor} -\DefaultVal{cyan}. - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Central symmetry} - -\defcom{pstSymO}% - {\OptArg{par}\Arg{$O$}\Arg{$M$}\Arg{$M'$}} - -Draw the symmetric point in relation to point $O$. The classical -parameter of point creation are usable here, and also for all the -following functions. - -\tabex{symcentrale} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Orthogonal (or axial) symmetry} - -\defcom{pstOrtSym}% - {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$M$}\Arg{$M'$}} - -Draw the symmetric point in relation to line $(AB)$. - -\tabex{symorthogonale} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Rotation} - -\defcom{pstRotation}% - {\OptArg{par}\Arg{$O$}\Arg{$M$}\Arg{$M'$}} - -Draw the image of $M$ by the rotation of center $O$ and angle given by -the parameter \param{RotAngle}. This later can be an angle specified -by three points. In such a case, the following function must be used: - -\defcom{pstAngleABC}{\Arg{$A$}\Arg{$B$}\Arg{$C$}} - -Never forget to use the rotation for a square or a equilateral -triangle. - -\tabex{rotation} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Translation} - -\defcom{pstTranslation}% - {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$M$}\Arg{$M'$}} - -Draw the translated of vector \Vecteur{AB} of the point -\Argsans{$M$}. Useful for drawing a parallel line. - -\tabex{translation} - -The parameter \param{DistCoef} can be used as a multiplicand -coefficient to modify the translation vector. - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Homothetie} - -\defcom{pstHomO}% - {\OptArg{par}\Arg{$O$}\Arg{$M$}\Arg{$M'$}} - -Draw $M'$ the image of $M$ by the homotethy of center $O$ and -coefficient specified with the parameter \param{HomCoef}. - -\tabex{homothetie} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Orthogonal projection} - -\defcom{pstProjection}% - {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$M$}\Arg{$M'$}} - -Project orthogonally the point $M$ on the line -$(AB)$. Useful for the altitude of a triangle. - -\tabex{projection} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Special object} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Midpoint} - -\defcom{pstMiddleAB}% - {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$I$}} - -Draw the midpoint $I$ of segment $[AB]$. - -\tabex{milieu} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Triangle center of gravity} - -\defcom{pstCGravABC}% - {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$C$}\Arg{$G$}} - -Draw the $ABC$ triangle centre of gravity $G$. - -\tabex{grav} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Centre of the circumcircle of a triangle} - -\defcom{pstCircleABC}{\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$C$}\Arg{$O$}} - -It's obvious. - -\tabex% - [@{}m{.35\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.627\linewidth}@{}]% - {ccirc} - -\cbstart - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Perpendicular bisector of a segment} - -\defcom{pstMediatorAB}{\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$I$}\Arg{$M$}} - -The perpendicular bisector of a segment is a line perpendicular to -this segment in its midpoint. The segment is $[AB]$, the midpoint $I$, -and $M$ is a point belonging to the perpendicular bisector line. It is -build by a rotation of $B$ of 90 degrees around $I$. This mean -that the order of $A$ and $B$ is important, it controls the position -of $M$. The command creates the two points $M$ end $I$. The -construction is controlled by the following parameters: - -\begin{itemize} -\item \param{CodeFig}, \param{CodeFigColor} et \param{SegmentSymbol} - for marking the right angle ; -\item \param{PointSymbol} et \param{PointName} for controlling the - drawing of the two points, each of them can be specified - separately with the parameters \param{...A} et \param{...B} ; -\item parameters controlling the line drawing. -\end{itemize} - -\tabex% - [@{}m{.35\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.627\linewidth}@{}]% - {mediator} -\cbend - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Bissectors of angles} - -\defcom{pstBissectBAC}{\OptArg{par}\Arg{$B$}\Arg{$A$}\Arg{$C$}\Arg{$N$}} - -\defcom{pstOutBissectBAC}{\OptArg{par}\Arg{$B$}\Arg{$A$}\Arg{$C$}\Arg{$N$}} - -there are two bisectors for a given geometric angle: the inside one and -the outside one; this is why there is two commands. The angle is -specified by three points specified in the trigonometric direction -(anti-clockwise). The result of the commands is the specific line and -a point belonging to this line. This point is built by a rotation of -point $B$. - -\tabex% - [@{}m{.35\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.627\linewidth}@{}]% - {bissec} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Intersections} - -Points can be defined by intersections. Three intersection types are -managed: - -\begin{itemize} -\item line-line; -\item line-circle; -\item circle-circle. -\end{itemize} - -An intersection can not exist: case of parallel lines. In such a case, -the point(s) are positioned at the origin. In fact, the user has to -manage the existence of these points. - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Line-Line} - -\defcom{pstInterLL}% - {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$C$}\Arg{$D$}\Arg{$M$}} - -Draw the intersection point between lines $(AB)$ and $(CD)$. - -\begin{description} -\item[basique] - - \tabex{interDD} - -\item[Horthocentre] - - \tabex% - [@{}m{.35\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.627\linewidth}@{}] - {orthocentre} - -\end{description} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Circle--Line} - -\defcom{pstInterLC}% - {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$O$}\Arg{$C$}% - \Arg{$M_1$}\Arg{$M_2$}} - -Draw the one or two intersection point(s) between the line $(AB)$ and -the circle of centre $O$ and with radius $OC$. - -The circle is specified with its center and either a point of its -circumference or with a radius specified with parameter \param{radius} -or its diameter specified with parameter \param{Diameter}. These two -parameters can be modify by coefficient \param{DistCoef}. - -\tabex - [@{}m{.4\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.5777\linewidth}@{}] - {interDC} - -In the case of a two points intersection, their position is not -specified, the user will have to check the result. It can arise that -after a modification the name of points can be swapped leading to a -wrong figure as shown in the following example: - -\hfill\rule[-\baselineskip]{0pt}{5cm+2\baselineskip} -\begin{pspicture}(5,5)\psgrid - \pstGeonode(2,2){O}\pstGeonode(3,1){A}% - \pstGeonode(1,2){B}\pstGeonode(1.1,4){C}% - \pstCircleOA{O}{A}% - \pstInterLC{B}{C}{O}{A}{D}{E} -\end{pspicture} -\hfill -\begin{pspicture}(5,5)\psgrid - \pstGeonode(2,2){O}\pstGeonode(3,1){A}% - \pstGeonode(1,2){B}\pstGeonode(.9,4){C}% - \pstCircleOA{O}{A}% - \pstInterLC{B}{C}{O}{A}{D}{E} -\end{pspicture} -\hspace*{\fill} - -This management is the same for the circle-circle intersection. - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Circle--Circle} - -\defcom{pstInterCC}% - {\OptArg{par}\Arg{$O_1$}\Arg{$B$}\Arg{$O_2$}\Arg{$C$}% - \Arg{$M_1$}\Arg{$M_2$}} - -This function is similar to the last one. The boolean parameters -\param{CodeFigA} et \param{CodeFigB} allow the drawing of the arcs -at the intersection. In order to get a coherence \param{CodeFig} allow -the drawing of both arcs. The boolean parameters \param{CodeFigAarc} and -\param{CodeFigBarc} specified the direction of these optional arcs: -trigonometric (by default) or clockwise. Here is a first example. - -\tabex{interCC} - -And a more complete one, which includes the special circle -specification using radius and diameter. For such specifications it -exists the parameters \param{RadiusA}, \param{RadiusB}, -\param{DiameterA} and \param{DiameterB}. - -\begin{center} - \rule[-.5cm]{0pt}{8cm} - \begin{pspicture}(-3,-4)(7,3)\psgrid - \input{Exemples/interCC_bis_in} - \end{pspicture} -\end{center} - -\verbatiminput{Exemples/interCC_bis_in} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Examples gallery} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Basic geometry} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Drawing of the bissector} - \nopagebreak[4] - -\tabex{gal_biss} - - -\cbstart - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{German right angle} - -In Germany, there is another convention for the right angle (remark -of U. Dirr). -\nopagebreak[4] - -\tabex{german_ra} - -\cbend - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Triangle lines} - -\begin{center} -\psset{unit=2cm} -\input{Exemples/remarq} -\end{center}\nopagebreak[4] - -\verbatiminput{Exemples/remarq_in} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Euler circle} - -\begin{center} -\psset{unit=2cm} -\input{Exemples/euler} -\end{center}\nopagebreak[4] - -\verbatiminput{Exemples/euler_in} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Orthocenter and hyperbola} - -The orthocenter of a triangle whose points are on the branches of the -hyperbola ${\mathscr H} : y=a/x$ belong to this hyperbola. -\nopagebreak[4] - -\begin{center} -\psset{unit=.5cm} -\input{Exemples/orthoethyper} -\end{center}\nopagebreak[4] - -\verbatiminput{Exemples/orthoethyper_in} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{17 sides regular polygon} - -Striking picture created by K. F. Gauss. -he also demonstrates that it is possible to build the polygons which -have $2^{2^p}+1$ sides, the following one has 257 sides! -\nopagebreak[4] - -%% Polygone à 17 côtés -\bgroup\centering -\psset{unit=1.5cm, CodeFig=true, RightAngleSize=.14, CodeFigColor=red, - CodeFigB=true, linestyle=dashed, dash=2mm 2mm} -\begin{pspicture}(-5.5,-5.5)(5.5,6)%\psgrid - \pstGeonode[PosAngle=-90](0,0){O}% - %% picture correct for x>0 and y<=0 for P_1!! - \pstGeonode[PosAngle=0](5,0){P_1}% - \pstCircleOA{O}{P_1}% - \pstSymO[PointSymbol=none, CodeFig=false]{O}{P_1}{PP_1}% - \ncline[linestyle=solid]{PP_1}{P_1}% - \pstRotation[RotAngle=90, PosAngle=90]{O}{P_1}{B}% - \pstRightAngle{B}{O}{PP_1}\ncline[linestyle=solid]{O}{B}% - \pstHomO[HomCoef=.25]{O}{B}{J}{J}% - \ncline{J}{P_1}% - \pstBissectBAC[PointSymbol=none]{O}{J}{P_1}{PE1}% - \pstBissectBAC[PointSymbol=none]{O}{J}{PE1}{PE2}% - \pstInterLL[PosAngle=-90]{O}{P_1}{J}{PE2}{E}% - \pstRotation[PosAngle=-90, RotAngle=-45, PointSymbol=none]{J}{E}{PF1}% - \pstInterLL[PosAngle=-90]{O}{P_1}{J}{PF1}{F}% - \pstMiddleAB[PointSymbol=none]{F}{P_1}{MFP1} - \pstCircleOA{MFP1}{P_1}% - \pstInterLC[PointSymbolB=none]{O}{B}{MFP1}{P_1}{K}{H}% - \pstCircleOA{E}{K}% - \pstInterLC{O}{P_1}{E}{K}{N_4}{N_6}% - \pstRotation[RotAngle=90, PointSymbol=none]{N_6}{E}{PP_6}{}% - \pstInterLC[PosAngleA=90, PosAngleB=-90, PointNameB=P_{13}]% - {N_6}{PP_6}{O}{P_1}{P_6}{P_13}% - \pstSegmentMark[SegmentSymbol=wedge]{N_6}{P_6}% - \pstSegmentMark[SegmentSymbol=wedge]{P_13}{N_6}% - \pstRotation[RotAngle=90, PointSymbol=none]{N_4}{E}{PP_4}{}% - \pstInterLC[PosAngleA=90, PosAngleB=-90, PointNameB=P_{15}]% - {N_4}{PP_4}{O}{P_1}{P_4}{P_15}% - \pstSegmentMark[SegmentSymbol=cup]{N_4}{P_4}% - \pstSegmentMark[SegmentSymbol=cup]{P_15}{N_4}% - \pstRightAngle{P_1}{N_6}{P_6}\pstRightAngle{P_1}{N_4}{P_4}% - \pstBissectBAC[PosAngle=90, linestyle=none]{P_4}{O}{P_6}{P_5}% - \pstOrtSym[PosAngle=-90, PointName=P_{14}]{O}{P_1}{P_5}{P_14}% - \pstInterCC[PosAngleB=90, PointSymbolA=none, - PointNameA=none]{O}{P_1}{P_4}{P_5}{H}{P_3}% - \pstOrtSym[PosAngle=-90, SegmentSymbol=pstslash, PointName=P_{16}]% - {O}{P_1}{P_3}{P_16}% - \pstInterCC[PosAngleB=90, PointSymbolA=none]{O}{P_1}{P_3}{P_4}{H}{P_2}% - \pstOrtSym[PosAngle=-90, SegmentSymbol=pstslashslashslash, - PointName=P_{17}]{O}{P_1}{P_2}{P_17}% - \pstInterCC[PosAngleA=90, PointSymbolB=none]{O}{P_1}{P_6}{P_5}{P_7}{H} - \pstOrtSym[PosAngle=-90, SegmentSymbol=circ, - PointName=P_{12}]{O}{P_1}{P_7}{P_12}% - \pstInterCC[PosAngleA=100, PointSymbolB=none]{O}{P_1}{P_7}{P_6}{P_8}{H} - \pstOrtSym[PosAngle=-100, SegmentSymbol=times, - PointName=P_{11}]{O}{P_1}{P_8}{P_11}% - \pstInterCC[PosAngleA=135, PointSymbolB=none]{O}{P_1}{P_8}{P_7}{P_9}{H} - \pstOrtSym[PosAngle=-135, SegmentSymbol=equiv, - PointName=P_{10}]{O}{P_1}{P_9}{P_10}% - \psline[linecolor=green, linestyle=solid]% - (P_1)(P_2)(P_3)(P_4)(P_5)(P_6)(P_7)(P_8)(P_9)% - (P_10)(P_11)(P_12)(P_13)(P_14)(P_15)(P_16)(P_17)(P_1)% -\end{pspicture} -\egroup - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Circles tangents} - -The drawing of the circle tangents which crosses a given point. -\nopagebreak[4] - -\begin{pspicture}(15,9)%\psgrid - %%% tangente à un cercle passant par un point - \pstGeonode(5, 5){O}% - \pstCircleOA[Radius=\pstDistVal{4}]{O}{}% - \pstGeonode(14,2){M}% - \pstMiddleAB[PointSymbol=none]{O}{M}{O'} - \pstInterCC[RadiusA=\pstDistVal{4}, DiameterB=\pstDistAB{O}{M}]% - {O}{}{O'}{}{A}{B}% - \psset{linecolor=red, linewidth=1.3pt, nodesep=-2} - \pstLineAB{M}{A}\pstLineAB{M}{B} -\end{pspicture} - -The drawing of the common tangent of two circles. -\nopagebreak[4] - -\begin{pspicture}(-2,0)(13,9)%\psgrid - %% tangente à deux cercles - \pstGeonode(9,3){O}\pstGeonode(3,6){O'}\psset{PointSymbol=none}% - \pstCircleOA[Radius=\pstDistVal{3}]{O}{}\pstCircleOA[Radius=\pstDistVal{1}]{O'}{} - \pstInterLC[Radius=\pstDistVal{3}]{O}{O'}{O}{}{M}{toto} - \pstInterLC[Radius=\pstDistVal{1}]{O}{O'}{O'}{}{M'}{toto} - \pstRotation[RotAngle=30]{O}{M}{N}% - \pstRotation[RotAngle=30]{O'}{M'}{N'}% - \pstInterLL[PointName=\Omega]{O}{O'}{N}{N'}{Omega} - \pstMiddleAB[PointSymbol=none]{O}{Omega}{I} - \pstInterCC{I}{O}{O}{M}{A}{B} - \psset{nodesepA=-1, nodesepB=-3, linecolor=blue, linewidth=1.3pt} - \pstLineAB[nodesep=-2]{A}{Omega}\pstLineAB[nodesep=-2]{B}{Omega} - \pstRotation[RotAngle=-150]{O'}{M'}{N''}% - \pstInterLL[PointName=\Omega']{O}{O'}{N}{N''}{Omega'} - \pstMiddleAB[PointSymbol=none]{O}{Omega'}{J} - \pstInterCC{J}{O}{O}{M}{A'}{B'} - \psset{nodesepA=-1, nodesepB=-3, linecolor=red} - \pstLineAB{A'}{Omega'}\pstLineAB{B'}{Omega'} -\end{pspicture} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Fermat's point} - -Drawing of Manuel Luque.\nopagebreak[4] - -\begin{pspicture}(-7,-6)(5,5)%\psgrid - \psset{PointSymbol=none} - \pstTriangle[PosAngleA=-160, PosAngleB=90, PosAngleC=-25]% - (-3,-2){B}(0,3){A}(2,-1){C}% - \psset{RotAngle=-60} - \pstRotation[PosAngle=-90]{B}{C}{A'} - \pstRotation{C}{A}{B'} - \pstRotation[PosAngle=160]{A}{B}{C'} - \pstLineAB{A}{B'} - \pstLineAB{C}{B'} - \pstLineAB{B}{A'} - \pstLineAB{C}{A'} - \pstLineAB{B}{C'} - \pstLineAB{A}{C'} - \pstCircleABC[linecolor=red]{A}{B}{C'}{O_1} - \pstCircleABC[linecolor=blue]{A}{C}{B'}{O_2} - \pstCircleABC[linecolor=Aquamarine]{A'}{C}{B}{O_3} - \pstInterCC[PointSymbolA=none]{O_1}{A}{O_2}{A}{E}{F} -\end{pspicture} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Escribed and inscribed circles of a triangle} - -%% cercles inscrit et exinscrits d'un triangle -\bgroup\centering -\psset{unit=1cm, dash=5mm 4mm, PointSymbolA=none, PointSymbolB=none} -\begin{pspicture}(-6,-5)(11,15)%\psgrid - \psframe(-6,-5)(11,15) - \pstTriangle[linewidth=2pt, PosAngleA=-75, PosAngleB=180, - PosAngleC=45, linecolor=red]% - (4,1){A}(0,3){B}(5,5){C}% - \psset{linecolor=blue} - \pstBissectBAC[PointSymbol=none]{C}{A}{B}{AB} - \pstBissectBAC[PointSymbol=none]{A}{B}{C}{BB} - \pstBissectBAC[PointSymbol=none]{B}{C}{A}{CB} - \pstInterLL{A}{AB}{B}{BB}{I} - \psset{linecolor=magenta, linestyle=dashed} - \pstProjection[PosAngle=-90]{A}{B}{I}{I_C} - \pstLineAB{I}{I_C}\pstRightAngle[linestyle=solid]{A}{I_C}{I} - \pstProjection{A}{C}{I}{I_B} - \pstLineAB{I}{I_B}\pstRightAngle[linestyle=solid]{C}{I_B}{I} - \pstProjection[PosAngle=80]{C}{B}{I}{I_A} - \pstLineAB{I}{IA}\pstRightAngle[linestyle=solid]{B}{I_A}{I} - \pstCircleOA[linecolor=yellow, linestyle=solid]{I}{I_A} - %% BISSECTRICES EXTÉRIEURES - \psset{linecolor=magenta, linestyle=none} - \pstOutBissectBAC[PointSymbol=none]{C}{A}{B}{AOB} - \pstOutBissectBAC[PointSymbol=none]{A}{B}{C}{BOB} - \pstOutBissectBAC[PointSymbol=none]{B}{C}{A}{COB} - \pstInterLL[PosAngle=-90]{A}{AOB}{B}{BOB}{I_1} - \pstInterLL{A}{AOB}{C}{COB}{I_2} - \pstInterLL[PosAngle=90]{C}{COB}{B}{BOB}{I_3} - \psset{linecolor=magenta, linestyle=dashed} - \pstProjection[PosAngle=50, PointName=I_{1C}]{A}{B}{I_1}{I1C} - \pstLineAB{I_1}{I1C}\pstRightAngle[linestyle=solid]{I_1}{I1C}{A} - \pstProjection[PointName=I_{1B}]{A}{C}{I_1}{I1B} - \pstLineAB{I_1}{I1B}\pstRightAngle[linestyle=solid]{A}{I1B}{I_1} - \pstProjection[PosAngle=110, PointName=I_{1A}]{C}{B}{I_1}{I1A} - \pstLineAB{I_1}{I1A}\pstRightAngle[linestyle=solid]{I_1}{I1A}{C} - \pstProjection[PointName=I_{2B}]{A}{C}{I_2}{I2B} - \pstLineAB{I_2}{I2B}\pstRightAngle[linestyle=solid]{A}{I2B}{I_2} - \pstProjection[PosAngle=-90, PointName=I_{2C}]{A}{B}{I_2}{I2C} - \pstLineAB{I_2}{I2C}\pstRightAngle[linestyle=solid]{I_2}{I2C}{A} - \pstProjection[PosAngle=90, PointName=I_{2A}]{B}{C}{I_2}{I2A} - \pstLineAB{I_2}{I2A}\pstRightAngle[linestyle=solid]{C}{I2A}{I_2} - \pstProjection[PosAngle=130, PointName=I_{3A}]{C}{B}{I_3}{I3A} - \pstLineAB{I_3}{I3A}\pstRightAngle[linestyle=solid]{C}{I3A}{I_3} - \pstProjection[PosAngle=-90, PointName=I_{3C}]{A}{B}{I_3}{I3C} - \pstLineAB{I_3}{I3C}\pstRightAngle[linestyle=solid]{A}{I3C}{I_3} - \pstProjection[PointName=I_{3B}]{C}{A}{I_3}{I3B} - \pstLineAB{I_3}{I3B}\pstRightAngle[linestyle=solid]{I_3}{I3B}{A} - \psset{linecolor=yellow, linestyle=solid} - \pstCircleOA{I_1}{I1C} - \pstCircleOA{I_2}{I2B} - \pstCircleOA{I_3}{I3A} - \psset{linecolor=red, linestyle=solid, nodesepA=-1, nodesepB=-1} - \pstLineAB{I1B}{I3B}\pstLineAB{I1A}{I2A}\pstLineAB{I2C}{I3C} -\end{pspicture} -\egroup - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Some locus points} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Parabola} - -\begin{minipage}[m]{.33\linewidth} -The parabola is the set of points which is at the same distance -between a point and a line. -\end{minipage} -\newcommand{\NbPt}{11} -\input{Exemples/parabole}\nopagebreak[4] - -\verbatiminput{Exemples/parabole_in} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Hyperbola} - -\begin{minipage}[m]{.33\linewidth} -The hyperbola is the set of points whose difference between their -distance of two points (the focus) is constant. -\end{minipage} -%% QQ DEFINITIONS -\newcommand{\Sommet}{1.4142135623}\newcommand{\PosFoyer}{2} -\newcommand{\HypAngle}{0} -\setcounter{i}{0}\newcounter{CoefDiv}\setcounter{CoefDiv}{20} -\newcounter{Inc}\setcounter{Inc}{1}\newcounter{n}\setcounter{n}{1} -\newcommand{\Ri}{% - \PosFoyer\space\Sommet\space% - sub \arabic{i}\space\arabic{CoefDiv}\space% - div add} -\newcommand{\Rii}{\Ri\space \Sommet\space 2 mul add .001 add} -\begin{pspicture}[.5](-4,-4)(4,4)%\psgrid - \pstGeonode[PosAngle=90](0,0){O}% - \pstGeonode(\PosFoyer;\HypAngle){F}% - \pstSymO[PosAngle=180]{O}{F}{F'}% - \pstLineAB{F}{F'} - %% TRACÉ DES ASYMPTOTES - %\psset{PointSymbol=none} - \pstCircleOA{O}{F} - \pstGeonode[PosAngle=-135](\Sommet;\HypAngle){S} - \pstRotation[RotAngle=90, PointSymbol=none]{S}{O}{B} - \pstInterLC[PosAngleA=90, PosAngleB=-90]{S}{B}{O}{F}{A_1}{A_2} - \pstLineAB[nodesepA=-3,nodesepB=-5]{A_1}{O} - \pstLineAB[nodesepA=-3,nodesepB=-5]{A_2}{O} - \pstMarkAngle[LabelSep=.8, MarkAngleRadius=.7, arrows=->]{F}{O}{A_1}{$\Psi$} - \ncline[linecolor=red]{A_1}{A_2} - \pstRightAngle[RightAngleSize=.15]{A_1}{S}{O} - \psset{PointName=none} - \whiledo{\value{n}<8}{% - \psset{RadiusA=\pstDistVal{\Ri},RadiusB=\pstDistVal{\Rii},PointSymbol=none} - \pstInterCC{F}{}{F'}{}{M\arabic{n}}{P\arabic{n}} - \pstInterCC{F'}{}{F}{}{M'\arabic{n}}{P'\arabic{n}} - \stepcounter{n}\addtocounter{i}{\value{Inc}}\addtocounter{Inc}{\value{Inc}} - } %% fin de whiledo - \psset{linecolor=blue} - \pstGenericCurve[GenCurvFirst=P1]{M}{1}{7}\pstGenericCurve{P}{1}{7} - \pstGenericCurve[GenCurvFirst=P'1]{M'}{1}{7}\pstGenericCurve{P'}{1}{7} -\end{pspicture} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Cycloid} - -The wheel rolls from $M$ to $A$. The circle points are on a -cycloids.\nopagebreak[4] - -\begin{center} -\input{Exemples/cyclo} -\end{center}\nopagebreak[4] - -\verbatiminput{Exemples/cyclo_in} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Hypocycloids (Astroid and Deltoid)} - -A wheel rolls inside a circle, and depending of the radius ratio, it -is an astroid, a deltoid and in the general case hypo-cycloids. -\nopagebreak[4] - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% ASTROIDE -\input{Exemples/hypocyclo} -%%%%%%%%%%%%%%%%%%%% -\begin{center} -\input{Exemples/astro}\input{Exemples/delto} -\end{center} - -\verbatiminput{Exemples/hypocyclo} -\verbatiminput{Exemples/astro_in} - -\cbstart - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Lines and circles envelope} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Conics} - -Let's consider a circle and a point $A$ not on the circle. The -set of all the mediator lines of segments defined by $A$ and the -circle points, create two conics depending of the position of $A$: - -\begin{itemize} -\item inside the circle: an hyperbola; -\item outside the circle: an ellipse. -\end{itemize} - -(figure of O. Reboux). - -\begin{center}\input{Exemples/envellipse}\end{center} - -\verbatiminput{Exemples/envellipse_in} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsubsection{Cardioid} - -The cardioid is defined by the circles centred on a circle and -crossing a given point. - -\begin{center}\input{Exemples/envcardi}\end{center} - -\verbatiminput{Exemples/envcardi_in} - -\cbend - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Homotethy and fractals} - -\tabex{fracthom} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{hyperbolic geometry: a triangle and its altitudes} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% Tracé de géodésique en géométrie hyperbolique -%% Attention ne fonctionne que si les points ne sont pas alignés avec O -%% Ceci est un cas particulier, je ne crois pas que les hauteurs -%% soient concourantes pour tous les triangles hyperboliques. -\begin{pspicture}(-5,-5)(5,5) - \psclip{\pscircle(0,0){4}}%\psgrid - %\newlength{\radius}\setlength{\radius}{0cm} - %\newcounter{rapport}\setcounter{i}{1} - %\whiledo{\value{i}<100}{ - % \setlength{\radius}{4cm*\value{i}} - % \setcounter{rapport}{\value{i}+1} - % \divide\radius by \arabic{rapport} - % \pscircle[linestyle=dotted, linecolor=gray]% - % (0, 0){\radius} - % \setcounter{i}{\value{i}*2} - % } - \pstGeonode(1, 2){M}\pstGeonode(-2,2){N}\pstGeonode(0,-2){P}% - \psset{DrawCirABC=false, PointSymbol=none}% - \pstGeonode(0,0){O}\pstGeonode(4,0){A}\pstCircleOA{O}{A}% - \pstHomO[HomCoef=\pstDistAB{O}{A} 2 mul \pstDistAB{O}{M} sub - \pstDistAB{O}{M} div]{O}{M}{M'}% - \pstHomO[HomCoef=\pstDistAB{O}{A} 2 mul \pstDistAB{O}{P} sub - \pstDistAB{O}{P} div]{O}{P}{P'}% - \pstHomO[HomCoef=\pstDistAB{O}{A} 2 mul \pstDistAB{O}{N} sub - \pstDistAB{O}{N} div]{O}{N}{N'}% - \psset{linecolor=green, linewidth=1.5pt}% - \pstCircleABC{M}{N}{M'}{OmegaMN}\pstArcOAB{OmegaMN}{N}{M}% - \pstCircleABC{M}{P}{M'}{OmegaMP}\pstArcOAB{OmegaMP}{M}{P}% - \pstCircleABC{N}{P}{P'}{OmegaNP}\pstArcOAB{OmegaNP}{P}{N}% - \psset{linecolor=blue} - %% la hauteur issue de M - \pstHomO[HomCoef=\pstDistAB{OmegaNP}{N} 2 mul \pstDistAB{OmegaNP}{M} sub - \pstDistAB{OmegaNP}{M} div]{OmegaNP}{M}{MH'} - \pstCircleABC{M}{M'}{MH'}{OmegaMH}\pstArcOAB{OmegaMH}{MH'}{M} - %% la hauteur issue de N - \pstHomO[HomCoef=\pstDistAB{OmegaMP}{M} 2 mul \pstDistAB{OmegaMP}{N} sub - \pstDistAB{OmegaMP}{N} div]{OmegaMP}{N}{NH'} - \pstCircleABC{N}{N'}{NH'}{OmegaNH}\pstArcOAB{OmegaNH}{N}{NH'} - %% la hauteur issue de P - \pstHomO[HomCoef=\pstDistAB{OmegaMN}{M} 2 mul \pstDistAB{OmegaMN}{P} sub - \pstDistAB{OmegaMN}{P} div]{OmegaMN}{P}{PH'} - \pstCircleABC{P}{P'}{PH'}{OmegaPH}\pstArcOAB{OmegaPH}{P}{PH'} - \endpsclip -\end{pspicture} - -\end{document} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \subsection{Une figure fractale : le flocon de \textsc{von Koch}} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% TENTATIVE DE FLOCON DE VON KOCH -\newcounter{nbvk}% -%%%%%%%%%%%%% -%% PILE LIFO% -\newcounter{lifon}\setcounter{lifon}{1}%% -\newcommand{\Push}[1]{%% - \expandafter\edef\csname lifocmd\roman{lifon}\endcsname{#1}%% - \addtocounter{lifon}{1}%% - }% -\newcommand{\Pop}{\csname lifocmd\roman{lifon}\endcsname}% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\newcounter{nombre}\setcounter{nombre}{0}% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\newcommand{\vonkoch}[3]{%% - \setcounter{nbvk}{#1}%% - \edef\Pointi{#2}\edef\Pointii{#3}%% - \ifnum\value{nbvk}=1% - \ncline{\Pointi}{\Pointii}% - \else % ELSE% - \addtocounter{nombre}{1}% - %% position des nouveauX points% - \pstHomO[HomCoef=1 3 div]{\Pointi}{\Pointii}{P\arabic{nombre}1}% - \pstHomO[HomCoef=2 3 div]{\Pointi}{\Pointii}{P\arabic{nombre}2}% - \pstRotation[RotAngle=60]{P\arabic{nombre}1}{P\arabic{nombre}2}{P\arabic{nombre}3}% - \addtocounter{nbvk}{-1}%% - \Push{\Pointi}\Push{\Pointii}\Push{\arabic{nombre}}\Push{\arabic{nbvk}}% - \vonkoch{\value{nbvk}}{\Pointi}{P\arabic{nombre}1}% - \addtocounter{lifon}{-1}\edef\nbvklocal{\Pop}% - \addtocounter{lifon}{-1}\edef\nombrelocal{\Pop}% - \addtocounter{lifon}{-1}\edef\Pointii{\Pop}% - \addtocounter{lifon}{-1}\edef\Pointi{\Pop}% - \addtocounter{lifon}{4}% - \vonkoch{\nbvklocal}{P\nombrelocal1}{P\nombrelocal3}% - \addtocounter{lifon}{-1}\edef\nbvklocal{\Pop}% - \addtocounter{lifon}{-1}\edef\nombrelocal{\Pop}% - \addtocounter{lifon}{-1}\edef\Pointii{\Pop}% - \addtocounter{lifon}{-1}\edef\Pointi{\Pop}% - \addtocounter{lifon}{4}% - \vonkoch{\nbvklocal}{P\nombrelocal3}{P\nombrelocal2}% - \addtocounter{lifon}{-1}\edef\nbvklocal{\Pop}% - \addtocounter{lifon}{-1}\edef\nombrelocal{\Pop}% - \addtocounter{lifon}{-1}\edef\Pointii{\Pop}% - \addtocounter{lifon}{-1}\edef\Pointi{\Pop}% - \addtocounter{lifon}{4}% - \vonkoch{\nbvklocal}{P\nombrelocal2}{\Pointii}% - \addtocounter{lifon}{-4}% - \fi% -}% -\psset{unit=2.5cm}% -\begin{pspicture}(-2,-2)(2,2)% - \psset{PointSymbol=none} - \pstGeonode(-2,-2){A}\pstGeonode(2,2){B}% - \pstRotation[RotAngle=60]{B}{A}{C}% - \vonkoch{5}{A}{B}\setcounter{nombre}{0}%%% - \vonkoch{4}{B}{C}\setcounter{nombre}{0}%%% - \vonkoch{4}{C}{A}%% -\end{pspicture}% -% -\end{document}% - -\begin{pspicture*}(-4,-5)(4,5)%\psgrid - \pstGeonode[PosAngle=-135](0,0){O} - \pstGeonode[PosAngle=-90](1,0.5){I} - \pstGeonode[PosAngle=-180](0.5,2){J} - \pstLineAB[nodesep=100]{O}{I} - \pstLineAB[nodesep=100]{O}{J} - \multips(-5,-2.5)(1,0.5){11}{\psline(0,-.15)(0,.15)}% - \multips(-5,-2.5)(1,0.5){11}{\psline[linestyle=dotted](-10,-40)(10,40)}% - \multips(-2,-8)(0.5,2){9}{\psline(-.15,0)(.15,0)}% - \multips(-2,-8)(0.5,2){9}{\psline[linestyle=dotted](-10,-5)(10,5)}% - %% huit points - \psset{PointSymbol=x} - \pstOIJGeonode(1,2){A}{O}{I}{J} - \pstOIJGeonode(-2,1){B}{O}{I}{J} - \pstOIJGeonode(-1,-1.5){C}{O}{I}{J} - \pstOIJGeonode(2,-1){D}{O}{I}{J} -\end{pspicture*} -% |