From 9dbbc9f2ead619557514c028316fcc130580cabf Mon Sep 17 00:00:00 2001 From: =?utf8?q?J=C3=BCrgen=20Gilg?= Date: Tue, 18 Oct 2011 23:44:19 +0200 Subject: [PATCH] Test subrepertoire en pst-anamorphosis, seulement pour tester, si j'ai SUCCESS... Keep your fingers crossed. All modified files of the documentation are in it... --- GILGTEST/eiffel.eps | 2715 +++++++++++++++++++++++ GILGTEST/pst-anamorphosis-doc-part1.pdf | Bin 0 -> 474646 bytes GILGTEST/pst-anamorphosis-doc-part1.tex | 1052 +++++++++ GILGTEST/pst-anamorphosis-doc-part2.pdf | Bin 0 -> 896879 bytes GILGTEST/pst-anamorphosis-doc-part2.tex | 623 ++++++ GILGTEST/pst-anamorphosis.eps | 912 ++++++++ 6 files changed, 5302 insertions(+) create mode 100644 GILGTEST/eiffel.eps create mode 100644 GILGTEST/pst-anamorphosis-doc-part1.pdf create mode 100644 GILGTEST/pst-anamorphosis-doc-part1.tex create mode 100644 GILGTEST/pst-anamorphosis-doc-part2.pdf create mode 100644 GILGTEST/pst-anamorphosis-doc-part2.tex create mode 100644 GILGTEST/pst-anamorphosis.eps diff --git a/GILGTEST/eiffel.eps b/GILGTEST/eiffel.eps new file mode 100644 index 0000000..d8a6e82 --- /dev/null +++ b/GILGTEST/eiffel.eps @@ -0,0 +1,2715 @@ +%!PS-Adobe-3.0 +%%Title: flattened PostScript generated from file: effel4.eps +%%Creator: 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+\def\epsRoot{C:/Dokumente und Einstellungen/Besitzer/Desktop/bergen/bergen/} + +\renewcommand{\ttdefault}{lmtt} + +\definecolor{syracuseGRIS}{HTML}{C1C1C1} +\definecolor{syracuseVERT}{HTML}{029235} + +\definecolor{sepia}{rgb}{1,0.8,0.8} +\definecolor{grisclair}{rgb}{0.8,0.8,0.8} +\definecolor{BleuCiel}{cmyk}{0.2,0,0,0} +\definecolor{OrangePale}{cmyk}{0,0.2,0.4,0} + +\lstset{% + language=[LaTeX]TeX,% + float=hbp,% + basicstyle=\ttfamily\small, % + texcsstyle=*\color{blue},% + identifierstyle=\color{black}, % + keywordstyle=\color{syracuseVERT}, % + otherkeywords={$, \{, \}, \[, \]}, + stringstyle=\color{syracuseVERT}, % + commentstyle=\color{syracuseVERT}, % + backgroundcolor=\color{syracuseGRIS!30},% + columns=flexible, % + tabsize=4, % + frame=single, % + %frame=shadowbox, % + %rulesepcolor=\color{syracuseGRIS!30},% + extendedchars=true, % + showspaces=false, % + showstringspaces=false, % + numbers=left, + numbersep=0.8em, + numberstyle=\tiny, % + breaklines=true, % + breakautoindent=true,% + captionpos=b,% + xleftmargin=1em,% + sensitive=true,% + morekeywords=[6]{pspicture,center},% + keywordstyle=[6]\color{FireBrick},% + %morekeywords=[7]{(,)},% + %keywordstyle=[7]\color{syracuseVERT} +} + +\renewcommand{\lstlistingname}{Source} +\renewcommand{\labelitemi}{$\bullet$} + + +\newcommand\cs[1]{\texttt{\char`\\#1}} +\newcommand\file[1]{\texttt{#1}} + +\parindent0pt +\parskip5pt + +\def\syracuseTitle{Les anamorphoses : pr\'{e}sentation th\'{e}orique} +%\def\syracuseGraphic{eiffel2} + + +\begin{document} + +%% === BEGIN == Page de garde ================================================= + +\thispagestyle{empty} + +\pstPutAbs(0,-29.7){% +\begin{pspicture}(0,0)(21,29.7) +\pspolygon[fillstyle=solid,linecolor=syracuseVERT,fillcolor=syracuseVERT](0,0)(10.5,14.85)(21,0) +\pspolygon[fillstyle=solid,linecolor=syracuseGRIS,fillcolor=syracuseGRIS](0,0)(21,29.7)(0,29.7) +\pspolygon[fillstyle=solid,linecolor=syracuseGRIS!50,fillcolor=syracuseGRIS!50](21,0)(10.5,14.85)(21,29.7) +\end{pspicture} +} +\pstPutAbs(2.5,-3.75){% +\includegraphics[scale=1]{pst-anamorphosis} +} +\pstPutAbs(2.5,-5.25){% +\LARGE \textbf{\syracuseTitle} +} +\pstPutAbs(2.5,-13.5){% +\begin{pspicture}(0,0)(8,8) +\rput(4,4){\includegraphics[height=8cm]{eiffel}} +%\rput(4,4){\includegraphics[height=8cm]{\syracuseGraphic}} +%\psframe(0,0)(8,8) +\end{pspicture} +} +\pstPutAbs(12.5,-15){% +\parbox{0.4\textwidth}{\Large\raggedleft + {\LARGE\textbf{Contributeurs}}\\[0.2cm] + J\"{u}rgen \textsc{Gilg}\\ + Manuel \textsc{Luque}\\ + Jean-Michel \textsc{Sarlat} +}} +\vfill +\begin{center} +\textcolor{white}{\textbf{\today}}\\[0.3cm] +\textcolor{white}{\url{http://melusine.eu.org/syracuse/G/pstricks/}}\\ +\includegraphics[scale=0.4]{logo_syracuse} +\end{center} + +%% == END == Page de garde ==================================================== + +\newpage + +\section{L'anamorphose cylindrique} + +On place \`{a} l'int\'{e}rieur du cylindre l'image telle qu'elle doit \^{e}tre vue par un observateur regardant dans le miroir cylindrique (on peut la placer \`{a} l'ext\'{e}rieur, mais il faut que les rayons lumineux rencontrent toujours le cylindre, il faut donc veiller aux dimensions). L'objet anamorphique est <<~l'objet d\'{e}form\'{e}~>> dont le miroir reconstituera les proportions r\'{e}elles. + +Objet et image ob\'{e}issent aux lois de la r\'{e}flexion de l'optique g\'{e}om\'{e}trique : +\begin{itemize} + \item rayon incident et rayon r\'{e}fl\'{e}chi appartiennent \`{a} un m\^{e}me plan ; + \item rayon incident et rayon r\'{e}fl\'{e}chi sont sym\'{e}triques par rapport \`{a} la normale au miroir au point d'incidence. +\end{itemize} + +\def\oeil{\psarc[linewidth=2pt](0,2.5){2.5}{215}{270}% +\psarc[linewidth=2pt](0,-2.5){2.5}{90}{140}% +\psarc(-2.5,0){1}{-30}{30}% +\psarc(0,0){1.75}{160}{200} +\psclip{% +\pscircle[linestyle=none](0,0){1.75}} +\pscircle[fillstyle=solid,fillcolor=lightgray](-2.5,0){0.9} +\endpsclip}% +\begin{center} +\psscalebox{0.7}{ +\begin{pspicture}(-2,-6)(15,6) +\psset{Xv=0,Yv=-100} +\newcommand\Rmirror{3} +\psset{viewpoint=-1 -1 1} +\ThreeDput[normal=0 0 1](0,0,0){% +\psaxes{->}(8,4) +\pscircle[doubleline=true]{\Rmirror} +\multido{\i=-2+1,\I=-2+1}{5}{% + \pnode(! \i\space -2){A} + \pnode(! \i\space 2){B} + \psline(A)(B) + \pslineA(A)(B) + \pnode(!-2 \I){A} + \pnode(!2 \I){B} + \pslineA(A)(B) + \psline(A)(B) + } +\input{LouisXIII.pst} +\input{ALouisXIII.pst} +\pstextA[fontsize=28,linecolor=red,fillstyle=solid,fillcolor=yellow!50](0,2){LouisXIII} + } +\FPmul{\largeur}{\Rmirror}{0.1745} +\newcount\n \n=135 + \loop + \FPmul{\tempa}{\FPpi}{\the\n} + \FPdiv{\angle}{\tempa}{180} + \FPsin{\SIN}{\angle}\FPcos{\COS}{\angle} + \FPmul{\xc}{\COS}{\Rmirror} + \FPmul{\yc}{\SIN}{\Rmirror} + \ThreeDput[normal={\COS} {\SIN} 0.0001](\xc,\yc,0){% + \psframe[fillstyle=vlines,hatchangle=0,linestyle=none,hatchcolor=BleuCiel,](0,0)(\largeur,5)} + \ifnum\n<315 \advance\n by 10 +\repeat +\ThreeDput[normal=0.707 -0.707 0.001](2.121,-2.121,0){% +\psline[linewidth=2\pslinewidth](0,5)} +\ThreeDput[normal=0.707 -0.707 0.001](-2.121,2.121,0){% +\psline[linewidth=2\pslinewidth](0,5)} +\ThreeDput[normal=0 0 1](0,0,5){% +\pscircle[doubleline=true]{\Rmirror} +} +\ThreeDput[normal=-1 0 0](0,0,0){% +\psaxes(15,5) +\rput{26.5}(21.5,10.8){\oeil} +\psline[linestyle=dashed,linecolor=blue](20,10)% +\psset{arrowsize=0.5,linecolor=red,linewidth=2\pslinewidth} +\psline{->}(3,1.5)(10,5) +\psline(3,1.5)(20,10) +\psline{->}(6,0)(4.5,0.75) +\psline(6,0)(3,1.5) +\uput[ur](20,10){\color{red}{$V$}} +\uput[u](3,1.5){\color{red}{$I$}} +\uput[u](6,0){\color{red}{$P'$}} +} +\end{pspicture} +} +\end{center} +L'image non d\'{e}form\'{e}e (celle qui est vue dans le miroir) est plac\'{e}e, dans cet exemple, au centre du miroir. Un rayon incident partant de l'objet anamorphique se r\'{e}fl\'{e}chit sur le miroir et apr\`{e}s r\'{e}flexion parvient \`{a} l'{\oe}il de notre observateur. L'observateur a l'illusion que le rayon provient du point image. Il faut donc reconstruire math\'{e}matiquement la marche d'un tel rayon lumineux en partant de l'image dans le miroir. + +L'observateur est suffisamment \'{e}loign\'{e} du miroir pour pouvoir \^{e}tre consid\'{e}r\'{e} comme ponctuel. + +Soit $P$ un point de l'image(not\'{e} $A'$ dans le sch\'{e}ma ci-apr\`{e}s), $V$ l'{\oe}il de l'observateur. Tra\c{c}ons un droite $(PV)$ et d\'{e}terminons le point d'intersection $I$ avec le cylindre : c'est le point d'incidence. +\[ +V(x_V,y_V,z_V)\quad\text{ et }\quad P(x_P,y_P,0) +\] +L'\'{e}quation param\'{e}trique de la droite $(PV)$ s'\'{e}crit $\overrightarrow{IV}=\rho\overrightarrow{PV}$: +\begin{equation}\label{eq:paracyl} +\left\lbrace + \begin{array}{lcl} + x_V-x_I&=&\rho(x_V-x_P)\\ + y_V-y_I&=&\rho(y_V-y_P)\\ + z_V-z_I&=&\rho(z_V-0) + \end{array} + \right. + \Longrightarrow + \left\lbrace + \begin{array}{lcl} + x_I&=&x_V(1-\rho)+\rho x_P\\ + y_I&=&y_V(1-\rho)+\rho y_P\\ + z_I&=&z_V(1-\rho) + \end{array} + \right. +\end{equation} +\begin{center} +\begin{pspicture}(-4,-0.5)(6.5,8) +\pnode(6,7){V} +\uput[0](V){$V$} +\pnode(3,6.5){S} +\pnode(-3,6.5){S'} +\pnode(3,2.8){I} +\pnode(1,0){P} +\pnode(5,0){P'} +\uput[45](P'){$P'$} +\pnode(3,0){G} +\pnode(-3,0){G'} +\pspolygon[fillstyle=solid,fillcolor=lightgray!30](G')(G)(S)(S') +\psaxes[ticks=none,labels=none]{->}(0,0)(-4,0)(6.5,7.5) +\uput[0](6.5,0){$x$} +\uput[90](0,7.5){$z$} +\uput[135](P){$P$} +\uput[-45](G){$G$} +\uput[135](I){$I$} +\psline(V)(P) +\psline(S)(G) +\rput(I){% +\psline[linestyle=dashed](3;0)% +\uput[0](3;0){$N$}% +\psarc[doubleline=true](0,0){1}{-90}{-54.46}% +\psarc[doubleline=true](0,0){1}{54.46}{90} +\uput[72](1.2;72){$\varepsilon$}% +\uput[-72](1.2;-72){$\varepsilon$}% +\rput{-90}(0,0){\psframe(0.3,0.3)} +} +\psline[linecolor=red](V)(I)(P') +\pcline[nodesepB=1.5,linecolor=red,arrowsize=0.175,arrowinset=0.075]{->>}(P')(I) +\pcline[nodesepB=2.5,linecolor=red,arrowsize=0.175,arrowinset=0.075]{->>}(I)(V) +\qdisk(I){2pt} +\qdisk(P){2pt} +\qdisk(V){2pt} +\qdisk(P'){2pt} +\qdisk(G){2pt} +\psline[linewidth=0.5\pslinewidth]{|<->|}(0,-0.2)(1,-0.2) +\uput[-90](0.5,-0.2){$r_P$} +\psline[linewidth=0.5\pslinewidth]{<->}(0,2.8)(3,2.8) +\uput[-90](1.5,2.8){$r_I=R$} +\end{pspicture} +\end{center} +Le point $I$ appartenant au cylindre, ses coordonn\'{e}es v\'{e}rifient la relation : +\begin{equation}\label{eq:cylindre} +x_I^2+y_I^2=R^2 +\end{equation} +(\ref{eq:paracyl}) en (\ref{eq:cylindre}) et apr\`{e}s d\'{e}veloppement, on obtient l'\'{e}quation du second degr\'{e} en $\rho$: +\begin{gather*} +\left(x_V(1-\rho)+\rho x_P\right)^2+\left(y_V(1-\rho)+\rho y_P\right)^2=R^2\\ +x_V^2(1-2\rho+\rho^2)+2(1-\rho)\rho x_Vx_P+\rho^2 x_P^2+y_V^2(1-2\rho+\rho^2)+2(1-\rho)\rho y_Vy_P+\rho^2 y_P^2=R^2\\ +(x_V^2+y_V^2-2x_Vx_P-2y_Vy_P)\rho^2+2(-x_V^2+x_Vx_P-y_V^2+y_Vy_P)\rho+x_V^2+y_V^2=R^2 +\end{gather*} +Comparaison avec +\[ +a\rho^2+2b'\rho+c=0 +\] +donne : +\[ +\left\lbrace + \begin{array}{lcl} + a&=&(x_V-x_P)^2+(y_V-y_P)^2\\ + 2b'&=&x_Vx_P+y_Vy_P-x_V^2-y_V^2\\ + c&=&x_V^2+y_V^2-R^2 + \end{array} + \right. +\] +La r\'{e}solution de cette \'{e}quation nous donne les solutions classiques: +\[ +\left\lbrace + \begin{array}{lcl} + \rho'&=&\dfrac{-b'+\sqrt{\Delta'}}{a}\\[0.5cm] + \rho''&=&\dfrac{-b'-\sqrt{\Delta'}}{a} + \end{array} + \right. + \qquad \Delta'=b'^2-ac +\] +On retiendra la plus petite valeur positive des deux, que par la suite j'appelle $\rho$. + +$(IV)$ repr\'{e}sente le rayon r\'{e}fl\'{e}chi par le miroir. Le rayon incident est d\'{e}fini par la droite sym\'{e}trique de $(IV)$ par rapport \`{a} la normale au miroir en $I$. Je cherche le sym\'{e}trique de $V$, nomm\'{e} $V'$ par rapport \`{a} cette normale $(IN)$. Ce point $V'$ remplit deux conditions : +\begin{enumerate} + \item $\overrightarrow{IV}+\overrightarrow{IV'}=k\overrightarrow{IN}$ + \item $\overrightarrow{VV'}.\overrightarrow{IN}=0$ +\end{enumerate} +La normale $(IN)$ a pour vecteur directeur $\overrightarrow{IN}(x_I,y_I,0)$ + +La premi\`{e}re condition se traduit par : +\[ +\left\lbrace + \begin{array}{lcl} + x_V-x_I+x_{V'}-x_I&=&kx_I\\ + y_V-x_I+y_{V'}-y_I&=&ky_I\\ + z_V-z_I+z_{V'}-z_I&=&0 + \end{array} + \right. + \Longrightarrow + \left\lbrace + \begin{array}{lcl} + x_{V'}&=&kx_I+2x_I-x_V\\ + y_{V'}&=&ky_I+2y_I-y_V\\ + z_{V'}&=&2z_I-z_V + \end{array} + \right. +\] +La deuxi\`{e}me par : +\[ +(x_{V'}-x_V)x_I+(y_{V'}-y_V)y_I=0 +\] +En rempla\c{c}ant $x_{V'}$ et $y_{V'}$ tir\'{e}s de la premi\`{e}re condition dans la deuxi\`{e}me : +\begin{gather*} +k(x_I^2+y_I^2)+2x_I^2-2x_Vx_I+2y_I^2-2y_Vy_I=0\\ +kR^2+2R^2=2(x_Vx_I+y_Vy_I)\\ +k+2=\dfrac{2}{R^2}(x_Vx_I+y_Vy_I) +\end{gather*} +Les coordonn\'{e}es de $V'$ s'en d\'{e}duisent : +\[ +\left\lbrace + \begin{array}{lcl} + x_{V'}&=&(k+2)x_I-x_V\\ + y_{V'}&=&(k+2)y_I-y_V\\ + z_{V'}&=&z_V(1-2\rho) + \end{array} + \right. +\] +Il reste \`{a} trouver l'intersection de $(IV')$ avec le plan horizontal $z=0$. + +\'Equation param\'{e}trique de $(IV')$, $M$ \'{e}tant un point courant : $\overrightarrow{MV'}=\alpha\overrightarrow{IV'}$ +\[ +\left\lbrace + \begin{array}{lcl} + x_{V'}-x&=&\alpha(x_{V'}-x_I)\\ + y_{V'}-y&=&\alpha(y_{V'}-y_I)\\ + z_{V'}-z&=&\alpha(z_{V'}-z_I) + \end{array} + \right. +\] +$z=0\Longrightarrow \alpha=\dfrac{z_{V'}}{z_{V'}-z_I}$ soit +\[ +\alpha=\dfrac{1-2\rho}{-\rho} +\] +En rempla\c{c}ant $\alpha$ par son expression, nous obtenons les coordonn\'{e}es du point $P'$ de l'objet anamorphique. +\[ +\left\lbrace + \begin{array}{lcl} + x_{P'}&=&x_{V'}-\alpha(x_{V'}-x_I)\\ + y_{P'}&=&y_{V'}-\alpha(y_{V'}-y_I) + \end{array} + \right. +\] +Cette s\'{e}rie de calculs doit \^{e}tre appliqu\'{e}e \`{a} tous les points de l'image <<~normale~>> afin d'obtenir l'objet anamorphique (d\'{e}form\'{e}) dont le miroir <<~redressera~>> la forme. + +On notera que \textit{la cote de l'observateur $z_V$ n'intervient pas}. On le comprend ais\'{e}ment en faisant un dessin du plan vertical passant par l'\oe{}il et l'axe du cylindre miroir. La position de la projection horizontale \'{e}tant fix\'{e}e $(x_V,y_V)$, quelle que soit la valeur de $z_V$, $A$ et $A'$ \'{e}tant sym\'{e}triques par rapport \`{a} la g\'{e}n\'{e}ratrice du miroir appartenant au plan vertical choisi, si $A'$ est donn\'{e} alors $A$ est fix\'{e} quelque soit~$z_V$. + +\newpage + +\section{L'anamorphose conique} + +Le principe est identique \`{a} celui de l'anamorphose cylindrique : imaginons un rayon lumineux provenant de l'objet <<~anamorphique~>>, se r\'{e}fl\'{e}chissant sur le miroir conique et parvenant \`{a} l'{\oe}il de l'observateur plac\'{e} au-dessus et dans l'axe du c\^{o}ne \`{a} une position suffisamment haute pour que l'observateur puisse \^{e}tre consid\'{e}r\'{e} comme ponctuel. Ainsi l'observateur aura l'illusion d'observer l'image reconstitu\'{e}e par le miroir conique. Image et objet sont dans le plan horizontal. + +\begin{center} +\psscalebox{0.7}{ +\begin{pspicture}(-7,-8)(7,9) +\newcommand\R{3} +\newcommand\hauteur{5} +\newcommand\pas{10} +\FPeval{\pasrad}{\pas*\FPpi/180} + \FPmul{\Largeur}{\R}{\pasrad} + \FPdiv{\largeur}{\Largeur}{2} +%Les coordonn\'{e}es de viewpoint +\def\vx{1}\def\vy{-0.8}\def\vz{1} +\FPeval{\RH}{(\R)/(\hauteur)} + \FParctan{\phy}{\RH} + \FPeval{\OH}{sin(\phy)*\hauteur} + \FPeval{\zH}{sin(\phy)*\OH} + \FPeval{\OK}{cos(\phy)*\OH} + \FPeval{\generatrice}{root(2,(\hauteur*\hauteur)+(\R*\R))} +\psset{viewpoint={\vx} {\vy} {\vz}} +\ThreeDput[normal=0 0 1](0,0,0){%% +\psframe*[linecolor=lightgray!25](-10,-10)(10,10) +\psgrid[subgriddiv=0,gridcolor=lightgray,gridlabels=0pt](-10,-10)(10,10) +\psaxes(0,0)(-10,-10)(10,10) +\pscircle[doubleline=true]{3} +\psset{type=conical} +\multido{\n=-1.50+0.50}{7}{% + \pnode(! \n\space -2.50){A} + \pnode(! \n\space -0.50){B} + \psline[linecolor=green](A)(B) + \pslineA[linecolor=green](A)(B) + } +\multido{\N=-2.50+0.50}{5}{% + \pnode(!-1.50 \N){A} + \pnode(!1.50 \N){B} + \pslineA[linecolor=green](A)(B) + \psline[linecolor=green](A)(B) + } +% +\multido{\n=-1.50+0.50}{7}{% + \pnode(! \n\space 2.50){A} + \pnode(! \n\space 0.50){B} + \psline[linecolor=green](A)(B) + \pslineA[linecolor=green](A)(B) + } +\multido{\N=2.50+-0.50}{5}{% + \pnode(!-1.50 \N){A} + \pnode(!1.50 \N){B} + \pslineA[linecolor=green](A)(B) + \psline[linecolor=green](A)(B) + } +\multido{\n=-1.50+0.50}{7}{% + \pnode(! \n\space -2.50){A} + \pnode(! \n\space -0.50){B} + \psline(A)(B) + \pslineA(A)(B) + } +\multido{\N=-2.50+0.50}{5}{% + \pnode(!-1.50 \N){A} + \pnode(!1.50 \N){B} + \pslineA(A)(B) + \psline(A)(B) + } +\multido{\n=-1.50+0.50}{7}{% + \pnode(! \n\space 2.50){A} + \pnode(! \n\space 0.50){B} + \psline(A)(B) + \pslineA(A)(B) + } +\multido{\N=2.50+-0.50}{5}{% + \pnode(!-1.50 \N){A} + \pnode(!1.50 \N){B} + \pslineA(A)(B) + \psline(A)(B) + } +\psanamorphosis[type=conical,scale=-0.7 -0.7](0,1.5){tiger.eps} +\pstextA[fontsize=15,fillcolor=green,scale=1 -1](0,-0.5){Anamorphose} +\pscircle[doubleline=true]{3}} +\newcount\n \n=0 + \loop + \FPmul{\tempa}{\FPpi}{\the\n} + \FPdiv{\angle}{\tempa}{180} + \FPsin{\SIN}{\angle}\FPcos{\COS}{\angle} + \FPmul{\xH}{\COS}{\OK} + \FPmul{\yH}{\SIN}{\OK} + \FPdiv{\grise}{\the\n}{360} + \FPsub{\gris}{1}{\grise} + \FPdiv{\teinte}{\grise}{2} +\definecolor{gris}{cmyk}{\teinte,\teinte,\teinte,0} +%tester les faces visibles +%le produit scalaire du vecteur viewpoint et du vecteur normal \`{a} la face >0 ? +\FPeval{\costest}{(\vx)*\xH+(\vy)*\yH+(\vz)*\zH} + \FPifpos{\costest} + \ThreeDput[normal={\xH} {\yH} \zH](0,0,\hauteur){% + \pspolygon[linecolor=BleuCiel]%[fillstyle=solid,fillcolor=gris,dimen=outer]% + (0,0)(-\largeur,-\generatrice)(\largeur,-\generatrice)}\else{}\fi + \FPifneg{\costest} + \ThreeDput[normal={\xH} {\yH} \zH](0,0,\hauteur){% + \psline[linestyle=dashed,linecolor=gray](0,0)(\largeur,-\generatrice)}\else{}\fi + \ifnum\n<360 \advance\n by \pas +\repeat +%Dessin des faces de dessus et de dessous +%\FPifneg\vz +%face de dessous +%\ThreeDput[normal=0 0 -1](0,0,0){% +%\pscircle[fillstyle=solid,fillcolor=lightgray](0,0){\R}\rput(0,0){\Large\textbf{\textsf{Dessous}}}}\fi +\ThreeDput[normal=1 0 0](0,0,0){% +\psaxes(1,10) +\rput{90}(0,12){\oeil}% +{\boldmath +\red +\pnode(0,10){V} +\uput[0](V){$V$} +\pnode(0,5){S} +\uput[0](S){$S$} +\pnode(-1.5,2.5){I} +\uput[180](I){$I$} +\qdisk(I){2pt} +\pnode(-2,0){P} +\uput[135](P){$P$} +\pnode(-4.545,0){P'} +\uput[135](P'){$P'$} +\psset{linecolor=red} +\psline(V)(I)(P') +\pcline[nodesepB=2,nodesepA=1,linecolor=red,arrowsize=0.175,arrowinset=0.075]{->}(P')(I) +\pcline[nodesepB=4,nodesepA=1,linecolor=red,arrowsize=0.175,arrowinset=0.075]{->}(I)(V) +\psline[linestyle=dashed](I)(P)}}% +\end{pspicture} +} +\end{center} +Il s'agit de d\'{e}terminer en premier, l'intersection de $(VP)$ avec le c\^{o}ne, qu'on appelle $I$ (point d'incidence). Les coordonn\'{e}es de $V$, $S$ et $P$ sont not\'{e}es : $V(0,0,z_V)$, $S(0,0,z_S)$ et $P(x_P,y_P,0)$. + +L'\'{e}quation param\'{e}trique de la droite $(PV)$ s'\'{e}crit $\overrightarrow{IV}=\lambda\overrightarrow{PV}$: +\[ +\left\lbrace + \begin{array}{lcl} + 0-x_I&=&\lambda(0-x_P)\\ + 0-y_I&=&\lambda(0-y_P)\\ + z_V-z_I&=&\lambda(z_V-0) + \end{array} + \right. + \Longrightarrow + \left\lbrace + \begin{array}{lcl} + x_I&=&\lambda x_P\\ + y_I&=&\lambda y_P\\ + z_I&=&(1-\lambda)z_V + \end{array} + \right. +\] +On pose : +\[ + r_I^2=x_I^2+y_I^2\quad\textrm{et}\quad r_P^2=x_P^2+y_P^2\quad\textrm{et}\quad |\overrightarrow{OG}|=R +\] +Le point $I$ appartenant au c\^{o}ne, ses coordonn\'{e}es v\'{e}rifient la relation (th\'{e}or\^{e}me de Thal\`{e}s): +\begin{align*} + \frac{R}{z_S}&=\frac{r_I}{z_S-z_I}\\ + \frac{R}{z_S}&=\frac{\lambda r_P}{z_S-(1-\lambda)z_P}\\ + \lambda&=\frac{R(z_S-z_V)}{r_Pz_S-R z_V} +\end{align*} +\begin{center} +\begin{pspicture}(-4,-1)(6.5,7.75) +\pnode(0,10){V} +\uput[0](V){$V$} +\pnode(0,5){S} +\uput[0](S){$S$} +\pnode(1.5,2.5){I} +\pnode(2,0){P} +\pnode(4.545,0){P'} +\uput[45](P'){$P'$} +\psdots[dotstyle=|](P') +\pnode(3,0){G} +\pnode(-3,0){G'} +\pspolygon[fillstyle=solid,fillcolor=lightgray!30](G')(G)(S) +\psaxes[ticks=none,labels=none]{->}(0,0)(-4,0)(5.5,10.25) +\uput[0](5.5,0){$x$} +\uput[90](0,10.25){$z$} +\uput[45](P){$P$} +\uput[-45](G){$G$} +\uput[70](I){$I$} +\psline(V)(P) +\psline(S)(G) +\rput(I){% +\psline[linestyle=dashed](4;30.96) +\uput[15.5](4;30.1){$N$} +\rput{-59.036}(0,0){\psframe(0.5,0.5)} +\psarc[doubleline=true](0,0){1}{101.31}{120.964} +%\psarc[doubleline=true](0,0){1}{-79.09}{-59.036} +\psarc[doubleline=true](0,0){1.2}{-59.036}{-39.38} +\pnode(1.2;-50){I1} +\pnode(1.2;112){I2} +\uput[-50](I1){$\varepsilon$} +\uput[112](I2){$\varepsilon$} +} +\rput(P'){\pswedge[fillstyle=solid,fillcolor=lightgray](0,0){1}{140.175}{180} +\uput[160](1;160){$\varepsilon'$}} +\rput(S){\psarc[linecolor=gray,linewidth=2\pslinewidth](0,0){1}{-90}{-59.036} +\uput[-75](1;-75){$\theta$}} +\rput(V){\psarc(0,0){2}{-90}{-78.69} +\uput[-80](2;-85){$\beta$}} +\psline[linecolor=red](V)(I)(P') +\pcline[nodesepB=1.5,linecolor=red,arrowsize=0.175,arrowinset=0.075]{->>}(P')(I) +\pcline[nodesepB=4,linecolor=red,arrowsize=0.175,arrowinset=0.075]{->>}(I)(V) +\qdisk(I){2pt} +\qdisk(P){2pt} +\qdisk(S){2pt} +\qdisk(V){2pt} +\qdisk(P'){2pt} +\qdisk(G){2pt} +\psline[linewidth=0.5\pslinewidth]{|<->|}(0,-0.2)(2,-0.2) +\uput[-90](1,-0.2){$r_P$} +\psline[linewidth=0.5\pslinewidth]{|<->|}(0,-0.8)(3,-0.8) +\uput[-90](1.5,-0.8){$R$} +\psline[linewidth=0.5\pslinewidth]{<->}(1.5,2.5)(0,2.5) +\uput[-90](0.75,2.5){$r_I$} +\psline[linewidth=0.5\pslinewidth]{<->|}(-0.2,0)(-0.2,5) +\uput[180](-0.2,2.5){$z_S$} +\psline[linewidth=0.5\pslinewidth]{<->|}(-1,0)(-1,10) +\uput[180](-1,5){$z_V$} +\psline[linewidth=0.5\pslinewidth,linestyle=dashed](1.5,2.5)(1.5,0) +\end{pspicture} +\end{center} +Pour construire le rayon incident, $(P'I)$, $(IV)$ est le rayon r\'{e}fl\'{e}chi, d\'{e}terminons~$\varepsilon'$. + +Un raisonnement g\'{e}om\'{e}trique \'{e}l\'{e}mentaire nous montre que : +\[ + \varepsilon'=90^\circ-2\theta+\beta +\] +avec +\[ + \beta=\arctan\frac{r_P}{z_V} +\] +et +\[ + \theta=\arctan\frac{R}{z_S}\quad\textrm{demi-angle au sommet du c\^{o}ne } +\] +Dans le plan horizontal, les coordonn\'{e}es de $P'$ sont : +\[ +\left\lbrace + \begin{array}{lcl} + x_{P'}&=&x_I+\dfrac{z_I}{\tan \varepsilon'}\\[0.5cm] + y_{P'}&=&y_I+\dfrac{z_I}{\tan \varepsilon'}\\ + \end{array} +\right. +\] + +\newpage + +\section{L'anamorphose sph\'{e}rique} + +On place \`{a} l'int\'{e}rieur de la demi-sph\`{e}re l'image telle qu'elle doit \^{e}tre vue par un observateur regardant dans le miroir sph\'{e}rique (on peut la placer \`{a} l'ext\'{e}rieur, mais il faut que les rayons lumineux rencontrent toujours la sph\`{e}re, il faut donc veiller aux dimensions). L'objet anamorphique est <<~l'objet d\'{e}form\'{e}~>> dont le miroir reconstituera les proportions r\'{e}elles.\par Objet et image ob\'{e}issent aux lois de la r\'{e}flexion de l'optique g\'{e}om\'{e}trique : +\begin{itemize} + \item rayon incident et rayon r\'{e}fl\'{e}chi appartiennent \`{a} un m\^{e}me plan ; + \item rayon incident et rayon r\'{e}fl\'{e}chi sont sym\'{e}triques par rapport \`{a} la normale au miroir au point d'incidence. +\end{itemize} +\begin{center} +\psscalebox{0.67}{ +\begin{pspicture}(-4,-7)(12,4.5) +\psset{lightsrc=viewpoint} +\psset{viewpoint=-100 -100 100,Decran=173.2} +\ThreeDput[normal=0 0 1](0,0,0){%% +\psframe*[linecolor=lightgray!25](-5,-14)(5,5) +\psgrid[subgriddiv=0,gridcolor=lightgray,gridlabels=0pt](-5,-14)(5,5) +\psaxes(0,0)(-5,-14)(5,5) +\psset{type=spherical,Rmirror=5} +\pscircle[doubleline=true]{5} +\multido{\n=-1.00+0.20}{11}{% + \pnode(! \n\space -4.80){A} + \pnode(! \n\space -3.40){B} + \psline(A)(B) + \pslineA(A)(B) + }% +\multido{\N=-4.80+0.20}{8}{% + \pnode(!-1.0 \N){A} + \pnode(!1.0 \N){B} + \pslineA(A)(B) + \psline(A)(B) + }% +\psanamorphosis[scale=0.4 0.4](0,-4){tiger.eps} +\pnode(0,-4){A} +\pnodeA(0,-4){A'} +%\psline(A)(A') + } +\ThreeDput[normal=-1 0 0](0,0,0){% +\pnode(15,5){OE} +\pnode(13.2,4.2){OE'} +\pnode(5,0.4545){I} +\rput{26.5}(OE){\oeil} +} +\psSolid[object=calottesphere,r=5,opacity=0.5, + ngrid=24 36,fillcolor=cyan!50,grid,hollow, + incolor=yellow,theta=90,phi=0,action=draw,linecolor=BleuCiel,linewidth=0.5pt](0,0,0) +\psline[linecolor=red](A')(I)(OE') +\pcline[nodesepB=2.5,linecolor=red,arrowsize=0.175,arrowinset=0.075]{->>}(I)(OE') +\pcline[nodesepB=1,linecolor=red,arrowsize=0.175,arrowinset=0.075]{->>}(A')(I) +\psline(I)(A) +\end{pspicture} +} +\end{center} +L'image non d\'{e}form\'{e}e (celle qui est vue dans le miroir) est plac\'{e}e, dans cet exemple, au centre du miroir. Un rayon incident partant de l'objet anamorphique se r\'{e}fl\'{e}chit sur le miroir et apr\`{e}s r\'{e}flexion parvient \`{a} l'{\oe}il de notre observateur. L'observateur a l'illusion que le rayon provient du point image. Il faut donc reconstruire math\'{e}matiquement la marche d'un tel rayon lumineux en partant de l'image dans le miroir. + +L'observateur est suffisamment \'{e}loign\'{e} du miroir pour pouvoir \^{e}tre consid\'{e}r\'{e} comme ponctuel. + +Soit $P$ un point de l'image, $V$ l'{\oe}il de l'observateur. Tra\c{c}ons un droite $PV$ et d\'{e}terminons le point d'intersection $I$ avec la sph\`{e}re : c'est le point d'incidence. +\[ +V(x_V,y_V,z_V)\quad\text{ et }\quad P(x_P,y_P,0) +\] +\begin{center} +\shorthandoff{!} +\begin{pspicture}(-4,-0.5)(8,5.5) +\pswedge[fillstyle=solid,fillcolor=lightgray!30](0,0){2}{0}{180} +\psaxes[labels=none,ticks=none]{->}(0,0)(-3,0)(7.5,5) +\uput[0](7.5,0){$x$} +\uput[90](0,5){$z$} +\pnode(1.5,0){P} +\pnode(0,0){O}\uput[dl](O){$O$} +\pnode(!/alpha 15 def /xI 2 alpha cos mul def /yI 2 alpha sin mul def /beta yI xI 1.5 sub atan def xI yI){I} +\psline(P)(I) +\rput(I){\pnode(!beta cos 5 mul beta sin 5 mul){V}} +\psline[linecolor=red](I)(V) +\rput(I){\psarc[doubleline=true](0,0){1}{!alpha}{!beta} +\psarc[doubleline=true](0,0){1.2}{!2 alpha mul beta sub}{!alpha}} +\uput{1.2}[!alpha beta add 2 div](I){$\varepsilon$} +\uput{1.4}[!3 alpha mul beta sub 2 div](I){$\varepsilon$} +\pnode(!yI neg xI 2 alpha mul beta sub tan mul add 2 alpha mul beta sub tan div 0){M} +\psline[linecolor=red](M)(I) +\pcline[nodesepB=0.5,linecolor=red,arrowsize=0.175,arrowinset=0.075]{->>}(M)(I) +\pcline[nodesepB=2,linecolor=red,arrowsize=0.175,arrowinset=0.075]{->>}(I)(V) +\psline[linewidth=0.5\pslinewidth]{<->}(O)(I) +\uput[75](1;15){$R$} +\psline[linewidth=0.5\pslinewidth]{<->}(O)(V) +\uput[75](3,2.7){$r_V$} +\psline[linewidth=0.5\pslinewidth]{|<->|}(0,-0.3)(1.5,-0.3) +\uput[-90](0.75,-0.3){$r_P$} +\rput(I){% +\psline[linestyle=dashed](5;15) +\uput[15](5;15){$N$} +\psline[linewidth=0.5\pslinewidth,linecolor=gray](1.5;105) +\uput[105](1.5;105){$T$} +\psline[linewidth=0.5\pslinewidth,linecolor=gray](1.25;-75) +\rput{-75}(0,0){\psframe(0.3,0.3)} +} +\uput[-45](P){$P$} +\uput[-45](M){$P'$} +\uput[70](I){$I$} +\uput[u](V){$V$} +\psdot(P) +\psdot(I) +\psdot(V) +\psdot(M) +\end{pspicture} +\end{center} +L'\'{e}quation param\'{e}trique de la droite $(PV)$ s'\'{e}crit $\overrightarrow{IV}=\lambda\overrightarrow{PV}$: +\begin{equation}\label{eq:para} +\left\lbrace + \begin{array}{lcl} + x_V-x_I&=&\lambda(x_V-x_P)\\ + y_V-y_I&=&\lambda(y_V-y_P)\\ + z_V-z_I&=&\lambda(z_V-0) + \end{array} + \right. + \Longrightarrow + \left\lbrace + \begin{array}{lcl} + x_I&=&x_V(1-\lambda)+\lambda x_P\\ + y_I&=&y_V(1-\lambda)+\lambda y_P\\ + z_I&=&z_V(1-\lambda) + \end{array} + \right. +\end{equation} +On pose : +\[ +r_V^2=x_V^2+y_V^2+z_V^2\quad\textrm{et}\quad r_P^2=x_P^2+y_P^2 +\] +Le point $I$ appartenant \`{a} la sph\`{e}re, ses coordonn\'{e}es v\'{e}rifient la relation : +\begin{equation}\label{eq:sphere} +x_I^2+y_I^2+z_I^2=R^2 +\end{equation} +(\ref{eq:para}) en (\ref{eq:sphere}) +\begin{gather*} +\left(x_V+\lambda(x_P-x_V)\right)^2+\left(y_V+\lambda(y_P-y_V)\right)^2+\left((1-\lambda)z_V\right)^2=R^2\\ +x_V^2+2\lambda x_V(x_P-x_V)+\lambda^2(x_P-x_V)^2+y_V^2+2\lambda y_V(y_P-y_V) +\lambda^2(y_P-y_V)^2+(1-2\lambda+\lambda^2)z_V^2=R^2 +\end{gather*} +Apr\`{e}s d\'{e}veloppement, on obtient l'\'{e}quation du second degr\'{e} en +$\lambda$: +\begin{gather*} +\lambda^2\left((x_P-x_V)^2+(y_P-y_V)^2+z_V^2\right)+2\lambda\left(x_V(x_P-x_V)+y_V(y_P-y_V)-z_V^2\right)+x_V^2+y_V^2+z_V^2- +R^2=0\\ +\lambda^2\left(x_V^2+y_V^2+z_V^2+x_P^2+y_P^2-2x_Px_V-2y_Py_V\right)+2\lambda\left(-x_V^2-y_V^2-z_V^2+x_Px_V+y_Py_V\right)+x_V^2+y_V^2+z_V^2- +R^2=0 +\end{gather*} +Comparaison avec +\[ +a\lambda^2+2b'\lambda+c=0 +\] +donne pour le coefficient $a$ de $\lambda^2$ : +\[ +a=x_V^2+y_V^2+z_V^2+x_P^2+y_P^2-2x_Px_V-2y_Py_V +\] +Pour le coefficient $2b'$ de $\lambda$ : +\[ +2b'=-x_V^2-y_V^2-z_V^2+x_Px_V+y_Py_V +\] +Pour le coefficient $c$ : +\[ +c=x_V^2+y_V^2+z_V^2-R^2 +\] +Alors +\[ +a\lambda^2+2b'\lambda+c=0 +\] +avec : +\[ +\left\lbrace + \begin{array}{lcl} + a&=&x_V^2+y_V^2+z_V^2+x_P^2+y_P^2-2x_Px_V-2y_Py_V\\ + 2b'&=&-x_V^2-y_V^2-z_V^2+x_Px_V+y_Py_V\\ + c&=&x_V^2+y_V^2+z_V^2-R^2 + \end{array} + \right. +\] +\[\left\lbrace + \begin{array}{lcl} + a&=&r_V^2+r_P^2-2x_Px_V-2y_Py_V\\ + 2b'&=&-r_V^2+x_Px_V+y_Py_V\\ + c&=&r_V^2-R^2 + \end{array} + \right. +\] +La r\'{e}solution de cette \'{e}quation nous donne les solutions classiques: +\[ +\left\lbrace + \begin{array}{lcl} + \lambda'&=&\dfrac{-b'+\sqrt{\Delta'}}{a}\\[0.5cm] + \lambda''&=&\dfrac{-b'-\sqrt{\Delta'}}{a} + \end{array} + \right. + \qquad \Delta'=b'^2-ac +\] +On retiendra la valeur positive.% et \texttt{Coeff1} dans le programme. + +$(IV)$ repr\'{e}sente le rayon r\'{e}fl\'{e}chi par le miroir. Le rayon incident est d\'{e}fini par la droite sym\'{e}trique de $(IV)$ par rapport \`{a} la normale au miroir en $I$. Je cherche le sym\'{e}trique de $V$, nomm\'{e} $V'$ par rapport \`{a} cette normale $(IN)$. Ce point $V'$ remplit deux conditions : +\begin{enumerate} + \item $\overrightarrow{IV}+\overrightarrow{IV'}=k\overrightarrow{IN}$ + \item $\overrightarrow{VV'}.\overrightarrow{IN}=0$ +\end{enumerate} +La normale $(IN)$ a pour vecteur directeur $\overrightarrow{IN}(x_I,y_I,z_I)$ + +La premi\`{e}re condition se traduit par : +\[ +\left\lbrace + \begin{array}{lcl} + x_V-x_I+x_{V'}-x_I&=&kx_I\\ + y_V-x_I+y_{V'}-y_I&=&ky_I\\ + z_V-z_I+z_{V'}-z_I&=&kz_I + \end{array} + \right. + \Longrightarrow + \left\lbrace + \begin{array}{lcl} + x_{V'}&=&kx_I+2x_I-x_V\\ + y_{V'}&=&ky_I+2y_I-y_V\\ + z_{V'}&=&kz_I+2z_I-z_V + \end{array} + \right. +\] +La deuxi\`{e}me par : +\[ +(x_{V'}-x_V)x_I+(y_{V'}-y_V)y_I+(z_{V'}-z_V)z_I=0 +\] +En rempla\c{c}ant $x_{V'}$, $y_{V'}$ et $z_{V'}$ tir\'{e}s de la premi\`{e}re condition dans la deuxi\`{e}me : +\begin{gather*} +k(x_I^2+y_I^2+y_I^2)+2x_I^2-2x_Vx_I+2y_I^2-2y_Vy_I+2z_I^2-2z_Vz_I=0\\ +kR^2+2R^2=2(x_Vx_I+y_Vy_I+z_Vz_I)\\ +k+2=\dfrac{2}{R^2}(x_Vx_I+y_Vy_I+z_Vz_I) +\end{gather*} +Les coordonn\'{e}es de $V'$ s'en d\'{e}duisent : +\[ +\left\lbrace + \begin{array}{lcl} + x_{V'}&=&(k+2)x_I-x_V\\ + y_{V'}&=&(k+2)y_I-y_V\\ + z_{V'}&=&(k+2)z_I-z_V + \end{array} + \right. +\] +Il reste \`{a} trouver l'intersection de $(IV')$ avec le plan horizontal $z=0$. + +\'Equation param\'{e}trique de $(IV')$, $M$ \'{e}tant un point courant : $\overrightarrow{MI}=\alpha\overrightarrow{V'I}$ +\[ +\left\lbrace + \begin{array}{lcl} + x_I-x&=&\alpha(x_I-x_{V'})\\ + y_I-y&=&\alpha(y_I-y_{V'})\\ + z_I-z&=&\alpha(z_I-z_{V'}) + \end{array} + \right. +\] +$z=0\Longrightarrow \alpha=\dfrac{z_{I}}{z_I-z_{V'}}$. + +En rempla\c{c}ant $\alpha$ par son expression, nous obtenons les coordonn\'{e}es du point $P'$ de l'objet anamorphique. +\[ +\left\lbrace + \begin{array}{lcl} + x_{P'}&=&x_I-\alpha(x_I-x_{V'})\\ + y_{P'}&=&y_I-\alpha(y_I-y_{V'})\\ + z_{P'}&=&0 + \end{array} + \right. +\] +Cette s\'{e}rie de calculs doit \^{e}tre appliqu\'{e}e \`{a} tous les points de l'image <<~normale~>> afin d'obtenir l'objet anamorphique (d\'{e}form\'{e}) dont le miroir <<~redressera~>> la forme. + +\textbf{Remarque} : l'image doit se former du c\^{o}t\'{e} de l'observateur \`{a} l'int\'{e}rieur du miroir, plus pr\`{e}s du bord du miroir que du centre. Si on d\'{e}place le point $P$ vers $O$, il arrive un moment o\`{u} le rayon r\'{e}fl\'{e}chi part au-dessus de l'horizontale et ne rencontre plus le plan horizontal. L'anamorphose n'est plus possible et pour les calculs c'est le CRASH ! + +\newpage + +\section{La perspective} + +Dans le livre de Jurgis Baltru\v{s}a\"{\i}tis\footnote{ \textit{Anamorphoses : les perspectives d\'{e}prav\'{e}es} en livre de poche chez Flammarion.}, on trouve le principe de la <<~\textit{costruzione legittima}~>> avec un sch\'{e}ma de L\'{e}onard de Vinci (1492) et des sch\'{e}mas anamorphiques de Niceron (1658). Je cite page 58 : +\begin{quote}\itshape +<<~Rappelons en quelques mots quels ont \'{e}t\'{e} le proc\'{e}d\'{e}s utilis\'{e}s par les artistes pour l'ordonnancement de leurs tableaux en perspective normale. La premi\`{e}re ligne trac\'{e}e est celle de l'horizon \`{a} la hauteur de l'\oe{}il. Deux points y sont ensuite fix\'{e}s : au milieu le point principal vers o\`{u} convergent toutes les lignes droites parall\`{e}les qui s'\'{e}loignent en profondeur ; sur la m\^{e}me horizontale et \`{a} la m\^{e}me distance du point principal que l'\oe{}il, en face de la composition -- le point de distance, vers lequel convergent les diagonales.~>> +\end{quote} +\begin{center} +\begin{pspicture}(-5,-3)(5,14.25) +\psset{ua=2,F=10,D=4,type=perspective} +\psframe*[linecolor=BleuCiel](-5,10)(5,13) +\psframe*[linecolor=OrangePale](-5,2)(5,10) +\psline[arrowinset=0,arrowsize=2mm]{->}(0,14) +\psline[arrowinset=0,arrowsize=2mm]{->}(7,0) +\uput[0](7,0){$x$} +\uput[90](0,14){$y$} +\psline(-5,2)(5,2) +\pnode(4,10){F'} +\pnode(0,10){F} +\uput[ul](F){${F}$} +\uput[ur](F'){${F'}$} +\rput(0,12){\multido{\n=22.5+45.0}{8}{% +\psline[linecolor=yellow](1;\n)}% +\pscircle[linestyle=none,fillstyle=gradient,gradmidpoint=0,gradend=yellow,gradbegin=gray]{0.5}} +\multido{\i=-2+1}{5}{% + \pnode(! \i\space -2){A} + \pnode(! \i\space 2){B} + \psline(A)(B) + \pslineA(A)(B) + }% +\multido{\i=-2+1}{5}{% + \pnode(!-2 \i){A} + \pnode(!2 \i){B} + \pslineA[linecolor=blue](A)(B) + \psline[linecolor=blue](A)(B) + }% +\pnode(0,0){O} +\pnodeA(0,0){O'} +\pnode(-2,2){A} +\pnodeA(-2,2){A'} +\pnode(2,2){B} +\pnodeA(2,2){B'} +\pnode(-2,-2){C} +\pnodeA(-2,-2){C'} +\pnode(2,-2){D} +\pnodeA(2,-2){D'} +\pnode(-1,-1){M1} +\pnodeA(-1,-1){M1'} +\pnode(-1,1){M2} +\pnodeA(-1,1){M2'} +\pnode(-1,0){N1} +\pnodeA(-1,0){N1'} +\pnode(-1,2){P} +\pnodeA(-1,2){P'} +\pnode(-2,0){N2} +\pnodeA(-2,0){N2'} +\pnode(-2,1){N3} +\pnodeA(-2,1){N3'} +\pnodeA(-2,-1){S'} +\pnode(0,2){Q} +\pnode(1,2){R} +\psline(A)(F)(B) +\psline[linecolor=red](A)(F') +\psline[linecolor=red](P)(F) +\psline[linecolor=lightgray](P)(F') +\psline[linecolor=lightgray](Q)(F') +\psline[linecolor=lightgray](R)(F') +\psline[linecolor=lightgray](S')(F') +\psline[linecolor=lightgray](N2')(F') +\psline[linecolor=lightgray](N3')(F') +\uput[dl](A){${A}$} +\uput[dr](B){${B}$} +\uput[ul](A'){${A'}$} +\uput[ur](B'){${B'}$} +\uput[dl](C){${C}$} +\uput[dr](D){${D}$} +\uput[ul](P){${P}$} +\uput[u](P'){${P'}$} +\uput[ul](C'){${C'}$} +\uput[ur](D'){${D'}$} +\uput[dl](O){${O}$} +\uput[dr](O'){${O'}$} +\psline{<->|}(-3,2)(-3,10) +\uput[l](-3,6){$f$} +\pcline{|<->|}(F)(F') +\uput[u](2,10){$e$} +\psdots(A)(B)(C)(D)(A')(B')(C')(D')(P)(P')(F)(F')(O)(O') +\psdots[linecolor=blue](N1)(N1') +\uput[dl](N1){$\blue {N_1}$} +\uput[dl](N1'){$\blue {N_1'}$} +\psdots[linecolor=gray](M2)(M2') +\uput[dr](M2){\tiny $\gray {(X,Y)}$} +\uput[dr](M2'){\tiny $\gray {(x',y')}$} +\psdots[linecolor=red](M1') +\uput[dr](M1'){\tiny $\red {(\alpha_1,\beta_1)}$} +\end{pspicture} +\end{center} + +\newpage + +Exemples : +\begin{itemize} + \item $A\longrightarrow A'$ + \item $B\longrightarrow B'$ + \item $C\longrightarrow C'$ + \item $D\longrightarrow D'$ + \item $O\longrightarrow O'$ + \item $M_1\longrightarrow M_1'$ + \item $M_2\longrightarrow M'_2$ +\end{itemize} +D\'{e}terminons les coordonn\'{e}es $(\alpha_1,\beta_1)$ de l'intersection de $(PF)$ avec $(AF')$. + +Posons que les coordonn\'{e}es des points essentiels sont : +\begin{itemize} + \item $F(0,f)$ + \item $F'(e,f)$ + \item $A(-a,a)$ + \item $B(a,a)$ + \item $C(a,-a)$ + \item $D(-a,-a)$ + \item $P(X,a)$ +\end{itemize} +\'Equation de $(AF')$ : +\[ +\frac{y-f}{x-e}=\frac{a-f}{-a-e}\Longrightarrow x(a-f)+y(a+e)-a(f+e)=0 +\] +\'Equation de $(PF)$ : +\[ +\frac{x-0}{y-f}=\frac{X-0}{a-f}\Longrightarrow x(a-f)-yX+fX=0 +\] +Intersection $(PF)\bigcap (AF')$ +\[ +\alpha_1=\frac{Xe}{X+a+e}\qquad\beta_1=\frac{a(f+e)+fX}{X+a+e} +\] +Si on prend maintenant, un point d'ordonn\'{e}e $Y\neq X$ par exemple $N_1$ dont l'image $N'_1$ se situe toujours sur $(PF)$, mais \`{a} l'intersection de $PF$ avec la parall\`{e}le \`{a} $x'Ox$ men\'{e}e par le point-image du point de coordonn\'{e}e $(Y,Y)$ (ici $O'$ qui est l'image de $O(0,0)$). + +Il s'agit de d\'{e}terminer l'intersection de $(PF)$ avec la droite d'\'{e}quation : +\[ +y=\beta_2=\frac{a(f+e)+fY}{Y+a+e} +\] +Apr\`{e}s calculs et simplifications, on trouve pour l'abscisse : +\[ +\alpha_2=\frac{Xe}{Y+a+e} +\] +En r\'{e}sum\'{e} si dans le rep\`{e}re $Oxy$, on appelle $({\red X},{\red Y})$ les coordonn\'{e}es d'un point-objet et $({\blue x'},{\blue y'})$ les coordonn\'{e}es du point image dans la transformation \textit{anamorphose oblique} ou \textit{perspective}, les formules qui permettent de passer de l'objet \`{a} l'image s'\'{e}crivent : +\[ +\left\lbrace + \begin{array}{lcl} + {\blue x'}&=&\displaystyle\frac{{\red X}e}{{\red Y}+a+e}\\[0.5cm] + {\blue y'}&=&\displaystyle\frac{a(f+e)+f{\red Y}}{{\red Y}+a+e} + \end{array} +\right. +\] +\end{document} \ No newline at end of file diff --git a/GILGTEST/pst-anamorphosis-doc-part2.pdf b/GILGTEST/pst-anamorphosis-doc-part2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a42015d788392f3a847a6842c15c99c68abd25e1 GIT binary patch literal 896879 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+\psset{ua=3,F=14,D=3,type=inverseperspective,linewidth=0.5\pslinewidth} +\multido{\i=-3+1}{7}{% + \pnode(! \i\space -3){A} + \pnode(! \i\space 3){B} + \psline(A)(B) + \pslineA(A)(B) + }% +\multido{\i=-3+1}{7}{% + \pnode(!-3 \i){A} + \pnode(!3 \i){B} + \pslineA(A)(B) + \psline(A)(B) + }% +\psanamorphosis{tiger.eps} +\psset{fillcolor={[cmyk]{0.2 0.55 0.85 0.0}},linecolor={[cmyk]{0.2 0.55 0.85 0.0}},fontsize=40,scale=1.32 1} +\pstextA(0,2){TIGRE*} +\pstextA(-2.48,1){I} +\pstextA(-2.48,0){G} +\pstextA(-2.48,-1){R} +\pstextA(-2.48,-2){E} +\pstextA(-2.48,-3){*} +\end{pspicture} +} +\end{center} +\begin{lstlisting} +\begin{pspicture}(-3,-4)(3,3) +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=6pt](-3,-3)(3,-6) +\psset{ua=3,F=14,D=3,type=inverseperspective,linewidth=0.5\pslinewidth} +\multido{\i=-3+1}{7}{% + \pnode(! \i\space -3){A} + \pnode(! \i\space 3){B} + \psline(A)(B) + \pslineA(A)(B) + }% +\multido{\i=-3+1}{7}{% + \pnode(!-3 \i){A} + \pnode(!3 \i){B} + \pslineA(A)(B) + \psline(A)(B) + }% +\psanamorphosis{tiger.eps} +\psset{fillcolor={[cmyk]{0.2 0.55 0.85 0.0}},linecolor={[cmyk]{0.2 0.55 0.85 0.0}},fontsize=40,scale=1.32 1} +\pstextA(0,2){TIGRE!} +\pstextA(-2.48,1){I} +\pstextA(-2.48,0){G} +\pstextA(-2.48,-1){R} +\pstextA(-2.48,-2){E} +\pstextA(-2.48,-3){!} +\end{pspicture} +\end{lstlisting} + +\section{Les fichiers pst} + +\begin{center} +\begin{pspicture}[showgrid](-5,-6)(5,3) +\rput(0,-1.3333){% +\psscalebox{0.6667}{% +\begin{pspicture}(-6,-8)(6,4) +\psset{Xv=0,Yv=-100} +\pscircle[doubleline=true,linecolor=black]{3} +\input{luckyluke.pst} +\AFP{luckyluke.pst} +\psset{linewidth=0.5\pslinewidth} +\multido{\n=-2.00+0.50}{9}{% + \pnode(! \n\space -2.00){A} + \pnode(! \n\space 2.00){B} + \psline(A)(B) + \pslineA(A)(B) + } +\multido{\N=-2.00+0.50}{9}{% + \pnode(!-2.0 \N){A} + \pnode(!2.0 \N){B} + \pslineA(A)(B) + \psline(A)(B) + } +\end{pspicture} +}} +\end{pspicture} +\end{center} +\begin{lstlisting} +\begin{pspicture}(-6,-8)(6,4) +\psset{Xv=0,Yv=-100} +\pscircle[doubleline=true,linecolor=black]{3} +\input{luckyluke.pst} +\AFP{luckyluke.pst} +\psset{linewidth=0.5\pslinewidth} +\multido{\n=-2.00+0.50}{9}{% + \pnode(! \n\space -2.00){A} + \pnode(! \n\space 2.00){B} + \psline(A)(B) + \pslineA(A)(B) + } +\multido{\N=-2.00+0.50}{9}{% + \pnode(!-2.0 \N){A} + \pnode(!2.0 \N){B} + \pslineA(A)(B) + \psline(A)(B) + } +\end{pspicture} +\end{lstlisting} +Les fichiers \textsf{.pst} sont des fichiers de commandes PSTricks obtenues par conversion d'un fichier \textsf{.eps} au moyen d'un script \'{e}crit par Jean-Michel \textsc{Sarlat}. + +L'anamorphose d'une telle image est obtenue \`{a} l'aide d'une seule commande \verb+\AFP{luckyluke.pst}+ (AFP $=$ \textbf{A}namorphose \`{a} partir d'un \textbf{F}ichier \textbf{P}STricks). De tels fichiers pourront donc \^{e}tre obtenues au moyen du script ou bien directement sur le serveur de \textsf{syracuse} (adresse \`{a} pr\'{e}ciser). + +En fonction de la complexit\'{e} du fichier (du nombre de lignes de commandes qu'il contient) il sera n\'{e}cessaire d'allouer un suppl\'{e}ment de m\'{e}moire \`{a} \TeX. Par exemple, sous Windows, \`{a} partir de MiK\TeX~2.8, Alexander \textsc{Grahn} donne la proc\'{e}dure \`{a} suivre dans la documentation de son package : `\textsf{animate}' +\begin{lstlisting} +1. Open a DOS command prompt window (execute `cmd.exe' via `Start'! `Run'). +2. At the DOS prompt, enter +initexmf --edit-config-file=latex +3. Type +main_memory=10000000 +into the editor window that opens, save the file and quit the editor. +4. To rebuild the format, enter +initexmf --dump=latex +5. Repeat steps 2-4 with config + +initexmf --edit-config-file=pdflatex +main_memory=10000000 +initexmf --dump=pdflatex + +initexmf --edit-config-file=xelatex +main_memory=10000000 +initexmf --dump=xelatex +\end{lstlisting} + +\section{Normalisation d'une image au format \textsf{.eps}} + +Il est souvent n\'{e}cessaire de ``pr\'{e}parer'' l'image \`{a} anamorphoser. Voici le d\'{e}tail de la proc\'{e}dure suivie \`{a} partir d'une image de Donald Duck captur\'{e}e sur un site de coloriage. + +\url{http://www.waouo.com/wp-content/uploads/2011/06/coloriagedonaldduck.jpg} + +\begin{enumerate} + \item Transformer l'image au format \textsf{bmp} avec \textsf{The Gimp} (par exemple). + \item Vectoriser l'image avec \textsf{Potrace}, en ligne de commande : \textsf{potrace donald.bmp -o donald2.eps} + \item Avec \textsf{pstoedit} convertir au format vectoriel ps simplifi\'{e} avec courbes. On peut op\'{e}rer en ligne de commande : \textsf{pstoedit -f ps donald2.eps donald1.eps} ou bien avec \textsf{GSview} dans le menu Edition, cliquer sur \textsf{Conversion vers un format vectoriel} et choisir \textsf{ps : Simplified Postscript with curves}, on enregistre sous le nom \textsf{donald1.eps}. + \item Avec votre \'{e}diteur de texte rechercher \`{a} la fin du fichier la ligne qui donne les dimensions de la bo\^{\i}te : \verb+%%BoundingBox: 4 7 453 640+ . + \item La hauteur de l'image eps est $640-7 =633$\,pts, c'est la plus grande dimension. Supposons que votre image finale doive poss\'{e}der une longueur de 4\,cm dans sa plus grande dimension. Sachant que 1\,cm vaut 28.45274\,pts, il faut donc lui appliquer un facteur d'\'{e}chelle \'{e}gal \`{a} $\dfrac{4\times28.45274}{633}$. +\item Ensuite il faut d\'{e}placer cette image pour que le centre de celle-ci soit \`{a} l'origine $(0,0)$, il faut donc appliquer la translation $-\dfrac{453+4}{2},\,-\dfrac{640+7}{2}$, ce qui convertit en postscript se traduira par l'ajout de ces deux lignes dans le fichier \textsf{donald1.eps}, au d\'{e}but du fichier juste apr\`{e}s la ligne : \verb+%%Page: 1 1+ +\begin{lstlisting} + 4 28.45274 mul 633 div dup scale + 4 453 add 2 div neg 640 7 add 2 div neg translate +\end{lstlisting} + On enregistre ce fichier. +\item \textsf{pstoedit} intervient \`{a} nouveau, avec les m\^{e}mes options, et on enregistre le fichier final sous le nom : \textsf{donald.eps}. +\end{enumerate} +\begin{center} +\begin{pspicture}[showgrid](-8,-4)(8,10) +\pscircle[doubleline=true]{3} +\psset{Yv=100,linestyle=dotted} +\psanamorphosis[scale=1 -1](0,0){donald.eps} +\multido{\n=-2.00+0.50}{9}{% + \pnode(! \n\space -2.00){A} + \pnode(! \n\space 2.00){B} + \psline(A)(B) + \pslineA(A)(B) + } +\multido{\N=-2.00+0.50}{9}{% + \pnode(!-2.0 \N){A} + \pnode(!2.0 \N){B} + \pslineA(A)(B) + \psline(A)(B) + } +\pstextA[fontsize=25,scale=1.5 -1,fillcolor=black](0,-2.5){Donald} +\end{pspicture} +\end{center} +\begin{lstlisting} +\begin{pspicture}[showgrid](-8,-4)(8,10) +\pscircle[doubleline=true]{3} +\psset{Yv=100,linestyle=dotted} +\psanamorphosis[scale=1 -1](0,0){donald.eps} +\multido{\n=-2.00+0.50}{9}{% + \pnode(! \n\space -2.00){A} + \pnode(! \n\space 2.00){B} + \psline(A)(B) + \pslineA(A)(B) + } +\multido{\N=-2.00+0.50}{9}{% + \pnode(!-2.0 \N){A} + \pnode(!2.0 \N){B} + \pslineA(A)(B) + \psline(A)(B) + } +\pstextA[fontsize=25,scale=1.5 -1,fillcolor=black](0,-2.5){Donald} +\end{pspicture} +\end{lstlisting} +\end{document} \ No newline at end of file diff --git a/GILGTEST/pst-anamorphosis.eps b/GILGTEST/pst-anamorphosis.eps new file mode 100644 index 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