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-\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}%
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-\input{ALouisXIII.pst}
-\pstextA[fontsize=28,linecolor=red,fillstyle=solid,fillcolor=yellow!50](0,2){LouisXIII}
- }
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-}
-\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){$V$}
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-\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.
-
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- \FPeval{\zH}{sin(\phy)*\OH}
- \FPeval{\OK}{cos(\phy)*\OH}
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- \pslineA(A)(B)
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-%le produit scalaire du vecteur viewpoint et du vecteur normal \`{a} la face >0 ?
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-{\boldmath
-\red
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-\psline[linestyle=dashed](I)(P)}}%
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-}
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-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*}
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-\uput[-90](1.5,-0.8){$R$}
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-\uput[-90](0.75,2.5){$r_I$}
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-\uput[180](-0.2,2.5){$z_S$}
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-\uput[180](-1,5){$z_V$}
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-\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}
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-}
-\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}
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-\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}
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-\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
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