sym_plan.tex

\documentclass{article}
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\usepackage{pst-solides3d}
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\pstheader{rhombicuboctahedron.pro}
\begin{document}
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%%%%% ### SymPlan ###
%% symétrie / plan ax+by+cz+d=0
/SymPlan {
12 dict begin
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   /y exch def
   /x exch def
args aload pop
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/c1 exch def
/b1 exch def
/a1 exch def
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y 2 b mul u mul sub
z 2 c mul u mul sub
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4 dict begin
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[T length 1 sub -1 0 {
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   for
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 ] def
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 def
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1 0.55409 rhombicuboctahedron
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                 /F PerBU2 solidgetfaces def
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def
/PerBUinitial {
PerBU  dup [0 1] solidputhuecolors {0 0 3 translatepoint3d} solidtransform
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PerBUinitial
PerBUSym
solidfuz
drawsolid**
2 newcube {0 0 3 translatepoint3d} solidtransform
drawsolid
/Cube2 { 2 newcube {0 0 3 translatepoint3d} solidtransform
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/SymCube {
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                  S F generesolid
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def
SymCube drawsolid
}
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\end{center}
Équation du plan : $\mathrm{a}x+\mathrm{b}y+\mathrm{c}z+\mathrm{d}=0$
 
Vecteur normal au plan : $\overrightarrow{n}(\mathrm{a},\mathrm{b},\mathrm{c})$
 
Un point $M(x,y,z)$ quelconque et son symétrique $M'(x',y',z')$ (orthogonalement) par rapport au plan.
 
$\overrightarrow{MM'}$ et $\overrightarrow{n}$ sont colinéaires :
 
\[
\frac{x'-x}{\mathrm{a}}=\frac{y'-y}{\mathrm{b}}=\frac{z'-z}{\mathrm{c}}=k
\]
Le milieu $I$ de $[MM']$ appartient au plan :
\[
\mathrm{a}(x+x')+\mathrm{b}(y+y')+c(z+z')+2d=0
\]
En posant $E=\mathrm{a}x+\mathrm{b}y+\mathrm{c}z+\mathrm{d}$, après calculs on obtient :
\[
\left\{
\begin{array}{rcl}
x'&=&x-\frac{2\mathrm{a}E}{\mathrm{a}^2+\mathrm{b}^2+c^2}\\[0.2cm]
y'&=&y-\frac{2\mathrm{b}E}{\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2}\\[0.2cm]
z'&=&z-\frac{2\mathrm{c}E}{a^2+\mathrm{b}^2+\mathrm{c}^2}
\end{array}
\right.
\]
\end{document}