427 lines
130 KiB
Tcl
427 lines
130 KiB
Tcl
<TeXmacs|1.99.11>
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<style|<tuple|base|french>>
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<\body>
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<doc-data|<doc-title|Notes techniques>>
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<\subsection>
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<name|2d> vs. <name|3d>
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</subsection>
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L'intensité d'une source omnidirectionnelle <name|2d> diminue en
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<math|<frac*|1|R>>, et non en <math|<frac*|1|R<rsup|2>>>
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<math|\<rightarrow\>> ce n'est pas une source <name|3d>-isotrope, mais
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plutôt une source focalisée dans la direction <math|z> (comme un
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phare-fente) ou un barre lumineuse infinie dans la direction <math|z>.
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Conjecture : les résultats de la simulation sont les mêmes que en <name|3d>
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avec des objets infinis suivant <math|z>.
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<subsection|Angles modulo <math|<with|math-condensed|true|2*\<pi\>>>>
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<\compact>
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De façon générale, on ne ramène jamais systématiquement les angles \S
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géométriques \T à <math|<around*|[|0,<with|math-condensed|true|2*\<pi\>>|[>>
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ou <math|<around*|[|-\<pi\>,+\<pi\>|[>>, car cela casse les potentielles
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relations d'ordres entre les angles (e.g.
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<math|-0.1*\<pi\>\<leqslant\>\<theta\>\<leqslant\>+0.1*\<pi\>> n'est
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<em|pas> équivalent à <math|1.9*\<pi\>\<leqslant\>\<theta\>\<leqslant\>0.1*\<pi\>>,
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qui n'est jamais vrai). Exceptions :
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<\itemize-dot>
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<item>angles d'incidences : <math|i\<in\><around*|[|-<frac|\<pi\>|2>,+<frac|\<pi\>|2>|]>>
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systématiquement
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<item>angle du rayon, ramené à <math|<around*|[|0,<with|math-condensed|true|2*\<pi\>>|[>>
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à chaque émission, pour éviter d'accumumuler des tours
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</itemize-dot>
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</compact>
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<subsection|Intersection segment / demi-droite>
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<\padded-center>
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</padded-center>
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Segment d'extrémités <math|A> et <math|B> : <math|A*s+<around*|(|1-s|)>*B>
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pour <math|s\<in\><around*|[|0,1|]>>.
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Demi-droite portée par <math|<wide|u|\<vect\>><rsub|\<alpha\>>> d'origine
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<math|O> : <math|O+t*<wide|u|\<vect\>><rsub|\<alpha\>>> pour
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<math|t\<in\><around*|[|0,\<infty\>|[>>.
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Intersection :
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<\equation*>
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\<exists\>s<rsub|<text|i>>,t<rsub|<text|i>><space|1em>:<space|1em>A*s<rsub|<text|i>>+<around*|(|1-s<rsub|<text|i>>|)>*B<separating-space|0.2em>=<separating-space|0.2em>O+t<rsub|<text|i>>*<wide|u|\<vect\>><rsub|\<theta\>><space|1em>\<Leftrightarrow\><space|1em><wide|B\<nospace\>A|\<vect\>>*s<rsub|<text|i>>+t<rsub|<text|i>>*<wide|u|\<vect\>><rsub|\<alpha\>>*<separating-space|0.2em>=<separating-space|0.2em><wide|B\<nospace\>O|\<vect\>>
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</equation*>
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<\equation*>
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\<Longleftrightarrow\><space|1em>\<exists\>s<rsub|<text|i>>\<in\><around*|[|0,1|]>,t<rsub|<text|i>>\<in\><around*|[|0,\<infty\>|[><space|1em>:<space|1em><bmatrix|<tformat|<table|<row|<cell|x<rsub|a>-x<rsub|b>>|<cell|-x<rsub|\<alpha\>>>>|<row|<cell|y<rsub|a>-y<rsub|b>>|<cell|-y<rsub|\<alpha\>>>>>>>*<bmatrix|<tformat|<table|<row|<cell|s<rsub|<text|i>>>>|<row|<cell|t<rsub|<text|i>>>>>>>=<bmatrix|<tformat|<table|<row|<cell|x<rsub|o>-x<rsub|b>>>|<row|<cell|y<rsub|o>-y<rsub|b>>>>>>
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</equation*>
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En ignorant le cas où le segment et la demi-droite sont parallèles, il
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suffit de résoudre ce système linéaire. Si <math|s\<nin\><around*|[|0,1|]>>
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ou si <math|t\<less\>0>, alors il n'y a pas intersection. Sinon, le point
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d'intersection est <math|P=O+t<rsub|<text|i>>*<wide|u|\<vect\>><rsub|\<alpha\>>>,
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et l'angle d'incidence est <math|i=\<alpha\>-\<theta\><rprime|'>>, où
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<math|\<theta\>=\<angle\><wide|B\<nospace\>A|\<vect\>>> et
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<math|\<theta\><rprime|'>=\<theta\>-<frac|\<pi\>|2>>. L'angle de la normale
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est <math|\<theta\><rsub|<text|normale>>=\<theta\><rprime|'>+\<pi\>>.
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<subsection|Construction d'un arc de cercle à partir de
|
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<math|<around*|(|A,B,R|)>>>
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<\wide-tabular>
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<tformat|<cwith|1|-1|1|-1|cell-valign|t>|<cwith|1|1|1|1|cell-rsep|20pt>|<table|<row|<\cell>
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Après translation de <math|-<wide|a|\<vect\>>> et rotation d'un angle
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<math|-\<theta\><rsub|0>>, on suppose que <math|A> est l'origine et que
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<math|B> est à la verticale de <math|A> (<math|x<rprime|'><rsub|B>=0>).
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Le centre <math|C> de l'arc de cercle voulu est l'intersection (de
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droite ici) des ceux cercles centrés en <math|A> et <math|B> de rayons
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<math|R>. On voit immédiatement que
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|
<math|y<rprime|'><rsub|C>=A\<nospace\>B/2>. Le centre <math|C> est
|
|
finalement défini par la solution négative de
|
|
<math|x<rprime|'><rsub|C><rsup|2>+y<rprime|'><rsub|C><rsup|2>=R<rsup|2>>,
|
|
c'est-à-dire
|
|
|
|
<\equation*>
|
|
x<rprime|'><rsub|C>=-<sqrt|R<rsup|2>-y<rprime|'><rsub|C><rsup|2>>=-<sqrt|R<rsup|2>-<around*|(|A\<nospace\>B/2|)><rsup|2>><space|1em><text|définie
|
|
si <math|A\<nospace\>B\<leqslant\>2*R>>
|
|
</equation*>
|
|
|
|
Pour obtenir <math|\<theta\><rprime|'><rsub|A>> et
|
|
<math|\<theta\><rprime|'><rsub|B>>, on a simplement
|
|
|
|
<\equation*>
|
|
tan<around*|(|\<mp\>\<theta\><rprime|'><rsub|A,B>|)>=<frac|y<rprime|'><rsub|C>|-x<rprime|'><rsub|C>>=1/<sqrt|<around*|(|2*R/A\<nospace\>B|)><rsup|2>-1>
|
|
</equation*>
|
|
</cell>|<\cell>
|
|
<image|<tuple|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200A30303030303033373433203030303030206E200A30303030303034343235203030303030206E200A30303030303034343032203030303030206E200A30303030303034353633203030303030206E200A30303030303034363931203030303030206E200A747261696C65720A3C3C202F53697A652031370A2020202F526F6F74203136203020520A2020202F496E666F203135203020520A3E3E0A7374617274787265660A343734340A2525454F460A>|pdf>||||>
|
|
</cell>>>>
|
|
</wide-tabular>
|
|
|
|
On obtient finalement <math|\<theta\><rsub|A,B>=\<theta\><rprime|'><rsub|A,B>+\<theta\><rsub|0>>
|
|
et <math|C> après rotation d'un angle <math|+\<theta\><rsub|0>> puis une
|
|
translation de <math|+<wide|a|\<vect\>>>.
|
|
|
|
<subsection|Intersection arc de cercle / demi-droite>
|
|
|
|
<\padded-center>
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</padded-center>
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On suppose que <math|\<theta\><rsub|A>\<less\>\<theta\><rsub|B>>. La
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distance <math|d=O\<nospace\>C> est non signée ici, et les angles ne sont
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pas absolus comme avec le segment, mais relatifs à l'axe
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<math|<wide|O\<nospace\>C|\<vect\>>> (on peut regarder la figure dans
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n'importe quel sens). L'arc est de rayon de courbure
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<math|R=C\<nospace\>A=C\<nospace\>B>. Clairement, si
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<math|<around*|\||\<alpha\>|\|>\<gtr\><frac|\<pi\>|2>> et <math|d\<gtr\>R>,
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il n'y a pas intersection.
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Pour le premier point d'intersection avec le disque (<math|P> ici),
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<\equation*>
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<wide*|d<phantom|<around*|\|||<right|.>>>tan
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\<alpha\>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|1>>+<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|2>>>=<wide*|<around*|(|+R*sin
|
|
\<theta\><rsub|1>|)>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|1>>>+<wide*|tan
|
|
\<alpha\>\<cdot\><around*|(|-R*cos \<theta\><rsub|1>|)>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|2>>><space|1em>\<Longleftrightarrow\><space|1em>b\<cdot\>sin<around*|(|\<alpha\>|)>=sin<around*|(|\<theta\><rsub|1>-\<alpha\>|)><space|1em><text|où><space|1em>b=<frac|d|R>
|
|
</equation*>
|
|
|
|
La demi-droite intersecte l'arc <math|<wide|A\<nospace\>B|\<invbreve\>>> si
|
|
<math|\<theta\><rsub|1>> est défini, donc si<\footnote>
|
|
On a alors <math|<around*|\||sin<around*|(|\<theta\><rsub|<text|t>>-\<alpha\><rsub|<text|t>>|)>|\|>=1>,
|
|
d'où <math|\<theta\><rsub|<text|t>>=\<alpha\><rsub|<text|t>>\<pm\><frac|\<pi\>|2>>,
|
|
d'où\
|
|
|
|
<\equation*>
|
|
cos \<theta\><rsub|<text|t>>=cos<around*|(|\<alpha\><rsub|<text|t>>\<pm\><tfrac|\<pi\>|2>|)>=\<mp\>sin<around*|(|\<alpha\><rsub|<text|t>>|)>=\<mp\><frac|R|d>
|
|
</equation*>
|
|
|
|
On peut retrouver cet angle <math|<math|\<theta\><rsub|<text|t>>>> en
|
|
écrivant la condition de tengentialité rayon/cercle au point <math|P> :
|
|
|
|
<\equation*>
|
|
<bmatrix|<tformat|<table|<row|<cell|R*cos
|
|
\<theta\><rsub|<text|t>>>>|<row|<cell|\<pm\>R*sin*\<theta\><rsub|<text|t>>>>>>><separating-space|0.2em>=<separating-space|0.2em>O+t*<rigid|<wide|n|\<vect\>><rsub|\<theta\><rsub|<text|t>>>><rsup|\<bot\>><separating-space|0.2em>=<separating-space|0.2em><bmatrix|<tformat|<table|<row|<cell|-d>>|<row|<cell|0>>>>>+t*<bmatrix|<tformat|<table|<row|<cell|sin
|
|
\<theta\><rsub|<text|t>>>>|<row|<cell|\<mp\>cos
|
|
\<theta\><rsub|<text|t>>>>>>>
|
|
</equation*>
|
|
|
|
Pour <math|y>, on a <math|R*sin \<theta\><rsub|<text|t>>=-t*cos
|
|
\<theta\><rsub|<text|t>>> donc <math|t=-R*sin
|
|
\<theta\><rsub|<text|t>>/cos \<theta\><rsub|<text|t>>>. Alors, pour
|
|
<math|x>, on a
|
|
|
|
<\equation*>
|
|
R*cos \<theta\><rsub|<text|t>>=-d+<around*|<left|(|-1>|-R*<frac|sin
|
|
\<theta\><rsub|<text|t>>|cos \<theta\><rsub|<text|t>>>|<right|)|-1>>*sin*\<theta\><rsub|<text|t>><space|1em>\<Longleftrightarrow\><space|1em>R*cos<rsup|2>
|
|
\<theta\><rsub|<text|t>>=-d*cos \<theta\><rsub|<text|t>>-R*sin<rsup|2>
|
|
\<theta\><rsub|<text|t>><space|1em>\<Longleftrightarrow\><space|1em>cos
|
|
\<theta\><rsub|<text|t>>=-<frac|R|d>*<around*|(|cos<rsup|2>+sin<rsup|2>|)>=-<frac|R|d>
|
|
</equation*>
|
|
|
|
Et puisque <math|sin<around*|(|arccos<around*|(|x|)>|)>=<sqrt|1-x<rsup|2>>>
|
|
est indépendant du signe de <math|x>, on peut tout aussi bien prendre
|
|
<math|cos \<theta\><rsub|<text|t>>=+R/d>.
|
|
</footnote>
|
|
|
|
<\wide-tabular>
|
|
<tformat|<cwith|1|-1|1|-1|cell-valign|t>|<table|<row|<\cell>
|
|
<\compact>
|
|
<\equation*>
|
|
b*<around*|\||\<space\>sin<around*|(|\<alpha\>|)>|\|>\<leqslant\>1<space|1em>\<Longleftrightarrow\><space|1em><around*|\||\<alpha\>|\|>\<less\>\<alpha\><rsub|<text|t>><separating-space|0.2em>=<separating-space|0.2em>arcsin<around*|(|R/d|)>
|
|
</equation*>
|
|
|
|
lorsque <math|d\<gtr\>R>. Pour <math|d\<less\>R> (<math|O> à
|
|
l'intérieur du cercle), il y a toujours une solution, évidemment.
|
|
</compact>
|
|
|
|
Pour le deuxième point d'intersection avec le disque, on a
|
|
|
|
<\equation*>
|
|
<wide*|d<phantom|<around*|\|||<right|.>>>tan
|
|
\<alpha\>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|1>>+<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|2>>>=<wide*|<around*|(|+R*sin
|
|
\<theta\><rsub|2>|)>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|1>+<decorated|2>+<decorated|3>>>-<wide*|tan
|
|
\<alpha\>\<cdot\><around*|(|+R*cos
|
|
\<theta\><rsub|2>|)>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|3>>>
|
|
</equation*>
|
|
|
|
<text-dots> ce qui revient au même :
|
|
<math|b\<cdot\>sin<around*|(|\<alpha\>|)>=sin<around*|(|\<theta\><rsub|2>-\<alpha\>|)>>.
|
|
|
|
Cette équation possède en effet, dans
|
|
<math|\<theta\>\<in\><around*|[|0,<with|math-condensed|true|2*\<pi\>>|[>>,
|
|
nos<space|3em><line-break>deux solutions (courbes
|
|
<math|\<theta\><around*|(|\<alpha\>|)>> bi-valuées).
|
|
|
|
Au final, les solution sont
|
|
|
|
<\equation*>
|
|
<tabular|<tformat|<table|<row|<cell|\<theta\><rsub|1>=-arcsin<around*|(|b\<cdot\>sin<around*|(|\<alpha\>|)>|)>+\<alpha\>+\<pi\>>>|<row|<cell|\<theta\><rsub|2>=+arcsin<around*|(|b\<cdot\>sin<around*|(|\<alpha\>|)>|)>+\<alpha\>>>>>>
|
|
</equation*>
|
|
|
|
Pour <math|d\<gtr\>R>, la solution correcte est
|
|
<math|\<theta\><rsub|1>> si <math|\<theta\><rsub|A>\<less\>\<theta\><rsub|1>\<less\>\<theta\><rsub|B>>,
|
|
et sinon, éventuellement <math|\<theta\><rsub|2>> si
|
|
<math|\<theta\><rsub|A>\<less\>\<theta\><rsub|2>\<less\>\<theta\><rsub|B>>.
|
|
Pour <math|d\<less\>R>, la seule solution correcte est
|
|
<math|\<theta\><rsub|2>> (et si on veut rester dans
|
|
<math|<around*|[|0,<with|math-condensed|true|2*\<pi\>>|[>>, on ajoute
|
|
<math|<with|math-condensed|true|2*\<pi\>>> à <math|\<theta\><rsub|2>>
|
|
pour <math|\<alpha\>\<less\>0>).
|
|
|
|
Les angles d'incidence respectifs sont
|
|
</cell>|<\cell>
|
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</cell>>>>
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</wide-tabular>
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<\equation*>
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i<rsub|1>=<around*|(|\<pi\>-\<theta\><rsub|1>|)>+\<alpha\><space|1em><text|et><space|1em><tfrac|\<pi\>|2>=\<alpha\>+<around*|(|-i<rsub|2>|)>+<around*|(|<tfrac|\<pi\>|2>-\<theta\><rsub|2>|)><separating-space|0.2em>\<Leftrightarrow\><separating-space|0.2em>i<rsub|2>=\<alpha\>-\<theta\><rsub|2>
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</equation*>
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<subsection|Réflexion et réfraction sur une interface
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<math|n<rsub|1>\<rightarrow\>n<rsub|2>>>
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|
Réflexion :
|
|
|
|
<\equation*>
|
|
<text|toujours,<space|1em>avec un angle><space|1em>i<rsub|<text|r>>=i<space|1em>\<Rightarrow\><space|1em>\<alpha\><rsub|<text|r>>=\<theta\><rsub|<text|normale>>-i
|
|
</equation*>
|
|
|
|
Réfraction (transmission) :
|
|
|
|
<\equation*>
|
|
n<rsub|1>*sin<around*|(|i|)>=n<rsub|2>*sin<around*|(|i<rsub|<text|t>>|)><space|2em><stack|<tformat|<cwith|1|-1|1|-1|cell-halign|l>|<table|<row|<cell|<text|toujours
|
|
lorsque <math|n<rsub|1>\<leqslant\>n<rsub|2>>>>>|<row|<cell|<text|pour
|
|
<math|i\<leqslant\>i<rsub|<text|cr>>=arcsin<around*|(|1/\<gamma\>|)>>
|
|
lorsque <math|n<rsub|1>\<gtr\>n<rsub|2>>>>>>>><space|1em><text|avec><space|1em>\<gamma\>\<assign\><frac|n<rsub|1>|n<rsub|2>>
|
|
</equation*>
|
|
|
|
<\equation*>
|
|
\<Rightarrow\><space|1em>\<alpha\><rsub|<text|t>><separating-space|0.2em>=<separating-space|0.2em>\<theta\><rsub|<text|anti-normale>>+i<rsub|<text|t>><separating-space|0.4em>=<separating-space|0.4em><around*|(|\<theta\><rsub|<text|normale>>+\<pi\>|)>+arcsin<around*|<left|(|-1>|\<gamma\>\<cdot\>sin<around*|(|i|)>|<right|)|-1>>
|
|
</equation*>
|
|
|
|
Coefficients de réflexion en puissance (c'est juste
|
|
<math|R=<around*|\||\<rho\>|\|><rsup|2>>) :
|
|
|
|
<\equation*>
|
|
R<rsub|<text|TE>><separating-space|0.2em>=<separating-space|0.2em><around*|\||<frac|n<rsub|1>*cos
|
|
i-n<rsub|2>*cos i<rsub|<text|t>>|n<rsub|1>*cos i+n<rsub|2>*cos
|
|
i<rsub|<text|t>>>|\|><rsup|2><space|1em><text|et><space|1em>R<rsub|<text|TM>><separating-space|0.2em>=<separating-space|0.2em><around*|\||<frac|n<rsub|2>*cos
|
|
i-n<rsub|1>*cos i<rsub|<text|t>>|n<rsub|2>*cos i+n<rsub|1>*cos
|
|
i<rsub|<text|t>>>|\|><rsup|2>
|
|
</equation*>
|
|
|
|
développés<\footnote>
|
|
<\equation*>
|
|
*i<rsub|<text|t>>=arcsin<around*|<left|(|-1>|<frac|n<rsub|1>|n<rsub|2>>*sin<around*|(|i|)>|<right|)|-1>><space|1em>\<Longrightarrow\><space|1em>cos<around*|(|*i<rsub|<text|t>>|)>=<choice|<tformat|<table|<row|<cell|<sqrt|1-\<gamma\><rsup|2>*sin<rsup|2>
|
|
i>>|<cell|<text|si <math|i\<leqslant\>i<rsub|<text|cr>>>>>>|<row|<cell|\<mathi\><sqrt|\<gamma\><rsup|2>*sin<rsup|2>
|
|
i-1>>|<cell|<text|si <math|i\<gtr\>i<rsub|<text|cr>>>>>>>>><space|1em><text|avec><space|1em>x=<frac|n<rsub|1>|n<rsub|2>>*sin<around*|(|i|)>
|
|
</equation*>
|
|
</footnote> :
|
|
|
|
<\equation*>
|
|
R<rsub|<text|TE>,<text|TM>>=<choice|<tformat|<table|<row|<cell|<around*|(|<frac|a-b|a+b>|)><rsup|2>>|<cell|<text|si
|
|
<math|i\<leqslant\>i<rsub|<text|cr>>>>>>|<row|<cell|1<space|1em><stack|<tformat|<table|<row|<cell|<very-small|<text|(refl.<nbsp>int.<nbsp>tot.)>>>>>>>>|<cell|<text|si
|
|
<math|i\<gtr\>i<rsub|<text|cr>>>>>>>>><space|1em><text|avec><space|1em><stack|<tformat|<cwith|1|-1|1|-1|cell-halign|l>|<table|<row|<cell|a<rsub|<text|TE>>=\<gamma\>*cos
|
|
i>>|<row|<cell|a<rsub|<text|TM>>=\<gamma\><rsup|-1>*cos
|
|
i>>>>><space|1em><text|et><space|1em>b=<sqrt|1-\<gamma\><rsup|2>*sin<rsup|2>
|
|
i>
|
|
</equation*>
|
|
|
|
Réflexion nulle (<math|a=b>) si :
|
|
|
|
<\itemize-dot>
|
|
<item><math|R<rsub|<text|TE>>=0<space|1em>\<Longleftrightarrow\><space|1em>cos<rsup|2>
|
|
i+sin<rsup|2> i=1=\<gamma\><rsup|2><space|2em>\<Longleftrightarrow\><space|1em>n<rsub|1>=n<rsub|2>>
|
|
|
|
<item><math|R<rsub|<text|TM>>=0<space|1em>\<Longleftrightarrow\><space|1em>cos<rsup|2>
|
|
i+\<gamma\><rsup|4>*sin<rsup|2> i=\<gamma\><rsup|2><space|1em>\<Longleftrightarrow\><space|1em>n<rsub|1>=n<rsub|2><space|1em><text|ou><space|1em>i=i<rsub|<text|Brewster>>=arctan<around*|<left|(|-1>|<frac|n<rsub|2>|n<rsub|1>>|<right|)|-1>>>
|
|
|
|
<small|<\eqnarray*>
|
|
<tformat|<table|<row|<cell|<text|car><space|3em>R<rsub|<text|TM>>=0>|<cell|\<Longleftrightarrow\>>|<cell|cos<rsup|2>+\<gamma\><rsup|4>*sin<rsup|2><separating-space|0.2em>=<separating-space|0.2em>\<gamma\><rsup|2><separating-space|0.2em>=<separating-space|0.2em>\<gamma\><rsup|2>*cos<rsup|2>+\<gamma\><rsup|2>*sin<rsup|2>>>|<row|<cell|>|<cell|\<Longleftrightarrow\>>|<cell|<around*|(|1-\<gamma\><rsup|2>|)>*cos<rsup|2>=\<gamma\><rsup|2>*<around*|(|1-\<gamma\><rsup|2>|)>*sin<rsup|2>>>|<row|<cell|>|<cell|\<Longleftrightarrow\>>|<cell|\<gamma\>=1<space|1em><text|ou><space|1em>tan<rsup|2>=\<gamma\><rsup|-2>>>>>
|
|
</eqnarray*>>
|
|
</itemize-dot>
|
|
|
|
Coefficients de transmission en puissance (ce n'est <em|pas><\footnote>
|
|
Coefficients de réflexion en amplitude :
|
|
|
|
<\equation*>
|
|
\<rho\><rsub|<text|TE>><separating-space|0.2em>=<separating-space|0.2em><frac|n<rsub|1>*cos
|
|
i-n<rsub|2>*cos i<rsub|<text|t>>|n<rsub|1>*cos i+n<rsub|2>*cos
|
|
i<rsub|<text|t>>><space|1em><text|et><space|1em>\<rho\><rsub|<text|TM>><separating-space|0.2em>=<separating-space|0.2em><frac|n<rsub|2>*cos
|
|
i-n<rsub|1>*cos i<rsub|<text|t>>|n<rsub|2>*cos i+n<rsub|1>*cos
|
|
i<rsub|<text|t>>>
|
|
</equation*>
|
|
|
|
développés :
|
|
|
|
<\equation*>
|
|
\<rho\><rsub|<text|TE>,<text|TM>>=<choice|<tformat|<table|<row|<cell|<frac|a-b|a+b>>|<cell|<text|si
|
|
<math|i\<leqslant\>i<rsub|<text|cr>>>>>>|<row|<cell|<frac|a-\<mathi\>*b|a+\<mathi\>*b><space|1em><stack|<tformat|<cwith|1|-1|1|-1|cell-lsep|0pt>|<cwith|1|-1|1|-1|cell-rsep|0pt>|<cwith|1|-1|1|-1|cell-bsep|0pt>|<cwith|1|-1|1|-1|cell-tsep|0pt>|<table|<row|<cell|<stack|<tformat|<table|<row|<cell|<very-small|<text|(de
|
|
norme 1)>>>>>>>>>>>>>|<cell|<text|si
|
|
<math|i\<gtr\>i<rsub|<text|cr>>>>>>>>><space|1em><text|avec><space|1em><stack|<tformat|<cwith|1|-1|1|-1|cell-halign|l>|<table|<row|<cell|a<rsub|<text|TE>>=\<gamma\>*cos
|
|
i>>|<row|<cell|a<rsub|<text|TM>>=\<gamma\><rsup|-1>*cos
|
|
i>>>>><space|1em><text|et><space|1em>b=<sqrt|<around*|\||1-\<gamma\><rsup|2>*sin<rsup|2>
|
|
i|\|>>
|
|
</equation*>
|
|
|
|
Coefficients de transmission en amplitude :
|
|
|
|
<\equation*>
|
|
\<tau\><rsub|<text|TE>><separating-space|0.2em>=<separating-space|0.2em><frac|2*n<rsub|1>*cos
|
|
i|n<rsub|1>*cos i+n<rsub|2>*cos i<rsub|<text|t>>><space|1em><text|et><space|1em>\<tau\><rsub|<text|TM>><separating-space|0.2em>=<separating-space|0.2em><frac|2*n<rsub|1>*cos
|
|
i|n<rsub|2>*cos i+n<rsub|1>*cos i<rsub|<text|t>>>
|
|
</equation*>
|
|
|
|
développés :
|
|
|
|
<\equation*>
|
|
\<tau\><rsub|<text|TE>,<text|TM>>=<choice|<tformat|<table|<row|<cell|<frac|c|a+b>>|<cell|<text|si
|
|
<math|i\<leqslant\>i<rsub|<text|cr>>>>>>|<row|<cell|<frac|c|a+\<mathi\>*b><space|1em><very-small|<text|(ampl.
|
|
plasmon)>>>|<cell|<text|si <math|i\<gtr\>i<rsub|<text|cr>>>>>>>>><space|1em><text|avec><space|1em><stack|<tformat|<cwith|1|-1|1|-1|cell-halign|l>|<table|<row|<cell|c<rsub|<text|TE>>=2*\<gamma\>*cos
|
|
i>>|<row|<cell|c<rsub|<text|TM>>=2*cos
|
|
i>>>>><space|2em>\<longrightarrow\><space|1em>T=<frac|n<rsub|2>*cos
|
|
i<rsub|<text|t>>|n<rsub|1>*cos i>*<around*|\||\<tau\>|\|><rsup|2>
|
|
</equation*>
|
|
</footnote> juste <math|T=<around*|\||\<tau\>|\|><rsup|2>>) :
|
|
|
|
<\equation*>
|
|
T<separating-space|0.2em>=<separating-space|0.2em>1-R
|
|
</equation*>
|
|
|
|
par conservation de l'énergie. Ici, l'histoire d'angle de vue de
|
|
l'interface ne rentre pas en compte, puisqu'on a seulement des rayons
|
|
<math|\<rightarrow\>> l'intégration spatiale est déjà faite, et on manipule
|
|
des <em|puissances>, pas des intensités. Cela est cohérent avec le fait
|
|
d'oublier la formule de l'éclairement <math|\<cal-E\>=I*cos \<theta\>>.
|
|
|
|
<subsection|Objets diffusants : BRDF>
|
|
|
|
<slink|https://en.wikipedia.org/wiki/Bidirectional_reflectance_distribution_function>
|
|
|
|
Propriété d'une <name|brdf> \S <name|2d> \T
|
|
<math|f<around*|(|i,i<rsub|<text|r>>|)>> :
|
|
|
|
<\itemize-dot>
|
|
<item>réciprocité : <math|f<around*|(|i,i<rsub|<text|r>>|)>=f<around*|(|i<rsub|<text|r>>,i|)>><space|1em>(crucial
|
|
pour avoir l'équivalence <name|2d> <math|\<leftrightarrow\>> <name|3d>
|
|
<math|z>-invar. : les rayons \S perdus \T hors du plan sont compensés par
|
|
tous rayons \S hors plan \T (venant des points <math|z\<neq\>0> des
|
|
sources) réfléchis dans le plan)
|
|
|
|
<item>positivité : <math|f<around*|(|i,i<rsub|<text|r>>|)>\<gtr\>0>
|
|
|
|
<item>conservation de l'énergie : <math|<smash|<with|math-display|true|\<forall\>i,<separating-space|0.2em><big|int><rsub|-<frac|\<pi\>|2>><rsup|<frac|\<pi\>|2>>\<mathd\>i<rsub|<text|r>>
|
|
f<around*|(|i,i<rsub|<text|r>>|)>=2\<pi\>>>> (les absorptions sont réglés
|
|
avec la variable d'albédo dédiée)
|
|
</itemize-dot>
|
|
|
|
\;
|
|
</body>
|
|
|
|
<\initial>
|
|
<\collection>
|
|
<associate|page-medium|paper>
|
|
</collection>
|
|
</initial>
|
|
|
|
<\references>
|
|
<\collection>
|
|
<associate|auto-1|<tuple|1|1>>
|
|
<associate|auto-2|<tuple|2|1>>
|
|
<associate|auto-3|<tuple|3|1>>
|
|
<associate|auto-4|<tuple|4|2>>
|
|
<associate|auto-5|<tuple|5|3>>
|
|
<associate|auto-6|<tuple|6|3>>
|
|
<associate|auto-7|<tuple|7|4>>
|
|
<associate|footnote-1|<tuple|1|2>>
|
|
<associate|footnote-2|<tuple|2|4>>
|
|
<associate|footnote-3|<tuple|3|4>>
|
|
<associate|footnr-1|<tuple|1|2>>
|
|
<associate|footnr-2|<tuple|2|4>>
|
|
<associate|footnr-3|<tuple|3|4>>
|
|
</collection>
|
|
</references>
|
|
|
|
<\auxiliary>
|
|
<\collection>
|
|
<\associate|toc>
|
|
<with|par-left|<quote|1tab>|1.<space|2spc><with|font-shape|<quote|small-caps>|2d>
|
|
vs. <with|font-shape|<quote|small-caps>|3d>
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
|
<no-break><pageref|auto-1>>
|
|
|
|
<with|par-left|<quote|1tab>|2.<space|2spc>Angles modulo
|
|
<with|font-family|<quote|rm>|<with|mode|<quote|math>|<with|math-condensed|<quote|true>|2*\<pi\>>>>
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
|
<no-break><pageref|auto-2>>
|
|
|
|
<with|par-left|<quote|1tab>|3.<space|2spc>Intersection segment /
|
|
demi-droite <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
|
<no-break><pageref|auto-3>>
|
|
|
|
<with|par-left|<quote|1tab>|4.<space|2spc>Intersection arc de cercle /
|
|
demi-droite <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
|
<no-break><pageref|auto-4>>
|
|
|
|
<with|par-left|<quote|1tab>|5.<space|2spc>Construction d'un arc de
|
|
cercle à partir de <with|font-family|<quote|rm>|<with|mode|<quote|math>|<around*|(|A,B,R|)>>>
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
|
<no-break><pageref|auto-5>>
|
|
|
|
<with|par-left|<quote|1tab>|6.<space|2spc>Réflexion et réfraction sur
|
|
une interface <with|font-family|<quote|rm>|<with|mode|<quote|math>|n<rsub|1>\<rightarrow\>n<rsub|2>>>
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
|
<no-break><pageref|auto-6>>
|
|
|
|
<with|par-left|<quote|1tab>|7.<space|2spc>Objets diffusants : BRDF
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
|
<no-break><pageref|auto-7>>
|
|
</associate>
|
|
</collection>
|
|
</auxiliary> |