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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Caustic (mathematics) ： ウィキペディア英語版 Caustic (mathematics) In differential geometry and geometric optics, a caustic is the envelope of rays either reflected or refracted by a manifold. It is related to the concept of caustics in optics. The ray's source may be a point (called the radiant) or parallel rays from a point at infinity, in which case a direction vector of the rays must be specified. More generally, especially as applied to symplectic geometry and singularity theory, a caustic is the critical value set of a Lagrangian mapping where is a Lagrangian immersion of a Lagrangian submanifold ''L'' into a symplectic manifold ''M'', and is a Lagrangian fibration of the symplectic manifold ''M''. The caustic is a subset of the Lagrangian fibration's base space ''B''. ==Catacaustic== A catacaustic is the reflective case. With a radiant, it is the evolute of the orthotomic of the radiant. The planar, parallel-source-rays case: suppose the direction vector is $(a,b)$ and the mirror curve is parametrised as $(u(t),v(t))$. The normal vector at a point is $(-v\text{'}(t),u\text{'}(t))$; the reflection of the direction vector is (normal needs special normalization) :$2\backslash mbox\_nd-d=\backslash frac-d=\backslash frac$ Having components of found reflected vector treat it as a tangent :$(x-u)(bu\text{'}^2-2au\text{'}v\text{'}-bv\text{'}^2)=(y-v)(av\text{'}^2-2bu\text{'}v\text{'}-au\text{'}^2).$ Using the simplest envelope form :$F(x,y,t)=(x-u)(bu\text{'}^2-2au\text{'}v\text{'}-bv\text{'}^2)-(y-v)(av\text{'}^2-2bu\text{'}v\text{'}-au\text{'}^2)$ $=x(bu\text{'}^2-2au\text{'}v\text{'}-bv\text{'}^2)$ -y(av'^2-2bu'v'-au'^2) +b(uv'^2-uu'^2-2vu'v') +a(-vu'^2+vv'^2+2uu'v') :$F\_t(x,y,t)=2x(bu\text{'}u\text{'}\text{'}-a(u\text{'}v\text{'}\text{'}+u\text{'}\text{'}v\text{'})-bv\text{'}v\text{'}\text{'})$ -2y(av'v''-b(u''v'+u'v'')-au'u'') +b( u'v'^2 +2uv'v'' -u'^3 -2uu'u'' -2u'v'^2 -2u''vv' -2u'vv'') +a(-v'u'^2 -2vu'u'' +v'^3 +2vv'v'' +2v'u'^2 +2v''uu' +2v'uu'') which may be unaesthetic, but $F=F\_t=0$ gives a linear system in $(x,y)$ and so it is elementary to obtain a parametrisation of the catacaustic. Cramer's rule would serve. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Caustic (mathematics)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.