Caustic (mathematics)
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In differential geometry a caustic is the envelope of rays either reflected or refracted by a manifold. Obviously it is related to the optical concept of caustics.
The ray's source may be a point (called the radiant) or infinity, in which case a direction vector must be specified.
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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 <math>(a,b)</math> and the mirror curve is parametrised as <math>(u(t),v(t))</math>. The normal vector at a point is <math>(-v'(t),u'(t))</math>; the reflection of the direction vector is
- <math>2\mbox{proj}_nd-d=2n\frac{n\cdot d}{n\cdot n}-d=\frac{
(av'^2-2bu'v'-au'^2,bu'^2-2au'v'-bv'^2) }{v'^2+u'^2}</math> so the reflected ray satisfies
- <math>(x-u)(bu'^2-2au'v'-bv'^2)=(y-v)(av'^2-2bu'v'-au'^2).</math>
Using the simplest envelope form
- <math>F(x,y,t)=(x-u)(bu'^2-2au'v'-bv'^2)-(y-v)(av'^2-2bu'v'-au'^2)</math> <math>=x(bu'^2-2au'v'-bv'^2)
-y(av'^2-2bu'v'-au'^2) +b(uv'^2-uu'^2-2vu'v') +a(-vu'^2+vv'^2+2uu'v')</math>
- <math>F_t(x,y,t)=2x(bu'u-a(u'v+uv')-bv'v)
-2y(av'v-b(uv'+u'v)-au'u) +b( u'v'^2 +2uv'v -u'^3 -2uu'u -2u'v'^2 -2uvv' -2u'vv) +a(-v'u'^2 -2vu'u +v'^3 +2vv'v +2v'u'^2 +2vuu' +2v'uu)</math> which looks horrid, but <math>F=F_t=0</math> gives a linear system in <math>(x,y)</math> and so it is elementary to obtain a parametrisation of the catacaustic. Cramer's rule would serve.
Example
Let the direction vector be (0,1) and the mirror be <math>(t,t^2).</math> Then
- <math>u'=1</math> <math>u=0</math> <math>v'=2t</math> <math>v=2</math> <math>a=0</math> <math>b=1</math>
- <math>F(x,y,t)=(x-t)(1-4t^2)+4t(y-t^2)=x(1-4t^2)+4ty-t</math>
- <math>F_t(x,y,t)=-8tx+4y-1</math>
and <math>F=F_t=0</math> has solution <math>(0,1/4)</math>; i.e., light entering a parabolic mirror parallel to its axis is reflected through the focus.
Diacaustic
A diacaustic is the refractive case. It is complicated by the need for another datum (refractive index) and the fact that refraction is not linear -- Snell's law is "ugly" in pure vector notation.