Caustic (mathematics) - 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 and the mirror curve is parametrised as . The normal vector at a point is ; the reflection of the direction vector is (normal needs special normalization)

2\mbox{proj}_nd-d=\frac{2n}{\sqrt{n\cdot n}}\frac{n\cdot d}{\sqrt{n\cdot n}}-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}

Having components of found reflected vector treat it as a tangent

Using the simplest envelope form

=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')
F_t(x,y,t)=2x(bu'u''-a(u'v''+u''v')-bv'v'') -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 gives a linear system in and so it is elementary to obtain a parametrisation of the catacaustic. Cramer's rule would serve.

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