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 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.

More generally, especially as applied to symplectic geometry and singularity theory, a caustic is the critical value set of a Lagrangian mapping (π ○ i) : L ↪ M ↠ B; where i : L ↪ M is a Lagrangian immersion of a Lagrangian submanifold L into a symplectic manifold M, and π : M ↠ B is a Lagrangian fibration of the symplectic manifold M. The caustic is a subset of the Lagrangian fibration's base space B.[1]

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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 (a,b) and the mirror curve is parametrised as (u(t),v(t)). The normal vector at a point is (-v'(t),u'(t)); 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%2Bu'^2}

Having components of found reflected vector treat it as a tangent

(x-u)(bu'^2-2au'v'-bv'^2)=(y-v)(av'^2-2bu'v'-au'^2).

Using the simplest envelope form

F(x,y,t)=(x-u)(bu'^2-2au'v'-bv'^2)-(y-v)(av'^2-2bu'v'-au'^2) =x(bu'^2-2au'v'-bv'^2) -y(av'^2-2bu'v'-au'^2) %2Bb(uv'^2-uu'^2-2vu'v') %2Ba(-vu'^2%2Bvv'^2%2B2uu'v')
F_t(x,y,t)=2x(bu'u''-a(u'v''%2Bu''v')-bv'v'') -2y(av'v''-b(u''v'%2Bu'v'')-au'u'') %2Bb( u'v'^2 %2B2uv'v'' -u'^3 -2uu'u'' -2u'v'^2 -2u''vv' -2u'vv'') %2Ba(-v'u'^2 -2vu'u'' %2Bv'^3 %2B2vv'v'' %2B2v'u'^2 %2B2v''uu' %2B2v'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.

Example

Let the direction vector be (0,1) and the mirror be (t,t^2). Then

u'=1   u''=0   v'=2t   v''=2   a=0   b=1
F(x,y,t)=(x-t)(1-4t^2)%2B4t(y-t^2)=x(1-4t^2)%2B4ty-t
F_t(x,y,t)=-8tx%2B4y-1

and F=F_t=0 has solution (0,1/4); i.e., light entering a parabolic mirror parallel to its axis is reflected through the focus.

References

  1. ^ Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985). The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1. Birkhäuser. ISBN 0817631879. 

See also

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