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 (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]

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)

$2\mbox{proj}_nd-d=\frac{2n}{\sqrt{n\cdot n}}\frac{n\cdot d}{\sqrt{n\cdot n}}-d=2n\frac{n\cdot d}{n\cdot n}-d=\frac{ (av'^2-2bu'v'-au'^2,bu'^2-2au'v'-bv'^2) }{v'^2+u'^2}$

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

$F(x,y,t)=(x-u)(bu'^2-2au'v'-bv'^2)-(y-v)(av'^2-2bu'v'-au'^2)$ $=x(bu'^2-2au'v'-bv'^2) -y(av'^2-2bu'v'-au'^2) +b(uv'^2-uu'^2-2vu'v') +a(-vu'^2+vv'^2+2uu'v')$

$F_t(x,y,t)=2x(bu'u''-a(u'v''+u''v')-bv'v'') -2y(av'v''-b(u''v'+u'v'')-au'u'') +b( u'v'^2 +2uv'v'' -u'^3 -2uu'u'' -2u'v'^2 -2u''vv' -2u'vv'') +a(-v'u'^2 -2vu'u'' +v'^3 +2vv'v'' +2v'u'^2 +2v''uu' +2v'uu'')$

which may be unaesthetic, but $F=F_t=0$ 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

$u'=1$   $u''=0$   $v'=2t$   $v''=2$   $a=0$   $b=1$

$F(x,y,t)=(x-t)(1-4t^2)+4t(y-t^2)=x(1-4t^2)+4ty-t$

$F_t(x,y,t)=-8tx+4y-1$

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 0-8176-3187-9.

See also

• Cut locus (Riemannian manifold)
• Last geometric statement of Jacobi

External links

• Weisstein, Eric W., "Caustic", MathWorld.