Caustic (mathematics)

For other uses, see Caustic (disambiguation).

Reflective Caustic generated from a circle and parallel rays

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.

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,t²). Then

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

F(x,y,t) = (x − t)(1 − 4t²) + 4t(y − t²) = x(1 − 4t²) + 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.

See also

• Cut locus (Riemannian manifold)

External links

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