caustic (mathematics)

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.

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

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} so the reflected ray satisfies

(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+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) which looks horrid, 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.

Diacaustic

A diacaustic is the refractive case. It is complicated by the need for another datum (refractive index) and the fact refraction is not linear -- Snell's law is "ugly" in pure vector notation.

External links

Mathworld

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