Caustic (mathematics)

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For other uses, see Caustic.

Reflective Caustic generated from a circle and parallel rays

In differential geometry 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.

1 Catacaustic
- 1.1 Example
2 Diacaustic
3 External links

[edit] 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)(b u' 2 - 2 a u' v' - b v' 2) = (y - v)(a v' 2 - 2 b u' v' - a u' 2).

Using the simplest envelope form

F (x, y, t) = (x - u)(b u' 2 - 2 a u' v' - b v' 2) - (y - v)(a v' 2 - 2 b u' v' - a u' 2)

= x (b u' 2 - 2 a u' v' - b v' 2) - y (a v' 2 - 2 b u' v' - a u' 2) + b (u v' 2 - u u' 2 - 2 v u' v') + a ( - v u' 2 + v v' 2 + 2 u u' v')

F t (x, y, t) = 2 x (b u' u'' - a (u' v'' + u'' v') - b v' v'') - 2 y (a v' v'' - b (u'' v' + u' v'') - a u' u'') + b (u' v' 2 + 2 u v' v'' - u' 3 - 2 u u' u'' - 2 u' v' 2 - 2 u'' v v' - 2 u' v v'') + a ( - v' u' 2 - 2 v u' u'' + v' 3 + 2 v v' v'' + 2 v' u' 2 + 2 v'' u u' + 2 v' u u'')

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.

[edit] Example

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

u' = 1

u'' = 0

v' = 2 t

v'' = 2

a = 0

b = 1

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

F t (x, y, t) = - 8 t x + 4 y - 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.

[edit] Diacaustic

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