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
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')
- Ft(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 looks horrid, but F = Ft = 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
- Ft(x,y,t) = - 8tx + 4y - 1
and F = Ft = 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.
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