Caustic (mathematics)

Concentration of light, especially sunlight, can burn. The word caustic, in fact, comes from the Greek καυστός, burnt, via the Latin causticus, burning. A common situation where caustics are visible is when light shines on a drinking glass. The glass casts a shadow, but also produces a curved region of bright light. In ideal circumstances (including perfectly parallel rays, as if from a point source at infinity), a nephroid-shaped patch of light can be produced. Rippling caustics are commonly formed when light shines through waves on a body of water. Another familiar caustic is the rainbow. Scattering of light by raindrops causes different wavelengths of light to be refracted into arcs of differing radius, producing the bow. ==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+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 -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. ==See also==