- Caustic (mathematics)
In
differential geometry a caustic is the envelope of rays either reflected orrefracted by amanifold . 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 theorthotomic 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):2mbox{proj}_nd-d=frac{2n}{sqrt{ncdot nfrac{ncdot d}{sqrt{ncdot n-d=2nfrac{ncdot d}{ncdot 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,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-1and 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 that refraction is not
linear --Snell's law is "ugly" in pure vector notation (unless the refractive index varies smoothly in space).External links
* [http://mathworld.wolfram.com/Caustic.html Mathworld]
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