Caustic (mathematics)

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.

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 $(-v\text{'}(t),u\text{'}(t))$; the reflection of the direction vector is (normal needs special normalization):$2mbox\{proj\}\_nd-d=frac\{2n\}\{sqrt\{ncdot\; nfrac\{ncdot\; d\}\{sqrt\{ncdot\; n-d=2nfrac\{ncdot\; d\}\{ncdot\; n\}-d=frac\{(av\text{'}^2-2bu\text{'}v\text{'}-au\text{'}^2,bu\text{'}^2-2au\text{'}v\text{'}-bv\text{'}^2)\}\{v\text{'}^2+u\text{'}^2\}$Having components of found reflected vector treat it as a tangent:$(x-u)(bu\text{'}^2-2au\text{'}v\text{'}-bv\text{'}^2)=(y-v)(av\text{'}^2-2bu\text{'}v\text{'}-au\text{'}^2).$Using the simplest envelope form:$F(x,y,t)=(x-u)(bu\text{'}^2-2au\text{'}v\text{'}-bv\text{'}^2)-(y-v)(av\text{'}^2-2bu\text{'}v\text{'}-au\text{'}^2)$ $=x(bu\text{'}^2-2au\text{'}v\text{'}-bv\text{'}^2)-y(av\text{'}^2-2bu\text{'}v\text{'}-au\text{'}^2)+b(uv\text{'}^2-uu\text{'}^2-2vu\text{'}v\text{'})+a(-vu\text{'}^2+vv\text{'}^2+2uu\text{'}v\text{'})$:$F\_t(x,y,t)=2x(bu\text{'}u"-a(u\text{'}v"+u"v\text{'})-bv\text{'}v")-2y(av\text{'}v"-b(u"v\text{'}+u\text{'}v")-au\text{'}u")+b(\; u\text{'}v\text{'}^2\; +2uv\text{'}v"\; -u\text{'}^3\; -2uu\text{'}u"\; -2u\text{'}v\text{'}^2\; -2u"vv\text{'}\; -2u\text{'}vv")+a(-v\text{'}u\text{'}^2\; -2vu\text{'}u"\; +v\text{'}^3\; +2vv\text{'}v"\; +2v\text{'}u\text{'}^2\; +2v"uu\text{'}\; +2v\text{'}uu")$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:$u\text{'}=1$ $u"=0$ $v\text{'}=2t$ $v"=2$ $a=0$ $b=1$:$F(x,y,t)=(x-t)(1-4t^2)+4t(y-t^2)=x(1-4t^2)+4ty-t$:$F\_t(x,y,t)=-8tx+4y-1$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 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]