Caustic (mathematics)

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 ${\displaystyle (a,b)}$ and the mirror curve is parametrised as ${\displaystyle (u(t),v(t))}$. The normal vector at a point is ${\displaystyle (-v^{\prime }(t),u^{\prime }(t))}$; the reflection of the direction vector is (normal needs special normalization)

${\displaystyle 2{\text{proj}}_{n}d-d=\frac{2n}{\sqrt{n\cdot n}}\frac{n\cdot d}{\sqrt{n\cdot n}}-d=2n\frac{n\cdot d}{n\cdot n}-d=\frac{(a{v}^{\prime 2}-2b{u}^{\prime }{v}^{\prime }-a{u}^{\prime 2},b{u}^{\prime 2}-2a{u}^{\prime }{v}^{\prime }-b{v}^{\prime 2})}{v^{\prime 2}+u^{\prime 2}}}$

Having components of found reflected vector treat it as a tangent

${\displaystyle (x-u)(b{u}^{\prime 2}-2a{u}^{\prime }{v}^{\prime }-b{v}^{\prime 2})=(y-v)(a{v}^{\prime 2}-2b{u}^{\prime }{v}^{\prime }-a{u}^{\prime 2}).}$

Using the simplest envelope form

${\displaystyle F(x,y,t)=(x-u)(b{u}^{\prime 2}-2a{u}^{\prime }{v}^{\prime }-b{v}^{\prime 2})-(y-v)(a{v}^{\prime 2}-2b{u}^{\prime }{v}^{\prime }-a{u}^{\prime 2})}$

${\displaystyle =x(b{u}^{\prime 2}-2a{u}^{\prime }{v}^{\prime }-b{v}^{\prime 2})-y(a{v}^{\prime 2}-2b{u}^{\prime }{v}^{\prime }-a{u}^{\prime 2})+b(u{v}^{\prime 2}-u{u}^{\prime 2}-2v{u}^{\prime }{v}^{\prime })+a(-v{u}^{\prime 2}+v{v}^{\prime 2}+2u{u}^{\prime }{v}^{\prime })}$

${\displaystyle F_t(x,y,t)=2x(b{u}^{\prime }{u}^{″}-a(u^{\prime }{v}^{″}+u^{″}{v}^{\prime })-b{v}^{\prime }{v}^{″})-2y(a{v}^{\prime }{v}^{″}-b(u^{″}{v}^{\prime }+u^{\prime }{v}^{″})-a{u}^{\prime }{u}^{″})}$

${\displaystyle +b(u^{\prime }{v}^{\prime 2}+2u{v}^{\prime }{v}^{″}-u^{\prime 3}-2u{u}^{\prime }{u}^{″}-2u^{\prime }{v}^{\prime 2}-2u^{″}v{v}^{\prime }-2u^{\prime }v{v}^{″})+a(-v^{\prime }{u}^{\prime 2}-2v{u}^{\prime }{u}^{″}+v^{\prime 3}+2v{v}^{\prime }{v}^{″}+2v^{\prime }{u}^{\prime 2}+2v^{″}u{u}^{\prime }+2v^{\prime }u{u}^{″})}$

which may be unaesthetic, but ${\displaystyle F=F_t=0}$ gives a linear system in ${\displaystyle (x,y)}$ 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 ${\displaystyle (t,t^2).}$ Then

${\displaystyle u^{\prime }=1}$ ${\displaystyle u^{″}=0}$ ${\displaystyle v^{\prime }=2t}$ ${\displaystyle v^{″}=2}$ ${\displaystyle a=0}$ ${\displaystyle b=1}$

${\displaystyle F(x,y,t)=(x-t)(1-4t^2)+4t(y-t^2)=x(1-4t^2)+4ty-t}$

${\displaystyle F_t(x,y,t)=-8tx+4y-1}$

and ${\displaystyle F=F_t=0}$ has solution ${\displaystyle (0,1/4)}$; i.e., light entering a parabolic mirror parallel to its axis is reflected through the focus.