# Caustic

The envelope of rays reflected or refracted by a given curve. The caustic of reflected rays is called a catacaustic, that of refracted rays — a diacaustic.

For example, the catacaustic of a parallel beam of rays reflected from a semi-circle is part of an epicycloid (see Fig. a); the diacaustic of a pencil of rays emanating from a point $A$ lying in a denser medium and refracted by a straight line is part of an astroid with cusp $A'$ the distance of which from the line is $1/n$ times that of the point $A$ (where $n$ is the refraction index) (see Fig. b).

#### References

 [1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)

One way of describing the caustic resulting from rays from a point $Q$ reflected by a curve $\gamma$ is as follows. Take a point $T$ on $\gamma$ and draw the tangent $l$; from $Q$ draw the perpendicular to this tangent and let $R$ be the point on the perpendicular on the other side of $l$ at the same distance from $l$ as $Q$. Then the caustic by reflection is the evolute of the curve described by $R$ as $T$ runs over $\gamma$. This result is due to A. Quetelet. It follows readily that the caustic by reflection of the circle is the Pascal limaçon. There is a similar result for a caustic by refraction.