Reflection in a Translation Invariant Surface

Abstract

We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by constructing explicitly the focal set of the reflected line congruence (2-parameter family of oriented lines in ${\Bbb{R}}^3$) with the aid of the natural complex structure on the space of all oriented affine lines.