ar X iv : m at h / 05 10 49 3 v 1 [ m at h . D G ] 2 3 O ct 2 00 5 REFLECTION IN A TRANSLATION INVARIANT SURFACE

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 R 3) with the aid of the natural complex structure on the space of all oriented affine lines. The purpose of this paper is to prove the following Theorem: Main Theorem: 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. In contrast to the focal surface, the reflected wavefront is not translation invariant , in general. There have been many investigations of generic focal sets of line congruences [1] [2] [5]. Rather than work in the generic setting, we compute the focal set explicitly in this special case. This we do by applying recent work on immersed surfaces in the space T of oriented affine lines in R 3 [3] [4]. The next section contains a summary of the background material on the complex geometry of T and the focal sets of arbitrary line congruences. It also details the reflection of a line congruence in an oriented surface in R 3. In Section 2 we solve the problem of reflection of a point source off an arbitrary translation invariant surface (Proposition 2). We then compute the focal set and thus prove the Main Theorem.