ar X iv : m at h / 04 11 18 9 v 1 [ m at h . D G ] 9 N ov 2 00 4 SYMMETRIES OF THE COFFEECUP CAUSTIC

It is shown that the caustic formed by the reflection of a plane wave off the inside of a cylinder is independent of the angle of incidence of the wave with the axis of the cylinder. Moreover, the caustic formed by the reflection of a point source off a cylinder is proven to be translation symmetric. This is done by constructing explicitly the focal sets of the reflected line congruences (2-parameter families of oriented lines in R 3) with the aid of the natural complex structure on the space of all oriented affine lines. The coffeecup caustic is the bright curve commonly observed on the top of a cup of coffee formed by a strong light reflected off the side of the cup. Mathematically, such caustics are attributed to the focal set of the 2-parameter family of oriented lines generated by the reflection of the incoming rays [1] [2] [5]. The purpose of this paper is to demonstrate that the coffeecup caustic enjoys an invariance that is not immediately obvious. In particular, we model the caustic by the reflection of a plane wave off a cylinder and show that the caustic is invariant under change of the angle of incidence of the wave. While the caustic itself is well-known, it is normally computed for horizontal rays only, and thus the extra symmetry is not detected. The symmetry can be traced back to the fact that the focal surface of the reflection of a point source off a cylinder is symmetric along the axis of the cylinder (even though the reflected line congruence itself has no such symmetry). We prove the preceding by explicitly constructing the focal sets involved as pa-rameterised sets in R 3. 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 reflection of line congruences. Section 2 discusses focal sets of arbitrary line congruences and how to compute them for a given parameterisation (Theorem 2). At the start of Section 3 we solve the problem of reflection of an arbitrary line congruence off a cylinder (Theorem 3). While we carry this out only for the cylinder , reflection in any quadric in R 3 can be similarly treated. We then specialise to reflection off the inside …