ar X iv : m at h / 04 11 18 9 v 2 [ m at h . D G ] 2 9 N ov 2 00 5 THE GEOMETRY OF FOCAL SETS

The space L of oriented lines, or rays, in R 3 is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral Kähler metric which is closely related to the Euclidean metric on R 3. In this paper we explore the relationship between the focal set of a line congruence (or 2-parameter family of oriented lines in R 3) and the geometry induced on the associated surface in L. The physical context of such sets is geometric optics in a homogeneous isotropic medium, and so, to illustrate the method, we compute the focal set of the k th reflection of a point source off the inside of a cylinder. The focal sets, which we explicitly parameterize, exhibit unexpected symmetries, and are found to fit well with observable phenomena. The space of oriented affine lines in R 3 , which we denote L, has an abundance of natural geometric structure. The purpose of this paper is to continue recent work [7] relating this structure to geometric optics in a homogeneous isotropic medium: the theory of light propagation under the assumption that the light travels along straight lines in R 3. The fundamental objects of study are 2-parameter families of oriented lines, or line congruences, which we view as surfaces in L. Thus we are lead to consider the geometry of immersed surfaces Σ ⊂ L. In the first instance, since L can be identified with the tangent space to the 2-sphere, there is the natural bundle map π : L → S 2. If π| Σ : Σ → S 2 is not an immersion, we say that Σ is flat. Otherwise, Σ can be described, at least locally, by sections of the canonical bundle. On the other hand, there is a natural symplectic structure Ω on L, and Σ ⊂ L is lagrangian with respect to this symplectic structure iff the line congruence admits a family of orthogonal surfaces in R 3. In geometric optics such surfaces are the wavefronts of the propagating light. As a wavefront evolves along the line congruence, if there is any focusing, the surface becomes singular. The points at which this occurs are referred to as focal points. In addition, L admits a natural complex structure J, which, together with the symplectic structure, forms a natural Kähler structure [8]. The metric G is of signature (+ + −−) and …