The Geometry of Focal Sets

Abstract

The space $\mathbb{L}$ of oriented lines, or rays, in ${\mathbb{R}}^3$ 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 ${\mathbb{R}}^3$. In this paper we explore the relationship between the focal set of a line congruence (or 2-parameter family of oriented lines in ${\mathbb{R}}^3$) and the geometry induced on the associated surface in $\mathbb{L}$. 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.

AMSTEX, 21 pages, 10 figures, 1 colour plate, significant revision of earlier version

References in corpus (2)

• An Indefinite Kaehler Metric on the Space of Oriented Lines
• Reflection of a wave off a surface