The Geometry of Focal Sets

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dc.creatorGuilfoyle, Brendan
dc.creatorKlingenberg, Wilhelm
dc.date2004-11-09
dc.date2005-11-29
dc.date.accessioned2026-07-07T10:19:15Z
dc.date.available2026-07-07T10:19:15Z
dc.descriptionThe space ${\Bbb{L}}$ of oriented lines, or rays, in ${\Bbb{R}}^3$ 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 ${\Bbb{R}}^3$. In this paper we explore the relationship between the focal set of a line congruence (or 2-parameter family of oriented lines in ${\Bbb{R}}^3$) and the geometry induced on the associated surface in ${\Bbb{L}}$. 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.
dc.descriptionAMSTEX, 21 pages, 10 figures, 1 colour plate, significant revision of earlier version
dc.identifierhttps://arxiv.org/abs/math/0411189
dc.identifierhttp://arxiv.org/abs/math/0411189
dc.identifierin Recent developments in pseudo-Riemannian Geometry, European Mathematical Society Publishing House, Zurich (2008) 149-178.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/174443
dc.subjectDifferential Geometry
dc.subjectMathematical Physics
dc.subject53A25; 78A05; 53C80
dc.titleThe Geometry of Focal Sets
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