A Minkowski-style theorem for focal functions of compact convex reflectors

Details A Minkowski-style theorem for focal functions of compact convex reflectors A Minkowski-style theorem for focal functions of compact convex reflectors

2006-12-26

arxiv.org/abs/math/0612757v1

Mathematics

Differential Geometry

15 pages

Scientific paper

This paper continues the study of a class of compact convex hypersurfaces in Euclidean space $R^{n+1}, ~n \geq 1$, which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In $R^3$ reflectors arise naturally in geometrical optics and are used in design of light reflectors and reflector antennas. They are also important in rendering problems in computer graphics. The notion of a focal function for reflectors plays a central role similar to that of the Minkowski support function for convex bodies. In this paper the basic question of when a given function is a focal function of a convex reflector is answered by establishing necessary and sufficient conditions. In addition, some smoothness properties of reflectors and of the associated directrix hypersurfaces are also etablished.

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