2006/12/26 by Vladimir Oliker, Oliker, Vladimir
Mathematics · #35J65 #52A20 #78A05 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:35J65 #msc:52A20 #msc:78A05
paper · pdf · doi:10.48550/arxiv.math/0612757
15 pages
arxiv created 2006/12/26 · arxiv updated 2009/12/01
This paper continues the study of a class of compact convex hypersurfaces in Euclidean space Rn+1, ~n ≥ 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 R3 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.