Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): Vladimir I. Oliker
Journal: Trans. Amer. Math. Soc. 360 (2008), 563-574.
MSC (2000): Primary 49K20, 35J65, 78A05
Posted: September 21, 2007
MathSciNet review: 2346462
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: This paper continues the study of a class of compact convex hypersurfaces in $\mathbf{R}^{n+1}, ~n \geq 1,$ which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In $\mathbf{R}^3$ 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.

References:

1.: I. J. Bakelman.
Convex Analysis and Nonlinear Geometric Elliptic Equations.
Springer-Verlag, Berlin, 1994. MR 1305147 (95k:35063)
2.: L. A. Caffarelli, C. Gutierrez, and Qingbo Huang.
On the regularity of reflector antennas.
Preprint, 2004.
3.: L.A. Caffarelli.
Allocation maps with general cost functions.
Partial Differential Equations and Applications, 177:29-35, 1996. MR 1371577 (97f:49055)
4.: L.A. Caffarelli, S. Kochengin, and V.I. Oliker.
On the numerical solution of the problem of reflector design with given far-field scattering data.
Contemporary Mathematics, 226:13-32, 1999. MR 1660740 (99i:65098)
5.: L.A. Caffarelli and V.I. Oliker.
Weak solutions of one inverse problem in geometric optics.
Unpublished manuscript, 1994.
6.: W. Gangbo and V. I. Oliker.
Existence of optimal maps in the reflector-type problems.
ESAIM: Control, Optimization and Calculus of Variations, 13:93-106, 2007. MR 2282103
7.: T. Glimm and V.I. Oliker.
Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem.
J. of Math. Sciences, 117(3):4096-4108, 2003. MR 2027449 (2004k:49101)
8.: T. Glimm and V.I. Oliker.
Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle.
Indiana Univ. Math. J., 53:1255-1278, 2004. MR 2104277 (2005f:49095)
9.: T. Hasanis and D. Koutroufiotis.
The characteristic mapping of a reflector.
J. of Geometry, 24:131-167, 1985. MR 793277 (86j:53004)
10.: L.V. Kantorovich and G.P. Akilov.
Functional Analysis, ch. VIII, §4, In Russian.
Nauka, Moscow, 1977, 2-nd revised edition. MR 0511615 (58:23465)
11.: V.I. Oliker.
On the geometry of convex reflectors.
PDE's, Submanifolds and Affine Differential Geometry, ed. by B. Opozda, U. Simon and M. Wiehe, Banach Center Publications, 57:155-169, 2002.
Errata: Banach Center Publications, v. 69(2005), 269-270. MR 1974709 (2004c:53103)
12.: G. Patow and X. Pueyo.
A survey of inverse surface design from light transport behavior specification.
Computer Graphics Forum, 24:773-789, 2005.
13.: R. Schneider.
Convex Bodies. The Brunn-Minkowski Theory.
Cambridge Univ. Press, Cambridge, 1993. MR 1216521 (94d:52007)
14.: Xu-Jia Wang.
On design of a reflector antenna II.
Calculus of Variations and PDE's, 20:329-341, 2004. MR 2062947 (2005f:78005)
15.: B. S. Westcott.
Shaped Reflector Antenna Design.
Research Studies Press, Letchworth, UK, 1983.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|