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Far-field reflector problem and intersection of paraboloids

Machado Manhães De Castro, Pedro; Mérigot, Quentin; Thibert, Boris (2016), Far-field reflector problem and intersection of paraboloids, Numerische Mathematik, 134, 2, p. 389–411. 10.1007/s00211-015-0780-z

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Type
Article accepté pour publication ou publié
Date
2016
Journal name
Numerische Mathematik
Volume
134
Number
2
Publisher
Springer
Pages
389–411
Publication identifier
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Author(s)
Machado Manhães De Castro, Pedro
Centro de Informatica UFPE [Recife] [CIn]
Mérigot, Quentin
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Thibert, Boris
Laboratoire Jean Kuntzmann [LJK]
Abstract (EN)
In this article, we study the intersection (or union) of the convex hull of N confocal paraboloids (or ellipsoids) of revolution. This study is motivated by a Minkowski-type problem arising in geometric optics. We show that in each of the four cases, the combinatorics is given by the intersection of a power diagram with the unit sphere. We prove the complexity is O(N) for the intersection of paraboloids and Omega(N^2) for the intersection and the union of ellipsoids. We provide an algorithm to compute these intersections using the exact geometric computation paradigm. This algorithm is optimal in the case of the intersection of ellipsoids and is used to solve numerically the far-field reflector problem.
Subjects / Keywords
Reflector problem; optimal transport; power diagram