Far-field reflector problem and intersection of paraboloids

Abstract : 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.
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Numerische Mathematik, Springer Verlag, 2016, 134 (2), pp.389-411. <http://link.springer.com/article/10.1007%2Fs00211-015-0780-z>. <10.1007/s00211-015-0780-z>
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https://hal.archives-ouvertes.fr/hal-00952720
Contributeur : Quentin Mérigot <>
Soumis le : jeudi 27 février 2014 - 13:51:35
Dernière modification le : vendredi 13 janvier 2017 - 01:02:25
Document(s) archivé(s) le : mardi 27 mai 2014 - 11:17:09

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Pedro Machado Manhães de Castro, Quentin Mérigot, Boris Thibert. Far-field reflector problem and intersection of paraboloids. Numerische Mathematik, Springer Verlag, 2016, 134 (2), pp.389-411. <http://link.springer.com/article/10.1007%2Fs00211-015-0780-z>. <10.1007/s00211-015-0780-z>. <hal-00952720>

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