Far-field reflector problem and intersection of paraboloids
Pedro Machado Manhães de Castro, Quentin Mérigot, Boris Thibert
Numerische Mathematik 134(2), pp. 389–411
Research account
Intersection of paraboloids and application to Minkowski-type problems
Related publications are listed separately below. The illustrated account describes their shared research context.
Proceedings of the thirtieth annual symposium on Computational geometry, pp. 308–317
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 O(n2) 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 numerically solve the far-field reflector problem.
The paper was presented at the Annual ACM Symposium on Computational Geometry 2014 (SoCG’14), and published in the Proceedings of SoCG 2014, here is the link. There is also a journal version of this work at Numerische Mathematik, and can be found here.