Far-field reflector problem and intersection of paraboloids

In this article, we propose a numerical approach to the far field reflector problem which is an inverse problem arising in geometric optics. Caffarelli et al. (Contemp Math 226:13-32, 1999) proposed an algorithm that involves the computation of the intersection of the convex hull of confocal paraboloids. We show that computing this intersection amounts to computing the intersection of a power diagram (a generalization of the Voronoi diagram) with the unit sphere. This allows us to provide an algorithm that computes efficiently the intersection of confocal paraboloids using the exact geometric computation paradigm. Furthermore, using an optimal transport formulation, we cast the far field reflector problem into a concave maximization problem. This allows us to numerically solve the far field reflector problem with up to 15k paraboloids. We also investigate other geometric optic problems that involve union of confocal paraboloids and also intersection and union of confocal ellipsoids. In all these cases, we show that the computation of these surfaces is equivalent to the computation of the intersection of a power diagram with the unit sphere.