Far-field reflector problem and intersection of paraboloids

Abstract

International audienceIn 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

Optimal transportation, modelling and numerical simulation

“…The method is tested by providing an image on a projection plane, computing the corresponding reflector, resimulating via ray-tracing, and finally comparing the two pictures: see Figure 2.2. Semi-discrete optimal transportation has been applied to the reflector problem in de Castro et al (2016). Equation (2.31) is simply the u 1 c-transform.…”

A Damped Newton Algorithm for Generated Jacobian Equations

“…Then the Damped newton Algorithm (Algorithm 1) converges toward a solution of (NF paral). In our implementation of the algorithm, the intersection of power diagrams with a paraboloid is computed using an algorithm presented in [13]. Once the diagram is computed, the function H and its differential DH are computed using the trapezoidal rule.…”

The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Ampère equations

“…In other words, it does not require generating a grid or a polyhedral tessellation of the manifolds, but only a suitable point cloud, which can be efficiently generated using Quasi-Monte Carlo methods. In the case of the round sphere various different numerical algorithms have previously been explored in the literature: see [30,70,72] for experimental work on the case of the cost function d(x, y) 2 and [18,28] for the case of the cost function − log |x − y|, as applied to the reflector antenna problem in geometric optics.…”