Far-field reflector problem and intersection of paraboloids

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 Minkowskitype 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 Ω(N) 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. Introduction The computation of intersection of half-spaces is a well-studied problem in computational geometry, which by duality is equivalent to the computation of a convex hull. Similarly, the computation of intersections or unions of spheres is also well studied and can be done by using power diagrams [2]. In this article, we study the computation and the complexity of the intersection of the convex hull of confocal paraboloids of revolution, showing that it is equivalent to intersecting a certain power diagram with the unit sphere. Union of convex hull of confocal paraboloids of revolution, and intersection or union of convex hull of confocal ellipsoids of revolution can be studied using the same tools. These studies are motivated by inverse problems similar to Minkowski problem that arise in geometric optics. We show how the algorithm we developed to compute the intersection of paraboloids are used to solve large instances of one of these problems. Minkowski-type problems. A theorem of Minkowski asserts that given a family of directions (yi)16i6N and a family of non-negative numbers (αi)16i6N , one can construct a convex polytope with exactlyN facets, such that the ith facet has exterior normal yi and area αi. Aurenhammer, Hoffman and Aronov [3] studied a variant of this problem involving power diagrams and showed its equivalence with the so-called constrained least-square matching problem. This article is motivated by yet another problem of Minkowski-type that arises in geometric optics, which is called the far-field reflector problem in the literature [9, 8]. Recall that a paraboloid of revolution is defined by three parameters: its focal point, its focal distance λ and its direction y. We assume that all paraboloids are focused at the origin, and we denote P (y, λ) the convex hull of a paraboloid of revolution with direction y and focal distance λ. We will say in the following that P (y, λ) is a solid paraboloid. Paraboloids of revolution have the well-known optical property that any ray of light emanating from the origin is reflected by the surface ∂P (y, λ) in the direction y. Assume first that one wants to send the light emited from the origin in N prescribed directions y1, . . . , yN . From the property of a paraboloid of revolution, this can be done by considering a surface made of pieces of paraboloids of revolution whose directions are among the (yi). In the far-field reflector problem, one would also like to prescribe the amount of light αi that is reflected in the direction yi. A theorem of Oliker-Caffarelli [9] ensures the existence of a solution to this problem: there exist unique (up to a common multiplicative 1 2 PEDRO MACHADO MANHÃES DE CASTRO, QUENTIN MÉRIGOT, AND BORIS THIBERT constant) focal distances λ1, . . . , λN such that the surface ∂(∩16i6NP (yi, λi)) reflects exactly the amount αi in each direction yi. Other types of inverse problems in geometric optics can be formulated as Minkowski-type problems involving the union of confocal solid paraboloids, and the union or intersection of confocal ellipsoids [17, 14]. Contributions. Motivated by these Minkowski-type problems, our goal is to compute the union and intersection of solid confocal paraboloids and ellipsoids of revolution. Using a radial parameterization, each of these computations is equivalent to the computation of a decomposition of the unit sphere into cells, that are not necessarily connected. Our contributions are the following: • We show that each of the four types of cells can be computed by intersecting a certain power diagram with the unit sphere (Propositions 1, 5 and 7). The approach is similar to the computation of union and intersection of balls using power diagrams in [2], or to the computation of multiplicatively weighted power diagrams in R using power diagrams in R [5]. • We show that the complexity bounds of these four diagram types are different. In the case of intersection of solid confocal paraboloids in R, the complexity of the intersection diagram is O(N) (Theorem 2). This is in contrast with the Ω(N) complexity of the intersection of a power diagram with a paraboloid in R [5]. In the case of the union and intersection of solid confocal ellipsoids, we recover this Ω(N) complexity (Theorem 8). Finally, the case of the union of paraboloids is very different from the case of the intersection of paraboloids. Indeed, in the latter case, the corresponding cells on the sphere are connected, while in the former case the number of connected component of a single cell can be Ω(N) (Proposition 6). The complexity of the diagram in this case is unknown. • In Section 3, we describe an algorithm for computing the intersection of a power diagram with the unit sphere. This algorithm uses the exact geometric computation paradigm and can be applied to the four types of unions and intersections. It is optimal for the union and intersection of ellipsoids, but its optimality for the case of intersection of paraboloids is open. • This algorithm is then used for the numerical resolution of the far-field reflector problem. Using a known optimal transport formulation [21, 11] and similar techniques to [3], we cast this problem into a concave maximization problem in Theorem 12. This allows us to solve instances with up to 15k paraboloids, improving by several order of magnitudes upon existing numerical implementations [8]. 1. Intersection of confocal paraboloids of revolution Because of their optical properties, finite intersections of solid paraboloids of revolutions with the same focal point play a crucial role in an inverse problem called the far-field reflector problem. This inverse problem is explained in more detail in Section 4. Here we study the computation and complexity of such an intersection when the focal point lies at the origin. We call this type of intersection a paraboloid intersection diagram. 1.1. Paraboloid intersection diagram. A paraboloid of revolution in R with focal point at the origin is uniquely defined by two parameters: its focal distance λ and its direction, described by a unit vector y. We denote the convex hull of such a paraboloid by P (y, λ). The boundary surface ∂P (y, λ) can be parameterized in spherical coordinates by the radial map INTERSECTION OF PARABOLOIDS AND APPLICATION TO MINKOWSKI-TYPE PROBLEMS 3 u ∈ Sd−1 7→ ρy,λ(u) u, where the function ρy,λ is defined by: (1.1) ρy,λ : u ∈ S 7→ λ 1− 〈y|u〉 . Given a family Y = (yi)16i6N of unit vectors and a family λ = (λi)16i6N of positive focal distances, the boundary of the intersection of the solid paraboloids (P (yi, λi))16i6N is parameterized in spherical coordinates by the function: (1.2) ρY,λ(u) := min 16i6N ρyi,λi(u) = min y∈Y λi 1− 〈yi|u〉 . Definition 1.1. The paraboloid intersection diagram associated to a family of solid paraboloids (P (yi, λi))16i6N is a decomposition of the unit sphere into N cells defined by: PIY (yi) := {u ∈ S; ∀j ∈ {1, . . . , N}, ρyi,λi(u) 6 ρyj ,λj (u)}. The paraboloid intersection diagram corresponds to the decomposition of the unit sphere given by the lower envelope of the functions (ρyi,λi)16i6N . 1.2. Power diagram formulation. We show in this section that each cell of the paraboloid intersection diagram is the intersection of a cell of a certain power diagram with the unit sphere. We first recall the definition of a power diagram. Let P = (pi)16i6N be a family of points in R d and (ωi)16i6N a family of weights. The power diagram is a decomposition of R d into N convex cells, called power cells, defined by PowP (pi) := { x ∈ R, ∀j ∈ {1, . . . , N} ‖x− pi‖ + ωi 6 ‖x− pj‖ + ωj }