Abstract:We propose two ways to compute the Delaunay triangulation of points on a sphere, or of rounded points close to a sphere, both based on the classic incremental algorithm initially designed for the plane. We use the so-called space of circles as mathematical background for this work. We present a fully robust implementation built upon existing generic algorithms provided by the cgal library. The eciency of the implementation is established by benchmarks. Nous proposons deux façons de calculer la triangulation de Delaunay d'un ensemble de points qui appartiennent soit à la sphère, soit à son voisinage. Ces deux méthodes reposent sur l'algorithme incrémental classique, tel qu'il a été créé à l'origine pour calculer les triangulations de Delaunay planaires. Le cadre mathématique classique justiant cette approche est rappelé, à l'aide de l'espace des cercles. Ces deux approches ont été implantées de façon robuste en s'appuyant sur les algorithmes génériques fournis par la bibliothèque CGAL. Des tests comparatifs montrent l'ecacité de nos implantations sur des jeux de données de taille variée.
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
We analyze, implement, and evaluate a distributionsensitive point location algorithm based on the classical Jump & Walk, called Keep, Jump, & Walk. For a batch of query points, the main idea is to use previous queries to improve the current one. In practice, Keep, Jump, & Walk is actually a very competitive method to locate points in a triangulation. Regarding point location in a Delaunay triangulation, we show how the Delaunay hierarchy can be used to answer, under some hypotheses, a query q with a O(log #(pq)) randomized expected complexity, where p is a previously located query and #(s) indicates the number of simplices crossed by the line segment s. The Delaunay hierarchy has O(n log n) time complexity and O(n) memory complexity in the plane, and under certain realistic hypotheses these complexities generalize to any finite dimension. Finally, we combine the good distribution-sensitive behavior of Keep, Jump, & Walk, and the good complexity of the Delaunay hierarchy, into a novel point location algorithm called Keep, Jump, & Climb. To the best of our knowledge, Keep, Jump, & Climb is the first practical distributionsensitive algorithm that works both in theory and in practice for Delaunay triangulations.
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