Intersection of paraboloids and application to Minkowski-type problems

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 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

State of the Art: Updating Delaunay Triangulations for Moving Points

This paper considers the problem of updating efficiently a two-dimensional Delaunay triangulation when vertices are moving. We investigate the three current state-of-the-art approaches to solve this problem: --1-- the use of kinetic data structures, --2-- the possibility of moving points from their initial to final position by deletion and insertion and --3-- the use of "almost" Delaunay structure that postpone the necessary modifications. Finally, we conclude with a global overview of the above-mentioned approaches while focusing on future works

On the Size of Some Trees Embedded in Rd

This paper extends the result of Steele [6,5] on the worst-case length of the Euclidean minimum spanning tree EMST and the Euclidean minimum insertion tree EMIT of a set of n points S contained in Rd. More precisely, we show that, if the weight w of an edge e is its Euclidean length to the power of α, the following quantities Σ_{e ∈ EMST} w(e) and Σ_{e ∈ EMIT} w(e) are both worst-case O(n^{1-α/d}), where d is the dimension and α, 0 < α < d, is the weight. Also, we analyze and compare the value of Σ_{e ∈ T} w(e) for some trees T embedded in Rd which are of interest in (but not limited to) the point location problem [2]

Walking Faster in a Triangulation

Point location in a triangulation is one of the most studied problems in computational geometry. For a single query, stochastic walk is a good practical strategy. In this work, we propose two approaches improving the performance of the stochastic walk. The first improvement is based on a relaxation of the exactness of the predicate, whereas the second is based on termination guessing.La localisation d'un point dans une triangulation est un des problèmes les plus étudiés en géométrie algorithmique. Pour un petit nombre de requêtes, la marche stochastique est une bonne stratégie en pratique. Dans ce travail, nous proposons deux idées qui améliorent les performances de la marche stochastique. La première est basée sur une relaxation de l'exactitude du prédicat d'orientation, tandis que la deuxième est basée sur lune tentative de divination de la longueur de cette marche

Practical Distribution-Sensitive Point Location in Triangulations

International audienceWe design, analyze, implement, and evaluate a distribution-sensitive 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 ac- tually a very competitive method to locate points in a triangulation. We also study some constant- memory distribution-sensitive point location algorithms, which work well in practice with the classical space-filling heuristic for fast point location. 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(nlogn) time complexity and O(n) memory complexity in the plane, and under certain realistic hypotheses these com- plexities 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 distribution-sensitive algorithm that works both in theory and in practice for Delaunay triangulations

Longueur moyenne de la marche de Vornoi dans une triangulation de Poisson-Delaunay en dimension dd

Let $X_n$ be a $d$ dimensional Poisson point process of intensity $n$.We prove that the expected length of the Voronoi path between twopoints at distance 1 in the Delaunay triangulation associated with $X_n$ is $\sqrt{\frac{2d}{\pi}}+O(d^{-\frac{1}{2}})$ for all $n\in\mathbb{N}$ and $d\rightarrow\infty$.In any dimension, we provide a precise interval containing the exactvalue, in 3D the expected length is between 1.4977 and 1.50007.Soit $X_n$ un processus ponctuel de Poisson d'intensité $n$ endimension $d$.Nous démontrons que l'espérance de la longueur du chemin de Voronoientre l'origine et un point à distance 1 dans la triangulation deDelaunay de $X_n$ est $\sqrt{\frac{2d}{\pi}}+O(d^{-\frac{1}{2}})$pour tout $n\in\mathbb{N}$ quand $d\rightarrow\infty$.Nous donnons des bornes inférieures et supérieures sur la bonne valeuren toute dimension, en 3D ces bornes sont 1.4977 et 1.50007

Expected Length of the Voronoi Path in a High Dimensional Poisson-Delaunay Triangulation

International audienceLet X be a d dimensional Poisson point process. We prove that the expected length of the Voronoi path between two points at distance 1 in the Delaunay triangulation associated with X is sqrt(2d/π) + O(d^(−1/2) when d → ∞. In any dimension, we also provide a precise interval containing the actual value; in 3D the expected length is between 1.4977 and 1.50007

Far-field reflector problem and intersection of paraboloids

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

Robust and Efficient Delaunay triangulations of points on or close to a sphere

We propose two approaches for computing 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. The space of circles gives the mathematical background for this work. We implemented the two approaches in a fully robust way, building upon existing generic algorithms provided by the cgal library. The effciency and scalability of the method is shown by benchmarks