Surface sampling and the intrinsic Voronoi diagram

Delaunay has shown that the Delaunay complex of a finite set of points P of Euclidean space R m triangulates the convex hull of P , provided that P satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay's genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stronger assumptions on P are required. A natural one is to assume that P is sufficiently dense. Although results in this direction have been claimed, we show that sample density alone is insufficient to ensure that the Delaunay complex triangulates a manifold of dimension greater than 2. Delaunay complex and Delaunay triangulationLet (M, d M ) be a metric space, and let P be a finite set of points of M. An empty ball is an open ball in the metric d M that contains no point from P . We say that an empty ball B is maximal if no other empty ball with the same centre properly contains B. A Delaunay ball is a maximal empty ball.A simplex σ is a Delaunay simplex if there exists some Delaunay ball B that circumscribes σ, i.e., such that the vertices of σ belong to ∂B ∩ P . The Delaunay complex is the set of Delaunay simplices, and is denoted Del M (P ). It is an abstract simplicial complex and so defines a topological space, |Del M (P )|, called its carrier . We say that DelThe Voronoi cell associated with p ∈ P is given byMore generally, a Voronoi face is the intersection of a set of Voronoi cells: given σ = {p 0 , . . . , p k } ⊂ P , we define the associated Voronoi face asIt follows that σ is a Delaunay simplex if and only if V M (σ) = ∅. In this case, every point in V M (σ) is the centre of a Delaunay ball for σ. Thus every Voronoi face corresponds to a Delaunay

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“…Density based sampling criteria for Delaunay triangulation of two dimensional manifolds has been validated [Lei99,DZM08].…”

Section: Discussionmentioning

confidence: 99%

“…Thus there would be a third Delaunay tetrahedron, {u, v, p, w} that shares τ as a face. In dimension 2 this problem does not arise [Lei99,DZM08]. The essential difference between dimension 2 and the higher dimensions can be observed by examining the topological intersection properties of spheres.…”

Section: A Qualitative Argumentmentioning

confidence: 99%

An Obstruction to Delaunay Triangulations in Riemannian Manifolds

Boissonnat

Dyer²,

Ghosh

et al. 2017

Discrete Comput Geom

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“…The idea is to approximate a Riemannian surface by the Delaunay triangulation of a dense set of points, and to use some recent results on intrinsic Voronoi diagrams on surfaces [16]. A/n |γ| G .…”

Section: From Discrete To Continuous Systolic Inequalitiesmentioning

confidence: 99%

Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

Verdière

Hubard

Mesmay

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and viceversa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions.Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length O(g 3/2 n 1/2 ) for any triangulated combinatorial surface of genus g with n triangles, and describe an O(gn)-time algorithm to compute such a decomposition.Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations. * Supported by the French ANR Blanc project ANR-12-BS02-005 (RDAM

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“…It was shown in [3,6] that given a sampling S of T , if ∀x ∈ T , ∃s i ∈ S, such that s i ∈ B(x, ρ m (x)), then V T (S) satisfies the closed ball property, where B(x, r) = {y ∈ T :…”

Section: Complexity Of V T (S) Onmentioning

confidence: 99%