Nice Point Sets Can Have Nasty Delaunay Triangulations

Erickson, Jeff

doi:10.1007/s00454-003-2927-4

Cited by 41 publications

(43 citation statements)

References 53 publications

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“…We chose the lower bound on 6 as ~ for making further calculations precise though any other constant greater than 1 will be equally valid for our theoretical analysis. Such sampling conditions have been studied in the context of surface reconstruction in [9,15].…”

Section: Sampling and Voronoi Geometrymentioning

confidence: 99%

Shape Dimension and Approximation from Samples

Dey¹,

Giesen²,

Goswami³

et al. 2003

Discrete and Computational Geometry

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There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary. Topological dimensions of shapes constitute an important global feature of them. We present a Voronoi based dimension detection algorithm that assigns a dimension to a sample point which is the topological dimension of the manifold it belongs to, Based on this dimension detection, we also present an algorithm to approximate shapes of arbitrary dimension from their samples. Our empirical results with data sets in three dimensions support our theory.

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Section: Sampling and Voronoi Geometrymentioning

confidence: 99%

Shape Dimension and Approximation from Samples

Dey¹,

Giesen²,

Goswami³

et al. 2003

Discrete and Computational Geometry

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“…Given that the spread has been used extensively to bound the complexity of Voronoi diagrams [Eri01,Eri05] (without Steiner points) and the pacing provides provably tighter results for mesh sizing than the spread, it might be possible to use pacing analysis to prove tighter per-instance bounds on the complexity of Voronoi diagrams.…”

Section: Some Directions For Future Workmentioning

confidence: 99%

New Bounds on the Size of Optimal Meshes

Sheehy

2012

Computer Graphics Forum

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The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing φ, then the number of vertices in an optimal mesh will be O(φ d n), where d is the input dimension. We give a new analysis of this integral showing that the output size is only Θ(n + n log φ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n).

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“…The minimum such distance in G (or more generally in any PSLG) is called the minimum feature size, denoted by mfs(G). See [Rup93,Dey07,HMP06,Eri03]. We call the ratio mfs(P ) mfs(G) the degradation of the decomposition of P into G.…”

Section: Introductionmentioning

confidence: 99%

Meshes Preserving Minimum Feature Size

Aloupis

Demaine

et al. 2012

Lecture Notes in Computer Science

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Abstract. The minimum feature size of a planar straight-line graph is the minimum distance between a vertex and a nonincident edge. When such a graph is partitioned into a mesh, the degradation is the ratio of original to final minimum feature size. For an n-vertex input, we give a triangulation (meshing) algorithm that limits degradation to only a constant factor, as long as Steiner points are allowed on the sides of triangles. If such Steiner points are not allowed, our algorithm realizes O(lg n) degradation. This addresses a 14-year-old open problem by Bern, Dobkin, and Eppstein.

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Nice Point Sets Can Have Nasty Delaunay Triangulations

Cited by 41 publications

References 53 publications

Shape Dimension and Approximation from Samples

Shape Dimension and Approximation from Samples

New Bounds on the Size of Optimal Meshes

Meshes Preserving Minimum Feature Size

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