Topic
Delaunay triangulation
About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.
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TL;DR: In this article, a new method of three-dimensional finite element mesh generation is presented, based on an extension of the Delaunay triangulation algorithm to three dimensions, using the mesh generator, problems are defined by means of simple threedimensional solid modeling commands.
Abstract: A new method of three-dimensional finite element mesh generation is presented in this paper. The method is based on an extension of the Delaunay triangulation algorithm to three dimensions, Using the mesh generator, problems are defined by means of simple three-dimensional solid modeling commands. Mesh generation is performed automatically from the objects entered. Various two and three-dimensional Delaunay mesh generation algorithms are also compared and insights to robust mesh generation provided.
73 citations
Posted Content•
TL;DR: In this paper, the complexity of Delaunay triangulations of sets of points in R^3 under certain practical geometric constraints was studied. And the authors showed that in the worst case, the triangulation of n points with spread D has complexity Omega(min{D^3, nD, n^2}) and O(min {D^4, n+2}).
Abstract: We consider the complexity of Delaunay triangulations of sets of points in R^3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of n points in R^3 with spread D has complexity Omega(min{D^3, nD, n^2}) and O(min{D^4, n^2}). For the case D = Theta(sqrt{n}), our lower bound construction consists of a uniform sample of a smooth convex surface with bounded curvature. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has near-quadratic complexity.
73 citations
TL;DR: In this paper, two shape parameters for Delaunay simplices are proposed which allow one to recognize slightly distorted tetrahedral and octahedral simplices, and these simplex types are found to be the basic building units in dense packings of hard and soft spheres.
Abstract: Two novel shape parameters for Delaunay simplices are proposed which allow one to recognize slightly distorted tetrahedral and octahedral simplices. These simplex types are found to be the basic building units in dense packings of hard and soft spheres.
73 citations
Book•
23 Feb 2004
TL;DR: This paper presents a meta-analysis of the distributional results for Cn,m-graphs using Voronoi Cells and Delaunay Triangularization as a model for Nearest Neighbor Prototypes, a type of graph designed to describe the relationship betweenNeighbor and Neighbourhood.
Abstract: Preface.Acknowledgments.1. Preliminaries.1.1 Graphs and Digraphs.1.2 Statistical Pattern Recognition.1.3 Statistical Issues.1.4 Applications.1.5 Further Reading.2. Computational Geometry.2.1 Introduction.2.2 Voronoi Cells and Delaunay Triangularization.2.3 Alpha Hulls.2.4 Minimum Spanning Trees.2.5 Further Reading.3. Neighborhood Graphs.3.1 Introduction.3.2 Nearest-Neighbor Graphs.3.3 k-Nearest Neighbor Graphs.3.4 Relative Neighborhood Graphs.3.5 Gabriel Graphs.3.6 Application: Nearest Neighbor Prototypes.3.7 Sphere of Influence Graphs.3.8 Other Relatives.3.9 Asymptotics.3.10 Further Reading.4. Class Cover Catch Digraphs.4.1 Catch Digraphs.4.2 Class Covers.4.3 Dominating Sets.4.4 Distributional Results for Cn,m-graphs.4.5 Characterizations.4.6 Scale Dimension.4.7 (alpha,beta) Graphs4.8 CCCD Classification.4.9 Homogeneous CCCDs.4.10 Vector Quantization.4.11 Random Walk Version.4.12 Further Reading.5. Cluster Catch Digraphs.5.1 Basic Definitions.5.2 Dominating Sets.5.3 Connected Components.5.4 Variable Metric Clustering.6. Computational Methods.6.1 Introduction.6.2 Kd-Trees.6.3 Class Cover Catch Digraphs.6.4 Cluster Catch Digraphs.6.5 Voroni Regions and Delaunay Triangularizations.6.6 Further Reading.References.Author Index.Subject Index.
73 citations
TL;DR: It is shown that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness, and this characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1-Tough.
Abstract: We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness. A graphG is1-tough if for any setP of vertices,c(G?P)≤|G|, wherec(G?P) is the number of components of the graph obtained by removingP and all attached edges fromG, and |G| is the number of vertices inG. This property arises in the study of Hamiltonian graphs: all Hamiltonian graphs are 1-tough, but not conversely. We also show that all Delaunay triangulationsT satisfy the following closely related property: for any setP of vertices the number of interior components ofT?P is at most |P|?2, where an interior component ofT?P is a component that contains no boundary vertex ofT. These appear to be the first nontrivial properties of a purely combinatorial nature to be established for Delaunay triangulations. We give examples to show that these bounds are best possible and are independent of one another. We also characterize the conditions under which a degenerate Delaunay triangulation can fail to be 1-tough. This characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1-tough. One consequence of the toughness results is that all Delaunay triangulations and all inscribable graphs have perfect matchings.
73 citations