Optimality of the Delaunay triangulation in ℝd

Rajan, V. T.

doi:10.1007/bf02574375

Cited by 164 publications

(100 citation statements)

References 18 publications

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“…We further define the maximal min-containment Bregman radius of a triangulation T ∈ T (S) as r mc (T ) = max τ ∈T r mc (τ ). The following result is an extension of a result due to Rajan for Delaunay triangulations [35].…”

Section: Theorem 7 the First-type Bregman Sphere Circumscribing Any Ssupporting

confidence: 59%

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Bregman Voronoi Diagrams

Boissonnat

Nielsen

Nock³

2010

Discrete Comput Geom

126

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The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define various variants of Voronoi diagrams depending on the class of objects, the distance function and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g., k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connection with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation.

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Section: Theorem 7 the First-type Bregman Sphere Circumscribing Any Ssupporting

confidence: 59%

“…The proof mimics Rajan's proof [35] for the case of Delaunay triangulations. We will now show that del F (S) is the geometric dual of vor F (S).…”

Section: Theorem 10 (Max-min-containment) For a Given Finite Set Of Psupporting

confidence: 55%

Bregman Voronoi Diagrams

Boissonnat

Nielsen

Nock³

2010

Discrete Comput Geom

126

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show abstract

“…Melissaratos [31] generalizes Rippa's result to higher dimensions. D'Azevedo and Simpson [13] show that a two-dimensional Delaunay triangulation minimizes the radius of the largest mincontainment circle of its simplices, and Rajan [36] generalizes this result to Delaunay triangulations and min-containment spheres of any dimensionality. The min-containment sphere of a simplex is the smallest hypersphere that encloses the simplex.…”

Section: Interpolation Criteria Optimized By Cdtsmentioning

confidence: 89%

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Shewchuk

2008

Twentieth Anniversary Volume:

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Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have "nicely shaped" triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions.The main contributions are rigorous, theory-tested definitions of constrained Delaunay triangulations and piecewise linear complexes (geometric domains that incorporate nonconvex faces with "internal" boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.

show abstract

“…Melissaratos [31] generalizes Rippa's result to higher dimensions. D'Azevedo and Simpson [13] show that a two-dimensional Delaunay triangulation minimizes the radius of the largest min-containment circle of its simplices, and Rajan [36] generalizes this result to Delaunay triangulations and min-containment spheres of any dimensionality. The min-containment sphere of a simplex is the smallest hypersphere that encloses the simplex.…”

Section: Interpolation Criteria Optimized By Cdtsmentioning

confidence: 90%

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Shewchuk

2007

Discrete Comput Geom

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Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have "nicely shaped" triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions.The main contributions are rigorous, theory-tested definitions of CDTs and piecewise linear complexes (geometric domains that incorporate nonconvex faces with "internal" boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.

show abstract

Optimality of the Delaunay triangulation in ℝd

Cited by 164 publications

References 18 publications

Bregman Voronoi Diagrams

Bregman Voronoi Diagrams

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

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