Curved Voronoi Diagrams

Boissonnat, Jean‐Daniel; Wormser, Camille; Yvinec, Mariette

doi:10.1007/978-3-540-33259-6_2

Cited by 77 publications

(79 citation statements)

References 19 publications

(26 reference statements)

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“…The fact that the α-Bregman complex has the same homotopy type as the union of the Bregman balls (Exercise 4.18) has been first observed by Edelsbrunner and Wagner [76]. A recent survey on affine and curved Voronoi diagrams can be found in [23].…”

Section: Bibliographical Notesmentioning

confidence: 87%

Geometric and Topological Inference

Boissonnat

Chazal

Yvinec

2018

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Section: Bibliographical Notesmentioning

confidence: 87%

Geometric and Topological Inference

Boissonnat

Chazal

Yvinec

2018

Self Cite

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“…This concept of space decomposition was extended to various kinds of geometric Voronoi sites, ambient spaces, and distance functions, such as power diagrams of circles in the plane, multiplicatively weighted Voronoi diagrams, additively weighted Voronoi diagrams, and more (e. g., [3,13,25,26,55]). An immediate extension is the creation of various Voronoi diagrams, embedded on twodimensional orientable parametric surfaces in general [50,56], and on the sphere in particular [53,54,58].…”

Section: Voronoi Diagrams On the Spherementioning

confidence: 99%

Arrangements on Parametric Surfaces II: Concretizations and Applications

et al. 2010

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Abstract. We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain two-dimensional orientable parametric surfaces in three-dimensional space. The fundamentals of the framework are described in a companion paper. Our work covers arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets, and documented the practical efficiency of our approach.Mathematics Subject Classification (2010). Primary 68U05; Secondary 14Q10.

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“…The presented diagrams are curved Voronoi diagrams [BWY06] requiring a higher-degree algebraic machinery.…”

Section: Voronoi Diagrams With Higher-degree Algebraic Bisectorsmentioning

confidence: 99%

Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space

Setter

Sharir

Halperin

2010

Transactions on Computational Science IX

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AcknowledgementsMany people had great influence on this thesis and its author during the research period. I deeply thank my advisor, Prof. Dan Halperin, for his help in guidance, support, and encouragement, and for introducing me the field of applied computational geometry.I wish to thank Efi Fogel and Eric Berberich for fruitful collaboration and for sharing priceless knowledge. Special thanks are given to Efi for his warm hospitality during fruitful Friday afternoons and for providing the basis for the player software, which enabled the creation of the 3D figures of this thesis. Special thanks are given to Eric for his admirable motivation and for sharing his insights through many rich discussions. I also thank Prof. Micha Sharir for his cooperation and help in theoretical parts of the thesis.I would also like to thank all other members of the applied computational geometry lab at the computer science school of Tel-Aviv University who provided support and useful suggestions. Special thanks are given to Ron Wein and to Michal Meyerovitch.I wish to thank all members of the algorithms group at the Max-Planck-Insitut für Informatik in Saarbrücken, Germany, for introducing and helping with the field of computational algebra, and for their hospitality. Special thanks are given to Michael Hemmer, to Eric Berberich, and to Michael Kerber.Work on this thesis has been supported in part by the Israel Science Foundation (grant no. 236/06), by the German-Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski-Minerva Center for Geometry at Tel Aviv University. AbstractWe present a general framework for computing two-dimensional Voronoi diagrams of different classes of sites under various distance functions. The framework is sufficiently general to support diagrams embedded on a family of two-dimensional parametric surfaces in R 3 . The computation of the diagrams is carried out through the construction of envelopes of surfaces in 3-space provided by Cgal (the Computational Geometry Algorithm Library). The construction of the envelopes follows a divide-and-conquer approach. A straightforward application of the divide-and-conquer approach for computing Voronoi diagrams yields algorithms that are inefficient in the worst case. We prove that through randomization the expected running time becomes near-optimal in the worst case. We show how to employ our framework to realize various types of Voronoi diagrams with different properties by providing implementations for a vast collection of commonly used Voronoi diagrams. We also show how to apply the new framework and other existing tools from Cgal to compute minimumwidth annuli of sets of disks, which requires the computation of two Voronoi diagrams of two different types, and of the overlay of the two diagrams. We do not assume general position. Namely, we handle degenerate input, and produce exact results.

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Curved Voronoi Diagrams

Cited by 77 publications

References 19 publications

Geometric and Topological Inference

Geometric and Topological Inference

Arrangements on Parametric Surfaces II: Concretizations and Applications

Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space

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