Divide-and-conquer for Voronoi diagrams revisited

Aichholzer, Oswin; Aigner, Wolfgang; Aurenhammer, Franz; Hackl, Thomas; Jüttler, Bert; Pilgerstorfer, Elisabeth; Rabl, Margot

doi:10.1145/1542362.1542401

Cited by 8 publications

(4 citation statements)

References 33 publications

(32 reference statements)

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“…with columns indexed by d 1 , d 2 , d 2 1 , d 1 d 2 + d 2 2 . We get two vectors in the kernel, the first yielding again d 1 + d 2 and a second one for λ 1 = −1, λ 2 = 0, λ 3 = λ 4 = 1, so we deduce that −d 1…”

Section: )mentioning

confidence: 88%

“…. , f n ) of polynomials f i of degree d i , we define f = ( f 1 2 + · · · + f n 2 ) 1 2 . An important property of this norm is that it stays invariant under an unitary change of coordinates.…”

Section: Notations and Preliminariesmentioning

confidence: 99%

“…Divide-and-conquer techniques are also utilized in Aichholzer et al [1], to propose a VD algorithm for general shapes. They exploit connections to medial axis and use a plane-sweep technique.…”

Section: Some Existing Workmentioning

confidence: 99%

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Robust algebraic methods for geometric computing

Mantzaflaris

2012

ACM Commun. Comput. Algebra

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Geometric computation in computer aided geometric design and solid modeling calls for solving non-linear polynomial systems in an approximate-yet-certified manner. We introduce new subdivision algorithms that tackle this fundamental problem. In particular, we generalize the univariate so-called continued fraction solver to general dimension. Fast bounding functions, unicity tests, projection and preconditioning are employed to speed up convergence. Apart from practical experiments, we provide theoretical bit complexity estimates, as well as bounds in the real RAM model, by means of real condition numbers.A main bottleneck for any real solving method is singular isolated points. We employ local inverse systems and certified numerical computations to provide certification criteria to treat singular solutions. In doing so, we are able to check existence and uniqueness of singularities of a given multiplicity structure using verification methods, based on interval arithmetic and fixed point theorems.Two major geometric applications are undertaken. First, the approximation of planar semialgebraic sets, commonly occurring in constraint geometric solving. We present an efficient algorithm to identify connected components and, for a given precision, to compute polygonal and isotopic approximation of the exact set.Second, we present an algebraic framework to compute generalized Voronoi diagrams, that is applicable to any diagram type in which the distance from a site can be expressed by a bi-variate polynomial function (anisotropic, power diagram etc). In cases where this is not possible (eg. Apollonius diagram, VD of ellipses and so on), we extend the theory to implicitly given distance functions. 237

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Section: )mentioning

confidence: 88%

Section: Notations and Preliminariesmentioning

confidence: 99%

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Robust algebraic methods for geometric computing

Mantzaflaris

2012

ACM Commun. Comput. Algebra

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“…However, when we solve practical problems, we usually deal with more complex geometrical shapes than just points or line segments. Normally these complex objects can be represented by parametric curves of arbitrary shape [Aic10a].…”

Section: Introductionmentioning

confidence: 99%

A fast approximation of the Voronoi diagram for a set of pairwise disjoint arcs

Kotsur¹,

Tereshchenko²,

Tereshchenko³

2018

CSRN

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We propose a method for fast approximation of the Voronoi diagram for a set of pairwise disjoint arcs on a plane. The arcs are represented by parameterized curves. A set of input curves is discretized into partition set, for which the Voronoi diagram is constructed. After merging corresponding Voronoi cells and removing redundant edges, the Voronoi graph is approximated by Bezier curves. We also propose the elaboration and optimization of the approximation. The total complexity of the algorithm is in the worst-case.

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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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Divide-and-conquer for Voronoi diagrams revisited

Cited by 8 publications

References 33 publications

Robust algebraic methods for geometric computing

Robust algebraic methods for geometric computing

A fast approximation of the Voronoi diagram for a set of pairwise disjoint arcs

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

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