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SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
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SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
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SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
SWAT '98 Proceedings of the 6th Scandinavian Workshop on Algorithm Theory
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STACS '94 Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science
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ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
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ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
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ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
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Transactions on Computational Science III
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ICMS'10 Proceedings of the Third international congress conference on Mathematical software
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Constructing two-dimensional Voronoi diagrams via divide-and-conquer of envelopes in space
Transactions on computational science IX
Constructing two-dimensional Voronoi diagrams via divide-and-conquer of envelopes in space
Transactions on computational science IX
Exact medial axis computation for circular arc boundaries
Proceedings of the 7th international conference on Curves and Surfaces
Anomalies in quasi-triangulations and beta-complexes of spherical atoms in molecules
Computer-Aided Design
Technical note: Voronoi diagrams of algebraic distance fields
Computer-Aided Design
Computer Aided Geometric Design
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We study the predicates involved in an efficient dynamic algorithm for computing the Apollonius diagram in the plane, also known as the additively weighted Voronoi diagram. We present a complete algorithmic analysis of these predicates, some of which are reduced to simpler and more easily computed primitives. This gives rise to an exact and efficient implementation of the algorithm, that handles all special cases. Among our tools we distinguish an inversion transformation and an infinitesimal perturbation for handling degeneracies. The implementation of the predicates requires certain algebraic operations. In studying the latter, we aim at minimizing the algebraic degree of the predicates and the number of arithmetic operations; this twofold optimization corresponds to reducing bit complexity. The proposed algorithms are based on static Sturm sequences. Multivariate resultants provide a deeper understanding of the predicates and are compared against our methods. We expect that our algebraic techniques are sufficiently powerful and general to be applied to a number of analogous geometric problems on curved objects. Their efficiency, and that of the overall implementation, are illustrated by a series of numerical experiments. Our approach can be immediately extended to the incremental construction of abstract Voronoi diagrams for various classes of objects.