Computer algebra: symbolic and algebraic computation (2nd ed.)
Computer algebra: symbolic and algebraic computation (2nd ed.)
A sweepline algorithm for Voronoi diagrams
SCG '86 Proceedings of the second annual symposium on Computational geometry
Power diagrams: properties, algorithms and applications
SIAM Journal on Computing
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Randomized incremental construction of abstract Voronoi diagrams
Computational Geometry: Theory and Applications
Algorithms in invariant theory
Algorithms in invariant theory
Matrices in elimination theory
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Computing a 3-dimensional cell in an arrangement of quadrics: exactly and actually!
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Voronoi diagram of a circle set from Voronoi diagram of a point set: topology
Computer Aided Geometric Design
Voronoi diagram of a circle set from Voronoi diagram of a point set: geometry
Computer Aided Geometric Design
On the combinatorial complexity of euclidean Voronoi cells and convex hulls of d-dimensional spheres
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Root comparison techniques applied to computing the additively weighted Voronoi diagram
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
SWAT '98 Proceedings of the 6th Scandinavian Workshop on Algorithm Theory
The Nature and Meaning of Perturbations in Geometric Computing
STACS '94 Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science
Dynamic Additively Weighted Voronoi Diagrams in 2D
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
High-Level Filtering for Arrangements of Conic Arcs
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
A Computational Basis for Conic Arcs and Boolean Operations on Conic Polygons
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Proximity structures for moving objects in constrained and unconstrained environments
Proximity structures for moving objects in constrained and unconstrained environments
Algebraic methods and arithmetic filtering for exact predicates on circle arcs
Computational Geometry: Theory and Applications
The predicates for the Voronoi diagram of ellipses
Proceedings of the twenty-second annual symposium on Computational geometry
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Voronoi diagram computations for planar NURBS curves
Proceedings of the 2008 ACM symposium on Solid and physical modeling
Real algebraic numbers and polynomial systems of small degree
Theoretical Computer Science
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
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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.