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
Randomized incremental construction of abstract Voronoi diagrams
Computational Geometry: Theory and Applications
MAPC: a library for efficient and exact manipulation of algebraic points and curves
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Robust Proximity Queries: An Illustration of Degree-Driven Algorithm Design
SIAM Journal on Computing
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
LEDA: a platform for combinatorial and geometric computing
LEDA: a platform for combinatorial and geometric computing
Robust Plane Sweep for Intersecting Segments
SIAM Journal on Computing
Voronoi diagram of a circle set from Voronoi diagram of a point set: topology
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
Dynamic Additively Weighted Voronoi Diagrams in 2D
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Algebraic methods and arithmetic filtering for exact predicates on circle arcs
Computational Geometry: Theory and Applications
Towards and open curved kernel
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
The predicates of the Apollonius diagram: algorithmic analysis and implementation
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
Predicates for line transversals to lines and line segments in three-dimensional space
Proceedings of the twenty-fourth annual symposium on Computational geometry
The Voronoi Diagram of Circles and Its Application to the Visualization of the Growth of Particles
Transactions on Computational Science III
The predicates of the Apollonius diagram: Algorithmic analysis and implementation
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
Convex hull and voronoi diagram of additively weighted points
ESA'05 Proceedings of the 13th annual European conference on Algorithms
| Hi-index | 0.00 |
This work examines algebraic techniques for comparing quadratic algebraic numbers, thus yielding methods for deciding key predicates in various geometric constructions. Our motivation and main application concerns a dynamic algorithm for computing the additively weighted Voronoi diagram in the plane. We propose effficient, exact, and complete methods, which are crucial for a fast and robust implementation of these predicates and the overall algorithm. Our first contribution is to minimize, on the one hand, the algebraic degree of the computed quantities, thus optimizing precision and, on the other hand, the total number of arithmetic operations. We focus on the hardest predicate, which involves quadratic polynomials, and detail the corresponding algorithms, which are based on polynomial Sturm sequences; ancillary tools include geometric invariants, multivariate resultants, and polynomial factorization. Our last contribution is a general and efficient implementation, which has been extensively tested in order to demonstrate the practical performance of our methods and the improvements achieved over existing approaches.