A Gröbner free alternative for polynomial system solving
Journal of Complexity
Geometric constraint solver using multivariate rational spline functions
Proceedings of the sixth ACM symposium on Solid modeling and applications
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
Voronoi diagrams of semi-algebraic sets
Voronoi diagrams of semi-algebraic sets
Precise Voronoi cell extraction of free-form rational planar closed curves
Proceedings of the 2005 ACM symposium on Solid and physical modeling
The Voronoi Diagram of Curved Objects
Discrete & 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
The predicates for the Voronoi diagram of ellipses
Proceedings of the twenty-second annual symposium on Computational geometry
EXACUS: efficient and exact algorithms for curves and surfaces
ESA'05 Proceedings of the 13th annual European conference on Algorithms
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We study the Voronoi diagram, under the Euclidean metric, of a set of ellipses, given in parametric representation. We use an efficient incremental algorithm and focus on the required predicates. The paper concentrates on InCircle, which is the hardest predicate: it decides the position of a query ellipse relative to the Voronoi circle of three given ellipses. We describe an exact, real-time, and complete implementation for InCircle, combining a certified numeric algorithm with algebraic computation. The numeric part leads to a real-time implementation for non-degenerate inputs. It relies on a geometric preprocessing that guarantees a unique solution in a box of parametric space, where a customized subdivision-based method approximates the Voronoi circle tracing the bisectors. Our subdivision method achieves quadratic convergence by exploiting the geometric characteristics of the problem. To achieve robustness, we develop interval-arithmetic techniques, based on the C++ package Alias. We switch to an algebraic approach for handling the degeneracies fast. Based on a different algebraic system to model InCircle, we apply real solving and resultant theory. The latter relies on certain symbolic routines which are efficiently implemented in Maple. Our approach readily generalizes to arbitrary conics. The paper concludes with experiments showing that most instances run in less than 0.1 sec, on a 2.6GHz Pentium-4, whereas degenerate cases may take up to 13 sec.