Voronoi diagrams and arrangements
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Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation
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Computer Aided Geometric Design
Curves and surfaces represented by polynomial support functions
Theoretical Computer Science
Voronoi diagram computations for planar NURBS curves
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Approximations of 2D and 3D generalized Voronoi diagrams
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Journal of Symbolic Computation
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
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Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
Divide-and-conquer for Voronoi diagrams revisited
Computational Geometry: Theory and Applications
Constructing two-dimensional Voronoi diagrams via divide-and-conquer of envelopes in space
Transactions on computational science IX
The offset to an algebraic curve and an application to conics
ICCSA'05 Proceedings of the 2005 international conference on Computational Science and its Applications - Volume Part I
A subdivision approach to planar semi-algebraic sets
GMP'10 Proceedings of the 6th international conference on Advances in Geometric Modeling and Processing
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We design and implement an efficient and certified algorithm for the computation of Voronoi diagrams (VDs) constrained to a given domain. Our framework is general and applicable to any VD-type where the distance field is given explicitly or implicitly by a polynomial, notably the anisotropic VD or VDs of non-punctual sites. We use the Bernstein form of polynomials and DeCasteljau's algorithm to subdivide the initial domain and isolate bisector, or domains that contain a Voronoi vertex. The efficiency of our algorithm is due to a filtering process, based on bounding the field over the subdivided domains. This allows to exclude functions (thus sites) that do not contribute locally to the lower envelope of the lifted diagram. The output is a polygonal description of each Voronoi cell, within any user-defined precision, isotopic to the exact VD. Correctness of the result is implied by the certified approximations of bisector branches, which are computed by existing methods for handling algebraic curves. First experiments with our C++ implementation, based on double precision arithmetic, demonstrate the adaptability of the algorithm.