Randomized incremental construction of abstract Voronoi diagrams
Computational Geometry: Theory and Applications
Journal of Computational and Applied Mathematics - Special issue on computational methods in computer graphics
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Discrete & Computational Geometry
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Voronoi diagram computations for planar NURBS curves
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On the asymptotic and practical complexity of solving bivariate systems over the reals
Journal of Symbolic Computation
Medial axis computation for planar free-form shapes
Computer-Aided Design
A subdivision approach to planar semi-algebraic sets
GMP'10 Proceedings of the 6th international conference on Advances in Geometric Modeling and Processing
Exact medial axis computation for circular arc boundaries
Proceedings of the 7th international conference on Curves and Surfaces
Yet another algorithm for generalized Voronoï Diagrams
Proceedings of the 27th Annual ACM Symposium on Applied Computing
Technical note: Voronoi diagrams of algebraic distance fields
Computer-Aided Design
Minimum area enclosure and alpha hull of a set of freeform planar closed curves
Computer-Aided Design
Computer Aided Geometric Design
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We examine the problem of computing exactly the Delaunay graph (and the dual Voronoi diagram) of a set of, possibly intersecting, smooth convex pseudo-circles in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) sites, every pair of which has at most two intersecting points. The Delaunay graph is constructed incrementally. Our first contribution is to propose robust end efficient algorithms for all required predicates, thus generalizing our earlier algorithms for ellipses, and we analyze their algebraic complexity, under the exact computation paradigm. Second, we focus on InCircle, which is the hardest predicate, and express it by a simple sparse 5 X 5 polynomial system, which allows for an efficient implementation by means of successive Sylvester resultants and a new factorization lemma. The third contribution is our cgal-based c++ software for the case of ellipses, which is the first exact implementation for the problem. Our code spends about 98 sec to construct the Delaunay graph of 128 non-intersecting ellipses, when few degeneracies occur. It is faster than the cgal segment Delaunay graph, when ellipses are approximated by k-gons for k 15.