Exact Voronoi diagram of smooth convex pseudo-circles: General predicates, and implementation for ellipses

Authors:
Ioannis Z. Emiris;Elias P. Tsigaridas;George M. Tzoumas
Affiliations:
National and Kapodistrian University of Athens, Greece;INRIA Paris-Rocquencourt, UPMC, Univ. Paris 06, LIP6, France;CNRS UMR 5158, LE2I, University of Burgundy, France
Venue:
Computer Aided Geometric Design
Year:
2013

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Abstract

We examine the problem of computing exactly the Voronoi diagram (via the dual Delaunay graph) 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 Voronoi diagram is constructed incrementally. Our first contribution is to propose robust and efficient algorithms, under the exact computation paradigm, for all required predicates, thus generalizing earlier algorithms for non-intersecting ellipses. Second, we focus on InCircle, which is the hardest predicate, and express it by a simple sparse 5x5 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 possibly intersecting ellipses, which is the first exact implementation for the problem. Our code spends about a minute to construct the Voronoi diagram of 200 ellipses, when few degeneracies occur. It is faster than the cgal segment Voronoi diagram, when ellipses are approximated by k-gons for k15, and a state-of-the-art implementation of the Voronoi diagram of points, when each ellipse is approximated by more than 1250 points.