Computational geometry for curved objects: Voronoi diagrams in the plane

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) closed curves, every pair of which has at most two intersection points. We propose robust end efficient algorithms for all required predicates under the exact computation paradigm, analyzing their algebraic complexity. To speed up the algebraic computations, we exploit geometric properties of the problem and provide a subdivision-based algorithm that exhibits quadratic convergence, allowing for real-time evaluations. Finally, we present a cgalbased c++ implementation for the case of ellipses, which is, to the best of our knowledge, the first exact implementation in non-linear computational geometry. Our code spends about 98 sec to construct the Delaunay graph of 128 non-intersecting ellipses, when no degeneracies occur. It is faster than the cgal segment Delaunay graph, when ellipses are approximated by k-gons for k > 15.