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.

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.

This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exterior point; κ2 decides the position of a query ellipse relative to an external bitangent line of 2 given ellipses; κ3 decides the position of a query ellipse relative to a Voronoi circle of 3 given ellipses; κ4 determines the type of conflict between a Voronoi edge, defined by 4 given ellipses, and a query ellipse. The paper is restricted to non-intersecting ellipses, but the extension to arbitrary ones is straightforward. The ellipses are input in parametric representation or constructively. For κ1 and κ2 we derive optimal algebraic conditions, solve them exactly and provide efficient implementations in C++. For κ3 we compute a tight bound on the number of complex tritangent circles and use the parametric form of the ellipses in order to design an exact subdivision-based algorithm, which is implemented on Maple. This approach essentially answers κ4 as well. We conclude with current work on optimizing κ3 and implementing it in C++.