ACS Algorithms for Complex Shapes with Certified Numerics and Topology Voronoi diagram of ellipses: cgal-based implementation

We present a C++ open-source implementation of an incremental algorithm for the computation of the Voronoi diagram of ellipses in the Euclidean plane. This is the first complete implementation, under the exact computation paradigm; it also computes the approximate diagram with any given precision. It is based on the cgal package for the Apollonius diagram. The ellipses are given in parametric representation, which allows us to develop a GUI for input / output. The software concerns non-intersecting ellipses, but it can be generalized. We implement an interval Newton solver, bivariate polynomial interpolation, and trivariate system resultant computation, and demonstrate the factorization of certain resultants so as to arrive at an efficient implementation of InCircle . As a ballmark estimate, our code spends about 99sec to construct the Voronoi diagram of 128 ellipses, when few degeneracies occur. It is faster than the cgal Voronoi diagram of polygons, when ellipses are approximated by k-gons for k > 15, but expectedly slower on circles than the cgal Apollonius package. Our software is connected to the algebraic library Synaps and, through it, to libraries mpfr and ntl, hence illustrating the concept of algebraic support for geometric computing; it is targeted for inclusion in the cgal library. In this direction, we present some experiments using the new Algebraic Kernel from MPI.