A cgal-based implementation for the Voronoi diagram of ellipses

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, for the given problem. It is based on the cgal package for the Apollonius diagram in the plane: exploiting the generic programming principle, our main additions concern the predicates. The ellipses are input in parametric representation, which allows us to develop a GUI for input and output. The software concerns non-intersecting ellipses, but the extension to the general case is possible. We develop practical algebraic methods, including an interval Newton solver, bivariate polynomial interpolation, and trivariate system resultant computation, and exploit the factorization of resultants in order to arrive at an efficient and exact C++ implementation of InCircle . The code is up to two orders of magnitude slower than the cgal Apollonius package on circles, which comes as no surprise. We also test different sets of ellipses; as a ballmark estimate, the current version spends about a second to insert a new ellipse, when few degeneracies occur. The software is connected to 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.