Arrangements on Surfaces of Genus One: Tori and Dupin Cyclides

An algorithm is presented to compute the exact arrangement induced by arbitrary algebraic surfaces on a parametrized ring Dupin cyclide, including the special case of the torus. The intersection of an algebraic surface of degree $n$ with a reference cyclide is represented as a real algebraic curve of bi-degree $(2n,2n)$ in the cyclide's two-dimensional parameter space. We use Eigenwillig and Kerber~\cite{ek-exact} to compute a planar arrangement of such curves and extend their approach to obtain more asymptotic information about curves approaching the boundary of the cyclide's parameter space. With that, we can base our implementation on a general software framework by Berberich~et.~al.~\cite{bfhmw-samtdaosafs-07} to construct the arrangement on the cyclide. Our contribution provides the demanded techniques to model the special topology of the reference surface of genus one. Our experiments show no combinatorial overhead of the framework, i.e., the overall performance is strongly coupled to the efficiency of the implementation for arrangements of algebraic plane curves