Exact Arrangements on Tori and Dupin Cyclides

An algorithm and implementation is presented to compute the exact arrangement
induced by arbitrary algebraic surfaces on a parametrized ring Dupin cyclide.
The family of Dupin cyclides contains as a special case 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 two-dimensional parameter space of the cyclide.
We use Eigenwillig and Kerber:
``Exact and Efficient 2D-Arrangements of Arbitrary Algebraic Curves'',
SODA~2008, 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 the general software framework
by Berberich~et.~al.: ``Sweeping and Maintaining Two-Dimensional
Arrangements on Surfaces: A First Step'', ESA~2007.
Our contribution provides the demanded techniques to model the special
geometry of surfaces intersecting a cyclide
and the special topology of the reference surface of genus one.
The contained implementation is complete and does not assume generic position.
Our experiments show that the combinatorial overhead of the framework
does not harm the efficiency of the method. Our experiments show that the
overall performance is strongly coupled to the efficiency of the
implementation for arrangements of algebraic plane curves.

(joint work with Eric Berberich)