Sweeping and Maintaining Two-dimensional Arrangements on Surfaces

We introduce a general framework for processing a set of curves defined on a continuous two-dimensional parametric surface, while sweeping the parameter space. A major goal of our work is to maximize code reuse in implementing algorithms that employ the prevalent sweep-line paradigm, and consequently to minimize the effort needed to extend the implementation of the paradigm to various surfaces and families of curves embedded on them. We show how the sweep-line paradigm is used to construct an arrangement of curves embedded on an orientable parametric surface, and explain how the arrangement package of {\sc cgal}, which previously handled only arrangements of bounded planar curves, is extended to handle curves embedded on a general surface. To the best of our knowledge, this is the first software implementation of generic algorithms that can handle arrangements on general parametric surfaces