Closed-form Generalized Winding Numbers of Rational Parametric Curves for Robust Containment Queries

We derive closed-form expressions for generalized winding numbers of rational parametric curves for robust containment queries. Given an oriented rational parametric curve and a query point, the generalized winding number can be reformulated to an integral of a rational polynomial. The key to computing the integral lies in using the residue theorem. Then, add up the contributions of each curve to obtain the generalized winding numbers of a set of rational parametric curves. Furthermore, the derivatives of generalized winding numbers are easily derived. Consequently, the expressions for generalized winding numbers are concise and computationally efficient, becoming faster than state-of-the-art methods. Moreover, the computational costs for various query points are almost the same.