Robust Containment Queries over Collections of Rational Parametric Curves via Generalized Winding Numbers

Point containment queries for regions bound by watertight geometric surfaces, i.e. closed and without self-intersections, can be evaluated straightforwardly with a number of well-studied algorithms. However, when such assumptions on domain geometry are not met, these methods are theoretically unfounded at best and practically unusable at worst. More robust classification schemes utilize generalized winding numbers, a mathematical construction that is indifferent to imperfections in the often human-defined geometric model. We extend this methodology to more general curved shapes, defining a robust containment query for regions whose boundary elements are defined by a collection of rational parametric curves. In doing so, we devise an algorithm that is stable and accurate at arbitrary points in space, circumventing the typical difficulties for queries that are arbitrarily close or coincident with the model. This is done by reducing the generalized winding number problem to an integer winding number problem, which is solved by approximating each curve with a polyline that provably has the same winding number at the point of interest. We demonstrate the improvements in computational complexity granted by this method over conventional techniques, as well as the robustness induced by its application