Caption: Overview of our method. We present an efficient winding number computation for parametric curves and extend it to periodic domains by lifting curves to the universal covering space. Using this approach, we enable fast and robust point containment queries on trimmed surfaces. First row: Illustration of the winding number computation (Section 4). Unlike Spainhour et al. [2024], we employ a more flexible ellipse bound, enabling linear-time Bézier evaluation and 𝑂(1) inclusion tests. Second row: By lifting curves to the universal covering space, we correctly compute winding numbers on periodic domains. The figure compares the winding number field computed in the covering space and in the parameter domain.

Abstract

Point containment queries on trimmed surfaces are fundamental to CAD modeling, solid geometry processing, and surface tessellation. Existing approaches such as ray casting suffer from robustness issues, while state-of-the-art winding number methods remain computationally expensive.

We propose a fast and robust method for point containment queries on trimmed surfaces, including those with periodic parameterizations. Our approach introduces a recursive winding number computation scheme that replaces the control-points bound with an ellipse-based bound for Bezier segments, avoiding costly curve subdivision. For periodic surfaces, we lift trimming curves to the universal covering space, allowing accurate winding number computation even for non-contractible loops and for loops whose p-curves appear open in the parameter domain due to periodic identification.

Experiments on large-scale 2D datasets and real-world CAD models show that our method achieves substantial speedups over existing winding number algorithms. For valid periodic trimming loops, the method is theoretically well-defined; its resilience to mild geometric noise and open-boundary defects is evaluated empirically. We further demonstrate its effectiveness in B-Rep model intersection and in robust tessellation of trimmed surfaces.