Point containment queries on trimmed surfaces are fundamental to CAD modeling, solid geometry processing, and surface tessellation. Existing approaches such as ray casting and generalized winding numbers often face limitations in robustness and computational efficiency. We propose a fast and numerically stable method for performing containment queries on trimmed surfaces, including those with periodic parameterizations. Our approach introduces a recursive winding number computation scheme that replaces costly curve subdivision with an ellipse-based bound for Bezier segments, enabling linear-time evaluation. For periodic surfaces, we lift trimming curves to the universal covering space, allowing accurate and consistent winding number computation even for non-contractible or discontinuous loops in parameter domain. Experiments show that our method achieves substantial speedups over existing winding-number algorithms while maintaining high robustness in the presence of geometric noise, open boundaries, and periodic topologies. We further demonstrate its effectiveness in processing real B-Rep models and in robust tessellation of trimmed surfaces.

翻译：裁剪曲面上的点包含性查询是CAD建模、实体几何处理和曲面网格剖分的基础。现有方法如光线投射和广义环绕数通常在鲁棒性和计算效率方面存在局限。我们提出了一种快速且数值稳定的方法，用于在裁剪曲面上执行包含性查询，包括具有周期性参数化的曲面。该方法引入了一种递归环绕数计算方案，用基于椭圆的贝塞尔段边界替代昂贵的曲线细分，实现了线性时间评估。对于周期性曲面，我们将裁剪曲线提升至通用覆盖空间，即使在参数域中存在不可缩或不连续环的情况下，也能实现精确且一致的环绕数计算。实验表明，该方法在保持对几何噪声、开放边界和周期性拓扑的高鲁棒性的同时，相比现有环绕数算法实现了显著加速。我们进一步展示了其在处理真实B-Rep模型和裁剪曲面鲁棒网格剖分中的有效性。