Fast and Robust Point Containment Queries on Trimmed Surface

TL;DR

This work tackles point containment queries on trimmed surfaces, a core operation in CAD and surface tessellation, by introducing a fast, numerically stable winding-number framework. A key contribution is an ellipse-based bound for Bézier segments that enables linear-time evaluation and a constant-time inclusion test, eliminating costly subdivision. The method furthermore extends to periodic parameterizations by lifting curves to the universal covering space, allowing robust handling of non-contractible or discontinuous loops; periodically extended curves are managed by finite, localized computations. Experiments show substantial speedups over prior winding-number approaches and robustness to geometric noise and open boundaries, with practical benefits demonstrated in B-Rep processing and robust tessellation of trimmed surfaces. Overall, the approach provides a theoretically grounded, practically efficient tool for containment queries in both simply connected and periodic CAD domains.

Abstract

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.

Figures (25)

• Figure 1: Containment queries on a trimmed torus. The opaque region represents the retained surface, while the translucent part is trimmed away. Points on the torus are randomly sampled; those inside the trimmed region are marked in red, and those outside are shown in green.
• Figure 2: Topological intersection between two faces. Extraneous portions of the intersection curve must be removed using point–surface containment queries.
• Figure 3: A periodic trimmed surface defined on a cylindrical geometry. The boundary consists of two separation loops, which do not form closed loops in the parameter domain.
• Figure 4: Ray casting and its failure cases. Left: The classic even–odd rule determines inside/outside classification by counting ray–boundary intersections. Right: Common failure cases include (a) invalid intersection points, (b) rays coincident with boundary segments, (c) open boundaries due to geometric errors, and (d) non-manifold configurations.
• Figure 5: Universal covering spaces of periodic domains. Contractible loops lift to closed loops, while non-contractible loops become periodically extended curves that repeat across tiles.

Theorems & Definitions (9)

• lemma 1: Ellipse Bound
• lemma 2: Termination
• proposition 1
• proposition 2
• proposition 3
• proposition 4
• proposition 5
• proposition 6
• proposition 7