Spatially Accelerated Winding Numbers for Curved Geometry

2026-05-18

AI Summary

This work addresses the challenge of efficiently and accurately determining inside–outside relationships for non-watertight, overlapping, or nested models involving NURBS curves and surfaces—a task at which existing fast generalized winding number (GWN) methods struggle. The paper presents the first extension of fast GWN to parametric surfaces, introducing a bounding volume hierarchy (BVH)-based framework that combines precomputed moment data for far-field Taylor expansion approximations with adaptive NURBS subdivision and direct evaluation in the near field. This hybrid approach preserves boundary accuracy while significantly accelerating computation. Experimental results demonstrate that the method achieves sublinear query complexity and high-precision inclusion tests on complex 2D and 3D NURBS models, substantially outperforming conventional direct evaluation in runtime efficiency.

Abstract

The generalized winding number (GWN) is a scalar field that supports robust containment queries on curved geometry, including non-watertight, overlapping, and nested boundary representations. While queries can be easily parallelized over samples, direct evaluation on parametric curves and surfaces remains costly for large and complex models. Fast, state-of-the-art GWN approaches leverage a spatial index to approximate the GWN, typically coupled with a Taylor expansion which approximates the GWN contribution for far clusters of geometric primitives. However, such methods operate only on discrete inputs such as triangle meshes and point clouds, and would introduce containment errors near boundaries if applied to curved input. We extend support for fast GWN evaluation over arbitrary collections of NURBS curves in 2D and trimmed NURBS patches in 3D via a Bounding Volume Hierarchy that stores efficiently precomputed moment data in the hierarchy nodes. When querying the hierarchy, approximations for far clusters are used alongside direct evaluation for nearby NURBS primitives, achieving sub-linear complexity while preserving the geometric features in the vicinity of the query point. Central to our performance improvements is an adaptive subdivision strategy for NURBS primitives during a preprocessing phase, creating better spatial partitions while retaining the same accuracy for containment decisions as a direct evaluation. We demonstrate the performance and accuracy of our approach across a large collection of 2D and 3D datasets.

Problem

• Generalized Winding Number
• NURBS
• spatial acceleration
• curved geometry
• containment queries

Innovation

• Generalized Winding Number
• NURBS
• Bounding Volume Hierarchy
• Adaptive Subdivision
• Spatial Acceleration