Spatially Accelerated Winding Numbers for Curved Geometry

TL;DR

This paper introduces a fast, accurate method for evaluating generalized winding numbers on curved geometries like NURBS, using a spatial hierarchy and adaptive subdivision to improve performance while maintaining precision.

Contribution

It extends fast winding number evaluation to NURBS and trimmed NURBS using a bounding volume hierarchy with precomputed moments, enabling efficient and accurate containment queries.

Findings

Achieves sub-linear complexity in winding number evaluation.

Maintains accuracy comparable to direct evaluation near boundaries.

Demonstrates improved performance on large 2D and 3D datasets.

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.