The Antipodal Method: Fast, Accurate, and Robust 3D Generalized Winding Numbers

2026-05-02

AI Summary

Existing 3D generalized winding number algorithms struggle to balance accuracy and efficiency, limiting their applicability in interactive and large-scale geometric processing. This work proposes a novel method that, for the first time, decomposes the winding number into a sum of ray-intersection counts and a boundary integral over the unit sphere. This formulation avoids costly surface integrals and spherical arrangements, naturally accommodating non-manifold geometries and complex topologies. The approach is applicable to both triangle meshes and parametric surfaces and leverages CPU/GPU parallelization for high performance. On the CPU, it achieves a 22× speedup over the fastest exact method and a 3× improvement over approximate alternatives. A GPU implementation enables up to 10⁹ queries per second, rendering at 120 FPS in 4K resolution, and accelerates parametric surface evaluation by 5.6×.

Abstract

Generalized winding numbers provide a robust measure of point insidedness for 3D surfaces - whether open, self-intersecting, or non-manifold - and are central to numerous geometry processing tasks. However, existing methods trade off between accuracy and computational efficiency, limiting their use in interactive and large-scale applications. We introduce a new formulation and algorithm for computing generalized winding numbers that is both fast and accurate to arbitrary precision, applicable to meshes and parametric surfaces. Our approach expresses the winding number as the sum of two intuitive geometric quantities: the signed number of ray-surface intersections and a boundary integral over the surface's projection onto the unit sphere. This insight leads to an efficient discretization that avoids expensive surface integrals and spherical arrangements. For meshes, our method achieves average speedups of $22\times$ on a CPU compared to the fastest precise methods and $3\times$ compared to the fastest approximation method, while maintaining full precision. On a GPU, for moderately complex meshes we reach a throughput of $10^9$ queries per second, or $4K$ generalized winding number slices at 120 FPS ($13\times$ faster than a naive GPU method). For parametric surfaces, our method is on average $5.6\times$ faster than the state-of-the-art method, with the same precision. Our method naturally handles complex topologies and non-manifold inputs. We extensively validate its accuracy, robustness, and time performance. Our code is available at https://github.com/MartensCedric/antipodal.

Problem

• generalized winding number
• computational efficiency
• accuracy
• 3D geometry processing
• non-manifold surfaces

Innovation

• generalized winding number
• antipodal method
• efficient geometric computation
• non-manifold surfaces
• GPU acceleration