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About this paper
Cedric Martens, Philip Trettner, Mikhail Bessmeltsev
The authors introduce a method for evaluating generalized winding numbers, which measure point-insidedness around oriented surfaces, including open meshes. It combines a signed ray-crossing count with a contribution computed along the surface boundary, avoiding whole-surface summation. On 1,417 open Thingi10K meshes, it achieves a 22.2× geometric-mean speedup over a hierarchical precise method, with double-precision accuracy on par; timings exclude preprocessing.
Key insights
- The reformulation of GWN into a combination of ray-surface intersections and a boundary integral avoids expensive surface integrals and complex spherical arrangements, simplifying computation.
- Robustness to complex geometries, including self-intersections and non-manifold inputs, is achieved through symbolic perturbation of query points and exact integer predicates for geometric tests.
- The discrete formulation for meshes naturally cancels out internal edge contributions, reducing computation to boundary segments, while for parametric surfaces, identifying the true surface boundary significantly optimizes integral evaluation.
Summary by alphaXiv.
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