article published yes Peiyuan Xie author 488e236c-6bad-11f0-9831-859175c78e8a Christian Hafner author 400429CC-F248-11E8-B48F-1D18A9856A87 Christopher J Wojtan author 3C61F1D2-F248-11E8-B48F-1D18A9856A870000-0001-6646-5546 GradSch department ChWo department Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena project We revisit the computation of 3D generalized winding numbers, a useful measure for inside-outside classification on triangle meshes with gaps, self-intersections, and open boundaries. At the core of our new method is an analytical reduction of the surface integral that defines the winding number, resulting in a single ray-mesh intersection test and an elementary sum over boundary edges per evaluation. This construction is orders of magnitude more efficient than the state of the art in practice, which we show in an extensive performance benchmark. Conveniently, the method also reduces to the best-available asymptotic complexity in the worst case, and it introduces no approximations apart from floating-point errors. Our algorithm is conceptually simple to understand, straightforward to implement and debug, and it works reliably even on extremely noisy and corrupt input geometry. https://research-explorer.ista.ac.at/download/22241/22249/2026_TransactionsGraphics_Xie.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Association for Computing Machinery2026 eng 0730-0301 1557-736810.1145/3811339 454 P. Xie, C. Hafner, C. Wojtan, ACM Transactions on Graphics 45 (2026). Xie, Peiyuan, et al. “Fast and Exact Winding Numbers for Triangle Meshes.” <i>ACM Transactions on Graphics</i>, vol. 45, no. 4, 41, Association for Computing Machinery, 2026, doi:<a href="https://doi.org/10.1145/3811339">10.1145/3811339</a>. P. Xie, C. Hafner, and C. Wojtan, “Fast and exact winding numbers for triangle meshes,” <i>ACM Transactions on Graphics</i>, vol. 45, no. 4. Association for Computing Machinery, 2026. Xie P, Hafner C, Wojtan C. Fast and exact winding numbers for triangle meshes. <i>ACM Transactions on Graphics</i>. 2026;45(4). doi:<a href="https://doi.org/10.1145/3811339">10.1145/3811339</a> Xie, P., Hafner, C., &#38; Wojtan, C. (2026). Fast and exact winding numbers for triangle meshes. <i>ACM Transactions on Graphics</i>. Association for Computing Machinery. <a href="https://doi.org/10.1145/3811339">https://doi.org/10.1145/3811339</a> Xie, Peiyuan, Christian Hafner, and Chris Wojtan. “Fast and Exact Winding Numbers for Triangle Meshes.” <i>ACM Transactions on Graphics</i>. Association for Computing Machinery, 2026. <a href="https://doi.org/10.1145/3811339">https://doi.org/10.1145/3811339</a>. Xie P, Hafner C, Wojtan C. 2026. Fast and exact winding numbers for triangle meshes. ACM Transactions on Graphics. 45(4), 41. 222412026-07-03T21:03:48Z2026-07-06T06:14:18Z