[{"department":[{"_id":"GradSch"},{"_id":"ChWo"}],"title":"Fast and exact winding numbers for triangle meshes","file":[{"content_type":"application/pdf","checksum":"7e36e69f377b680a893e65b620b43813","relation":"main_file","date_created":"2026-07-06T06:13:12Z","file_size":5212838,"success":1,"creator":"dernst","date_updated":"2026-07-06T06:13:12Z","file_name":"2026_TransactionsGraphics_Xie.pdf","file_id":"22249","access_level":"open_access"}],"quality_controlled":"1","project":[{"name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","grant_number":"101045083"}],"doi":"10.1145/3811339","das_tickbox":"0","author":[{"last_name":"Xie","id":"488e236c-6bad-11f0-9831-859175c78e8a","first_name":"Peiyuan","full_name":"Xie, Peiyuan"},{"first_name":"Christian","full_name":"Hafner, Christian","last_name":"Hafner","id":"400429CC-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Wojtan, Christopher J","first_name":"Christopher J","id":"3C61F1D2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6646-5546","last_name":"Wojtan"}],"day":"03","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"PlanS_conform":"1","date_published":"2026-07-03T00:00:00Z","corr_author":"1","article_processing_charge":"Yes (via OA deal)","publication_status":"published","year":"2026","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","acknowledgement":"We thank Sadashige Ishida and Ryusuke Sugimoto for their insightful discussions and proofreading and other members of the ISTA\r\nVisual Computing Group for their general feedback. This project was\r\nfunded in part by the European Research Council (ERC Consolidator\r\nGrant 101045083 CoDiNA).","scopus_import":"1","file_date_updated":"2026-07-06T06:13:12Z","issue":"4","abstract":[{"lang":"eng","text":"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."}],"OA_place":"publisher","_id":"22241","has_accepted_license":"1","type":"journal_article","language":[{"iso":"eng"}],"article_number":"41","month":"07","date_created":"2026-07-03T21:03:48Z","citation":{"ama":"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>","ieee":"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.","ista":"Xie P, Hafner C, Wojtan C. 2026. Fast and exact winding numbers for triangle meshes. ACM Transactions on Graphics. 45(4), 41.","apa":"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>","chicago":"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>.","short":"P. Xie, C. Hafner, C. Wojtan, ACM Transactions on Graphics 45 (2026).","mla":"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>."},"ddc":["000"],"intvolume":"        45","publication":"ACM Transactions on Graphics","publisher":"Association for Computing Machinery","OA_type":"hybrid","article_type":"original","publication_identifier":{"eissn":["1557-7368"],"issn":["0730-0301"]},"volume":45,"date_updated":"2026-07-06T06:14:18Z","oa":1,"supplementarymaterial":"no","oa_version":"Published Version","researchdata_availability":"no"}]