{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T11:35:59Z","timestamp":1786534559161,"version":"3.56.0"},"reference-count":70,"publisher":"Wiley","license":[{"start":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T00:00:00Z","timestamp":1786492800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T00:00:00Z","timestamp":1786492800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/100007065","name":"Nvidia","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100007065","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Computer Graphics Forum"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Cyclic animation is widely used in computer graphics and interactive content. It supports seamless playback in games, VR, and interactive simulation, where short clips must repeat smoothly over long durations. Converting an observed non\u2010cyclic trajectory into a seamless loop is challenging because the endpoint states of the observed sequence rarely match exactly, and the governing equations of the underlying system are often unavailable. We therefore propose an equation\u2010free framework that identifies a Koopman surrogate from the observed trajectory and computes a cyclic trajectory by applying a Fourier\u2010parameterized, time\u2010varying control input under a hard temporal periodicity constraint. The resulting formulation reduces cyclic synthesis to a linearly constrained quadratic program that can be solved efficiently through a structured KKT system. Our method is applicable to a diverse range of examples, including N\u2010body systems, cloth, deformable objects, shallow water, etc.<\/jats:p>","DOI":"10.1111\/cgf.70566","type":"journal-article","created":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T11:00:44Z","timestamp":1786532444000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Closing Trajectories: Equation\u2010Free Cyclic Animation via Koopman Surrogates"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-2463-1133","authenticated-orcid":false,"given":"Shixun","family":"Huang","sequence":"first","affiliation":[{"name":"University of Toronto Toronto Canada"},{"name":"University of British Columbia Vancouver Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-4309-862X","authenticated-orcid":false,"given":"Siyuan","family":"Chen","sequence":"additional","affiliation":[{"name":"University of British Columbia Vancouver Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2587-827X","authenticated-orcid":false,"given":"Yue","family":"Chang","sequence":"additional","affiliation":[{"name":"University of Toronto Toronto Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4989-6971","authenticated-orcid":false,"given":"Zhecheng","family":"Wang","sequence":"additional","affiliation":[{"name":"University of Toronto Toronto Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1919-5437","authenticated-orcid":false,"given":"Peter Yichen","family":"Chen","sequence":"additional","affiliation":[{"name":"University of British Columbia Vancouver Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,8,12]]},"reference":[{"key":"e_1_2_10_2_2","volume-title":"Proceedings of the Special Interest Group on Computer Graphics and Interactive Techniques Conference (SIGGRAPH '25)","author":"Abdal R.","year":"2025"},{"key":"e_1_2_10_2_3","unstructured":"doi:10.1145\/3721238.3730829. 2"},{"key":"e_1_2_10_3_2","volume-title":"ACM SIGGRAPH 2009 Papers","author":"Barbi\u010d J.","year":"2009"},{"key":"e_1_2_10_3_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/1576246.1531359. 2 4","DOI":"10.1145\/1576246.1531359"},{"key":"e_1_2_10_4_2","doi-asserted-by":"publisher","DOI":"10.1145\/3763310"},{"key":"e_1_2_10_4_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3763310. 2 8","DOI":"10.1145\/3763310"},{"key":"e_1_2_10_5_2","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073300"},{"key":"e_1_2_10_6_2","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185566"},{"key":"e_1_2_10_6_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/2185520.2185566. 2","DOI":"10.1145\/2185520.2185566"},{"key":"e_1_2_10_7_2","unstructured":"ChangY. et al.:Low\u2010rank koopman deformables with log\u2010linear time integration. arXiv:2602.07687. URL:https:\/\/arxiv.org\/abs\/2602.07687 arXiv:2602.07687 doi:10.48550\/arXiv.2602.07687. 2 3"},{"key":"e_1_2_10_8_2","doi-asserted-by":"publisher","DOI":"10.1145\/3731148"},{"key":"e_1_2_10_8_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3731148. 2","DOI":"10.1145\/3731148"},{"key":"e_1_2_10_9_2","unstructured":"ChenS. ChenY. PanuelosJ. BenchekrounO. ChangY. GrinspunE. WangZ.: Fast subspace fluid simulation with a temporally\u2010aware basis.ACM Transactions on Graphics(2025). 2 3"},{"key":"e_1_2_10_10_2","volume-title":"SIGGRAPH Asia 2023 Conference Papers","author":"Chang Y.","year":"2023"},{"key":"e_1_2_10_10_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3610548.3618158. 2","DOI":"10.1145\/3610548.3618158"},{"key":"e_1_2_10_11_2","volume-title":"ACM SIGGRAPH 2024 Conference Papers","author":"Chen Y.","year":"2024"},{"key":"e_1_2_10_11_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3641519.3657468. 2","DOI":"10.1145\/3641519.3657468"},{"key":"e_1_2_10_12_2","unstructured":"ChenP. Y. XiangJ. ChoD. H. ChangY. PershingG. A. MaiaH. T. ChiaramonteM. M. CarlbergK. T. GrinspunE.: CROM: Continuous reduced\u2010order modeling of PDEs using implicit neural representations. InThe Eleventh International Conference on Learning Representations(2023). URL:https:\/\/openreview.net\/forum?id=FUORz1tG8Og. 2"},{"key":"e_1_2_10_13_2","doi-asserted-by":"publisher","DOI":"10.1145\/1778765.1778843"},{"key":"e_1_2_10_14_2","volume-title":"ACM SIGGRAPH 2010 Papers","author":"Aguiar E.","year":"2010"},{"key":"e_1_2_10_14_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/1833349.1778843. 2","DOI":"10.1145\/1833349.1778843"},{"key":"e_1_2_10_15_2","doi-asserted-by":"publisher","DOI":"10.1145\/1230100.1230123"},{"key":"e_1_2_10_15_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/1230100.1230123. 2","DOI":"10.1145\/1230100.1230123"},{"key":"e_1_2_10_16_2","doi-asserted-by":"publisher","DOI":"10.1145\/3735126"},{"key":"e_1_2_10_17_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-020-2649-2"},{"key":"e_1_2_10_17_3","unstructured":"doi:10.1038\/s41586\u2010020\u20102649\u20102. 6"},{"key":"e_1_2_10_18_2","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2461922"},{"key":"e_1_2_10_18_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/2461912.2461922. 2 8","DOI":"10.1145\/2461912.2461922"},{"key":"e_1_2_10_19_2","doi-asserted-by":"publisher","DOI":"10.1145\/3606938"},{"key":"e_1_2_10_19_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3606938. 2 11 12","DOI":"10.1145\/3606938"},{"key":"e_1_2_10_20_2","doi-asserted-by":"publisher","DOI":"10.1145\/566654.566605"},{"key":"e_1_2_10_20_3","doi-asserted-by":"crossref","unstructured":"URL:https:\/\/doi.org\/10.1145\/566654.566605","DOI":"10.1145\/566654.566605"},{"key":"e_1_2_10_20_4","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/566654.566605. 2","DOI":"10.1145\/566654.566605"},{"key":"e_1_2_10_21_2","doi-asserted-by":"publisher","DOI":"10.1145\/1618452.1618469"},{"key":"e_1_2_10_21_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/1618452.1618469. 2","DOI":"10.1145\/1618452.1618469"},{"key":"e_1_2_10_22_2","first-page":"1","volume-title":"Proceedings of the Special Interest Group on Computer Graphics and Interactive Techniques Conference (SIGGRAPH '25)","author":"Kansy M.","year":"2025"},{"key":"e_1_2_10_22_3","unstructured":"doi:10.1145\/3721238.3730826. 2"},{"key":"e_1_2_10_23_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.17.5.315"},{"key":"e_1_2_10_23_3","unstructured":"arXiv:https:\/\/www.pnas.org\/doi\/pdf\/10.1073\/pnas.17.5.315"},{"key":"e_1_2_10_23_4","doi-asserted-by":"crossref","unstructured":"doi:10.1073\/pnas.17.5.315. 2 3","DOI":"10.1073\/pnas.17.5.315"},{"key":"e_1_2_10_24_2","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530157"},{"key":"e_1_2_10_24_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3528223.3530157. 2","DOI":"10.1145\/3528223.3530157"},{"key":"e_1_2_10_25_2","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459767"},{"key":"e_1_2_10_26_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.newast.2019.01.003"},{"key":"e_1_2_10_27_2","doi-asserted-by":"publisher","DOI":"10.1145\/3687760"},{"key":"e_1_2_10_27_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3687760. 2","DOI":"10.1145\/3687760"},{"key":"e_1_2_10_28_2","doi-asserted-by":"crossref","unstructured":"LiZ. TuckerR. SnavelyN. HolynskiA.: Generative image dynamics. InProceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition (CVPR)(June2024) pp.24142\u201324153. 2","DOI":"10.1109\/CVPR52733.2024.02279"},{"key":"e_1_2_10_29_2","unstructured":"MizuguchiM. BuchananJ. CalvertT.:Data driven motion transitions for interactive games. 2"},{"key":"e_1_2_10_30_2","doi-asserted-by":"publisher","DOI":"10.1145\/3658184"},{"key":"e_1_2_10_30_3","doi-asserted-by":"crossref","unstructured":"URL:https:\/\/doi.org\/10.1145\/3658184","DOI":"10.1145\/3658184"},{"key":"e_1_2_10_30_4","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3658184. 2","DOI":"10.1145\/3658184"},{"key":"e_1_2_10_31_2","doi-asserted-by":"publisher","DOI":"10.1145\/2508363.2508417"},{"key":"e_1_2_10_31_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/2508363.2508417. 2","DOI":"10.1145\/2508363.2508417"},{"key":"e_1_2_10_32_2","doi-asserted-by":"publisher","DOI":"10.1137\/15M1013857"},{"key":"e_1_2_10_32_3","doi-asserted-by":"crossref","unstructured":"arXiv:https:\/\/doi.org\/10.1137\/15M1013857","DOI":"10.1137\/15M1013857"},{"key":"e_1_2_10_32_4","doi-asserted-by":"crossref","unstructured":"doi:10.1137\/15M1013857. 2","DOI":"10.1137\/15M1013857"},{"key":"e_1_2_10_33_2","doi-asserted-by":"publisher","DOI":"10.1137\/16M1062296"},{"key":"e_1_2_10_33_3","doi-asserted-by":"crossref","unstructured":"arXiv:https:\/\/doi.org\/10.1137\/16M1062296","DOI":"10.1137\/16M1062296"},{"key":"e_1_2_10_33_4","doi-asserted-by":"crossref","unstructured":"doi:10.1137\/16M1062296. 2","DOI":"10.1137\/16M1062296"},{"key":"e_1_2_10_34_2","doi-asserted-by":"publisher","DOI":"10.1145\/545261.545279"},{"key":"e_1_2_10_34_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/545261.545279. 2","DOI":"10.1145\/545261.545279"},{"key":"e_1_2_10_35_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112010001217"},{"key":"e_1_2_10_36_2","doi-asserted-by":"publisher","DOI":"10.1145\/3180491"},{"key":"e_1_2_10_36_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3180491. 6","DOI":"10.1145\/3180491"},{"key":"e_1_2_10_37_2","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530178"},{"key":"e_1_2_10_37_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3528223.3530178. 2","DOI":"10.1145\/3528223.3530178"},{"key":"e_1_2_10_38_2","doi-asserted-by":"publisher","DOI":"10.1145\/3763306"},{"key":"e_1_2_10_38_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3763306. 2","DOI":"10.1145\/3763306"},{"key":"e_1_2_10_39_2","doi-asserted-by":"publisher","DOI":"10.1145\/378456.378507"},{"key":"e_1_2_10_39_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/378456.378507. 2","DOI":"10.1145\/378456.378507"}],"container-title":["Computer Graphics Forum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.70566","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/cgf.70566","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.70566","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T11:01:09Z","timestamp":1786532469000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1111\/cgf.70566"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,12]]},"references-count":70,"alternative-id":["10.1111\/cgf.70566"],"URL":"https:\/\/doi.org\/10.1111\/cgf.70566","archive":["Portico"],"relation":{},"ISSN":["0167-7055","1467-8659"],"issn-type":[{"value":"0167-7055","type":"print"},{"value":"1467-8659","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,12]]},"assertion":[{"value":"2026-08-12","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70566"}}