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The proposed approach effectively reduces the dimensionality of the problem by representing the motion trajectories of each vertex using continuous parametric B\u00e9zier splines with variable keyframe times. Because this mesh animation representation is continuous and fully differentiable, it can be optimized such that it follows the laws of physics under various constraints. The proposed method also integrates constraints, such as collisions and cyclic motion, making it suitable for real\u2010world applications where seamless looping and physical interactions are required. Leveraging projective dynamics, we further enhance the computational efficiency by decoupling the optimization into local parallelizable and global quadratic steps, enabling a fast and stable simulation. In addition, BlendSim is compatible with modern animation workflows and file formats, such as the glTF, making it practical way for authoring and transferring mesh animation.<\/jats:p>","DOI":"10.1111\/cgf.70068","type":"journal-article","created":{"date-parts":[[2025,4,18]],"date-time":"2025-04-18T04:32:36Z","timestamp":1744950756000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["BlendSim: Simulation on Parametric Blendshapes using Spacetime Projective Dynamics"],"prefix":"10.1111","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-5358-5579","authenticated-orcid":false,"given":"Yuhan","family":"Wu","sequence":"first","affiliation":[{"name":"The University of Tokyo  Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1251-970X","authenticated-orcid":false,"given":"Nobuyuki","family":"Umetani","sequence":"additional","affiliation":[{"name":"The University of Tokyo  Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,4,18]]},"reference":[{"key":"e_1_2_8_2_2","doi-asserted-by":"publisher","DOI":"10.1145\/3355089.3356569"},{"key":"e_1_2_8_3_2","doi-asserted-by":"crossref","unstructured":"Barbi\u010d Jernej daSilva Marco andPopovi\u0107 Jovan. \u201cDeformable object animation using reduced optimal control\u201d.ACM SIGGRAPH 2009 papers.2009 1\u201392\u20134 7 8 10.","DOI":"10.1145\/1576246.1531359"},{"key":"e_1_2_8_4_2","doi-asserted-by":"crossref","unstructured":"Bouaziz Sofien Martin Sebastian Liu Tiantian et al. \u201cProjective dynamics: Fusing constraint projections for fast simulation\u201d.Seminal Graphics Papers: Pushing the Boundaries Volume 2.2023 787\u20137972 4 6 8.","DOI":"10.1145\/3596711.3596794"},{"key":"e_1_2_8_5_2","article-title":"Tracks: toward directable thin shells","volume":"26","author":"Bergou Mikl\u00f3s","year":"2007","journal-title":"ACM Transactions on Graphics (TOG)"},{"key":"e_1_2_8_6_2","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276467"},{"key":"e_1_2_8_6_3","unstructured":"url:https:\/\/doi.org\/10.1145\/1276377.12764679."},{"key":"e_1_2_8_7_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12836"},{"key":"e_1_2_8_8_2","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185566"},{"key":"e_1_2_8_9_2","doi-asserted-by":"crossref","unstructured":"Benchekroun Otman Zhang Jiayi Eris Chaudhuri Siddhartha et al. \u201cFast Complementary Dynamics via Skinning Eigenmodes\u201d.arXiv preprint arXiv:2303.11886(2023) 4.","DOI":"10.1145\/3592404"},{"key":"e_1_2_8_10_2","doi-asserted-by":"publisher","DOI":"10.1145\/3203187"},{"key":"e_1_2_8_11_2","doi-asserted-by":"crossref","unstructured":"Cohen Michael F.\u201cInteractive spacetime control for animation\u201d.Proceedings of the 19th annual conference on Computer graphics and interactive techniques.1992 293\u20133023 10.","DOI":"10.1145\/133994.134083"},{"key":"e_1_2_8_12_2","first-page":"1","article-title":"Diffpd: Differentiable projective dynamics","volume":"41","author":"Du Tao","year":"2021","journal-title":"ACM Transactions on Graphics (TOG)"},{"key":"e_1_2_8_13_2","doi-asserted-by":"publisher","DOI":"10.1145\/3395208"},{"key":"e_1_2_8_14_2","doi-asserted-by":"publisher","DOI":"10.1145\/882262.882286"},{"key":"e_1_2_8_15_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00371-010-0422-5"},{"key":"e_1_2_8_16_2","doi-asserted-by":"crossref","unstructured":"Gleicher Michael. \u201cMotion editing with spacetime constraints\u201d.Proceedings of the 1997 symposium on Interactive 3D graphics.1997 139\u2010ff 3.","DOI":"10.1145\/253284.253321"},{"key":"e_1_2_8_17_2","article-title":"TetGen, a Delaunay\u2010based quality tetrahedral mesh generator","volume":"41","author":"Hang Si","year":"2015","journal-title":"ACM Trans. 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