{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T02:09:40Z","timestamp":1774058980880,"version":"3.50.1"},"reference-count":63,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2021,7,19]],"date-time":"2021-07-19T00:00:00Z","timestamp":1626652800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007911","name":"University of California, San Diego","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100007911","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006188","name":"Einstein Stiftung Berlin","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100006188","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009125","name":"National Center for Theoretical Sciences","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100009125","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006961","name":"California Institute of Technology","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100006961","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB 109"],"award-info":[{"award-number":["SFB 109"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2021,8,31]]},"abstract":"<jats:p>Smooth curves and surfaces can be characterized as minimizers of squared curvature bending energies subject to constraints. In the univariate case with an isometry (length) constraint this leads to classic non-linear splines. For surfaces, isometry is too rigid a constraint and instead one asks for minimizers of the Willmore (squared mean curvature) energy subject to a conformality constraint. We present an efficient algorithm for (conformally) constrained Willmore surfaces using triangle meshes of arbitrary topology with or without boundary. Our conformal class constraint is based on the discrete notion of conformal equivalence of triangle meshes. The resulting non-linear constrained optimization problem can be solved efficiently using the competitive gradient descent method together with appropriate Sobolev metrics. The surfaces can be represented either through point positions or differential coordinates. The latter enable the realization of abstract metric surfaces without an initial immersion. A versatile toolkit for extrinsic conformal geometry processing, suitable for the construction and manipulation of smooth surfaces, results through the inclusion of additional point, area, and volume constraints.<\/jats:p>","DOI":"10.1145\/3450626.3459759","type":"journal-article","created":{"date-parts":[[2021,7,20]],"date-time":"2021-07-20T00:04:26Z","timestamp":1626739466000},"page":"1-17","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":15,"title":["Constrained willmore surfaces"],"prefix":"10.1145","volume":"40","author":[{"given":"Yousuf","family":"Soliman","sequence":"first","affiliation":[{"name":"Caltech"}]},{"given":"Albert","family":"Chern","sequence":"additional","affiliation":[{"name":"UCSD"}]},{"given":"Olga","family":"Diamanti","sequence":"additional","affiliation":[{"name":"TU Graz, Austria"}]},{"given":"Felix","family":"Kn\u00f6ppel","sequence":"additional","affiliation":[{"name":"TU Berlin"}]},{"given":"Ulrich","family":"Pinkall","sequence":"additional","affiliation":[{"name":"TU Berlin, Germany"}]},{"given":"Peter","family":"Schr\u00f6der","sequence":"additional","affiliation":[{"name":"Caltech"}]}],"member":"320","published-online":{"date-parts":[[2021,7,19]]},"reference":[{"key":"e_1_2_2_1_1","article-title":"A Recursive Algebraic Coloring Technique for Hardware-Efficient Symmetric Sparse Matrix-Vector Multiplication","volume":"7","author":"Alappat Christie","year":"2020","unstructured":"Christie Alappat , Achim Basermann , Alan R. Bishop , Holger Fehske , Georg Hager , Olaf Schenk , Jonas Thies , and Gerhard Wellein . 2020 . A Recursive Algebraic Coloring Technique for Hardware-Efficient Symmetric Sparse Matrix-Vector Multiplication . ACM Trans. Par. Comput. 7 , 3 (2020), 19:1--37. Christie Alappat, Achim Basermann, Alan R. Bishop, Holger Fehske, Georg Hager, Olaf Schenk, Jonas Thies, and Gerhard Wellein. 2020. A Recursive Algebraic Coloring Technique for Hardware-Efficient Symmetric Sparse Matrix-Vector Multiplication. ACM Trans. Par. Comput. 7, 3 (2020), 19:1--37.","journal-title":"ACM Trans. Par. Comput."},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-93-07207-9"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1137\/16M1065379"},{"key":"e_1_2_2_4_1","volume-title":"Constrained Optimization and Lagrange Multiplier Methods","author":"Bertsekas Dimitri P.","unstructured":"Dimitri P. Bertsekas . 1996. Constrained Optimization and Lagrange Multiplier Methods . Athena Scientific . Dimitri P. Bertsekas. 1996. Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific."},{"key":"e_1_2_2_5_1","volume-title":"Vorlesungen \u00fcber Differentialgeometrie und geometrische Grundlagen von Einsteins Relativit\u00e4tstheorie III","author":"Blaschke Wilhelm","unstructured":"Wilhelm Blaschke . 1929. Vorlesungen \u00fcber Differentialgeometrie und geometrische Grundlagen von Einsteins Relativit\u00e4tstheorie III . Springer . Wilhelm Blaschke. 1929. Vorlesungen \u00fcber Differentialgeometrie und geometrische Grundlagen von Einsteins Relativit\u00e4tstheorie III. Springer."},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.2140\/gt.2015.19.2155"},{"key":"e_1_2_2_7_1","volume-title":"Proc. Symp. Geom. Proc. Eurographics, 101--110","author":"Alexander","unstructured":"Alexander I. Bobenko and Peter Schr\u00f6der. 2005. Discrete Willmore Flow . In Proc. Symp. Geom. Proc. Eurographics, 101--110 . Alexander I. Bobenko and Peter Schr\u00f6der. 2005. Discrete Willmore Flow. In Proc. Symp. Geom. Proc. Eurographics, 101--110."},{"key":"e_1_2_2_8_1","volume-title":"Circle Domains, Fuchsian and Schottky Uniformization. In Advances in Discrete Differential Geometry, Alexander I","author":"Bobenko Alexander I.","unstructured":"Alexander I. Bobenko , Stefan Sechelmann , and Boris Springborn . 2016. Discrete Conformal Maps: Boundary Value Problems , Circle Domains, Fuchsian and Schottky Uniformization. In Advances in Discrete Differential Geometry, Alexander I . Bobenko (Ed.). Springer , 1--56. Alexander I. Bobenko, Stefan Sechelmann, and Boris Springborn. 2016. Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization. In Advances in Discrete Differential Geometry, Alexander I. Bobenko (Ed.). Springer, 1--56."},{"key":"e_1_2_2_9_1","first-page":"71","article-title":"Constrained Willmore Tori in the 4-Sphere","volume":"86","author":"Bohle Christoph","year":"2010","unstructured":"Christoph Bohle . 2010 . Constrained Willmore Tori in the 4-Sphere . J. Diff. Geom. 86 , 1 (2010), 71 -- 132 . Christoph Bohle. 2010. Constrained Willmore Tori in the 4-Sphere. J. Diff. Geom. 86, 1 (2010), 71--132.","journal-title":"J. Diff. Geom."},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-007-0142-5"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1137\/17M1147615"},{"key":"e_1_2_2_12_1","doi-asserted-by":"crossref","unstructured":"Matthias Bollh\u00f6fer Olaf Schenk Radim Janalik Steve Hamm and Kiran Gullapalli. 2020. State-of-the-Art Sparse Direct Solvers. (2020) 3--33.  Matthias Bollh\u00f6fer Olaf Schenk Radim Janalik Steve Hamm and Kiran Gullapalli. 2020. State-of-the-Art Sparse Direct Solvers. (2020) 3--33.","DOI":"10.1007\/978-3-030-43736-7_1"},{"key":"e_1_2_2_13_1","volume-title":"Proc. Symp. Comp. Anim. 28--36","author":"Bridson Robert","year":"2003","unstructured":"Robert Bridson , S. Marino , and Ronald Fedkiw . 2003 . Simulation of Clothing with Folds and Wrinkles . In Proc. Symp. Comp. Anim. 28--36 . Robert Bridson, S. Marino, and Ronald Fedkiw. 2003. Simulation of Clothing with Folds and Wrinkles. In Proc. Symp. Comp. Anim. 28--36."},{"key":"e_1_2_2_14_1","unstructured":"Marcel Campen Ryan Capouellez Hanxiao Shen Leyi Zhu Daniele Panozzo and Denis Zorin. 2021. Efficient and Robust Discrete Conformal Equivalence with Boundary. (2021). arXiv:2104.04614.  Marcel Campen Ryan Capouellez Hanxiao Shen Leyi Zhu Daniele Panozzo and Denis Zorin. 2021. Efficient and Robust Discrete Conformal Equivalence with Boundary. (2021). arXiv:2104.04614."},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-5193(70)80032-7"},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/3197517.3201276"},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.4310\/jdg\/1080835659"},{"key":"e_1_2_2_18_1","first-page":"44","article-title":"Gradient-Domain Processing of Meshes","volume":"5","author":"Chuang Ming","year":"2016","unstructured":"Ming Chuang , Szymon Rusinkiewicz , and Misha Kazhdan . 2016 . Gradient-Domain Processing of Meshes . J. Comp. Graph. Tech. 5 , 4 (2016), 44 -- 55 . Ming Chuang, Szymon Rusinkiewicz, and Misha Kazhdan. 2016. Gradient-Domain Processing of Meshes. J. Comp. Graph. Tech. 5, 4 (2016), 44--55.","journal-title":"J. Comp. Graph. Tech."},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964999"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2461986"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/3375659"},{"key":"e_1_2_2_22_1","volume-title":"Oberwolfach Seminars","author":"Desbrun Mathieu","unstructured":"Mathieu Desbrun , Eva Kanso , and Yiying Tong . 2008. Discrete Differential Forms for Computational Modeling . In Discrete Differential Geometry, Alexander I. Bobenko, Peter Schr\u00f6der, John M. Sullivan, and G\u00fcnther M. Ziegler (Eds.). Oberwolfach Seminars , Vol. 38 . Birkh\u00e4user Verlag . Mathieu Desbrun, Eva Kanso, and Yiying Tong. 2008. Discrete Differential Forms for Computational Modeling. In Discrete Differential Geometry, Alexander I. Bobenko, Peter Schr\u00f6der, John M. Sullivan, and G\u00fcnther M. Ziegler (Eds.). Oberwolfach Seminars, Vol. 38. Birkh\u00e4user Verlag."},{"key":"e_1_2_2_23_1","volume-title":"Proc. Symp. Geom. Proc. Eurographics, 183--192","author":"Eckstein Ilja","year":"2007","unstructured":"Ilja Eckstein , Jean-Philippe Pons , Yiying Tong , C.-C. Jay Kuo , and Mathieu Desbrun . 2007 . Generalized Surface Flows for Mesh Processing . In Proc. Symp. Geom. Proc. Eurographics, 183--192 . Ilja Eckstein, Jean-Philippe Pons, Yiying Tong, C.-C. Jay Kuo, and Mathieu Desbrun. 2007. Generalized Surface Flows for Mesh Processing. In Proc. Symp. Geom. Proc. Eurographics, 183--192."},{"key":"e_1_2_2_24_1","unstructured":"George K. Francis. 1987. A Topological Picturebook. Springer.  George K. Francis. 1987. A Topological Picturebook. Springer."},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02567009"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1090\/tran\/7196"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459763"},{"key":"e_1_2_2_28_1","doi-asserted-by":"publisher","DOI":"10.4310\/jdg\/1304514973"},{"key":"e_1_2_2_29_1","volume-title":"Discrete Shells. In Proc. Symp. Comp. Anim. 62--67","author":"Grinspun Eitan","year":"2003","unstructured":"Eitan Grinspun , Anil Hirani , Mathieu Desbrun , and Peter Schr\u00f6der . 2003 . Discrete Shells. In Proc. Symp. Comp. Anim. 62--67 . Eitan Grinspun, Anil Hirani, Mathieu Desbrun, and Peter Schr\u00f6der. 2003. Discrete Shells. In Proc. Symp. Comp. Anim. 62--67."},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1145\/3369387"},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.1515\/znc-1973-11-1209"},{"key":"e_1_2_2_32_1","doi-asserted-by":"crossref","unstructured":"Lynn Heller. 2013. Constrained Willmore tori and elastic curves in 2-dimensional space forms. (2013). arXiv:1303.1445.  Lynn Heller. 2013. Constrained Willmore tori and elastic curves in 2-dimensional space forms. (2013). arXiv:1303.1445.","DOI":"10.4310\/CAG.2014.v22.n2.a6"},{"key":"e_1_2_2_33_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.difgeo.2015.03.003"},{"key":"e_1_2_2_34_1","unstructured":"Lynn Heller and Cheikh Birahim Ndiaye. 2019. First Explicit Constrained Willmore Minimizers of Non-Rectangular Conformal Class. (2019). arXiv:1710.00533.  Lynn Heller and Cheikh Birahim Ndiaye. 2019. First Explicit Constrained Willmore Minimizers of Non-Rectangular Conformal Class. (2019). arXiv:1710.00533."},{"key":"e_1_2_2_35_1","volume-title":"Discrete Surface Ricci Flow: Theory and Applications. In IMA International Conference on Mathematics of Surfaces. Springer, 209--232","author":"Jin Miao","year":"2007","unstructured":"Miao Jin , Junho Kim , and Xianfeng David Gu . 2007 . Discrete Surface Ricci Flow: Theory and Applications. In IMA International Conference on Mathematics of Surfaces. Springer, 209--232 . Miao Jin, Junho Kim, and Xianfeng David Gu. 2007. Discrete Surface Ricci Flow: Theory and Applications. In IMA International Conference on Mathematics of Surfaces. Springer, 209--232."},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1080\/16864360.2007.10738495"},{"key":"e_1_2_2_37_1","doi-asserted-by":"publisher","DOI":"10.1145\/2897824.2925944"},{"key":"e_1_2_2_38_1","first-page":"471","article-title":"Minimizers of the Willmore Functional under Fixed Conformal Class","volume":"93","author":"Kuwert Ernst","year":"2013","unstructured":"Ernst Kuwert and Reiner Sch\u00e4tzle . 2013 . Minimizers of the Willmore Functional under Fixed Conformal Class . J. Diff. Geom. 93 , 3 (2013), 471 -- 530 . Ernst Kuwert and Reiner Sch\u00e4tzle. 2013. Minimizers of the Willmore Functional under Fixed Conformal Class. J. Diff. Geom. 93, 3 (2013), 471--530.","journal-title":"J. Diff. Geom."},{"key":"e_1_2_2_39_1","volume-title":"Advances in Discrete Differential Geometry, Alexander I","author":"Lam Wai Yeung","unstructured":"Wai Yeung Lam and Ulrich Pinkall . 2016. Holomorphic Vector Fields and Quadatic Differentials on Planar Triangular Meshes . In Advances in Discrete Differential Geometry, Alexander I . Bobenko (Ed.). Springer , 241--265. Wai Yeung Lam and Ulrich Pinkall. 2016. Holomorphic Vector Fields and Quadatic Differentials on Planar Triangular Meshes. In Advances in Discrete Differential Geometry, Alexander I. Bobenko (Ed.). Springer, 241--265."},{"key":"e_1_2_2_40_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00208-016-1424-z"},{"key":"e_1_2_2_41_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2016.09.044"},{"key":"e_1_2_2_42_1","doi-asserted-by":"publisher","DOI":"10.1137\/120900186"},{"key":"e_1_2_2_43_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0219199704001501"},{"key":"e_1_2_2_44_1","unstructured":"Feng Luo Jian Sun and Tianqi Wu. 2020. Discrete Conformal Geometry of Polyhedral Surfaces and its Convergence. (2020). arXiv:2009.12706.  Feng Luo Jian Sun and Tianqi Wu. 2020. Discrete Conformal Geometry of Polyhedral Surfaces and its Convergence. (2020). arXiv:2009.12706."},{"key":"e_1_2_2_45_1","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2014.179.2.6"},{"key":"e_1_2_2_46_1","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160420503"},{"key":"e_1_2_2_47_1","volume-title":"Wright","author":"Nocedal Jorge","year":"2006","unstructured":"Jorge Nocedal and Stephen J . Wright . 2006 . Numerical Optimization (2 ed.). Springer . Jorge Nocedal and Stephen J. Wright. 2006. Numerical Optimization (2 ed.). Springer."},{"key":"e_1_2_2_48_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389060"},{"key":"e_1_2_2_49_1","volume-title":"A Discrete Version of Liouville's Theorem on Conformal Maps. Geom. Dedicata","author":"Pinkall Ulrich","year":"2021","unstructured":"Ulrich Pinkall and Boris Springborn . 2021. A Discrete Version of Liouville's Theorem on Conformal Maps. Geom. Dedicata ( 2021 ). Ulrich Pinkall and Boris Springborn. 2021. A Discrete Version of Liouville's Theorem on Conformal Maps. Geom. Dedicata (2021)."},{"key":"e_1_2_2_50_1","doi-asserted-by":"publisher","DOI":"10.1137\/0916082"},{"key":"e_1_2_2_51_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-008-0129-7"},{"key":"e_1_2_2_52_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02566840"},{"key":"e_1_2_2_53_1","unstructured":"Florian Sch\u00e4fer and Anima Anandkumar. 2019. Competitive Gradient Descent. (2019). arXiv:1905.12103v2.  Florian Sch\u00e4fer and Anima Anandkumar. 2019. Competitive Gradient Descent. (2019). arXiv:1905.12103v2."},{"key":"e_1_2_2_54_1","first-page":"375","article-title":"Conformally Constrained Willmore","volume":"6","author":"Sch\u00e4tzle Reiner Michael","year":"2013","unstructured":"Reiner Michael Sch\u00e4tzle . 2013 . Conformally Constrained Willmore Immersions. Adv. Calc. Var. 6 , 4 (2013), 375 -- 390 . Reiner Michael Sch\u00e4tzle. 2013. Conformally Constrained Willmore Immersions. Adv. Calc. Var. 6, 4 (2013), 375--390.","journal-title":"Immersions. Adv. Calc. Var."},{"key":"e_1_2_2_55_1","unstructured":"Henrik Schumacher. 2017. On H2-Gradient Flow for the Willmore Energy. (2017). arXiv:1703.06469v1.  Henrik Schumacher. 2017. On H 2 -Gradient Flow for the Willmore Energy. (2017). arXiv:1703.06469v1."},{"key":"e_1_2_2_56_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2006.00999.x"},{"key":"e_1_2_2_57_1","unstructured":"Boris Springborn. 2017. Hyperbolic Polyhedra and Discrete Uniformization. (2017). arXiv:1707.06848.  Boris Springborn. 2017. Hyperbolic Polyhedra and Discrete Uniformization. (2017). arXiv:1707.06848."},{"key":"e_1_2_2_58_1","doi-asserted-by":"publisher","DOI":"10.1145\/1360612.1360676"},{"key":"e_1_2_2_59_1","doi-asserted-by":"publisher","DOI":"10.1145\/2766915"},{"key":"e_1_2_2_60_1","doi-asserted-by":"publisher","DOI":"10.1145\/3272127.3275007"},{"key":"e_1_2_2_61_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2012.03173.x"},{"key":"e_1_2_2_62_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11390-009-9209-4"},{"key":"e_1_2_2_63_1","volume-title":"arXiv:2006.07859","author":"Yu Christopher","year":"2020","unstructured":"Christopher Yu , Henrik Schumacher , and Keenan Crane . 2020. Repulsive Curves . ( 2020 ). arXiv:2006.07859 . Christopher Yu, Henrik Schumacher, and Keenan Crane. 2020. Repulsive Curves. (2020). arXiv:2006.07859."}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3450626.3459759","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3450626.3459759","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T20:17:16Z","timestamp":1750191436000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3450626.3459759"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,19]]},"references-count":63,"aliases":["10.1145\/3476576.3476678"],"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,8,31]]}},"alternative-id":["10.1145\/3450626.3459759"],"URL":"https:\/\/doi.org\/10.1145\/3450626.3459759","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,19]]},"assertion":[{"value":"2021-07-19","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}