Constrained willmore surfaces
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...
Saved in:
| Published in | ACM transactions on graphics Vol. 40; no. 4; pp. 1 - 17 |
|---|---|
| Main Authors | , , , , , |
| Format | Journal Article |
| Language | English |
| Published |
New York, NY, USA
ACM
31.08.2021
|
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| ArticleNumber | 112 |
| Author | Soliman, Yousuf Diamanti, Olga Chern, Albert Pinkall, Ulrich Knöppel, Felix Schröder, Peter |
| Author_xml | – sequence: 1 givenname: Yousuf surname: Soliman fullname: Soliman, Yousuf organization: Caltech – sequence: 2 givenname: Albert surname: Chern fullname: Chern, Albert organization: UCSD – sequence: 3 givenname: Olga surname: Diamanti fullname: Diamanti, Olga organization: TU Graz, Austria – sequence: 4 givenname: Felix surname: Knöppel fullname: Knöppel, Felix organization: TU Berlin – sequence: 5 givenname: Ulrich surname: Pinkall fullname: Pinkall, Ulrich organization: TU Berlin, Germany – sequence: 6 givenname: Peter surname: Schröder fullname: Schröder, Peter organization: Caltech |
| BookMark | eNp9j01Lw0AQhhepYFo9C4LQP5B2JrMf7lGCX1Dwoucw3exCJB-yGxH_vSmNHjx4emFmnpd5lmLRD70X4hJhgyjVlqQCXejNlNYoeyIyVMrkhvTNQmRgCHIgwDOxTOkNALSUOhPX5dCnMXLT-3r92bRtN0S_Th8xsPPpXJwGbpO_mHMlXu_vXsrHfPf88FTe7nKeGsfcBCbnwt4ykbfIjp2tWVuuQ10EsoiFR0OGUUuDBewZp7msC6dCILS0EurY6-KQUvShcs3IYzP0h9faCqE6OFazYzU7Ttz2D_cem47j1z_E1ZFg1_0e_yy_AZIYXJQ |
| CitedBy_id | crossref_primary_10_1145_3592402 crossref_primary_10_1016_j_bpr_2022_100062 crossref_primary_10_1145_3687942 crossref_primary_10_1016_j_physleta_2023_128986 crossref_primary_10_1145_3478513_3480521 crossref_primary_10_1145_3604282 crossref_primary_10_1145_3658145 crossref_primary_10_1145_3618345 crossref_primary_10_1016_j_difgeo_2024_102200 crossref_primary_10_1016_j_jmaa_2023_127388 crossref_primary_10_1016_j_cagd_2024_102286 |
| Cites_doi | 10.1145/3369387 10.1137/16M1065379 10.1145/3450626.3459763 10.1007/BF02567009 10.1145/2461912.2461986 10.1007/BF02566840 10.1215/S0012-7094-93-07207-9 10.1007/s00526-007-0142-5 10.1016/S0022-5193(70)80032-7 10.1145/3272127.3275007 10.1090/tran/7196 10.2140/gt.2015.19.2155 10.1145/3197517.3201276 10.1016/j.cma.2016.09.044 10.1111/j.1467-8659.2006.00999.x 10.1145/1360612.1360676 10.1007/s00222-008-0129-7 10.1145/3375659 10.1137/0916082 10.4310/CAG.2014.v22.n2.a6 10.1142/S0219199704001501 10.1007/BF01389060 10.1111/j.1467-8659.2012.03173.x 10.1007/978-3-030-43736-7_1 10.1137/120900186 10.1145/2010324.1964999 10.4007/annals.2014.179.2.6 10.4310/jdg/1080835659 10.4310/jdg/1304514973 10.1016/j.difgeo.2015.03.003 10.1137/17M1147615 10.1515/znc-1973-11-1209 10.1002/cpa.3160420503 10.1080/16864360.2007.10738495 10.1007/s00208-016-1424-z 10.1145/2897824.2925944 10.1145/2766915 10.1007/s11390-009-9209-4 |
| ContentType | Journal Article |
| Copyright | Owner/Author |
| Copyright_xml | – notice: Owner/Author |
| DBID | AAYXX CITATION |
| DOI | 10.1145/3450626.3459759 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | CrossRef |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering |
| EISSN | 1557-7368 |
| EndPage | 17 |
| ExternalDocumentID | 10_1145_3450626_3459759 3459759 |
| GrantInformation_xml | – fundername: University of California, San Diego funderid: http://dx.doi.org/10.13039/100007911 – fundername: Deutsche Forschungsgemeinschaft grantid: SFB 109 funderid: http://dx.doi.org/10.13039/501100001659 – fundername: Einstein Stiftung Berlin funderid: http://dx.doi.org/10.13039/501100006188 – fundername: National Center for Theoretical Sciences funderid: http://dx.doi.org/10.13039/100009125 – fundername: California Institute of Technology funderid: http://dx.doi.org/10.13039/100006961 |
| GroupedDBID | --Z -DZ -~X .DC 23M 2FS 4.4 5GY 5VS 6J9 85S 8US AAKMM AALFJ AAYFX ABPPZ ACGFO ACGOD ACM ADBCU ADL ADMLS ADPZR AEBYY AENEX AENSD AFWIH AFWXC AIKLT ALMA_UNASSIGNED_HOLDINGS ASPBG AVWKF BDXCO CCLIF CS3 F5P FEDTE GUFHI HGAVV I07 LHSKQ P1C P2P PQQKQ RNS ROL TWZ UHB UPT W7O WH7 XSW ZCA ~02 AAYXX AEFXT AEJOY AETEA AKRVB CITATION |
| ID | FETCH-LOGICAL-a301t-7fa3ccfb9a33e91acac9da69adfd2f39112e1737a1647120ba1d2f4d2c5ff3193 |
| ISSN | 0730-0301 |
| IngestDate | Thu Apr 24 23:08:07 EDT 2025 Thu Jul 03 08:43:33 EDT 2025 Fri Feb 21 01:28:01 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 4 |
| Keywords | geometric flows willmore surfaces geometric modeling conformal geometry discrete differential geometry differential coordinates |
| Language | English |
| License | This work is licensed under a Creative Commons Attribution International 4.0 License. |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-a301t-7fa3ccfb9a33e91acac9da69adfd2f39112e1737a1647120ba1d2f4d2c5ff3193 |
| OpenAccessLink | https://dl.acm.org/doi/10.1145/3450626.3459759 |
| PageCount | 17 |
| ParticipantIDs | crossref_citationtrail_10_1145_3450626_3459759 crossref_primary_10_1145_3450626_3459759 acm_primary_3459759 |
| PublicationCentury | 2000 |
| PublicationDate | 2021-08-31 |
| PublicationDateYYYYMMDD | 2021-08-31 |
| PublicationDate_xml | – month: 08 year: 2021 text: 2021-08-31 day: 31 |
| PublicationDecade | 2020 |
| PublicationPlace | New York, NY, USA |
| PublicationPlace_xml | – name: New York, NY, USA |
| PublicationTitle | ACM transactions on graphics |
| PublicationTitleAbbrev | ACM TOG |
| PublicationYear | 2021 |
| Publisher | ACM |
| Publisher_xml | – name: ACM |
| References | Dimitri P. Bertsekas. 1996. Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific. Ming Chuang, Szymon Rusinkiewicz, and Misha Kazhdan. 2016. Gradient-Domain Processing of Meshes. J. Comp. Graph. Tech. 5, 4 (2016), 44--55. Feng Luo, Jian Sun, and Tianqi Wu. 2020. Discrete Conformal Geometry of Polyhedral Surfaces and its Convergence. (2020). arXiv:2009.12706. Wai Yeung Lam and Ulrich Pinkall. 2017. Isothermic Triangulated Surfaces. Math. Ann. 368, 1--2 (2017), 165--195. Florian Schäfer and Anima Anandkumar. 2019. Competitive Gradient Descent. (2019). arXiv:1905.12103v2. Tristan Rivière. 2008. Analysis Aspects of Willmore Surfaces. Invent. Math. 174, 1 (2008), 1--45. Reto A. Rüedy. 1971. Embeddings of Open Riemann Surfaces. Comm. Math. Helv. 46 (1971), 214--225. 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. Lynn Heller. 2015. Constrained Willmore and CMC Tori in the 3-Sphere. Diff. Geom. Appl. 40 (2015), 232--242. Christoph Bohle. 2010. Constrained Willmore Tori in the 4-Sphere. J. Diff. Geom. 86, 1 (2010), 71--132. Ulrich Pinkall and Boris Springborn. 2021. A Discrete Version of Liouville's Theorem on Conformal Maps. Geom. Dedicata (2021). Alexander I. Bobenko and Peter Schröder. 2005. Discrete Willmore Flow. In Proc. Symp. Geom. Proc. Eurographics, 101--110. Na Lei, Xiaopeng Zheng, Jian Jiang, Yu-Yao Lin, and David Xianfeng Gu. 2017. Quadrilateral and Hexahedral Mesh Generation Based on Surface Foliation Theory. Comp. Meth. Appl. Mech. & Eng. 316 (2017), 758--781. Boris Springborn, Peter Schröder, and Ulrich Pinkall. 2008. Conformal Equivalence of Triangle Meshes. ACM Trans. Graph. 27, 3 (2008), 77:1--11. Lok Ming Lui, Ka Chun Lam, Shing-Tung Yau, and Xianfeng Gu. 2014. Teichmuller Mapping (T-Map) and its Applications in Landmark Matching Registration. SIAM J. Img. Sci. 7, 1 (2014), 391--426. Anthony Gruber and Eugenio Aulisa. 2020. Computational P-Willmore Flow with Conformal Penalty. ACM Trans. Graph. 39, 5 (2020), 161:1--16. R. J. Renka and J. W. Neuberger. 1995. Minimal Surfaces and Sobolev Gradients. SIAM J. Sci. Comp. 16, 6 (1995), 1412--1427. Ulrich Pinkall. 1985. Hopf Tori in S3. Invent. Math. 81, 2 (1985), 379--386. Matthias Bollhöfer, Aryan Eftekhari, Simon Scheidegger, and Olaf Schenk. 2019. Large-scale Sparse Inverse Covariance Matrix Estimation. SIAM J. Sci. Comp. 41, 1 (2019), A380--A401. Amir Vaxman, Christian Müller, and Ofir Weber. 2015. Conformal Mesh Deformations with Möbius Transformations. ACM Trans. Graph. 34, 4 (2015), 55:1--11. Mina Konaković, Keenan Crane, Bailin Deng, Sofien Bouaziz, Daniel Piker, and Mark Pauly. 2016. Beyond Developable: Computational Design and Fabrication with Auxetic Materials. ACM Trans. Graph. 35, 4 (2016), 89:1--11. Albert Chern, Felix Knöppel, Ulrich Pinkall, and Peter Schröder. 2018. Shape from Metric. ACM Trans. Graph. 37, 4 (2018), 63:1--17. Christopher Yu, Henrik Schumacher, and Keenan Crane. 2020. Repulsive Curves. (2020). arXiv:2006.07859. Adriano M. Garsia. 1961. An Imbedding of Closed Riemann Surfaces in Euclidean Space. Comm. Math. Helv. 35 (1961), 93--110. Huabin Ge. 2018. Combinatorial Calabi Flows on Surfaces. Trans. Amer. Math. Soc. 370, 2 (2018), 1377--1391. Wilhelm Blaschke. 1929. Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie III. Springer. 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. Ernst Kuwert and Reiner Schätzle. 2013. Minimizers of the Willmore Functional under Fixed Conformal Class. J. Diff. Geom. 93, 3 (2013), 471--530. Boris Springborn. 2017. Hyperbolic Polyhedra and Discrete Uniformization. (2017). arXiv:1707.06848. Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2011. Spin Transformations of Discrete Surfaces. ACM Trans. Graph. 30, 4 (2011), 104:1--10. Fernando C. Marques and Andreé Neves. 2014. Min-Max Theory and the Willmore Conjecture. Ann. Math. 179, 2 (2014), 683--782. George K. Francis. 1987. A Topological Picturebook. Springer. Lynn Heller. 2013. Constrained Willmore tori and elastic curves in 2-dimensional space forms. (2013). arXiv:1303.1445. Christoph Bohle, G. Paul Peters, and Ulrich Pinkall. 2008. Constrained Willmore Surfaces. Calc. Var. 32 (2008), 263--277. 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. Lynn Heller and Cheikh Birahim Ndiaye. 2019. First Explicit Constrained Willmore Minimizers of Non-Rectangular Conformal Class. (2019). arXiv:1710.00533. Wei-Wei Xu and Kun Zhou. 2009. Gradient Domain Mesh Deformation-A Survey. J. Comp. Sci. Tech. 24, 1 (2009), 6--18. Jorge Nocedal and Stephen J. Wright. 2006. Numerical Optimization (2 ed.). Springer. Mikhail V. Babich and Alexander I. Bobenko. 1993. Willmore Tori with Umbilic Lines and Minimal Surfaces in Hyperbolic Space. Duke Math. J. 72, 1 (1993), 151--185. Mark Gillespie, Boris Springborn, and Keenan Crane. 2021. Discrete Conformal Equivalence of Triangle Meshes. ACM Trans. Graph. 40, 2 (2021). Robert Bridson, S. Marino, and Ronald Fedkiw. 2003. Simulation of Clothing with Folds and Wrinkles. In Proc. Symp. Comp. Anim. 28--36. David Glickenstein. 2011. Discrete Conformal Variations and Scalar Curvature on Piecewise Flat Two-and Three-Dimensional Manifolds. J. Diff. Geom. 87, 2 (2011), 201--238. David Mumford and Jayant Shah. 1989. Optimal Approximations by Piecewise Smooth Functions and assocaited Variational Problems. Comm. Pure Appl. Math. 42, 5 (1989), 577--685. 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. John W. Barrett, Harald Garcke, and Robert Nürnberg. 2016. Computational Parametric Willmore Flow with Spontaneous Curvature and Area Difference Elasticity Effects. SIAM J. Numer. Anal. 54, 3 (2016), 1732--1762. Bram Custers and Amir Vaxman. 2020. Subdivision Directional Fields. ACM Trans. Graph. 39, 2 (2020), 11:1--23. Pushkar Joshi and Carlo Séquin. 2007. Energy Minimizers for Curvature-Based Surface Functionals. Comp. Aid. Des. Appl. 4, 5 (2007), 607--617. 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. Olga Sorkine. 2006. Differential Representations for Mesh Processing. Comp. Graph. Forum 25, 4 (2006), 789--807. Mathieu Desbrun, Eva Kanso, and Yiying Tong. 2008. Discrete Differential Forms for Computational Modeling. In Discrete Differential Geometry, Alexander I. Bobenko, Peter Schröder, John M. Sullivan, and Günther M. Ziegler (Eds.). Oberwolfach Seminars, Vol. 38. Birkhäuser Verlag. Alexander I. Bobenko, Ulrich Pinkall, and Boris Springborn. 2015. Discrete Conformal Maps and Ideal Hyperbolic Polyhedra. Geom. & Top. 19, 4 (2015), 2155--2215. Ofir Weber, Ashish Myles, and Denis Zorin. 2012. Computing Extremal Quasiconformal Maps. Comp. Graph. Forum 31, 5 (2012), 1679--1689. Feng Luo. 2004. Combinatorial Yamabe Flow on Surfaces. Comm. Contemp. Math. 6, 5 (2004), 765--780. Henrik Schumacher. 2017. On H2-Gradient Flow for the Willmore Energy. (2017). arXiv:1703.06469v1. P. B. Canham. 1970. The Minimum Energy of Bending as a Possible Explanation of the Biconcave Shape of the Human Red Blood Cell. J. Theor. Biology 26, 1 (1970), 61--81. W. Helfrich. 1973. Elastic Properties of Lipid Bilayers: Theory and Possible Experiments. Z. Naturforsch. C 28, 11--12 (1973), 693--703. Eitan Grinspun, Anil Hirani, Mathieu Desbrun, and Peter Schröder. 2003. Discrete Shells. In Proc. Symp. Comp. Anim. 62--67. Reiner Michael Schätzle. 2013. Conformally Constrained Willmore Immersions. Adv. Calc. Var. 6, 4 (2013), 375--390. Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013. Robust Fairing via Conformal Curvature Flow. ACM Trans. Graph. 32, 4 (2013), 61:1--10. Matthias Bollhöfer, Olaf Schenk, Radim Janalik, Steve Hamm, and Kiran Gullapalli. 2020. State-of-the-Art Sparse Direct Solvers. (2020), 3--33. 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. Bennet Chow and Feng Luo. 2003. Combinatorial Ricci Flows on Surfaces. J. Diff. Geom. 63, 1 (2003), 97--129. Amir Vaxman, Christian Müller, and Ofir Weber. 2018. Canonical Möbius Subdivision. ACM Trans. Graph. 37, 6 (2018), 227:1--15. e_1_2_2_24_1 e_1_2_2_6_1 e_1_2_2_20_1 e_1_2_2_2_1 Blaschke Wilhelm (e_1_2_2_5_1) e_1_2_2_41_1 e_1_2_2_62_1 Yu Christopher (e_1_2_2_63_1) 2020 e_1_2_2_43_1 e_1_2_2_28_1 e_1_2_2_45_1 e_1_2_2_26_1 e_1_2_2_60_1 Jin Miao (e_1_2_2_35_1) 2007 Desbrun Mathieu (e_1_2_2_22_1) e_1_2_2_59_1 e_1_2_2_11_1 Alappat Christie (e_1_2_2_1_1) 2020; 7 Grinspun Eitan (e_1_2_2_29_1) 2003 e_1_2_2_30_1 e_1_2_2_51_1 e_1_2_2_19_1 e_1_2_2_32_1 e_1_2_2_53_1 e_1_2_2_17_1 e_1_2_2_34_1 e_1_2_2_55_1 e_1_2_2_15_1 e_1_2_2_36_1 e_1_2_2_57_1 Kuwert Ernst (e_1_2_2_38_1) 2013; 93 Lam Wai Yeung (e_1_2_2_39_1) Nocedal Jorge (e_1_2_2_47_1) 2006 e_1_2_2_25_1 e_1_2_2_48_1 e_1_2_2_21_1 e_1_2_2_3_1 Chuang Ming (e_1_2_2_18_1) 2016; 5 e_1_2_2_40_1 e_1_2_2_42_1 e_1_2_2_44_1 e_1_2_2_27_1 e_1_2_2_46_1 Alexander (e_1_2_2_7_1) Bridson Robert (e_1_2_2_13_1) 2003 e_1_2_2_61_1 Eckstein Ilja (e_1_2_2_23_1) 2007 Schätzle Reiner Michael (e_1_2_2_54_1) 2013; 6 e_1_2_2_14_1 e_1_2_2_37_1 e_1_2_2_12_1 e_1_2_2_10_1 Bertsekas Dimitri P. (e_1_2_2_4_1) Pinkall Ulrich (e_1_2_2_49_1) 2021 e_1_2_2_52_1 e_1_2_2_31_1 Bohle Christoph (e_1_2_2_9_1) 2010; 86 e_1_2_2_33_1 e_1_2_2_56_1 Bobenko Alexander I. (e_1_2_2_8_1) e_1_2_2_16_1 e_1_2_2_58_1 e_1_2_2_50_1 |
| References_xml | – reference: 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. – reference: Henrik Schumacher. 2017. On H2-Gradient Flow for the Willmore Energy. (2017). arXiv:1703.06469v1. – reference: Ofir Weber, Ashish Myles, and Denis Zorin. 2012. Computing Extremal Quasiconformal Maps. Comp. Graph. Forum 31, 5 (2012), 1679--1689. – reference: Mathieu Desbrun, Eva Kanso, and Yiying Tong. 2008. Discrete Differential Forms for Computational Modeling. In Discrete Differential Geometry, Alexander I. Bobenko, Peter Schröder, John M. Sullivan, and Günther M. Ziegler (Eds.). Oberwolfach Seminars, Vol. 38. Birkhäuser Verlag. – reference: 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. – reference: Jorge Nocedal and Stephen J. Wright. 2006. Numerical Optimization (2 ed.). Springer. – reference: Ulrich Pinkall. 1985. Hopf Tori in S3. Invent. Math. 81, 2 (1985), 379--386. – reference: Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2011. Spin Transformations of Discrete Surfaces. ACM Trans. Graph. 30, 4 (2011), 104:1--10. – reference: John W. Barrett, Harald Garcke, and Robert Nürnberg. 2016. Computational Parametric Willmore Flow with Spontaneous Curvature and Area Difference Elasticity Effects. SIAM J. Numer. Anal. 54, 3 (2016), 1732--1762. – reference: Tristan Rivière. 2008. Analysis Aspects of Willmore Surfaces. Invent. Math. 174, 1 (2008), 1--45. – reference: Huabin Ge. 2018. Combinatorial Calabi Flows on Surfaces. Trans. Amer. Math. Soc. 370, 2 (2018), 1377--1391. – reference: Eitan Grinspun, Anil Hirani, Mathieu Desbrun, and Peter Schröder. 2003. Discrete Shells. In Proc. Symp. Comp. Anim. 62--67. – reference: David Glickenstein. 2011. Discrete Conformal Variations and Scalar Curvature on Piecewise Flat Two-and Three-Dimensional Manifolds. J. Diff. Geom. 87, 2 (2011), 201--238. – reference: W. Helfrich. 1973. Elastic Properties of Lipid Bilayers: Theory and Possible Experiments. Z. Naturforsch. C 28, 11--12 (1973), 693--703. – reference: P. B. Canham. 1970. The Minimum Energy of Bending as a Possible Explanation of the Biconcave Shape of the Human Red Blood Cell. J. Theor. Biology 26, 1 (1970), 61--81. – reference: Feng Luo. 2004. Combinatorial Yamabe Flow on Surfaces. Comm. Contemp. Math. 6, 5 (2004), 765--780. – reference: Alexander I. Bobenko and Peter Schröder. 2005. Discrete Willmore Flow. In Proc. Symp. Geom. Proc. Eurographics, 101--110. – reference: David Mumford and Jayant Shah. 1989. Optimal Approximations by Piecewise Smooth Functions and assocaited Variational Problems. Comm. Pure Appl. Math. 42, 5 (1989), 577--685. – reference: Wai Yeung Lam and Ulrich Pinkall. 2017. Isothermic Triangulated Surfaces. Math. Ann. 368, 1--2 (2017), 165--195. – reference: Lynn Heller. 2013. Constrained Willmore tori and elastic curves in 2-dimensional space forms. (2013). arXiv:1303.1445. – reference: Reiner Michael Schätzle. 2013. Conformally Constrained Willmore Immersions. Adv. Calc. Var. 6, 4 (2013), 375--390. – reference: Amir Vaxman, Christian Müller, and Ofir Weber. 2015. Conformal Mesh Deformations with Möbius Transformations. ACM Trans. Graph. 34, 4 (2015), 55:1--11. – reference: Christoph Bohle. 2010. Constrained Willmore Tori in the 4-Sphere. J. Diff. Geom. 86, 1 (2010), 71--132. – reference: Albert Chern, Felix Knöppel, Ulrich Pinkall, and Peter Schröder. 2018. Shape from Metric. ACM Trans. Graph. 37, 4 (2018), 63:1--17. – reference: Bram Custers and Amir Vaxman. 2020. Subdivision Directional Fields. ACM Trans. Graph. 39, 2 (2020), 11:1--23. – reference: George K. Francis. 1987. A Topological Picturebook. Springer. – reference: Reto A. Rüedy. 1971. Embeddings of Open Riemann Surfaces. Comm. Math. Helv. 46 (1971), 214--225. – reference: Bennet Chow and Feng Luo. 2003. Combinatorial Ricci Flows on Surfaces. J. Diff. Geom. 63, 1 (2003), 97--129. – reference: Mina Konaković, Keenan Crane, Bailin Deng, Sofien Bouaziz, Daniel Piker, and Mark Pauly. 2016. Beyond Developable: Computational Design and Fabrication with Auxetic Materials. ACM Trans. Graph. 35, 4 (2016), 89:1--11. – reference: Pushkar Joshi and Carlo Séquin. 2007. Energy Minimizers for Curvature-Based Surface Functionals. Comp. Aid. Des. Appl. 4, 5 (2007), 607--617. – reference: Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013. Robust Fairing via Conformal Curvature Flow. ACM Trans. Graph. 32, 4 (2013), 61:1--10. – reference: Amir Vaxman, Christian Müller, and Ofir Weber. 2018. Canonical Möbius Subdivision. ACM Trans. Graph. 37, 6 (2018), 227:1--15. – reference: Fernando C. Marques and Andreé Neves. 2014. Min-Max Theory and the Willmore Conjecture. Ann. Math. 179, 2 (2014), 683--782. – reference: Boris Springborn, Peter Schröder, and Ulrich Pinkall. 2008. Conformal Equivalence of Triangle Meshes. ACM Trans. Graph. 27, 3 (2008), 77:1--11. – reference: Lynn Heller and Cheikh Birahim Ndiaye. 2019. First Explicit Constrained Willmore Minimizers of Non-Rectangular Conformal Class. (2019). arXiv:1710.00533. – reference: Anthony Gruber and Eugenio Aulisa. 2020. Computational P-Willmore Flow with Conformal Penalty. ACM Trans. Graph. 39, 5 (2020), 161:1--16. – reference: Florian Schäfer and Anima Anandkumar. 2019. Competitive Gradient Descent. (2019). arXiv:1905.12103v2. – reference: R. J. Renka and J. W. Neuberger. 1995. Minimal Surfaces and Sobolev Gradients. SIAM J. Sci. Comp. 16, 6 (1995), 1412--1427. – reference: Ulrich Pinkall and Boris Springborn. 2021. A Discrete Version of Liouville's Theorem on Conformal Maps. Geom. Dedicata (2021). – reference: Lynn Heller. 2015. Constrained Willmore and CMC Tori in the 3-Sphere. Diff. Geom. Appl. 40 (2015), 232--242. – reference: Olga Sorkine. 2006. Differential Representations for Mesh Processing. Comp. Graph. Forum 25, 4 (2006), 789--807. – reference: Feng Luo, Jian Sun, and Tianqi Wu. 2020. Discrete Conformal Geometry of Polyhedral Surfaces and its Convergence. (2020). arXiv:2009.12706. – reference: Mark Gillespie, Boris Springborn, and Keenan Crane. 2021. Discrete Conformal Equivalence of Triangle Meshes. ACM Trans. Graph. 40, 2 (2021). – reference: 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. – reference: Wilhelm Blaschke. 1929. Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie III. Springer. – reference: Christopher Yu, Henrik Schumacher, and Keenan Crane. 2020. Repulsive Curves. (2020). arXiv:2006.07859. – reference: 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. – reference: Matthias Bollhöfer, Aryan Eftekhari, Simon Scheidegger, and Olaf Schenk. 2019. Large-scale Sparse Inverse Covariance Matrix Estimation. SIAM J. Sci. Comp. 41, 1 (2019), A380--A401. – reference: Mikhail V. Babich and Alexander I. Bobenko. 1993. Willmore Tori with Umbilic Lines and Minimal Surfaces in Hyperbolic Space. Duke Math. J. 72, 1 (1993), 151--185. – reference: Na Lei, Xiaopeng Zheng, Jian Jiang, Yu-Yao Lin, and David Xianfeng Gu. 2017. Quadrilateral and Hexahedral Mesh Generation Based on Surface Foliation Theory. Comp. Meth. Appl. Mech. & Eng. 316 (2017), 758--781. – reference: Boris Springborn. 2017. Hyperbolic Polyhedra and Discrete Uniformization. (2017). arXiv:1707.06848. – reference: Matthias Bollhöfer, Olaf Schenk, Radim Janalik, Steve Hamm, and Kiran Gullapalli. 2020. State-of-the-Art Sparse Direct Solvers. (2020), 3--33. – reference: Ernst Kuwert and Reiner Schätzle. 2013. Minimizers of the Willmore Functional under Fixed Conformal Class. J. Diff. Geom. 93, 3 (2013), 471--530. – reference: Alexander I. Bobenko, Ulrich Pinkall, and Boris Springborn. 2015. Discrete Conformal Maps and Ideal Hyperbolic Polyhedra. Geom. & Top. 19, 4 (2015), 2155--2215. – reference: Christoph Bohle, G. Paul Peters, and Ulrich Pinkall. 2008. Constrained Willmore Surfaces. Calc. Var. 32 (2008), 263--277. – reference: Lok Ming Lui, Ka Chun Lam, Shing-Tung Yau, and Xianfeng Gu. 2014. Teichmuller Mapping (T-Map) and its Applications in Landmark Matching Registration. SIAM J. Img. Sci. 7, 1 (2014), 391--426. – reference: Robert Bridson, S. Marino, and Ronald Fedkiw. 2003. Simulation of Clothing with Folds and Wrinkles. In Proc. Symp. Comp. Anim. 28--36. – reference: Ming Chuang, Szymon Rusinkiewicz, and Misha Kazhdan. 2016. Gradient-Domain Processing of Meshes. J. Comp. Graph. Tech. 5, 4 (2016), 44--55. – reference: Dimitri P. Bertsekas. 1996. Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific. – reference: 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. – reference: Wei-Wei Xu and Kun Zhou. 2009. Gradient Domain Mesh Deformation-A Survey. J. Comp. Sci. Tech. 24, 1 (2009), 6--18. – reference: 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. – reference: Adriano M. Garsia. 1961. An Imbedding of Closed Riemann Surfaces in Euclidean Space. Comm. Math. Helv. 35 (1961), 93--110. – volume: 5 start-page: 44 year: 2016 ident: e_1_2_2_18_1 article-title: Gradient-Domain Processing of Meshes publication-title: J. Comp. Graph. Tech. – ident: e_1_2_2_30_1 doi: 10.1145/3369387 – ident: e_1_2_2_3_1 doi: 10.1137/16M1065379 – ident: e_1_2_2_27_1 doi: 10.1145/3450626.3459763 – ident: e_1_2_2_25_1 doi: 10.1007/BF02567009 – volume-title: Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie III ident: e_1_2_2_5_1 – volume: 6 start-page: 375 year: 2013 ident: e_1_2_2_54_1 article-title: Conformally Constrained Willmore publication-title: Immersions. Adv. Calc. Var. – ident: e_1_2_2_20_1 doi: 10.1145/2461912.2461986 – ident: e_1_2_2_52_1 doi: 10.1007/BF02566840 – ident: e_1_2_2_2_1 doi: 10.1215/S0012-7094-93-07207-9 – volume-title: Discrete Shells. In Proc. Symp. Comp. Anim. 62--67 year: 2003 ident: e_1_2_2_29_1 – ident: e_1_2_2_10_1 doi: 10.1007/s00526-007-0142-5 – ident: e_1_2_2_15_1 doi: 10.1016/S0022-5193(70)80032-7 – volume: 93 start-page: 471 year: 2013 ident: e_1_2_2_38_1 article-title: Minimizers of the Willmore Functional under Fixed Conformal Class publication-title: J. Diff. Geom. – volume-title: Wright year: 2006 ident: e_1_2_2_47_1 – ident: e_1_2_2_60_1 doi: 10.1145/3272127.3275007 – ident: e_1_2_2_26_1 doi: 10.1090/tran/7196 – ident: e_1_2_2_6_1 doi: 10.2140/gt.2015.19.2155 – ident: e_1_2_2_16_1 doi: 10.1145/3197517.3201276 – ident: e_1_2_2_41_1 doi: 10.1016/j.cma.2016.09.044 – ident: e_1_2_2_14_1 – ident: e_1_2_2_56_1 doi: 10.1111/j.1467-8659.2006.00999.x – ident: e_1_2_2_58_1 doi: 10.1145/1360612.1360676 – ident: e_1_2_2_51_1 doi: 10.1007/s00222-008-0129-7 – ident: e_1_2_2_55_1 – ident: e_1_2_2_21_1 doi: 10.1145/3375659 – volume: 7 year: 2020 ident: e_1_2_2_1_1 article-title: A Recursive Algebraic Coloring Technique for Hardware-Efficient Symmetric Sparse Matrix-Vector Multiplication publication-title: ACM Trans. Par. Comput. – volume-title: Oberwolfach Seminars ident: e_1_2_2_22_1 – ident: e_1_2_2_44_1 – ident: e_1_2_2_50_1 doi: 10.1137/0916082 – ident: e_1_2_2_32_1 doi: 10.4310/CAG.2014.v22.n2.a6 – ident: e_1_2_2_43_1 doi: 10.1142/S0219199704001501 – ident: e_1_2_2_48_1 doi: 10.1007/BF01389060 – ident: e_1_2_2_61_1 doi: 10.1111/j.1467-8659.2012.03173.x – volume-title: Discrete Surface Ricci Flow: Theory and Applications. In IMA International Conference on Mathematics of Surfaces. Springer, 209--232 year: 2007 ident: e_1_2_2_35_1 – volume-title: Advances in Discrete Differential Geometry, Alexander I ident: e_1_2_2_39_1 – volume-title: Constrained Optimization and Lagrange Multiplier Methods ident: e_1_2_2_4_1 – ident: e_1_2_2_12_1 doi: 10.1007/978-3-030-43736-7_1 – volume-title: Proc. Symp. Comp. Anim. 28--36 year: 2003 ident: e_1_2_2_13_1 – ident: e_1_2_2_42_1 doi: 10.1137/120900186 – ident: e_1_2_2_57_1 – volume: 86 start-page: 71 year: 2010 ident: e_1_2_2_9_1 article-title: Constrained Willmore Tori in the 4-Sphere publication-title: J. Diff. Geom. – ident: e_1_2_2_19_1 doi: 10.1145/2010324.1964999 – ident: e_1_2_2_45_1 doi: 10.4007/annals.2014.179.2.6 – ident: e_1_2_2_17_1 doi: 10.4310/jdg/1080835659 – volume-title: A Discrete Version of Liouville's Theorem on Conformal Maps. Geom. Dedicata year: 2021 ident: e_1_2_2_49_1 – ident: e_1_2_2_28_1 doi: 10.4310/jdg/1304514973 – volume-title: Proc. Symp. Geom. Proc. Eurographics, 101--110 ident: e_1_2_2_7_1 – ident: e_1_2_2_33_1 doi: 10.1016/j.difgeo.2015.03.003 – ident: e_1_2_2_11_1 doi: 10.1137/17M1147615 – ident: e_1_2_2_34_1 – ident: e_1_2_2_31_1 doi: 10.1515/znc-1973-11-1209 – ident: e_1_2_2_46_1 doi: 10.1002/cpa.3160420503 – ident: e_1_2_2_36_1 doi: 10.1080/16864360.2007.10738495 – ident: e_1_2_2_53_1 – ident: e_1_2_2_40_1 doi: 10.1007/s00208-016-1424-z – volume-title: Circle Domains, Fuchsian and Schottky Uniformization. In Advances in Discrete Differential Geometry, Alexander I ident: e_1_2_2_8_1 – volume-title: Proc. Symp. Geom. Proc. Eurographics, 183--192 year: 2007 ident: e_1_2_2_23_1 – ident: e_1_2_2_37_1 doi: 10.1145/2897824.2925944 – ident: e_1_2_2_24_1 – ident: e_1_2_2_59_1 doi: 10.1145/2766915 – ident: e_1_2_2_62_1 doi: 10.1007/s11390-009-9209-4 – volume-title: arXiv:2006.07859 year: 2020 ident: e_1_2_2_63_1 |
| SSID | ssj0006446 |
| Score | 2.4753783 |
| Snippet | Smooth curves and surfaces can be characterized as minimizers of squared curvature bending energies subject to constraints. In the univariate case with an... |
| SourceID | crossref acm |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 1 |
| SubjectTerms | Computer graphics Computing methodologies Discretization Mathematical analysis Mathematical optimization Mathematics of computing Mesh geometry models Numerical analysis Shape analysis Shape modeling |
| SubjectTermsDisplay | Computing methodologies -- Computer graphics -- Shape modeling -- Mesh geometry models Computing methodologies -- Computer graphics -- Shape modeling -- Shape analysis Mathematics of computing -- Mathematical analysis -- Mathematical optimization Mathematics of computing -- Mathematical analysis -- Numerical analysis -- Discretization |
| Title | Constrained willmore surfaces |
| URI | https://dl.acm.org/doi/10.1145/3450626.3459759 |
| Volume | 40 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3NT8IwFG8UL3owihpRMRw8mJjhtm6MHhEkREUPQOJt6brWkMxB-EiMf72vazfqB4l62UhTXta9t_fRvvd7CF3gSDACsY9lN6lteYJ4VgRmAz682GZcQl4RWSjcf2z0Rt7ds_-8apeaVZcsojp7_7Gu5D9chTHgq6yS_QNnC6IwAL-Bv3AFDsP1VzyW3TazHg9ZDnmSyKTZq_lyJmSelel2ttp92Qwi7wyeHRFkUNVGrvtgkoz1diiogPlSrE7-ucr4bUk8rCJNpjOmMF1lAzwlL4V6v0_l4ftNYzpVCQBdnozfzM0Ft9gtLXQQKABLRk3KXGgd6QdWgFU3nFyJKswlLSyeoREdw7SqKs3vStuT-BbY822IrupwJ4FGCf8Ej_3FbBXJhKq02g81gVAT2ERbLoQOoPu2Wp3-w6Cwz-ABZifY-dI04BOQuP7yDNJjYa-Gx2K4HsM9tKtjhlpLCcA-2uBpGe0YSJIHqGqIQi0XhVouCodo1L0dtnuW7nxhUXighRUIihkTEaEYc-JQRhmJaYPQWMSuwGCgXO4EOKASDc5x7Yg6MO7FLvOFAKWKj1ApnaT8GNUkuA58c0JQyjzRZJFPXUaYy2JBeExZBZVhjeFUYZvkL6-C6vmaQ6bB4uUiknDN266gy-IPOa01U09-P_UUba-k8gyVFrMlr4ITuIjONVc_AN7ZVQk |
| linkProvider | EBSCOhost |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Constrained+willmore+surfaces&rft.jtitle=ACM+transactions+on+graphics&rft.au=Soliman%2C+Yousuf&rft.au=Chern%2C+Albert&rft.au=Diamanti%2C+Olga&rft.au=Kn%C3%B6ppel%2C+Felix&rft.date=2021-08-31&rft.issn=0730-0301&rft.eissn=1557-7368&rft.volume=40&rft.issue=4&rft.spage=1&rft.epage=17&rft_id=info:doi/10.1145%2F3450626.3459759&rft.externalDBID=n%2Fa&rft.externalDocID=10_1145_3450626_3459759 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0730-0301&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0730-0301&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0730-0301&client=summon |