“Constrained willmore surfaces” by Soliman, Chern, Diamanti, Knöppel, Pinkall, et al. …

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.

1. 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.Google Scholar
2. 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.Google ScholarCross Ref
3. 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.Google ScholarDigital Library
4. Dimitri P. Bertsekas. 1996. Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific.Google Scholar
5. Wilhelm Blaschke. 1929. Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie III. Springer.Google Scholar
6. Alexander I. Bobenko, Ulrich Pinkall, and Boris Springborn. 2015. Discrete Conformal Maps and Ideal Hyperbolic Polyhedra. Geom. & Top. 19, 4 (2015), 2155–2215.Google ScholarCross Ref
7. Alexander I. Bobenko and Peter Schröder. 2005. Discrete Willmore Flow. In Proc. Symp. Geom. Proc. Eurographics, 101–110.Google Scholar
8. 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.Google Scholar
9. Christoph Bohle. 2010. Constrained Willmore Tori in the 4-Sphere. J. Diff. Geom. 86, 1 (2010), 71–132.Google Scholar
10. Christoph Bohle, G. Paul Peters, and Ulrich Pinkall. 2008. Constrained Willmore Surfaces. Calc. Var. 32 (2008), 263–277.Google ScholarCross Ref
11. 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.Google ScholarDigital Library
12. Matthias Bollhöfer, Olaf Schenk, Radim Janalik, Steve Hamm, and Kiran Gullapalli. 2020. State-of-the-Art Sparse Direct Solvers. (2020), 3–33.Google Scholar
13. Robert Bridson, S. Marino, and Ronald Fedkiw. 2003. Simulation of Clothing with Folds and Wrinkles. In Proc. Symp. Comp. Anim. 28–36.Google ScholarDigital Library
14. 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.Google Scholar
15. 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.Google ScholarCross Ref
16. Albert Chern, Felix Knöppel, Ulrich Pinkall, and Peter Schröder. 2018. Shape from Metric. ACM Trans. Graph. 37, 4 (2018), 63:1–17.Google ScholarDigital Library
17. Bennet Chow and Feng Luo. 2003. Combinatorial Ricci Flows on Surfaces. J. Diff. Geom. 63, 1 (2003), 97–129.Google ScholarCross Ref
18. Ming Chuang, Szymon Rusinkiewicz, and Misha Kazhdan. 2016. Gradient-Domain Processing of Meshes. J. Comp. Graph. Tech. 5, 4 (2016), 44–55.Google Scholar
19. Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2011. Spin Transformations of Discrete Surfaces. ACM Trans. Graph. 30, 4 (2011), 104:1–10.Google ScholarDigital Library
20. Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013. Robust Fairing via Conformal Curvature Flow. ACM Trans. Graph. 32, 4 (2013), 61:1–10.Google ScholarDigital Library
21. Bram Custers and Amir Vaxman. 2020. Subdivision Directional Fields. ACM Trans. Graph. 39, 2 (2020), 11:1–23.Google ScholarDigital Library
22. 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.Google Scholar
23. 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.Google Scholar
24. George K. Francis. 1987. A Topological Picturebook. Springer.Google Scholar
25. Adriano M. Garsia. 1961. An Imbedding of Closed Riemann Surfaces in Euclidean Space. Comm. Math. Helv. 35 (1961), 93–110.Google ScholarCross Ref
26. Huabin Ge. 2018. Combinatorial Calabi Flows on Surfaces. Trans. Amer. Math. Soc. 370, 2 (2018), 1377–1391.Google ScholarCross Ref
27. Mark Gillespie, Boris Springborn, and Keenan Crane. 2021. Discrete Conformal Equivalence of Triangle Meshes. ACM Trans. Graph. 40, 2 (2021).Google ScholarDigital Library
28. 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.Google ScholarCross Ref
29. Eitan Grinspun, Anil Hirani, Mathieu Desbrun, and Peter Schröder. 2003. Discrete Shells. In Proc. Symp. Comp. Anim. 62–67.Google ScholarDigital Library
30. Anthony Gruber and Eugenio Aulisa. 2020. Computational P-Willmore Flow with Conformal Penalty. ACM Trans. Graph. 39, 5 (2020), 161:1–16.Google ScholarDigital Library
31. W. Helfrich. 1973. Elastic Properties of Lipid Bilayers: Theory and Possible Experiments. Z. Naturforsch. C 28, 11–12 (1973), 693–703.Google ScholarCross Ref
32. Lynn Heller. 2013. Constrained Willmore tori and elastic curves in 2-dimensional space forms. (2013). arXiv:1303.1445.Google Scholar
33. Lynn Heller. 2015. Constrained Willmore and CMC Tori in the 3-Sphere. Diff. Geom. Appl. 40 (2015), 232–242.Google ScholarCross Ref
34. Lynn Heller and Cheikh Birahim Ndiaye. 2019. First Explicit Constrained Willmore Minimizers of Non-Rectangular Conformal Class. (2019). arXiv:1710.00533.Google Scholar
35. 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.Google Scholar
36. Pushkar Joshi and Carlo Séquin. 2007. Energy Minimizers for Curvature-Based Surface Functionals. Comp. Aid. Des. Appl. 4, 5 (2007), 607–617.Google ScholarCross Ref
37. 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.Google ScholarDigital Library
38. Ernst Kuwert and Reiner Schätzle. 2013. Minimizers of the Willmore Functional under Fixed Conformal Class. J. Diff. Geom. 93, 3 (2013), 471–530.Google Scholar
39. 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.Google Scholar
40. Wai Yeung Lam and Ulrich Pinkall. 2017. Isothermic Triangulated Surfaces. Math. Ann. 368, 1–2 (2017), 165–195.Google ScholarCross Ref
41. 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.Google ScholarCross Ref
42. 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.Google ScholarDigital Library
43. Feng Luo. 2004. Combinatorial Yamabe Flow on Surfaces. Comm. Contemp. Math. 6, 5 (2004), 765–780.Google ScholarCross Ref
44. Feng Luo, Jian Sun, and Tianqi Wu. 2020. Discrete Conformal Geometry of Polyhedral Surfaces and its Convergence. (2020). arXiv:2009.12706.Google Scholar
45. Fernando C. Marques and Andreé Neves. 2014. Min-Max Theory and the Willmore Conjecture. Ann. Math. 179, 2 (2014), 683–782.Google ScholarCross Ref
46. 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.Google ScholarCross Ref
47. Jorge Nocedal and Stephen J. Wright. 2006. Numerical Optimization (2 ed.). Springer.Google Scholar
48. Ulrich Pinkall. 1985. Hopf Tori in S3. Invent. Math. 81, 2 (1985), 379–386.Google ScholarCross Ref
49. Ulrich Pinkall and Boris Springborn. 2021. A Discrete Version of Liouville’s Theorem on Conformal Maps. Geom. Dedicata (2021).Google Scholar
50. R. J. Renka and J. W. Neuberger. 1995. Minimal Surfaces and Sobolev Gradients. SIAM J. Sci. Comp. 16, 6 (1995), 1412–1427.Google ScholarDigital Library
51. Tristan Rivière. 2008. Analysis Aspects of Willmore Surfaces. Invent. Math. 174, 1 (2008), 1–45.Google ScholarCross Ref
52. Reto A. Rüedy. 1971. Embeddings of Open Riemann Surfaces. Comm. Math. Helv. 46 (1971), 214–225.Google ScholarCross Ref
53. Florian Schäfer and Anima Anandkumar. 2019. Competitive Gradient Descent. (2019). arXiv:1905.12103v2.Google Scholar
54. Reiner Michael Schätzle. 2013. Conformally Constrained Willmore Immersions. Adv. Calc. Var. 6, 4 (2013), 375–390.Google Scholar
55. Henrik Schumacher. 2017. On H2-Gradient Flow for the Willmore Energy. (2017). arXiv:1703.06469v1.Google Scholar
56. Olga Sorkine. 2006. Differential Representations for Mesh Processing. Comp. Graph. Forum 25, 4 (2006), 789–807.Google ScholarCross Ref
57. Boris Springborn. 2017. Hyperbolic Polyhedra and Discrete Uniformization. (2017). arXiv:1707.06848.Google Scholar
58. Boris Springborn, Peter Schröder, and Ulrich Pinkall. 2008. Conformal Equivalence of Triangle Meshes. ACM Trans. Graph. 27, 3 (2008), 77:1–11.Google ScholarDigital Library
59. Amir Vaxman, Christian Müller, and Ofir Weber. 2015. Conformal Mesh Deformations with Möbius Transformations. ACM Trans. Graph. 34, 4 (2015), 55:1–11.Google ScholarDigital Library
60. Amir Vaxman, Christian Müller, and Ofir Weber. 2018. Canonical Möbius Subdivision. ACM Trans. Graph. 37, 6 (2018), 227:1–15.Google ScholarDigital Library
61. Ofir Weber, Ashish Myles, and Denis Zorin. 2012. Computing Extremal Quasiconformal Maps. Comp. Graph. Forum 31, 5 (2012), 1679–1689.Google ScholarDigital Library
62. Wei-Wei Xu and Kun Zhou. 2009. Gradient Domain Mesh Deformation-A Survey. J. Comp. Sci. Tech. 24, 1 (2009), 6–18.Google ScholarDigital Library
63. Christopher Yu, Henrik Schumacher, and Keenan Crane. 2020. Repulsive Curves. (2020). arXiv:2006.07859.Google Scholar