“Regular meshes from polygonal patterns”

  • ©Amir Vaxman, Christian Müller, and Ofir Weber




    Regular meshes from polygonal patterns


Session Title: Meshing



    We present a framework for designing shapes from diverse combinatorial patterns, where the vertex 1-rings and the faces are as rotationally symmetric as possible, and define such meshes as regular. Our algorithm computes the geometry that brings out the symmetries encoded in the combinatorics. We then allow designers and artists to envision and realize original meshes with great aesthetic qualities. Our method is general and applicable to meshes of arbitrary topology and connectivity, from triangle meshes to general polygonal meshes. The designer controls the result by manipulating and constraining vertex positions. We offer a novel characterization of regularity, using quaternionic ratios of mesh edges, and optimize meshes to be as regular as possible according to this characterization. Finally, we provide a mathematical analysis of these regular meshes, and show how they relate to concepts like the discrete Willmore energy and connectivity shapes.


    1. Sameer Agarwal, Keir Mierle, and Others. Ceres Solver. http://ceres-solver.org.Google Scholar
    2. Ergun Akleman, Vinod Srinivasan, and Esan Mandal. 2005. Remeshing Schemes for Semi-Regular Tilings. In Proc. SMI. IEEE Computer Society, 44–50. Google ScholarDigital Library
    3. de Blob. http://www.designboom.com/architecture/massimiliano-fuksas-de-blob-at-september-18-square/.Google Scholar
    4. Alexander I. Bobenko. 2005. A Conformal Energy for Simplicial Surfaces. Combinatorial and Computational Geom. (2005), 133–143.Google Scholar
    5. Alexander I. Bobenko, Tim Hoffmann, and Boris A. Springborn. 2006. Minimal surfaces from circle patterns: geometry from combinatorics. Ann. of Math. (2) 164, 1 (2006), 231–264.Google Scholar
    6. Alexander I. Bobenko, Helmut Pottmann, and Johannes Wallner. 2010. A curvature theory for discrete surfaces based on mesh parallelity. Math. Ann. 348, 1 (2010), 1–24. Google ScholarCross Ref
    7. Alexander I. Bobenko and Peter Schröder. 2005. Discrete Willmore Flow. In Proc. SGP. 101–110. Google ScholarDigital Library
    8. Mario Botsch, Leif Kobbelt, Mark Pauly, Pierre Alliez, and Bruno Levy. 2010. Polygon Mesh Processing.Google Scholar
    9. Sofien Bouaziz, Mario Deuss, Yuliy Schwartzburg, Thibaut Weise, and Mark Pauly. 2012. Shape-Up: Shaping Discrete Geometry with Projections. Comput. Graph. Forum 31, 5 (2012), 1657–1667. Google ScholarDigital Library
    10. Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013. Robust Fairing via Conformal Curvature Flow. ACM Trans. Graph. 32, 4 (2013). Eye. https://www.eyefilm.nl/. Google ScholarDigital Library
    11. Branko Grünbaum and G C Shephard. 1986. Tilings and Patterns. W. H. Freeman & Co., New York, NY, USA.Google Scholar
    12. Ewain Gwynne and Matvei Libine. 2012. On a Quaternionic Analogue of the Cross-Ratio. Adv. Appl. Clifford Algebr. 22, 4 (2012), 1041–1053. Google ScholarCross Ref
    13. Udo Hertrich-Jeromin. 2003. Introduction to Möbius differential geometry. London Mathematical Society Lecture Note Series, Vol. 300. Cambridge University Press. xii+413 pages. Google ScholarCross Ref
    14. Martin Isenburg, Stefan Gumhold, and Craig Gotsman. 2001. Connectivity Shapes. In Proc. Conf. on Visualization. IEEE Computer Society, 135–142. Google ScholarCross Ref
    15. Caigui Jiang, Chengcheng Tang, Amir Vaxman, Peter Wonka, and Helmut Pottmann. 2015. Polyhedral Patterns. ACM Trans. Graph. 34, 6 (2015), 172:1–172:12.Google ScholarDigital Library
    16. Hana Kouřimská, Lara Skuppin, and Boris Springborn. 2016. A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths. In Advances in Discrete Differential Geometry, Alexander I. Bobenko (Ed.). Springer Berlin Heidelberg, 177–195. Google ScholarCross Ref
    17. KREOD. http://www.kreod.com/.Google Scholar
    18. Yufei Li, Yang Liu, and Wenping Wang. 2015. Planar Hexagonal Meshing for Architecture. IEEE Trans. on Vis. and Comp. Graph. 21, 1 (2015), 95–106.Google ScholarCross Ref
    19. Christian Müller and Johannes Wallner. 2010. Oriented mixed area and discrete minimal surfaces. Discrete Comput. Geom. 43, 2 (2010), 303–320. Google ScholarDigital Library
    20. Museo Soumaya. http://www.soumaya.com.mx/.Google Scholar
    21. Jorge Nocedal and Stephen J. Wright. 2006. Numerical Optimization. Springer.Google Scholar
    22. Hao Pan, Yi-King Choi, Yang Liu, Wenchao Hu, Qiang Du, Konrad Polthier, Caiming Zhang, and Wenping Wang. 2012. Robust Modeling of Constant Mean Curvature Surfaces. ACM Trans. Graph. 31, 4 (2012), 85:1–85:11.Google ScholarDigital Library
    23. Ulrich Pinkall and Konrad Polthier. 1993. Computing discrete minimal surfaces and their conjugates. Experiment. Math. 2, 1 (1993), 15–36. Google ScholarCross Ref
    24. Helmut Pottmann, Michael Eigensatz, Amir Vaxman, and Johannes Wallner. 2015. Architectural Geometry. Computers and Graphics 47 (2015), 145–164. Google ScholarDigital Library
    25. Helmut Pottmann, Yang Liu, Johannes Wallner, Alexander Bobenko, and Wenping Wang. 2007. Geometry of Multi-layer Freeform Structures for Architecture. ACM Trans. Graph. 26, 3 (2007), 65:1–65:11.Google ScholarDigital Library
    26. Rinus Roelofs. http://www.rinusroelofs.nl/.Google Scholar
    27. Boris Springborn, Peter Schröder, and Ulrich Pinkall. 2008. Conformal Equivalence of Triangle Meshes. ACM Trans. Graph. 27, 3 (2008), 1–11. Google ScholarDigital Library
    28. Chengcheng Tang, Xiang Sun, Alexandra Gomes, Johannes Wallner, and Helmut Pottmann. 2014. Form-finding with Polyhedral Meshes Made Simple. ACM Trans. Graph. 33, 4 (2014), 70:1–70:9.Google ScholarDigital Library
    29. Marco Tarini, Kai Hormann, Paolo Cignoni, and Claudio Montani. 2004. PolyCube-Maps. ACM Trans. Graph. 23, 3 (2004), 853–860. Google ScholarDigital Library
    30. Amir Vaxman. 2016. libhedra: geometric processing and optimization of polygonal meshes. (2016). https://github.com/avaxman/libhedra.Google Scholar
    31. Amir Vaxman, Christian Müller, and Ofir Weber. 2015. Conformal mesh deformations with Möbius transformations. ACM Trans. Graph. 34, 4 (2015), 55:1–55:11.Google ScholarDigital Library

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