Developability of triangle meshes

Abstract

Developable surfaces are those that can be made by smoothly bending flat pieces without stretching or shearing. We introduce a definition of developability for triangle meshes which exactly captures two key properties of smooth developable surfaces, namely flattenability and presence of straight ruling lines. This definition provides a starting point for algorithms in developable surface modeling---we consider a variational approach that drives a given mesh toward developable pieces separated by regular seam curves. Computation amounts to gradient descent on an energy with support in the vertex star, without the need to explicitly cluster patches or identify seams. We briefly explore applications to developable design and manufacturing.

References

1. Miklos Bergou, Max Wardetzky, David Harmon, Denis Zorin, and Eitan Grinspun. 2006. A Quadratic Bending Model for Inextensible Surfaces. In Symposium on Geometry Processing, Alla Sheffer and Konrad Polthier (Eds.). The Eurographics Association.
2. J. Berkmann and T. Caelli. 1994. Computation of Surface Geometry and Segmentation using Covariance Techniques. IEEE Transactions on Pattern Analysis and Machine Intelligence 16, 11 (1994), 1114--1116.
3. Alexander Bobenko. 2008. Discrete Differential Geometry. Springer London, Limited.
4. Vincent Borrelli, Saïd Jabrane, Francis Lazarus, and Boris Thibert. 2012. Flat tori in three-dimensional space and convex integration. Proc. Natl. Acad. Sci. U. S. A. 109, 19 (2012), 7218--7223.
5. Davide P Cervone. 1996. Tight immersions of simplicial surfaces in three space. Topology 35, 4 (1996), 863 -- 873.
6. Lian Chang. 2015. The Software Behind Frank Gehry's Geometrically Complex Architecture. (2015). https://priceonomics.com/the-software-behind-frank-gehrys-geometrically/ Online (Priceonomics.com); posted May 12, 2015.
7. Chih-Hsing Chu and Jang-Ting Chen. 2005. Tool path planning for five-axis flank milling with developable surface approximation. The International Journal of Advanced Manufacturing Technology 29, 7 (12 Oct 2005), 707.
8. Keenan Crane and Max Wardetzky. 2017. A Glimpse Into Discrete Differential Geometry. Notices of the American Mathematical Society 64, 10 (November 2017), 1153--1159.
9. Philippe Decaudin, Dan Julius, Jamie Wither, Laurence Boissieux, Alla Sheffer, and Marie-Paule Cani. 2006. Virtual Garments: A Fully Geometric Approach for Clothing Design. Computer Graphics Forum (2006).
10. Levi H Dudte, Etienne Vouga, Tomohiro Tachi, and L Mahadevan. 2016. Programming curvature using origami tessellations. Nature materials (2016).
11. Jeff Erickson and Sariel Har-Peled. 2004. Optimally Cutting a Surface into a Disk. Discrete & Computational Geometry 31, 1 (2004), 37--59.
12. Michael Garland. 1999. Quadric-Based Polygonal Surface Simplification. Ph.D. Dissertation. Carnegie Mellon University.
13. Michael Garland and Paul S. Heckbert. 1997. Surface Simplification Using Quadric Error Metrics. In Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH '97). ACM Press/Addison-Wesley Publishing Co., New York, NY, USA, 209--216.
14. Ramy F. Harik, Hu Gong, and Alain Bernard. 2013. Review: 5-axis Flank Milling: A State-of-the-art Review. Comput. Aided Des. 45, 3 (March 2013), 796--808.
15. Wenzel Jakob, Marco Tarini, Daniele Panozzo, and Olga Sorkine-Hornung. 2015. Instant Field-aligned Meshes. ACM Trans. Graph. 34, 6, Article 189 (Oct. 2015), 15 pages.
16. Dan Julius, Vladislav Kraevoy, and Alla Sheffer. 2005. D-Charts: Quasi-Developable Mesh Segmentation. Computer Graphics Forum (2005).
17. Martin Kilian, Simon Flöry, Zhonggui Chen, Niloy J. Mitra, Alla Sheffer, and Helmut Pottmann. 2008. Curved Folding. ACM Trans. Graph. 27, 3, Article 75 (Aug. 2008), 9 pages.
18. J. Kopp. 2008. Efficient Numerical Diagonalization of Hermitian 3 X 3 Matrices. International Journal of Modern Physics C 19 (2008), 523--548.
19. Adrian S. Lewis and Michael L. Overton. 2013. Nonsmooth optimization via quasi-Newton methods. Math. Program. A, 141 (2013), 135--163.
20. Yang Liu, Helmut Pottmann, Johannes Wallner, Yong-Liang Yang, and Wenping Wang. 2006. Geometric Modeling with Conical Meshes and Developable Surfaces. ACM Trans. Graph. 25, 3 Quly 2006), 681--689.
21. Fady Massarwi, Craig Gotsman, and Gershon Elber. 2007. Papercraft Models using Generalized Cylinders. In Proceedings of the Pacific Conference on Computer Graphics and Applications, Pacific Graphics 2007, Maui, Hawaii, USA, October 29 - November 2, 2007. 148--157.
22. Jun Mitani and Hiromasa Suzuki. 2004. Making Papercraft Toys from Meshes Using Strip-based Approximate Unfolding. ACM Trans. Graph. 23, 3 (Aug. 2004), 259--263.
23. Jean-Marie Morvan and Boris Thibert. 2006. Unfolding of Surfaces. Discrete & Computational Geometry 36, 3 (01 Oct 2006), 393--418.
24. Rahul Narain, Tobias Pfaff, and James F. O'Brien. 2013. Folding and Crumpling Adaptive Sheets. ACM Transactions on Graphics 32, 4 (July 2013), 51:1--8. http://graphics.berkeley.edu/papers/Narain-FCA-2013-07/ Proceedings of ACM SIGGRAPH 2013, Anaheim.
25. Jorge Nocedal and Stephen J. Wright. 2006. Numerical Optimization (2nd ed.). Springer.
26. Martin Peternell. 2004. Developable Surface Fitting to Point Clouds. Comput. Aided Geom. Des. 21, 8 (Oct. 2004), 785--803.
27. Roi Poranne, Marco Tarini, Sandro Huber, Daniele Panozzo, and Olga Sorkine-Hornung. 2017. Autocuts: Simultaneous Distortion and Cut Optimization for UV Mapping. ACM Trans. on Graphics - Siggraph Asia 2017 36, 6 (2017). http://vcg.isti.cnr.it/Publications/2017/PTHPS17
28. Helmut Pottmann, Alexander Schiftner, Pengbo Bo, Heinz Schmiedhofer, Wenping Wang, Niccolo Baldassini, and Johannes Wallner. 2008. Freeform Surfaces from Single Curved Panels. ACM Trans. Graph. 27, 3, Article 76 (Aug. 2008), 10 pages.
29. Michael Rabinovich, Tim Hoffmann, and Olga Sorkine-Hornung. 2018. Discrete Geodesic Nets for Modeling Developable Surfaces. ACM Trans. Graph. 37, 2, Article 16 (Feb. 2018), 17 pages.
30. Kenneth Rose, Alla Sheffer, Jamie Wither, Marie-Paule Cani, and Boris Thibert. 2007. Developable Surfaces from Arbitrary Sketched Boundaries. In Proceedings of the Fifth Eurographics Symposium on Geometry Processing (SGP '07). Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, 163--172. http://dl.acm.org/citation.cfm?id=1281991.1282014
31. Robert Sauer. 1970. Differenzengeometrie. Springer.
32. Rohan Sawhney and Keenan Crane. 2017. Boundary First Flattening. CoRR abs/1704.06873 (2017). arXiv:1704.06873 http://arxiv.org/abs/1704.06873
33. Camille Schreck, Damien Rohmer, Stefanie Hahmann, Marie-Paule Cani, Shuo Jin, Charlie C.L. Wang, and Jean-Francis Bloch. 2015. Non-smooth developable geometry for interactively animating paper crumpling. ACM Transactions on Graphics 35, 1 (Dec. 2015), 10:1--10:18.
34. Idan Shatz, Ayellet Tal, and George Leifman. 2006. Paper craft models from meshes. The Visual Computer 22, 9--11 (2006), 825--834.
35. Justin Solomon, Etienne Vouga, Max Wardetzky, and Eitan Grinspun. 2012. Flexible Developable Surfaces. Computer Graphics Forum (2012).
36. Olga Sorkine, Daniel Cohen-Or, Rony Goldenthal, and Dani Lischinski. 2002. Bounded-distortion Piecewise Mesh Parameterization. In Proceedings of the Conference on Visualization '02 (VIS '02). IEEE Computer Society, Washington, DC, USA, 355--362. http://dl.acm.org/citation.cfm?id=602099.602154
37. Chengcheng Tang, Pengbo Bo, Johannes Wallner, and Helmut Pottmann. 2016. Interactive Design of Developable Surfaces. ACM Trans. Graph. 35, 2, Article 12 (Jan. 2016), 12 pages.
38. Boris Thibert, Jean-Pierre Gratier, and Jean-Marie Morvan. 2005. A direct method for modeling and unfolding developable surfaces and its application to the Ventura Basin (California). Journal of Structural Geology 27, 2 (2005), 303--316.
39. Charlie CL Wang and Kai Tang. 2004. Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches. The Visual Computer 20, 8 (2004), 521--539.
40. Yong-Liang Yang, Yi-Jun Yang, Helmut Pottmann, and Niloy J. Mitra. 2011. Shape Space Exploration of Constrained Meshes. ACM Trans. Graph. 30, 6, Article 124 (Dec. 2011), 12 pages.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

      • Mesh geometry models

• Mathematics of computing

  • Mathematical analysis

    • Numerical analysis

      • Discretization