“Curve-pleated structures” by Jiang, Mundilova, Rist, Wallner and Pottmann
Conference:
Type(s):
Title:
- Curve-pleated structures
Session/Category Title: Network
Presenter(s)/Author(s):
Moderator(s):
Abstract:
In this paper we study pleated structures generated by folding paper along curved creases. We discuss their properties and the special case of principal pleated structures. A discrete version of pleated structures is particularly interesting because of the rich geometric properties of the principal case, where we are able to establish a series of analogies between the smooth and discrete situations, as well as several equivalent characterizations of the principal property. These include being a conical mesh, and being flat-foldable. This structure-preserving discretization is the basis of computation and design. We propose a new method for designing pleated structures and reconstructing reference shapes as pleated structures: we first gain an overview of possible crease patterns by establishing a connection to pseudogeodesics, and then initialize and optimize a quad mesh so as to become a discrete pleated structure. We conclude by showing applications in design and reconstruction, including cases with combinatorial singularities. Our work is relevant to fabrication in so far as the offset properties of principal pleated structures allow us to construct curved sculptures of finite thickness.
References:
1. Thomas Barois, Loïc Tadrist, Catherine Quilliet, and Yoël Forterre. 2014. How a Curved Elastic Strip Opens. Phys. Rev. Letters 113 (2014), #214301, 1–5.Google Scholar
2. M. Ben Amar and Y. Pomeau. 1997. Crumpled paper. Proc. R. Soc. London Ser A. 453 (1997), 729–755.Google ScholarCross Ref
3. Alexander Bobenko and Yuri Suris. 2009. Discrete differential geometry: Integrable Structure. American Math. Soc.Google Scholar
4. Mario Botsch, Stephan Steinberg, Stephan Bischoff, and Leif Kobbelt. 2002. OpenMesh: A Generic and Efficient Polygon Mesh Data Structure. Proc. OpenSG Symposium.Google Scholar
5. Sebastien J.P. Callens and Amir A. Zadpoor. 2018. From flat sheets to curved geometries: Origami and kirigami approaches. Materials Today 21, 3 (2018), 241–264.Google ScholarCross Ref
6. Enrique Cerda, Sahraoui Chaieb, Francisco Melo, and L. Mahadevan. 1999. Conical dislocations in crumpling. Nature 401 (1999), 46–49.Google ScholarCross Ref
7. Enrique Cerda and L. Mahadevan. 1998. Conical surfaces and crescent singularities in crumpled sheets. Phys. Rev. Lett. 80 (1998), 2358–2361.Google ScholarCross Ref
8. Yan Chen, Rui Peng, and Zhong You. 2015. Origami of thick panels. Science 349, 6246 (2015), 396–400.Google Scholar
9. Eli Davis, Erik D. Demaine, Martin L. Demaine, and Jennifer Ramseyer. 2013. Reconstructing David Huffman’s Origami Tessellations. J. Mechanical Design 135 (2013), #111010, 1–7.Google Scholar
10. Erik Demaine, Martin Demaine, Vi Hart, Gregory Price, and Tomohiro Tachi. 2011b. (Non)existence of pleated folds: how paper folds between creases. Graphs and Combinatorics 27 (2011), 377–397.Google ScholarDigital Library
11. Erik Demaine, Martin Demaine, and Duks Koschitz. 2011a. Reconstructing David Huffman’s Legacy in Curved-Crease Folding. In Origami 5. A. K. Peters, 39–52.Google Scholar
12. Erik Demaine, Martin Demaine, Duks Koschitz, and Tomohiro Tachi. 2015b. A Review on curved Creases in Art, Design and Mathematics. Symmetry: Culture and Science 26 (2015), 145–161.Google Scholar
13. Erik D. Demaine, Martin L. Demaine, David A. Huffman, Duks Koschitz, and Tomohiro Tachi. 2015a. Characterization of Curved Creases and Rulings: Design and Analysis of Lens Tessellations. In Origami 6. Vol. 1. American Math. Soc, 209–230.Google Scholar
14. Erik D. Demaine, Martin L. Demaine, David A. Huffman, Duks Koschitz, and Tomohiro Tachi. 2018. Conic Crease Patterns with Reflecting Rule Lines. In Origami 7. Vol. 2. Tarquin, 574–590.Google Scholar
15. Erik D. Demaine, Martin L. Demaine, and Joseph S.B. Mitchell. 2000. Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami. Comp. Geom. 16 (2000), 3–21.Google ScholarDigital Library
16. Erik D. Demaine and Joseph O’Rourke. 2007. Geometric Folding Algorithms. Cambridge University Press.Google ScholarDigital Library
17. Erik D. Demaine and Tomohiro Tachi. 2017. Origamizer: A Practical Algorithm for Folding Any Polyhedron. In Proc. 33d Int. Symp. Comput. Geometry. #34, 1–15.Google Scholar
18. Marcelo Dias, Levi Dudte, L. Mahadevan, and Christian Santangelo. 2012. Geometric Mechanics of Curved Crease Origami. Phys. Rev. Lett. 109, 114301 (2012), 1–13.Google Scholar
19. Marcelo A. Dias and Basile Audoly. 2014. A non-linear rod model for folded elastic strips. J. Mechanics and Physics of Solids 62 (2014), 57–80.Google ScholarCross Ref
20. Marcelo A Dias and Basile Audoly. 2015. Wunderlich, meet Kirchhoff: A general and unified description of elastic ribbons and thin rods. J. Elasticity 119 (2015), 49–66.Google ScholarCross Ref
21. Marcelo A. Dias and Christian D. Santangelo. 2012. The shape and mechanics of curved-fold origami structures. Europhys. Lett. 100, 5 (2012), 54005.Google ScholarCross Ref
22. Manfredo do Carmo. 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall.Google Scholar
23. Yuntong Du, Changping Song, Jian Xiong, and Linzhi Wu. 2019. Fabrication and mechanical behaviors of carbon fiber reinforced composite foldcore based on curved-crease origami. Composites Sc. & Technology 174 (2019), 94–105.Google ScholarCross Ref
24. Levi H. Dudte, Etienne Vouga, Tomohiro Tachi, and L. Mahadevan. 2016. Programming curvature using origami tessellations. Nature Materials 15 (2016), 583–588.Google ScholarCross Ref
25. Thomas A. Evans, Robert J. Lang, Spencer P. Magleby, and Larry L. Howell. 2015a. Ridigly foldable Origami Twists. In Origami 6. Vol. 1. American Math. Soc, 119–130.Google Scholar
26. Thomas A. Evans, Robert J. Lang, Spencer P. Magleby, and Larry L. Howell. 2015b. Rigidly foldable origami gadgets and tessellations. Royal Society Open Science 2 (2015), #150067, 2–18.Google Scholar
27. William Frey. 2004. Modeling buckled developable surfaces by triangulation. Computer-Aided Des. 36, 4 (2004), 299–313.Google ScholarCross Ref
28. Dmitry Fuchs and Serge Tabachnikov. 1999. More on Paperfolding. Americal Math. Monthly 106 (1999), 27–35.Google ScholarCross Ref
29. Matthew Gardiner, Roland Aigner, Hideaki Ogawa, and Rachel Hanlon. 2018. Fold Mapping: Parametric Design of Origami Surfaces with Periodic Tessellations. In Origami 7. Vol. 1. Tarquin, 105–118.Google Scholar
30. Joseph Gattas and Zhong You. 2015. The behaviour of curved-crease foldcores under low-velocity impact loads. Int. J. Solids and Structures 53 (2015), 80–91.Google ScholarCross Ref
31. Amanda Ghassaei, Erik Demaine, and Neil Gershenfeld. 2018. Fast, Interactive Origami Simulation using GPU Computation. In Origami 7, Vol. 4. Tarquin, 1151–1166.Google Scholar
32. David A. Huffman. 1976. Curvature and creases: A primer on paper. IEEE Trans. Computers C-25 (1976), 1010–1019.Google Scholar
33. Ivan Izmestiev. 2017. Classification of flexible Kokotsakis polyhedra with quadrangular base. Int. Math. Res. Not. 3 (2017), 715–808.Google Scholar
34. Yannick L. Kergosien, Hironoba Gotoda, and Tosiyasu L. Kunii. 1994. Bending and Creasing Virtual Paper. Comput. Graph. Appl. 14 (1994), 40–48.Google ScholarDigital Library
35. Martin Kilian, Simon Flöry, Zhonggui Chen, Niloy Mitra, Alla Sheffer, and Helmut Pottmann. 2008. Curved Folding. ACM Trans. Graph. 27, 3 (2008), #75, 1–9.Google ScholarDigital Library
36. Martin Kilian, Aron Monszpart, and Niloy J. Mitra. 2017. String Actuated Curved Folded Surfaces. ACM Trans. Graph. 36, 3 (2017), #25, 1–13.Google ScholarDigital Library
37. Antonios Kokotsakis. 1933. Über bewegliche Polyeder. Math. Ann. 107 (1933), 627–647.Google ScholarCross Ref
38. 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
39. Mina Konaković-Luković, Pavle Konaković, and Mark Pauly. 2018. Computational Design of Deployable Auxetic Shells. In Advances in Architectural Geometry 2018, L. Hesselgren et al. (Eds.). Klein Publishing, 94–111.Google Scholar
40. Richard D. Koschitz. 2014. Computational design with curved creases: David Huffman’s approach to paperfolding. Ph.D. Dissertation. MIT.Google Scholar
41. Artur Lebée. 2015. From folds to structures, a review. Int. J. Space Structures 30, 2 (2015), 55–74.Google ScholarCross Ref
42. Ting-Uei Lee, Xiaochen Yang, Jiayao Ma, Yan Chen, and Joseph M. Gattas. 2019. Elastic buckling shape control of thin-walled cylinder using pre-embedded curved-crease origami patterns. Int. J. Mechanical Sciences 151 (2019), 322–330.Google ScholarCross Ref
43. 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 (2006), 681–689.Google ScholarDigital Library
44. Jun Mitani. 2009a. A Design Method for 3D Origami Based on Rotational Sweep. Computer-Aided Des. & Appl. 6 (2009), 69–79.Google ScholarCross Ref
45. Jun Mitani. 2009b. Origami Sphere (crease patterns). available from http://mitani.cs.tsukuba.ac.jp (accessed August 2019).Google Scholar
46. Jun Mitani. 2012. Column-shaped origami design based on mirror reflections. J. Geom. Graphics 16 (2012), 185–194.Google Scholar
47. Jun Mitani and Takeo Igarashi. 2011. Interactive Design of Planar Curved Folding by Reflection. In Pacific Graphics, Short Papers. Eurographics, 77–81.Google Scholar
48. Shuhei Miyashita, Isabella DiDio, Ishwarya Ananthabhotla, Byoungkwon An, Cynthia Sung, Slava Arabagi, and Daniela Rus. 2015. Folding Angle Regulation by Curved Crease Design for Self-Assembling Origami Propellers. J. Mechanisms Robotics 7 (2015), #021013, 1–8.Google Scholar
49. Pierre-Olivier Mouthuy, Michael Coulombier, Thomas Pardoen, Jean-Pierre Raskin, and Alain M. Jonas. 2012. Overcurvature describes the buckling and folding of rings from curved origami to foldable tents. Nature Comm. 3 (2012), #1290.Google Scholar
50. Klara Mundilova. 2019. On mathematical folding of curved crease origami: Sliding developables and parametrizations of folds into cylinders and cones. Computer-Aided Design 115 (2019), 34–41. Proc. SPM.Google ScholarCross Ref
51. Helmut Pottmann, Qixing Huang, Bailin Deng, Alexander Schiftner, Martin Kilian, Leonidas Guibas, and Johannes Wallner. 2010. Geodesic Patterns. ACM Trans. Graph. 29, 4 (2010), #43, 1–10.Google ScholarDigital Library
52. 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 (2008), #76, 1–10.Google ScholarDigital Library
53. Michael Rabinovich, Tim Hoffmann, and Olga Sorkine-Hornung. 2018a. Discrete Geodesic Nets for Modeling Developable Surfaces. ACM Trans. Graph. 37, 2 (2018), 16:1–16:17.Google ScholarDigital Library
54. Michael Rabinovich, Tim Hoffmann, and Olga Sorkine-Hornung. 2018b. The Shape Space of Discrete Orthogonal Geodesic Nets. ACM Trans. Graph. 37, 6 (2018), 228:1–228:17.Google ScholarDigital Library
55. Vlad A. Raducanu, Vasile D. Cojocaru, and Doina Raducanu. 2016. Structural Architectural Elements Made of Curved Folded Sheet Metal. In eCAADe 2016 – Complexity & Simplicity, A. Herneoja et al. (Eds.). Vol. 2. Univ. Oulu, 409–416.Google Scholar
56. Otto Röschel. 2019. Curved Folding with Pairs of Cones. In Proceedings 18th Int. Conf. Geometry and Graphics, L. Cocchiarella (Ed.). Springer, 381–388.Google ScholarCross Ref
57. Wolfgang Schief, Alexander Bobenko, and Tim Hoffmann. 2008. On the integrability of infinitesimal and finite deformations of polyhedral surfaces. In Discrete differential geometry, A. Bobenko et al. (Eds.). Oberwolfach Seminars, Vol. 38. Springer, 67–93.Google Scholar
58. Eike Schling, Denis Hitrec, and Rainer Barthel. 2018. Designing Grid Structures using Asymptotic Curve Networks. In Humanizing Digital Reality, K. de Rycke et al. (Eds.). Springer, 125–140. Proc. Design Modelling Symposium.Google Scholar
59. Keith A. Seffen. 2012. Compliant shell mechanisms. Phil. Trans. R. Soc. A 370 (2012), 2010–2026.Google ScholarCross Ref
60. Justin Solomon, Etienne Vouga, Max Wardetzky, and Eitan Grinspun. 2012. Flexible developable surfaces. Comput. Graph. Forum 31 (2012), 1567–1576. Proc. SGP.Google ScholarDigital Library
61. Keyao Song, Xiang Zhou, Shixi Zang, Hai Wang, and Zhong You. 2017. Design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns. Proc. R. Soc. A 473 (2017), #20170016, 1–18.Google ScholarCross Ref
62. Oded Stein, Eitan Grinspun, and Keenan Crane. 2018. Developability of triangle meshes. ACM Trans. Graph. 37, 4 (2018), #77, 1–14.Google ScholarDigital Library
63. Daniel M. Sussman, Yigil Cho, Toen Castle, Xingting Gong, Euiyeon Jung, Shu Yang, and Randall D. Kamien. 2015. Algorithmic lattice kirigami: A route to pluripotent materials. Proc. Nat. Ac. Sc. 112, 24 (2015), 7449–7453.Google ScholarCross Ref
64. Tomohiro Tachi. 2009. Generalization of rigid foldable quadrilateral mesh origami. J. Int. Ass. Shell & Spatial Structures 50 (2009), 173–179.Google Scholar
65. Tomohiro Tachi. 2010a. Geometric Considerations for the Design of Rigid Origami Structures. In Proc. IASS Symposium 2010. 771–782.Google Scholar
66. Tomohiro Tachi. 2010b. Origamizing polyhedral surfaces. IEEE Trans. Vis. Comp. Graphics 16, 2 (2010), 298–311.Google ScholarDigital Library
67. Tomohiro Tachi. 2013. Composite Rigid-Foldable Curved Origami Structure. In Proc. 1st Transformables Conf., F. Escrig and J. Sanchez (Eds.). Starbooks, Sevilla, 6 pp.Google Scholar
68. Tomohiro Tachi and Gregory Epps. 2011. Designing One-DOF Mechanisms for Architecture by Rationalizing Curved Folding. In Proc. ALGODE Symposium, Y. Ikeda (Ed.). Arch. Institute Japan, Tokyo, 14 pp. CD ROM.Google Scholar
69. Chengcheng Tang, Pengbo Bo, Johannes Wallner, and Helmut Pottmann. 2016. Interactive design of developable surfaces. ACM Trans. Graph. 35, 2 (2016), #12, 1–12.Google ScholarDigital Library
70. Sivan Toledo. 2003. Taucs, A Library of Sparse Linear Solvers. C library.Google Scholar
71. Zhiyan Wei, Zengcai Guo, Levi Dudte, Haiyi Liang, and L. Mahadevan. 2013. Geometric Mechanics of Periodic Pleated Origami. Phys. Rev. Lett. 110 (2013), #215501, 1–5.Google Scholar
72. Walter Wunderlich. 1950a. Pseudogeodätische Linien auf Kegelflächen. Sitzungsber. II, Österr. Akad. Wiss. 158 (1950), 75–105.Google Scholar
73. Walter Wunderlich. 1950b. Pseudogeodätische Linien auf Zylinderflächen. Sitzungsber. II, Österr. Akad. Wiss. 158 (1950), 61–73.Google Scholar
74. Walter Wunderlich. 1950c. Raumkurven, die pseudogeodätische Linien eines Zylinders und eines Kegels sind. Compos. Math. 8 (1950), 169–184.Google Scholar
75. Walter Wunderlich. 1950d. Raumkurven, die pseudogeodätische Linien zweier Kegel sind. Monatsh. Math. 54 (1950), 55–70.Google ScholarCross Ref
