“Interactive Design of Developable Surfaces” by Tang, Bo, Wallner and Pottmann

  • ©Chengcheng Tang, Pengbo Bo, Johannes Wallner, and Helmut Pottmann

Conference:


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Title:

    Interactive Design of Developable Surfaces

Session/Category Title: DEFORMABLE SURFACE DESIGN


Presenter(s)/Author(s):


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Abstract:


    We present a new approach to geometric modeling with developable surfaces and the design of curved-creased origami. We represent developables as splines and express the nonlinear conditions relating to developability and curved folds as quadratic equations. This allows us to utilize a constraint solver, which may be described as energy-guided projection onto the constraint manifold, and which is fast enough for interactive modeling. Further, a combined primal-dual surface representation enables us to robustly and quickly solve approximation problems.

References:


    1. Günter Aumann. 1991. Interpolation with developable Bézier patches. Comput. Aided Geom. Des. 8 (1991), 409–420. 
    2. Günter Aumann. 2003. A simple algorithm for designing developable Bézier surfaces. Comput. Aided Geom. Des. 20 (2003), 601–619. 
    3. Mohan Bodduluri and Bahram Ravani. 1993. Design of developable surfaces using duality between plane and point geometries. Comput. Aided Des. 25 (1993), 621–632.
    4. Dongren Chen and Guojin Wang. 2002. Developable Bézier function surface. Progr. Natural Sci. 12, 5 (2002), 383–387.
    5. Ming Chen and Kai Tang. 2010. A fully geometric approach for developable cloth deformation simulation. Vis. Comput. 26 (2010), 853–863. 
    6. Chih-Hsing Chu and Jang-Ting Chen. 2004. Geometric design of uniform developable B-spline surfaces. In Proc. DETC. Vol. 1. 431–436. Article DETC2004-57257.
    7. Chih-Hsing Chu and Carlo Séquin. 2002. Developable Bézier patches: properties and design. Comput. Aided Des. 34 (2002), 511–527.
    8. Carl de Boor. 1978. A Practical Guide to Splines. Springer.
    9. 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. 
    10. Erik Demaine, Martin Demaine, and Duks Koschitz. 2011a. Reconstructing David Huffman’s Legacy in Curved-Crease Folding. In Origami5, Patsy Wang-Iverson et al. (Eds.). A. K. Peters, 39–52.
    11. Marcelo Dias, Levi Dudte, L. Mahadevan, and Christian Santangelo. 2012. Geometric mechanics of curved crease origami. Phys. Rev. Lett. 109, 114301 (2012), 1–13.
    12. Manfredo do Carmo. 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall.
    13. William Frey. 2004. Modeling buckled developable surfaces by triangulation. Comput. Aided Des. 36, 4 (2004), 299–313.
    14. Josef Hoschek and Helmut Pottmann. 1995. Interpolation and approximation with developable B-spline surfaces. In Mathematical Methods for Curves and Surfaces, M. Dæhlen et al. (Eds.). 255–264.
    15. David A. Huffman. 1976. Curvature and creases: A primer on paper. IEEE Trans. Comput. 25 (1976), 1010–1019. 
    16. Dan Julius, Vladislav Kraevoy, and Alla Sheffer. 2005. D-Charts: Quasi-developable mesh segmentation. Comput. Graph. Forum 24, 3 (2005), 581–590. Proc. Eurographics.
    17. Yannick L. Kergosien, Hironobu Gotoda, and Tosiyasu L. Kunii. 1994. Bending and creasing virtual paper. Comput. Graph. Appl. 14 (1994), 40–48. 
    18. Martin Kilian, Simon Flöry, Zhonggui Chen, Niloy Mitra, Alla Sheffer, and Helmut Pottmann. 2008. Curved folding. ACM Trans. Graph. 27, 3, Article 75 (2008), 9 pages. Proc. SIGGRAPH. 
    19. Johann Lang and Otto Röschel. 1992. Developable (1, n)-Bézier surfaces. Comput. Aided Geom. Des. 9 (1992), 291–298. 
    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 (2006), 681–689. Proc. SIGGRAPH. 
    21. Takashi Maekawa and Julie S. Chalfant. 1998. Design and tessellation of B-spline developable surfaces. J. Mech. Des. 120 (1998), 453–461.
    22. Sandy Martedi and Hideo Saito. 2011. Foldable augmented papers with a relaxed constraint. In Proc. ISAS. IEEE, 127–131.
    23. Meher McArthur and Robert J. Lang. 2012. Folding Paper: The Infinite Possibilities of Origami. Int. Arts & Artists, Washington, DC.
    24. Neil Meredith and James Kotronis. 2012. Self-detailing and self-documenting systems for wood fabrication: The Burj Khalifa. In Advances in Architectural Geometry 2012, Lars Hesselgren et al. (Eds.). Springer, 185–198.
    25. Jun Mitani. 2012. Origami applications. http://mitani.cs.tsukuba.ac.jp/origami_application. (2012).
    26. Jun Mitani and Takeo Igarashi. 2011. Interactive design of planar curved folding by reflection. In Pacific Graphics, Short Papers. 77–81.
    27. Jun Mitani and Hiromasa Suzuki. 2004. Making papercraft toys from meshes using strip approximate unfolding. ACM Trans. Graph. 23, 3 (2004), 259–263. Proc. SIGGRAPH. 
    28. Rahul Narain, Tobias Pfaff, and James F. O’Brien. 2013. Folding and crumpling adaptive sheets. ACM Trans. Graph. 32, 4, Article 51 (2013), 8 pages. Proc. SIGGRAPH. 
    29. Francisco Pérez and José Antonio Suárez. 2007. Quasi-developable B-spline surfaces in ship hull design. Comp. Aided Geom. Design 39 (2007), 853–862.
    30. Mathieu Perriollat and Adrien Bartoli. 2012. A computational model of bounded developable surfaces with application to image-based three-dimensional reconstruction. Comput. Anim. Virt. Worlds 24, 5 (2012), 459–476.
    31. Martin Peternell. 2004. Developable surface fitting to point clouds. Comput. Aided Geom. Des. 21 (2004), 785–803. 
    32. Helmut Pottmann and Gerald Farin. 1995. Developable rational Bézier and B-spline surfaces. Comput. Aided Geom. Des. 12 (1995), 513–531. 
    33. 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 (2008), 10 pages. Proc. SIGGRAPH. 
    34. Helmut Pottmann and Johannes Wallner. 1999. Approximation algorithms for developable surfaces. Comput. Aided Geom. Des. 16 (1999), 539–556.
    35. Helmut Pottmann and Johannes Wallner. 2001. Computational Line Geometry. Springer. 
    36. Kenneth Rose, Alla Sheffer, Jamie Wither, Marie-Paul Cani, and Boris Thibert. 2007. Developable surfaces from arbitrary sketched boundaries. In Symposium Geometry Processing, Alexander Belyaev and Michael Garland (Eds.). 163–172. 
    37. Justin Solomon, Etienne Vouga, Max Wardetzky, and Eitan Grinspun. 2012. Flexible developable surfaces. Comp. Graph. Forum 31, 5 (2012), 1567–1576. Proc. Symposium Geometry Processing. 
    38. Tomohiro Tachi. 2010. Origamizing polyhedral surfaces. IEEE Trans. Vis. Comp. Graphics 16, 2 (2010), 298–311. 
    39. Tomohiro Tachi and Koryo Miura. 2012. Rigid-foldable cylinders and cells. J. Int. Assoc. Shell Spatial Structures 53, 4 (2012), 217–226.
    40. Chengcheng Tang, Xiang Sun, Alexandra Gomes, Johannes Wallner, and Helmut Pottmann. 2014. Form-finding with polyhedral meshes made simple. ACM Trans. Graph. 33, 4, Article 70 (2014), 9 pages. Proc. SIGGRAPH. 
    41. Charlie Wang and Kai Tang. 2004. Achieving developability of a polygonal surface by minimum deformation: A study of global and local optimization approaches. Vis. Comput. 20 (2004), 521–539. 
    42. K. Wang and Y. Chen. 2011. Folding a patterned cylinder by rigid origami. In Origami5, Patsy Wang-Iverson et al. (Eds.). A.K. Peters, 265–276.
    43. Wenping Wang, Helmut Pottmann, and Yang Liu. 2006. Fitting B-spline curves to point clouds by curvature-based squared distance minimization. ACM Trans. Graph. 25, 2 (2006), 214–238. 
    44. Hitoshi Yamauchi, Stefan Gumhold, Rhaleb Zayer, and Hans-Peter Seidel. 2005. Mesh segmentation driven by Gaussian curvature. Vis. Comput. 21 (2005), 659–668.

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