This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Origamizing Polyhedral Surfaces
March/April 2010 (vol. 16 no. 2)
pp. 298-311
Tomohiro Tachi, The University of Tokyo, Tokyo
This paper presents the first practical method for "origamizing” or obtaining the folding pattern that folds a single sheet of material into a given polyhedral surface without any cut. The basic idea is to tuck fold a planar paper to form a three-dimensional shape. The main contribution is to solve the inverse problem; the input is an arbitrary polyhedral surface and the output is the folding pattern. Our approach is to convert this problem into a problem of laying out the polygons of the surface on a planar paper by introducing the concept of tucking molecules. We investigate the equality and inequality conditions required for constructing a valid crease pattern. We propose an algorithm based on two-step mapping and edge splitting to solve these conditions. The two-step mapping precalculates linear equalities and separates them from other conditions. This allows an interactive manipulation of the crease pattern in the system implementation. We present the first system for designing three-dimensional origami, enabling a user can interactively design complex spatial origami models that have not been realizable thus far.

[1] E.D. Demaine and J. O'Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge Univ. Press, July 2007.
[2] J. Mitani and H. Suzuki, “Making Papercraft Toys from Meshes Using Strip-Based Approximate Unfolding,” ACM Trans. Graphics, vol. 23, no. 3, pp. 259-263, 2004.
[3] F. Massarwi, C. Gotsman, and G. Elber, “Papercraft Models Using Generalized Cylinders,” Proc. 15th Pacific Conf. Computer Graphics and Applications (PG '07), pp. 148-157, 2007.
[4] D. Julius, V. Kraevoy, and A. Sheffer, “D-Charts: Quasi-Developable Mesh Segmentation,” Proc. Ann. Conf. European Assoc. for Computer Graphics (Eurographics), pp. 581-590, 2005.
[5] C.C.L. Wang, “Towards Flattenable Mesh Surfaces,” Computer-Aided Design, vol. 40, no. 1, pp. 109-122, 2008.
[6] J. Subag and G. Elber, “Piecewise Developable Surface Approximation of General NURBS Surfaces with Global Error Bounds,” Proc. Int'l Conf. Geometric Modeling and Processing, pp. 143-156, 2006.
[7] Y. Liu, H. Pottmann, J. Wallner, Y.-L. Yang, and W. Wang, “Geometric Modeling with Conical Meshes and Developable Surfaces,” ACM Trans. Graphics, vol. 25, no. 3, pp. 681-689, 2006.
[8] H. Pottmann, A. Schiftner, P. Bo, H. Schmiedhofer, W. Wang, N. Baldassini, and J. Wallner, “Freeform Surfaces from Single Curved Panels,” ACM Trans. Graphics, vol. 27, no. 3, 2008.
[9] M. Wertheim, “Cones, Curves, Shells, Towers: He Made Paper Jump to Life,” The New York Times, http://www.theiff.org/pressNYThuffman.html , June 2004.
[10] R.D. Resch and H. Christiansen, “The Design and Analysis of Kinematic Folded Plate Systems,” Proc. Symp. for Folded Plate Structures, 1970.
[11] R.D. Resch, “Ron Resch Dot Com,” http:/www.ronresch.com/, 2009.
[12] J. Lobell, “The Milgo Experiment: An Interview with Haresh Lalvani,” Architectural Design, vol. 76, no. 4, pp. 52-61, 2006.
[13] RoboFold “Robofold,” http://www.robofold.comdesign.html, 2009.
[14] D. Huffman, “Curvature and Creases: A Primer on Paper,” IEEE Trans. Computers, vol. 25, no. 10, pp. 1010-1019, Oct. 1976.
[15] Y. Kergosien, H. Gotoda, and T. Kunii, “Bending and Creasing Virtual Paper,” IEEE Computer Graphics and Applications, vol. 14, no. 1, pp. 40-48, Jan. 1994.
[16] M. Kilian, S. Flöry, N.J. Mitra, and H. Pottmann, “Curved Folding,” ACM Trans. Graphics, vol. 27, no. 3, pp. 1-9, 2008.
[17] S.-M. Belcastro and T. Hull, “A Mathematical Model for Non-Flat Origami,” Proc. Third Int'l Meeting of Origami Math., Science, and Education (Origami3), pp. 39-51, 2002.
[18] T. Tachi, “Rigid Origami Simulator,” http://www.tsg.ne.jp/TTsoftware/, 2007.
[19] R. Burgoon, Z.J. Wood, and E. Grinspun, “Discrete Shells Origami,” Proc. Computers and Their Applications, pp. 180-187, 2006.
[20] M. Bern and B. Hayes, “The Complexity of Flat Origami,” Proc. Seventh Ann. ACM-SIAM Symp. Discrete Algorithms, pp. 175-183, 1996.
[21] T. Meguro, “The Method to Design Origami,” Origami Tanteidan Newspaper, 1991.
[22] E.D. Demaine, M.L. Demaine, and A. Lubiw, “Folding and Cutting Paper,” Proc. Revised Papers from the Japan Conf. Discrete and Computational Geometry (JCDCG '98), pp. 104-117, 1998.
[23] R.J. Lang, “A Computational Algorithm for Origami Design,” Proc. 12th Ann. Symp. Computational Geometry (SCG '96), pp. 98-105, 1996.
[24] M. Bern, E. Demaine, D. Eppstein, and B. Hayes, “A Disk-Packing Algorithm for an Origami Magic Trick,” Proc. Third Int'l Meeting of Origami Math., Science, and Education (Origami3), pp. 17-28, 2002.
[25] M. Bern and B. Hayes, “Origami Embedding of Piecewise-Linear Two-Manifolds,” Proc. Int'l Latin Am. Symp. Theoretical Informatics (LATIN), pp. 617-629, 2008.
[26] R.J. Lang, Tree Maker 4.0: A Program for Origami Design, http://www.langorigami.com/science/treemaker TreeMkr40.pdf, 1998.
[27] E.D. Demaine, M.L. Demaine, and J.S.B. Mitchell, “Folding Flat Silhouettes and Wrapping Polyhedral Packages: New Results in Computational Origami,” Computational Geometry: Theory and Applications, vol. 16, no. 1, pp. 3-21, 2000.
[28] M. Tanaka, “Possibility and Constructive Proof through Origami,” Hyogo Univ. J., pp. 75-82, 2006.
[29] T. Tachi, “3D Origami Design Based on Tucking Molecule,” Proc. Fourth Int'l Conf. Origami in Science, Math., and Education (Origami4), pp. 259-272, 2009.
[30] Tama Software “Pepakura Designer,” http://www.tamasoft. co.jppepakura-en/, 2004.
[31] T.K. Dey, “A New Technique to Compute Polygonal Schema for 2-Manifolds with Application To Null-Homotopy Detection,” Proc. ACM Ann. Symp. Computational Geometry (SoCG '94), pp. 277-284, 1994.
[32] F. Lazarus, M. Pocchiola, G. Vegter, and A. Verroust, “Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface,” Proc. ACM Ann. Symp. Computational Geometry (SoCG '01), pp. 80-89, 2001.
[33] X. Gu, S.J. Gortler, and H. Hoppe, “Geoemtry Images,” ACM Trans. Graphics, vol. 21, no. 3, pp. 355-361, 2002.
[34] W. Tutte, “How to Draw a Graph,” Proc. London Math. Soc., pp.743-768, 1963.
[35] A. Bateman, “Computer Tools and Algorithms for Origami Tessellation Design,” Proc. Third Int'l Meeting of Origami Math., Science, and Education (Origami3), pp. 121-127, 2002.
[36] E. Gjerde, Origami Tessellations: Awe Inspiring Geometric Designs. AK Peters, 2008.
[37] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Mesh Optimization,” Proc. ACM SIGGRAPH '93, pp. 19-26, 1993.
[38] M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, “Multiresolution Analysis of Arbitrary Meshes,” Proc. ACM SIGGRAPH '95, pp. 173-182, 1995.
[39] A.W.F. Lee, D. Dobkin, W. Sweldens, and P. Schroder, “Multiresolution Mesh Morphing,” Proc. ACM SIGGRAPH '99, pp. 343-350, 1999.
[40] P. Alliez, M. Meyer, and M. Desbrun, “Interactive Geometry Remeshing,” ACM Trans. Graphics, vol. 21, no. 3, pp. 347-354, 2002.
[41] G. Peyré and L. Cohen, “Geodesic Remeshing Using Front Propagation,” Proc. Variational and Level Set Methods in Computer Vision '03, pp. 33-40, 2003.
[42] S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J.C. Hart, “Spectral Surface Quadrangulation,” ACM Trans. Graphics, vol. 25, no. 3, pp. 1057-1066, 2006.
[43] D. Cohen-Steiner, P. Alliez, and M. Desbrun, “Variational Shape Approximation,” ACM Trans. Graphics, vol. 23, no. 3, pp. 905-914, 2004.

Index Terms:
Origami, origami design, developable surface, folding, computer-aided design.
Citation:
Tomohiro Tachi, "Origamizing Polyhedral Surfaces," IEEE Transactions on Visualization and Computer Graphics, vol. 16, no. 2, pp. 298-311, March-April 2010, doi:10.1109/TVCG.2009.67
Usage of this product signifies your acceptance of the Terms of Use.