Curved folding

ABSTRACT

Fascinating and elegant shapes may be folded from a single planar sheet of material without stretching, tearing or cutting, if one incorporates curved folds into the design. We present an optimization-based computational framework for design and digital reconstruction of surfaces which can be produced by curved folding. Our work not only contributes to applications in architecture and industrial design, but it also provides a new way to study the complex and largely unexplored phenomena arising in curved folding.

REFERENCES

• 1: Günter Aumann, Degree elevation and developable Bézier surfaces, Computer Aided Geometric Design, v.21 n.7, p.661-670, September 2004 [doi>10.1016/j.cagd.2004.04.007]
• 2: Bo, P., and Wang, W. 2007. Geodesic-controlled developable surfaces for modeling paper bending. Comp. Graphics Forum 26, 3, 365--374.
• 3: Bobenko, A., and Suris, Y., 2005. Discrete differential geometry. Consistency as integrability. Preprint, http://arxiv.org/abs/math.DG/0504358.
• 4: Mario Botsch , Mark Pauly , Markus Gross , Leif Kobbelt, PriMo: coupled prisms for intuitive surface modeling, Proceedings of the fourth Eurographics symposium on Geometry processing, June 26-28, 2006, Cagliari, Sardinia, Italy
• 5: Cerda, E., Chaieb, S., Melo, F., and Mahadevan, L. 1999. Conical dislocations in crumpling. Nature 401, 46--49.
• 6: Cerda, E., Mahadevan, L., and Pasini, J. M. 2004. The elements of draping. Proc. Nat. Acad. Sciences 101, 7, 1806--1810.
• 7: Chu, C. H., and Sequin, C. 2002. Developable Bézier patches: properties and design. Comp.-Aided Design 34, 511--528.
• 8: Timothy A. Davis , William W. Hager, Multiple-Rank Modifications of a Sparse Cholesky Factorization, SIAM Journal on Matrix Analysis and Applications, v.22 n.4, p.997-1013, 2000 [doi>10.1137/S0895479899357346]
• 9: Erik D. Demaine , Joseph O'Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, New York, NY, 2007
• 10: Desbrun, M., Polthier, K., and Schröder, P. 2005. Discrete Differential Geometry. Siggraph Course Notes.
• 11: Do Carmo, M. 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall.
• 12: Frey, W. 2004. Modeling buckled developable surfaces by triangulation. Comp.-Aided Design 36, 4, 299--313.
• 13: Glaeser, G., and Gruber, F. 2007. Developable surfaces in contemporary architecture. J. of Math. and the Arts 1, 1--15.
• 14: D. A. Huffman, Curvature and Creases: A Primer on Paper, IEEE Transactions on Computers, v.25 n.10, p.1010-1019, October 1976 [doi>10.1109/TC.1976.1674542]
• 15: Julius, D., Kraevoy, V., and Sheffer, A. 2005. D-charts: Quasi-developable mesh segmentation. Computer Graphics Forum 24, 3, 581--590. Proc. Eurographics 2005.
• 16: Yannick L. Kergosien , Hironobu Gotoda , Tosiyasu L. Kunii, Bending and Creasing Virtual Paper, IEEE Computer Graphics and Applications, v.14 n.1, p.40-48, January 1994 [doi>10.1109/38.250917]
• 17: Martin Kilian , Niloy J. Mitra , Helmut Pottmann, Geometric modeling in shape space, ACM Transactions on Graphics (TOG), v.26 n.3, July 2007
• 18: Yang Liu , Helmut Pottmann , Johannes Wallner , Yong-Liang Yang , Wenping Wang, Geometric modeling with conical meshes and developable surfaces, ACM Transactions on Graphics (TOG), v.25 n.3, July 2006
• 19: Fady Massarwi , Craig Gotsman , Gershon Elber, Papercraft Models using Generalized Cylinders, Proceedings of the 15th Pacific Conference on Computer Graphics and Applications, p.148-157, October 29-November 02, 2007 [doi>10.1109/PG.2007.46]
• 20: Jun Mitani , Hiromasa Suzuki, Making papercraft toys from meshes using strip-based approximate unfolding, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004
• 21: Yutaka Ohtake , Alexander Belyaev , Hans-Peter Seidel, Ridge-valley lines on meshes via implicit surface fitting, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004
• 22: Pérez, F., and Suárez, J. A. 2007. Quasi-developable B-spline surfaces in ship hull design. Comp.-Aided Design 39, 853--862.
• 23: Martin Peternell, Developable surface fitting to point clouds, Computer Aided Geometric Design, v.21 n.8, p.785-803, October 2004 [doi>10.1016/j.cagd.2004.07.008]
• 24: Computational Line Geometry, Springer-Verlag New York, Inc., Secaucus, NJ, 2001
• 25: Helmut Pottmann , Qi-Xing Huang , Yong-Liang Yang , Shi-Min Hu, Geometry and Convergence Analysis of Algorithms for Registration of 3D Shapes, International Journal of Computer Vision, v.67 n.3, p.277-296, May 2006 [doi>10.1007/s11263-006-5167-2]
• 26: Pottmann, H., Asperl, A., Hofer, M., and Kilian, A. 2007. Architectural Geometry. Bentley Institute Press.
• 27: Kenneth Rose , Alla Sheffer , Jamie Wither , Marie-Paule Cani , Boris Thibert, Developable surfaces from arbitrary sketched boundaries, Proceedings of the fifth Eurographics symposium on Geometry processing, July 04-06, 2007, Barcelona, Spain
• 28: Sauer, R. 1970. Differenzengeometrie. Springer.
• 29: Idan Shatz , Ayellet Tal , George Leifman, Paper craft models from meshes, The Visual Computer: International Journal of Computer Graphics, v.22 n.9, p.825-834, September 2006 [doi>10.1007/s00371-006-0067-6]
• 30: Alla Sheffer , Emil Praun , Kenneth Rose, Mesh parameterization methods and their applications, Foundations and Trends® in Computer Graphics and Vision, v.2 n.2, p.105-171, January 2006 [doi>10.1561/0600000011]
• 31: Shelden, D. 2002. Digital surface representation and the constructibility of Gehry's architecture. PhD thesis, M.I.T.
• 32: Subag, J., and Elber, G. 2006. Piecewise developable surface approximation of general NURBS surfaces with global error bounds. In Proc. Geometric Modeling and Processing. 143--156.
• 33: Charlie C. L. Wang , Kai Tang, Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches, The Visual Computer: International Journal of Computer Graphics, v.20 n.8-9, p.521-539, November 2004 [doi>10.1007/s00371-004-0256-0]
• 34: Charlie C. L. Wang, Towards flattenable mesh surfaces, Computer-Aided Design, v.40 n.1, p.109-122, January, 2008 [doi>10.1016/j.cad.2007.06.001]
• 35: Wertheim, M. 2004. Cones, Curves, Shells, Towers: He Made Paper Jump to Life. The New York Times, June 22.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Geometric algorithms, languages, and systems

Additional Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations

Keywords:
architectural geometry, computational differential geometry, computational origami, curved fold, developable surface, digital reconstruction, folding, industrial design, isometry