Helmut Pottmann
ABSTRACT
Geodesic curves in surfaces are not only minimizers of distance, but they are also the curves of zero geodesic (sideways) curvature. It turns out that this property makes patterns of geodesics the basic geometric entity when dealing with the cladding of a freeform surface with wooden panels which do not bend sideways. Likewise a geodesic is the favored shape of timber support elements in freeform architecture, for reasons of manufacturing and statics. Both problem areas are fundamental in freeform architecture, but so far only experimental solutions have been available. This paper provides a systematic treatment and shows how to design geodesic patterns in different ways: The evolution of geodesic curves is good for local studies and simple patterns; the level set formulation can deal with the global layout of multiple patterns of geodesics; finally geodesic vector fields allow us to interactively model geodesic patterns and perform surface segmentation into panelizable parts.
- Blaschke, W., and Bol, G. 1938. Geometrie der Gewebe. Springer.Google Scholar
- Caselles, V., Kimmel, R., and Sapiro, G. 1997. Geodesic active contours. Int. J. Comput. Vision 22, 1, 61--79. Google ScholarDigital Library
- Chen, J., and Han, Y. 1996. Shortest paths on a polyhedron. I. Computing shortest paths. Int. J. Comput. Geom. Appl. 6, 127--144.Google ScholarCross Ref
- Chen, Y., Davis, T. A., Hager, W. W., and Rajamanickam, S. 2008. Algorithm 887: Cholmod, supernodal sparse Cholesky factorization and update/downdate. ACM Trans. Math. Softw. 35, 3, #22, 1--14. Google ScholarDigital Library
- Chern, S. S. 1982. Web geometry. Bull. Amer. Math. Soc. 6, 1--8.Google ScholarCross Ref
- do Carmo, M. 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall.Google Scholar
- do Carmo, M. 1992. Riemannian Geometry. Birkhäuser.Google Scholar
- Geman, S., and McClure, D. E. 1987. Statistical methods for tomographic image reconstruction. Bull. Inst. Internat. Statist. 52, 4, 5--21.Google Scholar
- Graf, H., and Sauer, R. 1924. Über dreifache Geradensysteme. Sitz. Bayer. Akad. Math.-nat. Abt., 119--156.Google Scholar
- Huang, Q., Wicke, M., Adams, B., and Guibas, L. 2009. Shape decomposition using modal analysis. Comput. Graph. Forum 28, 2, 407--416.Google ScholarCross Ref
- Julius, D., Kraevoy, V., and Sheffer, A. 2005. D-charts: Quasi-developable mesh segmentation. Comput. Graph. Forum 24, 3, 581--590.Google ScholarCross Ref
- Kälberer, F., Nieser, M., and Polthier, K. 2007. Quad-Cover -- surface parameterization using branched coverings. Comput. Graph. Forum 26, 3, 375--384.Google ScholarCross Ref
- Kimmel, R., and Sethian, J. A. 1998. Computing geodesic paths on manifolds. PNAS 95, 8431--8435.Google ScholarCross Ref
- Levin, A., Fergus, R., Durand, F., and Freeman, W. T. 2007. Image and depth from a conventional camera with a coded aperture. ACM Trans. Graphics 26, 3, #70, 1--9. Google ScholarDigital Library
- Madsen, K., Nielsen, H. B., and Tingleff, O., 2004. Methods for non-linear least squares problems. Lecture Notes. http://http://www.imm.dtu.dk/courses/02611/nllsq.pdf.Google Scholar
- Mayrhofer, K. 1931. Sechseckgewebe aus Geodätischen. Monatsh. Math. Phys. 38, 401--404.Google ScholarCross Ref
- Müller, H. R. 1941. Über die Striktionslinien von Kurvenscharen. Monatsh. Math. Phys. 50, 101--110.Google ScholarCross Ref
- Natterer, J., Burger, N., and Müller, A. 2002. The roof structure "Expodach" at the world exhibition Hannover. In Proc. 5th Intl. Conf. Space Structures, 185--193.Google Scholar
- Osher, S. J., and Fedkiw, R. P. 2002. Level Set Methods and Dynamic Implicit Surfaces. Springer Verlag.Google Scholar
- Pauly, M., Keiser, R., and Gross, M. 2003. Multi-scale feature extraction on point-sampled surfaces. Comput. Graph. Forum 22, 3, 281--290.Google ScholarCross Ref
- Pirazzi, C., and Weinand, Y. 2006. Geodesic lines on free-form surfaces: optimized grids for timber rib shells. In Proc. World Conference on Timber Engineering. 7pp.Google Scholar
- Polthier, K., and Schmies, M. 1998. Straightest geodesics on polyhedral surfaces. In Mathematical Visualization, Springer, H.-C. Hege and K. Polthier, Eds., 391--409.Google Scholar
- Pottmann, H., Hofer, M., and Kilian, A., Eds. 2008. Advances in Architectural Geometry. Proc. of Vienna conference.Google Scholar
- Pottmann, H., Schiftner, A., Bo, P., Schmiedhofer, H., Wang, W., Baldassini, N., and Wallner, J. 2008. Freeform surfaces from single curved panels. ACM Trans. Graphics 27, 3, #76, 1--10. Google ScholarDigital Library
- Shelden, D. 2002. Digital surface representation and the constructibility of Gehry's architecture. PhD thesis, M.I.T.Google Scholar
- Spuybroek, L. 2004. NOX: Machining Architecture. Thames & Hudson.Google Scholar
- Sutton, D. 2007. Islamic Design: A Genius for Geometry. Walker Publ. Comp.Google Scholar
- Yamauchi, H., Gumhold, S., Zayer, R., and Seidel, H. P. 2005. Mesh segmentation driven by Gaussian curvature. Vis. Computer 21, 659--668.Google ScholarCross Ref
Index Terms
- Geodesic patterns
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