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Issue No.03 - May/June (2009 vol.15)
pp: 1
Kai Tang , Mech. Eng. Dept., Hong Kong Univ. of Sci. & Technol., Hong Kong
Ming Chen , Mech. Eng. Dept., Hong Kong Univ. of Sci. & Technol., Hong Kong
We present a new algorithm for finding a most "developable" smooth mesh surface to interpolate a given set of arbitrary points or space curves. Inspired by the recent progress in mesh editing that employs the concepts of preserving the Laplacian coordinates and handle-based shape editing, we formulate the interpolation problem as a mesh deformation process that transforms an initial developable mesh surface, such as a planar figure, to a final mesh surface that interpolates the given points and/or curves. During the deformation, the developability of the intermediate mesh is maintained by means of preserving the zero-valued Gaussian curvature on the mesh. To treat the high nonlinearity of the geometric constrains owing to the preservation of Gaussian curvature, we linearize those nonlinear constraints using Taylor expansion and eventually construct a sparse and over-determined linear system which is subsequently solved by a robust least-squares solution. By iteratively performing this procedure, the initial mesh is gradually and smoothly "dragged" to the given points and/or curves. The initial experimental data has shown some promising aspects of the proposed algorithm as a general quasi-developable surface interpolation tool.
Interpolation, Laplace equations, Iterative algorithms, Surface treatment, Robustness, Clothing, Costs, Minimization methods,Developable surface, Least squares methods, Computer-aided design, Surface fitting
Kai Tang, Ming Chen, "Quasi-Developable Mesh Surface Interpolation via Mesh Deformation", IEEE Transactions on Visualization & Computer Graphics, vol.15, no. 3, pp. 1, May/June 2009, doi:10.1109/TVCG.2008.192
[1] A. Hill, “Constructivism—The European Phenomenon,” Studio Int'l, vol. 171, pp. 140-147, 1966.
[2] T. Akgun, A. Koman, and E. Akleman, “Developable Sculptures of Ilhan Koman,” Proc. Bridges, 2006.
[3] O.K.-C. Au, “Differential Techniques for Scalable and Interactive Mesh Editing,” PhD dissertation, Computer Science, HKUST, 2007.
[4] O.K.-C. Au, H. Fu, C.-L. Tai, and D. Cohen-Or, “Handle-Aware Isolines for Scalable Shape Editing,” ACM Trans. Graphics, vol. 26, no. 3, pp. 83:1-83:10, 2007.
[5] M. doCarmo, Differential Geometry of Curves and Surfaces. Prentice Hall, 1976.
[6] H. Pottmann and J. Wallner, Computational Line Geometry. Springer, 2001.
[7] K. Rose, A. Sheffer, J. Wither, M.-P. Cani, and B. Thibert, “Developable Surfaces from Arbitrary Sketched Boundaries,” Proc. Fifth Eurographics Symp. Geometry Processing (SGP '07), pp.163-172, 2007.
[8] M. Peternell, “Developable Surface Fitting to Point Clouds,” Computer Aided Geometric Design, vol. 21, no. 8, pp. 785-803, 2004.
[9] H.-Y. Chen, I.-K. Lee, S. Leopoldseder, H. Pottmann, T. Randrup, and J. Wallner, “On Surface Approximation Using Developable Surfaces,” Graphical Models and Image Processing, vol. 61, no. 2, pp.110-124, 1999.
[10] G. Aumann, “Interpolation with Developable Bézier Patches,” Computer Aided Geometric Design, vol. 8, no. 5, pp. 409-420, 1991.
[11] G. Aumann, “A Simple Algorithm for Designing Developable Bézier Surfaces,” Computer Aided Geometric Design, vol. 20, nos. 8/9, pp. 601-619, 2003.
[12] G. Aumann, “Degree Elevation and Developable Bézier Surfaces,” Computer Aided Geometric Design, vol. 21, no. 7, pp. 661-670, 2004.
[13] C.H. Chu and C.H. Séquin, “Developable Bézier Patches: Properties and Design,” Computer-Aided Design, vol. 34, no. 7, pp. 511-527, 2002.
[14] J. Lang and O. Röschel, “Developable (1, n)-Bézier Surfaces,” Computer Aided Geometric Design, vol. 9, no. 4, pp. 291-298, 1992.
[15] H. Pottmann and G.E. Farin, “Developable Rational Bézier and B-spline Surfaces,” Computer Aided Geometric Design, vol. 12, no. 5, pp. 513-531, 1995.
[16] P. Bo and W. Wang, “Geodesic-Controlled Developable Surfaces for Modeling Paper Bending,” Computer Graphics Forum/Proc. Ann. Conf. European Assoc. for Computer Graphics (Eurographics '07), vol. 26, no. 3, pp. 329-338, 2007.
[17] W. Frey, “Boundary Triangulations Approximating Developable Surfaces That Interpolate a Closed Space Curve,” Int'l J. Foundations of Computer Science, vol. 13, pp. 285-302, 2002.
[18] W. Frey, “Modeling Buckled Developable Surface by Triangulation,” Computer-Aided Design, vol. 36, no. 4, pp. 299-313, 2004.
[19] C. Wang and K. Tang, “Optimal Boundary Triangulations of an Interpolating Ruled Surface,” ASME J. Computing and Information Science in Eng., vol. 5, no. 4, pp. 291-301, 2005.
[20] C.C.L. Wang and K. Tang, “Achieving Developability of a Polygonal Surface by Minimum Deformation: A Study of Global and Local Optimization Approaches,” The Visual Computer, vol. 20, nos. 8-9, pp. 521-539, 2004.
[21] 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, 2005.
[22] I. Shatz, A. Tal, and G. Leifman, “Paper Craft Models from Meshes,” The Visual Computer, vol. 22, no. 9, pp. 825-834, 2006.
[23] D. Julius, V. Kraevoy, and A. Sheffer, “D-Charts: Quasi-Developable Mesh Segmentation,” Computer Graphics Forum, vol. 24, no. 3, pp. 581-590, 2005.
[24] Y. Liu, H. Pottmann, J. Wallner, Y.-L. Yang, and W. Wang, “Geometric Modeling with Conical Meshes and Developable Surfaces,” ACM Trans. Graphics (Proc. ACM SIGGRAPH '06), vol. 25, no. 3, pp. 681-689, 2006.
[25] E. Magid, O. Soldea, and E. Rivlin, “A Comparison of Gaussian and Mean Curvature Estimation Methods on Triangular Meshes of Range Image Data,” Computer Vision and Image Understanding, vol. 107, no. 3, pp. 139-159, 2007.
[26] D. Terzopoulost, J. Platt, A. Barr, and K. Fleischert, “Elastically Deformable Models,” Proc. ACM SIGGRAPH, 1987.
[27] Y.J. Liu, K. Tang, and A. Joneija, “Modeling Dynamic Developable Meshes by Hamilton Principle,” Computer-Aided Design, vol. 39, no. 9, pp. 719-731, 2007.
[28] S.T. Tan, T.N. Wong, Y.F. Zhao, and W.J. Chen, “A Constrained Finite Element Method for Modeling Cloth Deformation,” The Visual Computer, vol. 15, no. 2, pp. 90-99, 1999.
[29] P. Volino, M. Courchesne, and N. Magnenat-Thalmann, “Versatile and Efficient Techniques for Simulating Cloth and Other Deformable Objects,” Proc. ACM SIGGRAPH, 1995.
[30] P. Volino and N. Magnenat-Thalmann, Virtual Clothing: Theory and Practice. Springer, 2000.
[31] P. Volino and N. Magnenat-Thalmann, “An Evolving System for Simulating Clothes on Virtual Actors,” IEEE Computer Graphics and Applications, vol. 16, no. 5, pp. 42-51, 1996.
[32] P. Volino and N. Magnenat-Thalmann, “Stop-and-Go Cloth Draping,” The Visual Computer, vol. 23, no. 8, pp. 669-677, 2007.
[33] R. Goldenthal, D. Harmon, R. Fattal, M. Bercovier, and E. Grinspun, “Efficient Simulation of Inextensible Cloth,” ACM Trans. Graphics (Proc. ACM SIGGRAPH '07), vol. 26, no. 3, 2007.
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