Abstract
Curvature-based surface energies are frequently used in mathematics, physics, thin plate and shell engineering, and membrane chemistry and biology studies. Invariance under rotations and shifts makes curvature-based energies very attractive for modeling various phenomena. In computer-aided geometric design, the Willmore surfaces and the so-called minimum variation surfaces (MVS) are widely used for shape modeling purposes. The Willmore surfaces are invariant w.r.t conformal transformations (Möbius or conformal invariance), and studied thoroughly in differential geometry and related disciplines. In contrast, the minimum variation surfaces are not conformal invariant. In this paper, we suggest a simple modification of the minimum variation energy and demonstrate that the resulting modified MVS enjoy Möbius invariance (so we call them conformal-invariant MVS or, shortly, CI-MVS).We also study connections of CI-MVS with the cyclides of Dupin. In addition, we consider several other conformal-invariant curve and surface energies involving curvatures and curvature derivatives. In particular, we show how filtering with a conformal-invariant curve energy can be used for detecting salient subsets of the principal curvature extremum curves used by Hosaka and co-workers for shape quality inspection purposes.
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References
Euler L. Additamentum ‘de curvis elasticis’. In: Methodus Inveniendi Lineas Curvas Maximi Minimive Probprietate Gaudentes, Lausanne. 1744
Levien R. The Elastica: A Mathematical History. Technical Report UCB/EECS-2008-103. EECS Department, University of California, Berkeley, 2008
Willmore T J. Note on embedded surfaces. Anal Stunt Ale Univ Sect Ia, 1965, 11B: 493–496
Willmore T J. Surfaces in conformal geometry. Ann Glob Anal Geom, 2000, 18: 255–264
White J H. A global invariant of conformal mappings in space. Proc Amer Math Soc, 1973, 38: 162–164
Hertrich-Jeromin U. Introduction to Möbius Differential Geometry. Cambridge: Cambridge University Press, 2003
Gravesen J, Upgstrup M. Constructing invariant fairness measures for surfaces. Adv Comput Math, 2002, 17: 67–88
Guven J. Conformally invariant bending energy for hypersurfaces. J Phys A-Math Theor, 2005, 38: 7943–7956
Bohle C, Peters G P, Pinkall U. Constrained Willmore surfaces. Calc Var Partial Differ Equ, 2008, 32: 263–277
Riviére T. Analysis aspects of Willmore surfaces. Invent Math, 2008, 174: 1–45
Marques F C, Neves A. Min-max theory and the Willmore conjecture. http://arxiv.org/abs/1202.6036, 2012
Bobenko A I, Schröder P. Discrete Willmore flow. In: Eurographics Symposium on Geometry Processing. Switzerland: Eurographics Association Aire-la-Ville, 2005. 101–110
Wardetzky M, Bergou M, Harmon D, et al. Discrete quadratic curvature energies. Comput Aided Geom Des, 2007, 24: 499–518
Moreton H P, Séquin C H. Surface design with minimum energy networks. In: Proceedings of ACM Symposium on Solid Modeling Foundations and CAD/CAM Applications. New York: ACM, 1991. 291–301
Moreton H P, Séquin C H. Functional optimization for fair surface design In: Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques. New York: ACM, 1992. 167–176
Joshi P, Séquin C. Energy minimizers for curvature-based surface functionals. Comput-Aided Des Appl, 2007, 4: 607–618
Xu G, Zhang Q. Minimal mean-curvature-variation surfaces and their applications in surface modeling. In: Proceedings of International Conference on Geometric Modeling and Processing. Berlin: Springer-Verlag, 2006. 357–370
Yoshizawa S, Belyaev A. Conformally invariant energies and minimum variation surfaces. In: Proceedings of Asian Conference on Design and Digital Engineering, Niseko, 2012. 20
Mann S, Dorst L. Geometric algebra: a computational framework for geometrical applications, part 2. IEEE Comput Graph Appl, 2002, 22: 58–67
Wareham R, Cameron J, Lasenby J. Applications of conformal geometric algebra in computer vision and graphics In: Proceedings of International Conference on Computer Algebra and Geometric Algebra with Applications. Berlin: Springer-Verlag, 2005. 329–349
Gu X, Yau S T. Surface classification using conformal structures. In: Proceedings of IEEE International Conference on Computer Vision. Washington DC: IEEE, 2003. 701–708
Mehra R, Tripathi P, Sheffer A, et al. Visibility of noisy point cloud data. Comput Graph, 2010, 34: 219–230
Bastl B, Jüttler B, Lávička M, et al. Curves and surfaces with rational chord length parameterization. Comput Aided Geom Des, 2012, 29: 231–241
Weatherburn C E. Differential Geometry of Three Dimensions, vol. I. Cambridge: Cambridge University Press, 1927
Struik D J. Lectures on Classical Differential Geometry. 2nd ed. New York: Dover Publications, 1988
Chandru V, Dutta D, Hoffmann C M. On the geometry of Dupin cyclides. Vis Comput, 1989, 5: 277–290
Foufou S, Garnier L. Dupin cyclide blends between quadric surfaces for shape modeling. Comput Graph Forum, 2004, 23: 321–330
Hosaka M. Modeling of Curves and Surfaces in CAD/CAM. Berlin: Springer, 1992
Yoshizawa S, Belyaev A, Yokota H, et al. Fast, robust, and faithful methods for detecting crest lines on meshes. Comput Aided Geom Des, 2008, 25: 545–560
Nealen A, Igarashi T, Sorkine O, et al. Fibermesh: designing freeform surfaces with 3-D curves. ACM Trans Graph, 2007, 26: 41
Wang C P. Surfaces in Möbius geometry. Nagoya Math J, 1992, 125: 53–72
Monga O, Benayoun S, Faugeras O D. From partial derivatives of 3-D density images to ridge lines. In: Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Champaign, 1992. 354–359
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Yoshizawa, S., Belyaev, A. Möbius-invariant curve and surface energies and their applications. Sci. China Inf. Sci. 56, 1–10 (2013). https://doi.org/10.1007/s11432-013-4997-0
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DOI: https://doi.org/10.1007/s11432-013-4997-0