On the convergence of metric and geometric properties of polyhedral surfaces. (English) Zbl 1125.52014
The main purpose of this paper is to analyze the behavior of some geometric properties of polyhedral surfaces converging to smooth surfaces. More precisely, the authors ask: if a sequence of triangulated polyhedral surfaces in \(R^3\) converges to a smooth surface, under what conditions do metric and geometric properties such as intrinsic distance, area, mean curvature, geodesics, Laplace-Beltrami operators converge too? It is demonstrated that if a sequence of polyhedral surfaces converges to a smooth surface in Hausdorff distance then the following conditions are equivalent: (i) convergence of normal fields, (ii) convergence of metric tensors, (ii) convergence of area, (iv) convergence of Laplace-Beltrami operators.
The convergence in (i)–(iv) is viewed in some particular senses; for instance, the Laplace-Beltrami operators are regarded as bounded linear maps of certain Sobolev spaces, the convergence is treated in some appropriate operator norm.
Besides, it is shown that if a sequence of polyhedral surfaces converges to a smooth surface in Hausdorff distance together with convergence of their normal fields, then uniform convergence of geodesics on compact sets, convergence of solutions to the Dirichlet problem and convergence of mean curvature vectors hold too. As consequence, if a sequence of polyhedral surfaces, which are minimal in the sense of Pinkall-Polthier, converges to a smooth surface in Hausdorff distance such that their normal fields converge, then the limit smooth surface is minimal in the classical sense.
Thus this paper provides a foundation to the modern discretization theory developed recently by Bobenko, Pinkall, Poltier, Hoffmann, Morvan and others.
The convergence in (i)–(iv) is viewed in some particular senses; for instance, the Laplace-Beltrami operators are regarded as bounded linear maps of certain Sobolev spaces, the convergence is treated in some appropriate operator norm.
Besides, it is shown that if a sequence of polyhedral surfaces converges to a smooth surface in Hausdorff distance together with convergence of their normal fields, then uniform convergence of geodesics on compact sets, convergence of solutions to the Dirichlet problem and convergence of mean curvature vectors hold too. As consequence, if a sequence of polyhedral surfaces, which are minimal in the sense of Pinkall-Polthier, converges to a smooth surface in Hausdorff distance such that their normal fields converge, then the limit smooth surface is minimal in the classical sense.
Thus this paper provides a foundation to the modern discretization theory developed recently by Bobenko, Pinkall, Poltier, Hoffmann, Morvan and others.
Reviewer: Vasyl Gorkaviy (Kharkov)
Software:
Surface EvolverReferences:
| [1] | Aleksandrov, A.D., Zalgaller, V.A.: Intrinsic Geometry of Surfaces, vol. 15 of Translation of Mathematical Monographs. AMS (1967) · Zbl 0146.44103 |
| [2] | Banchoff, T., Critical points and curvature for embedded polyhedra, J. Differ. Geom., 1, 257-268 (1967) · Zbl 0164.22903 |
| [3] | Bobenko, A. I.; Hoffmann, T.; Springborn, B. A., Minimal surfaces from circle patterns: geometry from combinatorics, Ann. Math., 164, 1, 231-264 (2006) · Zbl 1122.53003 · doi:10.4007/annals.2006.164.231 |
| [4] | Bobenko, A. I.; Pinkall, U., Discrete isothermic surfaces, J. Reine Angew. Math., 475, 187-208 (1996) · Zbl 0845.53005 |
| [5] | Braess, D., Finite Elements Theory, Fast Solvers and Applications in Solid Mechanics (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 0976.65099 |
| [6] | Brakke, K., The surface evolver, Exp. Math., 1, 2, 141-165 (1992) · Zbl 0769.49033 |
| [7] | Brehm, U.; Kühnel, W., Smooth approximation of polyhedral surfaces regarding curvatures, Geom. Dedicata, 12, 435-461 (1982) · Zbl 0483.53046 · doi:10.1007/BF00147585 |
| [8] | Cheeger, J.; Müller, W.; Schrader, R., Curvature of piecewise flat metrics, Comm. Math. Phys., 92, 405-454 (1984) · Zbl 0559.53028 · doi:10.1007/BF01210729 |
| [9] | Cohen-Steiner, D., Morvan, J.-M.: Restricted Delaunay triangulations and normal cycle. In: Proceedings of the Nineteenth Annual Symposium on Computational Geometry, pp. 312-321 (2003) · Zbl 1422.65051 |
| [10] | Duffin, R. J., Distributed and lumped networks, J. Math. Mech., 8, 793-825 (1959) · Zbl 0092.20501 |
| [11] | Dziuk G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Partial Differential Equations and Calculus of Variations, vol. 1357 of Lec. Notes Math., pp. 142-155. Springer (1988) · Zbl 0663.65114 |
| [12] | Dziuk, G., An algorithm for evolutionary surfaces, Num. Math., 58, 603-611 (1991) · Zbl 0714.65092 · doi:10.1007/BF01385643 |
| [13] | Dziuk, G.; Hutchinson, J. E., The discrete Plateau problem: convergence results, Math. Comput., 68, 519-546 (1999) · Zbl 1043.65126 · doi:10.1090/S0025-5718-99-01026-1 |
| [14] | Federer, H., Curvature measures, Trans. Am. Math., 93, 418-491 (1959) · Zbl 0089.38402 · doi:10.2307/1993504 |
| [15] | Fu, J. H.G., Convergence of curvatures in secant approximations, J. Differ. Geom., 37, 177-190 (1993) · Zbl 0794.53044 |
| [16] | Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations. Springer (1977) · Zbl 0361.35003 |
| [17] | Goodman-Strauss, C., Sullivan, J.M.: Cubic Polyhedra, http://arXiv.org.math/0205145 (2002) · Zbl 1048.52008 |
| [18] | Gromov M.: Structures métriques pour les variétés riemanniennes. Textes Mathématiques. Cedic/Fernand Nathan (1981) · Zbl 0509.53034 |
| [19] | Große-Brauckmann, K.; Polthier, K., Compact constant mean curvature surfaces with low genus, Exper. Math., 6, 1, 13-32 (1997) · Zbl 0898.53009 |
| [20] | Karcher, H.; Polthier, K., Construction of triply periodic minimal surfaces, Phil. Trans. Royal Soc. Lond., 354, 2077-2104 (1996) · Zbl 0869.53006 · doi:10.1098/rsta.1996.0093 |
| [21] | Meek, D. S.; Walton, D. J., On surface normal and Gaussian curvature approximations given data sampled from a smooth surface, Comput. Aided Geom. Design, 17, 6, 521-543 (2000) · Zbl 0945.68174 · doi:10.1016/S0167-8396(00)00006-6 |
| [22] | Mercat, C., Discrete Riemann surfaces and the Ising model, Commun. Math. Phys., 218, 1, 177-216 (2001) · Zbl 1043.82005 · doi:10.1007/s002200000348 |
| [23] | Morvan, J.-M.: Generalized curvatures. Preprint. · Zbl 1149.53001 |
| [24] | Morvan, J.-M.; Thibert, B., Approximation of the normal vector field and the area of a smooth surface, Discrete Comput. Geom., 32, 3, 383-400 (2004) · Zbl 1086.53003 · doi:10.1007/s00454-004-1096-4 |
| [25] | Oberknapp B., Polthier K.: An algorithm for discrete constant mean curvature surfaces. In: Hege, H.-C., Polthier, K. (eds.), Visualization and Mathematics. Springer (1997) |
| [26] | Pinkall, U.; Polthier, K., Computing discrete minimal surfaces and their conjugates, Exp. Math., 2, 15-36 (1993) · Zbl 0799.53008 |
| [27] | Polthier K.: Computational aspects of discrete minimal surfaces. In: Hoffman, D. (ed.), Global Theory of Minimal Surfaces. CMI/AMS (2005) · Zbl 1100.53014 |
| [28] | Polthier, K.: Unstable periodic discrete minimal surfaces. In: Hildebrandt, S., Karcher, H. (eds.) Geometric Analysis and Nonlinear Partial Differential Equations, pp. 127-143. Springer Verlag (2002) |
| [29] | Polthier, K.; Rossman, W., Index of discrete constant mean curvature surfaces, J. Reine Angew. Math., 549, 47-77 (2002) · Zbl 1014.53005 |
| [30] | Pozzi, P., The discrete Douglas problem: convergence results, IMA J. Num. Ana., 25, 42, 337-378 (2005) · Zbl 1075.65093 · doi:10.1093/imanum/drh019 |
| [31] | Reshetnyak Y.G.: Geometry IV. Non-regular Riemannian geometry. In: Encyclopaedia of Mathematical Sciences, vol. 70, Springer-Verlag, pp. 3-164 (1993) · Zbl 0777.00061 |
| [32] | Rossman, W., Infinite periodic discrete minimal surfaces without self-intersections, Balkan J. Geom. Appl., 10, 106-128 (2005) · Zbl 1119.49033 |
| [33] | Schramm, O., Circle patterns with the combinatorics of the square grid, Duke Math. J., 86, 347-389 (1997) · Zbl 1053.30525 · doi:10.1215/S0012-7094-97-08611-7 |
| [34] | Schwarz H.A.: Sur une définition erronée de l’aire d’une surface courbe. In: Gesammelte Mathematische Abhandlungen, vol. 2. Springer-Verlag, pp. 309-311 (1890) |
| [35] | Stone, D. A., Geodesics in piecewise linear manifolds, Trans. AMS, 215, 1-44 (1976) · Zbl 0328.53033 · doi:10.2307/1999713 |
| [36] | Thurston W.P. The Geometry and Topology of Three-manifolds. http://www.msri.org/publications/books/gt3m (1980) |
| [37] | Troyanov, M., Les surfaces euclidiennes à singularités coniques, Ens. Math., 32, 79-94 (1986) · Zbl 0611.53035 |
| [38] | Wardetzky M.: Discrete differential operators on polyhedral surfaces—convergence and approximation. Ph.D. thesis, Free University Berlin (2006) |
| [39] | Xu, G., Discrete Laplace-Beltrami operators and their convergence, Comput. Aided Geom. Design, 21, 8, 767-784 (2004) · Zbl 1069.58500 · doi:10.1016/j.cagd.2004.07.007 |
| [40] | Xu, G., Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces, Comput. Aided Geom. Design, 23, 2, 193-207 (2006) · Zbl 1083.65024 · doi:10.1016/j.cagd.2005.07.002 |
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