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Curvatures of Smooth and Discrete Surfaces

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Discrete Differential Geometry

Part of the book series: Oberwolfach Seminars ((OWS,volume 38))

Abstract

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss-Bonnet theorem and the mean-curvature force balance equation.

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References

  1. Thomas F. Banchoff, Critical points and curvature for embedded polyhedra, J. Differential Geometry 1 (1967), 245–256.

    MATH  MathSciNet  Google Scholar 

  2. _____, Critical points and curvature for embedded polyhedral surfaces, Amer. Math. Monthly 77 (1970), 475–485.

    Article  MATH  MathSciNet  Google Scholar 

  3. Philip L. Bowers and Monica K. Hurdal, Planar conformal mappings of piecewise flat surfaces, Visualization and Mathematics III (H.-C. Hege and K. Polthier, eds.), Springer, 2003, pp. 3–34.

    Google Scholar 

  4. Richard L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82 (1975), 246–251.

    Article  MATH  MathSciNet  Google Scholar 

  5. Ulrich Brehm and Wolfgang Kühnel, Smooth approximation of polyhedral surfaces regarding curvatures, Geom. Dedicata 12:4 (1982), 435–461.

    Article  MATH  MathSciNet  Google Scholar 

  6. Thomas F. Banchoff and Wolfgang Kühnel, Tight submanifolds, smooth and polyhedral, Tight and taut submanifolds (Berkeley, CA, 1994), Math. Sci. Res. Inst. Publ., vol. 32, Cambridge Univ. Press, Cambridge, 1997, pp. 51–118.

    Google Scholar 

  7. Alexander I. Bobenko, A conformal energy for simplicial surfaces, Combinatorial and computational geometry, Math. Sci. Res. Inst. Publ., vol. 52, Cambridge Univ. Press, Cambridge, 2005, pp. 135–145.

    Google Scholar 

  8. _____, Surfaces from circles, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 3–35; arXiv.org/0707.1318.

    Google Scholar 

  9. Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978.

    MATH  Google Scholar 

  10. _____, The Surface Evolver, Experiment. Math. 1:2 (1992), 141–165.

    Google Scholar 

  11. _____, The Surface Evolver, www.susqu.edu/brakke/evolver, online documentation, accessed September 2007.

    Google Scholar 

  12. Alexander I. Bobenko and Boris A. Springborn, A discrete Laplace-Beltrami operator for simplicial surfaces, preprint, 2005, arXiv:math.DG/0503219.

    Google Scholar 

  13. David Cohen-Steiner and Jean-Marie Morvan, Restricted Delaunay triangulations and normal cycle, SoCG’ 03: Proc. 19th Sympos. Comput. Geom., ACM Press, 2003, pp. 312–321.

    Google Scholar 

  14. _____, Second fundamental measure of geometric sets and local approximation of curvatures, J. Differential Geom. 74:3 (2006), 363–394.

    MATH  MathSciNet  Google Scholar 

  15. Klaus Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications, 57, Birkhäuser, Boston, MA, 2004.

    Google Scholar 

  16. Herbert Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491.

    Article  MATH  MathSciNet  Google Scholar 

  17. Robin Forman, Bochner’s method for cell complexes and combinatorial Ricci curvature, Discrete Comput. Geom. 29:3 (2003), 323–374.

    MATH  MathSciNet  Google Scholar 

  18. George Francis, John M. Sullivan, Robert B. Kusner, Kenneth A. Brakke, Chris Hartman, and Glenn Chappell, The minimax sphere eversion, Visualization and Mathematics (H.-C. Hege and K. Polthier, eds.), Springer, Heidelberg, 1997, pp. 3–20.

    Google Scholar 

  19. Joseph H.G. Fu, Curvature measures of subanalytic sets, Amer. J. Math. 116:4 (1994), 819–880.

    Article  MATH  MathSciNet  Google Scholar 

  20. Michael E. Gage and Richard S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23:1 (1986), 69–96.

    MATH  MathSciNet  Google Scholar 

  21. Karsten Große-Brauckmann, Robert B. Kusner, and John M. Sullivan, Triunduloids: Embedded constant mean curvature surfaces with three ends and genus zero, J. reine angew. Math. 564 (2003), 35–61; arXiv:math.DG/0102183.

    MATH  MathSciNet  Google Scholar 

  22. Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26:2 (1987), 285–314.

    MATH  MathSciNet  Google Scholar 

  23. David Hilbert and Stephan Cohn-Vossen, Anschauliche Geometrie, Springer, Berlin, 1932, second printing 1996.

    MATH  Google Scholar 

  24. Lucas Hsu, Robert B. Kusner, and John M. Sullivan, Minimizing the squared mean curvature integral for surfaces in space forms, Experimental Mathematics 1:3 (1992), 191–207.

    MATH  MathSciNet  Google Scholar 

  25. Tim Hoffmann, Discrete Hashimoto surfaces and a doubly discrete smoke-ring flow, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 95–115; arXiv.org/math.DG/0007150.

    Google Scholar 

  26. Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20:1 (1984), 237–266.

    MATH  MathSciNet  Google Scholar 

  27. Ivan Izmestiev, Robert B. Kusner, Günter Rote, Boris A. Springborn, and John M. Sullivan, Torus triangulations..., in preparation.

    Google Scholar 

  28. Tom Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108:520 (1994), x+90 pp.

    Google Scholar 

  29. Nicholas J. Korevaar, Robert B. Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30:2 (1989), 465–503.

    MATH  MathSciNet  Google Scholar 

  30. Liliya Kharevych, Boris A. Springborn, and Peter Schröder, Discrete conformal maps via circle patterns, ACM Trans. Graphics 25:2 (2006), 412–438.

    Article  Google Scholar 

  31. Robert B. Kusner, Bubbles, conservation laws, and balanced diagrams, Geometric analysis and computer graphics, Math. Sci. Res. Inst. Publ., vol. 17, Springer, New York, 1991, pp. 103–108.

    Google Scholar 

  32. Joel Langer and Ron Perline, Local geometric invariants of integrable evolution equations, J. Math. Phys. 35:4 (1994), 1732–1737.

    Article  MATH  MathSciNet  Google Scholar 

  33. Mark Meyer, Mathieu Desbrun, Peter Schröder, and Alan H. Barr, Discrete differentialgeometry operators for triangulated 2-manifolds, Visualization and Mathematics III (H.-C. Hege and K. Polthier, eds.), Springer, Berlin, 2003, pp. 35–57.

    Google Scholar 

  34. Ulrich Pinkall and Konrad Polthier, Computing discrete minimal surfaces and their conjugates, Experiment. Math. 2:1 (1993), 15–36.

    MATH  MathSciNet  Google Scholar 

  35. Ulrich Pinkall, Boris A. Springborn and Steffen Weißmann, A new doubly discrete analogue of smoke ring flow and the real time simulation of fluid flow, J. Phys. A 40 (2007), 12563–12576; arXiv.org/0708.0979.

    Article  MATH  MathSciNet  Google Scholar 

  36. Peter Schröder, What can we measure?, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 263–273.

    Google Scholar 

  37. Boris A. Springborn, A variational principle for weighted Delaunay triangulations and hyperideal polyhedra, preprint, 2006, arXiv.org/math.GT/0603097.

    Google Scholar 

  38. Jakob Steiner, Über parallele Flächen, (Monats-) Bericht Akad. Wiss. Berlin (1840), 114–118; Ges. Werke II, 2nd ed. (AMS/Chelsea, 1971), 171–176; bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=08-verh/1840&seite:int=114.

    Google Scholar 

  39. Kenneth Stephenson, Introduction to circle packing, Cambridge Univ. Press, Cambridge, 2005.

    MATH  Google Scholar 

  40. John M. Sullivan, Curves of finite total curvature, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 137–161; arXiv.org/math/0606007.

    Google Scholar 

  41. William P. Thurston, Shapes of polyhedra and triangulations of the sphere, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 511–549.

    Chapter  Google Scholar 

  42. Marc Troyanov, On the moduli space of singular euclidean surfaces, preprint, 2007, arXiv:math/0702666.

    Google Scholar 

  43. Ruud van Damme and Lyuba Alboul, Tight triangulations, Mathematical methods for curves and surfaces (Ulvik, 1994), Vanderbilt Univ. Press, Nashville, TN, 1995, pp. 517–526.

    Google Scholar 

  44. Max Wardetzky, Convergence of the cotangent formula: An overview, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 275–286.

    Google Scholar 

  45. Brian White, A local regularity theorem for mean curvature flow, Ann. of Math. (2) 161:3 (2005), 1487–1519.

    Article  MATH  MathSciNet  Google Scholar 

  46. Günter M. Ziegler, Polyhedral surfaces of high genus, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 191–213; arXiv.org/math/0412093.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, MA 3-2, Technische Universität Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    John M. Sullivan

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  1. John M. Sullivan
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Editors and Affiliations

  1. Institut für Mathematik, MA 8-3, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Alexander I. Bobenko

  2. Institut für Mathematik, MA 3-2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    John M. Sullivan

  3. Department of Computer Science, Caltech, MS 256-80, 1200 E. California Blvd., Pasadena, CA, 91125, USA

    Peter Schröder

  4. Institut für Mathematik, MA 6-2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Günter M. Ziegler

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Sullivan, J.M. (2008). Curvatures of Smooth and Discrete Surfaces. In: Bobenko, A.I., Sullivan, J.M., Schröder, P., Ziegler, G.M. (eds) Discrete Differential Geometry. Oberwolfach Seminars, vol 38. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8621-4_9

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