Convergence Analysis of Discrete Differential Geometry Operators over Surfaces

  • Zhiqiang Xu
  • Guoliang Xu
  • Jia-Guang Sun
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3604)


In this paper, we study the convergence property of several discrete schemes of the surface normal. We show that the arithmetic mean, area-weighted averaging, and angle-weighted averaging schemes have quadratic convergence rate for a special triangulation scenario of the surfaces. By constructing a counterexample, we also show that it is impossible to find a discrete scheme of normals that has quadratic convergence rate over any triangulated surface and hence give a negative answer for the open question raised by D.S.Meek and D.J. Walton. Moreover, we point out that one cannot build a discrete scheme for Gaussian curvature, mean curvature and Laplace-Beltrami operator that converges over any triangulated surface.


Convergence Analysis Gaussian Curvature Discretization Scheme Mesh Surface Negative Answer 
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  1. 1.
    Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: SIGGRAPH 1999, pp. 317–324 (1999)Google Scholar
  2. 2.
    Liu, G.H., Wong, Y.S., Zhang, Y.F., Loh, H.T.: Adaptive fairing of digitized point data with discrete curvature. Computer Aided Design 34(4), 309–320 (2002)CrossRefGoogle Scholar
  3. 3.
    Maltret, J.-L., Daniel, M.: Discrete curvatures and applications: a survey (2003) (preprint)Google Scholar
  4. 4.
    Mayer, U.F.: Numerical solutions for the surface diffusion flow in three space dimensions. Computational and Applied Mathematics (2001) (to appear)Google Scholar
  5. 5.
    Meek, D.S., Walton, D.J.: On surface normal and Gaussian curvature approximations given data sampled from a smooth surface. Computer Aided Geometric Design 17, 521–543 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Meyer, M., Desbrun, M., Schroder, P., Barr, A.: Discrete differential-geometry operator for triangulated 2-manifolds. In: Proc. VisMath 2002, Berlin, Germany (2002)Google Scholar
  7. 7.
    Taubin, G.: A signal processing approach to fair surface design. In: Proceedings SIGGRAPH 1995, pp. 351–385 (1995)Google Scholar
  8. 8.
    Wollmann, C.: Estimation of principal curvatures of approximated surfaces. Computer Aided Geometric Design 17, 621–630 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Xu, G.: Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces (submitted for publication)Google Scholar
  10. 10.
    Xu, G.: Convergence of discrete Laplace-Beltrami operator over surfaces. Computers and Mathematics with Applications 48, 347–360 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Xu, G.: Discrete Laplace-Beltrami operators and their convergence. Computer Aided Geometric Design 21, 767–784 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Zhiqiang Xu
    • 1
  • Guoliang Xu
    • 2
  • Jia-Guang Sun
    • 3
  1. 1.Department of Computer ScienceTsinghua UniversityBeijingChina
  2. 2.ICMSEC, LSEC, Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingChina
  3. 3.School of SoftwareTsinghua UniversityBeijingChina

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