Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
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)
Maltret, J.-L., Daniel, M.: Discrete curvatures and applications: a survey (2003) (preprint)
Mayer, U.F.: Numerical solutions for the surface diffusion flow in three space dimensions. Computational and Applied Mathematics (2001) (to appear)
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)
Meyer, M., Desbrun, M., Schroder, P., Barr, A.: Discrete differential-geometry operator for triangulated 2-manifolds. In: Proc. VisMath 2002, Berlin, Germany (2002)
Taubin, G.: A signal processing approach to fair surface design. In: Proceedings SIGGRAPH 1995, pp. 351–385 (1995)
Wollmann, C.: Estimation of principal curvatures of approximated surfaces. Computer Aided Geometric Design 17, 621–630 (2000)
Xu, G.: Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces (submitted for publication)
Xu, G.: Convergence of discrete Laplace-Beltrami operator over surfaces. Computers and Mathematics with Applications 48, 347–360 (2004)
Xu, G.: Discrete Laplace-Beltrami operators and their convergence. Computer Aided Geometric Design 21, 767–784 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Xu, Z., Xu, G., Sun, JG. (2005). Convergence Analysis of Discrete Differential Geometry Operators over Surfaces. In: Martin, R., Bez, H., Sabin, M. (eds) Mathematics of Surfaces XI. Lecture Notes in Computer Science, vol 3604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537908_27
Download citation
DOI: https://doi.org/10.1007/11537908_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28225-9
Online ISBN: 978-3-540-31835-4
eBook Packages: Computer ScienceComputer Science (R0)