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.
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© 2005 Springer-Verlag Berlin Heidelberg
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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
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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
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