Convergence Analysis of Discrete Differential Geometry Operators over Surfaces

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.