Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces

In this paper, we study the convergence property of a well known discretized scheme for approximating Gaussian curvature, derived from Gauss–Bonnet theorem, over triangulated surface. Suppose the triangulation is obtained from a sampling of a smooth parametric surface, we show theoretically that the approximation has quadratic convergence rate if the surface sampling satisfies the so-called parallelogram criterion. Numerical results which justify the theoretical analysis are also presented.