Angle deficit approximation of Gaussian curvature and its convergence over quadrilateral meshes

We propose a discrete approximation of Gaussian curvature over quadrilateral meshes using a linear combination of two angle deficits. Let gijgij and bijbij be the coefficients of the first and second fundamental forms of a smooth parametric surface FF. Suppose FF is sampled so that a surface mesh is obtained. Theoretically we show that for vertices of valence four, the considered two angle deficits are asymptotically equivalent to rational functions in gijgij and bijbij under some special conditions called the parallelogram criterion. Specifically, the numerators of the rational functions are homogenous polynomials of degree two in bijbij with closed form coefficients, and the denominators are g11g22−g122. Our discrete approximation of the Gaussian curvature derived from the combination of the angle deficits has quadratic convergence rate under the parallelogram criterion. Numerical results which justify the theoretical analysis are also presented.