Approximation of the curvature of Alexandrov surfaces using dual polyhedra

摘要

It is shown that a sequence of pairs of locally polar polyhedra makes it possible to construct piecewise-affine approximation to the spherical Gauss map and to construct convergent point-wise approximations to mean and Gauss curvatures of regular surfaces. Any DC surface (representable as a difference of convex functions, A. D. Alexandrov, 1949) possesses a second differential and a tangent paraboloid almost everywhere, hence approximation by dual polyhedra allows us to approximate the curvature of such surfaces.