We present a novel technique for utilizing the Gaussian curvature information in 3D nonrigid motion estimation in the absence of known correspondence. Differential-geometric constraints derived in the paper allow one to estimate parameters of the local affine motion model given the values of Gaussian curvature before and after motion. These constraints can be further combined with the previously known constraints based on the unit normals before and after motion. Our experiments demonstrate that the resulting hybrid algorithm is more accurate than each of its constituents and more accurate than the classical ICP algorithm. We also present a technique for curvilinear orthogonalization of quadratic Monge patches that is essential in our derivation and useful in other applications.