We will present a new theoretical method and experimental results for direct recovery of the principal shape descriptors (the curvatures and the principal curvature directions), and the surface itself by explicit integration of the Gauss map. The method does not rely on polygonal approximations, smoothing of the data, or model fitting. It is based on the observation that one can recover the surface restoring force from the Gauss map, and (i) applies to orientable surfaces of arbitrary topology (not necessarily closed); (ii) uses only first order linear differential equations; (iii) avoids the use of unstable computations; (iv) provides tools for filtering noise from the sampled data. The method is used for stable extraction of surfaces and surface shape invariants, in particular, in applications requiring accurate quantitative measurements.