Invariant smoothness measures for surfaces

Two novel smoothness measures for surfaces are presented in this paper. The second and third order smoothness are defined as the squared normal curvature and the squared variation in normal curvature integrated over all directions in the tangent plane. Both quantities are truly geometric in the sense that they are invariant with respect to the actual parametrization of the surface. All the same, all formulae are derived in terms of an arbitrary parametrization. In addition to providing a basis for variational surface construction, the second and third order smoothness can also be used for evaluation and assessment of the quality of an existing surface.