Surfaces parametrized by the normals

For a surface with non vanishing Gaussian curvature the Gauss map is regular and can be inverted. This makes it possible to use the normal as the parameter, and then it is trivial to calculate the normal and the Gauss map. This in turns makes it easy to calculate offsets, the principal curvatures, the principal directions, etc. Such a parametrization is not only a theoretical possibility but can be used concretely. One way of obtaining this parametrization is to specify the support function as a function of the normal, i.e., as a function on the unit sphere. The support function is the distance from the origin to the tangent plane and the surface is simply considered as the envelope of its family of tangent planes. Suppose we are given points and normals and we want a C ^k -surface interpolating these data. The data gives the value and gradients of the support function at certain points (the given normals) on the unit sphere, and the surface can be defined by determining the support function as a C ^k function interpolating the given values and gradients.