Efficient, fair interpolation using Catmull-Clark surfaces

We describe an efficient method for constructing a smooth surface that interpolates the vertices of a mesh of arbitrary topological type. Normal vectors can also be interpolated at an arbitrary subset of the vertices. The method improves on existing interpolation techniques in that it is fast, robust and general. Our approach is to compute a control mesh whose Catmull-Clark subdivision surface interpolates the given data and minimizes a smoothness or fairness measure of the surface. Following Celniker and Gossard, the norm we use is based on a linear combination of thin-plate and membrane energies. Even though Catmull-Clark surfaces do not possess closed-form parametrizations, we show that the relevant properties of the surfaces can be computed efficiently and without approximation. In particular, we show that (1) simple, exact interpolation conditions can be derived, and (2) the fairness norm and its derivatives can be computed exactly, without resort to numerical integration.