Models of die casts in manufacturing, level sets of functions, deposition fronts in simulation, pleasing shapes in animation, fast prototyping, and visualization of interactions in a virtual world have one thing in common: the need for efficient representations of smooth surfaces. At present, some of our understanding and use of scientific results is handicapped by the fact that the standard representation used for the task, tensor-product B-splines, can only model a small subset of these surfaces naturally and efficiently, namely surfaces that allow for a regular, checker-board arrangement of its pieces. Free-form surface splines extend the spline paradigm and techniques to cover also irregular arrangements. In extending, they preserve the intuitive generation of the surface from the control mesh by a process of cutting with hyperplanes, also known as corner cutting or subdivision. So far only first-order, tangent continuous surface splines exist. Extension to higher-order continuity is both of fundamental and practical interest. The fundamental aspect consists of a new theory that arises when surfaces from pieces using massive computing power in contrast to analyzing surfaces as in classical Gaussian differential geometry. The goal of this research is to develop a general framework for surface splines and to use the representation to optimize the shape of surfaces with respect to various cost functions. The immediate goal is the derivation of surface splines and to find tools to adjust the variation of curvature, an important quality criterion for surfaces. Since the challenge is considerably greater than the definition of first-order surface splines over irregular meshes, progress requires heavy use of symbolic computation and graphics for quality assessment.
