The representation of a fitting surface is fundamental to a wide variety of applications, and it is central to computer-aided design (CAD). This paper focuses on one aspect of generating surfaces: the use of local methods to develop energy-optimizing techniques using iterative subdivision algorithms. These algorithms involve iterative techniques, and generate a sequence of finer control nets that converge to a subdivision surface. Such surfaces can exhibit artifacts, and the surface generated may be unacceptable; for example, one algorithm of Catmull-Clark discussed in the paper cannot generate surfaces with convex extraordinary points in some cases. Hybrid surfaces can also be generated with both positive and negative Gauss curvatures. The control of the surface can be expressed in terms of the eigenvalues and eigenvectors near the extraordinary point. The paper focuses on the eigenanalysis and the optimization of free parameters to control the surface.

The paper begins with a thorough treatment of related methods based on this approach, and the role of local and global optimization. Subdivision methods are then presented, followed by the tuning method and the use of Fourier analysis. Energy minimization is then introduced, locality is proven, and the results are presented.

This is a highly interesting and well-done paper. It presents the context of the subdivision algorithm, and goes on to improvements in the method that address the deficiencies of prior methods, thus improving upon those results.