Polyhedral subdivision methods for free-form surfaces

Reviewer: Joshua Turner

This long paper provides some extensions to the literature on recursive subdivision surfaces, also known as Sabin-Doo surfaces [1]. These are procedural surfaces, defined on a polyhedral mesh that need not be rectangular. The mesh is refined by lopping off corners and edges until the mesh approaches a smooth surface. It has been proved that with the exception of certain anomalous regions, the Sabin-Doo surface can be represented as a collection of bi-quadratic B-Spline patches. However, there is no analytic representation for the entire surface. The advantage of Sabin-Doo surfaces over other free-form surfaces is that the designer is not required to piece together a number of surface patches by hand. This author addresses three deficiencies in the literature: First, the basic Sabin-Doo surface tends to draw away from the boundary of the original mesh as the refinement process proceeds. This makes it difficult to connect the surface to adjacent surfaces. The author gives an algorithm that compensates for the shrinking process by extending the boundary of the original mesh. However, it is evident that this algorithm may result in ill-formed boundary faces. Second, the author gives an algorithm for computing a Sabin-Doo surface that interpolates a given set of points. There are many ways to do this, and it is not evident that the author's approach is most likely to yield an acceptable surface, particularly in the presence of error in the placement of the given points. Finally, an algorithm for intersecting two Sabin-Doo surfaces is provided. This is based on the use of bounding boxes and recursive subdivision. No basis is given for the appraisal of the performance. In addition to the areas addressed by this paper, it should be noted that the lack of an analytical representation would make it difficult to integrate Sabin-Doo surfaces with functional surfaces in a general-purpose design system.