Abstract:  Free form subdivision surfaces are generated from a base mesh of arbitrary topology through an iterative process that smooths the mesh while progressively increasing its density. A subdivision surface scheme consists of two major components: A) a subdivision scheme in the regular setting and B) a set of extraordinary vertex rules.

In the first quarter of the talk, we shall review in a leisurely fashion how the two components work together to produce the kind of surfaces we see in some of the Pixar animation movies.

Due to its elegant mathematical connection with splines and wavelets, subdivision scheme in the regular setting is very well-studied; and can be used to generate arbitrarily smooth functions. This actually means that one can create subdivision surfaces as smooth as we wish -- as long as we stay away from those unavoidable extraordinary vertices. In the vicinity of extraordinary vertices, one is faced with a relatively more subtle smoothness theory. In particular, it is known that there are no 'natural' C^2 schemes 'most of the time'. (In computer aided geometric design it is desirable to have curvature continuous surfaces.)

We shed some light to this notorious C^2 problem by showing that there actually exists a surprisingly simple C^2 scheme in the case of valence 3 vertex. This also allows for some special applications pertaining to genus 0 surfaces.