A unified framework for primal/dual quadrilateral subdivision schemes

Most commonly used subdivision schemes are of primal type, i.e., they split faces. Examples include the schemes of Catmull-Clark (quadrilaterals) and Loop (triangles). In contrast, dual subdivision schemes such as Doo-Sabin, are based on vertex splits. Triangle based subdivision does not admit primal and dual schemes as the latter are based on hexagons. Quadrilateral schemes on the other hand come in both primal and dual varieties allowing for the possibility of a unified treatment and common implementation. In this paper we consider the construction of an increasing sequence of alternating primal/dual quadrilateral subdivision schemes based on a simple averaging approach. Beginning with a vertex split step we successively construct variants of Doo-Sabin and Catmull-Clark schemes followed by novel schemes generalizing B-splines of bi-degree up to nine. We prove the schemes to be C at extraordinary points, and analyze the behavior of the schemes as the number of averaging steps increases. We discuss a number of implementation issues common to all quadrilateral schemes; in particular we describe a simple algorithm for adaptive subdivision of dual schemes. Because of the unified construction framework and common algorithmic treatment of primal and dual quadrilateral subdivision schemes it is straightforward to support multiple schemes in the same application. This is useful for more flexible geometric modeling as well as in p-versions of the Subdivision Element Method.