Case Studies of C ^k₁ -Subdivision Algorithms

In this chapter, we formally introduce and scrutinize three of the most popular subdivision algorithms, namely the Catmull–Clark algorithm [CC78], the Doo–Sabin algorithm [DS78], and Simplest subdivision¹ [PR97]. Besides the algorithms in their original form, it is instructive to consider certain variants. We selectively modify a subset of weights to obtain a variety of algorithms that is rich enough to illustrate the relevance of the theory developed so far. In particular, we show that a double subdominant eigenvalue is neither necessary nor sufficient for a C ^k₁ -algorithm: First, there are variants of the Doo-Sabin algorithm with a double subdominant eigenvalue, which provably fail to be C ¹₁ because the Jacobian determinant ^×Dψ of the characteristic ring changes sign. Second, for valence n = 3, Simplest subdivision reveals an eightfold subdominant eigenvalue, but due to the appropriate structure of Jordan blocks, it is still C ¹₁ . In all cases, the algorithms are symmetric so that the conditions of Theorem 5.24/105 can be used for the analysis.