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.
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© 2008 Springer-Verlag Berlin Heidelberg
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(2008). Case Studies of C k1 -Subdivision Algorithms. In: Subdivision Surfaces. Geometry and Computing, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76406-9_6
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DOI: https://doi.org/10.1007/978-3-540-76406-9_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-76405-2
Online ISBN: 978-3-540-76406-9
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