Shape Analysis and C ^k₂ -Algorithms

In the preceding chapters, we have studied first order properties of subdivision surfaces in the vicinity of an extraordinary point. Now we look at second order properties, such as the Gaussian curvature or the embedded Weingarten map, which characterize shape. To simplify the setup, we assume k ≥ 2 throughout. That is, second order partial derivatives of the patches x ^m _j exist and satisfy the contact conditions (4.7/62) and (4.8/62) between neighboring and consecutive segments. However, most concepts are equally useful in situations where the second order partial derivatives are well defined only almost everywhere. In particular, all piecewise polynomial algorithms, such as Doo-Sabin type algorithms or Simplest subdivision, can be analyzed following the ideas to be developed now.

In Sect. 7.1/126, we apply the higher-order differential geometric concepts of Chap. 2/15 to subdivision surfaces and derive asymptotic expansions for the fundamental forms, the embedded Weingarten map, and the principal curvatures. In particular, we determine limit exponents for L^p-integrability of principal curvatures in terms of the leading eigenvalues of the subdivision matrix. The central ring will play a key role, just as the characteristic ring for the study for first order properties.

In Sect. 7.2/134, we can leverage the concepts to characterize fundamental shape properties. To this end, the well-known notions of ellipticity and hyperbolicity are generalized in three different ways to cover the special situation in a vicinity of the central point. Properties of the central ring reflect the local behavior, while the Fourier index F(μ) of the subsubdominant eigenvalue μ of the subdivision matrix is closely related to the variety of producible shapes. In particular, F(μ) ⊃ {0, 2, n − 2} is necessary to avoid undue restrictions. Further, we introduce shape charts as a tool for summarizing, in a single image, information about the entirety of producible shape.

Conditions for C ^k₂ -algorithms are discussed in Sect. 7.3/140. Following Theorem 2.14/28, curvature continuity is equivalent to convergence of the embedded Weingarten map. This implies that the subsubdominant eigenvalue μ must be the square of the subdominant eigenvalue λ, and the subsubdominant eigenrings must be quadratic polynomials in the components of the characteristic ring. These extremely restrictive conditions explain the difficulties encountered when trying to construct C ^k₂ -algorithms. In particular, they lead to a lower bound on the degree of piecewise polynomial schemes, which rules out all schemes generalizing uniform B-spline subdivision, such as the Catmull-Clark algorithm.

Section 7.4/145 presents hitherto unpublished material concerning a general principle for the construction of C ^k₂ -algorithms, called the PTER-framework. This acronym refers to the four building blocks: projection, turn-back, extension, and reparametrization. The important special case of Guided subdivision, which inspired that development, is presented in Sect. 7.5/149.