Subdivision Surfaces: C2 schemes and generalized control nets.

After a brief introduction in Chapter 1, in Chapter 2 we introduce and discuss a new basis for C2 splines of orders seven–eight. Cardinal cubic B-splines are among the generating elements of this basis, which allows to decompose the space of polynomials of high degree into the direct sum of the subspace of cubic splines, and some “details”, whose purpose is to allow for curvature continuity at extraordinary points in the bivariate setting. Masks for binary subdivision are provided. We also prove convergence rates of the cubic part of the spline under repeated refinement. We show how it is possible to change from B-spline representations to this basis. Besides this main topic of the chapter, we point out new insights into polynomial subdivision in the regular setting. The analysis leads to techniques of a general nature that allow to deduce convergence rates for generalized control structures toward the limit curve, or surface. The third chapter centers on the characteristic map of a subdivision scheme. We present a method by which characteristic maps to arbitrary eigenvalues 0 < lambda < 1 can be constructed, which is, for instance, needed for the PTER-scheme. Further, a solution to verifying injectivity of a characteristic map for infinitely many valencies is presented and executed at hand of a sample characteristic map. In Chapter 4 we construct and test C2-subdivision schemes based on the PTER-principle by minimizing quadratic functionals. We discuss some selected differential operators that can be used, and example surfaces, as well as generating splines derived by them. Convergence rates of control nets have been studied extensively only in recent years. Chapter 5 further develops the concept of extraordinary proxies from the book Subdivision Surfaces. Proxies abstract the relevant properties that make control nets converge to the limit surface. Parametric and Hausdorff distances are estimated, with sharpness established for each. We continue by analyzing convergence speed of unit normals in the vicinity of extraordinary points. Finally, we conclude by pointing out how slow convergence—of distance or of normals—can be circumvented in situations where the Catmull- Clark algorithm is still used. This also provides a new perspective on using control-nets as approximations to the limit surface