Approximation and Linear Independence

In this chapter, we elaborate on two aspects of subdivision which, besides smoothness, are very important for applications — convergence of sequences of so-called proxy surfaces, such as control polyhedra, and linear independence of generating splines.

In Sect. 8.1/157, we consider a sequence {x̌^k} _k of proxy surfaces to a subdivision surface x. For example, piecewise linear proxy surfaces arise as ‘control polyhedra’ in whatever sence, or as a sequence of finer and finer piecewise linear interpolants of x. The analysis to be developed is, however, sufficiently general to cover cases where the proxy surfaces consist of non-linear pieces, for instance, when approximating x by an increasing, but finite number of polynomial patches. We derive upper bounds on the parametric and geometric distance between x̌^k and x, which are asymptotically sharp up to constants as k → ∞. Our results show that the rate of convergence of the geometric distance, which is crucial for applications in Computer Graphics, depends on the subsubdominant eigenvalue μ.

In Sect. 8.2/169, we consider the question of local and global linear independence of the generating splines B = [b₀,…, b _ℓ̄ ]. This topic is closely related to the existence and uniqueness of solutions of approximation problems in spaces of subdivision surfaces, such as interpolation or fairing. We show that local linear independence cannot be expected if the valence n is high, and that even global linear independence is lost in special situation, like Catmull-Clark subdivision for a control net with the combinatorial structure of a cube.