Analysis of uniform binary subdivision schemes for curve design

Abstract

The paper analyses the convergence of sequences of control polygons produced by a binary subdivision scheme of the form

$$\begin{array}{*{20}c} {f_{2i}^{k + 1} = \sum\limits_{j = 0}^m {a_j f_{i + j}^k } ,} & {f_{2i + 1}^{k + 1} = \sum\limits_{j = 0}^m {b_j f_{i + j}^k ,} } & {i \in Z,k = 0,1,2,....} \\ \end{array}$$

The convergence of the control polygons to aC° curve is analysed in terms of the convergence to zero of a derived scheme for the differencesf ^k _i+1 −f ^k _i . The analysis of the smoothness of the limit curve is reduced to the convergence analysis of “differentiated” schemes which correspond to divided differences off ^k_i ∶i∈ Z with respect to the diadic parametrizationt ^k _i =i/2^k. The inverse process of “integration” provides schemes with limit curves having additional orders of smoothness.