Analysis of Convergence and Smoothness by the Formalism of Laurent Polynomials
In order to design “good” subdivision schemes, tools for analyzing the convergence and smoothness of a scheme, given its mask, are needed.
A Laurent polynomial, encompassing all the available information on a subdivision scheme to be analysed, (a finite set of real numbers), is the basis of the analysis. By simple algebraic operations on such a polynomial, sufficient conditions for convergence of the subdivision scheme, and for the smoothness of the limit curves/surfaces generated by the subdivision scheme, can be checked rather automatically. The chapter concentrates on univariate subdivision schemes, (schemes for curve design) because of the simplicity of this case, and only hints on possible extensions to the bivariate case (schemes for surface design). The analysis is then demonstrated on schemes from the first two chapters of this volume.
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