Exact evaluation of a class of nonstationary approximating subdivision algorithms and related applications

For a class of nonstationary approximating subdivision schemes generalizing the cubic B-spline scheme, we derive closed-form expressions to describe the evolution of each initial control vertex during the whole subdivision process, and we find explicit formulas for exact evaluation of the limit curve and its first derivative at any dyadic-rational parameter value. These results provide, for the first time in the subdivision literature, an exact evaluation of limits obtained by nonstationary approximating subdivision algorithms. As will be shown at the end of this paper, the achievement of closed-form expressions for the limit stencils of a nonstationary approximating scheme has immediate benefits in the context of geometric modelling. Moreover, it marks a first step forward towards the development of further new theoretical results aimed at extending the use of nonstationary subdivision algorithms to new fields of applications.