Repeated local operations and associated interpolation properties of dual 2n-point subdivision schemes

In this paper we first derive a recursive relation of the generating functions of a family of dual 2n-point subdivision schemes. Based on the recursive relation we design repeated local operations for implementing the 2n-point subdivision schemes. Associated interpolation properties of the limit curve sequence of the dual 2n-point subdivision schemes when n tends to infinity are then investigated. Based on the repeated local operations, we further prove that the limit curves of the family of the dual 2n-point subdivision scheme sequence approach a circle that interpolates all initial control points as n approaches infinity, provided that the initial control points form a regular control polygon. Other interpolation properties show that the limit curve interpolates all closed initial control points with odd points or with even points but satisfying an extra condition, and interpolates all newly inserted vertices of an original closed polygon, when n approaches infinity. Some numerical examples are provided to illustrate the validity of our theoretic analyses.