The Constructions And Applications Of Subdivision Schemes

Subdivision methods have become a powerful tool in Computer Aided Geometric Design because of its high efficiency, simple expression and low-cost computation, and have been widely applied in many areas. Thus, this thesis gives some basic introduction to some subdivision schemes and presents several subdivision schemes.Firstly, we construct the binary and ternary m-point (m>1) approximating subdivision schemes with the uniform B-spline basis functions. Using the symbol of subdivision scheme, we discuss Cn continuity of these schemes. Some examples are given to show the process of the generating limit curves.We also present a non-stationary binary three point approximating subdivision scheme with an initial parameter which can generate a wide variety of C3-continuous limit curves with different initial control parameter α0. The well-known results by Dyn and Levin (1995) have been used to prove the C3-continuity of the scheme. Lots of examples are provided to show the advantage of the scheme. We find that when α0â†'+∞, the generated C3-continuous limit curves tend to approximate the initial control polygon extremely and keep high smoothness at the same time, which is beyond reach of the stationary subdivision scheme.