The Constructions And Applications Of Ternary Subdivision

Subdivision provides a new approach to construct the complex curves and surfaces in computer aided geometry design.This approach is very unique since it is not only able to produce curves and surfaces of any topology,but also generate smooth limiting curves fast.Hussan first proposed a ternary 4-point subdivision scheme,and fully analyzed the convergence and continuity of the scheme.Two kinds of ternary schemes are proposed in this thesis.Firstly,a stationary subdivision scheme is presented,which can generate the curve of high continuity.Then a ternary four point non-stationary scheme is proposed.Since the non-stationary scheme is asymptotically equivalent to the stationary scheme,they both can generate limiting curves with3 C continuity.In addition,we analyzed the symmetry of subdivision scheme basis function,width of support interval and reproduction.Examples are given to illustrate the limiting curves generated by the scheme.Secondly,a unified ternary arbitrary point scheme is proposed,which is based on the relationship between subdivision scheme matrix and difference matrix.Some exiting schemes can be included in the proposed scheme.Moreover,four cases of asymmetrical schemes are presented according to the characters of the scheme,the relevant properties such as continuity and the support interval are discussed.Finally,examples are given to illustrate that the presented schemes can be used to construct the complicated curves.