Ternary Subdivision Scheme H (?) Lder Continuity And Its Geometric Nature

Subdivisions scheme is one of the topics that have been widely studied in CAGD. We start with an initial mesh of vertices, and in each step, we insert new vertices, calculated as linear combinations of the existing ones, and we connect theme with edges, producing a refined mesh. Two basic questions naturally arising in the study of subdivision schemes are: what weight should we choose for the linear combinations, and what are the properties of a scheme given a choice of weights. In this paper, we will concentrate our attention on the Holder continutiy of subdivision schemes.Four chapters are contained in the present thesis. In chapter 1, we introduce the history of research of subdivision scheme and the main result of our work. In chapter 2, we study the Holder continuity of a ternary subdivision scheme. For this purpose, we first introduce the definition of ternary subdivision scheme and the basic property of the scheme. Then we present a method to calculate the Holder exponent of a ternary subdivision scheme. we can prove S^∞P⁰∈C^N+α, by first deriving 1/2S_N+1 and finding the interger m, such that||(1/3S_N+1)^m||_∞< 1. Then for every k > m (k∈N), S^∞P⁰∈C^N+αk, ||(1/3S_N+1)^k(x_||_∞= 3^-kαk. As an application, we reconstruct the Hassan ternary subdivision scheme. Then we derived the Holder exponent against the parameterμ,It's easy to find the optimal regularity of Hassan ternary subdivision scheme isC^2.183.In chapter 3, we study the ternary subdivision scheme through a geometric approach. The main properties of the scheme is constant reproducing property, positive preserving property, monotonicity preserving property, convexity preserving property, polynomial reproducing property. Then we study the continuity of the scheme through Fourier analysis. In the last chapter, we conclude our work and forecast the further research direction.