Research On Binary Subdivision Method

Subdivision method is a method which starts from the initial control polygons to generate smooth curves through ceaseless iterations.It has been widely used in geometric modeling because of its simple expression and low-cost computations.In this dissertation,the basic principles of curve subdivision are introduced and some subdivision schemes are presented.Firstly,a rational four-point binary approximation subdivision scheme is proposed,and the convergence andkC continuity of the scheme are analyzed.A shape preserving scheme can be obtained when the parameter takes a specific value.For some control polygons without sharp angles,the limit curve generated by this shape preserving scheme is very close to the initial control polygon.Secondly,a Laurent polynomial with multi parameters is proposed according to the relationship between the generating polynomials and Laurent polynomials.This new polynomial can not only generate a family of classic subdivision schemes,but also be used to construct asymmetric subdivision schemes.The convergence and kC continuity of the four-point and the five-point schemes are analyzed respectively,and the approximation effect of the symmetric scheme is compared with that of the asymmetric scheme in the special case of the five-point scheme.