Blossoms Method Of Beta-splines

Blossoms method as a basic tool for studying spline theories,is the fundamental of the research on node insertion,subdivision and higher order,as well as many other algorithms.This paper mainly studies the blossoms method in Beta-spline space,According to the definition of basis functions and the conditions of geometric continuity,this paper respectively deduces the explicit basis functions expressions for the quadratic Beta-splines,cubic Beta-splines and quartic Beta-splines under the Beta-spline space.All these can be classified into single node and multiple nodes.And the different images of basis functions are mapped by using mathematical software MATLAB.Further,this paper has put forward the definition of Beta-spline curves,introducing the nature of the Beta-spline curves on the basis of their basis functions.In combination with Beta-splines dual spaces,dual functional and polya polynomial,blossoms formula of the Beta-splines curves are given,which can be applied in low-degree Beta-splines curves to calculate and get the specific blossoms results of quadratic Beta-splines and cubic Beta-splines.Besides this paper will emphasize on the case of the multiple nodes,verify the relationship between blossoms and control vertices,and present the rationality of blossoms method.Then,this paper generalizes the blossoms method to rational forms,and researches the blossoms of rational Beta-spline curves.Therefore,the blossoms formula of rational Beta-spline curves can be proposed by algebraic method,and the blossoms results of quadratic rational Beta-spline curves is derived and calculated in detail.Subsequently,the blossoms method to rational Beta-spline curves is applied to the instance,and the specific blossoms forms of quadratic rational Beta-spline curves are obtained.