Degree Reduction Of The Said-Ball Curves And Surfaces

This paper is concerned about the degree reduction of the Said-Ball curves and surfaces. There are totally four chapters in this paper.In Chapter one the author surveys the significances and background of the degree reduction techniques for curves and surfaces.Chapter two is the preparation knowledge of this paper, which firstly covers the current definition of the Said-Ball curves as well as their properties, and secondly finds for the first time the matrix representation of such curves one degree lower based on the degree elevation formula of the Said-Ball curves, and finally offers the general formulae of derivatives of arbitrary orders at endpoints by differentiation.In Chapter three the author offers two methods of multi-degree reduction of the Said-Ball curves, one of which is proposed by using the theory of generalized inverse matrix theory, another of which is proposed by minimizing L2 norm distance function between two Said-Ball curves, each of which reduces a given degree n Said-Ball curve to a degree m Said-Ball curve. In the process of degree reduction, two cases are considered respectively, one of which is about the case without conditions of endpoint interpolations, the other of which is about the case with conditions of endpoint interpolations of higher orders. The explicit formulas of control points of the reduced Said-Ball curves are obtained.In the last chapter, the above two methods are generalized to the tensor product of the Said-Ball surfaces, each of which reduces a given degree nxm tensor product Said-Ball surface to an n1 xm1 one. In the process of degree reduction for the tensor product Said-Ball surfaces, we not only discuss the case with constraints of corner interpolations, but also the case without that.In this paper, some numerical examples are given to demonstrate the effects of our methods.