Research On A Uniform B-spline Of Order Four With Multi-shape-parameter

Computer aided geometric design (CAGD) serves the technique of computer aided design (CAD) and is a powerful tool to provide mathematic models for the design of CAD. The spline theories and methods have been the important basic theories and central contents of the technique of CAGD. In this paper we give a kind of uniform B-spline basis function of four-order with two independent and three independent shape-parameters and make a lot of researches on this kind of uniform B-spline.Firstly we introduce the development history and background of parameter curves and surfaces modeling with shape parameter and the properties of the cubic uniform B-spline function. The cubic uniform B-spline curve has some good properties in geometric modeling, such as simple shape, flexibility and wide application in many fields. Then we give a class of uniform B-spline of 4-order and 5-degree with multi-parameter in shape, in the third chapter and the forth chapter we give the definitions of the uniform B-spline function of four-order based on two independent shape parameters and the uniform B-spline function of four-order in three independent shape parameters. We prove these uniform B-spline basis functions with three or four shape parameters are positive in some cases and the cubic uniform B-spline basis is a special kind of these multi-shape parameter functions. Then the uniform B-spline curves of four-order with three or four shape parameters possess the same geometric properties and terminal point properties of the cubic uniform B-spline curve. We canalso prove that the generated curves are C~2 continuous and G~2 continuous. Bychanging the value of the parameter we can get different curves in different shapes and locations. So the methods of designing curves will be more flexible and various. It is difficult to design surface with these uniform B-spline functions of four-order with multi-shape parameter. How to use parameters more effectively will be another important problem to study in the future.