The Study Of The Generalized Bezier Curves And Generalized B-splines Curves

The present dissertation is concerned with the construction of the generalized Bernstein basis for the space Pu,v=span{1,t,u(t),v(t)}, and gives out the construction method. Then we get the generalized Bezier curves and generalized B-splines with shape parameters. Theses generalized curves can represent circle and elliptic exactly, which can achieve the higher approximation accuracy. This dissertation consists of six chapters:In chapter â… , we review the modeling methods of the curves and surfaces in the CAGD, and then we introduce briefly the history and present situation of the curves and surfaces with shape parameters.In chapter â…¡, we introduce mainly the definition and properties of the Bernstein basis and Bezier curves.Chapter â…¢ roots in our research. In this chapter, the construction method of generalized Bernstein basis and Bezier curves for space Pu,v=span{1,t,u(t),v(t)} is presented, and then we provide the specific expression of the generalized cubic Bernstein basis.In chapter IV, we construct the generalized Bernstein basis for spaces: respectively on the basis of the theory of chapter â…¢. And then the properties of the generalized Bernstein basis are researched respectively. We study the properties and the splicing conditions of the generalized Bezier curves based on the previous generalized Bernstein basis. Finally, some applications in curve design are presented. Theses generalized curves can represent circle and elliptic exactly.In chapter V, we give a method of transforming the generalized Bernstein basis to generalized B-splines. The specific expression of the generalized B-splines is provided too. We also define the generalized B-splines curves based on the generalized B-splines and provide some applications in curve design by the generalized B-splines curves.In chapter VI, we give a summary and outlook. We briefly make a summing up of the main contents of the thesis, and propose some suggestions to the future study.