Research On The Theory And The Application In Modeling Of C-B-Spline Curves & Surfaces

Curve/surface modeling is an important content of CAGD and CG, and also is the most crucial part of CAD\CAM. Curve/surface modeling is applied on three manufacturings that are aviation, shipbuilding and auto. It also relates to the field of medical diagnosis, bioengineering. The primary methods of curve/surface modeling are Bezier and NURBS at present, which exist some flaws on engineering curve/surface modeling. C-B-spline theory that is developed by Jiwen Zhang presents conic exactly and inherits some advantages of B-spline. C-B-spline is applied on engineering surface, which contains quadratic curves and surfaces.In this thesis, some key issues of C-B-spline theory are systematically and deeply investigated. Firstly, CAGD technology and their theories and methods of surface modeling are briefly introduced, as well as the advantages and disadvantages of the surface modeling with C-B-spline for engineering. Not only the various forms of C-curves and surfaces are presented, but also the theories, general technology and relationships with C-Ferguson, C-Bezier and C-B-spline. Secondly, the theories of C-B-spline curve are systematically studied. First of all, the relation between C-B-spline curve and the control parameter a is obtained based on the analysis of the derivative C-B-spline basis functions. Furthermore, the thesis presents a new method constrained optimization based on changing the control points of the curves, which can modify control points of the original C-B-spline curves to satisfy the given constraints and modify the shape of the curves optimally. Moreover, it presents the subdivision algorithm and G~1 condition of C-B-spline curves. The G~1 and G~2 continuous conditions between C-B-spline curves and cubic uniform B-spline curves are also given. In the end, the relationships with three forms of C-B-spline curves are introduced, and the proof about control polygon converges to its C-B-spline is presented. Finally, it researches the theory of C-B-spline surface. It contains the algorithm for subdivision and joining in different direction of C-B-spline surface with equal a. The entire algorithms are carried out. At last, it is the study of constructing general cylinders, circular cylindrical surface, circular conical surfaces and surface of revolution.