Study On The Geometric Modeling By Generalized Ball Curves And Surfaces

Free curves/ surfaces defined in different topological structure have different expressions in CAGD, and generalized Ball curves are more efficient than Bezier curves in calculation, the degree elevation and reduction. Based on the transformation between the different curves/ surfaces, and the interval (disc) arithmetical and the degree reduction of the curves and surfaces, this dissertation focuses on the research about the geometrical modeling of the generalized Ball curves/ surfaces. The achieved main results are as follows:1. By means of the dual functionals of the generalized Ball basis of Wang-Said type, we investigated the basis transformation formula between Bernstein basis and WSGB basis, and at the same time, worked out the basis transformation formulae between Bernstein basis and Wang-Ball and Said-Ball basis. We discussed the explicit subdivision algorithms for WSGB curves of degree n = 2m + 1, including the subdivision algorithm for Wang-Ball curves with L = m-1 and the subdivision algorithm for Said-Ball curves with L = 0. In the last part, we also constructed the envelope algorithm of the WSGB curves. As a result, the curves of higher degrees can be generated from the curves of lower degrees with envelope technique.2. By introducing a family of generalized Ball bases and generalized Ball surfaces with position parameter H and using the relation between the bases of adjacent surfaces, this dissertation presents a new algorithm for conversion of Said-Ball surfaces and Bezier surfaces on triangular domain, which is easy for programming.3. Considering the fact that orthogonal Tchebyshev polynomials, which produce the least zero-deviation approximation, play important roles in the degree reduction of curves and surfaces, we worked out a recursive conversion algorithm between the Tchebyshev polynomials and Bernstein polynomials instead of computing the inverse matrices in the process of degree reduction of Bezier curves and surfaces as required in many current existing methods, which proves more stable and more economical than other methods.4. The boundary of interval Ball curves of degree n is discussed and two algorithms for the degree reduction of interval Ball curves are given by means of the linear programming method and best approximation method respectively. If the interval curves obtained by linear programming method fail to be controlled within the expected error, then the subdivision techniques can be used.5. The basic concepts of disk Said-Ball curves are given in a form suitable for the study. The control points of disk Said-Ball curves are disks, rather than rectangles as in the interval Said-Ball curves. To solve the degree reduction problem of disk Said-Ball curves, we design an algorithm involving both cases: non-interpolating endpoints and interpolating endpoints of disk Said-Ball curves. Some examples are provided to demonstrate the efficiency of the algorithm.6. The degree reduction of generalized Ball curves of Wang-Said type by perturbation and the best uniform approximation are respectively discussed, the approximation error and relative approximation error are analyzed, and the comparison between these two methods is made by some examples.