Research On The Curves And Surfaces In The Hybrid Polynomial Space And Generalized Ball Curves

This dissertation summaries our researches on the two important kinds of curves and surfaces---curves and surfaces in the hybrid polynomial space and generalized Ball curves, which include the bi-cubic C-Hermite surfaces based on algebraic trigonometric polynomial, H-Bezier curves and surfaces based on algebraic hyperbolic polynomial, Wang-Bezier type generalized Ball curves (WBGB curves for short) and Bezier-Said-Wang type generalized Ball curves (BSWGB curves for short), respectively. The main results in this dissertation are outlined as follows:1、 Based on the analysis of the properties of C-Hermite polynomials, Bi-cubic C-Hermite surface has been constructed in this dissertation. Furthermore, the applications of bi-cubic C-Hermite surface in geometric modeling and image interpolation are also given, and experimental results demonstrate that the presented method has good effects.2、The studies on the H-Bezier curves and surfaces are summaried as follows:· A subdivision algorithm of cubic H-Bezier curves is put forward, which serves to determine the control parameters and control points of the two subcurves pt*(t)(0≤t≤t*) and pα-t*(t)(0≤t≤α-t*) subdivided by any point p(t*)(0≤t*≤a) of cubic H-Bezier curves. The connection conditions of cubic H-Bezier curves and cubic Bezier curves are worked out and the applications of cubic H-Bezier curves in the surface modeling are given. The obtained results, which are simple and intuitionistic, can effectively improve the shape representation and control of cubic H-Bezier curves.· This dissertation gives an approximation method of multidegree reduction of H-Bezier curves by making use of the elevation property of H-Bezier curves, and the theory of generalized inverse matrix. The degree reduction error bound is estimated, the relationship between the reduction of H-Bezier curves and that of Bezier curves is established, and the method of multidegree reduction of tensor H-Bezier surfaces is also got. Numerical examples are given to show that the presented method has good approximation effects, and hence the theory of H-Bezier curves is enriched.· The applications of the quasi-Legendre basis defined in the algebraic hyperbolic space in the inversion approximation and offsetting are given in this dissertation. Inversion approximation is constructed by using the blending of polynomial and hyperbolic functions, and the experimental results show that the approximation method is effective; We present an approach to approximate the offset curves of the H-Bezier curve based on the ideal of the approximation for the normal curve. The algebraic approximation algorithms which can obtain the control points of the approximation curves directly are simple, intuitive and of high precision. · Generalized H-Bezier surfaces are defined. On the basis of the properties of generalized H-Bezier surfaces, the study on the α-equaled H-Bezier surfaces is focused. The conditions for geometric continuity and the algorithm for subdivision are given, and some specific surfaces are constructed.3、A subdivision algorithm of Bezier-Said-Wang type generalized Ball curves is presented. This method gives an explicit subdivision matrix by using the dual bases of BSWGB bases, which is different from the traditional method where an algorithm is needed to convert BSWGB bases into power bases and solve the inverse matrix. This algorithm gives the unifying representation of subdivision matrices of the existing generalized Ball curves, which can be used in solving the subdivision of this kind of curves. Numerical examples are also given to show the effectiveness of our methods.4、The degree reduction of generalized Ball curves of Wang-Bezier type is discussed by perturbation and the best uniform approximation respectively. The approximation error is given. The experimental results demonstrate that the effect of the best uniform approximation is better than that of perturbation. If we cannot get the error in advance, we can subdivide the original curve firstly, and then reduce the degree piecewisely. The degree reduction method of WBGB curves enriches the theory system of generalizen Ball curves effectively.