Application Research Of Dual Basis And Geometric Iteration Method In CAGD

When study the properties of a space, people often transfer the problem into the corresponding dual space if it is too difficult to discuss in the original one, dual bases which are the bases functions of the dual space possess many excellent properties. Recently, dual bases based on inner product attract widely attention. With the help ofthe given bases’ dual bases, it can realize least square approximation of square integrable function, and the result is linear combination of the corresponding bases which means no transformation is needed. Scattered data interpolation and approximation problems in CAGD attract more and more attention recently. Recently, one new method which is called progressive iterative approximation is put forward to sovle scattered data fitting. Compared with traditional interpolation method, progressive iterative approximation method has clear geometric meaning, thus it is also called geometric iteration method. It also has advantages such as no need for solving linear system of equations and easy programming.As we know, most of papers are focused on the theoretical research of dual basis and progressive iterative method. However, there is few paper which focuses on application study in CAGD. Based on these, this thesis focuses on the following works: firstly, we put forward a new offset approximation algorithm which is based on the progressive iterative approximation method and most of the recent offset approximation results, The effectiveness of the proposed algorithm is verified by using a large number of numerical examples which we both take iterative error and the number of control points into account. Secondly, we carry out series of work in geometric approach by taking dual bases of generalized B-splines as tools. We propose a new construction method of dual generalized B-splines, nd the coefficient matrix is lower triangular invertible matrix. Then we further study the geometric application of dual generalized B-splines, such as function approximation, offset approximation, degree reduction, etc. Considering the need of complex modeling, finally, we discuss the G2 smooth blending conditions of generalized B-spline curves and put forward the G2 smooth blending condtions of bicubic generalized B-splines surfaces with the aid of dual generalized B-splines.