Geometric Iterative Approximation And Free-form Deformation Of Parametric Curves And Surfaces

B-spline curve is one of the most widely used curves in CAGD; it not only possesses most of advantages of Bezier curves, but also has excellent properties such as local regulation. It is a powerful tool for free form curves and surfaces. Therefore, B-spline curve has been widely studied. Based on B-spline curve, generalized B-spline curves with core functions come into being and provide more flexible design recently.Interaction design of curves usually need insert nodes to maintain the shape of the curves, but the redundant data are introduced at the same time. It is necessary to study nodes removal and maintain the smooth of curves simultaneously. Progressive iterative approximation method has been widely used in the approximation of curves and surfaces. By adjusting the control points iteratively, it not only avoids solving linear equations, but also has obvious geometric meaning. Recently, it has been effectively applied in many fields and demonstrates its value. In computer graphics, parametric surface deformation has always been a hot issue. However, the existing techniques need a strong theoretical knowledge and a better controllability in control area. In order to make the deformation operation more convenient, it is necessary to provide a new method with obvious geometric meaning and convenient operation. In view of the above research, this paper has done the following work:1. A geometric iteration algorithm is presented to fit large scale data points. We give two kinds of weights in the iterative process, and discuss the iteration influence of these two kinds of weights respectively. The limit curve of the iteration is weighted least square approximation of original data points. Moreover, with the flexibility of core functions of generalized B-splines, different core functions can be chosen for fitting different set of data points.2. Generalized B-spline dual basis is constructed. According to the properties of the dual basis, an algorithm for the nodes removal the generalized B-spline curves is presented. Single node or multiple nodes can be removed at one time, and the jump values at each node are calculated. Smoothing of curves can be achieved by removing the nodes with large jump values.3. Considering parametric surface deformation, a polynomial extension function of the rectangular area is constructed. By investigating the partial derivatives of deformed region boundaries, a parametric surface deformation algorithm with good interaction and geometric meaning is presented. The ideal effects can be obtained by flexibly adjusting and controlling the deformation parameters for arbitrary parametric surfaces.