The PIA Algorithm Of B-spline Approximation With Geometric Feature Constraints

B-spline curve or surface approximation is an important problem in geometric modelling and the study of science and engineering. Progressive iterative approximations (abbreviated to PIA) is an iterative process for the data fitting, which has attracted much attention in recent years. The main idea of the PIA algorithm is to adjust the control vertices constantly, and use the new control vertices to generate the corresponding curve or surface. The resulting limit curve or surface will interpolate the data points eventually. Existing relevant literatures have proved that the whole iterative process of algorithm is convergent. The PIA algorithm has many advantages, such as locality, convergence and self-adaptive, etc.But the research results of the PIA algorithm mainly focus on the problem for interpolating the data points recently. The resulting curve or surface may be unnecessary twist and have wave phenomenon. In this paper, we propose a PIA algorithm for B-spline curve approximation with tangent constraint and a PIA algorithm for B-spline surface approximation with normal constraint. By doing that, we will achieve the better control to the shape of the curve or surface.Firstly, we propose a PIA algorithm for B-spline curve approximation with tangent constraint. On the one hand the tangent vector, curvature and other geometric characteristics of the discrete data points are fully applied to the approximation problem. Tangent constraint can avoid unnecessary fluctuations, and obtain the better approximation effect. On the other hand, the number of the selecting feature points which is to be the control vertices is less than the number of the data points, so the PIA algorithm can be used for the approximation of the mass of discrete data points. The steps in the process of each iteration of the algorithm are independent, so the algorithm is easy to be applied to the parallel computing, which may improve the computational efficiency. Some examples are given to show the validity of the algorithm.Secondly, based on the PIA algorithm for B-spline curve approximation with tangent constraint, we present a PIA algorithm for B-spline surface approximation with normal constraint. In the algorithm the geometrical characteristics of the data points are considered to apply to surface approximation problem. By adding the geometric constraints such as normal vector and curvature in the iterative process of this PIA algorithm, the shape of the approximate surface can be controlled better. Compared to the traditional method of PIA, which only consider data point parameter error, this algorithm can avoid unnecessary fluctuations and obtain better approximation effect Some examples are given to show the validity of the algorithm.Finally, we consider the convergence rate of the PIA algorithm. By fixing control vertices and changing weight factor, we improve the convergence speed of the PIA algorithm of B-spline curve approximation with tangent constraint and B-spline surface approximation with normal constraint and propose the algorithms for the convergence acceleration We give the analysis and comparison of the experimental results. The experimental results show the validity of the algorithms.