The PIA Algorithm Of The B-spline Curve Interpolation With Prescribed Tangent Vector

In recent years, progressive iterative approximation has attracted much attention (abbreviated to PIA), which is an iterative process for the data fitting. The idea of the PIA algorithm is that a sequence of curves or surfaces is generated by adjusting their control points gradually according to the iterative formulas, and the limit curve or surface of the sequence will interpolate the data points eventually. The PIA algorithm have many advatantages, such as with NURBS expression, convexity preserving, possessing locality, convergence and self-adaptive, etc. So it has wide applications in Computer Aided Geometric Design and the other relevant fields.Recently, the research results of the PIA algorithm mainly focus on the problem for interpolateing the data points. In this paper, we consider the problem for the data points with tangent constraint in order to control the shape of the curve preferably. We study the problem that B-spline curve interpolates the data points and their tangent vectors based on PIA, and obtain some results.Firstly, the PIA algorithm for uniform cubic B-spline curve which interpolates the data points with prescribed tangent vectors is proposed. The idea of this algorithm is that starting with an initial B-spline curve which takes the given data points as the control points with even indexes and takes the end points of the tangent vectors as the control points with odd indexes, then by adjusting its control points gradually with iterative formulas, we obtain a sequence of curves. The limit curve of the sequence will interpolate the data points with prescribed tangent vectors. According to the idea of the algorithm, the iterative formulas of the PIA algorithm are derived, which make the uniform cubic B-spline curve interpolating the data points and their tangent vectors. The convergence of algorithm is proved.Secondly, for the nonuniformly distributed data points, the interpolation effect of uniform cubic B-spline curve is sometime dissatisfactory, so the PIA algorithm for nonuniform cubic B-spline curve which interpolates the data points with prescribed tangent vectors is proposed. The idea of the algorithm is the same as the one of the algorithm for uniform cubic B-spline curve. A new parameterization method for the data points is presented. The iterative formulas are derived based on the idea of the algorithm and the iterative matrix is achieved. The convergence of the iterative scheme is proved.Finally, considering the convergence rate of the algorithm, the weighted PIA algorithm is further studied for nonuniform cubic B-spline curve. The iterative formulas are derived and the convergence of the algorithm is proved.For the above three algorithms, some examples are presented to illustrate the validity of the algorithms.