散乱数据光顺拟合的SOR方法

针对CAGD中散乱数据光顺拟合的一般模型的求解问题，采用罚函数方法处理几何约束。根据最优性条件，将反映曲面光顺性的泛函极小化问题，离散化为曲面参数域网格点上的九点差分格式。得到了关于拟合曲面在网格点上函数值的线性方程组，并证明了该线性方程组的系数矩阵对称正定的性质．保证了采用超松弛法求解线性方程组的收敛性。为了验证所提出方法的有效性，对空间散乱分布的14个数据点，当模型参数取不同值时。分别进行了拟合。试验结果表明，用超松弛法能够简单快速实现散乱数据点的光顺拟合。

SOR method for smoothing of scattered data

Given a set of scattered three-dimensional points, we propsed the Successive Over Relaxation Method(SOR) for surface fitting and smoothing scattered data. The problem was actually the minimum one of quadratic functionalizing about surface smoothness. Nine-point difference schemes of surface functional values at parameter gridding points were obtained according to optimal condition. We treated the constraints with penalty function method after quadratic functional being discrete. The property of symmetric and positive coefficient matrix of linear equations was proved. Then the SOR method converged for any choice of initial vector. The method can be used in the field where smoothing shape is desired by interpolating or approximating a given scattered point set. Numerical examples from simulated and real data were presented to show the efficiency of the new method.