基于高斯分布的插值型细分方法

Interpolatory subdivision scheme based on Gaussian distribution

为了通过设置初始点的协方差矩阵调整极限曲线(曲面)形状,提出基于高斯分布的插值型细分方法。首先,介绍了高斯分布的几何特征。然后,根据形状需求,设定初始点处的切、法向和特征值大小,并在每一点设置相应的初始协方差矩阵。最后,在每一步细分中,由旧点和旧点处的协方差矩阵计算新点和新点处的协方差矩阵,形成新的带有协方差矩阵的细分多边形(网格),依次迭代得到极限曲线(曲面)。数值算例表明,相比线性插值细分算法,所提算法易于控制极限曲线(曲面)的形状。在实际运用中,该算法能得到曲率较好的插值曲线,同时也可以根据需求构造尖点、折痕等特征。

In order to adjust the shape of limit curve(surface)by setting covariance matrix of initial points,an interpolatory subdivision scheme based on Gaussian distribution was proposed.Firstly,introduction of the geometric characteristic of Gaussian distribution.Secondly,determination of the tangent,normal and eigenvalues at the initial point appropriately according to the shape requirements,and the corresponding covariance matrix is set at each point.Finally,in each step of subdivision,new points and their covariance matrix are calculated by old points and their covariance matrix,and a new subdivision polygon(grid)with covariance matrix is formed.Limit curve(surface)is obtained by iteration.Numerical examples show that,compared with the linear interpolatory subdivision scheme,the shape of the limit curve(surface)is easy to be controlled via the proposed scheme.In practical applications,this scheme can obtain the interpolatory curve with good curvature,and can also construct the characteristics of cusp and crease according to the demand.