基于PIA的非均匀三次B样条曲线Hermite插值  

Hermite Interpolation of Non-Uniform Cubic B-Spline Curve Based on PIA

作　　者：吴硕琳 李亚娟[1] 邓重阳[1] WU Shuo-Lin;LI Ya-Juan;DENG Chong-Yang(School of Science,Hangzhou Dianzi University,Hang zhou 310018)

机构地区：[1]杭州电子科技大学理学院,杭州310018

出　　处：《计算机学报》2023年第11期2463-2475,共13页Chinese Journal of Computers

基　　金：国家自然科学基金(61872121);浙江省重点研发计划项目(2021C0018)资助。

摘　　要：提出基于渐进迭代逼近(Progressive Iteration Approximation,PIA)的非均匀三次B样条曲线Hermite插值算法.首先,以给定数据点作为初始控制顶点,采用累加弦长法得到节点序列,通过构造误差向量更新控制顶点,迭代生成插值数据点的非均匀三次B样条曲线.当需要同时插值数据点和单位切向时,在每个节点区间上插入一个节点;当需要同时插值数据点、单位切向和曲率向量时,在每个节点区间上插入两个节点;更新初始控制顶点,进而迭代得到插值B样条曲线.理论分析表明算法是收敛的.数值算例结果说明,与均匀三次B样条曲线插值算法相比,当相邻数据点间距离变化程度越大时,该算法的收敛速度越快,在相同误差条件下迭代次数更少.We propose an algorithm for Hermite interpolation by non-uniform cubic B-spline based pro-gressive iteration approximation(PIA).Firstly,interpolate given data points.Data points that need interpo-lation are used as known conditions,and apply cumulative chord length parameterization to define the knot vector.We use these points that need to be interpolated as the initial control points.The number of initial control points are n+3,and the initial non-uniform cubic B-spline curve is obtained by these n+3 control points.The error vectors are constructed by the errors between the initial control points and the corre-sponding points on the initial non-uniform cubic B-spline curve.And then update the control points by us-ing the constructed error vectors.The B-spline curve is obtained by using the updated control points.Con-stantly update the above process of iteration.The non-uniform cubic B-spline curve of interpolation data points is generated,which the specified iteration stop condition is the average distance between the control points and the corresponding points on the curve.The iteration is stopped when the average distance be-tween the two is less than the specified error.Secondly,data points and tangent vectors need to be interpo-lated simultaneously.Based on the interpolation of the node vector of given data points,we insert one knot into each knot interval by dichotomy method and update node sequence.The n+3 new control points ob-tained by the interpolation given data points algorithm are used to calculate the initial control points after increasing the degree of freedom.The initial non-uniform cubic B-spline curve is obtained by increasing the number of initial control points to 2n+4.The error vectors are constructed by controlling the errors be-tween the control points and the corresponding points on the initial non-uniform cubic B-spline curve.The control points are updated by using the constructed error vectors.The B-spline curve is obtained by using the updated control points.The above process is iterat

关 键 词：非均匀三次B样条曲线 迭代算法 HERMITE插值 渐进迭代逼近 控制顶点 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]