G^1保形多项式插值曲线  

G^1 Shape-Preserving Polynomial Interpolation Curves

作　　者：严兰兰[1] 韩旭里[2] 李水平[1] YAN Lanlan;HAN Xuli;LI Shuiping(School of Science, East China University of Technology, Nanchang Jiangxi 330013, China;School of Mathematics and Statistics, Central South University, Changsha Hunan 410083, China)

机构地区：[1]东华理工大学理学院,江西南昌330013 [2]中南大学数学与统计学院,湖南长沙410083

出　　处：《图学学报》2017年第2期144-154,共11页Journal of Graphics

基　　金：国家自然科学基金项目(11261003;11271376;60970097);江西省自然科学基金项目(20161BAB211028);江西省教育厅项目(GJJ160558)

摘　　要：针对构造一种具有保形性的多项式插值曲线。首先证明了文献中一组含参数的3次多项式函数为一组全正基,然后借助该全正基定义了一种含两个局部形状参数的分段插值多项式曲线。该曲线在分段连接点处G^1连续。分别给出了插值曲线保正、保单调、保凸的充分条件。这些条件制约了两个局部形状参数之间的关系。通过转化,不管插值曲线保持数据点的哪种形状特征,每一段都依然存在两个独立的形状参数。当数据点既是正的又单调时,只需考虑保单调条件,就可得到既保单调又保正的插值曲线;当数据点既单调又为凸时,只需考虑保凸条件,就可得到既保凸又保单调的插值曲线;当数据点既是正的又单调且为凸时,只需考虑保凸条件,就可得到同时保正、保单调、保凸的插值曲线。证明了插值曲线的有界性并给出了误差估计。This article aims to construct a shape-preserving polynomial interpolation curve.Firstly,a set of cubic polynomial functions with parameters in literature is proved to be a totally positive basis.With this basis,we then define a piecewise interpolation polynomial curve with two local shape parameters.The curve has G1continuity at the join points.The sufficient conditions for the interpolation curve to be positivity-preserving,monotonicity-preserving and convexity-preserving are given.These conditions restrict the relationship between the two local shape parameters.By transformation,no matter what kind of shape characteristic the interpolation curve of the data points keeps,each segment still has two independent shape parameters.When the data points are both positive and monotonous,just considering the monotonicity-preserving conditions,we can obtain the interpolation curve not only monotonicity-preserving but also positivity-preserving.When the data points are both monotonous and convex,just considering the convexity-preserving conditions,we can obtain the interpolation curve not only convexity-preserving but also monotonicity-preserving.When the data points are positive,monotonous and convex,just considering the convexity-preserving conditions,we can get the interpolation curve with positivity-preserving,monotonicity-preserving,and convexity-preserving simultaneously.The interpolation curve is proved to be bounded and its error is estimated.

关 键 词：插值曲线 形状参数 保正 保单调 保凸 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术]