您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2021, Vol. 56 ›› Issue (6): 81-94.doi: 10.6040/j.issn.1671-9352.0.2021.136

• • 上一篇    

带形状参数的三角β-B曲线的渐进迭代逼近

王曾珍,刘华勇*,查东东   

  1. 安徽建筑大学数理学院, 安徽 合肥 230601
  • 发布日期:2021-06-03
  • 作者简介:王曾珍(1996— ),女,硕士研究生,研究方向为计算机图形学. E-mail:wangz_z1014@163.com*通信作者简介:刘华勇(1972— ),男,硕士,教授,硕士生导师,研究方向为计算机图形学与计算数学. E-mail:aiaiwj@126.com
  • 基金资助:
    安徽省高等学校自然科学研究资助项目(KJ201012);安徽建筑大学质量工程资助项目(2020jy22);安徽省级质量工程资助项目(2020kfkc163)

Progressive-iterative approximation by the triangular β-B curves with shape parameter

WANG Zeng-zhen, LIU Hua-yong*, ZHA Dong-dong   

  1. Anhui Jianzhu University, Hefei 230601, Anhui, China
  • Published:2021-06-03

PDF (PC)

219

摘要： 为加快渐进迭代逼近法收敛速度,克服一般B样条曲线不能表示圆或椭圆等曲线的缺陷,基于β-B曲线探讨其(加权)渐进迭代逼近法。根据所选β-B基函数求得曲线(加权)渐进迭代逼近法的迭代矩阵,基于谱半径最小、收敛速度最快的结论,推导出β-B曲线迭代速度最快时的最优形状参数β和加权渐进迭代逼近法的最优权值w;然后分别对其进行收敛性分析;最后给出数值实例分析形状参数取不同值时的迭代速度和迭代误差,所得实验结果证实了取最优形状参数和最优权值时收敛速度最快的结论。

关键词: β-B曲线, 渐进迭代逼近, 谱半径, 收敛速度

Abstract: In order to accelerate the convergence speed of the progressive-iterative approximation method and overcome the defect that the general B-spline curves cannot represent curves such as circles or ellipses, this paper discusses its(weighted)progressive-iterative approximation based on β-B curves. Compared with general cubic uniform polynomial B-spline curve, the former has better continuity. The iterative matrix of the(weighted)progressive-iterative approximation is obtained according to the selected β-B basis function. Based on the conclusion that the spectral radius is the smallest and the convergence speed is the fastest,the optimal shape parameters of the(weighted)progressive-iterative approximation and the optimal weight w of the weighted progressive-pterative approximation method are derived. Then this paper does convergence analysis on them respectively. Finally, numerical examples are given to analyze the iteration speed and iteration error when the shape parameters are different. The experimental results show that the convergence speed is the fastest when the shape parameter and weight are optimal.

Key words: β-B curve, progressive-iterative approximation, spectral radius, convergence rate

中图分类号: 

  • O241.5
[1] 齐东旭,田自贤,张玉心,等. 曲线拟合的数值磨光方法[J]. 数学学报,1975,18(3):173-184. QI Dongxu, TIAN Zixian, ZHANG Yuxin, et al. The method of numeric polish in curve fitting[J]. Acta Mathematica Sinica, 1975, 18(3):173-184.
[2] DE BOOR C. How does Agees smoothing method work?[R/OL]. 1979[2021-04-26]. https://ftp.cs.wisc.edu/Approx/agee.pdf.
[3] DELGADO J, PENA J M. Progressive iterative approximation and bases with the fastest convergence rates[J]. Computer Aided Geometric Design, 2007, 24(1):10-18.
[4] LIN Hongwei, BAO Hujun, WANG Guojin. Totally positive bases and progressive iteration approximation[J]. Computer & Mathematics with Applications, 2005, 50(3/4):575-586.
[5] LU Lizheng. Weighted progressive iteration approximation and convergence analysis[J]. Computer Aided Geometric Design, 2010, 27(2):129-137.
[6] 刘晓艳, 邓重阳. 非均匀三次B样条曲线插值的Jacobi-PIA算法[J]. 计算机辅助设计与图形学学报,2015, 27(3):484-491. LIU Xiaoyan, DENG Chongyang. Jacobi-PIA algorithm for non-uniform cubic B-spline curve interpolation[J]. Journal of Computer-Aided Design & Computer Graphics, 2015, 27(3):484-491.
[7] 张莉, 陆中华, 赵林, 等. 带多权值局部插值型的几何迭代法[J]. 计算机辅助设计与图形学学报, 2018, 30(9):1699-1704. ZHANG Li, LU Zhonghua, ZHAO Lin, et al. Local interpolation type of geometric iterative method with multiple weights[J]. Journal of Computer-Aided Design & Computer Graphics, 2018, 30(9):1699-1704.
[8] 韩旭里, 刘圣军. 三次均匀B样条的扩展[J].计算机辅助设计与图形学学报, 2003, 15(5):576-578. HAN Xuli, LIU Shengjun. An extension of the cubic uniform B-spline curve[J]. Journal of Computer-Aided Design & Computer Graphics, 2003, 15(5):576-578.
[9] 刘成志, 韩旭里, 李军成. 三次均匀B样条扩展曲线的渐进迭代逼近法[J]. 计算机辅助设计与图形学学报, 2019, 31(6):899-910. LIU Chengzhi, HAN Xuli, LI Juncheng. Progressive-iterative approximation by extension of cubic uniform B-spline curves[J]. Journal of Computer-Aided Design & Computer Graphics, 2019, 31(6):899-910.
[10] Barsky B A. Computer graphics and geometric modeling using beta-splines[J]. New York: Springer-Verlag, 1988:156.
[11] 严兰兰. 带形状参数的三角曲线曲面[J]. 东华理工大学学报(自然科学版), 2012, 35(2):197-200. YAN Lanlan. Trigonometric curves and surfaces with shape parameters[J]. Journal of East China Institute of Technology(Natural Science), 2012, 35(2):197-200.
[12] 蔺宏伟.几何迭代法及其应用综述[J]. 计算机辅助设计与图形学学报, 2015, 27(4):582-589. LIN Hongwei. Survey on geometric iterative methods with applications[J]. Journal of Computer-Aided Design & Computer Graphics, 2015, 27(4):582-589.
[13] 杨胜良. 三对角矩阵的特征值及其应用[J]. 数学的实践与认识,2010,40(3):155-160. YANG Shengliang. Eigenvalue of tridiagonal matric and its applications[J]. Mathematics in Practice and Theory, 2010, 40(3):155-160.
[14] CARNICER J M, DELGADO J, PENA J M. Richardson method and totally nonnegative linera systems[J]. Linear Algebra and its Applications, 2010, 433(11/12):2010-2017.
[1] 卢鹏丽,刘文智. 图的广义距离谱[J]. 《山东大学学报(理学版)》, 2020, 55(9): 19-28.
[2] 陈倩竹,胡海平. 基于最优解距离估计的光滑凸极小化的一阶算法[J]. 《山东大学学报(理学版)》, 2020, 55(4): 102-107.
[3] 刘婷婷,陈志勇,李晓琴*,杨文志. 随机变量序列的Berry-Esseen界[J]. 山东大学学报（理学版）, 2014, 49(03): 101-106.
[4] 薛秋芳1,2,高兴宝1*,刘晓光1. H-矩阵基于外推GaussSeidel迭代法的几个等价条件[J]. J4, 2013, 48(4): 65-71.
[5] 刘晓光,畅大为*. 亏秩线性方程组PSD迭代法的最优参数[J]. J4, 2011, 46(12): 13-18.
[6] 邹黎敏1, 姜友谊1， 胡兴凯2. 关于矩阵Frobenius范数的一个猜想[J]. J4, 2010, 45(4): 48-50.
[7] 冯立华,于桂海 . 与图谱有关的一个图兰定理[J]. J4, 2008, 43(6): 31-33 .
[8] 张新东,王秋华 . 避免二阶导数计算的Newton迭代法的一个改进[J]. J4, 2007, 42(7): 72-76 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发