基于4点分段的FC^(3)连续组合曲线  

FC^(3) curve of continuous combination based on four points segmentation

作　　者：严兰兰 魏子华 揭梦柔 马金位 YAN Lanlan;WEI Zihua;JIE Mengrou;MA Jinwei(College of Science,East China University of Technology,Nanchang 330013,China)

机构地区：[1]东华理工大学理学院,南昌330013

出　　处：《中国科技论文》2023年第3期340-350,共11页China Sciencepaper

基　　金：国家自然科学基金资助项目(11261003,11761008);江西省自然科学基金资助项目(20161BAB211028);江西省教育厅科技项目(GJJ160558)。

摘　　要：为了保留B样条方法的局部控制性和自动光滑性,同时提升曲线的连续阶,并赋予曲线独立于控制顶点和节点矢量的形状调整自由度,提出了一种与3次B样条曲线结构相同的新曲线。首先,在代数多项式空间上,以幂基为基底,预设调配函数的初步表达式,其中包含若干待定系数;然后,根据欲定义的曲线的主要性质,反推出调配函数需满足的基本性质,并据此建立关于调配函数中待定系数的方程组;之后,借助MATLAB软件编程计算,确定待定系数之间的关系,从而确定调配函数的原始表达式,其中包含自由参数;最后,对原始表达式进行整理,使其中的自由参数具有明确的几何意义,从而确定调配函数的最终表达式。基于该调配函数,采用与3次B样条曲线相同的分段定义方式,构造了一种基于4点分段的组合曲线。该曲线不仅继承了B样条曲线的诸多优点,而且突破了3次B样条曲线的连续阶,同时拥有了灵活的局部形状调整参数。另外,对曲线中形状参数的取值范围进行扩展,即可构造插值于局部或全部控制顶点的曲线。与3次B样条方法相比,新方法既可以满足形状设计中对连续性方面更高的要求,又可以满足对形状调整以及逼近与插值方面更多的要求。In order to keep the local controllability and automatic smoothness of B-spline method,improve the continuity order of the cubic B-spline curve,and give curves the freedom of shape adjustment independent to control points and knot vector,a new curve with the same structure as the cubic B-spline curve was proposed.Firstly,the preliminary expression of the blending function,which contains several undetermined coefficients,was given in algebraic polynomial space and based on the power basis.Then,according to the main properties of the curve to be defined,the basic properties that the blending functions need to satisfy were derived,and the equations of the undetermined coefficients in the blending function were established accordingly.With the help of MATLAB programming,the relationship between the undetermined coefficients was determined,so as to determine the original expression of the blending function,which contains free parameters.Finally,the original expression was arranged to make the free parameters have a clear geometric meaning and give the final expression of the blending function.Based on this blending function,a combined curve based on four points segmentation was constructed by using the same segmental definition method as the cubic B-spline curve.This curve not only inherits many advantages of B-spline curve,but also breaks through the continuity order of the cubic B-spline curve,and has local shape parameters which can adjust the shape conveniently and flexibly.In addition,a curve interpolated to part or all control points can be constructed by extending the value range of shape parameters in the curve.Compared with the cubic B-spline method,the new method can not only meet the higher requirements of continuity in shape design,but can also meet more requirements in shape adjustment and approximation and interpolation.

关 键 词：曲线设计 B样条曲线 参数连续性 弗朗内特标架连续性 形状调整 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术]