三次Bézier曲线曲面的新扩展  

New Extension of Cubic Bézier Curves and Surfaces

作　　者：涂超 严兰兰 徐梦豪 TU Chao;YAN Lanlan;XU Menghao(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,江西南昌330013

出　　处：《湖南科技大学学报（自然科学版）》2022年第3期113-124,共12页Journal of Hunan University of Science And Technology：Natural Science Edition

基　　金：国家自然科学基金资助项目(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资助(GJJ160558)。

摘　　要：通过将五次Bernstein基函数进行重新组合,构造由4个含单参数的多项式形成的调配函数,并由之定义结构与三次Bézier曲线曲面相同的新曲线曲面.新曲线不仅继承了Bézier曲线的一系列基本性质,而且在控制顶点给定的前提下,通过形状参数来调整曲线对控制多边形的逼近程度;更特别的是,在常规的C^(2)光滑拼接条件下,新曲线之间可以自动达到C^(2)∩FC^(3)连续,在G^(2)光滑拼接条件下,可以自动达到G^(3)连续.为了使形状参数的选取有迹可循,给出使曲线弧长、曲率、曲率变化率近似最小时,参数的计算公式.新曲面具有与新曲线对应的诸多优点.By recombining the quintic Bernstein basis functions,a configuration function formed by four polynomials with one parameter is constructed,and a new surface with the same structure as the cubic Bézier curve and surface is defined.The new curve can not only inherit a series of basic properties of the Bézier curve,but also adjusts the degree of approximation of the curve to the control polygon through the shape parameters under the premise of the given control vertex.What’s more,under the normalC^(2)smooth connection conditions,the new curves can automatically reachC^(2)∩FC^(3)continuity,while under the condition ofG^(2)smooth connection conditions,it can automatically reachG^(3)continuity.In order to make the selection of the shape parameter traceable,the formulas for calculating the parameter are given to minimize the approximate arc length,curvature and rate of change of curvature.The new surface has many advantages corresponding to the new curve.

关 键 词：BÉZIER曲线曲面 BERNSTEIN基函数 形状参数 光滑拼接 C~2∩FC~3连续 G~3连续 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术]