构造曲率单调的组合二次h-Bézier曲线  

Constructing Combined Quadratic h-Bezier Curves with Monotone Curvature

作　　者：李林[1] 解滨[2] 韩力文[1,3,4] LI Lin;XIE Bin;HAN Liwen(School of Mathematics Sciences,Hebei Normal University,Shijiazhuang 050024,China;College of Computer and Cyber Security,Hebei Normal University,Shijiazhuang 050024,China;Hebei Key Laboratory of Computational Mathematics and Applications,Shijiazhuang 050024,China;Hebei International Joint Research Center for Mathematics and Interdisciplinary Science,Shijiazhuang 050024,China)

机构地区：[1]河北师范大学数学科学学院,石家庄050024 [2]河北师范大学计算机与网络空间安全学院,石家庄050024 [3]河北省计算数学与应用重点实验室,石家庄050024 [4]河北省数学与交叉科学国际联合研究中心,石家庄050024

出　　处：《计算机科学》2023年第7期119-128,共10页Computer Science

基　　金：国家自然科学基金(62076088);河北省自然科学基金(A2018205103);河北师范大学科研基金资助项目(L2020Z02)。

摘　　要：h-Bézier曲线(h>0)又被称为Pólya曲线,它具有与经典Bézier曲线(h=0)一致的诸多优良性质。为此,文中研究了二次h-Bézier曲线具有单调曲率的充要条件及其构造算法。首先,讨论二次h-Bézier曲线曲率极值的存在性,得到曲线具有单调曲率的充要条件;通过引入曲率临界圆,给出判断二次h-Bézier曲线曲率单调性的几何方法,即检查二次h-Bézier曲线的中间控制点是否在曲率临界圆上或圆内;并由此得到构造具有单调曲率的二次h-Bézier曲线的两种算法,通过调节形状参数h可保证曲线具有单调递减或单调递增的曲率。其次,研究两条二次h-Bézier曲线的光滑拼接,基于对二次h-Bézier曲线性质的分析,选择第二条曲线在肩点处与第一条曲线的端点实现拼接,得到G~2拼接的充要条件;讨论参数对拼接曲线形状的影响。最后,构造出同时满足G~2拼接、曲率单调递减(或单调递增)的组合二次h-Bézier曲线。数值实例显示了组合二次h-Bézier曲线的造型优势和灵活性。The h-Bézier curves(h>0),also known as Pólya curves,share many excellent properties with classical Bézier curves(h=0).In this paper,the necessary and sufficient conditions for quadratic h-Bézier curves with monotonic curvature and its construction algorithm are studied.First,the sufficient and necessary conditions for quadratic h-Bézier curves with monotone curvature are obtained by discussing the existence of the extremes of the curves.By introducing curvature critical circles of the curvature,the monotony for the curvature of the quadratic h-Bézier curve can be directly verified by checking whether the middle control point of the curve is on or inside the curvature critical circle.Two algorithms for construct quadratic h-Bezier curves with monotonic curvature are obtained,ensuring that the curves can have monotonically decreasing or increasing curvature by adjusting the shape parameter h.Secondly,the G2 smooth blending of two quadratic h-Bézier curves is studied.Based on the analysis of the properties of the quadratic h-Bézier curves,the shoulder point of the second curve is selected to join with the end point of the first curve,and the necessary and sufficient conditions are obtained and the influence of parameters on the shape of the blending curve is discussed.Finally,the combined quadratic h-Bézier curve with decreasing(or increasing)is constructed.The numerical examples show the modeling advantage and flexibility of the combined quadratic h-Bézier curve.

关 键 词：h-Bézier曲线 单调曲率 曲率临界圆 G~2拼接 肩点 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]