利用距离与内能极小平滑链接Bézier曲线  

Smoothing linked Bézier curves by distance and internal energy minimization

作　　者：李军成 刘成志 刘珊君 Li Juncheng;Liu Chengzhi;Liu Shanjun(College of Mathematics and Finance,Hunan University of Humanities,Science and Technology,Loudi 417000,China)

机构地区：[1]湖南人文科技学院数学与金融学院,娄底417000

出　　处：《中国图象图形学报》2021年第10期2450-2461,共12页Journal of Image and Graphics

基　　金：湖南省自然科学基金项目(2021JJ30373);湖南省教育厅科研项目(18A415)。

摘　　要：目的对于满足低阶连续的链接Bézier曲线,提高曲线之间的连续性以达到平滑的目的,需要对曲线的控制顶点进行相应调整。因此,可根据具体的目标对需要调整的控制顶点进行优化选取,使得平滑后的链接曲线满足相应的要求。针对这一问题,给出了3种目标下优化调整控制顶点的方法。方法首先对讨论的问题进行描述,分别指出链接Bézier曲线从C^(0)连续平滑为C^(1)连续和从C^(1)连续平滑为C^(2)连续两种情形需调整的控制顶点;然后分别给出两种情形下,以新旧控制顶点距离极小为目标、曲线内能极小为目标、新旧控制顶点距离与曲线内能同时极小为目标,对链接Bézier曲线进行平滑的方法,最后对3种极小化方法进行对比,并指出了不同方法的适用场合。结果数值算例表明,距离极小化方法调整后的控制顶点偏离原控制顶点的距离相对较小,适合于控制顶点取自于实物时的应用场合;内能极小化方法获得的链接曲线内能相对较小,适合于要求曲线能量尽可能小的应用场合;距离与内能同时极小化方法兼顾了新旧控制顶点的距离和链接曲线的内能,适合于对两个目标都有要求的应用场合。结论提出的方法为链接Bézier曲线的平滑提供了3种有效手段,且易于实现,对其他类型链接曲线的平滑具有参考价值。Objective The Bézier curve is a widely used tool in the representation of parametric curves.At present,most computer aided design(CAD)systems take the Bézier curve as a basic module.When using the Bézier curve for geometric modeling,it is often necessary to link multiple curves to meet the needs of designing complex curves.In order to satisfy the continuity between the linked Bézier curves,the control points that satisfy the corresponding continuity conditions need to be selected in advance.For linked Bézier curves with low order continuity,the control points of the curves can be adjusted to improve the continuity so as to smooth the linked curves.In theory,the continuity between linked Bézier curves can be improved by arbitrarily adjusting the control points of the curves to make the corresponding continuity conditions hold.However,this kind of adjustment of control points without specific objectives is often unable to meet the needs of practical applications.Given that the continuity can be improved by adjusting the control points of the linked Bézier curves,the control points that need to be adjusted can be optimized according to some specific targets,so that the smooth linked curves meet the corresponding requirements.In practical applications,if the control points of the curves are taken from the real objects,it is often hoped that the distance between the new control points and the original control points is as small as possible.Hence,the minimum distance between the new control points and the original control points can be used as the target to optimize the control points that need to be adjusted.In addition,energy minimization has become a common method for constructing curves and surfaces in CAD and related fields.Thus,the minimum energy can be used as a target to optimize the control points that need to be adjusted.In this study,three methods for smoothing the linked Bézier curves from C;to C;and from C;to C;by distance and internal energy minimization are given.Method First,the problems to be d

关 键 词：BÉZIER曲线 链接 平滑 控制顶点优化 距离极小 内能极小 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术]