插值给定数据点的四次PH曲线构造  

Construction of quartic PH curves via interpolating given points

作　　者：方林聪[1] 阳诚砖 邸文钰 刘芳[1] Fang Lincong;Yang Chengzhuan;Di Wenyu;Liu Fang(School of Information Management and Artificial Intelligence,Zhejiang University of Finance and Economics,Hangzhou 310018,China;School of Economics and Management,Tongji University,Shanghai 200082,China)

机构地区：[1]浙江财经大学信息管理与人工智能学院,杭州310018 [2]同济大学经济与管理学院,上海200082

出　　处：《中国图象图形学报》2020年第7期1473-1480,共8页Journal of Image and Graphics

基　　金：浙江省自然科学基金项目(LY18F020023,LQ19F020003,LY19F020011)。

摘　　要：目的PH(Pythagorean hodograph)曲线由于具备有理等距曲线、弧长可精确计算等优良的几何性质,广泛应用于数控加工和路径规划等方面。曲线插值是曲线构造的主要手段之一,虽然对PH曲线的Hermite插值方法进行了广泛研究,但插值给定数据点的构造方法仍有待突破,为推广四次PH曲线的应用范围,提出了一种新的四次PH曲线的3点插值问题解决方法。方法从四次PH曲线的代数充分必要条件出发,在该曲线的Bézier控制多边形中引入辅助控制顶点,指出其中实参数的几何意义,该实参数可作为形状调节因子对构造曲线进行交互。对给定的3个平面型值点进行参数化确定相应的参数值;通过对四次PH曲线一阶导数积分得到曲线的显式表达,其中包含一个待定复常量,将给定的约束点代入曲线的显式表达式得到关于待定复常量的一元二次复方程,求解该复方程并反求Bézier控制顶点得到符合约束条件的四次PH曲线。结果实验对通过构造插值给定数据点的四次PH曲线进行比较,当形状调节因此改变时,曲线形状可进行有效交互。每次交互得到两条四次PH曲线,通过弧长、弯曲能量、绝对旋转数的计算得到最优曲线,并构造得到PH曲线的等距线。结论本文方法给定的形状调节参数具有明确的代数意义和几何意义,本文方法易于实现,可有效进行交互。Objective The problem of interpolating three distinct planar points using quartic Pythagorean hodograph(PH)curves is studied.PH curves comprise an important class of polynomial curves that form a mathematical foundation of most current computer-aided design(CAD)tools.By incorporating special algebraic structures into their hodograph curves,PH curves exhibit many advanced properties over ordinary polynomial parametric curves.These properties include polynomial arc-length functions and rational offset curves.Thus,PH curves are considered a sophisticated solution for a variety of difficult issues arising in applications(e.g.,tool paths)in the fields of computer numerically controlled machining and realtime motion control.For example,the arc-length of a PH curve can be computed without numerical integration,accelerating algorithms for numerically controlled machining.The offsets of a PH curve can also be represented exactly rather than being approximated in CAD systems.Thus,analyzing and manipulating PH curves are of considerable practical value in CAD and other applications.PH curves can be represented as widely used Bézier curves.The most intuitive and efficient method for constructing PH curves is by adjusting the control points of Bézier curves under conditions that guarantee PH properties.Therefore,a variety of methods for identifying PH curves are developed.Another important application of PH curves is the geometric construction of these curves.Considerable work has been conducted on Hermite interpolation with different degrees of PH curves.However,methods for interpolating three or more planar points have been rarely studied.Method The necessary and sufficient condition for a planar curve to be a PH curve is a form of a product of complex polynomials,and a Bézier curve and its first derivative are Bernstein polynomials,which are a form of the sum of Bernstein basis functions.We derive a system of complex nonlinear equations by considering the compatibility of the two forms.The geometric meanings of the coef

关 键 词：计算机辅助设计 BÉZIER曲线 控制多边形 等距曲线 四次PH曲线 插值 

分 类 号：TP391.41[自动化与计算机技术—计算机应用技术]