三次PH曲线的曲率单调性及过渡曲线构造  

On Curvature Monotony for a PH Cubic Curve and Constructing Transition Curve

作　　者：郑志浩[1] 汪国昭[1] 

机构地区：[1]浙江大学数学系,杭州310027

出　　处：《计算机辅助设计与图形学学报》2014年第8期1219-1224,共6页Journal of Computer-Aided Design & Computer Graphics

基　　金：国家自然科学基金(61272300)

摘　　要：已有的用于构造过渡曲线的PH曲线均为五次,不仅次数高,而且相关参数的确定要涉及复杂的非线性方程组.通过引入次数较低的三次PH曲线,以避免在CAGD中出现以上问题.首先利用三次PH曲线控制多边形的边角几何特征,结合其曲率的导数表达式得到三次PH曲线曲率单调递增的充分条件;基于该条件,讨论如何使用三次PH曲线来构造相包含关系的圆弧间G2连续过渡曲线,分别计算了圆心距的取值范围以及证明了过渡曲线存在的唯一性,并给出曲线生成的算法步骤.通过实例与已有方法进行比较,其结果表明了该方法的有效性和实用性.The PH curves used to construct transition curves are all quintic,a PH quintic curve not only has higher degree but also the determination of its relevant parameters involves the complicated system of non-linear equations.PH cubic curves with lower degree are introduced to avoid the above problems in CAGD.Firstly,a sufficient condition for the curvature of a PH cubic curve to be monotonous is derived by using the geometric property of the legs and the angles of PH cubic curve control polygon and combining with the expression of its curvature derivative.Based on the condition,we discuss how to construct transition curve of G^2 contact,between two circles with one circle inside the other by PH cubic curve.We compute an attainable range of the distance between the centers of two circles and prove that the transition curve is unique respectively,the algorithm steps for constructing transition curve are given.Compared with the existing methods in examples,the results show that our method is effective and practical.

关 键 词：三次PH曲线 曲率单调 过渡曲线 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]