Mbius变换下四次有理抛物-PH曲线的C^2 Hermite插值  

C^2 Hermite interpolation based on quartic rational parabolic-PH curves by using Mbius transformation

作　　者：郭宇[1] 江平[1] 王剑敏 刘植[1] Guo Yu;Jiang Ping;Wang Jianmin;Liu Zhi(School of Mathematics,Hefei University of Technology,Hefei 230000,China)

机构地区：[1]合肥工业大学数学学院,合肥230000

出　　处：《中国图象图形学报》2019年第1期96-102,共7页Journal of Image and Graphics

基　　金：国家自然科学基金项目(11471093)~~

摘　　要：目的曲线插值问题在机器人设计、机械工业、航天工业等诸多现代工业领域都有广泛的应用,而已知端点数据的Hermite插值是计算机辅助几何设计中一种常用的曲线构造方法,本文讨论了一种偶数次有理等距曲线,即四次抛物-PH曲线的C2Hermite插值问题。方法基于Mbius变换引入参数,利用复分析的方法构造了四次有理抛物-PH曲线的C^2 Hermite插值,给出了具体插值算法及相应的Bézier曲线表示和控制顶点的表达式。结果通过给出"合理"的端点插值数据,以数值实例表明了该算法的有效性,所得12条插值曲线中,结合最小绝对旋转数和弹性弯曲能量最小化两种准则给出了判定满足插值条件最优曲线的选择方法,并以具体实例说明了与其他插值方法的对比分析结果。结论本文构造了Mbius变换下的四次有理抛物-PH曲线的C^2 Hermite插值,在保证曲线次数较低的情况下,达到了连续性更高的插值条件,计算更为简单,插值效果明显,较之传统奇数次PH曲线具有更加自然的几何形状,对偶数次PH曲线的相关研究具有一定意义。Objective The offset curve, also known as the parallel curve, refers to the locus of points along the normal vector direction with distance d. In recent years, the offset curve has played an important role in many fields and is widely applied in computer-aided geometric design (CAGD). In general, the arc length and offset curve of the polynomial curve have no rational form, and the offset-rational (OR) curve is a special polynomial parameter curve with exactly rational offset curves. The special properties of the curve have attracted the attention of many researchers. In recent years, the interpolation problem of OR curves has been widely studied. The problem of curve interpolation is widely used in many modern industrial fields, such as robot design, machinery industry, and space industry. The Hermite interpolation of given endpoint is a common method to construct a curve in CAGD. The C^2 Hermite interpolation problem of the quartic parabolic-PH curve, which is an even order of offset rational curve is discussed in this paper. Method Based on the parameters introduced by M bius transformation, a bijective linear fractional transformation, the C^2 Hermite interpolation of quartic rational parabolic-PH curve is constructed through complex analysis. The data HC^2={R0,R1,T0,T1,V0,V1} to be interpolated is given with R0 and R1 referring to the two end points, T0 and T1 for the tangent vectors at R0 and R1, and V0 and V1 for the second tangent vectors at R0 and R1. By appropriate transformation, rotation, and scaling, making R0=0 and R1=1, we can further obtain the interpolation conditions for C^2 curves after M bius transformation. This paper shows a concrete construction method of quartic rational parabolic-PH curves for C^2 Hermite interpolation, whose tangents have three orders. By supposing the expression of r(t),F(t),G(t), the first-and second-order derivative of the curve can be obtained. The corresponding expression of the control points and the Bézier curve can be obtained by using the integral relation form

关 键 词：MOBIUS变换 有理等距曲线 C^2 HERMITE插值 绝对旋转数 弹性弯曲能量 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]