Print

发布时间: 2019-01-16
摘要点击次数:
全文下载次数:
DOI: 10.11834/jig.180318
2019 | Volume 24 | Number 1




    计算机图形学    




  <<上一篇 




  下一篇>> 





Möbius变换下四次有理抛物-PH曲线的${{\rm{C}}^{\rm{2}}} $ Hermite插值
expand article info 郭宇, 江平, 王剑敏, 刘植
合肥工业大学数学学院, 合肥 230000

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${{\rm{C}}^{\rm{2}}} $ Hermite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${{\rm{C}}^{\rm{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${{\rm{C}}^{\rm{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。

关键词

Möbius变换; 有理等距曲线; ${{\rm{C}}^{\rm{2}}} $Hermite插值; 绝对旋转数; 弹性弯曲能量

${{\rm{C}}^{\rm{2}}} $ Hermite interpolation based on quartic rational parabolic-PH curves by using Möbius transformation
expand article info Guo Yu, Jiang Ping, Wang Jianmin, Liu Zhi
School of Mathematics, Hefei University of Technology, Hefei 230000, China
Supported by: National Natural Science Foundation of China (11471093)

Abstract

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 ${{\rm{C}}^{\rm{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 ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation of quartic rational parabolic-PH curve is constructed through complex analysis. The data $ H_C^2 = \{ {R_0},{R_1},{T_0},{T_1},{V_0},{V_1}\} $ 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 ${{\rm{C}}^{\rm{2}}} $ curves after M bius transformation. This paper shows a concrete construction method of quartic rational parabolic-PH curves for ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation, whose tangents have three orders. By supposing the expression of $r\left( t \right) $, $F\left( t \right) $, $G\left( t \right) $, 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 formula. The exact value of the parameter are calculated by the ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation condition of the curve. Then, the quartic rational parabolic-PH curves formed by the M bius transformation are finally constructed. Result By providing a set of "reasonable" endpoints to be interpolated, we can obtain 12 ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation curves from the transformed quartic polynomial parabolic-PH curve under the initial interpolation condition and further obtain the ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation curves of the 12 quartic rational parabolic-PH curves under the initial interpolation condition. Numerical examples show the effectiveness of the algorithm. It is not clear and convenient to choose the appropriate interpolation curve from the 12 interpolation curves. We need to select the curves that satisfy the interpolated condition and can elastically handle the inflection points. Other interpolation curves may have cusp points, node points, closed loops, or obviously inconsistent with geometric design requirements. By combining the minimum absolute rotation number and the elastic bending energy minimization, the selection method for determining the optimal curve satisfying the interpolation condition is put forward. When the absolute rotation number and the elastic bending energy of the interpolated curve are minimized, the optimal curve is often obtained, which has better smoothness and natural shape that meet the needs of geometric design. The examples illustrate that the traditional quartic parabolic-PH curve can construct $ {{\rm{C}}^1}$ Hermite interpolation curves. However, the constraint of interpolation condition does not allow the direct construction of a curve with higher continuity. For traditional quintic PH curve, we cannot directly construct a curve with a continuity higher than $ {{\rm{C}}^1}$, whereas through M bius transformation, we can achieve ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation, which has a higher continuity than the traditional method. For the same set of given data, we construct the ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation curve from quintic rational PH curve and quartic rational parabolic-PH curve. Compared with the 18 quintic rational PH curves, we can achieve the optimal curve from the 12 quartic rational parabolic-PH curves with lower elastic bending energy. Hence, the quartic rational parabolic-PH curves constructed by our method have more natural geometry than the traditional quintic rational PH curves. Although parabolic-PH curves with eight degree can be used to construct ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation curves, the solution is complex, and the computation is large. Hence, through analysis and comparison, the quartic rational parabolic-PH curve presented in this paper has a simpler computation than quintic PH curves and parabolic-PH curves with eight degree. The interpolation results of the quartic rational parabolic-PH curve is more obvious, and the optimal curve best meets the requirement for the geometric design. Conclusion The use of ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation of quartic rational parabolic-PH curves constructed by the introduction of M bius transformation not only ensures low degree of interpolation curve but also ensures a higher continuity of interpolation conditions. It makes the calculation simpler and the interpolation effect more obvious compared with the traditional PH curve with odd number of order. Related research on the sub PH curve is of certain significance. This report is significant for the study of PH curves with even number of degree.

Key words

Möbius transformation; offset-rational curves; ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation; absolute rotation number; elastic bending energy

0 引言

等距曲线也称为平行线,是指曲线沿法向量方向距离为$d $的点的轨迹[1]。近年来,等距曲线在机器人行走轨迹、3维数控机床及铁路公路线性设计等众多领域有着重要作用,为计算机辅助几何设计(CAGD)的研究热点。

一般情况下,多项式曲线的弧长和等距线不具备有理形式,而OR(offset-rational)曲线作为一类具有精确有理等距曲线的特殊多项式参数曲线,其特有的优良性质引起了众多研究者的注意。1990年Farouki等人[2]首次提出了PH(pythagorean hodographs)曲线,并给出了平面参数曲线为OR曲线的一个充分条件,随后Farouki等人[3-4]又对PH曲线插值问题等做了大量研究,吕伟等人[5-6]又利用复分析、重新参数化和代数几何技术给出了一般参数多项式曲线为OR曲线的充分必要条件。2001年,陈国栋等人[7]利用八次抛物-PH曲线构造了OR曲线的${{\rm{C}}^{\rm{2}}} $ Hermite插值,但是由于曲线次数较高,计算较为复杂。2010年,Bartoň等人[8]提出运用Möbius变换描述有理曲线的构造。2012年,Lee等人[9]利用Möbius变换引入参数构造了三次PH曲线${{\rm{G}}^1} $ Hermite插值。截止目前,学者们对于奇数次曲线的研究已经涌现出大量的研究成果,但是很少涉及偶数次曲线的插值问题,近年来,桂校生等人[10]提出了四次抛物-PH曲线的$ {{\rm{C}}^1}$ Hermite插值,张威等人[11]研究了四次PH曲线的渐开线及其几何Hermite螺线插值,但是对偶数次曲线的其他类型插值鲜有研究。

本文主要研究了一类特殊的具有精确有理等距曲线的偶数次OR曲线的插值,使其达到连续性更高的插值条件,且同时具有曲线次数更低、插值效果更好等特点,即四次有理抛物-PH曲线的${{\rm{C}}^{\rm{2}}} $ Hermite插值,并说明了最优插值曲线具有曲线能量低、光顺性和自然形态特征好等特点,满足几何设计的需求。

1 预备知识

利用复分析的方法简述一般参数多项式曲线为OR曲线的充分必要条件及其分类情况。

$\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}:P\left( t \right) = \left( {x\left( t \right), y\left( t \right)} \right) = \sum\limits_{i = 0}^n {} {P_i}B_i^n\left( t \right)$$ n$次已适当参数化的平面多项式曲线,其在复平面上的相应表示为$ Z\left( t \right) = \sum\limits_{i = 0}^n {} {Z_i}B_i^n\left( t \right), {Z_i} = {\rm{ }}{x_i} + {\rm{i}}{y_i}$是控制顶点${\mathit{\boldsymbol{P}}_i} = \left( {{x_i}, {y_i}} \right) $对应的复数表示,$ \left\{ {B_i^n\left( t \right)} \right\}_{i = 0}^n$$ n$次Bernstein多项式基函数。

引理1[6]   $ Z(t)$ 为OR曲线的充要条件,即

$ Z'\left( t \right) = r\left( t \right)F\left( t \right){G^2}\left( t \right) $ (1)

式中,$r\left( t \right) $是实系数多项式,$F\left( t \right) $$G\left( t \right) $为复系数多项式,且$F\left( t \right) $的次数deg($F\left( t \right) $)≤1。

对于OR曲线可分为以下3种:

1) $F\left( t \right) $恒为常数(可设为1)时,为PH曲线;

2) $G\left( t \right) $恒为常数(可设为1)时,为抛物型曲线;

3) $F\left( t \right) $$G\left( t \right) $均不为常数时,其速端曲线$Z\prime (t) $由抛物型曲线速端曲线与PH曲线速端曲线的结合,称抛物-PH曲线。

在拓展的复平面$ {\mathit{\boldsymbol{C}}_\infty } = \mathit{\boldsymbol{C}} \cup \left\{ \infty \right\}$上,Möbius变换$\mathit{\Phi} \left( z \right) $是双射线性分式变换,即$\mathit{\Phi} \left( z \right) = \frac{{az + b}}{{cz + d}} $$ a, b, c, d$为复数,且$ad - bc \ne 0 $。其逆变换与一阶导数分别为

$ {\mathit{\Phi }^{ - 1}}\left( z \right) = \frac{{dz - b}}{{ - cz + a}} $

$ \mathit{\Phi '}\left( z \right) = \frac{{ad - bc}}{{{{\left( {cz + d} \right)}^2}}} $

定理1   若$\mathit{\Phi} \left( z \right) $是Möbius变换,$Z(t) $ 是多项式抛物-PH曲线,则$ s\left( t \right) = \left( {\mathit{\Phi} \circ Z} \right)(t)$是有理抛物-PH曲线,$\left( {\mathit{\Phi} \circ Z} \right)(t) $)表示$Z(t) $经过$\mathit{\Phi} \left( z \right) $的变换。

证明:$Z(t) $是多项式抛物-PH曲线,由引理1可设:$Z\prime \left( t \right){\rm{ }} = {\rm{ }}{r_1}\left( t \right){F_1}\left( t \right)G_1^2\left( t \right), {r_1}\left( t \right)$是实系数多项式,$ {F_1}\left( t \right)$${G_1}\left( t \right) $为复系数多项式且${\rm{deg}}\left( {{F_1}\left( t \right)} \right) \le 1$,故

$ \begin{array}{*{20}{c}} {s'\left( t \right) = \mathit{\Phi '}\left( {Z\left( t \right)} \right)Z'\left( t \right) = }\\ {\frac{{ad - bc}}{{{{\left( {cz + d} \right)}^2}}}{r_1}\left( t \right){F_1}\left( t \right)G_1^2\left( t \right)} \end{array} $

$ G_2^2\left( t \right){\rm{ }} =\frac{{ad - bc}}{{{{\left( {cz + d} \right)}^2}}}G_1^2\left( t \right)$,即$ s\prime \left( t \right) = {r_1}\left( t \right){F_1}\left( t \right)G_2^2\left( t \right), {G_2}\left( t \right)$为复系数多项式,故由引理1可得,$ s(t)$ 为抛物-PH曲线。

又由于$s\prime \left( t \right) = \mathit{\Phi} \prime \left( {Z\left( t \right)} \right)Z\prime (t) $,可得

$ \begin{array}{*{20}{c}} {\left| {s'\left( t \right)} \right| = \frac{{\left| {ad - bc} \right|}}{{{{\left| {cZ\left( t \right) + d} \right|}^2}}}\left| {Z'\left( t \right)} \right| = }\\ {\frac{{\left| {ad - bc} \right|}}{{{\mathop{\rm Re}\nolimits} {{\left( {cZ\left( t \right) + d} \right)}^2} + {\mathop{\rm Im}\nolimits} {{\left( {cZ\left( t \right) + d} \right)}^2}}}\left| {Z'\left( t \right)} \right|} \end{array} $

$ s(t)$为有理曲线。

综上所述,定理1得证。

引理2    $\mathit{\Phi} \left( z \right) $是Möbius变换,且$ \mathit{\Phi} \left( 0 \right) = 0$$ \mathit{\Phi} \left( 1 \right) = 1$,则存在非零的复数$ \alpha $使得

$ \mathit{\Phi }\left( z \right) = \frac{{\alpha z}}{{\left( {\alpha - 1} \right)z + 1}} $

2 插值问题的描述

定义1  给定平面上${{\rm{C}}^{\rm{2}}} $ Hermite插值数据$\mathit{\boldsymbol{H}}_C^2 = \left\{ {{R_0}, {R_1}, {T_0}, {T_1}, {V_0}, {V_1}} \right\}, {R_0}、{R_1} $表示始点与终点,${{T_0}、{T_1}} $表示${R_0}、{R_1} $处的一阶导数,$ {{V_0}、{V_1}}$表示${R_0}、{R_1} $处的二阶导数,构造多项式曲线$Z(t) $和Möbius变换$\mathit{\Phi} \left( z \right) $满足

$ \left\{ \begin{array}{l} \left( {\mathit{\Phi } \circ Z} \right)\left( 0 \right) = {R_0}\\ {\left( {\mathit{\Phi } \circ Z} \right)^\prime }\left( 0 \right) = {T_0},{\left( {\mathit{\Phi } \circ Z} \right)^{\prime \prime }}\left( 0 \right) = {V_0}\\ \left( {\mathit{\Phi } \circ Z} \right)\left( 1 \right) = {R_1}\\ {\left( {\mathit{\Phi } \circ Z} \right)^\prime }\left( 1 \right) = {T_1},{\left( {\mathit{\Phi } \circ Z} \right)^{\prime \prime }}\left( 1 \right) = {V_1} \end{array} \right. $ (2)

通过合适的变换、旋转、比例缩放,可以得到$ {R_0} = 0, {R_1} = 1$且Möbius变换为$\mathit{\Phi} \left( z \right) $$ = \frac{{\alpha z}}{{\left( {\alpha - 1} \right)z + 1}} $$ \alpha $为非零的复数。固定${R_0} = 0, {R_1} = 1 $,对于复数$ \alpha $,在Möbius变换下曲线的${{\rm{C}}^{\rm{2}}} $ Hermite插值条件由式(2)可化为

$ \left\{ \begin{array}{l} Z\left( 0 \right) = 0,\;\;\;\;\;\int_0^1 {Z'\left( t \right){\rm{d}}t} = 1\\ Z'\left( 0 \right) = \frac{1}{\alpha }{T_0},\;\;\;\;Z'\left( 1 \right) = \alpha {T_1}\\ Z''\left( 0 \right) = \frac{{{V_0}\alpha + 2\left( {\alpha - 1} \right)T_0^2}}{{{\alpha ^2}}}\\ Z''\left( 1 \right) = {V_1}\alpha + 2\alpha \left( {\alpha - 1} \right)T_1^2 \end{array} \right. $ (3)

$ \alpha = 1$,Möbius变换$\mathit{\Phi} \left( z \right) $是恒等变换。

3 四次有理抛物-PH曲线${{\rm{C}}^{\rm{2}}} $ Hermite插值的构造

给定两个端点$ {R_0}、{R_1}$,以${{T_0}、{T_1}} $表示$ {R_0}、{R_1}$处的一阶导数,以${{V_0}、{V_1}} $表示$ {R_0}、{R_1}$处的二阶导数,用四次抛物-PH曲线来进行${{\rm{C}}^{\rm{2}}} $ Hermite插值,即曲线导矢为三次。由式(1)知,可设

$ r\left( t \right) = 1 $

$ F\left( t \right) = a\left( {1 - t} \right) + bt $

$ G\left( t \right) = \left( {1 - t} \right) + ct $

式中,$a, b, c $为待定复系数。即

$ Z'\left( t \right) = \left[ {a\left( {1 - t} \right) + bt} \right]{\left[ {\left( {1 - t} \right) + ct} \right]^2} $ (4)

$ \begin{array}{*{20}{c}} {Z''\left( t \right) = \left( {b - a} \right){{\left[ {\left( {1 - t} \right) + ct} \right]}^2} + }\\ {2\left[ {a\left( {1 - t} \right) + bt} \right]\left[ {\left( {1 - t} \right) + ct} \right]\left( {c - 1} \right)} \end{array} $ (5)

将式(4)展开,并利用积分关系式

$ \begin{array}{*{20}{c}} {\int_0^1 {\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right){{\left( {1 - t} \right)}^{n - k}}{t^k}{\rm{d}}t} = }\\ {\frac{1}{{n + 1}}\sum\limits_{j = k + 1}^{n + 1} {\left( \begin{array}{l} n + 1\\ j \end{array} \right){{\left( {1 - t} \right)}^{n + 1 - j}}{t^j}} } \end{array} $ (6)

对式(4)进行积分可得$Z\left( t \right) = \sum\limits_{i = 0}^4 {} {Z_i}B_i^4(t)$的表达式,此时初始点${R_0} $对应上述积分常数,从而求得该曲线Bézier表示的控制顶点为

$ \begin{array}{*{20}{c}} {\left( {{Z_0},{Z_1},{Z_2},{Z_3},{Z_4}} \right) = }\\ {\left[ \begin{array}{l} {R_0},{Z_0} + \frac{a}{4},{Z_1} + \frac{{2ac + b}}{{12}},\\ {Z_2} + \frac{{a{c^2} + 2bc}}{{12}},{Z_3} + \frac{{b{c^2}}}{4} \end{array} \right]} \end{array} $ (7)

由式(3)(4)(5)(7)可得,对于经过Möbius变换的曲线${{\rm{C}}^{\rm{2}}} $ Hermite插值条件有

$ \left\{ \begin{array}{l} \frac{\alpha }{4} + \frac{{2ac + b}}{{12}} + \frac{{a{c^2} + 2bc}}{{12}} + \frac{{b{c^2}}}{4} = 1\\ Z'\left( 0 \right) = a = \frac{1}{\alpha }{T_0}\\ Z'\left( 1 \right) = b{c^2} = \alpha {T_1}\\ Z''\left( 0 \right) = b - a + 2a\left( {c - 1} \right) = \\ \frac{{{V_0}\alpha + 2\left( {\alpha - 1} \right)T_0^2}}{{{\alpha ^2}}}\\ Z''\left( 1 \right) = \left( {b - a} \right){c^2} + 2bc\left( {c - 1} \right) = \\ {V_1}\alpha = 2\alpha \left( {\alpha - 1} \right)T_1^2 \end{array} \right. $ (8)

方程组式(8)可化简为

$ \left\{ \begin{array}{l} \frac{a}{4} + \frac{{2ac + b}}{{12}} + \frac{{a{c^2} + 2bc}}{{12}} + \frac{{b{c^2}}}{4} = 1\\ b - a + 2a\left( {c - 1} \right) = \frac{{a{V_0} + 2a{T_0}\left( {{T_0} - a} \right)}}{{{T_0}}}\\ \left( {b - a} \right)c + 2b\left( {c - 1} \right) = \frac{{bc}}{{{T_1}}}{V_1} + 2bc\left( {b{c^2} - {T_1}} \right) \end{array} \right. $ (9)

利用方程组式(9)可以求出对应的复系数$ a, b, c$。由此得到四次多项式抛物-PH曲线的Bézier曲线表示,即

$ \begin{array}{*{20}{c}} {Z\left( t \right) = \sum\limits_{i = 0}^4 {{Z_i}B_i^4\left( t \right)} = }\\ {{Z_0}{{\left( {1 - t} \right)}^4} + 4{Z_1}{{\left( {1 - t} \right)}^3}t + }\\ {6{Z_2}{{\left( {1 - t} \right)}^2}{t^2} + 4{Z_3}\left( {1 - t} \right){t^3} + {Z_4}{t^4} = }\\ {{Z_0}{{\left( {1 - t} \right)}^4} + 4\left( {{Z_0} + \frac{a}{4}} \right){{\left( {1 - t} \right)}^3}t + }\\ {6\left( {{Z_1} + \frac{{2ac + b}}{{12}}} \right){{\left( {1 - t} \right)}^2}{t^2} + }\\ {4\left( {{Z_2} + \frac{{a{c^2} + 2bc}}{{12}}} \right)\left( {1 - t} \right){t^3} + }\\ {\left( {{Z_3} + \frac{{b{c^2}}}{4}} \right){t^4}} \end{array} $ (10)

由定理1、引理2,便得到经过变换的四次有理抛物-PH曲线$\mathit{\Phi}(Z(t)) $,即

$ \mathit{\Phi }\left( {Z\left( t \right)} \right) = \frac{{\alpha Z\left( t \right)}}{{\left( {\alpha - 1} \right)Z\left( t \right) + 1}} $ (11)

式中,$ \alpha = \frac{{{T_0}}}{a}$$ \alpha = \frac{{b{c^2}}}{{{T_1}}}$$Z(t) $由式(10)确定。

综上所述可得出定理2。

定理2   给定平面上${{\rm{C}}^{\rm{2}}} $ Hermite插值数据$\mathit{\boldsymbol{H}}_C^2 = \left\{ {{R_0}, {R_1}, {T_0}, {T_1}, {V_0}, {V_1}} \right\} $,则四次有理抛物-PH曲线的${{\rm{C}}^{\rm{2}}} $ Hermite插值曲线为$\mathit{\Phi}(Z(t)) $ $ = \frac{{\alpha Z\left( t \right)}}{{\left( {\alpha - 1} \right)Z\left( t \right) + 1}}$,参数$ \alpha = \frac{{{T_0}}}{a}$$\alpha = \frac{{b{c^2}}}{{{T_1}}} $$Z(t) $由式(7)(10)确定,复系数$ a, b, c$由方程组式(9)确定。

4 数值实例

若给定两个端点${R_0} $(0, 0),${{R_1}} $ (1, 0),相应的一阶导数${T_0} = 1 - 2{\rm{i}}, {T_1} = 2 + {\rm{i}} $和二阶导数${V_0} = 1 + {\rm{i}}, {V_1} = 1 + 2{\rm{i}} $,即$ \mathit{\boldsymbol{H}}_C^2 = \{ 0, 1, 1 - 2{\rm{i}}, 2 + {\rm{i}}, 1 + {\rm{i}}, 1 + 2{\rm{i}}\} $。按照上述算法,应用数学工具MATLAB可以求出$a, b, c $的所有12组复根, 即

$ \begin{array}{l} a = \left\{ {3.583\;3 - 1.360\;5{\rm{i}};{\rm{3}}{\rm{.558}}\;{\rm{2}} - 0.630\;8{\rm{i}};} \right.\\ \;\;\;\;3.564\;6 - 3.106\;6{\rm{i}};1.635\;5 + 0.646\;7{\rm{i}};\\ \;\;\;\;4.049\;7 - 2.989\;8{\rm{i}};3.182\;7 - 0.664\;4{\rm{i}};\\ \;\;\;\;2.205\;2 - 2.651\;2{\rm{i;2}}{\rm{.437}}\;{\rm{4 - 2}}{\rm{.263}}\;{\rm{8i;}}\\ \;\;\;\;0.500\;3 + 0.759\;1{\rm{i}};0.136\;3 - 0.141\;2{\rm{i;}}\\ \;\;\;\;\left. {0.125\;5 + 0.558\;5{\rm{i;0}}{\rm{.321}}\;{\rm{4 - 0}}{\rm{.055}}\;{\rm{9i}}} \right\} \end{array} $

$ \begin{array}{l} b = \left\{ {1.202\;7 - 0.253\;8{\rm{i;0}}{\rm{.147}}\;{\rm{4}} - {\rm{1}}{\rm{.537}}\;{\rm{2i;}}} \right.\\ \;\;\;\;\;\; - 1.030\;7 + 1.428\;6{\rm{i;7}}{\rm{.421}}\;{\rm{6}} - 5.107\;9{\rm{i;}}\\ \;\;\;\;\;\; - 4.478\;0 + 14.719\;2{\rm{i;}} - 4.719\;1 - 0.136\;1{\rm{i;}}\\ \;\;\;\;\;\;0.166\;7 - 0.506\;2{\rm{i;1}}{\rm{.069}}\;{\rm{9}} + 4.686\;7{\rm{i}};\\ \;\;\;\;\;\;5.096\;6 + 0.061\;6{\rm{i}}; - 0.385\;9 - 1.262\;4{\rm{i}};\\ \;\;\;\;\;\;\left. {0.082\;4 + 0.084\;4{\rm{i; - 0}}{\rm{.073}}\;{\rm{5 + 0}}{\rm{.323}}\;{\rm{2i}}} \right\} \end{array} $

$ \begin{array}{l} c = \left\{ { - 1.341\;7 - 0.364\;2{\rm{i}}; - 1.215\;4 - 0.863\;3{\rm{i;}}} \right.\\ \;\;\;\;\; - 0.983\;2 + 1.364\;3{\rm{i}}; - 0.663\;7 - 0.220\;3{\rm{i}};\\ \;\;\;\;\; - 0.423\;4 + 0.377\;8{\rm{i}}; - 0.076\;6 - 0.866\;8{\rm{i;}}\\ \;\;\;\;\;0.122\;9 + 0.979\;5{\rm{i}};{\rm{0}}{\rm{.324}}\;{\rm{1}} - 0.061\;8{\rm{i;}}\\ \;\;\;\;\;0.328\;9 - 0.137\;3{\rm{i}};0.632\;3 + 1.382\;4{\rm{i}};\\ \;\;\;\;\;\left. {2.186\;8 - 2.204\;4{\rm{i;2}}{\rm{.274}}\;{\rm{3}} - 2.112\;8{\rm{i}}} \right\} \end{array} $

由此将每一组$a, b, c $代入式(7)可求出其余控制点$ {Z_1}、{Z_2}、{Z_3}$,再分别将这些控制点及$a, b, c $代入式(10), 可求出12条四次多项式抛物-PH曲线在变换后的初始插值条件下的${{\rm{C}}^{\rm{2}}} $ Hermite插值曲线$Z(t) $,如图 1所示。进一步将$Z(t) $代入式(11), 可求出12条四次有理抛物-PH曲线在初始插值条件下的${{\rm{C}}^{\rm{2}}} $ Hermite插值曲线$\mathit{\Phi}(Z(t)) $,如图 2所示。

图 1 四次多项式抛物-PH曲线${{\rm{C}}^{\rm{2}}} $ Hermite插值曲线$Z(t) $
Fig. 1 Constructing ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation curves $Z(t) $ by quartic polynomial parabolic-PH curve
图 2 四次有理抛物-PH曲线${{\rm{C}}^{\rm{2}}} $ Hermite插值曲线$\mathit{\Phi}(Z(t)) $
Fig. 2 Constructing ${{\rm{C}}^{\rm{2}}} $ Hermite interpolation curves $\mathit{\Phi}(Z(t)) $ by quartic rational parabolic-PH curve

5 最优插值形状的选取

根据给定的插值条件和“合理”的插值数据,可以得到若干条插值曲线,然而仅仅从观察的角度来选择合适的插值曲线不够明确和方便。需要从中选取满足插值数据且能灵活处理拐点的曲线,而其他插值曲线可能出现尖点、结点、闭环或明显不符合几何设计的需求。

1995年,Farouki等人[3]基于复分析的方法构造了五次PH曲线的满足$ {{\rm{C}}^1}$ Hermite插值数据,并利用“最小绝对旋转数”来选取最优解。1996年,Farouki又提出采用“最小弹性弯曲能量”来选取最佳曲线[4]

绝对旋转数$ R = \frac{1}{{{\rm{2 \mathsf{ π} }}}}\int_0^1 {} |\kappa \left( t \right)||Z\prime (t)|{\rm{d}}t$

弹性弯曲能量$ E = \int_0^1 {} {\kappa ^2}\left( t \right)|Z\prime \left( t \right)|{\rm{d}}t$

式中,$\kappa \left( t \right) $为曲线曲率。

对于基于Möbius变换的四次有理抛物-PH曲线的${{\rm{C}}^{\rm{2}}} $ Hermite插值而言:

绝对旋转数为

$ \begin{array}{*{20}{c}} {R\left( {\left( {\mathit{\Phi } \circ Z} \right)\left( t \right)} \right) = }\\ {\frac{1}{{2{\rm{ \mathsf{ π} }}}}\int_0^1 {\left| {\kappa \left( t \right)} \right|\left| {{{\left( {\mathit{\Phi } \circ Z} \right)}^\prime }\left( t \right)} \right|{\rm{d}}t} } \end{array} $ (12)

弹性弯曲能量为

$ E\left( {\left( {\mathit{\Phi } \circ Z} \right)\left( t \right)} \right) = \int_0^1 {{\kappa ^2}\left( t \right)\left| {{{\left( {\mathit{\Phi } \circ Z} \right)}^\prime }\left( t \right)} \right|{\rm{d}}t} $ (13)

式中,$ \kappa \left( t \right)$$\left( {\mathit{\Phi} \circ Z} \right)(t) $)的曲率,即

$ \kappa \left( t \right) = \frac{{\left| {{{\left( {\mathit{\Phi } \circ Z} \right)}^\prime }\left( t \right) \times {{\left( {\mathit{\Phi } \circ Z} \right)}^\prime }^\prime \left( t \right)} \right|}}{{{{\left| {{{\left( {\mathit{\Phi } \circ Z} \right)}^\prime }\left( t \right)} \right|}^3}}} $

对于第4节实例中所得12条四次有理抛物-PH曲线${{\rm{C}}^{\rm{2}}} $ Hermite插值曲线$\mathit{\Phi}(Z(t)) $,可利用式(12)分别算得对应的曲线绝对旋转数如表 1所示。

表 1 图 2中12条曲线$\mathit{\Phi}(Z(t)) $对应的绝对旋转数
Table 1 Absolute rotation number for 12 curves $\mathit{\Phi}(Z(t)) $ in Fig. 2

下载CSV
1 2 3 4 5 6 7 8 9 10 11 12
R 0.988 9 0.872 5 1.286 6 0.950 1 1.316 3 0.858 1 0.473 7 0.387 5 0.400 7 0.355 7 0.449 8 0.386 8

表 1中可知,若插值曲线的绝对旋转数相对较大时,往往存在尖点、结点或闭环等明显不符合几何设计的情形,而具有相对较小的绝对旋转数的曲线往往其几何形状能够满足设计要求,12条曲线中第10条曲线的绝对旋转数最小,对应图 1图 2中标示的粗线。通过绝对旋转数可有效地排除奇异情况,然而并不足以给出插值曲线的最佳选择。为此,进一步利用式(13)计算曲线的弹性弯曲能量来选择最佳插值曲线,如表 2所示。

表 2 图 2中12条曲线$\mathit{\Phi}(Z(t)) $对应的弹性弯曲能量
Table 2 Elastic bending energy for 12 curves $\mathit{\Phi}(Z(t)) $ in Fig. 2

下载CSV
1 2 3 4 5 6 7 8 9 10 11 12
E 822.36 80.212 154.12 209.99 89.828 130.43 6.313 9 4.968 8 4.934 1 4.416 8 5.970 2 4.930 6

表 1表 2可知,第10条插值曲线的绝对旋转数和弹性弯曲能量同时达到最小,对应图 1图 2中标示的粗线,具有相对的光顺性和较好自然形状特征,满足几何设计的需求。

对于传统的四次抛物-PH曲线,可以构造出$ {{\rm{C}}^1}$ Hermite插值,但由于插值条件的限定,使其无法直接构造出连续性更高的插值曲线。而对于传统五次PH曲线而言,可设$ r\left( t \right) = 1, G\left( t \right) = a{\left( {1 - t} \right)^2} + b\left( {1 - t} \right)t + c{t^2}, a, b, c$为待定复系数,此时所得五次PH曲线$Z(t) $$Z\prime \left( t \right) = {\left( {a{{\left( {1 - t} \right)}^2} + b\left( {1 - t} \right)t + c{t^2}} \right)^2} $ ,通过Möbius变换可使其达到连续性较高的${{\rm{C}}^{\rm{2}}} $ Hermite插值[12]

给定$ \mathit{\boldsymbol{H}}_C^2 = \{ 0, 1, - 2 - 2{\rm{i}}, 2 + {\rm{i}}, 1 + {\rm{i}}, 1 + 2{\rm{i}}\} $,分别在Möbius变换下构造出五次有理PH曲线和四次有理抛物-PH曲线C2 Hermite插值曲线,其最优曲线所对应的弹性弯曲能量如表 3所示。

表 3 最优曲线所对应的弹性弯曲能量对比
Table 3 Comparison of elastic bending energy corresponding to the optimal curve

下载CSV
最优曲线对应的E值
五次有理PH曲线 1.33
四次有理抛物-PH曲线 0.839 9

表 3可知,利用本文方法构造的12条四次有理抛物-PH曲线中最优曲线的弹性弯曲能量比得到的18条五次有理PH曲线中最优曲线的弹性弯曲能量更小,数值实例表明利用本文方法构造的四次有理抛物-PH曲线具有比传统五次有理PH曲线更自然的几何形状。

另外,用八次抛物-PH曲线也可以直接构造${{\rm{C}}^{\rm{2}}} $ Hermite插值,可设

$ r\left( t \right) = 1,F\left( t \right) = a\left( {1 - t} \right) + bt $

$ G\left( t \right) = {\left( {1 - t} \right)^3} + c{\left( {1 - t} \right)^2}t + d\left( {1 - t} \right){t^2} + e{t^3} $

式中,$a, b, c, d, e $为待定复系数,即$Z\prime \left( t \right) = \left[ {a\left( {1 - t} \right) + bt} \right]{\left[ {{{\left( {1 - t} \right)}^3} + c{{\left( {1 - t} \right)}^2}t + d\left( {1 - t} \right){t^2} + e{t^3}} \right]^2} $但曲线次数高,计算过程复杂。

通过分析对比可知,本文提出的四次有理抛物-PH曲线通过Möbius变换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,较之五次PH曲线和八次抛物-PH曲线次数更低,具有更小的计算量,且达到插值条件的同时,插值效果明显,能够得出满足几何设计需求的最优曲线。

6 结论

本文通过Möbius变换引入参数,利用复分析的方法构造了四次抛物-PH曲线的${{\rm{C}}^{\rm{2}}} $ Hermite插值,可以通过调整给出“合理”的端点插值数据,使所得12条插值曲线中至少存在一条满足几何设计需求的曲线,并结合最小绝对旋转数和弹性弯曲能量最小化两种准则选择出最优曲线。本文构造的Möbius变换下的四次有理抛物-PH曲线的${{\rm{C}}^{\rm{2}}} $ Hermite插值与现有研究成果相比较,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,对进一步探索研究偶数次PH曲线具有一定意义。

参考文献

  • [1] Wang G J, Wang G Z, Zheng J M. Computer Aided Geometric Design[M]. Beijing: Higher Education Press, 2001. [ 王国瑾, 汪国昭, 郑建民. 计算机辅助几何设计[M]. 北京: 高等教育出版社, 2001.]
  • [2] Farouki R T, Sakkalis T. Pythagorean Hodographs[M]. New York: IBM Corp, 1990.
  • [3] Farouki R T, Neff C A. Hermite Interpolation by Pythagorean Hodograph Quintics[M]. Washington: American Mathematical Society, 1995.
  • [4] Farouki R T. The elastic bending energy of Pythagorean-hodograph curves[J]. Computer Aided Geometric Design, 1996, 13(3): 227–241. [DOI:10.1016/0167-8396(95)00024-0]
  • [5] Zheng J M. A rational parametrization of the offset curve[J]. Chinese Science Bulletin, 1994(20): 1915. [郑建民. 关于等距曲线的有理参数化[J]. 科学通报, 1994(20): 1915. ] [DOI:10.3321/j.issn:0023-074X.1994.20.027]
  • [6] Lv W. Offset-rational parametric plane curves[J]. Computer Aided Geometric Design, 1995, 12(6): 601–616. [DOI:10.1016/0167-8396(94)00036-R]
  • [7] Chen G D, Wang G J. ${{\rm{C}}} ^{{2}}$ hermite interpolation by offset-rational curves[J]. Journal of Engineering Graphics, 2000, 21(3): 64–69. [陈国栋, 王国瑾. 有理等距曲线的${{\rm{C}}} ^{{2}} $ Hermite插值[J]. 工程图学学报, 2000, 21(3): 64–69. ] [DOI:10.3969/j.issn.1003-0158.2000.03.012]
  • [8] Bartoň M, Jüttler B, Wang W P. Construction of rational curves with rational rotation-minimizing frames via Möbius transformations[C]//Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces. Berlin: Springer-Verlag, 2010, 5862: 15-25.
  • [9] Lee S, Lee H C, Lee M R, et al. Hermite interpolation using Möbius transformations of planar Pythagorean-Hodograph cubics[J]. Abstract and Applied Analysis, 2012, 2012: 560246.
  • [10] Gui X S. The construction of quartic pythagorean hodograph curve[D]. Hefei: Hefei University of Technology, 2010. [桂校生.四次Pythagorean Hodograph速端曲线的构造[D].合肥: 合肥工业大学, 2010.] http://cdmd.cnki.com.cn/Article/CDMD-10359-2010246511.htm
  • [11] Zhang W, Wang G J. Involutes of quartic PH curves and their geometric Hermite interpolation[J]. Journal of Computer-Aided Design & Computer Graphics, 2011, 23(2): 216–222. [张威, 王国瑾. 四次PH曲线的渐开线及其几何Hermite螺线插值[J]. 计算机辅助设计与图形学学报, 2011, 23(2): 216–222. ]
  • [12] Ding C. Research on the issues and application of the PH curves[D]. Hefei: Hefei University of Technology, 2015. [丁晨. PH曲线的若干问题及应用研究[D].合肥: 合肥工业大学, 2015.] http://cdmd.cnki.com.cn/Article/CDMD-10359-1015722589.htm