吳偉棟，楊勛年

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一類代數(shù)-三角函數(shù)表示的空間PH曲線及其應(yīng)用

吳偉棟1,2，楊勛年1

(1. 浙江大學(xué)數(shù)學(xué)科學(xué)學(xué)院，浙江 杭州 310027；2.山東理工大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，山東 淄博 255049)

在代數(shù)-三角函數(shù)空間Ω=span{1,···,θ+1, sin, cos,sin, ···,θcos}定義了一類空間曲線。通過選取合適的積分核函數(shù)，該曲線在-平面上的投影具有內(nèi)蘊(yùn)表示或整條曲線是PH曲線。曲線的笛卡爾坐標(biāo)可由預(yù)定義的核函數(shù)通過積分計(jì)算得到。此外，給出了不同核函數(shù)表示的積分曲線的Hermite插值算法。對(duì)給定的邊界條件，積分核函數(shù)系數(shù)可通過求解方程組得到。最后，利用PH曲線設(shè)計(jì)了一族標(biāo)架，并用于構(gòu)造有理形式的掃掠曲面。實(shí)驗(yàn)表明，分片定義的掃掠曲面在脊線處1連續(xù)，在其余連接處達(dá)到近似1連續(xù)。

幾何Hermite插值；混合函數(shù)空間；PH曲線；標(biāo)架

給定邊界數(shù)據(jù)構(gòu)造光順曲線是曲線曲面造型中的重要課題。傳統(tǒng)的曲線造型方法采用Bézier、NURBS曲線，用戶可以通過調(diào)整曲線的控制頂點(diǎn)改變曲線的形狀，便于交互，且計(jì)算方便。但是難以控制曲率的變化。對(duì)于平面曲線造型，為了滿足產(chǎn)品設(shè)計(jì)中的美觀需求，具有單調(diào)曲率的各式各樣的螺線被用于1Hermite插值[1-5]或者2Hermite插值[6-9]。此外，HARADA等[10]提出一類美觀曲線(也稱log-aesthetic曲線)，這類曲線具有線性對(duì)數(shù)曲率圖。隨后，MIURA等[11-12]推導(dǎo)出曲線的表達(dá)式。YOSHIDA和SAITO[13]將曲線表示成關(guān)于切向角的函數(shù)形式，進(jìn)一步分析美觀曲線的性質(zhì)，并給出了一種構(gòu)造1連續(xù)的曲線段的方法。WU和YANG[14]利用曲線的內(nèi)蘊(yùn)方程，將曲線的曲率半徑表示為多項(xiàng)式的形式，定義了一類平面曲線。這類曲線的弧長是關(guān)于原參數(shù)的多項(xiàng)式函數(shù)，且等距線具有有理形式。

在數(shù)控加工領(lǐng)域，比如設(shè)計(jì)刀具的運(yùn)動(dòng)軌跡時(shí)，往往需要多項(xiàng)式曲線的弧長和等距線具備有理形式。為此，F(xiàn)AROUKI和SAKKALIS[15]引入了畢達(dá)哥拉斯速端(Pythagorean hodograph, PH)平面曲線，具有多項(xiàng)式形式的弧長和有理形式的等距線。隨后，F(xiàn)AROUKI和SAKKALIS[16]將平面PH曲線推廣到空間PH曲線。相較于PH曲線定義在多項(xiàng)式空間上，ROMANI等[17]給出一類定義在代數(shù)-三角函數(shù)空間上的平面曲線，PH曲線可看作這類曲線的子集。類似于Bézier曲線的表示形式，這類曲線可以寫成B-basis和控制點(diǎn)的組合形式。在過去的二十幾年，涌現(xiàn)出大量的PH曲線插值算法。由于奇數(shù)次的PH曲線是本原的，曲線插值算法通常采用三次[18-24]或五次PH曲線[25-29]。

此外，空間PH曲線可以用于構(gòu)造有理掃掠曲面(sweep surface)[30-31]和剛體運(yùn)動(dòng)[32]，在幾何造型中具有重要應(yīng)用。特別地，有理形式的掃掠曲面是最受歡迎的。但是軌跡線和截面線具有有理形式并不能保證掃掠曲面是有理的，相應(yīng)的正交標(biāo)架也必須是有理的。目前應(yīng)用最多的3種標(biāo)架就是Frenet標(biāo)架、Euler-Rodrigues標(biāo)架(簡(jiǎn)稱ERF)和旋轉(zhuǎn)最小標(biāo)架(rotation-minimizing frame, RMF)。最為熟悉的是Frenet標(biāo)架[33-34]，但是其有些不可忽視的缺點(diǎn)：在曲線的拐點(diǎn)(曲率為零的點(diǎn))處沒有定義；主、副法向量會(huì)繞著切向量發(fā)生不必要的旋轉(zhuǎn)；對(duì)于一般的多項(xiàng)式或有理多項(xiàng)式曲線，標(biāo)架不具有理形式。為了避免Frenet標(biāo)架的缺點(diǎn)，CHOI和HAN[35]提出了Euler-Rodrigues標(biāo)架。ERF雖然不是幾何內(nèi)蘊(yùn)的，但是具有有理形式，在拐點(diǎn)處不是奇異的。為了使活動(dòng)標(biāo)架不會(huì)繞著軌跡線的切向量發(fā)生旋轉(zhuǎn)，BISHOP[36]提出了旋轉(zhuǎn)最小標(biāo)架。KLOK[37]將其刻畫為常微分方程的解，并將其用于構(gòu)造掃掠曲面。由于難以精確計(jì)算旋轉(zhuǎn)最小標(biāo)架，往往采用逼近方法來計(jì)算RMF[38-39]。由于PH曲線具有很多優(yōu)良性質(zhì)，往往采用PH曲線插值算法來構(gòu)造逼近曲線，并為PH曲線計(jì)算標(biāo)架。特別地，有理形式的標(biāo)架在CAD中是非常重要的，可以與大部分CAD系統(tǒng)的表示方式兼容，便于計(jì)算。由于空間PH曲線具有有理切向量，構(gòu)造有理旋轉(zhuǎn)最小標(biāo)架(rational rotation-minimizing frames, RRMF)受到廣泛的關(guān)注[40-45]。利用具有RRMF的五次PH曲線，F(xiàn)AROUKI等[43]提出一種1Hermite插值算法，其算法只適合部分插值數(shù)據(jù)。隨后，對(duì)于任意次數(shù)的PH曲線，F(xiàn)AROUKI和SAKKALIS[44]給出了RRMF存在的充分必要條件。截至2016年，F(xiàn)AROUKI[45]從基本理論、算法和應(yīng)用方面總結(jié)了具有RRMF的曲線的發(fā)展。

1 一類代數(shù)-三角函數(shù)表示的空間PH曲線

空間曲線在平面上的投影曲線是內(nèi)蘊(yùn)定義的，投影曲線的笛卡爾坐標(biāo)可以通過曲線的內(nèi)蘊(yùn)方程獲得，即

圖1 ρ0(θ)和ρ1(θ)定義的積分曲線

經(jīng)過簡(jiǎn)單計(jì)算，很容易得到曲線的一階、二階導(dǎo)矢及一些幾何量的表示。

(1) 一階導(dǎo)矢為

(2) 二階導(dǎo)矢為

(3) 導(dǎo)矢模長為

(4) 空間曲線的弧長為

(5)平面投影曲線的弧長為

則

2 G1Hermite插值

圖20()和1()為二次多項(xiàng)式的空間PH曲線(0=0.01(1–)+0.04,1=0.5(1–)+0.3,=/(12π),∈[0,12π])

圖3 G1 Hermite插值

令

其中

式(3)等價(jià)于

其中

圖4 G1 Hermite 插值(P1=(0,0,0), P2=(7.1, 4.5, 2), T1=, T2=,其中u(t)=u0(1–t)+ u1t, v(t)=v0(1–t)2+2v1(1–t)t+, , u0=2.2813, u1=3.4795, v0=0.5385, v1=–2.2567, v2=2.5079)

3 基于空間PH曲線插值的掃掠曲面造型

在計(jì)算機(jī)動(dòng)畫、路徑規(guī)劃、掃掠曲面構(gòu)造等諸多應(yīng)用中，計(jì)算空間曲線的正交標(biāo)架是一項(xiàng)重要的工作[32]。有理形式的標(biāo)架在CAD中是非常重要的，可以與大部分CAD系統(tǒng)的表示方式兼容，便于計(jì)算。本文提出的空間PH曲線的導(dǎo)矢模長是多項(xiàng)式，所以曲線具有有理單位切向量、多項(xiàng)式弧長。這些良好的性質(zhì)可用于構(gòu)造有理形式的標(biāo)架，進(jìn)而構(gòu)造有理形式的掃掠曲面。

3.1 定義一族標(biāo)架

3.2 掃掠曲面(Swept surface)

有理形式標(biāo)架的一個(gè)重要應(yīng)用是構(gòu)造有理形式的掃掠曲面。具有顯式表示的曲面可以被精確計(jì)算，減小計(jì)算的復(fù)雜性，且可與CAD系統(tǒng)相兼容。此外，有理形式的掃掠曲面在數(shù)控機(jī)床中也有重要應(yīng)用。

圖5 標(biāo)架和掃掠曲面

圖6 形狀扭曲的掃掠面

圖7 設(shè)計(jì)一族標(biāo)架，構(gòu)造掃掠曲面并刻畫剛體運(yùn)動(dòng)

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A Family of Spacial PH Curves Represented by Algebraic-Trigonometric Functions and Their Applications

WU Weidong1,2, YANG Xunnian1

(1. School of Mathematical Sciences, Zhejiang University, Hangzhou Zhejiang 310027, China;2. School of Mathematics and Statistics, Shandong University of Technology, Zibo Shandong 255049, China)

A class of spacial curves are defined over the algebraic-trigonometric space Ω=span{1,···,θ+1, sin, cos,sin, ···,θcos}. By choosing proper integral kernels, the projection of the spacial integral curve on the-plane has intrinsic definition, or the whole spacial curve is a PH curve. The Cartesian coordinates of the curve can be explicitly evaluated by integrals of the predefined kernels. Besides, techniques of interpolation of integral curves with different integral kernels have been studied. Given the boundary data, the coefficients within the kernel functions are obtained by solving a system. Finally, we use PH curves to design a family of frames, which can be used to construct a rational swept surface. Experimental results show that the piecewise swept surface is1continuous at the ridge line and is nearly1continuous at the remaining junctions.

geometric Hermite interpolation; mixed space of functions; PH curves; frame

TP 391

10.11996/JG.j.2095-302X.2018020295

A

2095-302X(2018)02-0295-09

2017-07-07；

2017-09-02

國家自然科學(xué)基金項(xiàng)目(11290142)

吳偉棟(1988-)，女，山東淄博人，講師，博士。主要研究方向?yàn)橛?jì)算機(jī)輔助幾何設(shè)計(jì)。E-mail：wuweidong.happy@163.com