一類代數(shù)-三角函數(shù)表示的空間PH曲線及其應用

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曲線及其應用

吳偉棟1,2，楊勛年1

(1. 浙江大學數(shù)學科學學院，浙江 杭州 310027；2.山東理工大學數(shù)學與統(tǒng)計學院，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)蘊表示或整條曲線是PH曲線。曲線的笛卡爾坐標可由預定義的核函數(shù)通過積分計算得到。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曲線設計了一族標架，并用于構造有理形式的掃掠曲面。實驗表明，分片定義的掃掠曲面在脊線處1連續(xù)，在其余連接處達到近似1連續(xù)。

幾何Hermite插值；混合函數(shù)空間；PH曲線；標架

給定邊界數(shù)據(jù)構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)整曲線的控制頂點改變曲線的形狀，便于交互，且計算方便。但是難以控制曲率的變化。對于平面曲線造型，為了滿足產(chǎn)品設計中的美觀需求，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數(shù)曲率圖。隨后，MIURA等[11-12]推導出曲線的表達式。YOSHIDA和SAITO[13]將曲線表示成關于切向角的函數(shù)形式，進一步分析美觀曲線的性質(zhì)，并給出了一種構造1連續(xù)的曲線段的方法。WU和YANG[14]利用曲線的內(nèi)蘊方程，將曲線的曲率半徑表示為多項式的形式，定義了一類平面曲線。這類曲線的弧長是關于原參數(shù)的多項式函數(shù)，且等距線具有有理形式。

在數(shù)控加工領域，比如設計刀具的運動軌跡時，往往需要多項式曲線的弧長和等距線具備有理形式。為此，F(xiàn)AROUKI和SAKKALIS[15]引入了畢達哥拉斯速端(Pythagorean hodograph, PH)平面曲線，具有多項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曲線可以用于構造有理掃掠曲面(sweep surface)[30-31]和剛體運動[32]，在幾何造型中具有重要應用。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，相應的正交標架也必須是有理的。目前應用最多的3種標架就是Frenet標架、Euler-Rodrigues標架(簡稱ERF)和旋轉最小標架(rotation-minimizing frame, RMF)。最為熟悉的是Frenet標架[33-34]，但是其有些不可忽視的缺點：在曲線的拐點(曲率為零的點)處沒有定義；主、副法向量會繞著切向量發(fā)生不必要的旋轉；對于一般的多項式或有理多項式曲線，標架不具有理形式。為了避免Frenet標架的缺點，CHOI和HAN[35]提出了Euler-Rodrigues標架。ERF雖然不是幾何內(nèi)蘊的，但是具有有理形式，在拐點處不是奇異的。為了使活動標架不會繞著軌跡線的切向量發(fā)生旋轉，BISHOP[36]提出了旋轉最小標架。KLOK[37]將其刻畫為常微分方程的解，并將其用于構造掃掠曲面。由于難以精確計算旋轉最小標架，往往采用逼近方法來計算RMF[38-39]。由于PH曲線具有很多優(yōu)良性質(zhì)，往往采用PH曲線插值算法來構造逼近曲線，并為PH曲線計算標架。特別地，有理形式的標架在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，便于計算。由于空間PH曲線具有有理切向量，構造有理旋轉最小標架(rational rotation-minimizing frames, RRMF)受到廣泛的關注[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、算法和應用方面總結了具有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)蘊定義的，投影曲線的笛卡爾坐標可以通過曲線的內(nèi)蘊方程獲得，即

圖1 ρ0(θ)和ρ1(θ)定義的積分曲線

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

(1) 一階導矢為

(2) 二階導矢為

(3) 導矢模長為

(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

在計算機動畫、路徑規(guī)劃、掃掠曲面構造等諸多應用中，計算空間曲線的正交標架是一項重要的工作[32]。有理形式的標架在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，便于計算。本文提出的空間PH曲線的導矢模長是多項式，所以曲線具有有理單位切向量、多項式弧長。這些良好的性質(zhì)可用于構造有理形式的標架，進而構造有理形式的掃掠曲面。

3.1 定義一族標架

3.2 掃掠曲面(Swept surface)

有理形式標架的一個重要應用是構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算，減小計算的復雜性，且可與CAD系統(tǒng)相兼容。此外，有理形式的掃掠曲面在數(shù)控機床中也有重要應用。

圖5 標架和掃掠曲面

圖6 形狀扭曲的掃掠面

圖7 設計一族標架，構造掃掠曲面并刻畫剛體運動

4 結束語

[1] Meek D S, Walton D J. Clothoid spline transition spirals [J]. Mathematics of Computation, 1992, 59(199): 117-133.

[2] HELD M, SPIELBERGER C. A smooth spiral tool path for high speed machining of 2D pockets [J]. Computer- Aided Design, 2009, 41(7): 539-550.

[3] Walton D J, Meek D S.1interpolation with a single Cornu spiral segment [J]. Journal of Computational and Applied Mathematics, 2009, 223(1): 86-96.

[4] Meek D S, Saito T, Walton D J, et al. Planar two-point Hermite interpolating log-aesthetic spirals [J]. Journal of Computational and Applied Mathematics, 2012, 236(17): 4485-4493.

[5] Yang X. Geometric Hermite interpolation by logarithmic arc splines [J]. Computer Aided Geometric Design, 2014, 31(9): 701-711.

[6] Goodman T N T, Meek D S, Walton D J. An involute spiral that matches2Hermite data in the plane [J]. Computer Aided Geometric Design, 2009, 26(7): 733-756.

[7] Ait-Haddou R, Biard L.2approximation of an offset curve by Tschirnhausen quartics [C]// Mathematical Methods for Curves and Surfaces. Nashville: Vanderbilt University Press, 1994: 1-10.

[8] Kurnosenko A. Applying inversion to construct planar, rational spirals that satisfy two-point2Hermite data [J]. Computer Aided Geometric Design, 2010, 27(3): 262-280.

[9] Kurnosenko A I. Two-point2Hermite interpolation with spirals by inversion of hyperbola [J]. Computer Aided Geometric Design, 2010, 27(6): 474-481.

[10] Harada T, Yoshimoto F, Moriyama M. An aesthetic curve in the field of industrial design [C]// IEEE Symposium on Visual Languages. New York: IEEE Press, 1999: 38-47.

[11] Miura K T. A general equation of aesthetic curves and its self-affinity [J]. Computer-Aided Design and Applications, 2006, 3(1-4): 457-464.

[12] Miura K T, Sone J, Yamashita A, et al. Derivation of a general formula of aesthetic curves [C]// Proceedings of the Eighth International Conference on Humans and Computers (HC2005). Abingdon: Taylor & Francis Group, 2005: 166-171.

[13] Yoshida N, Saito T. Interactive aesthetic curve segments [J]. The Visual Computer, 2006, 22(9-11): 896-905.

[14] Wu W, Yang X. Geometric Hermite interpolation by a family of intrinsically defined planar curves [J]. Computer-Aided Design, 2016, 77: 86-97.

[15] Farouki R T, Sakkalis T. Pythagorean hodographs [J]. IBM Journal of Research and Development, 1990, 34(5): 736-752.

[16] Farouki R T, Sakkalis T. Pythagorean-hodograph space curves [J]. Advances in Computational Mathematics, 1994, 2(1): 41-66.

[17] Romani L, Saini L, Albrecht G. Algebraic-trigonometric Pythagorean-hodograph curves and their use for Hermite interpolation [J]. Advances in Computational Mathematics, 2014, 40(5): 977-1010.

[18] Meek D S, Walton D J. Geometric Hermite interpolation with Tschirnhausen cubics [J]. Journal of Computational and Applied Mathematics, 1997, 81(2): 299-309.

[19] Byrtus M, Bastl B.1Hermite interpolation by PH cubics revisited [J]. Computer Aided Geometric Design, 2010, 27(8): 622-630.

[20] Jakli? G, Kozak J, Krajnc M, et al. On interpolation by Planar cubic2Pythagorean-hodograph spline curves [J]. Mathematics of Computation, 2010, 79(269): 305-326.

[21] Jüttler B, M?urer C. Cubic Pythagorean-hodograph spline curves and applications to sweep surface modeling [J]. Computer-Aided Design, 1999, 31(1): 73-83.

[22] Pelosi F, Farouki R T, Manni C, et al. Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics [J]. Advances in Computational Mathematics, 2005, 22(4): 325-352.

[23] Kwon S H. Solvability of1Hermite interpolation by spatial Pythagorean-hodograph cubics and its selection scheme [J]. Computer Aided Geometric Design, 2010, 27(2): 138-149.

[24] Jakli? G, Kozak J, Krajnc M, et al. An approach to geometric interpolation by Pythagorean-hodograph curves [J]. Advances in Computational Mathematics, 2012, 37(1): 123-150.

[25] Farouki R T, Neff C A. Hermite interpolation by Pythagorean-hodograph quintics [J]. Mathematics of Computation, 1995, 64(212): 1589-1609.

[26] Farouki R T, Manni C, Sestini A. Shape- preserving interpolation by1and2PH quintic splines [J]. IMA Journal of Numerical Analysis, 2003, 23(2): 175-195.

[27] Pelosi F, Sampoli M L, Farouki R T, et al. A control polygon scheme for design of planar2PH quintic spline curves [J]. Computer Aided Geometric Design, 2007, 24(1): 28-52.

[28] Farouki R T, Giannelli C, Manni C, et al. Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures [J]. Computer Aided Geometric Design, 2008, 25(4-5): 274-297.

[29] Farouki R T, Manni C, Sestini A. Spatial2PH quintic splines [C]//Curve and Surface Design. Saint Malo: Nashboro Press, 2008: 147-156.

[30] Jüttler B. Generating rational frames of space curves via Hermite interpolation with Pythagorean hodograph cubic splines [J]. Geometric Modelling and Processing, 1998, 98: 83-106.

[31] M?urer C, Jüttler B. Rational approximation of rotation minimizing frames using Pythagorean- hodograph cubics [J]. Journal for Geometry and Graphics, 1999, 3(2): 141-159.

[32] Krajnc M, Vitrih V. Motion design with Euler–Rodrigues frames of quintic Pythagorean- hodograph curves [J]. Mathematics and Computers in Simulation, 2012, 82(9): 1696-1711.

[33] Wagner M G, Ravani B. Curves with rational Frenet-serret motion [J]. Computer Aided Geometric Design, 1997, 15(1): 79-101.

[34] Kreyszig E. Differential geometry [M]. New York: Dover Publications, 1991: 12-20.

[35] Choi H I, Han C Y. Euler-rodrigues frames on spatial Pythagorean-hodograph curves [J]. Computer Aided Geometric Design, 2002, 19(8): 603-620.

[36] Bishop R L. There is more than one way to frame a curve [J]. The American Mathematical Monthly, 1975, 82(3): 246-251.

[37] Klok F. Two moving coordinate frames for sweeping along a 3D trajectory [J]. Computer Aided Geometric Design, 1986, 3(3): 217-229.

[38] Farouki R T, Han C Y. Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves [J]. Computer Aided Geometric Design, 2003, 20(7): 435-454.

[39] Wang W, Jüttler B, Zheng D, et al. Computation of rotation minimizing frames [J]. ACM Transactions on Graphics, 2008, 27(1): 1-18

[40] Farouki R T, Giannelli C, Manni C, et al. Quintic space curves with rational rotation-minimizing frames [J]. Computer Aided Geometric Design, 2009, 26(5): 580-592.

[41] Farouki R T. Quaternion and Hopf map characterizations for the existence of rational rotation- minimizing frames on quintic space curves [J]. Advances in Computational Mathematics, 2010, 33(3): 331-348.

[42] Barton M, Jüttler B, Wang W.Construction of rational curves with rational rotation-minimizing frames via m?bius transformations [C]//International Conference on Mathematical Methods for Curves and Surfaces. Berlin: Springer Press, 2008: 15-25.

[43] Farouki R T, Giannelli C, Sestini A. Geometric design using space curves with rational rotation-minimizing frames [C]//International Conference on Mathematical Methods for Curves and Surfaces. Berlin: Springer Press, 2008: 194-208.

[44] Farouki R T, Sakkalis T. Rational rotation-minimizing frames on polynomial space curves of arbitrary degree [J]. Journal of Symbolic Computation, 2010, 45(8): 844-856.

[45] FAROUKI R T. Rational rotation-minimizing frames-recent advances and open problems [J]. Applied Mathematics and Computation, 2016, 272: 80-91.

[46] Hui K C, Tan S T. Display techniques and boundary evaluation of a Sweep-CSG modeler [J]. The Visual Computer, 1991, 8(1): 18-34.

[47] Xu G, Hui K, Ge W, et al. Direct manipulation of free-form deformation using curve-pairs [J]. Computer-Aided Design, 2013, 45(3): 605-614.

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

國家自然科學基金項目(11290142)

吳偉棟(1988-)，女，山東淄博人，講師，博士。主要研究方向為計算機輔助幾何設計。E-mail：wuweidong.happy@163.com