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1 IAENG Internatonal Journal of Appled Mathematcs, 45:4, IJAM_45_4_18 Degree Reducton of Ds Ratonal Bézer Curves Usng Mult-obectve Optmzaton Technques Mao Sh Abstract In ths paper, we start by ntroducng a novel ds ratonal Bézer based on parallel proecton, whose propertes are also dscussed. Then applyng weghted least squares, multobectve optmzaton technques and constraned quadratc programmng, we acheve mult-degree reducton of ths nd of ds ratonal Bézer curve. The paper also gves error estmaton and shows some numercal examples to llustrate the valdty of theoretcal reasonng. Index Terms Ds ratonal Bézer curve, Mult-degree reducton, Weghted least squares, Constraned quadratc programmng, Mult-obectve optmzaton methods. I. INTRODUCTION BECAUSE operatons of geometrc obects n current Computer-Aded Desgn systems are based on floatng pont arthmetc, representatons of geometrc obects are naccurate and geometrcal computatons are approxmate. In order to deal wth ths problem, nterval arthmetc s used n the felds. In 1992, Sederberg and Farou [1] formally ntroduced the concept of nterval Bézer curve that can transfer a complete descrpton of approxmaton errors along wth the curves to applcatons n other systems. Inspred by Sederberg s wor, Hu et al. [2] [3] [4] researched the algorthms for curve and surface ntersectons and sold modelng. Chen and Lou [5] dscussed the problem of boundng nterval Bézer curve wth lower degree nterval Bézer curve. However, as Chen ponted out [5], nterval curve possesses two shortcomngs: nterval generally enlarge rapdly n a computatonal process and rectangular ntervals are not rotatonally symmetrc. To overcome these shortcomngs, Ln and Rone [6] appled a ds to replace a rectangle. The correspondng nterval curve are called ds Bézer curve. Snce nterval Bézer curve can t represent conc precsely, Hu et al. [3] [4] ntroduced nterval non-unform ratonal B-splnes INURBS curve based on perspectve proecton. In 211, usng parallel proecton, the frst author of [7] defned a novel ds ratonal Bézer curve, whose error radus functons are Bézer polynomal functons. One of the mportant theme for ratonal Bézer curve s degree reducton. Ths problem arses because of the lmt of mum degree for polynomal and the need of data compresson [8]. In 1983, Farn [9] descrbed a degree reducton method for ratonal Bézer curve for nteractve nterpolaton and approxmaton. Later, Sederberg and Chang [1] acheved one degree reducton based on perturbng the Manuscrpt receved Aprl 29, 215; revsed September 22, 215. Ths wor s supported by the Natural Scence Basc Research Plan n Shaanx Provnce of Chna No. 213JM14, the Fundamental Research Funds for the Central Unverstes No. GK and GK21431 and the Natonal Natural Scence Foundaton of Chna No and Mao Sh s wth the School of Mathematcs and Informaton Scence of Shaanx Normal Unversty, X an 7162, Chna e-mal: shmao@snnu.edu.cn numerator and denomnator polynomals of a ratonal curve such that the best lnear common dvsor s canceled. Chen [11] appled the shfted Chebyshev polynomals to acheve the degree reducton of ratonal Bézer wth C, -contnuty at end ponts. A defcency of above methods s that reducedweghts may be negatve. n 21, Ca and Wang [12] researched C r,s -contnuty at end ponts usng the Steepest Descent algorthm. In fact, the degree reducton of ratonal Bézer curve s a vector-valued optmzaton problem, so the mult-obectve optmzaton method s used n ths paper. For the detals about mult-obectve optmzaton method, the reader can see [13] or [16]. On the other hand, for the degree reducton of the error radus curve, we can transform t nto solvng a constraned quadratc programmng problem. Ths paper has the followng structure: To ensure the structural ntegrty of ths paper, n secton 2, we revew the defnton of ds ratonal Bézer curve and ts propertes. In secton 3, we propose an effcent algorthm to the problem of degree reducton of ratonal ds Bézer curve. In secton 4, some examples are provded. II. DISK RATIONAL BÉZIER CURVES A. Ds ratonal arthmetc A ds n the plane s defned to be the set q x, y r {x R 2 x q r, r R + }, whose centrc pont s q and radus s r. For any two dss q x, y r, 1, 2, the two operatons are defned as follow q q x, y r, R, 1, 2, 1 q 1 + q 2 x 1 + x 2, y 1 + y 2 r1+r 2. 2 Equatons 1 and 2 can be generalzed as n n n q x, y n r. 3 In homogeneous coordnates, a ds can be defned as P ω ωx, ωy, ω ωr {x ω ωx, ωy, ω R 3 x ω P ω ωr}. Applyng the perspectve proecton H to the ds P ω can yelds a correspondng ratonal ds n plane ω 1. That s q HP ω X ω, Y x, y r. 4 ω R ω Advance onlne publcaton: 14 November 215

2 IAENG Internatonal Journal of Appled Mathematcs, 45:4, IJAM_45_4_18 In addton, the ds can also be represented by the homogeneous coordnates P ω ωx, ωy, ω r {x ω ωx, ωy, ω R 3 x ω P ω r}. Applyng the oblque proecton I to the ds P ω can yelds another correspondng ratonal ds n plane ω 1. That s q IP ω X ω, Y x, y r. 5 ω r Both equatons 4 and 5 obvously agree wth operaton 3. Based on equatons 3 and 4, a nd of ds ratonal Bézer curve can be defned and has been researched n [3] [4]. However, usng oblque proecton I, we can defne a novel nd of ds ratonal curve and ts propertes, such as end nterpolaton, affne nvarant etc., are smlar to the classc ds ratonal Bézer curve. B. Ds ratonal Bézer curves A ds ratonal Bézer curve of degree n wth control ds ponts p x, y r and correspondng weghts ω R +,,..., n, s defned by pt [pt; rt] n p ω B nt n n ; r B n ω B nt t, or be wrtten n the bass form B n t pt [pt; rt] [ n ] n p R n t; r B n t, n t 1 t n, t 1,,..., n, are Bernsten polynomals, R n t ω B nt n ω B n, 1,..., n, t, are the ratonal bass functons. pt and rt are respectvely called the center curve and the radus of the ds ratonal Bézer curve pt. C. Propertes of ds ratonal Bézer curves A ds ratonal Bézer curve satsfes the followng propertes. End nterpolaton: p p and p1 p n. Affne nvarant: Let A be an affne transformaton for example, a rotaton, reflecton, translaton, or scalng, then n ω p B nt n ω Ap B nt A n ω B nt n ω B nt Convex hull: The ds ratonal Bézer curve les n the convex hull of the control dss. Snce the convex hull of control dss p,, 1,..., n, are the set of all convex combnatons n α p, α R nt and n Rn t 1, and the property s desred. Non-unform convergence: Snce t,, 1 t,, lm ω Rn t 1 < t < 1,,..., n, + 1 t 1, n, t 1, n, the ds ratonal Bézer curve converges non-unformly on colsed nterval [, 1], De Castelau algorthm: For any t [, 1], pt can be computed as follows: p ω 1 1 t p 1 ω ω r 1 tω 1 1 tr 1 + t ω 1 +1 p 1 ω +1, + tω 1 +1, + tr 1 +1, 1,..., n and,..., n. Subdvson: Let c, 1 be a real number. Then pt can be subdvded nto two segments: n p cω cbn t c n n ; r ω cbn t c cbn t c, t c, pt n p n n n c t 1, cωn n cb n t c 1 c ; ωn n cb n t c 1 c n rn n cb n t c 1 c Degree elevaton: A ds ratonal Bézer curve pt of degree m can be represented as a ds ratonal Bézer curve of degree m + s as follows m p ω B mt pt m ω B mt and ˆω ˆp 1ˆω ˆr, 1,..., m + s. m+s m+s mnm,, s ˆp ˆω B m+s mnm,, s mnm, ˆω B m+s t, s t, t 1, s m ω, +s s m p ω, +s s m r, 6 +s, Advance onlne publcaton: 14 November 215

3 IAENG Internatonal Journal of Appled Mathematcs, 45:4, IJAM_45_4_18 Exact degree reducton: A degree n ds ratonal Bézer curve pt represents exactly a degree m m < n ds ratoanl Bézer curve ˇpt wth control dss ˇp and weghts ˇω R +,, 1,..., m, f only f the followng equatons are satsfed and mnm,, n mnm,, n n +n, 1,..., n + m, r mnm,, n+m, 1,..., n. For the center curve, by ˇω ω p n +n ˇω ω ˇp, pt ˇpt, we have n m p ω B n t ˇω B m t 7 n m m ř n, 8 n m ˇp ˇω B n t ω B m t, whch, after some rearrangement of the equaton, gves m+n mnm, m n ˇω ω p B m+n t, m m+n mnm,, m +n n ω ˇω ˇp B m+n t. +n Comparng coeffcents of le terms on both sdes of the equaton, ths establshes the equaton 7. The proof for the radus 8 s straghtforward by equaton 6. By equaton 7, t s clear that f n C mnm,, m +n ˇω ω p ω ˇω ˇp, 9,..., µ 1, and n + n ν + 1,..., n + m, then the curves pt and ˇpt satsfy C µ,ν - contnuty. III. DEGREE REDUCTION OF DISK RATIONAL BÉZIER CURVES The problem of degree reducton of ds Ratonal Bézers curve can be stated as follows: Gven a degree n ds ratonal Bézer curve pt, fnd a degree m < n ds ratonal Bézer curve ˇpt such that ˇpt s a closure of pt. The above problem can be decomposed nto two parts. A Degree reducton approxmaton of center curve. Usng weghted least squares, the degree reducton of ratonal Bézer curve wth C µ,ν - contnuty can be expressed as the followng mathematcal formula mn ρtpt ˇpt2 dt s.t. C,,..., µ 1, 1 C, n + n ν + 1,..., n + m, ˇω >,,..., m, C are gven by equaton 9. Specfcally, let n pt qt ω p B nt ωt n, ω B nt ˇpt ˇqt ˇωt m ˇω ˇp B mt m ˇω B mt and ρt ωtˇωt 2. The obectve functons of equaton 1 can be wrtten as f 1, f 2 H A B C D E and F ρtpt ˇpt 2 dt ˇωtqt ωtˇqt 2 dt 2n+2m 1 2m + 2n + 1 2m 2n 2m+2n mnm,, m mnn,, n mnm,, m mnn,, n mnm,, m mnn,, n mn2m,, 2n H A B 2C D + E F, ˇω ˇω, 2m n n ω p ω p, 2n ˇω ˇp ˇω, 2m n n ω p ω, 2n 2m ˇω ˇp ˇω ˇp n n ω ω. 2n Many methods can be used to solve the above equaton. The weghted-sum-of-obectve-functons method [13] s used n ths paper. That s, a new obectve functon s f 1 2 f 1 + f 2. Accordngly, we use the fmncon procedure of MATLAB to solve the nonlnear programmng and the algorthm opton s the nteror-pont method [14] [15]. B Degree reducton approxmaton of error radus curve The problem of degree reducton approxmaton of error radus curve can be expressed as the followng formula mn řt rt 2 2 s.t. řt rt + dstpt, ˇpt 11 ř >,, 1,..., m. Advance onlne publcaton: 14 November 215

4 IAENG Internatonal Journal of Appled Mathematcs, 45:4, IJAM_45_4_18 TABLE I ERROR AND WEIGHT COMPARISONS OF THE SEDERBERG S, CHEN S AND OUR METHODS Method Constrant Weghts Error Sederberg s method N/A , , , Chen s method C, , , , Ca and Wang s method C,.3719,1.8149,.5698, Our method C, , ,2.5549, dstpt, ˇpt, t [, 1], s the Hausdorff dstance between the curve ˇpt and the curve pt. In practce, the equaton 11 can be further smplfed as quadratc programmng. For the constrant functon, elevatng the degree of the error radus curve řt from m to n by equaton 8, we have m n řt ř B m t ˆr B n t, ˆr mnm,, n+m n m m ř n. Then one of a suffcent condton to satsfy the equaton 11 can be stated as d { ˆr > r + d,, 1,..., n, mn pt pt, ˇpt ˇpt mn ˇpt ˇpt, pt pt pt ˇpt, pt pt } 12 and pt ˇpt s dscrete Eucldean dstance, pt and ˇpt,, 1,, M, are dscrete sample ponts on curves pt and ˇpt. For the obectve functon, we have rt řt 2 2 rt řt 2 dt ř 2 tdt 2 rtřtdt + m m m n ř ř H 2 ř r S + r 2 tdt n n r r G, H m 2m+1 +, S 2m m n m+n+1+n + and G n n 2n+1 +. The thrd term n n 2n r r G s constant and can be omtted. So the problem of degree reducton of error radus functon can be transformed to fnd the optmal soluton of the followng problem: m m mn řř H 2 m n řr S s.t. ˆr r + d,, 1,..., n, ř >,, 1,..., m. Smlarty to the equaton 12, the error between curves rt and řt s defned as e { mn rt rt, řt řt mn řt řt, rt rt rt řt, rt rt } rt and řt,, 1,, M, are dscrete sample ponts on curves rt and řt. IV. EXAMPLES Example 1 Also example n [1]. Gven a 4 degree ratonal Bézer curve wth control ponts,, 2, 2, 3,, 4, 2, 4, and assocated weghts 1, 4, 2, 1, 1, to fnd a 1-degree reduced ratonal Bézer curve to a approxmate the orgnal curve. See Table 1 for comparsons of approxmaton error, and Fg. 1 for llustraton. Although our method s error s larger than Ca and Wang s method [12], we fnd that our resultng curve may be better than others by Fg. 2 and Fg. 3. Example 2. Gven a ds ratonal Bézer pt of eght degree wth control dss 6, , 8.6, 25.4, 2.3, 3 1,35, , 4.2, 25 2, 37.5, , 47.2, 8.1.8,65.1, , 71.5, 25.5 and assocated weghts 1.88, 1.68, 1.63, 1.73, 1.79, 2.18, 1.24, 1.8, 1.9. The best 3-degree reducton curve satsfyng C 1,1 -contnuty wth the gven curve has control dss 6., , , , , , , , , , 71.5, , and assocated weghts , 1.581, , ,.4741, See Fg. 4 and Fg. 5. The error dstances of center curve and the error radus curve are.2577 and.3938, respectvely See Fg. 6 and Fg. 7. V. CONCLUSION In ths paper, we dscussed the problem of degree reducton of ds ratonal Bézer curves and proposed an effcent method to solve the problem. Theoretc results and experments show that the proposed algorthm produces s very effectve. The dea presented n ths paper can be easly generalzed to solve the degree reducton problem of ds ratonal Bézer surfaces and NURBS curves and surfaces. ACKNOWLEDGEMENTS The authors would le to express ther sncere thans to anonymous revewers for the valuable suggestons and comments. Advance onlne publcaton: 14 November 215

5 IAENG Internatonal Journal of Appled Mathematcs, 45:4, IJAM_45_4_ C C3.4.2 C4 C1:The gven curve.2 C2:Sederberg s method C3:Chen s method C4:Ca and Wang s method.4 C5:Our method Fg. 1. Comparson of four degree reducton methods. For nterpretaton of the reference to color n ths fgure legend, the reader s referred to the web verson of ths artcle..1 C C5 C C1:The gven curve C2:Sederbergs method C3:Chens method C4:Ca and Wangs method C5:Our method C1:The gven curve C2:Sederbergs method C3:Chens method C4:Ca and Wangs method C5:Our method C4 C5.5.6 C Fg. 2. The photomcrograph of Fg. 1 on the nterval [1.3, 2.2]. For Fg. 3. The photomcrograph of Fg. 1 on the nterval [3.3, 4.1]. nterpretaton of the reference to color n ths fgure legend, the reader s For nterpretaton of the reference to color n ths fgure legend, referred to the web verson of ths artcle. the reader s referred to the web verson of ths artcle. REFERENCES [1] T. W. Sederberg, R. T. farou, Approxmaton by nterval Bézer curves, IEEE Computer Graphcs and ts Applcatons 1992,152, pp [2] C. Y. Hu, T. Maeawa, E. C. Sherbrooe, N.M. Patralas, Robust nterval algorthm for curve ntersectons, Computer-Aded Desgn, 1996, , pp [3] C. Y. Hu, N. M. Patralas, X. Z. Ye, Robust nterval sold modelng- Part I: Representatons, Computer-Aded Desgn, 1996, 281, pp [4] C. Y. Hu, N. M. Patralas, X. Z. Ye, Robust nterval sold modelng- Part II: Boudary evaluaton, Computer-Aded Desgn, 1996,128, pp [5] F. L. Chen, W. P. Lou, Degree reducton of nterval Bézer curves, Computer-Aded Desgn,322, pp [6] Q. Ln, J. Rone. Ds Bézer curves. Computer Aded Geometrc Desgn , pp [7] M. Sh, Z. Ye, B. Kang, Ds ratonal Bézer curves, Journal of Computer-Aded Desgn and Computer Graphcs, , pp n chnese. [8] F. L. Chen, W. Yang, Degree reducton of ds Bézer curves, Com- Advance onlne publcaton: 14 November 215

6 IAENG Internatonal Journal of Appled Mathematcs, 45:4, IJAM_45_4_ C1:The gven curves C2:The resultng curves Fg. 4. A degree 8 ds ratonal Bézer curve wth blac and ts three-degree reduced curve wth bluewth C 1,1 contnuty at two endponts. For nterpretaton of the reference to color n ths fgure legend, the reader s referred to the web verson of ths artcle C1: Error of center curves C2: Error of error curves C2 C C1:The gven curves C2:The resultng curves Fg. 5. The photomcrograph of Fg. 1 on the nterval [9., 24.]. For Fg. 6. The correspondng error dstance curve of center ratonal Bézer curve nterpretaton of the reference to color n ths fgure legend, the reader s and error curve. For nterpretaton of the reference to color n ths fgure referred to the web verson of ths artcle. legend, the reader s referred to the web verson of ths artcle. puter Aded Geometrc Desgn, 21 24, pp [9] G.Farn. Algorthms for ratonal Bézer curves. Computer-Aded Desgn,1983,152, pp [1] T. W. Sederberg,G. Z. Chang. Best lnear common dvsors for approxmate degree reducton. Computer-Aded Desgn,1993,253, pp [11] F.L. Chen. Constraned best lnear common dvsors and degree reducton for ratonalcurves. Numercal Mathematcs A Journal of Chnese Unverstes Computer Ade Geomterc Desgn Album,7,1993, pp n chnese. [12] H.J. Ca, G.J. Wang. Constraned approxmaton of ratonal Bzer curves based on a matrx expresson of ts end ponts contnuty condton. Computer-Aded Desgn, 21,426: [13] Y. Collette, P. Sarry. Multobectve Optmzaton: Prncples and Case Studes. Sprnger 23. [14] M. H. Wrght. The nteror-pont revoluton n optmzaton: hstory, recent developments, and lastng consequences. Bulletn of the Amercan Mathematcal Socety, 2442, pp [15] J. Nocedal,S.J.Wrght. Numercal Optmzaton 2nd. Sprnger 26. [16] H. Yano,K Matsu. Random Fuzzy Multobectve Lnear Programmng Through Probablty Maxmzaton and Its Applcaton to Farm Plannng. IAENG Internatonal Journal of Appled Mathematcs,213,432: Advance onlne publcaton: 14 November 215