Approximate merging of a pair of BeÂzier curves

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1 COMPUTER-AIDED DESIGN Computer-Aded Desgn 33 (1) 15±136 Approxmate mergng of a par of BeÂzer curves Sh-Mn Hu a,b, *, Rou-Feng Tong c, Tao Ju a,b, Ja-Guang Sun a,b a Natona CAD Engneerng Center, Tsnghua Unversty, Beng 184, Peope's Repubc of Chna b Department of Computer Scence and Technoogy, Tsnghua Unversty, Beng 184, Peope's Repubc of Chna c Department of Computer Scence, Zheang Unversty, Hangzhou 317, Peope's Repubc of Chna Receved 18 September 1999; receved n revsed form 4 Apr ; accepted 9 Apr Abstract Ths paper deas wth the mergng probem,.e. to approxmate two adacent BeÂzer curves by a snge BeÂzer curve. A nove approach for approxmate mergng s ntroduced n the paper by usng the constraned optmzaton method. The basc dea of ths method s to nd condtons for the precse mergng of BeÂzer curves rst, and then compute the constraned optmzaton souton by movng the contro ponts. ªDscreteº coef cent norm n L sense and ªsquared dfference ntegraº norm are used n our method. Contnuty at the endponts of curves are consdered n the mergng process, and approxmate mergng wth ponts constrants are aso dscussed. Further, t s shown that the degree eevaton of orgna BeÂzer curves w reduce the mergng error. q 1 Esever Scence Ltd. A rghts reserved. Keywords: BeÂzer curve; Approxmate converson; Mergng 1. Introducton and probem statement Parametrc poynoma representatons are wdey used n CAD systems whch mode free-form curves and surfaces. Bernsten±Bezer, Schoenberg-B-spne and Hermte± Coons type bass functons are frequenty used n dfferent systems. Wth the avaabty of a fast growng varety of modeng systems the demand has rsen for exchangng curve and surface descrptons between varous CAD systems [1]. The genera am when transferrng geometrc nformaton from one system to another s to ensure a hgh degree of accuracy, the east possbe oss of nformaton and a sma amount of geometrc data for communcaton. Converson from one poynoma base to another can be acheved by drect matrx mutpcaton. For reducng the amount of communcatng data, approxmate converson s consdered by Hoschek []. As mentoned by Hoschek, the approxmate converson ncudes the foowng. ² Degree reducton: to nd a parametrc curve of degree n to approxmate the gven curve of degree m n, m : ² Mergng: to merge as many as possbe curve segments of degree M to one curve segment of degree N M # N : Degree reducton methods of BeÂzer and Ba curves have * Correspondng author. Te.: ; fax: E-ma address: shmn@tsnghua.edu.cn (S.-M. Hu). been extensvey nvestgated by Watkns and Worsey [3], Lachance [4], Eck [5,6], Bogack et a. [7] and Hu et a. [8,9] etc., degree reducton of B-spne curves has been consdered by Peg et a. [1], degree reducton of parametrc surfaces aso has been consdered by Hu et a. [11,1]. However mergng of BeÂzer curves s st an nterestng and open probem. Probem. For two adacent BeÂzer curves P(u) and Q(v) # u; v # 1 wth correspondng contro ponts P and Q ˆ ; 1; ¼; n ; mergng of P(u) and Q(v) s a process that amounts to ndng an n degree BeÂzer curve R(t) wth contro ponts R ˆ ; 1; ¼n ; such that a sutabe dstance functon d R; R between R(t) and 8 >< R t ˆ >: P B n t Q B n t 1 # t # # t # 1 s mnmzed on the nterva [, 1], where s a subdvson parameter. The basc dea of ths artce s to nd condtons for precse mergng of BeÂzer curves. We modfy contro ponts of two BeÂzer curves such that the mod ed curves satsfy the precse mergng condtons, then new contro ponts of /1/$ - see front matter q 1 Esever Scence Ltd. A rghts reserved. PII: S1-4485()83-X

2 16 S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15±136 the merged curve can be obtaned by an extrapoaton agorthm. The rest of the paper s organzed as foows. Two constraned optmzaton methods wth dfferent optmzaton crtera are ntroduced, both of whch acheve mergng of BeÂzer curves by sovng near equaton systems. An exstence proof of the system of near equatons s gven n Secton 3. Condtons for matchng orgna endponts and dervatves are consdered n Secton 4, and mergng wth ponts constrants s deat wth n Secton 5. Fnay, we dscuss approxmate mergng wth degree eevaton n Secton 6.. Constraned optmzaton method for mergng.1. Condtons of precse mergng Consder two n degree BeÂzer curves P(u) and Q(v), we rst gve condtons of precse mergng. If P(u) and Q(v) can be merged precsey, ths s to say, there exsts an n degree BeÂzer curve R t ˆPn R B n t such that R t ˆ R t ( R t s de ned as n Eq. (1)), so we have P u u Note that uˆ1ˆ Q v v ˆ ; 1; ¼n: vˆ u ˆ t ; v ˆ t 1 ; by the dervatve formua of BeÂzer curves, Eq. () yeds 1 n n D P n ˆ 1 n 1 n D Q ; where D s the dfference operator de ned as DP ˆ P 11 P ; so we have the foowng theorem. Theorem 1. Two n degree BeÂzer curves P(u) and Q(v) can be merged precsey, f and ony f there exsts m m. ; such that D P n ˆ m D Q for ˆ ; 1; ¼; n: Proof. 1. Necessty. If P(u) and Q(v) can be merged precsey, from Eq. (3) we have D P n ˆ m D Q for ˆ ; 1; ¼; n; where m ˆ 1 : Therefore condton (4) s necessary.. Suf cency. Frst, we represent BeÂzer curves n the 3 4 A exponenta form. For any n degree BeÂzer curve S(t) wth contro ponts S ˆ ; 1; ¼; n we have S t ˆXn n t D S ˆ Xn n t 1 D S n : 5 Note that Eq. (5) hods true for any t t $ : Next we construct a new BeÂzer curve R(t), whose contro ponts R ˆ ; 1; ¼; n are de ned by R ˆ X P B 1 ; where ˆ m m 1 1 and B t s the degree BeÂzer base functon. We w see that P(u) and Q(v) can be precsey merged nto R(t). We cacuate D k R k ˆ ; 1; ¼n as " D k R ˆ Xk k P B 1 # ˆ ˆ Xk X 4 k P X k 1 B ˆ 1 k 3 1 k 5 ˆ Xk k P 1 k 1 k ˆ k Dk P : 6 Substtute Eq. (6) nto exponenta form (5), for t # t # ; we have R t ˆXn n t D R ˆ Xn t n D P ˆ P t Gven condton (4), for t # t # 1 ; we have R t ˆXn t n D P ˆ Xn ˆ Xn t 1 n D P n t 1 n m D Q ˆ Q t 1 Accordng to Eq. (1), P(u) and Q(v) can be precsey merged nto R(t) wth as the subdvson parameter. Ths competes the proof of Theorem 1... Constraned optmzaton method (1) We now consder the approxmate mergng probem

3 stated n the prevous Secton.1. In order to obtan the merged BeÂzer curve, we w expore two dfferent optmzaton crtera n the foowng dscusson. Frst, we use the ªdscreteº coef cent norm,.e. we set d E t ; F t ˆ P n E F ; where E(t) and F(t) are n degree BeÂzer curves wth contro ponts E and F, s the Eucdean norm. Mnmzaton of the norm w resut n the east dfference between correspondng contro ponts of the two curves. For BeÂzer curves P(u) and Q(v), perturbaton e ˆ e x ; e y ; ez Š T and d ˆ d x ; d y ; dz Š T can be obtaned for contro ponts P of P(u) and Q of Q(v), respectvey, such that the mod ed curves ^P u ˆXn ^P B n u ˆXn S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15± D d ˆ X 1 k d k ; kˆ k 1 by settng P 1 e B n u # u # 1 L e x ; L e y ; L e z ; L d x ; to be zero, we have e 1 Xn d Xn ˆn ˆ 1 n m 1 L d y ; L d z ; L L ; x y and L z ˆ ˆ ; 1; ¼; n; n ˆ ˆ ; 1; ¼; n; 11 1 ^Q v ˆXn ^Q B n v ˆXn Q 1 d B n v # v # 1 sats es the condtons of Theorem 1,.e., D ^P n ˆ m D ^Q : Then, ^P u and ^Q v can be precsey merged nto abeâzer curve R(t) of degree n, whch s consdered to be the approxmate merged curve of P(u) and Q(v). We determne e ; d ˆ ; 1; ¼; n by settng the optmzaton crteron O(e, d )as O e ; d ˆXn e 1 d ˆMn and Lagrange functon s de ned by L ˆ Xn 1 Xn e 1 d D P n 1 e n m D Q 1 d Š; where ˆ x ; y ; z Š are Lagrange mutpers. In order to obtan an approprate vaue of m, we de ne a sequence m ˆ 1; ; ¼; n such that D P n ˆ m D Q Accordng to Theorem 1, the average vaue of the sequence w be a reasonabe choce for parameter m. Thus we take m as m ˆ ˆ1 Note that D e n ˆ n m Xn kˆn ˆ ˆ1 1 nk 7 s 1 D P n D Q n C : 8 A e k ; n k 9 D P n 1 D e n m D Q m D d ˆ ˆ ; 1; ¼; n: 13 Substtutng Eqs. (11) and (1) nto Eq. (13), produces the system of near equatons " # 1 X m 1 mn ; ˆ k k kˆ ˆ D P n m D Q ˆ ; 1; ¼; n: 14 Eq. (14) can be further smp ed as " # m 1 ˆ D P n m D Q ˆ ˆ ; 1; ¼; n: (15) So ˆ ; 1; ¼; n can be obtaned by sovng the above system of near equatons, and e ; d ˆ ; 1; ¼; n can be obtaned from Eqs. (11) and (1). The exstng proof of souton of the near equaton system (15) w be gven n Secton Constraned optmzaton method () We can aso use the ªsquared dfference ntegraº norm, such that d E t ; F t ˆ Z1 E t F t dt; where E(t) and F(t) are n degree BeÂzer curves. The norm can be vewed geometrcay as the squared area between two BeÂzer curves. Therefore, mnmzaton of the norm w resut n the east encosng area between the two curves. We determne perturbatons e ; d ˆ ; 1; ¼; n ; de ned n Secton., by settng the optmzaton crteron O(e, d ) as " Z1 # O e ; d ˆ B n t e 1 Xn B n t d dt ˆ Mn

4 18 S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15±136 (a) Fg. 1. Compute new contro ponts by extrapoaton. and Lagrange functon s de ned by " Z1 # L ˆ B n t e 1 Xn B n t d dt 1 Xn D P n 1 e n m D Q 1 d Š 16 where ˆ x ; y ; z Š are Lagrange mutpers and m s de ned as n Secton.. By settng L e x ; L e y ; L e z ; L d x ; L ; d y L d z ; L L ; x y and L z (b) (c) to be zero, we have Xn e N 1 Xn 1 n ˆn Xn d N Xn m 1 ˆ ˆ ˆ ; 1; ¼; n; n 17 ˆ ˆ ; 1; ¼; n; 18 D P n 1 D e n m D Q m D d ˆ ˆ ; 1; ¼; n: 19 where N s de ned as N ˆ Z1 B n t B n t dt ; ˆ ; 1; ¼; n: Thus e, d and ˆ ; 1; ¼; n can be obtaned by sovng the near equaton system consstng of Eqs. (17)±(19). The exstence proof w be gven n Secton 3. (d).4. Cacuaton of new contro ponts R We are now n a poston to gve a recursve agorthm to compute new contro ponts of the merged curve R(t). Snce contro ponts of ^P u and ^Q v can be derved by subdvdng R(t) att ˆ ˆ m= 1 1 m (see Fg. 1), we have the foowng agorthm to compute new contro ponts by usng extrapoaton. Agorthm Set ˆ m= 1 1 m Set P ˆ P 1 e for ˆ ; 1; ¼; n For ˆ 1; ; ¼; n For ˆ ; 1 1; ¼; n Fg.. Mergng of two cubc BeÂzer curves. P ˆ 1 1= P 1 1 Next Next R k ˆ P k k for k ˆ ; 1; ¼; n: 1 1=P 1 Exampes. In Fg., we demonstrate mergng of cubc BeÂzer curves par usng the constraned optmzaton methods dscussed so far. Exampes (a) and (c) adopt the rst constraned optmzaton method, whe (b) and (d) use the

5 S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15± second optmzaton method n mergng. The orgna curves and contro poygons are shown as sod nes, whe the merged curves and assocated poygons are shown as dashed nes. We can see that the rst method works better n (a), whe the second method acheves ncer resuts n (d). 3. An exstence proof of souton In ths secton, we rst gve the exstence proof of system (15). By rewrtng Eqs. (7) and (13) n the matrx form, we get L ˆ h T h 1 T Ah 1 B and Ah 1 B ˆ 1 n whch hˆ e ; e 1 ; ¼;e n ;d ;d 1 ; ¼; d n T ;ˆ ; 1 ; ¼; n T ; A ˆ a n11 n1 de nte matrx, thus t's non-snguar. Ths competes the exstence proof of the system (15). Smary, by rewrtng Eq. (16) n matrx form, we get L ˆ h T Ch 1 T Ah 1 B : 5 here C s a symmetrc matrx de ned as C ˆ c n1 n1 n whch 8 ; for # # n and n, # n 1 1; >< or # # n and n, # n 1 1 c ˆ ; N ; for # ; # n >: N n1 n1 ; for n, ; # n 1 1 where N are de ned n Eq. (). Accordng to Eq. (16), h T Ch s aways postve for any h h ± ; therefore C s postve de nte. Dfferentatng Eq. (5) wth respect to h, we get (Eqs. (17) and (18) n matrx form) Ch 1 A T ˆ : 6 B ˆ b ; b 1 ; ¼; b n Š T ; Eq. (19) can aso be wrtten n matrx form where Ah 1 B ˆ : 8 ; for # 1, n or n 1 1 1, # n 1 1 >< 1 n ; for # ; # n; 1 $ n a ˆ n >: 1 n1 m ; for # # n; n 1 1 # # n n 1 7 b ˆ D P n m D Q : Dfferentatng Eq. (1) wth respect to h, we obtan (Eqs. (11) and (1) n the matrx form) h 1 A T ˆ 3 By substtutng Eq. (3) nto Eq. (), we obtan (Eq. (15) n matrx form) 1 AAT ˆ B: 4 Note that 1 ¼ p p ¼ ¼ p p p p ¼ ¼ p p p p p p ¼... A ˆ... ] ] p ¼ p p p p p p ¼ p p p ¼ p p p p p p ¼ p p C A p p p ¼ p p p p p p ¼ p p p n11 n1 s obvousy a row-fu-rank matrx. So 1 AAT s a postve Substtutng Eq. (6) nto Eq. (7), we obtan 1 AC1 A T ˆ B 8 because A s a row-fu-rank matrx and C 1 s symmetrc postve de nte, so 1 AC1 A T s non-snguar [13]. By substtutng Eq. (8) nto Eq. (6), we obtan h as h 1 C 1 A T AC 1 A T 1 B ˆ Ths competes the exstence proof of the near equaton system consstng of Eqs. (17)±(19). 4. Condtons for matchng orgna endponts and dervatves To match orgna endponts P and Q n, two constrants e ˆ ; d n ˆ shoud be added. Take the rst constraned optmzaton method for exampe, by dscusson smar to

6 13 S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15±136 (a) (b) Fg. 3. Mergng of two quartc BeÂzer curves wth endponts matchng. that n Secton., we obtan a system of near equatons 8 " # m 1 ˆ D P n m D Q ˆ >< ˆ ; 1; ¼; n 1 " # 1 X 1 1 m 1n mn ;n 1 n ˆ D n P m n D n Q >: ˆ kˆ k k Then, ˆ ; 1; ¼; n can be obtaned, and e ; d ˆ ; 1; ¼; n can be obtaned as foows 8 e ˆ e ˆ 1 >< ˆn d ˆ 1 1 ˆ >: d n ˆ 1 n ˆ 1; ¼; n n m ˆ ; 1; ¼; n 1 Further, constrants e 1 ˆ and d n1 ˆ coud be added to the above equaton system so that the merged curve woud match the orgna dervatves at the eft and rght ends of the par of BeÂzer curves. The system of near equatons woud become 8 " # m 1 ˆ D P n m D Q ˆ >< ˆ ; 1; ¼; n " # 1 X 1 1 m 1 mn ;n ˆ D P m D Q ˆ kˆ k k >: ˆ n 1; n Then e ; d ˆ ; 1; ¼; n can be obtaned as foows 8 e ˆ e 1 ˆ e ˆ 1 1 n ˆ ; ¼; n >< ˆn n d ˆ 1 1 m ˆ ; 1; ¼; n ˆ >: d n ˆ d n1 ˆ Smary, for the second constraned optmzaton method, matchng of orgna endponts and dervatves can be acheved by addng constrants e ˆ e 1 ˆ and d n ˆ d n1 ˆ to Lagrange functon (16). Perturbatons e, d can then be obtaned by sovng the correspondng system of near equatons. Exampes. In Fg. 3, we ustrate mergng resuts of two quartc BeÂzer curves usng the rst optmzaton method. Endponts matchng s mpemented n (a), and matchng of dervatves at endponts s acheved n (b). The orgna curves and contro poygons are shown as sod nes, and the merged curves and assocated poygons are shown as dashed nes. 5. Approxmate mergng wth ponts constrants A more genera probem of matchng endponts s for the merged curve to pass through target ponts, not ony endponts on the orgna BeÂzer curves. Ths can be acheved by addng ponts constraned condtons to the Lagrange functon. Gven that B n u P ˆ Xn B n v k Q ˆ Xn B n u P 1 e ˆ ; 1; ¼; ; B n v k Q 1 d k ˆ ; 1; ¼; m; where u ˆ ; 1; ¼; and v k ˆ ; 1; ¼; m are parameters of dfferent target ponts on P(u) and Q(v), respectvey, we have B n u e ˆ ˆ ; 1; ¼; ; 9

7 S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15± (a) (b) Note that unke the equaton system n Secton.3, the above near system s not sovabe when the tota number of constraned condtons n Eqs. (13), (9) and (3) exceeds the number of constraned varabes (e and d ). Generay speakng, the tota number of dfferent ponts constrants on P(u) and Q(v) 1 m 1 shoud not be arger than n 1 1 for a sovabe souton. Exampe. Ponts constrants can be used n preservng the geometrc detas of the orgna par of BeÂzer curves. In Fg. 4(a), two quntc BeÂzer curves are merged usng the second optmzaton method wth endponts matchng,.e. u ˆ :; v ˆ 1:; whe (b) ustrates the mergng resuts after three more ponts constrants at u ˆ :5; u 1 ˆ 1:; v ˆ :5 are added. 6. Constraned optmzaton wth degree eevaton Fg. 4. Mergng of two quntc BeÂzer curves wth ponts constrants. B n v k d ˆ k ˆ ; 1; ¼; m: 3 By addng the above-constraned condtons, we rewrte the Lagrange functon as L ˆ O e ; d 1 Xn 1 X n111 ˆ D P n 1 e n m D Q 1 d Š 1 Xm n11k1 kˆ B n u e B n v k d ; (31) where O(e, d ) s the approprate optmzaton crteron. Dfferentatng Eq. (31) wth respect to e ; d ˆ ; 1; ¼; n and ˆ ; 1; ¼; n 1 1 m 1 ; usng smar technques descrbed n Sectons. and.3, we obtan a near system of 3n 1 m equatons consstng of Eqs. (13), (9), (3) and O e ; d e 1 Xn ˆn 1 n n 1 X ˆ n111 B n u ˆ ˆ ; 1; ¼; n; (3) O e ; d d Xn ˆ m 1 1 Xm ˆ n111 B n v ˆ ˆ ; 1; ¼; n: 33 In ths secton, we w dscuss constraned optmzaton wth degree eevaton. Sometmes, two adacent BeÂzer curves are not sutabe to be approxmated by a new BeÂzer curve wth the same degree. For exampe, a cubc BeÂzer curve can have at most one n ecton pont, so mergng a par of BeÂzer curves that each has an n ecton pont nto a new cubc BeÂzer curve w not be a good dea. A reasonabe choce n ths case s to merge two cubc curves nto a curve of degree 4 or 5. In effect, better approxmaton can be acheved when two n degree BeÂzer curves P(u) and Q(v) are merged nto a new BeÂzer curve after the degree eevaton. We can thnk of the ncrease n approxmaton precson n ths way. Constraned optmzaton can be vewed as a two-step process: rsty, a set of BeÂzer curve pars ^P u and ^Q v are obtaned such that each par can be precsey merged nto a BeÂzer curve; secondy, the curves par wth the east optmzaton norm s ªchosenº and merged as the na approxmate merged curve. Snce each par of ^P u and ^Q v obtaned n the rst step can be eevated to a hgher degree and st satsfy the condton of precse mergng, the set of curve pars obtaned n the rst step of constraned optmzaton of degree n s ony a subset of those obtaned n the rst step for degree n 1 1 or above. In other words, optmzaton method w have more egbe curve pars to ªchoose fromº after degree eevaton. Consequenty better approxmate merged curve w be obtaned. We ustrate ths process by usng the second optmzaton method. Frst, we prove a theorem about degree eevaton. Theorem. P(u) s a n degree BeÂzer curve wth contro ponts P ˆ ; 1; ¼; n ; and D s the dfference operator, we have D P p n11 ˆ 1 D P n 1 1 n ˆ ; 1; ¼; n 1 1; 34

8 13 S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15±136 (a) (b) (c) (d) (e) (f) Fg. 5. Mergng of two quartc BeÂzer curves wth degree eevaton. D P p ˆ 1 D P n 1 1 ˆ ; 1; ¼; n where P p ˆ ; 1; ¼; n 1 1 are n 1 1 degree contro ponts after degree eevaton of P(u). Proof. Note that P p ˆ 1 P n n 1 1 P 1 ˆ ; 1; ¼; n 1 1; 36

9 S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15± (a) (b) (c) (d) (e) (f) Fg. 6. Mergng of two quartc BeÂzer curves wth degree eevaton.

10 134 S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15±136 Tabe 1 Comparson of mergng error after degree eevaton and D P p n11 ˆ X D P p ˆ X kˆ kˆ 1 k 1 k P p n111k ˆ ; 1; ¼; n 1 1; k 37 P p k ˆ ; 1; ¼; n 1 1: 38 k Substtutng Eq. (36) nto Eqs. (37) and (38) woud yed Eqs. (34) and (35). Ths competes the proof of Theorem. Theorem guarantees that the vaue of m (de ned n Eq. (8)) does not change after degree eevaton. Thus mergng methods dscussed n ths paper w proceed wth the same parameter m for BeÂzer curves par after degree eevaton. Gven two adacent BeÂzer curves of degree n, P(u) and Q(v), we rst consder the approxmate merged curve R(t) usng the second optmzaton method. Here, we de ne the mergng error E as n Secton.3, such that E R t ; m ˆ m E R t ; m Z1 Degree 4 Degree 5 Degree 6 Exampe (a) Exampe (d) ^P u P u Z1 du 1 ^Q v Q v dv; 39 where ^P u and ^Q v can be precsey merged nto R(t) wth subdvson parameter m= 1 1 m : After the degree eevaton of BeÂzer curves P(u) and Q(v), we obtan two new BeÂzer curves P p (u) and Q p (v) of degree n 1 1: Smary, we obtan n 1 1 degree BeÂzer curves ^P p u and ^Q p v after degree eevaton of ^P u and ^Q v : Therefore BeÂzer curves ^P p u and ^Q p v can aso be precsey merged nto R p (t), whch s the n 1 1 degree BeÂzer curve eevated from R(t). Examne the mergng Tabe Comparson of mergng error after degree eevaton m E R t ; m Degree 4 Degree 5 Degree 6 Exampe (a) Exampe (d) error of R p (t) wth Eq. (39), we have E R p t ; m ˆ ˆ Z1 Z1 ^P p u P p u Z1 du 1 ^Q p v Q p v dv ^P u P u Z1 du 1 ^Q v Q v dv ˆ E R t ; m On the other hand, for n 1 1 degree BeÂzer curves P p (u) and Q p (v), we compute an approxmate merged curve R t of degree n 1 1 wth parameter m usng the second optmzaton method. Accordng to the optmzaton crteron, we have E R t ; m ˆMn # E R p t ; m ˆE R t ; m 4 From Eq. (4) we can concude that R t resuts n ess mergng error E than ts n degree counterpart R(t), and thus acheves better approxmaton after degree eevaton. Smar resuts can be obtaned for the rst optmzaton method, wth the de nton of mergng error E as E R t ; m ˆXn e 1 d ; where e and d are perturbatons of contro ponts of ^P u and ^Q v from P(u) andq(v). Practca exampes of mergng wth both methods are provded n Fgs. 5 and 6. Exampes. Two mergng exampes of quartc cases are presented n Fg. 5 by usng the rst optmzaton method. Two orgna quartc BeÂzer curves are merged nto a new quartc BeÂzer curve each n (a) and (d), and are then merged nto BeÂzer curves of degree 5 and 6, respectvey, n (b), (c) and (e), (f) after degree eevaton. The orgna curves and contro poygons are shown as sod nes, and the merged curves and assocated poygons are shown as dashed nes. Tabe 1 contans the comparson resuts of mergng error of each exampe presented n Fg. 5. From Tabe 1 we can see that the mergng error E(R(t),m) decreases each tme after degree eevaton. Exampes. Two mergng exampes of quartc cases are presented n Fg. 6 by usng the second optmzaton method. Two orgna quartc BeÂzer curves are merged nto a new quartc BeÂzer curve each n (a) and (d), and are then merged nto BeÂzer curves of degree 5 and 6, respectvey, n (b), (c) and (e), (f) after degree eevaton. The orgna curves and contro poygons are shown as sod nes, and the merged curves and assocated poygons are shown as dashed nes. Tabe contans resuts of mergng error of each exampe

11 S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15± presented n Fg. 6. From Tabe we can see that mergng error E(R(t),m) decreases each tme after degree eevaton. 7. Remarks Remark 1. In genera, the dscusson n ths paper can appy to a pars of curves wth G onng. However, Theorem 1 prescrbes that the merged curve shoud have at east G 1 contnuty at the connectng pont. Therefore we recommend usng the methods presented n the paper to pars of BeÂzer curves wth at east G 1 onng n order to avod dscrepancy at connectng pont. Remark. Strcty speakng, the subdvson parameter m n Eq. (7) shoud be dependent on e ; d ˆ ; 1; ¼; n ; such that m ˆ P n 1 e n P n1 e n1 : Q 1 1 d 1 Q d Under such a choce, the Lagrange functon L w get too compcated. So we take m as n Eq. (8), where m s ndependent on e ; d ˆ ; 1; ¼; n : However, a better choce m can be obtaned from m so that the mergng error w get smaer. By usng m de ned n Eq. (8), the merged curve R(t) of P(u) and Q(v) can be computed rsty as n Secton. We hope to obtan better parameter m. Wth the unknown parameter m, we subdvde the merged curve R(t) nto two BeÂzer curves wth contro ponts S and T ˆ ; 1; ¼; n ; and de ne e ; d ˆ ; 1; ¼; n as e ˆ S P ˆ X ˆ d ˆ T Q ˆ Xn ˆ B m R P ˆ ; 1; ¼; n 41 B n m R 1 Q ˆ ; 1; ¼; n 4 Snce we wsh to obtan m such that optmzaton crteron O be mnmzed, we have do e ; d dm ˆ 43 where O e ; d s the approprate optmzaton crteron. Substtutng Eqs. (41) and (4) nto Eq. (43) w produce a non-near equaton of m. Practca agorthms can be derved to cacuate m by usng Newton's method. Remark 3. Mergng methods dscussed n ths paper can be further extended for cases of mutpe BeÂzer curves. Gven a sequence of consecutve BeÂzer curves, our obectve now s to obtan a new sequence of consecutve BeÂzer curves that best approxmates the orgna BeÂzer curves set. We ntend to mnmze the number of curves n the new sequence wth the condton that the mergng error stays wthn a requred error range E. It can proceed as foows. Frst we group the nta sequence nto adacent pars, eavng the snge BeÂzer curve at the end f the number of curves s odd. Then for each par, we appy optmzaton method wth endponts matchng and examne the mergng error. If the error exceeds a spec ed error bound e, the par w not be merged and becomes two curves n the resut sequence. Otherwse the par w be merged and a merged curves from these pars form a new sequence of BeÂzer curves (not necessary consecutve) for the next round of groupng. Repeat the process unt no two adacent BeÂzer curves can be found n the sequence, and we nay obtan the resut sequence of BeÂzer curves. Note that error bound of each step of mergng e must be carefuy spec ed such that accumuated mergng error of the process meets the requrement E. Acknowedgements Ths work was supported by the Natura Scence Foundaton of Chna (proect number 6994). We thank Prof. Tong-Guang Jn and Prof. Guo-Zhao Wang for encouragng ths work, and Dr Wen-Shen Xu for hs usefu suggestons. We especay thank the revewers for ther hepfu comments. References [1] Dannenberg L, Nowack H. Approxmate converson of surface representatons wth poynoma bases. Computer-Aded Geometrc Desgn 1985;:13±31. [] Hoschek J. Approxmate converson of spne curves. Computer- Aded Geometrc Desgn 1987;4:59±66. [3] Watkns MA, Worsey AJ. Degree reducton of BeÂzer curves. Computer-Aded Desgn 1988;:398±45. [4] Lachance MA. Chebyshev economzaton for parametrc surfaces. Computer-Aded Geometrc Desgn 1988;5:19±8. [5] Eck M. Degree reducton of BeÂzer curves. Computer-Aded Geometrc Desgn 1993;1:37±51. [6] Eck M. Least square degree reducton of BeÂzer curves. Computer- Aded Desgn 1995;7:845±51. [7] Bogack P, Wensten SE, Xu Y. Degree reducton of BeÂzer curves by unform approxmaton wth endpont nterpoaton. Computer-Aded Desgn 1995;7:651±61. [8] Hu Sh-Mn, Jn Tong-Guang. Approxmate degree reducton of BeÂzer curves. Proceedngs of Symposum on Computatona Geometry, Hed n Hangzhou, Chna, 199. p. 11±6 (aso n: Tsnghua Scence and Technoogy 1998;3:997±1). [9] Hu Sh-Mn, Wang Guo-Zhao, Jn Tong-Guang. Propertes of two types of generazed Ba curves. Computer-Aded Desgn 1996;8:18±. [1] Peg L. Agorthm for degree reducton of B-spne curves. Computer-Aded Desgn 1995;7:11±1. [11] Hu Sh-Mn, Zheng Guo-Qn, Sun Ja-Guang. Approxmate degree reducton of rectanguar Bezer surfaces. Chnese Journa of Software Research 1997;4:353±61. [1] Hu Sh-Mn, Zuo Zheng, Sun Ja-Guang. Approxmate degree reducton of tranguar Bezer surfaces. Tsnghua Scence and Technoogy 1998;3:11±4. [13] Zhhao Cao, Zhang Yude, et a. Matrx computaton and equaton sovng. Chna: Advanced Educaton Press, 1984.

12 136 S.-M. Hu et a. / Computer-Aded Desgn 33 (1) 15±136 Sh-Mn Hu s an assocate professor n the Department of Computer Scence and Technoogy at Tsnghua Unversty. He receved the BS n mathematcs from the Jn Unversty of Chna n 199, the MS and the PhD n CAGD and Computer Graphcs from Zheang Unversty of Chna n 1993 and 1996, he nshed hs postdoctora research at Tsnghua Unversty n Hs current research nterests are n computer-aded geometrc desgn, computer graphcs and content-based mages ndexng. Tao Ju s an Undergraduate at Tsnghua Unversty for hs degree both n Computer Scence and Engsh Language. Hs current research nterests are n computer-aded geometrc desgn. Ruo-Feng Tong s an assocate professor n the Department of Computer Scence at Zheang Unversty. He receved the BS n Apped Mathematcs from Zheang Unversty of Chna n 1991, and the PhD n CAGD and Computer Graphcs from the Zheang unversty of Chna n Hs current research nterests are n computer-aded geometrc desgn and computer graphcs. Ja-Guang Sun s a professor n the Department of Computer Scence and Technoogy at Tsnghua Unversty. He s aso the Drector of Natona CAD Engneerng Center at Tsnghua Unversty and Academcan of the Chnese Academy of Engneerng. He receved the BS n Computer Scence from Tsnghua Unversty n 197. Hs current research nterests are n computer-aded geometrc desgn, computer graphcs and Product Data Management.

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