Computer Aded Geometrc Desg 19 (2002 365 377 wwwelsevercom/locate/comad Optmal mult-degree reducto of Bézer curves wth costrats of edpots cotuty Guo-Dog Che, Guo-J Wag State Key Laboratory of CAD&CG, Isttute of Computer Images ad Graphcs, Zhejag Uversty, Hagzhou 310027, Cha Abstract Gve a Bézer curve of degree, the problem of optmal mult-degree reducto (degree reducto of more tha oe degree by a Bézer curve of degree m(m< 1 wth costrats of edpots cotuty s vestgated Wth respect to L 2 orm, ths paper presets oe approxmate method (MDR by L 2 that gves a explct soluto to deal wth t The method has good propertes of edpots terpolato: cotuty of ay r, s (r, s 0 orders ca be preserved at two edpots respectvely The method the paper performs mult-degree reducto at oe tme ad does ot eed the stepwse computg Whe appled to the mult-degree reducto wth edpots cotuty of ay orders, the MDR by L 2 obtas the best least squares approxmato Comparso wth aother method of mult-degree reducto (MDR by L, whch acheves the early best uform approxmato wth respect to L orm, s also gve The approxmate effect of the MDR by L 2 s better tha that of the MDR by L Explct approxmate error aalyss of the mult-degree reducto methods s preseted 2002 Publshed by Elsever Scece BV Keywords: Degree reducto; Bézer curve; Optmal approxmato; Edpot cotuty 1 Itroducto The exchagg of product model data betwee varous CAD/CAM systems s ofte eeded However the represetato schemes of parametrc curves ad surfaces are vared dfferet geometrc modelg systems Such as, the maxmum degree, whch dfferet computer systems ca deal wth, vares qute dramatcally Therefore for the data commucato betwee dverse CAD/CAM systems, curves of hgh degree must be approxmated by curves of lower degree due to varato the maxmum degree allowed * Correspodg author E-mal address: wgj@mathzjueduc (G-J Wag 0167-8396/02/$ see frot matter 2002 Publshed by Elsever Scece BV PII: S0167-8396(0200093-6
366 G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 Thus the problem of how to optmally approxmate a gve parametrc curve by a lower degree curve wth a certa error boud has arse CAGD I recet years, may methods have bee used to reduce the degree of Bézer curves The problem of degree reducto s vewed as the verse process of degree elevato (Forrest, 1972; Far, 1983; Pegl ad Tller, 1995 I geeral, degree reducto s ot exactly possble cotract to the reverse process of degree elevato Thus degree reducto approxmato of parametrc curves ad surfaces has bee wdely studed Dscrete pots ad dervatve formato of orgal curve are used degree reducto approxmato (Daeberg ad Nowack, 1985; Hoschek, 1987 The degree reducto of Bézer curves ca also bee doe by usg Chebyshev polyomals approxmato (Watks ad Worsey, 1988; Lachace, 1988 A smple geometrc costructve method of degree reducto wth costraed Chebyshev polyomals s preseted (Eck, 1993, whle a least squares method of degree reducto wth costraed Legedre polyomals s preseted (Eck, 1995 Usg coverso of bases betwee Chebyshev ad Berste bases, a method of degree reducto wth the reducto matrx s developed (Bogack et al, 1995 From the practcal pot of vew, whe trasmttg geometrc formato from oe system to aother, t s our geeral am to esure a hgh degree of accuracy ad the least possble loss of geometrc formato Moreover degree reducto schemes ofte eed to be combed wth the subdvso algorthm, e, a hgh degree curve s approxmated by a umber of lower degree curve segmets ad cotuty betwee adjacet lower degree curve segmets should be mataed Ufortuately, all methods kow up-to-ow have some dsadvatages Frst, they have o explct solutos for optmal mult-degree reducto wth costrats of edpots cotuty of hgh order ad have to be determed by umerc algorthms such as Remes-type algorthm Secodly, for the multdegree reducto, most methods eed stepwse approxmato ad hece a lot of tme for computg s spet Thrdly, most methods geeral caot acheve the optmal approxmato ay more The am of ths paper s to fd out the method of optmal mult-degree reducto wth edpots cotuty of hgh order Based o the verse of degree elevato ad orthogoal polyomal approxmato theory, a method called MDR by L 2 s preseted ths paper, whch gves a explct soluto ad has the optmal precso for mult-degree reducto wth costras of edpot cotuty wth respect to L 2 orm The method ca perform the degree reducto of more tha oe degree at a tme ad avod the stepwse computg The geometrc terpolato formato of edpots betwee the orgal curve ad the degree-reduced curve ca be preserved, e, cotuty of ay r, s (r, s 0 orders ca be preserved at two edpots respectvely I ths paper aother method called MDR by L s troduced, whch also gves a explct soluto for mult-degree reducto wth respect to L orm ad whch s compared wth the MDR by L 2 For the costrats of edpot cotuty of ay orders, the MDR by L 2 ca obta the best least squares approxmato of mult-degree reducto ad the MDR by L acheves early best uform approxmato The approxmate effect of the MDR by L 2 s obvously better tha that of the MDR by L ad the computatoal examples also dsplay t The orgazato of the paper s as follows The secod secto s prelmares I the thrd secto the best least square mult-degree reducto of Bézer curves wth costras of edpots cotuty (MDR by L 2 ad the approxmate error are preseted The fourth secto compares the MDR by L 2 wth the MDR by L Secto fve presets the cocluso
G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 367 2 Prelmares I ths paper, Π deotes all real polyomals of degree at most The deotato deotes the Eucldea vector orm v v,v, add (, deotes as the dstace fucto wth respect to L orm Gve a degree Bézer curve P (t P B (t, t [0, 1], (1 B (t ( t (1 t s the Berste polyomal of degree ad {P } are the cotrol pots The problem of degree reducto s to fd out a Bézer curve Q(t, t [0, 1], ofdegreem(m< such that a sutable dstace fucto d(p, Q s mmzed Obvously the approxmate result of degree reducto wll vary much accordg to the chose dstace fucto I ths paper, we use the least squares (L 2 orm ad the uform (L orm to measure the approxmate error The dstace fuctos betwee P (t ad Q(t wth respect to L 2 ad L orms o the terval [a,b] are defed as follows respectvely: b d 2 (P, Q P (t Q(t 2 dt, (2 a d (P, Q max P (t Q(t (3 t [a,b] Now the optmal mult-degree reducto wth costras of edpot cotuty ca be descrbed as follows Defto 1 Gve a degree Bézer curve P (t, approxmate t optmally by a degree m(m< 1 Bézer curve Q m (t wth respect to dfferet dstace fucto ad avod the stepwse computg, by cotuty of ay r, s (r, s 0 orders should be preserved at two edpots respectvely Ths problem s called optmal mult-degree reducto wth costras of edpot cotuty The key of the optmal mult-degree reducto the paper s the costrats of edpot cotuty ad the mmzato of the approxmate error dstace fucto, e, fd the best least squares ad best uform approxmato wth respect to the dstace fuctos d 2 ad d, respectvely We frst preset the followg property of Berste polyomals: Lemma 1 Berste polyomal B m (t of degree m, 0 t 1, ca be represeted by + m B m (t,j Bj (t, > m, 0, 1,,m, (4,j j ( m ( m j /( j (5
368 G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 Proof From the defto of Berste polyomals, we have ( ( m m B m (t (1 t m t (1 t m t (1 t + t m m ( ( m m (1 t m t (1 t m j t j j j0 m (( m j0 ( m j /( + j ( + m (1 t (+j t +j + j j,j B j (t I the followg theorem, a part of the cotrol pots of the approxmate degree reduced Bézer curve are frstly derved to satsfy the geometrc terpolato costrats of two edpots Theorem 1 Gve a degree Bézer curve P (t P B (t, t [0, 1], ad let r + s<m< 1, the the curve ca be expressed as P (t Q(t r f ad oly f Eqs (7 (9 s satsfed: s 1 Q B m (t + P I B (t + r+1 m m s Q B m (t, (6 Q 0 1 Q m 1 Q m j 0,0 b m,m (m, ( P 0, Q j 1 j,j P, 1 m j, j Whe m s>+ r m, ( P j P j j 1 j 1 max(0,j ( m m, j Q m max(0,j ( m,j Q, j 1, 2,,r,, j 1, 2,,s r P I j P j,j Q, j r + 1,r + 2,,+ r m, P I j P j, m s 1 >+ r m; j + r m + 1,+ r m + 2,,m s 1, s P I j P j m,j Q m, j m s,m s + 1,, s 1 (7 (8
G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 369 Whe m s + r m, r P I j P j,j Q, m s 1 >r; j r + 1,r + 2,,m s 1, P I j P j r,j Q s m,j Q m, j m s,m s + 1,,+ r m, s P I j P j m,j Q m, m s 1 >r; j + r m + 1,+ r m + 2,, s 1 (9 Where {Q } r, {Q } m m s are the part cotrol pots of degree reduced curve Q m(t of degree m, are the ukow accessoral cotrol pots, ad the followg two equatos must be satsfed {P I } s 1 r+1 d λ Q m (0 dt λ dλ P (0 dt λ, λ 0, 1,,r, d µ Q m (1 dt µ dµ P (1 dt µ, µ 0, 1,,s (10 Proof By Lemma 1, we have B m (t + m j,j Bj (t Substtute t to the rght sde of Eq (6 ad express t matrx form Let,j 0 (j < or j > m,the (Q 0, Q 1,,Q r 0,0 0,1 0, m 1,1 1, m + ( P I r+1, P I r+2,,p I s 1 r,r B r+1 B r+2 B s 1 b(m, 1, m+1 r,r+1 r, m+r + (Q m s,,q m 1, Q m m s,m s m s, s m 1,m 1 m 1, 1 b m,m (m, m, 1 (P 0, P 1,,P r, P r+1,,p s 1, P s,,p B 0 B 1 B m, B m s B 1 B B 0 B 1 B +r m wth the lear depedet property of Berste bases {B }, the ecessary ad suffcet codtos of Eqs (7 (9 ca be derved I addto, from Lemma 1 ad Eq (7, we have,
370 G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 m Q B m (t m + m j,j Q Bj (t r s 1 P j Bj (t + j0 jr+1 [ m(j,m max(0,j ( m ],j Q Bj (t + j s P j B j (t The Eqs (10 ca be derved from the above equato ad the symmetrcal property of the Bézer curves That s the ed of the proof Remark 1 The process theorem 1 essece s the part verse process of degree elevato (Far, 1991 Remark 2 Let m r + s + 1ad{Q } m s show as (7 The the approxmate degree reduced curve Q m (t of degree m wth cotrol pots {Q } m s the smplest Hermte terpolat, whch preserves the terpolato of r, s orders at two edpots repectvely I fact all cotrol pots {Q } m ca be derved from terpolato codtos Remark 3 Let m 1, r s [(m 1/2] Whem s odd {Q } m s show as (7 ad whe m s eve, {Q } m/2 1 ad {Q } m m/2+1 are show as (7 ad Q m/2 ( Q L s + QR r /2, Q L s P r+1 (r + 1Q r (r + 1, Q R r P m s ( m + sq m s m s The the approxmate curve Q m (t wth cotrol pots {Q } m s the degree reduced Bézer curve the paper of (Pegl ad Tller, 1995 3 Best least squares mult-degree reducto (MDR by L 2 I the approxmate theory, the orthogoal polyomal bass fuctos are always used to solve least squares approxmate problems We wll use costraed Jacob polyomals to mmze the least squares dstace fucto The Jacob polyomals J (r,s (x (Szego, 1975 are orthogoal o the doma ad ca be explctly represeted Berste forms as ( +r ( +s ( J (r,s (x ( 1 + x + 1 ( B, 0, 1,, (11 2 x [ 1, 1] ad r, s > 1 These Jacob polyomals are orthogoal o [ 1, 1] wth respect to the weght fucto w (r,s (x (1 + x r (1 x s Defe the costraed Jacob polyomals J,r,s as J,r,s (x (1 + x r+1 (1 x s+1 J (2r+2,2s+2 r s 2 (x, r + s + 2,r + s + 3, (12
G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 371 It forms the orthogoal bass o [ 1, 1] wth respect to the weght w(x 1, ad has roots of multplcty r + 1, s + 1atx 1, +1, respectvely If the fucto f(x (x [ 1, 1] ca be represeted as f(x (1 + x r+1 (1 x s+1 f(x,the Π, the polyomal J (r,s (x a J,r,s (x (13 r+s+2 s the best least squares approxmato of degree to f(x o x [ 1, 1], >r+ s + 1, a 1 δ 1 1 J,r,s (xf (x dx, δ The least squares approxmate error s 1 d 2 (f, J f(x J (x 2 dt 1 1 1 1 1 f 2 (t dt ( J,r,s (x 2 dx r+s+2 δ a 2 (14 If f(xs defed o the terval [a,b], we ca trasform t to the [ 1, 1] terval by a lear trasform as t (2x b a/(b a From the propertes of Berste polyomals ad Jacob polyomals, we have Lemma 2 For k 0, 1,, let J (r,s k (2t 1 (0 t 1 be the Jacob polyomal of degree k The the lear relato betwee Jacob polyomals ad Berste polyomals {B (t} (0 t 1 ca be expressed the matrx form as follows: J (r,s J (r,s L (r,s B, B ( L (r,s 1J (r,s E (r,s J (r,s, (15 ( J (r,s 0 (2t 1, J (r,s 1 (2t 1,,J (r,s (2t 1 T, (16 B ( B0 (t, B 1 (t,, B (t T, (17 L (r,s ( m(j,k L (r,s k,j (+1 (+1, (( ( /( k + r k + s k L(r,s k,j ( 1 k+ b (k,,j, (18 k ad b (k,,j s show as Eq (5 max(0,j+k Proof By (11, the Jacob polyomals ca be represeted Berste form as (( ( /( + r + s J (r,s (2t 1 ( 1 + B (t, 0, 1,, o the terval [ 1, 1] The from Lemma 1 ad the lear depedet property of Jacob bases ad Berste bases, the cocluso ca be easly derved Now we preset the best least squares mult-degree reducto (MDR by L 2 as follows
372 G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 Deote P I (t s 1 r+1 P I B (t The the problem of optmal approxmato of P (t wth costrats of edpots cotuty of ay r, s (r, s 0,r + s< 1 orders s equal to fdg out the optmal approxmato of P I (t wthout costrats of edpots cotuty By the propertes of Berste polyomals, we have s 1 r+1 P I B (t (1 ts+1 t r+1 N (r + s + 2, Deote P II N N ( P II P I r+1+ II ( P 0, P II 1,, P II N, P II P II P II BN (t (1 t s+1 t r+1 P II N (t, (19 r + 1 + The from (19, there s P I (t (1 ts+1 t r+1 P II N (t (1 ts+1 t r+1 P II N B N O the other had, by Lemma 2, there s P II N B N P II N E(2r+2,2s+2 N N N Suppose r + s<m 1, deote M m (r + s + 2, the t s easy to kow that J (2r+2,2s+2 P III N /( N, 0, 1,,N, 0, 1,,N, P II N (t N P III N J (2r+2,2s+2 N III ( P 0, P III 1 J (r,s m (t (1 ts+1 III t r+1 P M J (2r+2,2s+2 M s the best least squares approxmato of degree to J (r,s (t P I (t (1 ts+1 III t r+1 P N J (2r+2,2s+2 N o the terval [ 1, 1] III,, P N P II N E(2r+2,2s+2 N N, Now we try to derve the correspodg Berste form of (1 t s+1 t there s P III M J (2r+2,2s+2 M P III M L(2r+2,2s+2 M M B M P IV M B M, P IV IV M ( P 0, P IV 1,, P IV M P III M L(2r+2,2s+2 M M The (1 t s+1 III t r+1 P M J (2r+2,2s+2 M (1 t s+1 t r+1 P m s 1 IV M B M Q B m (t, ( M Q P IV r 1 r 1 /( m r+1 P II BN (t r+1 P III M J (2r+2,2s+2 M By Lemma 2,, r + 1,r + 2,,m s 1 (20
G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 373 The best least squares approxmate error ca be derved as follows: ( ε 2 d 2 P (t, Q m (t ( 1/2 d 2 ( J (r,s (t, J (r,s m (t δ P III 2 (21 m+1 Legedre polyomals are specal cases of Jocob polyomals Therefore, covertg a polyomal Bézer bass to Jacob bass s very smlar to that from Bézer to Legedre Ad as for the detaled covertg process, oe ca refer to (L ad Zhag, 1998 Ths covertg process s smple ad stable To sum up, we ca obta the followg theorem: Theorem 2 Gve a degree Bézer curve P (t, whe{q } m s show as (7 ad (20, m< 1, the mult-degree reduced Bézer curve Q m (t m Q B m (t of degree m s ts best least squares approxmato, by the cotuty of r, s (r, s 0,r + s<m 1 orders ca be preserved at two edpots, respectvely The best least squares error of approxmate mult-degree reducto s ε 2 As compared wth most degree reducto methods, such as the methods by Eck (1993, 1995, the MDR by L 2 possesses a seres of advatages Frst, It avods stepwse approxmato for the mult-degree reducto so that the computg tme ca correspodgly be decreased ad there are o accumulatve calculatve errors Secodly, t ca satsfy costrat codtos of edpot cotuty of ay orders at the same tme of mult-degree reducto, thus the degree reducto computato ca combe wth the subdvso algorthm to crease the precso Furthermore, t acheves the optmal approxmato usg the orthogoal polyomals ad acqures the explct soluto of degree reducto curves Example 1 Let P 12 (t be a Bézer curve of degree 12 wth cotrol pots ( 14, 8, ( 10, 5, ( 7, 5, ( 5, 7, (1, 3, ( 3, 5, (1, 11, (4, 9, (7, 7, (10, 4, (12, 9, (14, 11, (19, 0 The best least squares 3-degree reductos wth dfferet edpot costrats are show Fg 1 Fg 2 shows the 6-degree reducto wth C 1 cotuty after oe subdvso As ca be see from the example, the MDR by L 2 the paper obtas the good approxmato of mult-degree reducto ad ca be combed wth the subdvso algorthm effectvely 4 Comparso betwee the MDR by L 2 ad the MDR by L I the paper (Che ad Wag, 2000, we preset oe smple method (MDR by L to the mult-degree reducto wth respect to L orm ad the MDR by L obtas the early best uform approxmato I the followg we troduce the MDR by L brefly ad compare t wth the MDR by L 2 ths paper by some umercal examples Chebyshev polyomals are oe of the classcal orthogoal polyomals that have bee studed extesvely Let T (x deote the Chebyshev polyomal of degree, whch s defed by T (x cos( arccos x ( 1 x 1 The we ca obta the followg explct Berste represetato (Eck, 1993 T (x ( 1 + (( 2 2 /( B ( x + 1 2, 0, 1, (22
374 G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 (a (b Fg 1 Reducto from degree 12 (sold to degree 9 (dash (a (r, s (2, 2 (b(r, s (3, 3 (a (b Fg 2 Reducto from degree 12 (sold to degree 6 (dash ((r, s (1, 1 (a Wthout subdvso (b Wth oe subdvso Oe of the mportat propertes of Chebyshev polyomals T (x s the so-called equoscllatg property, e, that the Chebyshev polyomals have + 1 extremal values ( 1 at x cos(π/, 0, 1,, Chebyshev polyomals have bee used wdely degree reducto the past The costraed Chebyshev polyomal s frstly troduced to deal wth degree reducto wth costrats of edpots cotuty (Lachace, 1988 Ufortuately, these costraed Chebyshev polyomals have o explct represetatos except certa specal cases ad have to be determed umercally by a modfed Remez algorthm (Davs, 1963
G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 375 From the propertes of Berste polyomals ad Chebyshev polyomals, we have Lemma 3 For k 0, 1,,, the lear relato betwee Chebyshev polyomals {T (2t 1} (0 t 1 ad Berste polyomals {B (t} (0 t 1 ca be expressed matrx form as follows: T C B, B C 1 T A T, (23 T ( T 0 (2t 1, T 1 (2t 1,,T (2t 1 T, (24 C (C k,j (+1 (+1, C k,j ad b (k,,j s show as Eq (5 m(j,k max(0,j+k ( 1 k+ b (k,,j ( /( 2k k, (25 2 The proof of Lemma 3 s smlar to the proof of Lemma 2 Thus the MDR by L ca be descrbed as follows Deote P I (t (1 ts+1 t r+1 P II N (t (1 ts+1 t r+1 P II II N ( P 0, P II 1,, P II N, P II P II, The by (19 ad Lemma 3, there s P II N B N P II N A N N T N P III N T N Let r + s<m 1, deote M m (r + s + 2, P III N III ( P 0, P III 1 N P II BN (t (1 t s+1 t r+1 P II N B N, 0, 1,,N III,, P N P II N A N N By Chebyshev polyomal approxmato theory (Fox ad Parker, 1968, P III M T M s the early best uform approxmato of P II N B N amog all polyomals of degree M o the terval [ 1, 1] Now we derve the correspodg Berste form of (1 t s+1 III tr+1 P M T M By Lemma 3, there s (1 t s+1 III t r+1 P M T M (1 t s+1 t r+1 P m s 1 IV M B M Q B m (t, P IV M IV ( P 0, P IV 1,, P IV M P III ( /( M m r 1 Q P IV r 1 M C M M, r+1, r + 1,r + 2,,m s 1 (26
376 G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 (a (b Fg 3 The comparso betwee the MDR by L 2 (dash ad the MDR by L (dot (the sold curve s the orgal curve (a Reducto from degree 12 to 6 ((r, s (0, 0 (b Reducto from degree 12 to 8 ((r, s (2, 2 The error boud of early best uform approxmato s preseted as follows: ( d P (t, Q m (t ( ε max (1 t s+1 t r+1 N P III 0 t 1 M+1 (r + 1r+1 (s + 1 s+1 N P III (r + s + 2 r+s+2 (27 M+1 Therefore we ca obta the followg theorem: Theorem 3 Let Q m (t m Q B m (t,{q } m s show as (7 ad (26, the the mult-degree reduced Bézer curve Q m (t of degree m(m< 1 ca approxmate the gve degree Bézer curve P (t wth that the cotuty of r, s (r, s 0,r + s<m 1 orders ca be preserved at two edpots, respectvely It s the early best uform approxmato uder the codtos of edpot terpolao ad the correspodg error boud of degree reducto approxmato s ε Fg 3 presets the comparso of the MDR by L wth the MDR by L 2 usg the put curve the Example 1 of Secto 3 Obvously, the MDR by L 2 s better tha the early best uform approxmate method (MDR by L uder the codtos of edpot cotuty 5 Coclusos By usg the costraed Jacob orthogoal polyomal bass fuctos, ths paper derves oe best least squares approxmato method for mult-degree reducto of Bézer curves wth costrats of edpots cotuty The approxmate degree reduced Bézer curve s show as a explct soluto
G-D Che, G-J Wag / Computer Aded Geometrc Desg 19 (2002 365 377 377 form ad ca preserve cotuty of ay r, s (r, s 0 orders at two edpots, respectvely The error boud s gve ad the degree of accuracy for the approxmato s optmal wth respect to L 2 orm accordg to the tradtoal approxmate theory The method the paper avods stepwse computg for the mult-degree reducto so that the computg tme ca obvously be reduced The method ths paper ca be effectvely combed wth the subdvso algorthm for the approxmato of mult-degree reducto wth a prescrbed error tolerace Ackowledgemets Ths work s supported by the Natoal Natural Scece Foudato of Cha (No 60173034 ad the Foudato of State Key Basc Research 973 Item (No G1998030600 Refereces Bogack, P, Weste, S, Xu, Y, 1995 Degree reduto of Bézer curves by uform approxmato wth edpot terpolato Computer-Aded Desg 27 (9, 651 661 Che, G-D, Wag, G-J, 2000 Multdegree reducto of Bézer curves wth codtos of edpot terpolatos J Software 11 (9, 1202 1206 I Chese Daeberg, L, Nowack, H, 1985 Approxmate coverso of surface represetatos wth polyomal bases Computer Aded Geometrc Desg 2 (2, 123 132 Davs, PJ, 1963 Iterpolato ad Approxmato Dover, New York Eck, M, 1993 Degree reducto of Bézer curves Computer Aded Geometrc Desg 10 (4, 237 257 Eck, M, 1995 Least squares degree reducto of Bézer curves Computer-Aded Desg 27 (11, 845 851 Far, G, 1983 Algorthms for ratoal Bézer curves Computer-Aded Desg 15 (2, 73 79 Far, G, 1991 Curves ad Surfaces for Computer Aded Geometrc Desg, A Practcal Gude Academc Press, New York Forrest, AR, 1972 Iteractve terpolato ad approxmato by Bézer curve Computer J 15 (1, 71 79 Fox, L, Parker, IB, 1968 Chebyshev Polyomals Numercal Aalyss Oxford Uversty Press, Lodo Hoschek, J, 1987 Approxmato of sple curves Computer Aded Geometrc Desg 4 (1, 59 66 Lachace, MA, 1988 Chebyshev ecoomzato for parametrc surfaces Computer Aded Geometrc Desg 5 (3, 195 208 L, Y-M, Zhag, X-Y, 1998 Bass coverso amog Bézer, Tchebyshev ad Legedre Computer Aded Geometrc Desg 15, 637 642 Pegl, L, Tller, W, 1995 Algorthm for degree reducto of B-sple curves Computer-Aded Desg 27 (2, 101 110 Szego, G, 1975 Orthogoal Polyomals, 4th ed Amerca Mathematcal Socety, Provdece, RI Watks, M, Worsey, A, 1988 Degree reducto for Bézer curves Computer-Aded Desg 20 (7, 398 405