ANZIAM J. 45(003), 195 05 DEGREE REDUCTION OF BÉZIER CURVES USING CONSTRAINED CHEBYSHEV POLYNOMIALS OF THE SECOND KIND YOUNG JOON AHN 1 (Receved 3 August, 001; revsed 7 June, 00) Abstract In ths paper a constraned Chebyshev polynomal of the second nd wth C 1 -contnuty s proposed as an error functon for degree reducton of Bézer curves wth a C 1 -constrant at both endponts. A sharp upper bound of the L norm for a constraned Chebyshev polynomal of the second nd wth C 1 -contnuty can be obtaned explctly along wth ts coeffcents, whle those of the constraned Chebyshev polynomal whch provdes the best C 1 -constraned degree reducton are obtaned numercally. The representatons n closed form for the coeffcents and the error bound are very useful to the users of Computer Graphcs or CAD/CAM systems. Usng the error bound n the closed form, a smple subdvson scheme for C 1 -constraned degree reducton wthn a gven tolerance s presented. As an llustraton, our method s appled to C 1 -constraned degree reducton of a plane Bézer curve, and the numercal result s compared vsually to that of the best degree reducton method. 1. Introducton Degree reducton of Bézer curves s one of the most mportant problems n CAGD (Computer Aded Geometrc Desgn) or CAD/CAM. In general, degree reducton cannot be done exactly, leadng to approxmaton problems. Much effort has been drected at dealng wth these problems n the past twenty years. Most publcatons focus on partcular aspects of the problems, such as best degree reducton [7, 13], C -constrants [1,, 10, 14], the L p -norm [3, 8, 11, 16, 17, 18] and smple algorthms [1, 19, 1,, 3]. It s well-nown that the error functon of the best degree reducton n the unform (L ) norm s the Chebyshev polynomal up to the leadng coeffcent. But, n many 1 Department of Mathematcs Educaton, Chosun Unversty, Gwangju 501 759, Korea; e-mal: ahn@chosun.ac.r. c Australan Mathematcal Socety 003, Seral-fee code 1446-8735/03 195
196 Young Joon Ahn [] actual CAD/CAM systems, t s requred [] that the approxmate curve s contnuous to order 0 at each juncton pont of consecutve curve segments. In order to reduce the degree of Bézer curves wth C 0, C 1 or C -constrants at both endponts, constraned Chebyshev polynomals are necessary as the error functons, whch are the best approxmate monc polynomals to zero [13, 14] for each case. In the C 0 - constrant case, constraned Chebyshev polynomals can be expressed n terms of classcal Chebyshev polynomals, but the other cases can be obtaned numercally usng the (modfed) Remes algorthm [4, ]. Recently, Km and Ahn [10] proposed a good C 1 -constraneddegree reducton method usng constraned Jacob polynomals. The method gves the coeffcents of the polynomals explctly, and also presents the L norm of the polynomals n closed form for even degree. In ths paper, we propose another method for degree reducton wth C 1 -constrants usng properly modfed Chebyshev polynomals of the second nd. It s not the best C 1 -constraned degree reducton and ts unform error bound s also larger than the L norm of the constraned Jacob polynomal wth C 1 -contnuty numercally [10]. But our method presents the explct form of a sharp error bound of the L norm for all degrees along wth the coeffcents. We also present a smple subdvson scheme usng the unform error bound of our method n closed form. We apply our method to the degree reducton of a plane Bézer curve and compare the numercal result to that of the best degree reducton method. The outlne of ths paper s as follows. In Secton, we ntroduce our method of C 1 -constraned degree reducton usng constraned Chebyshev polynomals of the second nd wth C 1 -contnuty. We also present explctly ther unform error bound and the control ponts n Bézer form, whch are useful n actual CAD/CAM systems. Usng the unform error bound we gve the subdvson scheme for the C 1 -constraned degree reducton wthn a specfed tolerance. In Secton 3, we gve an example of the C 1 -constraned degree reducton of a plane Bézer curve of degree seven usng our method, and compare ts result to that of the best degree reducton by plottng the graphs of the degree reduced Bézer curves. In Secton 4, we summarse our wor.. Degree reducton wth C 1 -constrants In ths secton, we ntroduce our method for degree reducton of a Bézer curve wth a C 1 -constrant at both endponts. It s well-nown [14] that the best C 1 - constraned degree reducton has the constraned Chebyshev polynomal wth C 1 -contnuty as an error functon, but the polynomal can be obtaned numercally usng the Remes algorthm. A good C 1 -constraned degree reducton usng the constraned Jacob polynomal wth C 1 -contnuty [10] has the error bound explctly for even degree. We use constraned Chebyshev polynomal of the second nd wth C 1 -contnuty
[3] Degree reducton of Bézer curves 197 as an error functon for C 1 -constraned degree reducton, snce the polynomal has explct representaton n terms of Bézer coeffcents and unform error bound for all degrees. The followng are some well-nown propertes for Chebyshev polynomals of the second nd whch we shall call on later n the paper. PROPERTY.1 (Referto [6, 0]). The Chebyshev polynomal of the second nd, U n.x/ = sn..n + 1//= sn./, = arccos x, ofdegreen, has the followng propertes: (a) U n.x/ has leadng coeffcent n ; (b) The zeros of U n.x/ are x = cos.³=.n + 1//, = 1;:::;n, and the largest zero of U n.x/ s cos.³=.n + 1// whch s denoted by ¼ n n ths paper; (c) n U n.x/ has the smallest L norm on [ 1; 1] amongst all monc polynomals weghted by 1 x ; (d) U n.x/ 1= 1 x for x [ 1; 1]. (e) U n.x/ = n =0 0n..x 1/=/,where 0 n = 4n ( n + 1= n )( )/( ) n + + 1 n + 1 : n DEFINITION.. We defne the constraned Chebyshev polynomal of the second nd wth C 1 -contnuty for n 4by E n.t/ = t.t 1/U n.¼ n t ¼ n /.4¼ n / n for t [0; 1],wheremu n = cos.³=.n 1// s the largest zero of U n.x/. THEOREM.3. The constraned Chebyshev polynomal of the second nd wth C 1 - contnuty E n.t/ s a monc polynomal of degree n, has double zeros at t = 0; 1, and ts unform norm s bounded by E n. / L [0;1] In partcular, equalty holds for even n. 1 4 n 1 cos n.³=.n 1// : (.1) PROOF. By Property.1 (a), U n.¼ n t ¼ n / has leadng coeffcent.4¼ n / n so that E n.t/ s a monc polynomal. By Property.1 (b), U n.x/ has a zero at ±¼ n, so that E n.t/ has double zeros at both endponts t = 0; 1.
198 Young Joon Ahn [4] 1 n =8 n =10 0 n =9 n =7 n =5 n =6 n =4 10 0:5 1 FIGURE 1. Constraned Chebyshev polynomals of the second nd wthc 1 -contnuty, 4 n 1 cos n.³=.n 1//E n.t/, unformsed by the error bound, for 4 n 10, are plotted by dashed lnes for odd degree n and by sold lnes for even n. By Property.1 (d), we have E n.t/ for all t [0; 1]. Snce0<¼ n < 1, we get 1 t.1 t/.4¼ n / n 1.¼n t ¼ n / ; 4t.1 t/ =.1.t 1/ /.1 ¼ n.t 1/ / for all t [0; 1]. Thus E n.t/ 1.4¼ n / 1 n 4 = 1 4 n 1 cos n.³=.n 1// for all t [0; 1], and (.1) s satsfed. Partcularly, for even n, E n.1=/ = 1 U n.0/ 4.4¼ n / = 1 n 4 n 1 cos n.³=.n 1// yelds that the equalty n (.1) holds. 1 ¼n.t 1/ As an llustraton, we plot the constraned Chebyshev polynomal of the second nd wth C 1 -contnuty, 4 n 1 cos n.³=.n 1//E n.t/, unformsed by the error bound n Theorem.3, for 4 n 10, n Fgure 1. As shown n Table 1, we compare the unform error bound of E n.t/ whch s obtanable explctly wth the unform
[5] Degree reducton of Bézer curves 199 TABLE 1. The unform error norms of the best degree reducton wth C 1 -constrant T n./.t/ obtaned by the Remes algorthm, and the unform error bound of our method E n.t/ obtaned explctly. degree n Cn. / L [0;1] bound of E n. / L [0;1] 4 6:500 10 6:500 10 5 8:9443 10 3 1:1049 10 6 1:6546 10 3 :797 10 3 7 3:4009 10 4 5:0117 10 4 8 7:3959 10 5 1:1411 10 4 9 1:6648 10 5 :6559 10 5 10 3:8340 10 6 6:744 10 6 11 8:971 10 7 1:4981 10 6 1 :138 10 7 3:6051 10 7 13 5:074 10 8 8:777 10 8 14 1:196 10 8 :18 10 8 15 :948 10 9 5:18 10 9 16 7:1577 10 10 1:689 10 9 17 1:7440 10 10 3:1148 10 10 18 4:614 10 11 7:66 10 11 19 1:0440 10 11 1:8878 10 11 0 :568 10 1 4:6581 10 1 norm of the constraned Chebyshev polynomals wth C 1 -contnuty C n.t/ of leadng coeffcent 1 whch are obtaned numercally by the Remes algorthm [14]. (The superscrpt on C n.t/ means double zeros at both endponts [10, 14].) Now, usng the constraned Chebyshev polynomal of the second nd wth C 1 - contnuty E n.t/, we descrbe our method of C 1 -constraned degree reducton of a Bézer curve f.t/ = n b B n.t/ of degree n, wherebn.t/ = ( n ) t.1 t/ n and b,for = 0;:::;n, are Bernsten polynomals and Bézer coeffcents (or control ponts), respectvely. PROPOSITION.4. Let f.t/ = n b B n.t/ be the gven Bézer curve of degree n havng control ponts b. Then the Bézer curve f.t/ of degree (less than or equal to) n 1 gven by f.t/ := f.t/ 1 n b 0 E n.t/ s a degree reducton of f.t/ wth C 1 -constrant at both endponts, and ts error norm s bounded by f. / f. / L [0;1] = 1 n b 0 E n. / L [0;1] 1 n b 0 4 n 1 cos n.³=.n 1// ; where the nth forward dfference 1 n b 0 = n. 1/( n ) bn s equal to the leadng coeffcent of the polynomal f.t/ of degree n. PROOF. Snce the n-th degree polynomals f.t/ and 1 n b 0 E n.t/ have the same
00 Young Joon Ahn [6] leadng coeffcent, f.t/ s a polynomal of degree less than or equal to n 1. The -th order dervatve E./.t/ = 0for = 0; 1, at t = 0; 1, yelds that n f./.t/ = f./.t/ and f.t/ s a degree reducton of f.t/ wth a C 1 -constrant at both endponts. The error bound s easly obtaned from Theorem.3. The followng proposton gves the closed form of the control ponts of the constraned Chebyshev polynomals of the second nd wth C 1 -contnuty n Bézer form, whch s needed to calculate f.t/ as a Bézer curve or a segment of splne n CAD/CAM systems. PROPOSITION.5. The constraned Chebyshev polynomal of the second nd wth C 1 -contnuty of degree n n Bézer form s gven by n E n.t/ = c B n.t/; where c 0 = c 1 = c n 1 = c n = 0 and ( 1 n n 3= )( n+ 1 n c = ( n 3 )( n ) j=0 = j n = ;:::;n. )( n j 1 j)( j ) ( ¼n 1 ) j ¼ j+ n n ; (.) PROOF. By Defnton. and Property.1 (e), we have E n.t/ n a power bass t.t 1/ n ( ) E n.t/ = 0 n ¼n t ¼ n 1 1.4¼ n / n =0 n t.t 1/ ( )( ) = 0 n ¼n 1 j.4¼ n / n ¼ j n j t j =0 j=0 t.t 1/ n n ( )( ) = 0 n ¼n 1 j 4 n ¼ j+ n n t j ; j j=0 = j for n. Usng the transformaton from a power bass of degree m nto a Bernsten bass ( m m m j ) v j t j j = ( m ) v j B m.t/; we obtan E n.t/ = t.t 1/ 4 n n j=0 n j=0 = j ( n j j ( n j=0 ) ( )( ) 0 n ¼n 1 j ) j ¼ j+ n n B n.t/:
[7] Degree reducton of Bézer curves 01 By the equaton n n 1 t.t 1/ u B n.t/ = =1 ( n 1 ) ( n ) u 1 B n.t/ we have the Bézer form of the constraned Chebyshev polynomal of the second nd wth C 1 -contnuty E n.t/ = 1 4 n n 1 1 n =1 j=0 = j ( n j 1 j ) ( )( ( n 0 ) n ¼n 1 j ) j ¼ j+ n n B n.t/: (.3) Thus (.), for = ;:::;n, follows from (.3) and Property.1 (e), and c 0 = c 1 = c n 1 = c n = 0 follows from the fact that E n.t/ has double zeros at t = 0; 1. The fact that c 1 = c n 1 = 0 can be also proved by calculatng (.3) as follows: c 1 = c n 1 = 1 n4 n 1 n4 n n =0 0 n n n j=0 = j ( ) ¼n 1 ¼ n n = ( )( ¼n 1 0 n n 1 = U n.4¼ n / n n.¼ n / = 0: =0 j 1 n.4¼ n / n U n. ¼ n / = 0; ) j ¼ j+ n n Usng the proposton above we represent the degree reducton f.t/ of f.t/ n Bézer form: n 1 f.t/ = b B n 1.t/; where b, = 0;:::;n 1, are the Bézer coeffcents of f.t/ of degree n 1. PROPOSITION.6. The Bézer coeffcents b of f.t/ are gven by recursvely. b 0 1 n b 0 c 0 f = 0; b = n (b 1 n b 0 c n ) n b 1 f = 1;:::;n 1; PROOF. See Km and Ahn [10].
0 Young Joon Ahn [8] In the practcal applcaton, for a gven tolerance t s necessary to subdvde the Bézer curve f.t/ of degree n nto peces n order to approxmate each pece by the lower degree Bézer curve wthn the tolerance. In the followng theorem, we show how many subdvsons are requred so that each pecewse degree reducton usng our method has error less than the gven tolerance. THEOREM.7. For a gven tolerance ", the Bézer curve f.t/ = n b B n.t/ must be subdvded nto segments so that the degree reducton for each segment has unform error less than ",where s gven by ( ) 1 n 1=n b 0 = (.4) " 4 n 1 cos n.³=.n 1// and x denotes the smallest nteger larger than x. PROOF. Snce each subdvded segment f j.t/ = f.t= + j=/, j = 0;:::; 1, has leadng coeffcent.1=/ n 1 n b 0, the unform error of the degree reducton for each segment s equal to.1=/ n 1 n b 0 E n. / L [0;1]. In order to satsfy that the unform error bound be less than ",(.4) holds by Theorem.3. 3. Example In ths secton, we apply our method to reduce the degree of a plane Bézer curve of degree seven. Let the Bézer curve be gven by [7] f.t/ = 7 b B 7.t/; where the b s are.0; 0/,.:5; 0/,.:3; 1/,.1;:5/,.1; :75/,.1:7;:5/,.1:5; :5/,.; :5/, n order, as shown n Fgure. The degree reducton method usng a constraned Chebyshev polynomal of the second nd wth C 1 -contnuty yelds the approxmate Bézer curve f.t/ of degree sx, f.t/ = f.t/ 1 7 b 0 E 7.t/ = 6 b B 6.t/; where 1 7 b 0 =.4:4; 64:5/ and the b s are.0; 0/,.0:583; 0/,.0:3; 1:043/,.1; 0:119/,.1:678; 0:193/,.1:417; 0:5/,.:; 0:5/, n order. The unform error bound for the degree reducton s gven by f. / f. / L [0;1].4:4; 64:5/ E 7. / L [0;1] 0:0336:
[9] Degree reducton of Bézer curves 03 4 4 4 4 4 4 4 gven f (t) best μ f best (t) our method μ f(t) FIGURE. The best degree reducton and our method: the gven Bézer curve f.t/, the best degree reducton f best.t/ and our method f.t/ are plotted by sold lnes, dashed lnes, and dashed lnes wth crosses, respectvely. The boxes, trangles and crcles are the control ponts of each Bézer curve, n order. On the other hand, the best degree reducton by the constraned Chebyshev polynomal yelds f best.t/ = f.t/.4:4; 64:5/C.t/ = 6 v 7 B 6.t/, where the v s are.0; 0/,.0:583333; 0/,.0:337096; 1:00389/,.1; 0:11875/,.1:669; 0:153889/,.1:41667; 0:5/,.:; 0:5/, n order. We compare the graph of our degree reducton f.t/ to that of f best.t/ by plottng the Bézer curves and ther control ponts n Fgure. Letthe errortolerance " be gven by 0:001. By Theorem.7, the curve f.t/ must be subdvded nto two peces because = ( ).4:4; 6:5/ 1=7 1:65 =: 4 6.cos.³=6// 5 0:001 By subdvdng f.t/ at t = 1= ntotwobézer segments as shown n Fgure 3, the unform error bound s gven by 1.4:4; 64:5/ 0:00063 < 0:001; 7 4 6 cos.³=6/ 5 and the degree reducton wth C 1 -constrant for each Bézer segment s acheved wthn the specfed tolerance.
04 Young Joon Ahn [10] gven f (t) frst segment second segment FIGURE 3. Degree reducton usng the subdvson scheme: the degree reductons usng our method E n.t/ for the frst and second segments are plotted by dashed lnes and dashed lnes wth crosses, respectvely. The crcles and boxes are the control ponts of the degree reducton for each subdvson Bézer segment, n order. 4. Comments In ths paper we presented a method for degree reducton of Bézer curves wth the C 1 -constrant at both endponts havng the explct form of the unform error bound. The constraned Chebyshev polynomal of the second nd wth C 1 -contnuty E n.t/ proposed n ths paper has the mnmum L error bound among the monc polynomals whch have zeros of multplcty two at both endponts t = 0; 1, and for whch the unform error bounds are nown explctly. Even f E n.t/ s not the best degree reducton wth a C 1 -constrant and ts error bound s larger than that of the constraned Jacob polynomal wth C 1 -contnuty proposed by Km and Ahn [10], t s more useful than those polynomals snce no numercal calculatons are needed. That s to say, the Bézer coeffcents and the unform error bound of E n.t/ n explct form can be obtaned for any degree. We also gave a smple subdvson scheme and the numercal results for an example usng our degree reducton method. Acnowledgements The author s very grateful to the Edtor and anonymous referees for ther valuable comments. Ths study was supported by research funds from Chosun Unversty, 003.
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