q-bernstein polynomials and Bézier curves

Transcription

1 Jounal of Computatonal and Appled Mathematcs 151 (2003) 1-12 q-bensten polynomals and Béze cuves Hall Ouç a, and Geoge M. Phllps b a Depatment of Mathematcs, Dokuz Eylül Unvesty Fen Edebyat Fakültes, Tınaztepe Kampüsü Buca İzm, Tukey E-mal: hall.ouc@deu.edu.t b Mathematcal Insttute, Unvesty of St Andews Noth Haugh, St Andews, Ffe KY16 9SS, Scotland E-mal: gmp@st-and.ac.uk Receved 1 Decembe 2001; eceved n evsed fom 22 July 2002 Abstact We defne q-bensten polynomals, whch genealze the classcal Bensten polynomals, and show that the dffeence of two consecutve q-bensten polynomals of a functon f can be expessed n tems of second ode dvded dffeences of f. It s also shown that the appoxmaton to a convex functon by ts q-bensten polynomals s one sded. A paametc cuve s epesented usng a genealzed Bensten bass and the concept of total postvty s appled to nvestgate the shape popetes of the cuve. We study the natue of degee elevaton and degee educton fo ths bass and show that degee elevaton s vaaton dmnshng, as fo the classcal Bensten bass. Key Wods: Genealzed Bensten polynomal; shape pesevng; total postvty; degee elevaton AMS Subect Classfccaton: Pmay 65D17, seconday 41A10 1. Intoducton When epesentng a paametc cuve o suface t s mpotant whch bass s used f we wsh to peseve the shape of the cuve o suface. Fo these easons the Bensten-Béze cuve and suface epesentaton play a sgnfcant ole n CAGD. See, fo example, [5, 11. In ths pape we genealze some of the vey well known Béze cuve technques by usng a genealzaton of the Bensten bass, called the q-bensten bass. The Béze cuve s eteved when we set the paamete q to the value 1. Ths pape s oganzed as follows. Fst we defne a one-paamete famly of genealzed Bensten polynomals (called q-bensten polynomals) fom whch we ecove the classcal Bensten polynomals when Coespondng autho

2 2 Jounal of Computatonal and Appled Mathematcs 151 (2003) 1-12 we set q = 1. We pove that the appoxmaton to a convex functon by ts q-bensten polynomals s one sded. Then we show that the dffeence of two consecutve q-bensten polynomals has a epesentaton nvolvng second ode dvded dffeences. We descbe some of the shape pesevng popetes whch the genealzed Bensten polynomals shae wth the classcal countepats. The connecton between the powe bass, the Bensten bass and the q-bensten bass s evealed by devng the tansfomaton matces. We then constuct paametc cuves usng the q-bensten bass and dscuss shape popetes usng the concept of total postvty. Fnally, we pesent a degee elevaton algothm fo q-bensten paametc cuves and show that ths pocess s vaaton dmnshng, as n the classcal case. The q-bensten polynomals wee defned as follows by the second autho [15: 1 n B n (f; x) = f x (1 q s x), (1.1) =0 whee an empty poduct denotes 1, the paamete q s a postve eal numbe and f = f([/[n). Hee [ denotes a q ntege, defned by [ = s=0 { (1 q )/(1 q), q 1,, q = 1. The q-bnomal coeffcent [ n, whch s also called a Gaussan polynomal, s defned as [ n [n [n 1 [n + 1 = [ [ 1 [1 fo n 1, and has the value 1 when = 0 and the value zeo othewse. Note that ths educes to the usual bnomal coeffcent when we set q = 1. It satsfes the ecuence elatons [ [ [ n n 1 n 1 = q n + (1.2) 1 and [ n = and t can easly be vefed by nducton on n that (1 x)(1 qx) (1 q n 1 x) = [ [ n 1 n 1 + q, (1.3) 1 ( 1) q ( 1)/2 =0 x. (1.4) The q-bnomal coeffcent can be ntepeted combnatoally as the geneatng functon fo countng estcted pattons. We may wte [ (n ) n = p(n,, )q, =0

3 H. Ouç, G.M. Phllps / q-bensten polynomals and Béze cuves 3 whee p(n,, ) s the numbe of pattons of wth at most pats each not exceedng n. It s also elated (see [1) to the poblem of countng the numbe of subspaces ove a fnte feld. We note that B n, defned by (1.1), s a monotone lnea opeato fo any 0 < q 1 and B n epoduces lnea functons, that s B n (ax + b; x) = ax + b, a, b R. It also satsfes the end pont ntepolaton condtons B n (f; 0) = f(0) and B n (f; 1) = f(1). It s shown n [14 that (1.1) may be evaluated by the followng de Castelau type algothm: Gven: f [0 0, f [0 1,..., f n [0 fo m = 1 to n do fo = 0 to n m do f [m := (q q m 1 x)f [m 1 etun etun + xf [m 1 +1 Note that wth q = 1 we ecove the ognal de Castelau algothm. Wth q = 1 and any pont b R 2, R 3, ths algothm has a nce geometc ntepetaton whch s called subdvson, see [5. The effect of ntoducng the paamete q nto the de Castelau algothm can be seen n Fgue 1 n whch sufaces ae made by evolvng the cuves about an appopate vetcal axs. Fgue 1: Seven contol ponts and change of q, q = 0.8, q = 1. The genealzed Bensten polynomal B n (f; x) defned by (1.1) shaes the well known shape pesevng popetes of the classcal Bensten polynomal. Thus when the functon f s convex then (see [12) B n 1 (f; x) B n (f; x) fo n 2 and any 0 < q 1. In addton, t behaves n a vey nce way when we vay the paamete q : t s poved n [10 that B n(f; x) B q n(f; x) fo any 0 < q 1. As a consequence of ths we can show that the appoxmaton to a convex functon by ts q-bensten polynomal s one sded. Theoem 1.1 If f s a convex functon on [0, 1 then B n (f; x) f(x) fo 0 < q 1.

4 4 Jounal of Computatonal and Appled Mathematcs 151 (2003) 1-12 Poof Let l(x) = ax+b be any lne. Also let l be tangent at an abtay pont t [0, 1 so that l(t) = f(t) and f l 0. Usng B n (ax + b; x) = ax + b and the fact that B n s a monotone lnea opeato, we see that B n (f l) = B n f l 0. Thus, B n (f; t) l(t) = f(t) at any tangent pont t. By contnuty we deduce that B n f f. Theoem 1.2 Fo n = 2, 3,... we have n 2 x(1 x) [ n 2 B n 1 (f; x) B n (f; x) = q n+ 1 [n 1[n =0 [ n 2 [ [ + 1 [ + 1 f,, x (1 q s x). (1.5) [n 1 [n [n 1 Poof It s shown n [12 that the dffeence of consecutve q-bensten polynomals can be wtten as n 1 1 n B n 1 (f; x) B n (f; x) = x a (1 q s x), (1.6) whee a = =1 ( ) [n [ f + q [n [n 1 [n f s=0 s=1 ( ) [ 1 f [n 1 Let us evaluate the dvded dffeence of f at the ponts [ 1 the symmetc fom fo the dvded dffeences we obtan [ [ 1 f [n 1, [ [n, [ = [n 1 [n 1 2 [n q 2 2 [n f ( ) [ 1 [n 1 Fom (1.8) and (1.7) we see that a = qn+ 2 [n 1[n [n 1, [ [n [n 1 2 [n 2 q n+ 2 [[n f ( ) [. (1.7) [n [ and [n 1. Usng ( ) [ [n + [n ( ) 12 [n [ q n+ 2 [ f [n 1 [ 2 [ 1 f 1 [n 1, [ [n, and also t follows fom (1.6) that n 1 x(1 x) [ n 2 B n 1 (f; x) B n (f; x) = q n+ 2 [n 1[n 1 f =1 [ [n 1 [ [ 1 [n 1, [ n 1 [n, [ x 1 (1 q s x). [n 1 s=1 (1.8) Shftng the lmts of the latte equaton completes the poof. The ecent study [13 nvestgates convegence popetes of (1.1) as well convegence of ts teates and ts Boolean sums.

5 H. Ouç, G.M. Phllps / q-bensten polynomals and Béze cuves 5 2. Totally postve bases and the shape of cuves The bass functons whch appea n (1.1), B n (x) = 1 n x (1 q x), = 0, 1,..., n, (2.1) =0 satsfy the followng ecuence elatons whch can be deduced usng (1.2) and (1.3) espectvely, and B n (x) = q n xb 1 n 1 (x) + (1 qn 1 x)b n 1 (x), It can also be easly vefed that B n (x) = xb 1 n 1 (x) + (q q n 1 x)b n 1 (x). B m (x)b n (q m x) = [ m q (m ) [ m+n + B+ m+n (x). It s shown n [10 that the bass (2.1) povdes a nomalzed totally postve bass (NTP) fo 0 < q 1 on the nteval [0, 1 fo P n, the space of polynomals of degee not exceedng n. When q = 1, (2.1) s smply the classcal Bensten bass. NTP bases such as the genealzed Ball bass, the Bensten bass, and the B-splne bass, have an mpotant ole n geometc desgn whch we wll menton below. Let us ecall that a matx s sad to be totally postve (TP) f all ts mnos ae non-negatve. It s poved n [2 that a fnte matx s TP f and only f t s a poduct of 1-banded matces wth non-negatve elements. We say that a sequence Φ = (φ 0,..., φ n ) of eal-valued functons on an nteval I s TP f, fo any ponts x 0 < < x n n I, the collocaton matx (φ (x )),=0 n s TP. If Φ s TP and n =0 φ = 1 (so that ts collocaton matx s stochastc), we say that Φ s a NTP bass. Totally postve tansfomatons have a vaaton dmnshng popety, defned as follows : f T s a totally postve matx and v s any vecto fo whch Tv s defned, then S (Tv) S (v) (see [9), whee S (v) denotes the numbe of stct sgn changes n the (eal) sequence of elements of the vecto v. Smlaly, fo a eal-valued functon f on an nteval I we defne S (f) to be the numbe of sgn changes of f, that s S (f) = sup S (f(x 0 ),..., f(x m )) whee the supemum s taken ove all nceasng sequences (x 0,..., x m ) n I fo all m. Thus fom ths and the vaaton dmnshng popety we have S (B n (f; x)) S (f(0), f([1/[n),..., f(1)) S (f). Ths, togethe wth the fact that genealzed Bensten polynomals epoduce lnea functons, mples that when the functon f s monotonc so s B n (f; x)

6 6 Jounal of Computatonal and Appled Mathematcs 151 (2003) 1-12 and when t s convex so s B n (f; x) fo any 0 < q 1. Consequently, the opeato B n peseves the shape of the functon f on [0, 1, fo any 0 < q 1. Let us defne the paametc cuve P(t) by P(t) = (p 1 (t), p 2 (t)) = b B n (t), 0 t 1, (2.2) =0 whee b = (x, y ) R 2, = 0,..., n. We wll wte p(b 0,..., b n ) to denote the polygonal ac whch ons up the ponts b = (x, y ), = 0,..., n, usng pecewse lnea ntepolaton. Snce the genealzed Bensten bass s a nomalzed totally postve bass fo P n and 0 < q 1, P(t) s a convex combnaton of the ponts b 0,..., b n. Thus P(t) must le n the convex hull of the contol ponts fo all 0 t 1. Anothe consequence s the vaaton dmnshng popety. Theoem 2.1 The numbe of tmes any staght lne l cosses the cuve P(t) defned by (2.2) s no moe than the numbe of tmes t cosses p(b 0,..., b n ). Ths s ndeed tue fo any NTP bass Φ, and s poved n [9. It s obvous, on compang the numbe of sgn changes of l = ax + by + c wth that of P n (2.2), that we have ( ) S (ap 1 + bp 2 + c) = S (ax + by + c)b n (t) S (ax 0 + by 0 + c,..., ax n + by n + c), whch gves the desed esult. It follows fom ths that f the polygonal ac p(b 0,..., b n ) s monotonc n a gven decton then so s the cuve P. Moeove, f p(b 0,..., b n ) s convex, then any staght lne cosses t at most twce. Hence the cuve P cosses any lne at most twce whch mples that P s convex. Thus the shape of the cuve (2.2) mmcs the shape of the contol polygon p(b 0,..., b n ). Snce both Ψ = (B0 n(x),..., Bn n(x)) and the powe bass Φ = (1, x,..., x n ) fom a bass fo the space of polynomals P n, we may fnd the tansfomaton matx M such that Φ T = MΨ T. Snce n (x) = 1 we have n x = k k=0 k=0 Bn k x +k n k 1 t=0 (1 q t x). On [ shftng the lmts of the sum and the poduct above, usng (1.4) and wtng n [ k = n [ k [ k / n, we obtan [ k x = Bk n (x), = 0,..., n. (2.3) k= The matx M has the entes m,k = [ k = [n! [k! [k! [n!

7 H. Ouç, G.M. Phllps / q-bensten polynomals and Béze cuves 7 We may wte M = ATB such that A s a dagonal matx wth a, = [n! [n!, B s a dagonal matx wth b k,k = [k! and T s a Toepltz matx wth t,k = 1/([k!). Obvously the matces A and B ae TP fo any q > 0. Wth a lttle wok on the matx T we can vefy that t can be wtten as a poduct of 1-banded matces such that T = T (1) T (2) T (n) and the elements of each facto ae { 1, = k, t (),k = q k [k, = k 1, k. Thus T s TP fo any q > 0. Snce the poduct of TP matces s a TP matx we conclude that M s a TP matx. We also nvet the matx M to obtan coespondng coeffcents n Ψ T = M 1 Φ T. In a smla way, usng (1.4), we have [ [ n k B n (x) = ( 1) k q (k )(k 1)/2 x k. (2.4) k k= Thus the nvese of M has elements m 1,k = ( 1)k q (k )(k 1)/2 k [ k. Let us take φ = ( n ) x (1 x) n, = 0, 1,..., n. In the case of Ψ T = MΦ T, the elements of M satsfy m, = ( n ) (1 q) S(n 1, ), whee S(n, ) s the sum of ( n ) possble poducts of dstnct factos chosen fom the set {[1, [2,..., [n}. Note that S(n, ) satsfes the followng ecuence elaton S(n, ) = S(n 1, ) + [ns(n 1, 1), as can easly be vefed fom ts geneatng functon (1 + x)(1 + [2x) (1 + [nx) = S(n, )x. Snce any polynomal cuve can be expessed n tems of both bases, P(t) = b B n (t) = p φ (t), =0 the tansfomaton matx M also povdes the elatonshp between the contol ponts of P, (b 0,..., b n ) T = M T (p 0,..., p n ) T. It can be shown that the matx M obtaned above s TP fo any 0 < q 1 and M T s stochastc. Moeove t can be wtten as a poduct of 1-banded matces as follows : M = A n,q T 1,q... T n 1,q (A n,1 ) 1, =0 =0

8 8 Jounal of Computatonal and Appled Mathematcs 151 (2003) 1-12 whee A n,q s the dagonal matx wth elements a n,q, 1-banded matx such that t k,q, = 1, =, 1 q n k, = 1, 0 < k n 1, 0, othewse. = [ n and T k,q s the Ths class of matces s of patcula nteest n geometc desgn, when applyng a cone cuttng algothm whch s defned by a 1-banded TP stochastc matx. It s shown n [8 that a matx whch s nonsngula, TP and stochastc can be wtten as a poduct of 1-banded matces of the same type that descbes a cone cuttng algothm. It s also shown n [3, by usng ths technque to obtan the Béze polygon, that the Bensten bass has optmal shape pesevng popetes among all NTP bases fo P n. 3. Degee elevaton and educton One may wsh to ncease the flexblty of a gven cuve, usng the technque of degee elevaton. A degee elevaton algothm calculates a new set of contol ponts by choosng a convex combnaton of the old set of contol ponts whch etans the old end ponts. Fo ths pupose the denttes (1 q n t)b n (t) = [n + 1 [n + 1 B n+1 (t) (3.1) and q n t B n (t) = ( 1 ) [n [n + 1 pove useful. These follow mmedately fom (2.1). Theoem 3.1 Let P(t) = n =0 b B n (t), 0 t 1. Then =0 B+1 n+1 (t) (3.2) n+ P(t) = b B n+ (t), (3.3) whee, fo n 3 and = 0, 1,..., n +, [ [ n b = q ( )(n ) b. (3.4) =0 + Poof Ths can be poved by nducton on, as follows. We wte the gven cuve as P(t) = (1 q n t)p(t) + q n tp(t) and apply the followng ecusve algothm on degee elevated ponts b = ( 1 ) [n + b 1 [n b 1 [n + [n + { = 1, 2,... = 0, 1,..., n + (3.5)

9 H. Ouç, G.M. Phllps / q-bensten polynomals and Béze cuves 9 whee b 0 = b. On wtng ( ) [ [n + n / = qn+ and [n + 1 =0 + [n + [n + [ n+ 1 = 1 we can smplfy the ght sde of (3.5) to gve [ ( n q + [ [ 1 b = q ( )(n ) ) b. + + The q-bnomal coeffcents n the numeato of the latte expesson ae combned usng (1.2) to gve [. Ths vefes (3.4), and thus (3.3) holds. When q s eplaced by 1 above, we obtan the well known degee elevaton pocess fo Béze cuves. (See [5, 11, 6.) We obseve fom (3.5) that each new pont s obtaned by a convex combnaton of the two pevous ponts. Ths suggests the followng. Let b denote the vecto such that b T = [b 0,..., b n, whee the elements ae the contol vetces defned above. We also defne b as the vecto whose elements ae the contol vetces b, = 0, 1,..., n+, geneated by epeatng the degee elevaton pocess tmes. Theoem 3.2 Defne the cuve n+ E P(t) = b B n+ (t) =0 obtaned by tmes degee elevaton. Then, fo 0 < q 1, the numbe of tmes the cuve E P cosses any staght lne l s bounded by the numbe of tmes the polygon p(b 0,..., b n+) cosses l. Poof Let T,n be the tansfomaton matx such that b = T,n b. It s enough to show that T,n s a TP matx. Once agan we use nducton on to pove that T,n s a poduct of 1-banded postve matces wth the elements [ B (), T,n, [ = q( )(n ) n (3.6) + Thus T,n, s zeo unless 0. We note that the elements T,n, ae the coeffcents whch appea n (3.4). Now, the esult holds fo = 0 snce T 0,n s smply the (n+1) (n+1) dentty matx. Let B () denote the (n++1) (n+) 1-banded postve matx such that + 1 = q ( )(n+ ), fo 0 1. (3.7) + Then T,n = B () B ( 1) B (1). Let V = B (+1) T,n. Explctly ths yelds V, = n++1 k=0 B (+1),k T,n k,.,

10 10 Jounal of Computatonal and Appled Mathematcs 151 (2003) 1-12 We see fom (3.7) that B (+1),k s nonzeo only fo k = 1 and k =. Thus V, = B (+1), 1 T,n 1, + B(+1), T,n,. Hence, wth + 1 n (3.7) and (3.6), we obtan [ [ n [ V, = q (n+ +1)+( 1 )(n ) 1 + q ( )(n ) and hence V, = q ( )(n ) ++1 Usng the dentty (1.2) we obtan V, = q ( )(n ) ++1 (q ++1 [ [ [ ) +. 1 = T +1,n,, thus completng the poof. Thus the degee elevaton pocess fo the cuve wth the q-bensten bass s vaaton dmnshng. Ths has the followng consequences. If the contol polygon p(b 0,..., b n ) s monotonc n the y decton, so s the degee elevated polygon p(b 0,..., b n+). If the contol polygon p(b 0,..., b n ) s convex, so s the degee elevated polygon p(b 0,..., b n+). The nvese pocess of degee elevaton, whch s called degee educton, ams to epesent a gven cuve of degee n by one of degee n 1. In geneal, exact degee educton s not possble. We eque the q-dffeence fom of the Bensten polynomals, [ n B n (f; x) = f 0 x, =0 whee f = 1 f +1 q 1 1 f, and an nducton agument shows that [ f = ( 1) k q k(k 1)/2 f + k. (3.8) k k=0 We deduce that a q-béze cuve of degee n wth contol ponts b 0,..., b n has a degee n 1 epesentaton f and only f n b 0 = 0. Thus, fom (3.8) we have [ n n b 0 = ( 1) q ( 1)/2 b n = 0. =0 In ths case, n ode to fnd the new ponts b 0,..., b n 1 fo the q-béze epesentaton of degee n 1 we use the degee elevaton fomulas (3.1) and (3.2) so that n 1 b B n (t) = =0 =0 ( ( [n b B n (t) + 1 [n ) ) [n 1 B n [n +1(t).

11 H. Ouç, G.M. Phllps / q-bensten polynomals and Béze cuves 11 On compang the coeffcents of the bass functons B n (t), we obtan ( ) [n [n b = b + 1 b 1, = 0, 1,..., n 1, [n [n fom whch we obtan b = [n [n b ( ) [n [n 1 b 1, = 0, 1,..., n 1. (3.9) Ths appoxmaton s fom the left of the contol polygon, takng b 0 = b 0. When s eplaced by n n (3.9) we have an appoxmaton fom the ght sde, wth b n 1 = b n, b n 1 = [n [n [ b n [ n, = 0, 1,..., n 1. (3.10) [n [ b It s well known that when q = 1, as the numbe of degee elevated ponts the degee elevated polygon p(b 0,..., b n+ ) tends to the ognal cuve P(t) defned n the Theoem 3.1. wth the ate O( 1 n ), see [4, 16, 7. Acknowledgement The authos would lke to thank the efeee fo valuable emaks whch helped them mpove the pesentaton of ths study. Refeences [1 G. E. Andews, The Theoy of Pattons, Cambdge Unvesty Pess, Cambdge [2 C. de Boo and A. Pnkus, The appoxmaton of a totally postve band matx by a stctly banded totally postve one, Lnea Alg. Appl. 42 (1982), [3 J. M. Cance and J. M. Pẽna, Shape pesevng epesentatons and optmalty of the Bensten bass, Adv. n Comput. Math. 1 (1993) [4 E. Cohen, L.L. Schumake, Rates of convegence of contol polygons, Comp. Aded Geom. Desgn 2 (1985) [5 G. E. Fan, Cuves and Sufaces fo Compute-Aded Geometc Desgn. A Pactcal Gude, Academc Pess, San Dego [6 R. T. Faouk and V.T. Raan, On the numecal condton of polynomals n Bensten fom, Comp. Aded Geom. Desgn 4 (1987) [7 M.S. Floate and T. Lyche, Asymptotc convegence of degee-asng, Adv. n Comput. Math. 12 (2000)

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