Computer Aded Geometrc Desg 21 2004 151 164 wwwelsevercom/locate/cagd A ufed matrx represetato for degree reducto of Bézer curves Hask Suwoo a,,1, Namyog Lee b a Departmet of Mathematcs, Kokuk Uversty, Chugju, 380-701, South Korea b Departmet of Mathematcs, Mesota State Uversty, Makato, MN 56001, USA Receved 25 October 2002; receved revsed form 28 July 2003; accepted 29 July 2003 Abstract I ths paper, we show the degree reducto of Bézer curves a matrx represetato Most degree reducto algorthms have bee descrbed as a set of recursve equatos whch are based o the verse problem of degree elevato However, degree elevato ca be easly expressed terms of matrces Motvated by ths observato, we represet most well kow degree reducto algorthms a ufed matrx form I ths way, we ca smply express the process of degree reducto ad acheve greater sght for may kow results 2003 Elsever BV All rghts reserved Keywords: Bézer curve; Degree reducto; Geeralzed verse; Matrx algebra 1 Itroducto I geeral, degree reducto of Bézer curves addresses the followg problem: Problem 1 Let {b } Rs be a gve set of cotrol pots whch defe the Bézer curve x t = b B t, t [0, 1], * Correspodg author E-mal addresses: suwoo@kkuackr H Suwoo, amyoglee@msuedu N Lee 1 Preset address: Departmet of Mathematcs, Kokuk Uversty, Chugju, 380-701, South Korea 0167-8396/$ see frot matter 2003 Elsever BV All rghts reserved do:101016/jcagd200307007
152 H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 terms of Berste polyomals B t = t 1 t of degree The fd aother pot set { b } m Rs defg the approxmatve Bézer curve m x m t = b B m t, t [0, 1], of lower degree m<so that a sutable dstace fucto dx, x m betwee x ad x m s mmzed o the terval [0, 1] The problem of degree reducto of Bézer curves was tally proposed as a verse process of degree elevato the works of Forrest 1972 ad Far 1983 where most of the algorthms were descrbed terms of two dfferet recursve extrapolato formulas of a verse of degree elevato Accordgly, we ca classfy these methods by the ature of the recursos used: 1 Combg both formulas by takg the left half ad the rght half: Forrest 1972, Pegl ad Tller 1995 2 Takg a weghted average: Far 1983 3 L -orm approxmato usg costraed Chebyshev polyomals: Watks ad Worsey 1988, Lachace 1988, Eck 1993 4 Least squares approxmato usg costraed Legedre polyomals: Eck 1995, Lee ad Park 1997 The purpose of ths paper s to descrbe the process of degree reducto of Bézer curves by meas of matrx represetato By troducg the Moore Perose geeralzed verse matrx, we ca descrbe most degree reducto algorthms by a ufed matrx represetato I ths way, we smplfy the degree reducto of Bézer curves ad clarfy the relatoshp amog other degree reducto algorthms For smplcty of otato, we restrct ourselves to the case of m = 1 ad oly cosder a sglevalued fucto case Now, we restate the degree reducto problem as follows: Problem 2 Let {b } be a gve set of real umbers whch defes the Bézer fucto x t = b B t, t [0, 1] The, fd aother set of real coeffcets, { b } 1, such that 1 x 1 t = b B 1 t, t [0, 1], defes a approxmato to x t for whch a sutable dstace fucto, dx, x 1, s mmzed o the terval [0, 1] The orgazato of ths paper s as follows: We troduce some useful results from matrx algebra Secto 2, partcular, the cocept of geeralzed verse I Secto 3, a matrx represetato s gve for the geeral costructo of degree reducto for Bézer curves Matrx represetatos for other well kow degree reducto methods are gve Secto 4
H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 153 2 Some basc propertes of matrces The degree elevato matrx s well kow ad s of a very smple form However, sce the exact degree reducto s ot possble, the exact degree reducto matrx does ot exst But, by troducg a geeralzed verse matrx, we ca fd a degree reducto process a matrx form for each method descrbed Secto 1 A dscusso of geeralzed verses for matrces ca be foud Rao ad Mtra 1971 ad Searle 1982 Now, we recall ther deftos ad some of ther useful propertes Cosder a lear system Ax = b If A s a square matrx ad s osgular, the the lear system has a uque soluto as x = A 1 bbut,fa s sgular or f A s ot a square matrx, the the lear system has o soluto or has ftely may solutos I ether case, there s o verse matrx of A But, whe the lear system has ftely may solutos, we ca exted the cocept of verse matrx to a geeralzed verse Defto 1 Let A be a m rectagular matrx The a m matrx G s called a geeralzed verse of A f AGA = A ad s deoted by G = A Clearly, f A 1 exsts, the A = A 1 s the uque geeralzed verse of A But f ot, there exst ftely may geeralzed verses Lemma 1 For a gve lear system Ax = b, f the system s cosstet, the x = A b s a soluto Furthermore, the geeral solutos are gve by x = A b + I A Az, for ay vector z If we mpose some addtoal codtos, we ca get a uque geeralzed verse, amely the Moore Perose verse The defto of the Moore Perose verse s as follows: Defto 2 For ay matrx A there exsts a uque matrx G satsfyg the followg four codtos: AGA = A, GAG = G, AG s symmetrc, v GA s symmetrc, ad s deoted by G = A The Moore Perose verse has a beautful applcato the study of a cosstet lear system Ax = b Sce there s o exact soluto to the cosstet lear system, we cosder least squares solutos whch mmze Ax b 2 A vector x s called a least squares soluto to Ax = b f Ax b 2 Ax b 2 for all x A vector x s called a mmal least squares soluto to Ax = b f x s a least squares soluto to Ax = b ad x 2 s the smallest amog all least squares solutos
154 H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 Wth the help of matrx algebra t s easy to see that ay soluto to the ormal equatos A t Ax = A t b would be a least squares soluto As see the ext lemma, the mmal least squares soluto ca be obtaed by usg the Moore Perose verse Lemma 2 x = A b s the mmal least squares soluto to Ax = b 3 The geeral costructo I ths secto, we descrbe, frst, the geeral process of degree reducto of Bézer curves terms of recursve formula The we wll represet the procedure a ufed matrx represetato Also some useful results about matrx represetato wll be gve The well kow procedure of elevatg the degree of a Bézer curve from 1to ca be wrtte terms of the cotrol pots as b = 1 b 1 + b for = 0,, 1 We solve ths overdetermed system for the ukows { b } 1 twce: frst by eglectg the last equato =, ad secod by gorg the frst equato = 0 ad obtag the recursvely defed pots b I = 1 b b I 1 for = 0,, 1 2 ad b 1 II = 1 b b II for =,,1 3 whch are dstgushed by upper dces I ad II Based o ths costructo, the key dea of the degree reducto process s to wrte the ukow pots b of the fal degree reduced curve x 1 as a lear combato of b I ad b II, b = 1 λ b I + λ II b for = 0,, 1 4 Now, vew of 4, the problem of computg the pots { b } s shfted to the problem of computg factors {λ } 1 These ukow factors were troduced by Eck 1993 However the earler kow degree reducto procedures, for example, Forrest 1972 ad Far 1983, ca be expressed of the form 4 Ths observato eables us to fd a ufed approach of degree reducto of Bézer curves Now we express the procedure of degree reducto matrx-vector form Frst, we rewrte the degree elevato equato 1 matrx-vector otato as follows b = A b, where b =[b 0 b ] t ad b =[ b 0 b 1 ] t wth the degree elevato matrx A = a j beg a + 1 matrx gve by /, = j, a j = /, = j + 1, 6 0, otherwse, 5
H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 155 for = 0,,ad j = 0,, 1 Note that, A has the form A = 1 1 1 1 1 7 Note also that Eck 1993 ad Bruet et al 1996, show that the two coeffcets of b I ad b II, Eqs 2 ad 3, ca be rewrtte as, ad b I = 1 b II 1 1 j j j=0 = 1+1 1 j=+1 b j 1 j b j j for = 0,, 1 Usg these two formulas, we ca easly fd two + 1 matrces L = l j ad U = u j as follows { 1 l j = +j j, for = 0,, 1; j = 0,,, 1 10 0, elsewhere, ad { u j = 1 +j+1 j, for = 0,, 1; j = + 1,,, 1 11 0, elsewhere Note that L s a lower tragular matrx ad U s a upper tragular matrx such that b I = Lb ad b II = Ub 12 where b I =[ b 0 I b 1 I ]t ad b II =[ b 0 II b 1 II ]t Now, as oted earler, the key dea of the degree reducto process s to wrte the ukow pots b of the fal degree reduced curve x 1 as a lear combato of b I ad b II Furthermore, b ca be represeted matrx form: b = I D b I + D b II = { I DL + DU } b = Gb 13 where D = dagλ 0,,λ 1 s a dagoal matrx ad G = I DL + DU, whch s actually a geeralzed verse of the degree elevato matrx Lemma 3 Both L ad U defed 10 ad 11 are geeralzed verses of the degree elevato matrx A Furthermore LA = UA= I where I deotes a detty matrx Proof We wll prove oly that LA = I It follows that ALA = A, whch mples that L s a geeralzed verse of A 8 9
156 H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 Sce there are oly two ozero elemets each colum of A, thej th elemet of LA s LA j = l k a kj = l j a jj + l,j+1 a j+1,j k=0 Now f j>,thealll j = 0 ad ths mples LA j = 0 If j<,the LA j = 1 +j j 1 j + 1 +j+1 j+1 1 j + 1 = 1 +j 1 j j + 1 = 0 1 j j + 1 Fally f = j,thel,+1 = 0 ad therefore LA = l a = 1 + 1 = 1 The result ow follows wth the case of UA= I beg proved smlarly Lemma 3 gves us the followg result Corollary of Lemma 3 For ay matrx D, G = L DL U s a geeralzed verse of A Though G = L DL U s a geeralzed verse of A for ay matrx D, we are terested D beg a dagoal matrx For ay dagoal matrx D λ = dagλ 0,λ 1,,λ 1, let us defe A λ = A λ 0,λ 1,,λ 1 = L D λ L U 14 The A λ s also a geeralzed verse of A We have wrtte A λ = L D λl U stead of A λ = I D λl + D λ U because Lb ad L Ub are more meagful whe we compute A λ b I fact, L Ub = 1 b 0 h 15 where h = h wth h = 1 1 1 for = 0,, 1, ad the th forward dfferece b 0 s defed by b 0 = 1 +j j j=0 b j From Eq 15, oe ca easly see that b I = b II f ad oly f b 0 = 0 For ths reaso, we assume that b 0 0 for the rest of ths paper Note that, b I ad b II represet the degree reduced Bézer curves If we multply A to both vectors, the we obta artfcally degree elevated Bézer curves whose cotrol pots are gve by the followg lemma Lemma 4 Let b I = ALb ad b II = AUbThe b I = b b 0 e ad b II = b + 1 +1 b 0 e 1 where e = 0,,0, 1, 0,,0 t
H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 157 Proof From the deftos of matrces A, L, adu Eqs 6, 10, ad 11, respectvely, we ca easly see that 0 d 1 d I AL = 1 0, AU = 0 I 1 c 0 c 1 0 where c j = 1 +j j for j = 0,, 1addj = 1 j+1 j for j = 1,, Hece, we have b 0 b 0 ALb = b 1 1 = b 1 j=0 1+j j bj b b 0 ad j=1 1j+1 j bj b 0 + 1 +1 b 0 AUb = b 1 = b 1 b b Ths completes the proof Ths lemma s useful descrbg the relatoshp betwee x t ad x I 1 t, or betwee x t ad x II 1 t We wll dscuss ths at the ed of ths secto Smlarly, f b = A λ b ad c = AA λ b, the two Bézer curves, b ad c, represet the same curves geometrcally ad parametrcally That s, 1 x 1 t = b B 1 t = c B t 16 I the followg theorem we ca obta the explct form of c ad thus the error betwee b ad AA λ b Theorem 1 b AA λ b = 1 λ 0 1 + λ λ 1 1 λ 1 b 0 Proof Usg the fact ALb = b b 0 e,wehave AA λ b = A L D λ L U b = ALb AD λ L Ub = b b 0 e AD λ L Ub
158 H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 Hece b AA λ b = b 0 e + AD λ L Ub O the other had, AD λ L Ub = 1 Ths completes the theorem 1 1 1 λ 0 = 1 + λ λ 1 λ 1 1 1 b 0 1 λ 0 1 + 1 λ b 0 λ 1 Whle Theorem 1 gves us the error I AA λ, the ext theorem gves a potwse represetato of the error betwee the orgal curve ad the degree-reduced curve Theorem 2 By lettg λ 1 = 0 ad λ = 1, we have x t x 1 t = b 0 1 + λ λ 1 B t As a cosequece of Theorem 2, we ca easly see the relatoshp betwee x t ad x 1 t by vestgatg the dagoal elemets Now, we descrbe more propertes of b I ad b II Sceb I = A b I, both b I ad b I represet the same Bézer curves as do b II ad b II Thats, 1 x I 1 t = 1 x II 1 t = b I B 1 t = b II B 1 t = b I B t, 17 b II B t 18 Hece wth the help of Lemma 4, we have x t x I 1 t = b 0 B t, 19 x t x II 1 t = 1 b 0 B0 t 20 Sce B 0 = 0adB 0 1 = 0, x t ad x I 1 t meet at t = 0, ad x t ad x II 1 t meet at t = 1 Also the errors are gve by d x, x I 1 = d x, x 1 II = b 0 21
H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 159 ad d 2 x, x I 1 = d2 x, x II b 0 1 =, 22 2 + 1 where d f, g = max ft gt ad d 2 f, g = t [0,1] 1 0 ft gt 2 dt 4 Matrx represetatos of degree reducto of Bézer curves We may descrbe the problem of degree reducto as that of fdg a sutable vector b from b such that A b = b where A s the degree elevato matrx defed 5 As we descrbed the prevous secto, the vector b, the cotrol pots of degree reduced Bézer curve, ca be represeted the form b = A λ b 23 where A λ = L D λl U for a dagoal matrx D λ = dagλ 0,,λ 1 Usg the results Secto 3, we ow llustrate the use of matrx represetatos each of several well kow methods 41 Forrest s work Forrest 1972 proposed to combe both formulas by takg the left half of the coeffcets from 2 ad the rght half of the coeffcets from 3 as b = b I = 0, 1,, b = b II =, 2, 1 If s odd, the mdpot s defed by b = 1 2 b I + b II wth = 1/2 Hece Forrest s method s gve by { A A Forrest = 0,,0, 1,,1, for eve, A 0,,0, 1 2, 1,,1, for odd 24 For eve, { 1 f = /2, λ λ 1 = 0 otherwse, ad for odd, { 1/2 f = 1/2,+ 1/2, λ λ 1 = 0 otherwse
160 H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 Thus, by Theorem 2, x t x 1 t ca be represeted as follows: 1 /2 b 0 /2 B /2 t, for eve, x t x 1 t = 1 1/2 b 0 { B 2 1/2 1/2 t B+1/2 t}, for odd From the defto of the Berste polyomal, B t = t 1 t, t s easly see that for eve, x t x 1 t = b 0 t /2 1 t /2 ad for odd, x t x 1 t b 0 = 2 because B 1/2 t B +1/2 t = t 1/2 1 t 1/2 1 2t t 1/2 1 t 1/2 1 2t 1/2 Hece, the error s gve by max x t x 1 t { 1 2 b 0 f s eve, 1 0 t 1 2 1 4 1 4 1/2 b 0 f s odd, gudg the same the result of Park ad Cho 1994 25 42 Weghted average: Far 1983 Far 1983 proposed the method by takg a weghted average of the form b = 1 b I 1 + II b, for = 0,, 1 1 Hece Far s method s gve by A Far = 1 0, A 1, 2 1,,1 26 Sce λ λ 1 = 1 1 for = 1,, 2adλ 0 = 1 λ 1 = 0 Thus, by Theorem 1, we have 0 b AA λ b = 1 1 + 1 b 0 27 0 Also, by Theorem 2, we have x t x 1 t = b 0 1 1 + 1 B =1 t 28
H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 161 Note that 1 =1 1 + B t = { t 1 t+ 1 t1 t } 29 Therefore, the error boud s gve by max x t x 1 t b 0 = 0 t 1 1 max 1 1 + t 1 t 0 t 1 =1 = b 0 1 max t 1 t+ 1 t1 t 0 t 1 2 b 0 1 max t 1 t 0 t 1 = 2 b 2 0 1 + 1 Ths result s dfferet from the result of Park ad Cho 1994 We obta better error boud for the degree reducto of Far s method because the terms, whe = 0ad =, do ot appear the summato of 28 43 L -orm approxmato: Eck 1993 May authors have used the Chebyshev polyomals to fd the degree reducto of Bézer curves, such as the earler works by Watks ad Worsey 1988, Lachace 1988, ad Eck 1993 The Chebyshev polyomal of degree m s defed by T m t = cosm arccos t, t [ 1, 1], 30 ad ca be represeted usg a Berste bass by m t + 1 2m / m T m t = c,m B m where c,m = 1 m+ 31 2 2 The ma dea the works cted above s to express a Bézer curve x t terms of Chebyshev polyomals shfted to the terval [0, 1] as x t = a T 2t 1, t [0, 1], 32 ad trucate the hghest coeffcet 32 to obta the best L -orm approxmato as 1 x 1 t = a T 2t 1, t [0, 1] 33 Immedately, from Eqs 32 ad 33, we have x t x 1 t = a T 2t 1 34
162 H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 or equvaletly, matrx-vector otato, we have I AA λ b = a h 35 where A λ = L D λl U for some dagoal matrx D λ ad h =[c 0, c, ] t SceLA = I,by multplyg L to both sdes, we obta D λ L Ub = a Lh Sce 1 D λ L Ub = 1 λ 1 b 0 ad Lh = 1 j=0 1j j cj,, we have λ = a 1 +j c b 0 j j, = a 2 b j=0 0 2j j=0 for = 0,, 1 36 By Theorem 1, the last row of Eq 35 ca be wrtte as 1 λ 1 b 0 = a c, = a 37 By substtutg λ 1 36 ad 37, ad by usg the detty 2 j=0 2j = 2 2 1,wehave a = b 0 2 38 2 1 Therefore, the dagoal elemets are gve by λ = 1 2 for = 0,, 1, 39 2 2 1 2j j=0 whch are deduced for best L -approxmato usg Chebyshev polyomals, ad thus recover Eck s results of 1993 44 The least squares approxmato: Eck 1995 Eck 1995 foud the best L 2 orm degree reducto of a Bézer curve by usg so-called costraed Legedre polyomals whch are explctly kow We ow derve explct expressos for the dagoal elemets {λ } 1 Theorem 3 Eck, 1995 Cosder a curve x wth b 0 0 Iff the factors {λ } 1 λ = 1 2 +2α j=0 j α j + α are gve by the the cotrol pots { b } 1 determe the curve x 1 that mmzes d 2 x, x 1 wth the addtoal costrats for t 0 = 0 ad t 0 = 1 ad 2α d r dt x t r = dr t=t0 dt x 1t r, 0 r α 1 41 t=t0 40
H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 163 Eck 1995 showed that the dagoal elemets are the soluto of the followg equato: b AA λ b = a h where h =[h 0 h ] t ad 1 h = 1 + 43 α + α We ca solve Eq 42 by usg almost the same procedure as descrbed the prevous secto By troducg α, we obta the C α 1 -cotuous degree reduced Bézer curve at the two edpots t = 0adt = 1 Note that we have b AA λ b Theorem 1 The followg result shows the relatoshp betwee cotuty at the edpots ad the vector b AA λ b Theorem 4 Let e = e where e = b AA λ bif e = 0, = 0,,α 1, the x t ad x 1 t meet C α 1 -cotuously at t = 0If e = 0, = 0,,α 1, the x t ad x 1 t meet C α 1 -cotuously at t = 1 Sce h = h = 0, for = 0,,α 1, 43, we see that e = e = 0, for = 0,,α 1 Hece we have C α 1 -cotuous degree reduced Bézer curve at both of two edpots Fally, the best costraed Legedre polyomals appear for the best costraed L 2 -approxmato from Eck s results of 1995 Ad as a specal case, the best L 2 -approxmato ad thus the choce of the λ,wheα = 0, whch geerate the matrx A λ that s fact the Moore Perose verse accordg to Lee ad Park 1997 42 5 Cocluso I ths paper, we have descrbed the degree reducto of Bézer curves terms of matrx-vector represetato Wth the help of matrx algebra, we ca easly express the procedures matrx-vector form We have also preseted a ufed matrx represetato for these methods usg two dfferet recursve extrapolato formulas for a verse of degree elevato As a applcato, we descrbed ts uses may well kow degree reducto results Secto 4 I some case, we obta better error bouds for the degree reducto Ackowledgemets The authors would lke to thak the edtor ad the referee for ther valuable commets
164 H Suwoo, N Lee / Computer Aded Geometrc Desg 21 2004 151 164 Refereces Bruett, G, Schreber, T, Brau, J, 1996 The geometry of optmal degree reducto of Bézer curves Computer Aded Geometrc Desg 13, 773 788 Far, G, 1983 Algorthms for ratoal Bézer curves Computer-Aded Desg 15, 73 77 Forrest, AR, 1972 Iteractve terpolato ad approxmato by Bézer polyomals Computer J 15, 71 79 Eck, M, 1993 Degree reducto of Bézer curves Computer Aded Geometrc Desg 10, 237 251 Eck, M, 1995 Least squares degree reducto of Bézer curves Computer-Aded Desg 27, 845 851 Lachace, MA, 1988 Chebyshev ecoomzato for parametrc surfaces Computer Aded Geometrc Desg 5, 195 208 Lee, B-G, Park, Y, 1997 Dstace for Bézer curves ad degree reducto Bull Austral Math Soc 56, 507 515 Park, Y, Cho, UJ, 1994 The error aalyss for degree reducto of Bézer curves Computers Math Appl 27, 1 6 Pegl, L, Tller, W, 1995 Algorthm for degree reducto of B-sple curves Computer-Aded Desg 27, 101 110 Searle, SR, 1982 Matrx Algebra Useful for Statstcs Wley, New York Rao, CR, Mtra, SK, 1971 Geeralzed Iverse Matrces ad ts Applcatos Wley, New York Watks, MA, Worsey, AJ, 1988 Degree reducto of Bézer curves Computer-Aded Desg 20, 398 405