The univariate Bernstein-Bézier form

Transcription

1 Jorg s Splne Lecture Notes The unvarate Bernsten-Bézer form 1 The unvarate Bernsten-Bézer form The Bernsten bass functons of degree d on the nterval [0, 1] are ( ) d B d, : u (1 u) d u. When the degree s understood, we drop the subscrpt d. A polynomal of degree d n Bernsten (Bézer) form has the representaton c()b. If c() =f(/d) for some contnuous functon f then the polynomal s called Bernsten polynomal of f. ThsspecaltypeofpolynomalnBernstenformpopular n analysswll onlybe of margnal nterest n thefollowng. We are nterested n the polynomal pece traced out when u [0, 1] (otherwse see reparametrzaton below). a Symmetry: The equvalent representaton B (u, v) := d!!j! u v j, where u + v =1,+ j = d, shows the symmetry B (v) =B j (u). b Parttonofunty andpostvty B =1, 1 B (u) 0 for u [0..1]. Hence the polynomal s n the convex hull of the coeffcents for u [0, 1] and the form s affnely nvarant for all u. ForA 1,A 2 R A 2 + A 1 c()b (u) = (A 2 + A 1 c())b (u).

2 Jorg s Splne Lecture Notes The unvarate Bernsten-Bézer form 2..1 Exercse [3]: Prove that d B (u) =1,and1 B (u) Exercse [5]: Prove that B d, has ts maxmum over [0, 1] at the Grevlle abscssa x := /d...3 Exercse [5]: Draw the degree 3 bass functons B for =0..3 on the nterval [0, 1]. c Recurrence: The recurrence B d+1, (u) =(1 u)b d, (u)+ub d +1, 1 (u). s the bass for evaluatng polynomals n Bernsten form. The recurrence s consstent wth the defnton of the Bernsten form: ( ) d d +1 B d +1, (u) = d +1 vvd u 1 u ( ) ( ) d d +1 d = d +1 vvd u + d +1 vd+1 u 1 u = vb d, (u)+ub d+1, 1 (u)..4 Exercse [2]: Gve an alternatve nterpretaton n terms of probabltes. The correspondng nested multplcaton for evaluatng p d = d c()b at x s called De Casteljau s algorthm: for =0:d, p(d, ) :=c() for l =1:d, end for =0:d l, end p d (x) =p(0, 0). p(d l, ) =(1 x)p(d l +1, ) + xp(d l, +1)

3 Jorg s Splne Lecture Notes The unvarate Bernsten-Bézer form 3 De Casteljau s algorthm actually does more than generate the value. The dentty B d, (xu) = B k, (x)b d k,k (u) k mples that the restrcton of p to the nterval [0,x] s gven by p [0,x] (u) := a()b d, (xu) ( ) = a()b k, (x) B d k,k (u) = = k= ( k ) a()b k, (x) B d k,k (u) k=0 p(k, 0)B d k,k (u). k=0 d Dfferentaton The dervatve of a polynomal n Bernsten form must be wrtable as a polyno- Dfferentaton = mal of one degree less n Bernsten form: Dfferencng coeffcents ( ) d 1 D c()b d, = b()b d 1,..5 Exercse [5]: Show that the coeffcents of the dervatve are b() :=d( c( +1) c())...6 Example: The second dervatve of the quadratc polynomal p 2 =3B 0,2 +5B 1,1 +8B 2,0

4 Jorg s Splne Lecture Notes The unvarate Bernsten-Bézer form 4 s p 2 (u) =3(1 u) (1 u)u +8u 2 Dp 2 (u) =2(5 3)B 0,1 +2(8 5)B 1,0 =4(1 u)+6u D 2 p 2 (u) =1 (6 4)B 0,0 =2. Indeed, applyng the generc rules of dfferentaton to p 2,weget D 2 p 2 =3 2+5 ( 4) + 8 2=2...7 Exercse [5]: Compute coeffcents c(j) such that ( ) d 2 D 2 c()b d, = c(j)b d 2 j,j. j=0 e Integraton..8 Exercse [10]: Show that Integraton = Summng coeffcents 1 0 c()bd, = c()/(d +1). f Multplcaton..9 Exercse [10]: Show that d 1 d 2 c 1 ()B d1, c 2 ()B d2, = where d = d 1 + d 2,and c() = = ( d1 )( d2 1 ( d c 1 ( 1 )c 2 ( 2 ). ) 2 ) c()b d, Multplcaton = Collectng coeffcents wth equal ndex sums

5 Jorg s Splne Lecture Notes The unvarate Bernsten-Bézer form 5 Hnt: Show that for v := 1 u, j = d B j, := v j u, B1,j 1 B2,j 2 = B 1+ 2,j 1+j 2 Multplcaton by (1 u)+u s called degree-rasng. Usesaredataconverson and approxmate evaluaton...10 Exercse [5]: Wrte out the coeffcents of a quadratc polynomal that s degree-rased to a cubc polynomal...11 Exercse [5]: Show that the coeffcents of the degree-rased polynomal are convex combnatons of pars of coeffcents of the orgnal polynomal...12 Exercse [2]: Do degree-rasng and dfferentaton commute? g Hermte Interpolaton..13 Exercse [1]: Show that B (0) = { 1 f =0 0 else, B (1) = { 1 f = d 0 else Interpolaton = Matchng dfferences of coeffcents Hence for p d = c()b, h p d (0) = c(0), p d (1) = c(d), Dp d (0) = d(c(1) c(0)), Dp d (1) = d(c(d) c(d 1)) Controlpolygon The ponts etc. (x k,c(k)) wth Grevlle abscssae x k := k d are called control ponts. Thecontrol polylne l of p s a broken lne connectng the control ponts. Its kth segment l [xk,x k+1 ] on the nterval [x k,x k+1 ],sdefned by l [xk,x k+1 ](u) :=c(k) x k+1 u u x k + c(k +1). x k+1 x k x k+1 x k..14 Exercse [5]: Show that x k has to be the Grevlle abscssa f the control

6 Jorg s Splne Lecture Notes The unvarate Bernsten-Bézer form 6 polylne of a lnear polynomal s to agree wth ts graph. The control polylne s central to reasonng about nonlnearty and curved geometry due to convex hull property, the Hermte Interpolaton property and the followng theorem. Theorem [NPL 98] The dstance from the unvarate, scalar-valued, degree d polynomal p to ts control polylne l s bounded as where p(t) l(t),[0,1] N(d) 2 c 2 c := max c() 2c( +1)+c( +2) 0 d 2 N(d) := d/2 d/2 2d Subdvson and approxmateevaluaton Representng the polynomal pece p(u),u [0, 1] as two peces p 1 (u),u [0, 1/2] and p 2 (u),u [1/2, 1] has the advantage that the control polygons of p 1 and p 2 (on the fner subdvson) approxmate the graph of the functon more closely than the control polygon of p. In partcular, the theorem of the prevous subsecton mples Subdvson=Reparametrzaton p [0,x] (t) l [0,x] (t),[0,x] x 2 N(d) 2 c and m-fold subdvson results n the bound x 2m N(d) 2 a where x := max{x, 1 x}. That s, the control polylne converges to the graph of the functon lke x 2m...15 Exercse [15]: (a) Show that the control polygon of a degree-rased polynomal s obtaned from that of the orgnal polynomal by cuttng off corners (lnear nterpolaton). (b) Next show that the r-fold degree-rased representaton of the polynomal c()b has coeffcents b = j c j( n j) K(r,, j) for some K that converges for ncreasng r by Strlng s formula to t j (1 t) d j where t = /(d + r). (c) Use ths to show the varaton dmnushng property: a lne ntersects the graph of a Bernsten polynomal at most as often as t ntersects the control polylne.

7 Jorg s Splne Lecture Notes The unvarate Bernsten-Bézer form 7 j Converson topowerform..16 Exercse [10]: Check that B d, =1, ( d )B d,(u) =u, lnear precson ( d )2 B d, (u) =(1 1/d)u 2 + u/d. and that =k ( ) B d, (t) = k ( ) d t k. k k Weerstrass approxmatontheorem Defne the modulus of contnuty of f: ω(f,δ) :=lub b a δ f(b) f(a)

8 Jorg s Splne Lecture Notes The unvarate Bernsten-Bézer form 8 where lub means least upper bound. Choose δ =1/ d and let d then any contnuous functon f can be approxmated arbtrarly closely by polynomals: f f(j/d)b j [0,1] (f(u) f(j/d))b j u j/d δ + u j/d >δ (f(u) f(j/d))b j ω(f,δ)+ (1 + u j/d /δ)b j ω(f,δ) > ω(f,δ)[2 + > (u j/d) 2 /δ 2 B j ] = ω(f,δ)[2 + (u 2 2u(j/d)+(j/d) 2 )B j /δ 2 ] = ω(f,δ)[2 + (u 2 2u 2 +(1 1/d)u 2 + u/d)/δ 2 ] = ω(f,δ)[2 + u(1 u)/(dδ 2 )] ω(f,δ)[2 + 1/4] 0 as d..17 Exercse [2]: Show that polynomals are dense n L 2. Hnt: Use that the contnuous functons are dense n L 2.