Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0

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1 Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the form p(x a +a 1 x+ +a n x n a x, a R. We wll denote by π n the lnear (vector space of all such polynomals. The actual degree of p s the largest for whch a s non-zero. The functons 1,x,...,x n form a bass for π n, known as the monomal bass, and the dmenson of the space π n s therefore n+1. Bernsten polynomals are an alternatve bass for π n, and are used to construct Bézer curves. The -th Bernsten polynomal of degree n s ( n B n (x x (1 x n, (1 where n and The frst few examples are ( n n!!(n!. B (x 1, B 1 (x 1 x, B 1 1(x x, 1

2 Fgure 1: The cubc Bernsten polynomals B 2 (x (1 x 2, B 2 1(x 2x(1 x, B 2 2(x x 2, B 3 (x (1 x 3, B 3 1(x 3x(1 x 2, B 3 2(x 3x 2 (1 x, B 3 3(x x 3. The cubc ones are shown n Fgure 1. These polynomals are defned for all x R, but are usually restrcted to x [,1] n practce. They have varous mportant propertes. They are lnearly ndependent, for f c x (1 x n, x (,1, then, by dvdng by (1 x n and lettng y x/(1 x, we see that c y, y >, whch mples that c c 1 c n. Snce there are n+1 Bernsten polynomals of degree n, they do ndeed form a bass for π n. They are symmetrc n the sense that B n (x B n n (1 x. They are postve for x n the open nterval (,1 and at the endponts, { { B n 1 ; ( and B n,...,n 1; (1 1,...,n, 1 n. (2 2

3 They form a partton of unty: by the bnomal theorem, B n (x (x+(1 x n 1 n 1. They satsfy the recurson formula, B n (x xb n 1 1 (x+(1 xbn 1 (x, (3 whch follows from the defnton (1 and the bnomal dentty ( ( ( n n 1 n In (3 and elsewhere we defne B n 1 B n n+1. The computaton of the Bernsten polynomals up to degree n can be arranged n the followng trangular scheme, wth each column beng computed from the prevous column, startng from the left: 2 Bézer curves 1 B B 1 B 2 B n B 1 1 B 2 1 B n 1 B 2 2 B n B n n When modellng geometry n some Eucldean space R d we often model a curve, or part of a curve, as a parametrc polynomal, p(t a t, a R d. (4 Snce the Bernsten polynomals of degree n form a bass for π n, p can also be represented n the Bernsten form, p(t c B n (u, c R d, (5 3

4 Fgure 2: A cubc Bézer curve wth respect to any nterval [a,b], where u s the local varable u t a b a, (6 correspondng to t. If t [a,b] then u [,1] and vce versa. A polynomal curve expressed n ths form s known as a Bézer curve and the ponts c are known as the control ponts of p. The curve s usually restrcted to the parameter doman (parameter nterval [a, b], but s well defned also for t outsde [a, b]. The polygon formed by connectng the sequence of control ponts c,c 1,...,c n s known as the control polygon of p. Fgure 2 shows a cubc Bézer curve wth ts control polygon. To a large extent the shape of a Bézer curve relects the shape of ts control polygon, whch s why t s a popular choce for desgnng geometry n an nteractve graphcal envronment. As the user moves the control ponts nteractvely, the shape of the Bézer curve tends to change n an ntutve and predctable way. Varous propertes of Bézer curves follow from propertes of the Bernsten polynomals. For example, from (2, we obtan the endpont property of Bézer curves, p(a c, p(b c n. Snce the Bernsten polynomals sum to one, every pont p(t s an affne combnaton of the control ponts c,...,c n. From ths t follows that Bézer curves are affnely nvarant,.e., f Φ s an affne map n R d then the mapped curve Φ(p has control ponts Φ(c. So see ths, recall that an affne map has the form Φ(x Ax+b, x R d, for some matrx A of dmenson d d and a vector b of length d. Snce the 4

5 Bernsten polynomals sum to one, Φ(p(t Ac B n (u+b (Ac +bb n (u Φ(c B n (u. Snce the Bernsten polynomals are non-negatve n [, 1], the pont p(t, wth t [a,b], s a convex combnaton of the control ponts c,...,c n, and so p, restrcted to [a,b], les n the convex hull of ts control ponts: C : { λ c : λ,...,λ n, λ 1}. By treatng each of the d coordnates of p separately, a smlar reasonng shows that p restrcted to [a,b] also les n the boundng box [α 1,β 1 ] [α 2,β 2 ] [α d,β d ], where, f the pont c has coordnates c 1,...,c d, α k mn n c k and β k max n c k, k 1,...,d. Boundng boxes are used n varous algorthms, and are easer to compute than convex hulls. 3 The de Casteljau algorthm One way of computng a pont p(t of the Bézer curve p s frst to evaluate the Bernsten polynomals B n at u and then use the formula (5. Another, more drect method s de Casteljau s algorthm. We frst set c c, and then for each r 1,...,n, let c r (1 uc r 1 +uc r 1 +1,,1,...,n r. (7 The last pont computed n ths algorthm s the pont on the curve: p(t c n. We can show ths by showng more generally that for any r,1,...,n, n r p(t c r B n r (u. (8 5

6 Fgure 3: de Casteljau algorthm, u.3 Ths equaton clearly holds for r, and we can use nducton on r for the general case. Suppose (8 holds for some r n 1. Then applyng the recurson formula (3 to the rght hand sde gves n r p(t c r n r 1 ( ub n r 1 1 (u+(1 ub n r 1 (u ( uc r +1 +(1 uc r B n r 1 (u. Ths shows that (8 also holds wth r replaced by r + 1, and therefore, by nducton, for all r,1,...,n. Lke the recursve algorthm for computng Bernsten polynomals, the de Casteljau algorthm can be vewed as a trangular scheme, here arranged row-wse, wth each row beng computed from the row above: c c 1 c 2 c n c 1 c 1 1 c 1 n c n 1 c1 n 1 c n Fgure 3 llustrates the algorthm appled to the Bézer curve of Fgure 2 wth u.3. A numercal example wth n 3 s a, b 1, t u 2/3, d 1 and [c c 1 c 2 c 3 ] [4 4 18]. The three steps of the algorthm are then: [4 4 18] [4/3 8/3 4/3] [2/9 88/9] [196/27]. 6

7 The pont c r of the de Casteljau algorthm s tself the result of applyng r steps of the algorthm to the subsequence of control ponts c,c +1,...c +r, and so c r, vewed as a functon of t, s tself a Bézer curve of degree r: c r c r (t r c +j Bj(u. r (9 4 Dervatves To fnd the frst dervatve of p n (5, we apply the chan rule to make the converson d dt 1 d Ldu, where L b a. We also use a formula for the dervatve of a Bernsten polynomal. Dfferentatng (1 wth respect to x and usng the product rule gves d dx Bn (x n ( B n 1 1 (x Bn 1 (x. (1 From these observatons we fnd that the dervatve of the Bézer curve n (5 wth respect to ts parameter t s ( p (t n n 1 n 1 c +1 B n 1 (u c B n 1 (u. (11 L Thus one way of expressng the dervatve s as p (t n L where denotes the forward dfference operator, n 1 c B n 1 (u, (12 c c +1 c. Ths means that the dervatve of p s tself a Bézer curve wth respect to [a,b] wth control ponts, whch we now vew as vectors, n c /L. For each 7

8 t [a,b], the tangent vector p (t les n the convex cone of the vectors c,.e., n n 1 C : { λ c : λ,...,λ n 1 }. and by the endpont property of Bézer curves, p (a n L c and p (b n L c n 1. An alternatve way of expressng the dervatve s n terms of the ntermedate ponts (9 of the de Casteljau algorthm. Settng r n 1 n (9, t follows from (11 that p (t n L 5 Hgher dervatves ( c n 1 1 (t c n 1 (t n L cn 1 (t. (13 Applyng the dervatve formula (12 repeatedly leads to p (r (t dr dt rp(t n! (n r!l r n r r c B n r (u, for any r 1,...,n, where r s the r-th forward dfference operator r c r 1 c +1 r 1 c. At the endponts of the curve, the r-th dervatve depends only on the frst or last r+1 control ponts: p (r ( n! (n r!l r r c and p (r (1 n! (n r!l r r c n r. Alternatvely, n terms of the ntermedate de Casteljau ponts, dfferentatng (13 repeatedly gves p (r (t n! (n r!l r r c n r (t. 8

9 6 Integraton Integratng the dervatve formula (1 over x [, 1] gves ( 1 B n (1 B n ( n B n 1 1 (xdx 1 B n 1 (x dx, and snce the left hand sde s zero for 1,...,n 1, we deduce that 1 B n 1 1 (xdx 1 B n 1 (x dx. Thus the ntegral over [,1] of each Bernsten polynomal of the same degree s constant. Snce the Bernsten polynomals of degree n sum to one and there are n+1 of them, 1 B n (xdx 1 n+1. It follows that the ntegral of p n (5, normalzed by the dfference L b a, s 1 b p(tdt c +c 1 + +c n, L n+1 a whch s the barycentre of the control ponts c,...,c n. 7 Converson to Bézer form Sometmes we need to convert a polynomal from monomal form to Bézer form and vce versa. For smplcty we wll treat only the canoncal case that a and b 1 n (5, n whch case u t. Suppose we start wth the monomal form (4 and want to convert t to the Bézer form (5. We use the fact that n ( n 1 (1 t+t n t j (1 t n j, j 9

10 to show that p(t a n a ( n j ( n j j t +j (1 t n j t j (1 t n j, j ( n a t j (1 t n j, j and therefore c j 1 ( n j j ( n a. j Conversely, suppose we want to convert the Bézer form (5 to the monomal form (4. To do ths, observe that n ( n (1 t n ( 1 j t j, j and so p(t ( n ( n c j ( ( n n c j n j j ( n c ( n j ( 1 j t +j ( 1 j t j, ( 1 j t j, and t follows that a j j ( n ( n j ( 1 j c. 1

11 Ths can be alternatvely wrtten as a j j ( n j ( j ( 1 j c ( n j c. j 11