Lecture 21: Bezier Approximation and de Casteljau s Algorithm. and thou shalt be near unto me Genesis 45:10

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1 Lecure 2: Bezier Approximaion and de Caeljau Algorihm and hou hal be near uno me Genei 45:0. Inroducion In Lecure 20, we ued inerpolaion o pecify hape. Bu inerpolaion i no alway a good way o decribe he conour of a curve or urface. To accuraely reproduce complicaed hape, we may need o inerpolae lo of daa. Polynomial inerpolaion for many poin i impracical becaue he degree of he inerpolan can ge exremely high leading o low and numerically unable compuaion. Alo polynomial inerpolan may ocillae unnecearily and fail o reproduce he deired hape (ee Figure ). Thu, even if we were o pecify more and more poin, here i no guaranee ha he polynomial inerpolan would converge o he curve or urface we wih o repreen. Daa Polynomial Inerpolan Figure : Lagrange inerpolaion. Noice he ocillaion in he inerpolaing polynomial curve, even hough here i no ocillaion in he original daa poin. Spline inerpolaion -- ha i, inerpolaion by piecewie polynomial funcion -- i beer compuaionally becaue pline allow u o keep he degree low. Bu inerpolaing pline may ill ocillae unnecearily and fail o reproduce he deired hape. Our approach here i raher o abandon inerpolaion alogeher and o ake a very differen approach o decribing he hape of a curve or urface. Given a relaively mall collecion of poin in affine pace, we are going o inveigae mehod for generaing polynomial curve and urface ha approximae he hape decribed by hee poin. We hall no ini ha our curve and urface go hrough hee poin, bu we hall ini ha hee curve and urface come near o he poin and capure in ome mahemaically precie way he hape defined by hee poin. A uual we begin wih cheme for curve and laer exend our echnique o urface.

2 De Caeljau Algorihm Le u reurn for a momen o where we began our inveigaion of polynomial curve and urface: Lagrange inerpolaion and Neville algorihm. Recall ha Neville algorihm (Figure 2) i a dynamic programming procedure for compuing poin along a polynomial inerpolan. We are going o ar our inveigaion of approximaion cheme by uing he ame baic riangular rucure bu implifying he compuaion along he edge. The imple hing -- one migh almo ay he only hing -- we know how o do i linear inerpolaion. All our inerpolaion procedure, and epecially Neville algorihm, are baed on hi imple idea or ome varian hereof. Wha make Neville algorihm he lea bi complicaed i ha we perform a differen linear inerpolaion a each node of he diagram. To ake he ame riangular rucure and make he evaluaion algorihm a eay a poible, we will perform he ame linear inerpolaion a each node. Thi idea generae he algorihm repreened in Figure 3. 3 P 023 () 0 P 02 () P 23 () P 0 () P 2 () P 23 () P 0 P P 2 P 3 Figure 2: Neville algorihm (unnormalized) for cubic polynomial inerpolaion. b b B() a b a b a a b a b a P 0 P P 2 P 3 Figure 3: The de Caeljau algorihm for a cubic Bezier curve B() in he inerval [a, b]. The label on every edge mu be normalized by dividing by b a, o ha he label along arrow enering each node um o one. 2

3 b Thi algorihm repreened in Figure 3 i called de Caeljau evaluaion algorihm, and he curve ha emerge a he apex of hi diagram are called Bezier curve. Inermediae node marked and alo repreen Bezier curve, bu of lower degree. Thu he de Caeljau algorihm i a dynamic programming algorihm for compuing poin on a Bezier curve. Typically, for reaon ha will become clear in he nex ecion, Bezier curve are rericed o he inerval [a, b]. Uually, for impliciy, we ake a = 0 and b =, bu here are cae, a we hall ee laer on, where i i ueful o allow a and b o be arbirary a long a b > a. Noice ha when a = 0 and b =, no normalizaion i required. The de Caeljau algorihm ha he following elegan geomeric inerpreaion. Since each node repreen a linear inerpolaion, each node ymbolize a poin on he line egmen joining he wo poin whoe arrow poin ino he node. Drawing all hee line egmen generae he relli in Figure 4. a b P 0 P a P 2 b a b b b a a B() a a Figure 4: Geomeric conrucion algorihm for a poin on a cubic Bezier curve baed on a geomeric inerpreaion of he de Caeljau evaluaion algorihm. A he parameer, each line egmen in he relli i pli in he raio ( a)/(b ). b P 3 We are going o udy he geomeric characeriic of curve generaed by de Caeljau algorihm. We begin wih ome imple feaure; in ubequen lecure we will develop ome more advanced mahemaical properie of hi approximaion cheme. 3. Elemenary Properie of Bezier Curve Bezier curve have he following elemenary properie:. Polynomial Paramerizaion 2. Affine Invariance 3. Convex Hull 4. Symmery 5. Inerpolaion of End Poin 3

4 Below we briefly dicu and derive each of hee properie in urn, and we explain a well why hee feaure are imporan for compuer graphic. Mo of hee properie can be proved by direc obervaion or by eay inducive argumen uing he de Caeljau algorihm. To e up hee inducive argumen, le B[P 0,..., P n ]() denoe he Bezier curve over he inerval [a, b] wih affine conrol poin P 0,..., P n. Then he la age of he de Caeljau algorihm can be wrien a B[P 0,..., P n ]() = b b a B[P 0,..., P n ]() + a b a B[P,..., P n ](). (3.) We are now ready o proceed wih our derivaion.. Polynomial Paramerizaion In he de Caeljau algorihm, he only operaion we perform involving he funcion along he edge are addiion and muliplicaion (ee Figure 3). Since he funcion along he edge are linear polynomial, i follow ha a Bezier curve wih n + conrol poin i a polynomial curve of degree n becaue here are n level from he conrol poin a he bae o he curve a he apex of he riangle. (Thi reul alo follow by an eay inducion from Equaion (3.)). Since Bezier curve are polynomial curve, all he ool we know for polynomial apply. 2. Affine Invariance A curve i aid o be affinely invarian if i coni of a collecion of poin in affine pace. Equivalenly, a curve cheme i aid o be affinely invarian if applying an affine ranformaion o he conrol poin ranform every poin on he curve by he ame affine ranformaion. Affine invariance i an eay conequence of he de Caeljau algorihm. ince by linear inerpolaion every node in he algorihm i affinely invarian (ee Figure 5,6). (Thi reul alo follow by an eay inducion from Equaion (3.)). Affine invariance i a crucial feaure for any curve cheme becaue i aer ha he curve i independen of he choice of he coordinae yem. Thi propery i eenial for a good approximaion cheme, ince in a ypical geomeric model many differen coordinae yem are preen. Affine invariance guaranee ha he curve will be he ame no maer which coordinae yem i invoked. 4

5 b B() + v a b P 0 + v b + v +v a b +v a + v + v a b a b a P + v P 2 + v P 3 + v Figure 5: Tranlaion invariance. Tranlaing each conrol poin by a vecor v ranlae every poin on he curve by he ame vecor v. b ( B(),) M a b (P 0,) M (,) M (,) M b a b a (,) M (,) M (,) M a b a b (P,) M (P 2,) M a (P 3,) M Figure 6: Affine invariance. Tranforming each conrol poin by an affine map M ranform every poin on he curve by he ame affine map M. 3. Convex Hull Propery A e S of poin in affine pace i aid o be convex if whenever P and Q are poin in S he enire line egmen from P o Q lie in S (ee Figure 7). The inerecion S of a collecion of convex e {S i } i a convex e becaue if P and Q are poin in S, hey mu alo be poin in each of he e S i. Since, by aumpion, he e S i are convex, he enire line egmen from P o Q lie in each e S i. Hence he enire line egmen from P o Q lie in he inerecion S, o S oo i convex. The convex hull of a collecion of poin in affine pace i he inerecion of all he convex e conaining he poin. Since he inerecion of convex e i a convex e, he convex hull i he malle convex e conaining he poin. For wo poin, he convex hull i he line egmen 5

6 joining he poin. For hree non-collinear poin, he convex hull i he riangle whoe verice are he hree poin. The convex hull of a finie collecion of poin in he plane can be found mechanically by placing a nail a each poin, reching a rubber band o ha i inerior conain all he nail, and hen releaing he rubber band. When he rubber band come o re on he nail, he inerior of he rubber band i he convex hull of he poin. P Q P Q (a) Convex Se (b) Non-Convex Se Figure 7: In a convex e (a) he line egmen joining any wo poin in he e lie enirely wihin he e. In a non-convex e (b) par of he line egmen joining wo poin in he e may lie ouide he e. Since he convex hull of wo poin i he line egmen joining he wo poin, ConvexHull{P 0, P } = {c 0 P 0 + c P c 0 + c = and c 0,c 0}. More generally i follow by a imple inducive argumen (ee Exercie ) ha n n ConvexHull{P 0,..., P n } = { c k P k c k = and c k 0}. k= 0 k =0 Bezier curve alway lie in he convex hull of heir conrol poin. Tha i, B[P 0,..., P n ]() ConvexHull{P 0,..., P n }. Again we can prove hi aerion by a imple inducive argumen. Fir recall ha, by convenion, we alway reric he Bezier curve B[P 0,..., P n ]() in (3.) o he parameer inerval a b. Wih hi rericion, he convex hull propery i cerainly rue for a Bezier curve wih only wo conrol poin ince, by conrucion, hi curve i he line egmen joining he wo conrol poin. More generally uppoe ha hi reul i valid for Bezier curve wih n conrol poin. By (3.), B[P 0,..., P n ]() lie on he line egmen joining he poin B[P 0,..., P n ]() and B[P,..., P n ](), and by he inducive hypohei B[P 0,..., P n ]() and B[P,..., P n ]() boh lie in he convex hull of he poin P 0,..., P n. Bu if wo poin lie in a convex e, he enire line egmen joining hem alo lie in he e; hu he enire Bezier curve B[P 0,..., P n ](), a b, mu lie in ConvexHull{P 0,..., P n } 6

7 0 2. The convex hull propery i imporan becaue i conrain Bezier curve o lie in he proximiy of heir conrol poin. Thi propery i a vial feaure for an approximaion cheme. Scieni and engineer no only require curve ha approximae he hape defined by heir conrol poin, hey alo demand curve ha lie in he ame region of pace a heir conrol poin. To be ueful in compuer graphic, he curve mu be viible on he graphic erminal. The convex hull propery guaranee ha if all he conrol poin are viible on he graphic erminal, hen he enire curve i viible a well. The rericion a b on he parameer i here preciely o guaranee he convex hull propery. 4. Symmery Replacing by a + b revere he order of he parameer domain. A he parameer varie from a o b, he curve B[P 0,..., P n ](a + b ) ravere he ame poin a B[P 0,..., P n ]() bu in he direcion from b o a raher han from a o b. Thu B[P 0,..., P n ](a + b ) i eenially he ame curve a B[P 0,..., P n ]() bu wih oppoie orienaion. Similarly, revering he order of he conrol poin of a Bezier curve generae he ame Bezier curve bu wih oppoie orienaion. Analyically hi mean ha, B[P n,..., P 0 ]() = B[P 0,..., P n ](a + b ) a b. (3.2) Thi ymmery propery i wha profeional engineer and mo naive uer would naurally expec of a imple approximaion cheme, o i i graifying o ee ha i hold for all Bezier curve. To prove (3.2), imply replace by a + b in he de Caeljau diagram and oberve ha he new diagram i he mirror image of he de Caeljau diagram for B[P n,..., P 0 ]() (ee Figure 8). B() a a b a b b a b a b a b P 0 P P 2 P 3 Figure 8: The de Caeljau algorihm for B[P 0,..., P n ](a + b ). Compare o Figure 3 wih he conrol poin in revere order. 7

8 5. Inerpolaion of End Poin Unlike Lagrange polynomial, Bezier curve generally do no inerpolae all heir conrol poin. Bu Bezier curve alway inerpolae heir fir and la conrol poin. In fac, B[P 0,..., P n ](a) = P 0 B[P 0,..., P n ](b) = P n. The fir reul follow eaily from eing = a in de Caeljau algorihm and oberving ha all he label on lef poining arrow become zero while all he label on righ poining arrow become one. If k 0, hen any pah from P k o he apex of he riangle mu ravere a lea one lef poining arrow, o here i no conribuion from P k o he value of he curve a = a. On he oher hand, when = a all he label on he ingle pah from P 0 o he apex of he riangle are one. Hence B[P 0,..., P n ](a) = P 0. A imilar argumen for = b how ha B[P 0,..., P n ](b) = P n. Again an eay inducive argumen baed on (3.) yield he ame reul. Inerpolaing end poin i imporan becaue we ofen wan o connec wo curve. To aure ha wo Bezier curve join a heir end poin, all we need o do i o make ure ha he fir conrol poin of he econd curve i he ame a he la conrol poin of he fir curve. Thi device inure coninuiy. In he nex ecion, we hall develop echnique for guaraneeing higher order moohne beween adjacen Bezier curve, bu before we can perform hi analyi we need o know how o differeniae Bezier curve. 4. Differeniaion Algorihm for Bezier Curve To compue he derivaive of a Bezier curve, we need o differeniae he de Caeljau algorihm. Differeniaing he de Caeljau algorihm for a degree n Bezier curve i acually quie eay: imply differeniae he label -- ha i, replace and -- on any level of he de Caeljau algorihm and muliply he oupu by n (ee Figure 9). We hall defer he proof of hi differeniaion algorihm for Bezier curve ill Lecure 23. In hi ecion we will udy he conequence of hi differeniaion algorihm. There are everal hing o noice abou Figure 9. Fir, by running only he lowe level of he differeniaion algorihm, we ee ha, up o a conan muliple, we can hink of he derivaive of a cubic Bezier curve wih conrol poin {P k } a a quadraic Bezier polynomial wih conrol vecor {P k+ P k } (ee Figure 0). 8

9 0 2 B () = 3 _ P 0 P P 2 P 3 Figure 9: Compuing he fir derivaive of a cubic Bezier curve wih conrol poin P 0,P, P 2,P 3 by differeniaing he label on he boom level of he de Caeljau algorihm. To ge he correc derivaive B (), we mu muliply he oupu of hi algorihm by n = 3. B () = 3 _ P P 0 P2 P P 3 P 2 Figure 0: The derivaive of a cubic Bezier curve wih conrol poin {P k } i, up o a conan muliple, a quadraic Bezier polynomial wih conrol vecor {P k+ P k }. In general, up o a conan muliple, he derivaive of a Bezier curve wih conrol poin {P k } i a Bezier polynomial of one lower degree wih conrol vecor {P k+ P k }. Therefore, by inducion, we can expre he kh derivaive of a degree n Bezier curve a a Bezier polynomial of degree n k. Moreover, o compue he kh derivaive of a Bezier curve, we can differeniae he label on any k n! level he de Caeljau algorihm and muliply he oupu by (ee Figure ). (n k)! 9

10 0 2 B () = 6 _ P 0 P P 2 P 3 Figure : Compuing he econd derivaive of a cubic Bezier curve wih conrol poin P 0,P, P 2,P 3 by differeniaing he label on he boom wo level of he de Caeljau algorihm. Here o ge he correc econd derivaive B (), we mu muliply he oupu of hi algorihm by 6. We prefaced our dicuion of differeniaion by aying ha we waned o be able o join wo Bezier curve ogeher moohly. To do o, we need o calculae derivaive a heir end poin. Le B() be a Bezier curve wih conrol poin P 0,K,P n. Subiuing = 0 or = ino he differeniaion algorihm, we ee ha: B(0) = P 0 B() = P n B (0) = n(p P 0 ) B () = n(p n P n ) B (0) = n(n )(P 2 2P + P 0 ) B () = n(n )(P n 2P n + P n 2 ) In general, i follow by inducion on k ha he kh derivaive a = 0 depend only on he fir k + conrol poin, and he kh derivaive a = depend only on he la k + conrol poin. Suppoe hen ha we are given a Bezier curve wih conrol poin P 0,K,P n and we wan o conruc anoher Bezier curve wih conrol poin Q 0,K,Q n ha mee he fir curve and mache i fir r derivaive a i end poin. From he reul in he previou paragraph, we find ha he conrol poin Q 0,K,Q n mu aify he following conrain: r = 0 : Q 0 = P n r = : Q Q 0 = P n P n Q = P n + (P n P n ) r = 2 : Q 2 2Q + Q 0 = P n 2P n + P n 2 Q 2 = P n 2 + 4(P n P n ) and o on for higher value of r. Each addiional derivaive allow u o olve for one addiional conrol poin. Clearly we could go on in hi manner olving for one poin a a ime. An alernaive approach ha avoid all hi ediou compuaion will be preened in Lecure 22. 0

11 5. Tenor Produc Bezier Surface A enor produc Bezier pach i a recangular urface pach generaed by a recangular array of conrol poin. Connecing conrol poin wih adjacen indice by raigh line generae a conrol polyhedron ha conrol he hape of he Bezier pach in much he ame way ha he Bezier conrol polygon conrol he hape of a Bezier curve (ee Figure 2 and 3). In paricular, dragging a conrol poin pull he urface pach in he ame general direcion a he conrol poin. = d P 03 P 3 P 23 P 33 = a = b P 02 P 2 P 22 P 32 P 0 P P 2 P 3 = c P 00 P 0 P 20 P 30 (a) Domain -- Recangle (b) Range -- Recangular Array of Poin Figure 2: Daa for a bicubic enor produc Bezier pach. Noice ha he domain i imply a recangle and ha, unlike Lagrange inerpolaion, he domain ha no node and no grid. Compare o Figure 8 in Lecure 20. Figure 3: A bicubic enor produc Bezier urface wih i conrol polyhedron, formed by connecing conrol poin wih adjacen indice. The conrol poin are he ame a hoe for he Lagrange urface in Figure 0 of Lecure 20. To conruc a enor produc Bezier pach B(,) from a recangular array of conrol poin {P ij }, we proceed in a manner imilar o he conrucion of a enor produc Lagrange urface,

12 excep ha Lagrange curve and Neville algorihm are replaced by Bezier curve and de Caeljau algorihm. Le P i () = B[P i,0,k,p i,n ]() i = 0,K,m, (4.) be he Bezier curve for he conrol poin P i,0,k,p i,n. For each fixed value of, le B(,) = B[P 0 (),K, P m ()]() (4.2) be he Bezier curve for he conrol poin P 0 (),K, P m (). Then a varie from a o b and varie from c o d, he curve B(,) weep ou a urface (ee Figure 4,5). Thi urface i called a enor produc Bezier pach. Equaion (4.), (4.2) and Figure 4,5 ugge he following evaluaion algorihm for enor produc Bezier pache: fir ue de Caeljau algorihm m + ime o compue he poin a he parameer along he degree n Bezier curve P 0 (),..., P m () ; hen ue de Caeljau algorihm one more ime o compue he poin a he parameer along he degree m Bezier curve wih conrol poin P 0 (),..., P m () (ee Figure 6). P 3 P 23 P 33 P 03 P 2 P 22 P 02 P 0 P 0 () P 00 P P 0 B(, ) P 2 () P ( ) P 2 P 20 P 30 P 3 ( ) Figure 4: Conrucion of poin on a bicubic enor produc Bezier urface B(, ). Fir he Bezier curve P i (), i = 0,..., 3 are conruced from he conrol poin P i0, P i, P i2, P i3. Then for a fixed value of, he Bezier curve B(, ) i conruced uing he poin P 0 (), P (), P 2 (), P 3 () a conrol poin. In general, he Bezier urface B(, ) doe no inerpolae i conrol poin. Compare o Figure 9 of Lecure 20 for bicubic Lagrange inerpolaion. P 32 P 3 2

13 Figure 5: The bicubic Bezier pach in Figure 3 along wih i cubic Bezier conrol curve. Noice ha only he boundary conrol curve are inerpolaed by he urface. Compare o he enor produc Lagrange urface in Figure 0 of Lecure 20. B(,) P 0 () P () P 2 () P 00 P 0 P 02 P 0 P P 2 P 20 P 2 P 22 Figure 6: De Caeljau evaluaion algorihm for a biquadraic Bezier pach. The hree lower riangle repreen Bezier curve in he direcion, and he upper riangle blend hee curve in he direcion. Compare o Figure of Lecure Neville algorihm for a biquadraic Lagrange inerpolaing pach. 3

14 Alernaively, inead of aring wih he Bezier curve P i () = B[P i,0,k,p i,n ]() i = 0,K,m, we could begin wih he Bezier curve P j () = B[P0, j,k,p m, j ]() j = 0,K,n, and for each fixed value of le B(,) = B[P 0 (),K,Pn ()]() be he Bezier curve for he conrol poin P 0 (),K,Pn (). Again a varie from a o b and varie from c o d, he curve B(,) weep ou a urface. We hall how in a laer lecure ha hi urface i idenical o he enor produc Bezier pach in our previou conrucion. Thu we have he following alernaive evaluaion algorihm for enor produc Bezier pache: fir ue de Caeljau algorihm n + ime o compue he poin a he parameer along he degree m Bezier curve P 0 (),...,Pn (); hen ue de Caeljau algorihm one more ime o compue he poin a he parameer along he degree n Bezier curve wih conrol poin P 0 (),...,Pn () (ee Figure 7). When m < n, hen hi algorihm i more efficien han he previou algorihm. B(,) P 0 () P () P 2 () P 00 P 0 P 20 P 0 P P 2 P 02 P 2 P 22 Figure 7: An alernaive verion of he de Caeljau evaluaion algorihm for a biquadraic Bezier pach. The hree lower riangle repreen Bezier curve in he direcion, and he upper riangle blend hee curve in he direcion. Compare o Figure 6. 4

15 Tenor produc Bezier pache inheri many of he characeriic properie of Bezier curve: hey are affine invarian, lie in he convex hull of heir conrol poin, and inerpolae heir corner conrol poin (ee Exercie 6-8). Thee properie follow eaily from Figure 4,6 and he correponding properie of Bezier curve. Moreover he boundarie of a enor produc Bezier pach are he Bezier curve deermined by heir boundary conrol poin, ince B(0,) = P 0 () = B[P 0,0,K,P 0,n ]() and B(,) = P m () = B[P m,0,k,p m,n ]() B(,0) = P 0 () = B[P0,0,K,P m,0 ]() B(,) = P n () = B[P0,n,K,P m,n ](). I follow ha alhough enor produc Bezier pache do no generally inerpolae heir conrol poin, hey alway inerpolae he four corner poin P 00,P m0, P 0n,P mn. To compue he parial derivaive of a Bezier pach, we can apply our procedure for differeniaing he de Caeljau algorihm for Bezier curve. Conider Figure 6. We can compue B / imply by differeniaing he de Caeljau algorihm for each of he Bezier curve P 0 (),K, P m () a he bae of he diagram. Similarly, we can compue B / by differeniaing any one of he upper level of he diagram and muliplying he reul by he degree in (ee Figure 8). Symmeric reul hold for differeniaing he algorihm in Figure 7: imply revere he role of and. The normal vecor N o a Bezier pach i given by N = B B. B B = 2 _ P 0 () P () P 2 () P 0 () P () P 2 () Figure 8: Compuing he parial derivaive of a biquadraic Bezier pach by applying he procedure for differeniaing he de Caeljau algorihm for Bezier curve. To find B / (lef) imply differeniae he de Caeljau algorihm for each of he Bezier curve P 0 (), P (), P 2 (). To find B / (righ) imply differeniae one level of he de Caeljau algorihm i and muliply he reul by he degree in. 5

16 Exercie:. Le P 0,..., P n be a collecion of poin in an affine pace. Prove ha n n ConvexHull{P 0,..., P n } = { c k P k c k = and c k 0}. k= 0 k =0 2. Prove ha B[P 0,..., P n ](b) = P n. 3. Show ha Lagrange inerpolaing polynomial curve are affine invarian. 4. Give an example o how ha a degree n Lagrange inerpolaing polynomial wih node 0 < < L < n doe no necearily aify he convex hull propery on he inerval [ 0, n ]. 5. Give an example o how ha Bezier curve do no necearily reproduce polynomial curve. Tha i, give an example o how ha if here i a degree n polynomial P() uch ha P k = P( k ), hen i doe no necearily follow ha B[P 0,K,P n ]() = P(). 6. Show ha enor produc Bezier pache are affine invarian. 7. Show ha every enor produc Bezier pach lie in he convex hull of i conrol poin. 8. Show ha every enor produc Bezier pach inerpolae i corner conrol poin. 9. Le P i, j, 0 i m, 0 j n, be a recangular array of conrol poin, and le B[P i, j ](,), 0,, denoe he correponding enor produc Bezier pach of bidegree (m, n). Show ha he urface pach B[P i, j ](,) ha he following ymmery properie: a. B[P j,i ](,) = B[P i, j ](,) ; b. B[P i,n j ](,) = B[P i, j ](, ) ; c. B[P m i, j ](,) = B[P i, j ](,). 0. Le P i, j, 0 i m, 0 j n, be a recangular array of conrol poin, and le P i () = B[P i,0,k,p i,n ]() i = 0,K,m, be a equence of Bezier curve. Show how o combine Neville algorihm and de Caeljau algorihm o generae a urface C(,) ha inerpolae he curve P 0 (),...,P m () a he parameer value 0,..., m. 6

17 . Given poin and derivaive daa (R 0,v 0 ),...,(R n,v n ), explain how o place Bezier conrol poin o generae a C piecewie cubic curve o inerpolae hi daa. 2. The formula for he uni angen U(), he curvaure K(), and he orion T() of a parameric curve P() are given by i. U() = P () P () ii. K() = P () P () P () 3 iii. T() = P () ( P () P ()) P () P () 2 a. Compue he uni angen, curvaure, and orion of a Bezier curve a = 0,. b. Find condiion on he conrol poin of a Bezier curve C() o ha i mache a given Bezier curve B() wih coninuou uni angen, curvaure, and orion. 7