Introduction. Question: Why do we need new forms of parametric curves? Answer: Those parametric curves discussed are not very geometric.

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1 Itrodctio Qestio: Why do we eed ew forms of parametric crves? Aswer: Those parametric crves discssed are ot very geometric.

2 Itrodctio Give sch a parametric form, it is difficlt to kow the derlyig geometry it represets withot some frther aalysis. It is almost impossible to predict the chage of shape if oe or more coefficiet are modified.

3 Itrodctio I practice, desigers or sers sally do ot care abot the derlyig mathematics ad eqatio.

4 Itrodctio A system that spports sers to desig crves mst be: 1. Ititive: We expect that every step. 2. Flexible: The system shold provide the sers with more cotrol for desigig ad editig the shape of a crve. 3. Easy: The way of creatig ad editig a crve shold be easy.

5 Itrodctio 4. Uified Approach: The way of represetig, creatig ad editig differet types of crves (e,g., lies, coic sectios ad cbic crves) mst be the same. 5. Ivariat: The represeted crve will ot chage its geometry der geometric trasformatio (traslatio, rotatio, )

6 Itrodctio Bézier, B-splie ad NURBS crves advatage: 1. A ser layots a set of cotrol poits for the system. 2. A ser ca chage the positios of some cotrol poits ad some other characteristics for modifyig the shape of crve.

7 Itrodctio 3. If ecessary, a ser ca add cotrol poits. 4. They are very geometric, ititive. 5. The trasitio from crve to srface will ot case mch difficlty.

8 Bézier Crves

9 Bézier Crves Bézier splies are: splie approximatio method; sefl ad coveiet for crve ad srface desig; easy to implemet; available i Cad system, graphic package, drawig ad paitig packages.

10 Bézier Crves I geeral, a Bézier crve sectio ca be fitted to ay mber of cotrol poits. The mber of cotrol poits to be approximated ad their relative positio determie the degree of the Bézier polyomial.

11 Bézier Crves Give +1 cotrol poit positios: p ( x, y, zk 0 k k k y These coordiate poits ca be bleded to prodced the followig positio vector C(), which describes the path of a approximatig Bézier polyomial fctio betwee P 0 ad P. C( ) p B k k, k0 ) ( ), 0 1

12 Properties of Bézier Crves

13 Properties of a Bézier Crve C( ) p k Bk, k0 ( ), The degree of a Bézier crve defied by +1 cotrol poits is : Parabola Crve Cbic Crve Cbic Crve Cbic Crve

14 Properties of a Bézier Crve 2. The crve passes thogh the first ad the last cotrol poit C() passes throgh P 0 ad P.

15 Properties of a Bézier Crve 3. Bézier crves are taget to their first ad last edges of cotrol polylie

16 Properties of a Bézier Crve 4. The Bézier crve lies completely i the covex hll of the give cotrol poits. Note that ot all cotrol poits are o the bodary of the covex hll. For example, cotrol poits 3, 4, 5, 6, 8 ad 9 are i the iterior. The crve, except for the first two edpoits, lies completely i the covex hll.

17 Properties of a Bézier Crve 5. Movig cotrol poits:

18 Properties of a Bézier Crve 5. Movig cotrol poits:

19 Bézier Crves 6. The poit that correspods to o the Bézier crve is the "weighted" average of all cotrol poits, where the weights are the coefficiets B k, (). C( ) p k Bk, k0 ( ), 0 1

20 Desig Techiqes Usig Bézier Crve (Weights) 7. Mltiple cotrol poits at a sigle coordiate positio gives more weight to that positio.

21 Desig Techiqes Usig Bézier Crve (Closed Crves) 8. Closed Bézier crves are geerated by specifyig the first ad the last cotrol poits at the same positio Note: Bézier crves are polyomials which caot represet circles ad ellipses.

22 Properties of a Bézier Crve 9. If a affie trasformatio is applied to a Bézier crve, the reslt ca be costrcted from the affie images of its cotrol poits.

23 Costrctio of Bézier Crves

24 Bézier Crves Give +1 cotrol poit positios: 1 0 ), ( ) (, 0 B k k p k C ),, ( k y k k z y x p 0 k k k k k C B ) (1 ), ( ) (, )!!(! ), ( k k k C The Bézier bledig fctios are the Berstei polyomials: The C(,k) are the biomial coefficiets:

25 Properties of a Bézier Crve 1 0 ), ( ) (, 0 B k k p k C k k k k C B ) (1 ), ( ) (, 10. All basis fctios are positive ad their sm is always 1 k B k 0, 1 ) (

26 Example Cbic Bézier Crves k B ( ) C(, k) (1 ) k, Cbic Bézier crves are geerated with for cotrol poits. The for bledig fctios for cbic Bézier crves (=3): k B B B B 0,3 1,3 2,3 3,3 ( ) ( ) ( ) ( ) ( ) 3 3(1 ) 2 (1 ) B 1,3

27 Desig Techiqes Usig Bézier Crve (Complicated crves)

28 Desig Techiqes Usig Bézier Crve (Complicated crves) Whe complicated crves are to be geerated, they ca be formed by piecig several Bézier sectios of lower degree together. Piecig together smaller sectios gives s better cotrol over the shape of the crve i small regio.

29 Desig Techiqes Usig Bézier Crve (Complicated crves) Sice Bézier crves pass throgh edpoits; it is easy to match crve sectios (C 0 cotiity) Zero order cotiity: P 0=P 2

30 Desig Techiqes Usig Bézier Crve (Complicated crves) Sice the taget to the crve at a edpoit is alog the lie joiig that edpoit to the adjacet cotrol poit;

31 Desig Techiqes Usig Bézier Crve (Complicated crves) To obtai C 1 cotiity betwee crve sectios, we ca pick cotrol poits P 0 ad P 1 of a ew sectio to be alog the same straight lie as cotrol poits P -1 ad P of the previos sectio First order cotiity: P 1, P 2, ad P 1 colliear.

32 Desig Techiqes Usig Bézier Crve (Complicated crves) This relatio states that to achieve C 1 cotiity at the joiig poit the ratio of the legth of the last leg of the first crve (i.e., p m - p m-1 ) ad the legth of the first leg of the secod crve (i.e., q 1 - q 0 ) mst be /m. Sice the degrees m ad are fixed, we ca adjst the positios of p m-1 or q 1 o the same lie so that the above relatio is satisfied

33 Desig Techiqes Usig Bézier Crve (Complicated crves) The left crve is of degree 4, while the right crve is of degree 7. Bt, the ratio of the last leg of the left crve ad the first leg of the secod crve seems ear 1 rather tha 7/4=1.75. To achieve C 1 cotiity, we shold icrease (resp., decrease) the legth of the last (resp. first) leg of the left (resp., right). However, they are G1 cotios

34 Cbic Bézier Crves

35 Cbic Bézier Crves Cbic Bézier crves gives reasoable desig flexibility while avoidig the icreased calclatios eeded with higher order polyomials.

36 Cbic Bézier Crves 1 0 ), ( ) (, 0 B k k p k C k k k k C B ) (1 ), ( ) (, Cbic Bézier crves are geerated with for cotrol poits. The for bledig fctios for cbic Bézier crves (=3): 3 3,3 2 2,3 2 1,3 3 0,3 ) ( ) (1 3 ) ( ) (1 3 ) ( ) (1 ) ( B B B B

37 Cbic Bézier Crves At =0, B 0,3 =1, ad at =1, B 3,3 =1. ths, the crve will always pass throgh cotrol poits P 0 ad P 3. The fctios B 1,3 ad B 2,3, iflece the shape of the crve at itermediate vales of parameter, so that the resltig crve teds toward poits P 1 ad P 3. At =1/3, B 1,3 is maximm, ad at =2/3, B 2,3 is maximm. B B B B 0,3 1,3 2,3 3,3 ( ) ( ) ( ) ( ) (1 ) (1 ) (1 )

38 Cbic Bézier Crves At the ed positios of the cbic Bézier crve, The parametric first ad secod derivatives are: With C 1 ad C 2 cotiity betwee sectios, ad by expadig the polyomial expressios for the bledig fctios: the cbic Bézier poit fctio i the matrix form: ) 3( (1) ), 2 3( 0) ( p p C p p C ) 2 6( (1) ), 2 6( 0) ( p p p C p p p C ) ( p p p p M C Bez M Bez

39 Fidig a poit o a Bézier Crve: De Castelja's Algorithm

40 Fidig a poit o a Bézier Crve A simple way to fid the poit C() o the crve for a particlar is 1. to plg ito every basis fctio 2. Compte the prodct of each basis fctio ad its correspodig cotrol poit 3. Add them together. C( ) p k Bk, k0 ( ), 0 1 B k, ( ) C(, k) k (1 ) k

41 Fidig a poit o a Bézier Crve De Castelja's Algoritm The fdametal cocept of de Castelja's algoritm is to choose a poit C i lie segmet AB sch that C divides the lie segmet AB i a ratio of :1-.

42 Fidig a poit o a Bézier Crve De Castelja's Algoritm The vector from A to B is B-A. is a ratio i the rage of 0 ad 1, poit C is located at (B-A). Takig the positio of A ito cosideratio, poit C is A+(B-A)=(1-)A+B

43 De Castelja's Algoritm Castelja's algorithm: we wat to fid C(), where is i [0,1]. Startig with the first polylie, , se the formla to fid a poit 1i o the leg from 0i to 0(i+1) that divides the lie segmet i a ratio of :1-. we ill obtai poit 10,11,12,,1(-1), they defid a ew polylie of -1 legs.

44 De Castelja's Algoritm Castelja's algorithm: we wat to fid C(), where is i [0,1]. Startig with the first polylie, , se the formla to fid a poit 1i o the leg from 0i to 0(i+1) that divides the lie segmet i a ratio of :1-. we ill obtai poit 10,11,12,,1(-1), they defid a ew polylie of -1 legs.

45 De Castelja's Algoritm Apply the procedre to this ew polylie ad we shall get a third polylie of -1 poits ,2(- 2) ad -2 legs.

46 De Castelja's Algoritm Apply the procedre to this ew polylie ad we shall get a forth polylie of -1 poits ,3(- 3) ad -3 legs.

47 De Castelja's Algoritm From this forth polylie, we have the fifth oe of two poits 40 ad 41.

48 De Castelja's Algoritm Do it oce more, ad we have 50, the poit C(0.4) o the crve. De Castelja proved that this is the poit C() o the crve that correspods to.

49 De Castelja's Algoritm Actal Comptio From the iitial colm, colm 0, we compte colm 1; from colm 1 we obtai colm 2 ad so o. After applicatios we shall arrive at a sigle poit 0 ad this is the poit o the crve.

50 Sbdivisio a Bézier Crve

51 Sbdivisio a Bézier Crve Give s set of +1 cotrol poits P 0,p 1,P 2,,P ad a parameter vale i the rage of 0 ad 1, we wat to fid two sets of +1 cotrol poits Q 0,Q 1,Q 2,..,Q ad R 0,R 1,R 2,,R sch that the Bézier crve defide by Q i s (resp. R i s) is the piece of the origial Bézier crve o [0,] (resp., [,1]).

52 Sbdivisio a Bézier Crve Left polylie cosists of poits P 00 =P 0,P 10,P 20,P 30,P 40,P 50 ad P 60 =C(). Right polylie cosist of poits P 60 =C(),P 51,P 42,P 33,P 24,P 15 ad P 06 =P 6.

53 Sbdivisio a Bézier Crve

54 Sbdivisio a Bézier Crve

55 Sbdivisio a Bézier Crve Note that sice the lie segmet defied by 50 ad 51 is taget to the crve at poit 60, the last leg of the left crve (i.e, poit 50 to poit 60) is taget to the left crve, ad the first leg o the right crve (i.e, poit 60 to poit 51) is taget to the right crve.

56 Sbdivisio a Bézier Crve Why Do we eed crve Sbdivisio? Used for: Comptatig the itersectio of two Bézier crves Rederig Bézier crves Makig crve desig easier.