Curves and Surfaces. Chap. 8 Intro. to Computer Graphics, Spring 2009, Y. G. Shin

Transcription

1 Curves and Surfaces Cha. 8 Inro. o Comuer Grahics, Sring 9, Y. G. Shin

2 Reresenaion of Curves and Surfaces Key words surface modeling arameric surface coninuiy, conrol oins basis funcions Bezier curve B-sline curve

3 Why we need surface models? All shaes can be described in erms of oins. Bu, i is imracical o enumerae he oins ha comrise a shae We define shae indirecly hrough exressions ha relae cerain roeries of oins ha comrise hem.

4 inrinsic and exrinsic roeries Inrinsic roeries CFG roeries B has four sides All four sides of B have equal lengh All four angles of B are 9 o,... Exrinsic roeries Coordinae deenden geomery B has wo horizonal sides Two verical sides verices of B are a x, y,( x, y ( x, y, and ( x, ( 4 y 4 y B x

5 inrinsic and exrinsic roeries shae definiions ha use exrinsic roeries (of he shae are deenden on he coordinae sysem used. a line: y, x 7 Axis deendency 7

6 inrinsic and exrinsic roeries shae definiions ha use inrinsic roeries (of he shae are axis-indeenden. (7,, ( (,, ( ( ( y x y x y y x x y x

7 Axis Indeendence A mahemaical reresenaion of a line/curve is axis indeenden if is shae deends on only he relaive osiion of he oins defining is characerisic vecors and is indeenden of he coordinae sysem used.

8 Curve & Surface Models Exlici/imlici Parameric/non-arameric Aroximaion olygon mesh : a collecion of edges, verices, and olygons

9 Nonarameric exlici reresenaion x x y f(x successive values of y can be obained by lugging in successive values of x. easy o generae olygons or line segmens single-valued funcion y r x y r x

10 Nonarameric imlici reresenaion f ( x, y, z Define curves as soluion of equaion sysem E.g., a circle: x y r

11 Nonarameric imlici reresenaion algebraic quadric surfaces : f is a olynomial of degree <,, ( k jz hy gx fxz eyz dxy cz by ax z y x f : : : : z y x araboloid z y x corn y x cylinder z y x shere

12 Nonarameric imlici reresenaion Coefficiens deermine geomeric roeries Can reresen closed or muli-valued curves Easy o classify oin-membershi Hard o render (have o solve non-linear equaion sysem shere : x y z

13 Parameric Curve ( ( ( a a a a z a a a a y a a a a x

14 Parameric Curve (Examle,sin(u (cos(u Q(u circle uni : blening funcions, oins :conrol P P, y y ( y x x ( x,y (x P o,y (x P from line π π

15 Parameric Curve Characerisics Simle o render evaluae arameer funcion Can reresen closed or muli-valued curves Curve or surface can be easily ranslaed or roaed. Comosie curves and surfaces can be formed by iecewise descriions. Hard o check wheher a oin lies on curve have o comue he inverse maing from (x, y o

16 Parameric Curve Characerisics No infinie sloe roblem arameric form : Q'( u ( sin( u π / π,cos( u π / π, imlici form :, x y z a x, y, he arameric derivaive is (,/ π, imlici form f'( xyz,, x/ y

17 Parameric Curve Characerisics No uni form e.g., a circle wih radius cenered a he origin x y cosθ sinθ y x -

18 Tangen line o a curve he sraigh line ha gives he curve's sloe a a oin deduced from he derivaive of he curve a he oin X(, y( derivaive vecor (x (,y (

19 Piecewise Polynomial Curves Cu curve ino segmens and reresen each segmen as a olynomial curve Bu how do we guaranee smoohness a he joins? ( coninuiy roblem non-smooh smooh

20 Coninuiy imlies a noion of smoohness a he connecion oins Parameric coninuiy We view he curve or surface as a funcion raher han a shae. Maching he arameric derivaives of adjoining curve secions a heir common boundary You need a aramerizaion

21 Coninuiy (Parameric coninuiy C C C : a curve is coninuous if i can be drawn wihou lifing he encil from he aer. (x,y,z-values of he wo curves agree. :he derivaive curve is also coninuous, i.e., ( dx / d, dy / d, dz / d agree a heir juncion. :he direcion and magniude of [ (] d d Q are equal a he join oin

22 Coninuiy Geomeric coninuiy Geomeric coninuiy can be defined using only he shae of he curve. Geomeric smoohness indeenden of aramerizaion G : joining wo segmens a a common end oin ( C G : a curve' s angen direcion changes coninuously (direcion equal, bu necessarily he magniude

23 The order of olynomial curves A olynomial of order k ( number of coefficiens degree k ( k P u c cu cu cu k In comuer grahics, usually degree Sufficien flexibiliy w/o much cos The cubic is he lowes degree olynomial ha gives C and C coninuiy

24 Curve models Curve fiing echniques (inerolaion echniques ass hrough each and every daa oin linear aroximaion, naural cubic sline Curve fairing echniques (aroximaion echniques few if any oins on he curve ass hrough each and every daa oins Hermie curve, Bezier curve, B-sline curve

25 Secifying Curves Conrol Poins A se of oins ha influence he curve's shae Knos Conrol oins ha lie on he curve Subinerval endoins kno oins Conrol oins Convex hull Inerolae Aroximae

26 Polynomial Inerolaion Linear inerolaion wih a olynomial of degree one Inu: wo oins Ouu: Linear olynomial (, x x ( a a (, x a a x a a x a x a x

27 Polynomial Inerolaion Quadraic inerolaion wih a olynomial of degree wo (, x (, x (, x a a a x a a a x a a a x x ( a a a a x a x a x

28 Polynomial Inerolaion Polynomial inerolaion of degree n (, x (, x ( n, xn a x n n n n a x n n a n n n xn n x( a a a n Too heavy comuaion!

29 Naural Sline Curves Use a iecewise cubic olynomial for inerolaing a comlex curve. C n- coninuiy can be achieved from slines of degree n Moivaed by lofman s sline Long narrow sri of wood or lasic Shaed by lead weighs (called ducks

30 Naural Cubic Slines a sline curve, C(u, comosed of cubic olynomials ha inerolae he oins P, P,, P n on he inerval [, n]. divide he inerval [, n ] ino n inervals [ u i, u i ], for i o n. The numbers are called knos. u i P P P P P n u u u u u n

31 each cubic sline curve is deermined by he osiion vecors, angen vecors and he arameer values C coninuiy a he joins he olynomial coefficiens are deenden on all n conrol oins a change in any one segmen affecs he enire curve more comuaional ime. (no local conrollabiliy Cubic Slines o for ( ' ' ( ' ' o for ( ' ( ' o for ( and ( n i u q u q n i u q u q n i u q u q i i i i i i i i i i i i i i

32 Secifying Curves a olynomial of order k ( degree k I is inconvenien o reresen he curve direcly using he coefficiens c i ( k P u c cu cu cu k he relaionshi beween he shae of he curve and he coefficiens is no clear or inuiive rearrange he olynomial form ino conrol oins and basis funcions (GEOMETRIC FORM

33 Hermie Curves Parameric curves Defined by wo end oins wih he derivaive of he curve a hese oins P( P'( P( P'(

34 Hermie Curves (cubic olynomial P( P'( P( P( P'( P'( P'( P( P( P'( P( P( P( P(,a,a (a where z(] y( [x( P( i i i i a a a a a a a a a a a a a a a a a a a a a a z( a a a a y( a a a a x(

35 Hermie Curves 4 4 ( ( ( ( '( '( ( ( ( ( ( B B B B P B P B P B P B P ( B i P(,P(,P'(,P'( : blending funcions : geomeric coefficiens P( P'( P'( P(

36 Hermie Curves [ ] G B ( G M T P' P' P P P( H H H H H G M :Hermie basis marix :Hermie geomery vecor

37 (kh Order Polynomials Polynomials including degree k forms a vecor sace P k Secify a curve P( as a osiion in he vecor sace P k via he coordinae,, and he basis (,,,, k ( k k B(, i k: basis funcions i,, k : conrol oins k P ( B( i i i

38 Proeries shared by mos useful bases Convex hull roery if k i B( i and basis funcions are no negaive over he inerval hey are defined hen any oin on he curve is a weighed average of is conrol oins.

39 Proeries shared by mos useful bases no oins on he curve lies ouside he olygon formed by joining he conrol oins ogeher inexensive means for calculaing he bound of a curve or surface in sace

40 Proeries shared by mos useful bases Affine invariance - any linear ransformaion or ranslaion of he conrol oins defines a new curve ha is he jus he ransformaion or ranslaion of he original curve. (Persecive ransform is no affine. Variaion diminishing - he curve does no oscillae abou any sraigh line more ofen han he conrol oin olygon How smooh a curve is!

41 Bezier Curves Develoed by Pierre Bézier in he 97's for CAD/CAM oeraions. (PosScri drawing model Reresen a olynomial segmen as n P ( J (, J i ni, i J ( C (, ( ni ni, n i i ni are he Bernsein funcions basis or blending funcion of degree n used o scale or blend he conrol oins

42 Bezier blending funcions Noe ha n i J n,i ( convex hull roery

43 Bezier Curves (examle,,,,,,,,, ( ( ( ( Thus, ( ( ( ( ( ( (. n Since here are four verices,, ( ( find he Bezier curve. (,, and (4, (,, (,, Given J J J J P J J J J J P n i i n

44 Bezier Curves (Marix Form [ ] [ ] [ ] n,n n, n, T n B 6 P( J J J B where G G B G M T P(

45 Bezier Curves The Bezier curve of order n (degree n has n conrol oins. We can hink a Bezier curve as a weighed average of all of is conrol oins Linear (n : P( (- P P Quadraic (n : Cubic (n : P( P( (- [(- P P ] [(- P P ] P( (- P ( P ( ( ( P

46 Bezier Curves

47 A curve ha is made of several Bézier curves is called a comosie Bézier curve or a Bézier sline curve. Tangenial coninuiy beween Bezier segmens : Coninuiy condiions creae resricions on conrol oins Bezier Curves ( ( R R k Q Q C coninuiy ( ( ( '( '( Q Q Q Q R Q R R R Q Q R Q

48 Bezier Sline Curves C coninuous uniform B-sline of wo cubic Bezier segmens V( and W( wih he conrol oins (V,V,V,V and (W,W,W,W. For cubic Bezier sline: V'( (V V, V'( (V V, V"( 6(V V V, V"( 6(V V V Coninuiy a he juncion oin W V. Coninuiy of he firs derivaive W'( V'( W W V V W V V i.e. W deends on V & V Coninuiy of he second derivaive W"( V"( W W W V V V W W - (V - V Only one conrol oin W of he Bezier curve W is really free.

49 Characerisics of Bezier Curves Convex hull Affine invariance Variaion diminishing The degree of he olynomial defining he curve segmen is one less han he number of defining conrol oins. In CAGD alicaions, a curve may have a so comlicaed shae ha i canno be reresened by a single Bézier cubic curve Global conrol (disadv. : change a conrol oin affecs he coninuiy of he curve.

50 The de Caseljau Algorihm Evaluaion of he Bezier curve funcion Reeaed linear inerolaion Examle of a quadraic (degree Bezier curve b b b b b b conrol oins inerolae.

51 The de Caseljau Algorihm b b b b he oin on he curve Degree and.5 b b b reeaing he rocedure b b b

52 Parameric Surface exend D arameric reresenaion increase he number of arameers from one o wo, (s, in order o address each oin in he D saces. exress he D srucure of he curved D surface by inroducing a arameer z coordinae, z(s,, i.e., a ach x f ( s,, x y f ( s,, y s, z f ( s,. z

53 Bicubic Bezier Surface Bezier ach:6 conrol oins define one ach ease of ineraciviy & reresenaion [ ][ ] P B where v v v BPB u u u v P(u, T j i j j n n j i i n n i j n s s i n s, ( (, P( P

54 B-Sline Curves n Q( u PB k kd, ( u k P k : an inu se of n conrol oins B k,d : blending funcion of degree d- The olynomial curve has degree d- and C d- coninuiy over he range of u For n conrol oins, he curve is described wih n blending funcions

55 B-Sline Curves n Q( u PB k kd, ( u k A cubic b-sline which consiss of hree curve segmens Q( u P B ( u i i k i k,4 k Q Q 4 Q 5 P P P P P P4 5 n5, d4 5 conrol oins B,4 B, 4 B, 4 B, 4 B4, 4 B 5, 4 u u u 9 u 5 blending funcions wih degree 4-54 knos

56 B-Slines Curves Kno vecor : a se of subinerval endoins in nondecreasing sequence U { u, u,..., u n d Uniform kno vecor : he vecor for which u i u i cons e.g., [,,,,4,5,6] Oen Uniform kno vecor are uniform kno vecor which has d - equal kno values a each end e.g., [,,,,,,4,4,4,], d, n5 Non-uniform kno vecor : he vecor for which u i < u i }

57 B-Sline Basis Funcions Cox-deBoor Algorihm generae he basis funcions recursively Kno vecor: U { u, u,..., u m } B ( u, if u u u k, k k, oherwise uu u u B ( u B ( u B ( u k k d kd, kd, k, d uk d uk uk duk

58 Uniform B-Sline Basis Funcions Knos are saced a equal inervals of arameer. e.g., {,,,,4,5,6} B, B, B, B, B4, 4 5 B 5, 6 u B ( i, u B, B, B, B, B 4, B, B, B, B,4 B,4 B,4 B, B ub ( u ( u B ( u,,, uu u u B ( u B ( u B ( u k k d kd, kd, k, d uk d uk uk duk

59 Uniform B-Sline Basis Funcions B, B, B, B, B4, B 5, u B, B, B, B, B 4, B, B, B, B, Bkd, is nonzero on [ uk, u k d B,4 B,4 B,4 B,4 B, 4 B, 4 B, 4 u u u u u u 4 5 u6 u7

60 B-Sline Basis Funcions The meaning of coefficiens? uu u u B ( u B ( u B ( u k k d kd, kd, k, d uk d uk uk duk uk u k u k d u u k u k d uk uk d u u u k d k u u k d B k,d (u is a linear combinaion of B k,d- (u and B k,d- (u wih wo coefficiens, boh linear in u.

61 Uniform B-Slines Basis Funcions B B, ( u ( u u <, oherwise, u / u u.5 u< ( u / u < oherwise u u < B ( u u u < u < oherwise u /6 u < ( u u u 4 / 6 u< B u u u u u,4( ( / 6 < (4 u / 6 u < 4 oherwise Once we have B ( u B ( u B ( u i, d id,, d

62 Uniform B-slines B ( i, u consan B ( i, u linear ui u i ui u i u i B ( i, u B ( i,4 u quadraic cubic ui u i u i u i u i u i u i u i u i 4

63 Uniform cubic B-sline basis funcions When d4 and U{,,,,4,5,6,7,8,9} Bell-shaed basis funcion Each blending funcion B k,d is defined d4 over subinervals saring a kno value u k B,4 B, 4 B, 4 B, 4 B4, 4 B 5, 4 u u 9 Parameric range of curve u

64 Inerolaion n f( u B i id, ( u i Consan n f( u B i i,( u i Linear n f( u B i i,( u i To ge he value of B u. (. B (.,, P should conribue more han P! Bu NOT

65 Inerolaion Modify f(u n f( u B i id, ( u i n ( (, * f u B i isd, u i s.5 d d d 4 Consan f * n (. B (. i i i.5, I s jus a neares neighbor inerolaion

66 Inerolaion Linear n n * f( u B i id, ( u f ( u B i i,( u i i B ik, Bi, k In D, i will be a ri-linear inerolaion I s jus a bi-linear (. (. inerolaion (. B (. B (. * f u B, B 4,, 4,.9. 4

67 Uniform Cubic Inerolaion Funcion n ( ( * f u B i i,4 u i 4 B,4 B, 4 B, 4 B, 4 B4, 4 B 5, u f(. B (. B (. B (. B (.,4,4 4,4 5,4 B (. B (. B (. B (.,4,4 4,4 5,4 Noe ha,, 8, 9 are no used for f( u!!

68 Uniform Cubic B-slines i-h cubic segmen Q( u B ( u i i k i k k Q Q 4 Q 5 P P P P P4 P5 B,4 B, 4 B, 4 B, 4 B4, 4 B 5, 4 k : local conrol oin index u : local conrol arameer, A cubic B-sline is a series of m- curve segmens, Q, Q 4,, Q, m Aroximae a series of m conrol oins P, P,, Pm, m Wih m4 kno values u u u 9 u

69 Uniform Cubic B-Slines Curves Each conrol oin is associaed wih a unique blending funcion. (Local conrol Each conrol oin affecs he shae of a curve only over a range of a arameer values, d curve secions, where is associaed basis funcion is nonzero. Q Q 4 Q 5 P P P P P4 P5 B,4 B, 4 B, 4 B, 4 B4, 4 B 5, 4 u u u9

70 Uniform B-slines n Qu ( B k kd, ( u k Uniform, Quadraic B-Slines Le dn, we need nd7 kno values: {,,,,4,5,6}. Ge blending funcions using Cox-deBoor Algorihm u u B,( u B, ( u B, ( u Read ex book!!

71 Uniform B-slines(Examle The curve is defined from u d- o u n 4 We can ge saring and ending osiions (boundary condiion of he curve: Q (, Qend ( begin by alying u and u4 o he Q(u. In general, weighed average of d- conrol oins. Derivaives a he saring and ending osiion Q' begin, Q' end

72 Uniform Cubic B-slines Using a general cubic olynomial exression and he following boundary condiions: [ ] i u u u Q (u ( Q'( ( Q'( 4 ( 6 Q( 4 ( 6 Q( We can ge a marix formulaion:

73 Convex Hull Proery of B-Slines Curves B-sline curve of degree d- mus lie wihin he union of all such convex hulls formed by aking d successive defining olygon verices.

74 Uniform Cubic B-slines The effec of mulile conrol oins inerolae conrol oins bu he loss of coninuiy. muliliciy G G G coninuous coninuous coninuous

75 Non-uniform B-slines Non-uniform inerval of kno values To ermi he sline o inerolae conrol oins by insering mulile knos

76 Non-uniform B-slines kno vecor: [,,,,,,,,,] nine segmen: Q, Q, Q, Q, Q, and Q Q, Q, and Q Q, Q Q,, 8 are reduced o a single oin are defined over he range u kno vecor [,,,,,,,] Bezier curve 8 P, P conrol oins

77 B-Sline Surfaces Given he following informaion: a se of m rows and n column conrol oins i,j, where < i< m, <j<n; a kno vecor of h knos in he u-direcion, U u, u, u,..., u ( h a kno vecor of k knos in he v-direcion, V ( v, v, v,..., v k he degree in he u-direcion; and he degree q in he v-direcion;

78 B-Sline Surfaces The B-sline surface defined by hese informaion is he following: Q( u, v m i n j B i, ( u B j, q ( v ij

79 B-Sline Surfaces The coefficien of conrol oin i,j is he roduc of wo onedimensional B-sline basis funcions, one in he u-direcion, B i, (u, and he oher in he v-direcion, B j,q (v. All of hese roducs are wo-dimensional B-sline funcions. The following figures show he basis funcions of conrol oins,,,,,,,,,4 and,5

80 NURBS NURBS(non-uniform raional B-sline Adding some relaive weigh o he conrol oin for exra conrol faciliy Can reresen more various curves such as circles and cylinders More useful for inerolaion Invarian w.r. a rojecive ransformaion

81 NURBS P w i i i i i i i i n wb i i ik, ( u Bik, ( uw i o i Pu ( R ( n i,k u n wb ( u B ( u w w i ( wx, wy, wz, w i ik, jk, i i j n i i ik, ( u w for all i R ( u B ( u i ik, i, k i weigh R exra shae arameer w increase curve is ulled oward conrol oin Pi

82 Drawing Curves Forward-differencing mehod : o lo a curve or a surface, a olynomial mus be evaluaed a successive values wih fixed incremens. For P a b c d (, Pi P(/ i n a(/ i n b(/ i n c(/ i n d Pi Pi a{(( i / n ( i/ n } b{(( i / n ( i/ n } c{(( i / n ( i/ n} a b c, i (i i (i n n n c b, i, i, i 6( i n n 6a, i, i, i n

83 Drawing Curves Recursive subdivision sos when he conrol oins ge sufficienly close o he curve need flaness es Bezier curve - divide he conrol oins

84 Drawing Bezier Curves / ( / / 4 ( / ( / 4 ( / / ( Q S Q Q S S Q Q S R S S R R Q Q R R Q Q R Q R

85 Comarison of Surface