Computer Graphics. Shi-Min Hu. Tsinghua University

Transcription

1 Computer Graphcs Sh-M Hu Tsghua Uversty

2 Bézer Curves ad Surfaces arametrc curves ad surface: Some Cocepts Bézer Cuvres: Cocept ad propertes Bézer surfaces: Rectagular ad Tragular Coverso of Rectagular ad Tragular Bézer surfaces

3 arametrc curves ad surface I Graphcs we usually desg a scee ad the geerate realstc mage by usg rederg equato. It s ecessary to troduce geometrc modelg for scee desg. How to represet 3D shapes models computer?

4 Hstory of Geometrc Modelg Surface Modelg : I 962 erre Bézer a egeer of Frech Reault Car compay propose a ew kd of curve represetato ad fally developed a system UNISURF for car surface desg 972.

5 erre Etee Bezer was bor o September 90 ars. So ad gradso of egeers He etered the Ecole Supereure d'electrcte ad eart a secod degree electrcal egeerg 93. I years later he receved hs DSc degree mathematcs from the Uversty of ars. I 933 aged 23 Bezer etered Reault ad worked for ths compay for 42 years. Bezer's academc career bega 968 whe he became rofessor of roducto Egeerg at the Coservatore Natoal des Arts et Meters. He held ths posto utl 979.

6 Sold Modelg : I973 Ia Brad of Cambrdge Uversty developed a sold modelg system for desgg egeerg parts. Ia Brad preseted hs dssertato Desgg wth volumes ths work beg demostrated wth the BUILD- system.

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8 How to represet a curve? There are three major types of object represetato: Explct represetato: the explct form of a curve 2D gves the value of oe varable the depedet varable terms of the other the depedet varable. I x y space we may wrte y f x For the le we usually wrte y mx h

9 Implct represetato: I two dmesos a mplct curve ca be represeted by the equato For the le For the crcle f x y 0 ax by c x y r 0

10 arametrc form: The parametrc form of a curve expresses the value of each spatal varable for pots o the curve terms of a depedet varable t the parameter. I 3D we have three explct fuctos x y z x t y t z t

11 Oe of the advatages of the parametrc form s that t s the same two ad three dmesos. I the former case we smply drop the equato for z. A useful represetato of the parametrc form s to vsualze the locus of pots t [ x t y t z t] T beg draw as t vares.

12 We ca thk of the dervatve dx t dt d t dy t dt dt dz t dt As the velocty wth whch the curve s traced out ad pots drecto taget to the curve.

13 arametrc polyomal curves arametrc curves are ot uque. A gve curve or surface ca be represeted may ways but we shall fd that parametrc forms whch the fuctos are polyomals t. A polyomal parametrc curve of degree 3 s of form 3 2 t at bt ct d It was kow as Ferguso curve ad was used for arplae desg earler USA.

14 But Ferguso curve s ot straghtforward eve the coeffcets a b c d are gve t s stll hard to mage the shape of the curve.

15 Bézer Cuvres: Cocept ad propertes erre Bézer proposed a method to represet a curve terms of coected vectors. V t f t A 0 0 f t t d t! dt t

16 Curve ca be desged teractvely? Curve are geerated accordg to the cotrol et those coected vectors We may chage vectors to modfy the curve See demo: curvesystem.exe

17 However such a defto s utellgble 972 Forrest publshed hs famous paper Computer Aded Desg joural he poted out that Bézer curve ca be defed terms of pots wth help of Berste olyomals. Lag Youdog Chag Gegzhe Lu Dgyua Bézer was passed away 999 CAGD publshed a Specal ssue for hm 200

18 CAGD specal ssue

19 Defto Gve cotrol pots 0 Bezer curve ca be defed as: t B 0 t t [0] Where B t s -th Berste polyomal of degree! B t Ct t t t!! 0...

20 Degree two Degree three Degree four Three Bezer curves

21 roperty of Berste polyomal No-egatve B Ed pot B B 0 t 0 t 0 t 0 2 ; otherswse otherswse

22 Uty 0 B t t 0 roof: Accordg to Bomal Theorem we have B t Ct t [ t t] 0 0

23 Symmetry roof: B t B t ] [ t B t t C t t C t B

24 Recursve Ths meas that Berste olyomal of degree s a lear combato of two Berste olyomal of degree t tb t B t t B t tb t B t t t tc t t C t t t C C t C t t B

25 Dervato B Maxmum t [ B t B t] 0 ; B t has a uque local maxmum o the terval [0] at x = /

26 degree rasg formula t B t B t B t B t tb t B t B t

27 Itegral 0 B t

28 roperty of Bezer curve Ed pot propertes osto of ed pot Accordg to the ed posto s property of Berste polyomal We have 0 0 So the start pot ad ed pot of Bezer curve cocde wth start pot ad ed pot of the cotrol polygo.

29 Taget Vector Sce We have ' t 0 [ B t B t] '0 0 ' Ths mples the taget vector of the curve at the start ad ed pots s the same wth the drecto of the frst ad last edges of the cotrol polygo

30 Secod Dervatve So we have By the Curvature formula we have 2 '' t B t " 2 2 " k 3 2 k

31 Dfferece form of k-th Dervatve k t! k! k 0 B k t t [0] The hgh-order forward dfferece vector s recursvely defed by low-order forward dfferece vector: 0 k k k k

32 Symmetry * The curve wth cotrol pots 0... rema the shape of the curve t but wth opposte drecto. * * C t B t B t B t B t t [0] 0

33 Covex hull By ad 0 B t roperty 0 t 0 t 0 B The curve t s sde the covex hull of the cotrol polygo

34 Geometrc varace It meas that some geometry property does t chage wth Coordates varyg. The posto ad shape of Bezer curve are depedet o vertex of cotrol polygo but ot the Coordates.

35 de Casteljau Algorthm I dustry applcato t s requred to evaluate a pot o the curve at parameter t. We dd t evaluate the value by the Bezer Curve Equato but use a recursve algorthm whch s umercal stable.

36 lease ote t B t t B t tb t t t B t

37 We have the recursve formula for evaluato of Bézer curve by t 0 k 0 k k k k 2... t t 0... k

38 Whe = 3 the recursve procedure s llustrated by the followg fgure:

39 / 3 3

40 Geometrc cotuty I CAD applcatos t s ot ecouraged to desg a curve by hgh degree Bézer curve. It s commo to use lower degree Bézer curves wth smooth coectos. Ca we use tradtoal cocept of cotuty? t V V0 V0 t 0 t 3 V V0 2 V V0 V0 t t 2 3 3

41 lease ote 3 V V V V 0 Ths meas the fucto s ot cotuous. But the fucto s actually a straght le. Such a fact mples tradtoal cocept of cotuous s ot sutable for descrbg smoothess of shapes CAD ad Graphcs. Ths s why we use Geometrc cotuty Lu ad Lag 2 3 0

42 Gve two Bezer Curves defed by cotrol pots ad Q j 0 m 0 Two curves share a ed pot ad : a b Q Q j j j j respectvely a a Q b Q b Q Qt

43 cotuous cotuous ad - = Q 0 Q are collear cotuous curvature cotuous 0 ' " 2 " Q 0 Q Q Q 0 G 0 Q G 2 G

44 Degree rasg/elevato Degree rasg meas that addg cotrol pots to rase the degree of the Bézer curves but the shape ad drecto of the curve rema uchaged. Degree rasg creases the flexblty of shape cotrol. After degree rasg cotrol pots are chaged. How to rasg the degree of a polyomal? Bézer curves?

45 lease ote t B t t t B t 0 0 B t 0 We have the degree rasg formula * 0

46 The above formula llustrates The ew cotrol pots are lear combato of the old cotrol pots wth Coeffcet /+. The ew cotrol polygo s sde the covex hull of old cotrol polygo. The ew cotrol polygo s earer to the curve. Demo: curve-system

47 Degree reducto Degree reducto s the opposte of degree rasg Ca we reduce the degree of a polyomal wthout chage of shapes? Degree reducto s to fd a curve defed by ew cotrol pots wth mmum error * 0

48 Suppose s result of degree rasg *: We ca get two recursve formula * * # # 0 * *

49 The we have two kds of degree reducto schemes Forrest 972 Far ˆ * # * # ˆ

50 Referece for accurate degree reducto M. A. Watks ad A. J. Worsey Degree reducto of Bézer curves Computer Aded Desg Approxmate degree reducto of Bezer curves Tsghua Scece ad Techology No was reported atoal CAGD coferece 993 Degree reducto of B- sple curves Computer Aded Geometrc Desg 200 Vol. 3 NO

51 Bezer surface Rectagular Bézer Surface Suppose j 0 ; j 0 m s m cotrol pots a degree m rectagular Bezer surface ca be defed the form of tesor product u v m 0 j0 m j j j where B u C u u ad B v C v v are m Berste polyomal. m jb m u Bj v u v[0]

52 Bézer Surfaces by matrx represetato v B v B v B u B u B u B v u m m m m m m m

53 ropertes of rectagular Bézer Surface hold smlar propertes of Bézer curves : The four corers of cotrol et are also the corers of the Bézer surface. The tragles m o 0 gve taget plae at 4 corers. 0 0 m mm m m0 m0 m

54 Geometrc varace Symmetry Covex hull u v v u0 0 30

55 Geometrc cotuty Gve two degree m* Bézer surfaces wth cotrol pots j ad Q j : u v Q u v m 0 j0 m 0 j0 B j Q j m B m u B u B j j v v u v[0] 0 Q0 00 u v Q u v 0 Q00 Q Q0

56 Codtos of G 0 cotuous: v Q0 v.e. Q0 0 m Codtos of G cotuous: Q u 0 v v v v v u v

57 de Casteljau algorthm De Casteljau algorthm of Bézer curves ca be exteded for surfaces. Gve cotrol pots j 0 m; j 0 we have 0 j0 ad parameter uv. u v B u B v uv [0] mk l k l m. j m j 00

58 are defed by followg recursve formulas Or ; 2 2 0; m l k u u l k v v l k k k l j l j j l k j 2 0 ; m l k v v m l k u u l k m l j m l j k j k j j l k j kl j

59 Straghtforward terpretato of de Casteljau algorthm m m j j j m j j m j u B v B v B u B v u

60 Tragular Bézer surfaces Tragular Bézer surfaces are defed over tragles ot squares. Barycetrc coordates uvw are used for the defto of tragular Bézer surfaces.

61 u v j 0 v u v j v u u0 u v j wk 0 v w u v j wk u v0 u0 w

62 Berste Base Fucto:! j k B j k u v w u v w u v w[0]! j! k! Where +j+k= ad jk>=0. There are 2 degree Berste Base 2 fuctos.

63 B v 2 2 B0 2vw B0 2uv B B 2 2uw w 0 B u

64 ropertes of o-egatve ad uty Recursve: 0 j k j k j k B u v w B u v w w v u wb w v u vb w v u ub w v u B k j k j k j k j

65 Defto of tragular Bézer surfaces u v w B u v w jk 0 j0 j k j k u v w 0 u v w B u v w j k j k

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67 de Casteljau Algorthm u v j k j k j k j k w

68 Coverso of Rectagular ad Tragular Bézer surfaces Because the rectagular ad the tragular patches use dfferet base fucto Whe they are used the same CAD system there wll be some problem. So we troduce the coverso betwee them. A tragular surface ca be represeted as oe degeerate rectagular surfaces or three o-degeerate rectagular surfaces

69 To a degeerate rectagular surface T T T A A A

70 s a degree rasg operator: 0 A k k k k k k k k k k k k A Sh-M Hu Coverso betwee tragular ad rectagular Bezer patches Computer Aded Geometrc Desg

71 To three rectagular surfaces Doma decomposto V 0 a c D D 3 D d D 3 D 2 D 2 3 O 00 b U

72 some operator Idety operator: Shftg operator: Dfferece operator: Wth help of those operators we ca rewrte tragular Bézer surfaces as I E : IT j T j : E Tj T j E2Tj T j : Tj T j Tj 2Tj T j T u v ue ve u v I T 2 00 u v I T 2 00 T j

73 The cotrol pots defed over obtaed by D ca be

74 CAGD 996

75 ACM SIGGRAH opup: Automatc aper Archtectures from 3D Models Gve a 3D archtectural model wth user-specfed backdrop ad groud left our algorthm automatcally creates a paper archtecture approxmatg the model md-rght wth the plaar layout md-left whch ca be physcally egeered ad popped-up rght.

76 Fdg Approxmately Repeated Scee Elemets for Image Edtg We propose a ovel framework where smple user put the form of 8 scrbbles are used to gude detecto ad extracto of such repeated 9 elemets

77 Thak You!