Curves and Curved Surfaces. Where there is matter, there is geometry. Johannes Kepler

Transcription

1 Cures and Cured Surfaces Where here s maer, here s geomery. Johannes Keler

2 Paramerc cure Bezer cure Herme cure Kochanek-Barels Slnes Paramerc surface Bezer ach Bezer rangle Oulne Subdson Surfaces Chaken s Algorhm Loo Modfed buerfly Camull clark Har renderng usng essellaed cubc cures

3 Cured surface s. Trangle mesh Effcency of reresenaon Memory for sorage Effcency of renderng LOD for real-me renderng If bole neck n ransformaon and lghng, generae coarser mesh for renderng Effcen n collson deecon Conrol ons are lmed

4 Paramerc cure Inerolaon Lnear Inerolaon Conrol Pons

5 Bezer cure Perre Eenne Bezer (9~999) 'Seen A. Coons' award, 985

6 Conrol Pons Cubc Bezer cure Fnd he on on he cure as a funcon of arameer : () P P P P ( ) [,] Cure

7 de Caseljau Algorhm Recurse lnear nerolaon for () In he case of a cubc Bezer cure, sar wh four ons

8 de Caseljau Algorhm X() q q q q Ler Ler Ler,,,,,, q q Ler, a, b a b

9 de Caseljau Algorhm q r q r r Ler Ler, q, q, q, q r q

10 de Caseljau Algorhm Ler, r, r r r

11 Bezer Cure

12 Recurse Lnear Inerolaon,,,,,, q q q Ler Ler Ler,,,, q q r q q r Ler Ler,, r r Ler q q q r r

13 Eandng he Lers,,,,,,,,,,,, r r q q r q q r q q q Ler Ler Ler Ler Ler Ler P P P P

14 Bernsen Polynomals!!! n n n n B B B B B B n n

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16 The cure s nsde a cone hull of he conrol ons

17 4

18 Cubc Equaon Form 6 6 d c b a d c b a

19 Cubc Mar Form 6 d c b a d c b a 6 d c b a d c b a

20 Cubc Mar Form z y z y z y z y 6 6

21 Mar Form C G B Bez Bez z y z y z y z y 6

22 Deraes Fndng he derae (angen) of a cure s easy: c b a d c b a d d d c b a d c b a d d

23 Tangens The derae of a cure reresens he angen ecor o he cure a some on d d

24 Raonal Bezer cures Wegh w ( ) n n w B n w B ( ) n ( )

25 Pecewse Bezer cure,,,, 4, 5, 6 C C - =c( 4 - )

26 C G C

27 Herme Cure Inerolaon Smoohly nerolae beween key-ons objec moemen n keyframe anmaon camera conrol a secal form of he herme cures KB-slnes cures wh conrol oer enson, connuy and bas

28 Cubc Herm Inerolaon a herme cure P : he sar on of he cure m : he angen (e.g. drecon and seed) o how he cure leaes he sar on P : he endon of he cure m : he angen (e.g. drecon and seed) o how he cures mees he endon m m

29 Cubc Herm Inerolaon 4 Herme bass funcons ) ( ) ( ) ( ) ( ) ( m m h h h h

30 Cubc Herm Inerolaon Mar algebra Vecor S: The nerolaon-on and 's owers u o : Vecor C: The arameers of our herme cure: Mar h: The mar form of he 4 herme olynomals: S C m m

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32 The Kochanek-Barels Slnes nroduced by D. Kochanek and R. Barels n 984 ge anmaors more conrol oer keyframe anmaon. nroduced hree conrol-alues for each keyframe on Three conrol-alues Tenson: How sharly does he cure bend? Bas: Wha s he drecon of he cure as asses hrough he keyon? Connuy: How rad s he change n seed and drecon?

33 a= a= a=- The Kochanek-Barels Slnes Tenson Bas ) ( ) ( ) ( ) ( ) ( ) ( )) ( ) (( a b a b m b b m a m b< b> P - P + P m

34 The Kochanek-Barels Slnes ) ( ) ( ) ( ) ( c c d c c s ;,, ;,, ;, s d c d s c d s c The "ncomng" Tangen The "ougong" Tangen Connuy ) ( ) )( )( ( ) ( ) )( )( ( ) ( ) )( )( ( ) ( ) )( )( ( c b a c b a d c b a c b a s

35 Bezer aches Bezer aches are a sraghforward eenson o Bezer cures Insead of he cure beng arameerzed by a sngle arable, we use wo arables, u and By defnon, we choose o hae u and range from o and we say ha an u-angen crossed wh a -angen wll reresen he normal for he fron of he surface, u, n,,

36 Bezer aches Blnear Inerolaon k k k ( u, ) ( u)( ) u( ) ( u), j, j, j k, j u k, j

37 Bezer aches ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), (,,, m m n j n j m m j m n j j n j m j m m n j n j n j j n j m m u B B q u B u u j n m u B B B u B u Bernsen olynomal ach

38 Bezer aches Deraes and normals u u u u n B u B n u B u B m u u n j j j n j m m n j j j n j m m ), ( ), ( ), ( ) )( ( ) ( ), ( ) )( ( ) ( ), (,,,,

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40 Cured Surfaces The Bezer surface s a ye of aramerc surface A aramerc surface s a surface ha can be aramerzed by wo arables, u and Paramerc surfaces hae a recangular oology In comuer grahcs, aramerc surfaces are somemes called aches, cured surfaces, or jus surfaces

41 Conrol Mesh of bcubc surface Consder a bcubc Bezer surface (bcubc means ha s a cubc funcon n boh he u and arameers) A cubc cure has 4 conrol ons, and a bcubc surface has a grd of 44 conrol ons, hrough u 6 7

42 Surface Ealuaon The bcubc surface 4 cures along he u arameer 4 cures along he arameer locaon of he surface for some (u, ) ar frs sole each of he 4 u-cures for he secfed alue of u, hen ealuae Alernaely, f we frs sole he 4 -cures and o creae a new cure whch we hen ealuae a u a rey sraghforward way o mlemen smooh surfaces wh lle more han wha s needed o mlemen cures (.,.6) u

43 Mar Form We saw he mar form for a D Bezer cure s C G B Bez Bez z y z y z y z y 6

44 Mar Form To smlfy noaon for surfaces, we wll defne a mar equaon for each of he, y, and z comonens, nsead of combnng hem no a sngle equaon as for cures For eamle, o ealuae he comonen of a Bezer cure, we can use: Bez c g B 6

45 Mar Form To ealuae he comonen of 4 cures smulaneously, we can combne 4 cures no a 44 mar To ealuae a surface, we ealuae he 4 cures, and use hem o make a new cure whch s hen ealuaed Ths can be wren n a comac mar form:, u u u u u u T T Bez Bez B G B

46 Mar Form, u u u u u u u u T Bez Bez T z T y T B G B C C C C T Bez Bez G B B C coeffcens of he bcubc equaon for G conrol ons BBez : Bezer bass u and are he ecors formed from he eonens of s and

47 Tangens To comue he s and angen ecors a some (s, ) locaon, we can use T z T y T T z T y T d u d u d u du du du u C C C C C C d u u du u u u u

48 Normals To comue he normal of he surface a some locaon (u, ), we comue he wo angens a ha locaon and hen ake her cross roduc Usually, s normalzed as well n n * u * n * n u n u

49 Bezer Surface Proeres Lke Bezer cures, Bezer surfaces rean he cone hull roery any on on he acual surface wll fall whn he cone hull of he conrol ons Wh Bezer cures wll nerolae (ass hrough) he frs and las conrol ons, aromae he oher conrol ons Wh Bezer surfaces he 4 corners wll nerolae and he oher ons n he conrol mesh are only aromaed The 4 boundares of he Bezer surface are jus Bezer cures defned by he ons on he edges of he surface By machng hese ons, wo Bezer surfaces can be conneced recsely

50 Bezer rangle Pons on he cure are arameerzed wh Barycenrc coordnaes. Gen a arameer doman bounded by,, and, a on nsde he doman can be descrbed as a weghng of he : Gen hs arameerzaon, he bass funcons requred are he - dmensonal Bernsen olynomals, whch are defned as he fnal equaon for he cure:

51 Bezer rangle,,,,,,,, ) ( ), ( l k j l k j l k j l k j u u u

52 N-Paches

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54

55 Connuy

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57 Effcen essellaon Tessellaon To creae rangles on he surface To acually use aramerc surfaces n a real me renderng cone Smles way Samle he u-sace unformly

58 Dfferencng schemes Forward dfference k fk fk k k k k k k... n n n k k k

59 Adae essellaon Unform essellaon ges good resuls f he samlng rae s hgh enough Howeer, n some regons here may no be as grea a need for hgh essellaon as n oher regons.

60 AdaeTessellae(,q,r) :esspq= no curetessenough(,q); :essqr= no curetessenough(q,r); :essrp= no curetessenough(r,); 4:f(essPQand essqrand esstp) 5:AdaeTessellae(,(+q)/,(+r)/); 6:AdaeTessellae(q,(q+r)/,(q+)/); 7:AdaeTessellae(r,(r+)/,(r+q)/); 8:AdaeTessellae((+q)/,(q+r)/,(r+)/); 9:else f(esspqand essqr) :AdaeTessellae(,(+q)/,r); :AdaeTessellae((+q)/,(q+r)/,r); :AdaeTessellae((+q)/,q,(q+r)/); :else f(esspq) 4:AdaeTessellae(,(+q)/,r); 5:AdaeTessellae(q,r,(+q)/); 6:else f(no rtessenough(,q,r)) 7:AdaeTessellae((+q+r)/,,q); 8:AdaeTessellae((+q+r)/,q,r); 9:AdaeTessellae((+q+r)/,r,); :end;

61 Degeneracy measure

62 If l s large enough, c s added o he essellaon

63 Oher facors used for on-he fly essellaon Insde he ew frusum Fron facng Occuyng a large area n screen sace Close o he slhouee of he objec Illumnaed by a sgnfcan amoun of secular lghng

64 n

65 Fraconal Tessellaon

66

67 Fraconal Tessellaon for rangles

68 Dslaced erran renderng usng adae fraconal essellaon

69 Subdson Surfaces Aroach Lm Cure Surface hrough an Ierae Refnemen Process.

70 Subdson n D Same aroach works n D

71 Tyes of Subdson Inerolang Schemes Lm Surfaces/Cure wll ass hrough orgnal se of daa ons. Aromang Schemes Lm Surface wll no necessarly ass hrough he orgnal se of daa ons.

72 Chaken s Algorhm P P Q P P Q P P Q P P Q P P Q P P Q P P Q P P Q Aly Ieraed Funcon Sysem Lm Cure Surface P P P P Q Q Q Q Q 4 Q 5 Quadrac b-slne

73 Chaken s Algorhm aromaon Q Q P ( w)( P P ) w( P P ) Inerolaon Fgure.4,-.5

74 Aromaon mechansm Udang he esng ons Mdons on a lne segmen Cubc B-slne

75

76 Subdson surfaces Camull Clark scheme

77 Refnemen and smoohng Surface subdson Two hase rocess

78 D Surfaces Loo Subdsons Loo Subdsons Works on rangular meshes an Aromang Scheme Guaraneed o be smooh eerywhere ece a eraordnary erces.

79 Loo Subdsons 4 n

80 Ordnary and Eraordnary Loo Subdson Valence 6 Camull-Clark Subdson Valence 4 Subddng a mesh does no add eraordnary erces. Make u rules for eraordnary erces ha kee he surface smooh.

81 Loo Subdson Masks n n n n n n n n n n n n n cos 4 64 n n Edge Mask Insde 8 )... ( ) ( k k k k k k n k k n k n k n

82 Loo Subdson Boundares Subdson Mask for Boundary Condons Edge Rule Vere Rule

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84 Loo Secal rules for creases and eraordnary erces o roduce angen lane connuous surfaces on eher sde of he crease.

85 Loo Pecewse C -connuous eenson [Hoe 94] Eenson of he Loo s scheme. corner dar crease

86

87 Modfed Buerfly Scheme Subdson Each edge s sl Each face s sl no four Rules are defned for comung he slng ere of each edge Basc rule for a unform regon Slng an edge wh endons ha hae degree 6 As before, all new neror erces wll hae degree 6

88 Buerfly mask

89 Modfed Buerfly Scheme e e e e e N- e N- e N- N j N j N e N e e e e N e e e N j 4 cos cos 4 :, 4 : 5: :, 8 : :,, 8 :, 4 : 4 : :, :, 5 :, 4 : : Weghs: s he eraordnary ere

90 Modfed Buerfly Scheme The buerfly scheme mus be modfed o deal wh edges wh an endon of degree 6 comue new ere based on only he neghbors of he eraordnary ere If an edge has wo eraordnary endons, aerage he resuls from each endon o ge he new endon Boundares sl edges on he boundary, bu do no moe he new ere lace a he edge mdon Verces on he boundary wll robably be eraordnary

91 Loo Buerfly

92

93 Subdson 4 n m n m

94 Subdson

95 Subdson

96 Camull-Clark Subdson (978) n n f FACE 4 f f e EDGE VERTEX j j j j f n e n n n Quadrlaeral meshes Bcubc B-slne surface

97 Masks for odd erces Camull-Clark

98

99 Bum mang an llusonary rck ha changes he normal n each el and he shadng. Slhouee of objec looks he same wh or whou bum mang Dslacemen mang Surface s dslaced Along he drecon of he normal. Dslaced Subdson

100 Dslaced Subdson

101

102 Subdson Surfaces n Characer Anmaon Tony Derose, Mchael Kass, Ten Troung, (SIGGRAPH 98)