CS 428: Fall Introduction to. Geometric Transformations (continued) Andrew Nealen, Rutgers, /20/2010 1

Transcription

1 CS 428: Fall 2 Inroducion o Compuer Graphic Geomeric Tranformaion (coninued) Andrew Nealen, Ruger, 2 9/2/2

2 Tranlaion Tranlaion are affine ranformaion The linear par i he ideni mari The 44 mari for he ranlaion b vecor (,, ) i given a /2/2 2 Andrew Nealen, Ruger, 2

3 Scaling, hearing and roaion Affine ranformaion caling, hearing and roaion leave he origin invarian Their ranlaion componen i ero Thee are purel linear ranformaion 33 marice would uffice, if we were onl inereed in hee Andrew Nealen, Ruger, 2 9/2/2 3

4 Scaling, hearing and roaion Homogeneou form a a2 a3 a2 a22 a23 a 3 a32 a33 The image of he bai vecor (,,), (,, ),(,, ) define he linear ranformaion 3 3 A : R a R A a implificaion, vecor are wrien ranpoed in he e Andrew Nealen, Ruger, 2 (,, ) 9/2/2 4

5 Scaling, hearing and roaion Mulipling he canonical coordinae ae from he righ how he image of he bai vecor in he column of he mari a a 2 a 3 a a2 a22 a23 a2 a 3 a32 a33 a3 a a 2 a 3 a 3 a2 a22 a23 a23 a 3 a32 a33 a33 a a a 2 3 a a a Andrew Nealen, Ruger, 2 a a a a a a So, hi linear ranformaion i given b /2/2 5

6 Scaling anioropic Scaling Smodifie he bai vecor a S((,,,) ) (,, ) S((,, ) ) (, 2, ) S((,, ) ) (,, ) S((,, ) ) (,, 3 ) Reuling in he following 33 linear and 44 homogeneou ranformaion /2/2 6 Andrew Nealen, Ruger, 2

7 Scaling ioropic The pecial cae 2 3 mean equal (ioropic) caling for all coordinae ae The homogeneou mari ha he form 9/2/2 7 Andrew Nealen, Ruger, 2

8 Shearing Shearing SHmodifie he bai vecor a SH((,, ) ) (,, 3 ) SH((,, ) ) ( 2,, 4 ) SH((,, ) ) (,, ) SH((,, ) ) ( 5, 6, ) Reuling in he following 33 linear and 44 homogeneou ranformaion /2/2 8 Andrew Nealen, Ruger, 2

9 Linear ranformaion in 3D can be ued o compue affine ranformaion in 2D Homogeneou coordinae Geomeric inerpreaion ranformaion in 2D Affineranlaion in 2D become linearhear in 3D wihin he w plane (!) Andrew Nealen, Ruger, 2 9/2/

10 Roaion Roaion R wih angle abou he -ai modifie he bai vecor a R ((,, ) ) ( co, in, ) R ((,, ) ) (-in, co, ) R ((,, ) ) (,, ) Reuling in he following 33 linear and 44 homogeneou ranformaion co in in co co in in co 9/2/2 Andrew Nealen, Ruger, 2

11 Roaion The following urning angle are poiive in a righ handed coordinae em Andrew Nealen, Ruger, 2 9/2/2

12 Roaion For roaion R abou he -and -ai Angle abou he -ai in co Angle abou he -ai co in co in in co 9/2/2 2 Andrew Nealen, Ruger, 2

13 Roaion abou an arbirar ai Roaion R(,,)abou he normalied vecor r (,,) wih angle R R - r r r R () R (,,) R - R ()R Andrew Nealen, Ruger, 2 9/2/2 3

14 Roaion abou an arbirar ai Compuing R Define orhonormalbai (r,,) Fir bai vecor i r Second bai vecor i orhogonal o r: r r e e Third bai vecor r Andrew Nealen, Ruger, 2 r or ( if r ) r R R - r e R () e r e 9/2/2 4 r

15 Roaion abou an arbirar ai Compuing R Wrie vecor (r,,)ino he column of he ranformaion mari T-mari i orhogonal and ranform e r, e, e. (hi i R - ) For orhogonal marice Ahe following hold A - A Therefore: Ri conruced b wriing he vecor (r,,)ino he row of he mari Andrew Nealen, Ruger, 2 9/2/2 5

16 Roaion abou an arbirar ai Compuing R For clockwie roaion abou he vecor (,,) b he angle, uing horhand in(), cco()und -co() he reuling mari i given a ( ), 2 2 2,, c c c R 9/2/2 6 Andrew Nealen, Ruger, 2

17 Roaion abou an arbirar poin Ai of roaion hrough a poin differen from he origin Move cener of roaion o he origin Perform roaion a previoul decribed Move cener of roaion back r T T - R(r) r r Andrew Nealen, Ruger, 2 9/2/2 7

18 Roaion abou an arbirar poin Eample Roaion in poiive direcion abou an ai hrough he poin (,, )b angle The ai of roaion i he -direcion in hi eample p p co in in co 9/2/2 8 Andrew Nealen, Ruger, 2

19 Euler angle Ai angle (previou lide) i preferred over Euler angle Gimballock! Andrew Nealen, Ruger, 2 9/2/2 9

20 Ecurion/aide: quaernion 4-dimenional analog o comple number Muliplicaion of comple number can decribe orienaion and roaion in 2D Comple number c a+ ib c e iθ Θ Muliplicaion repreen a imilari ranformaion c c 2 c c 2 i e ( Θ +Θ ) 2 i θ + θ 2 c c 2 θ 2 c θ iθ c e c c e iθ Andrew Nealen, Ruger, 2 9/2/2 2

21 Ecurion/aide: quaernion Definiion Three imaginar number: i,j,k q a + bi + cj + dk Muliplicaion rule i 2 j 2 k 2 - ij -ji k jk -kj i ki -ik j Careful: muliplicaion i no commuaive! Andrew Nealen, Ruger, 2 9/2/2 2

22 Ecurion/aide: quaernion Properie Quaernion can be pli ino real and imaginar Par q r (, v) + v i + v 2 j + v 3 k Muliplicaion r r r r r r q q ( v v v + v + v ) Conjugae Norm 2 2 2, 2 2 v2 q r, ( v) q + v + v2 + v 2 3 Andrew Nealen, Ruger, 2 9/2/2 22

23 Roaion and quaernion Poin in pace can be repreened a purel imaginar quaernion q p (,p) p i + p 2 j + p 3 k Roaion of pabou he origin q p q r q p q r -, where q i i a uni quaernion Invere For uni quaernion (a well a for comple number) q q The invere of a uni quaernion i equal o i conjugae Andrew Nealen, Ruger, 2 9/2/2 23

24 Roaion and quaernion Uni quaernion are iomorph o orienaion Uni quaernion can be epreed a ( ) q r co( ),in( )v wih uni vecor v r q r i equivalen o a roaion of angle 2 abou he ai q p q r q p q r - Andrew Nealen, Ruger, 2 9/2/2 24

25 Compoiion of ranformaion We can compoe he baic operaion Andrew Nealen, Ruger, 2 9/2/2 25

26 Compoiion of ranformaion In general, ranformaion do no commue! Andrew Nealen, Ruger, 2 9/2/2 26

27 Compoiion of ranformaion In general, ranformaion do no commue! Andrew Nealen, Ruger, 2 9/2/2 27

28 Compoiion of ranformaion In general, ranformaion do no commue! Andrew Nealen, Ruger, 2 9/2/2 28

29 Compoiion of ranformaion Onl commue in general An wo ranlaion Two roaion around he ame ai An wo cale Roaion and uniform cale Andrew Nealen, Ruger, 2 9/2/2 29

30 How i hi implemened? Tranform poin + vecor Original geomer ( poiion in local coord) i lef unchanged! Compuaion wih ranformed verion Ue hape repreenaion baed on poin and vecor Thee are preerved under affine ranformaion Andrew Nealen, Ruger, 2 9/2/2 3

31 How i hi implemened? Line egmen Affine ranformaion map line o line So ju ranform he verice (poin) and connec he ranformed poin Andrew Nealen, Ruger, 2 9/2/2 3

32 How i hi implemened? Curve and urface work oo Work ince hape i buil uing muliple linear inerpolaion (ranformed curve curve produced uing ranformed poin) Some nonlinear deformaion work hi wa Andrew Nealen, Ruger, 2 9/2/2 32

33 In OpenGL Mainain he curren affine ranformaion Thi i impl a ingle 44 mari All pecified poin (uing glvere( )) are ranformed b hi mari OpenGL provide ranformaion funcion for modifing hi mari Andrew Nealen, Ruger, 2 9/2/2 33

34 In OpenGL Mainain he curren affine ranformaion Mari ack (incl. puh and pop operaion) o mainain a li of marice Top mari i curren modelview mari Andrew Nealen, Ruger, 2 9/2/2 34