CS 112 Transformations II. Slide 1

Transcription

1 CS 112 Trasformatios II Slide 1

2 Compositio of Trasformatios Example: A poit P is first traslated ad the rotated. Traslatio matrix T, Rotatio Matrix R. After Traslatio: P = TP After Rotatio: P =RP =RTP Example: A poit is first rotated ad the traslated. After Rotatio: P = RP After Traslatio: P =TP =TRP Sice matrix multiplicatio is ot commutative, RTP = TRP Slide 2

3 Compositio of Trasformatios R T R T RTP TRP Slide 3

4 Scalig About a poit (0,2) (2,2) (0,4) (4,4) (0,0) (2,0) (0,0) (4,0) Scalig about origi -> Origi is fixed with trasformatio Slide 4

5 Scalig About a poit (0,2) (2,2) (-1,3) (3,3) (1,1) (1,1) (0,0) (2,0) (-1,-1) (3,-1) Scalig about ceter -> Ceter is fixed with trasformatio Slide 5

6 Doe by cocateatio (0,2) (2,2) (1,1) (-1,1) (1,1) (0,0) (2,0) (1,1) (-1,-1) (1,-1) Traslate so that ceter coicides with origi - T(-1,-1). Slide 6

7 Doe by cocateatio (0,2) (2,2) (-2,2) (2,2) (1,1) (0,0) (2,0) (1,1) (-2,-2) (2,-2) Scale the poits about the ceter S(2,2) Slide 7

8 Doe by cocateatio (0,2) (2,2) (-1,3) (3,3) (1,1) (1,1) (0,0) (2,0) (-1,-1) (3,-1) Traslate it back by reverse parameters T(1,1) Total Trasformatio: T(1,1) S(2,2) T(-1,-1) P Slide 8

9 Rotatio about a fixed poit z-axis rotatio of θ about its ceter P f Traslate by P f : T(-P f ) Rotate about z-axis : R z (θ) Traslate back by P f : T(P f ) Total Trasformatio M = T(P f )R z (θ)t(-p f ) Slide 9

10 Rotatio About a Arbitrary Axis Axis give by two poits P2 P 1 (startig poit) ad P 2 (edig poit) P 1 (x 1, y 1, z 1 ) ad P 2 (x 2, y 2, z 2 ) P1 Aticlockwise agle of rotatio is θ Rotate all poits to aroud P 1 P 2 by θ Z Slide 10

11 Rotatio about a Arbitrary Axis Make P 1 P 2 coicide with Z- axis Traslate P 1 to origi: T(- x 1,-y 1,-z 1 ) Coicides oe poit of the axis with origi P1 P2 Z Slide 11

12 Rotatio about a Arbitrary Axis Make P 1 P 2 coicide with Z- axis Traslate P 1 to origi: T(- x 1,-y 1,-z 1 ) Coicides oe poit of the axis with origi Rotate shifted axis to coicide with Z axis P1 P2 Z Slide 12

13 Rotatio about a Arbitrary Axis Make P 1 P 2 coicide with Z- axis Traslate P 1 to origi: T(- x 1,-y 1,-z 1 ) Coicides oe poit of the axis with origi Rotate shifted axis to coicide with Z axis R 1 : Rotate about to lie o Z plae Z P1 P2 Slide 13

14 Rotatio about a Arbitrary Axis Make P 1 P 2 coicide with Z- axis Traslate P 1 to origi: T(- x 1,-y 1,-z 1 ) Coicides oe poit of the axis with origi Rotate shifted axis to coicide with Z axis R 1 : Rotate about to lie o Z plae R 2 : Rotate about to lie o Z axis Z P1 P2 Slide 14

15 Rotatio about a Arbitrary Axis Make P 1 P 2 coicide with Z- axis Traslate P 1 to origi: T(- x 1,-y 1,-z 1 ) Coicides oe poit of the axis with origi Rotate shifted axis to coicide with Z axis R 1 : Rotate about to lie o Z plae R 2 : Rotate about to lie o Z axis Z P2 P1 Slide 15

16 Rotatio about a Arbitrary Axis Make the axis P 1 P 2 coicide with the Z-axis Traslatio to move P 1 to the origi: T(-x 1,-y 1,-z 1 ) Coicides oe poit of the axis with origi Rotatio to coicide the shifted axis with Z axis R 1 : Rotatio aroud such that the axis lies o the Z plae. R 2 : Rotatio aroud such that the axis coicides with the Z axis R 3 : Rotate the scee aroud the Z axis by a agle θ Iverse trasformatios of R 2, R 1 ad T 1 to brig back the axis to the origial positio M = T -1 R 1-1 R 2-1 R 3 R 2 R 1 T Slide 16

17 Traslatio After traslatio P 2 P 2 Axis V = P 2 P 1 = (x 1 -x 2,y 1 -y 2,z 1 -z 2 ) P 1 u P 1 u = V V = (a, b, c) Z Slide 17

18 Rotatio about axis Rotate u about so that it coicides with Z plae u = (0, b, c) u = (a, b, c) α Z α u = (a, 0, d) R 1 = Project u o Z plae : u (0, b, c) α is the agle made by u with Z axis Cos α = c/ b 2 +c 2 = c/d Si α = b/d c/d -b/d 0 0 b/d c/d Slide 18

19 Rotatio about axis Rotate u about so that it coicides with Z axis Cos β = d/ a 2 +d 2 = d/ a 2 +b 2 +c 2 = d Si β = a Z β u = (a, 0, d) R 2 = d 0 -a a 0 d Slide 19

20 Rotatio about Z axis Rotate by θ about Z axis R 3 = cosθ siθ 0 0 siθ cosθ Slide 20

21 M = T -1 R 1-1 R 2-1 R 3 (θ) R 2 (β) R 1 (α) T = T -1 R x -1 R y -1 R z (θ) R y (β) R x (α) T = T -1 R x (-α) R y (-β) R z (θ) R y (β) R x (α) T Slide 21

22 Faster Way Faster way to fid R 2 R 1 u x, u y, u z are uit vectors i the,, Z directio u = u z u y Set up a coordiate system where u = u z u z = u u u y = u z x u z x u x = u y x u z u x u x1 u x2 u x3 0 Z R 1-1 R 2-1 = R -1 R = R 2 R 1 = u y1 u y2 u y3 0 u z1 u z2 u z Slide 22

23 Rigid ad Affie Trasformatios Rigid (Does ot deform the object) Preserves agles ad legths Rotatio ad traslatio Affie (Deforms i a restricted maer) Preserves colliearity ad ratio of legths Agles may ot be preserved Scalig ad shear are affie but ot rigid Ca be expressed as a combiatio of rotatio, traslatio, scalig ad shear Slide 23

24 Trasformatios Modelview Trasformatio geerates modelview matrix (GL_MODELVIEW) Projectio Trasformatio geerates projectio matrix (GL_PROJECTION) Premultiply modelview with projectio ad apply it to all the vertices of the model Slide 24

25 Coordiate Systems ou Say: A poit P is first traslated ad the rotated. ou Write: P = RTP (write Rotatio first, the traslatio, the the poit) Right to Left: Global Coordiate System Left to Right: Local Coordiate System Results of both are same Sice matrix multiplicatio is associative Just the iterpretatio is differet. Slide 25

26 Local/Global Coordiate Systems GCS: Right to Left: poit is first traslated ad the rotated Local Local Local Local LCS: Left to Right: coordiate first rotated ad the traslated Slide 26

27 Local / Global Coordiate Systems GCS: Right to Left: poit is first scaled ad the rotated LCS: Left to Right: coordiate first rotated ad the scaled Slide 27

28 Coordiate Systems for Modelview OpeGL follows LOCAL COORDINATE SSTEM glloadidetity() gltraslate( ) glrotate( ) glscale( ) DrawModel() Meas: TRS.P (ou issue trasformatio commads i the order your write!!) Slide 28

29 Loadig, Pushig ad Poppig glloadmatrix(myarray) If it is easier to set up the matrix yourself, like shear glpushmatrix(), glpopmatrix() glpushmatrix(); gltraslatef( ); glscalef( ); glpopmatrix(); Slide 29

30 OpeGL Stack Fuctio 1 ( ) glloadidetity() gltraslate( ) glrotate( ) DrawModel(All objs) Fuctio 2( ) Fuctio 2 ( ) glpushmatrix( ) glscale( ) DrawModel(Obj A) DrawModel(Obj B) glpopmatrix( ) Slide 30