CS 543: Computer Graphics Lecture 4 (Part I): 3D Affine transforms. Emmanuel Agu

Transcription

1 CS 543: Coputer Graphics Lecture 4 (Part I): 3D Affine transfors Eanuel Agu

2 Introduction to Transforations Introduce 3D affine transforation: Position (translation) Sie (scaling) Orientation (rotation) Shapes (shear) Previousl developed 2D (,) Now, etend to 3D or (,,) case Etend transfor atrices to 3D Enable transforation of points b ultiplication

3 Point Representation Previousl, point in 2D as colun atri Now, etending to 3D, add -coponent: or P P P P

4 Transfors in 3D 2D: 33 atri ultiplication 3D: 44 atri ultiplication: hoogenous coordinates Recall: transfor object = transfor each vertice General for: M P P P M Q Q Q Xfor of P

5 Recall: 33 2D Translation Matri t t t t * Previousl, 2D :

6 44 3D Translation Matri t t t t t t * Now, 3D : Where: = t. = + t, etc OpenGL: gltranslated(t,t,t)

7 2D Scaling Scale: Alter object sie b scaling factor (s, s ). i.e =. S =. S S S (4,4) (2,2) S = 2, S = 2 (,) (2,2)

8 Recall: 33 2D Scaling Matri S S S S

9 44 3D Scaling Matri S S S S S Eaple: If S = S = S =.5 Can scale: big cube (sides = ) to sall cube ( sides =.5) 2D: square, 3D cube OpenGL: glscaled(s,s,s)

10 Eaple: OpenGL Table Leg // define table leg // void tableleg(double thick, double len){ glscaled(thick, len, thick); glutsolidcube(.); }

11 Recall: 33 2D Rotation Matri (,) (, ) r ) cos( ) sin( ) sin( ) cos( ) cos( ) sin( ) sin( ) cos(

12 Rotating in 3D Cannot do indless conversion like before Wh? Rotate about what ais? 3D rotation: about a defined ais Different Xfor atri for: Rotation about -ais Rotation about -ais Rotation about -ais New terinolog X-roll: rotation about -ais Y-roll: rotation about -ais Z-roll: rotation about -ais

13 Rotating in 3D New terinolog X-roll: rotation about -ais Y-roll: rotation about -ais Z-roll: rotation about -ais Which wa is +ve rotation Look in ve direction (into +ve arrow) CCW is +ve rotation +

14 Rotating in 3D

15 Rotating in 3D For a rotation angle, about an ais Define: cos c sin s c s s c R A -roll: OpenGL: glrotated(,,,)

16 Rotating in 3D c s s c R A -roll: c s s c R A -roll: Rules: Rotate row, colun int. is Rest of row/col is c,s in rect pattern OpenGL: glrotated(,,,) OpenGL: glrotated(,,,)

17 Eaple: Rotating in 3D c s s c Q Q: Using -roll equation, rotate P = (3,,4) b 3 degrees: A: c = cos(3) =.866, s = sin(3) =.5, and E.g. first line: 3.c s +. = 4.6

18 Matri Multiplication Code Q: Write C code to Multipl point P = (P, P, P, ) b a 44 atri shown below to give new point Q = (Q,Q,Q, ). i.e. P P P M Q Q Q M where

19 Matri Multiplication Code Outline of solution: Declare P,Q as arra: Double P[4], Q[4]; Declare transfor atri as 2-diensional arra Double M[4][4]; Reeber: C indees fro, not Long wa: Q Q Q P P M P Write out equations line b line epression for Q[i] E.g. Q[] = P[]*M[][] + P[]*M[][] + P[2]*M[][2] + P[3]*M[][3] Cute wa: Use indeing, sa i for outer loop, j for inner loop

20 Matri Multiplication Code Using loops looks like: for(i=;i<4;i++) { tep = ; for(j=;j<4;j++) { tep += P[j]*M[i][j]; } Q[i] = tep; } Test atrice code rigorousl Use known results (or b hand) and plug into our code

21 3D Rotation About Arbitrar Ais Arbitrar rotation ais (r, r, r) opengl: rotate(, r, r, r) Without opengl: a little hair!! Iportant: read Hill and Kelle, pg (r, r, r)

22 3D Rotation About Arbitrar Ais Can copose arbitrar rotation as cobination of: X-roll Y-roll Z-roll M R ( 3) R ( 2) R ( )

23 3D Rotation About Arbitrar Ais Classic: use Euler s theore Euler s theore: an sequence of rotations = one rotation about soe ais Our approach: Want to rotate about the ais u through origin and arbitrar point Use two rotations to align u and -ais Do -roll through angle Negate two previous rotations to de-align u and -ais

24 3D Rotation About Arbitrar Ais R u ( ) R ( ) R ( ) R ( ) R ( ) R ( )

25 Coposing Transforation Coposing transforation appling several transfors in succession to for one overall transforation Eaple: M X M2 X M3 X P where M, M2, M3 are transfor atrices applied to P Be careful with the order Matri ultiplication is not coutative

26 References Hill, chapter 5.3