CS 543: Computer Graphics. 3D Transformations

Transcription

1 CS 543: Coputer Graphics 3D Transforations Robert W. Lindean Associate Professor Interactive Media Gae Developent Departent of Coputer Science Worcester Poltechnic Institute (with lots of help fro Prof. Eanuel Agu :-)

2 Introduction to Transforations A transforation changes an objects Sie (scaling) Position (translation) Orientation (rotation) Shape (shear) Previousl developed 2D or (, ) Now we etend to 3D (,, ) case Transfor object b appling sequence of atri ultiplications to 3D object vertices R.W. Lindean - WPI Dept. of Coputer Science 2

3 R.W. Lindean - WPI Dept. of Coputer Science 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 in hoogenous coordinates Recall Transfor object transfor each verte General for: M Transfor of P Q Q Q P P M P R.W. Lindean - WPI Dept. of Coputer Science 4

5 R.W. Lindean - WPI Dept. of Coputer Science 5 Recall: 33 2D Translation Matri Previousl, in 2D t t + t t *

6 R.W. Lindean - WPI Dept. of Coputer Science D Translation Matri Now, in 3D OpenGL: gltranslated( t, t, t ); Where: * + * + * + t * + t, etc. t t t + 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) R.W. Lindean - WPI Dept. of Coputer Science 7

8 R.W. Lindean - WPI Dept. of Coputer Science 8 Recall: 33 2D Scaling Matri S S S S

9 R.W. Lindean - WPI Dept. of Coputer Science D Scaling Matri 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 ); S S S S S

10 Eaple: OpenGL Table Leg // define table leg // void tableleg( double thick, double len ) { glpushmatri( ); gltranslated(, ( len *.5 ), ); glscaled( thick, len, thick ); glutsolidcube(. ); glpopmatri( ); } R.W. Lindean - WPI Dept. of Coputer Science

11 R.W. Lindean - WPI Dept. of Coputer Science Recall: 33 2D Rotation Matri (,) (, ) θ φ r ) cos( ) sin( ) sin( ) cos( θ θ θ θ ) cos( ) sin( ) sin( ) cos( θ θ θ θ

12 Rotating in 3D Cannot do indless conversion like before Wh? Rotate about what ais? 3D rotation: about a defined ais Different transfor atri for: Rotation about -ais Rotation about -ais Rotation about -ais New terinolog Pitch: rotation about -ais Yaw: rotation about -ais Roll: rotation about -ais R.W. Lindean - WPI Dept. of Coputer Science 2

13 Recall: Right-Handed Coordinates To deterine positive rotations Make a fist with our right hand, and stick thub up in the air (CCW) +Y +X +Z R.W. Lindean - WPI Dept. of Coputer Science 3

14 Rotating in 3D (cont.) R.W. Lindean - WPI Dept. of Coputer Science 4

15 Rotating in 3D (cont.) For a rotation angle, β about an ais Define c cos( β ) s sin( β ) An -rot: OpenGL: R ( β ) glrotated( ß,,, ); c s R.W. Lindean - WPI Dept. of Coputer Science 5 c s

16 Rotating in 3D (cont.) c ( ) ( ) cos β s sin β A -rot: OpenGL: glrotated( ß,,, ); A -rot: OpenGL: glrotated( ß,,, ); Rules: Rotation (row, col) is c, s in rectangular pattern Rest of rows cols. are R R ( β ) ( β ) c s c s s c s c R.W. Lindean - WPI Dept. of Coputer Science 6

17 R.W. Lindean - WPI Dept. of Coputer Science 7 Eaple: Rotating in 3D Q: Using -rot. equation, rotate P (3,, 4) b 3 degrees A: c cos(3).866, s sin(3).5, and e.g., first line: 3*c + * + 4*s + * c s s c Q

18 R.W. Lindean - WPI Dept. of Coputer Science 8 Matri Multiplication Code Q: Write C code to Multipl point P (P, P, P, ) b the 44 atri shown below to give new point Q (Q,Q,Q, ) P P P M Q Q Q M

19 Matri Multiplication Code (cont.) Outline of solution: Declare P, Q as arras: double P[4], Q[4]; Declare transfor atri as two-diensional arra double M[4][4]; Reeber: C/C++ indees fro, not Long wa Write out line b line epressions for Q[i] 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 R.W. Lindean - WPI Dept. of Coputer Science 9

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 atri code rigorousl Use known results (or b hand) and plug into our code R.W. Lindean - WPI Dept. of Coputer Science 2

21 3D Rotation About Arbitrar Ais Arbitrar rotation ais (r, r, r) OpenGL: rotate(θ, r, r, r) Without OpenGL: a little hair Iportant: read Hill pp (r, r, r) R.W. Lindean - WPI Dept. of Coputer Science 2

22 3D Rotation About Arbitrar Ais Can copose arbitrar rotation as cobination of X-rot Y-rot Z-rot M R ( β3) R ( β 2) R ( β) R.W. Lindean - WPI Dept. of Coputer Science 22

23 3D Rotation About Arbitrar Ais Want to rotate β degrees about an ais u that passes through origin and an arbitrar point Classic: Euler s theore An sequence of rotations one rotation about soe ais Our approach: Use two rotations to align u and -ais Do -rot through angle β Negate two previous rotations to de-align u and -ais R.W. Lindean - WPI Dept. of Coputer Science 23

24 3D Rotation About Arbitrar Ais R u ( β ) R ( θ ) R ( φ) R ( β ) R ( φ ) R ( θ ) R.W. Lindean - WPI Dept. of Coputer Science 24

25 Coposing Transforations 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 R.W. Lindean - WPI Dept. of Coputer Science 25