w = COI EYE view direction vector u = w ( 010,, ) cross product with y-axis v = w u up vector

Transcription

1 . w COI EYE view direction vector u w ( 00,, ) cross product with -ais v w u up vector (EQ ) Computer Animation: Algorithms and Techniques 29

2 up vector view vector observer center of interest 30 Computer Animation: Algorithms and Techniques

3 Object Space Object transformed into world space Yon Clipping Distance Hither Clipping Distance Up Vector Observer Angle of View View Vector Center of Interest World Space View Frustum FIGURE 7. Object to world space transformation and the view frustum in world space. Computer Animation: Algorithms and Techniques 3

4 Object Space World Space lines of sight emanating from observer Ee Space ee at negative infinit parallel lines of sight Image Space Screen Space FIGURE 8. Displa pipeline showing transformation between spaces. 32 Computer Animation: Algorithms and Techniques

5 . virtual frame buffer Object Space ra constructed through piel center World Space Screen Space FIGURE 9. Transformation through spaces using ra casting. Computer Animation: Algorithms and Techniques 33

6 --- w w ---,, --- [,,, w] w (EQ 2). (,, ) [,,, ] (EQ 3) (EQ 4) (EQ 5). P' M M 2 M 3 M 4 M 5 M 6 P M M M 2 M 3 M 4 M 5 M 6 P' MP (EQ 6) P' PM T 6 M T 5 M T 4 M T 3 M T T 2 M M T M T 6 M T 5 M T 4 M T 3 M T T 2 M P' PM T (EQ 7) ' ' ' abcd e f g h i j k m 000 (EQ 8) t t t 00t 00t 00t 000 (EQ 9) 34 Computer Animation: Algorithms and Techniques

7 S S S S S S (EQ 0) S S S -- S S -- (EQ ) ' ' ' cosθ sinθ 0 0 sinθ cosθ (EQ 2) ' ' ' cosθ 0 sinθ sinθ 0 cosθ (EQ 3) ' ' ' cosθ sinθ 0 0 sinθ cosθ (EQ 4) Computer Animation: Algorithms and Techniques 35

8 up vector (23,-4,40) (20, -0, 35) a) Object space definition b) World space position and orientation of aircraft FIGURE 0. Desired Position and Orientation. 36 Computer Animation: Algorithms and Techniques

9 . Y Z -4 5 ψ (-4,5) FIGURE. Projection of desired orientation vector onto - plane. Computer Animation: Algorithms and Techniques 37

10 . Z 3 5 φ (3,5) X FIGURE 2. Projection of desired orientation vector onto - plane. 38 Computer Animation: Algorithms and Techniques

11 Y X,, - global coordinate sstem Z X,Y,Z - deisred orientation defined b unit coordinate sstem FIGURE 3. Global coordinate sstem and unit coordinate sstem to be transformed. Computer Animation: Algorithms and Techniques 39

12 X M Y M Z M X X X M 0 0 Y Y Y M 0 0 Z Z Z M 0 0 (EQ 5) X Y Z X Y Z X Y Z M X Y Z X Y Z M (EQ 6) X Y Z ' ' ' A A 2 A 3 A 4 A 2 A 22 A 23 A 24 A 3 A 32 A 33 A (EQ 7) 40 Computer Animation: Algorithms and Techniques

13 . (r,0,0) FIGURE 4. Translation of moon out to its initial position on the -ais. Computer Animation: Algorithms and Techniques 4

14 3 (r,0,0) 2 for each point P of the moon { P P } R d -ais rotation of 5 degrees repeat until (done) { for each point P of the moon { P R d *P } record a frame of the animation } FIGURE 5. Rotation b appling incremental rotation matrices to points. 42 Computer Animation: Algorithms and Techniques

15 3 R identit matri R d -ais rotation of 5 degrees 2 repeat until (done) { for each point P of the moon { P R*P } record a frame of the animation R R*R d } (r,0,0) FIGURE 6. Rotation b incrementall updating the rotation matri. Computer Animation: Algorithms and Techniques 43

16 3 (r,0,0) 2 0 repeat until (done) { R -ais rotation matri of degrees for each point P of the moon { P R*P } record a frame of the animation +5 } FIGURE 7. Rotation b forming the rotation matri new for each frame. 44 Computer Animation: Algorithms and Techniques

17 . Step :Normalie one of the vectors the original unit orthogonal vectors have ceased to be orthogonal from each other due to repeated transformations Step 2: Form vector perpendicular (orthogonal) to the vector just normalied and to one of the other two original vectors b taking cross product of the two. Normalie it. Step 3: Form the final orthogonal vector b taking the cross product of the two just generated. Normalie it. FIGURE 8. Orthonormaliation. Computer Animation: Algorithms and Techniques 45

18 a) Positive 90 degree -ais rotation b) Negative 90 degree -ais rotation c) Half wa between orientation representations FIGURE 9. Direct interpolation of transformation matri values can result in nonsense. 46 Computer Animation: Algorithms and Techniques

19 . FIGURE 20. Fied angle representation. Computer Animation: Algorithms and Techniques 47

20 a) Original definition b) (0,90,0) orientation FIGURE 2. Fied angle representation of (0,90,0). 48 Computer Animation: Algorithms and Techniques

21 . a) (+/-ε,90,0) orientation b) (0,90+/-ε,0) orientation c) (0,90,+/-ε) orientation FIGURE 22. Effect of slightl altering values of fied angle representation (0,90,0). Computer Animation: Algorithms and Techniques 49

22 (0,90,0) orientation (90,45,90) orientation; the object lies in the - plane FIGURE 23. Eample orientations to interpolate. 50 Computer Animation: Algorithms and Techniques

23 . aw roll Y X pitch Z Global coordinate sstem Local Coordinate sstem attached to object FIGURE 24. Euler angle representation. Computer Animation: Algorithms and Techniques 5

24 R '( β)r ( α) R ( α)r ( β)r ( α)r ( α) R ( α)r ( β) R ''()R γ '( β)r ( α) R ( α)r ( β)r ()R γ ( α)r ( β)r ( β)r ( α) R ( α)r ( β)r () γ (EQ 8) (EQ 9) 52 Computer Animation: Algorithms and Techniques

25 orientation A Y orientation B Y θ angle and ais of rotation X X Z Z FIGURE 25. Euler s Rotation Theorem implies that, for an two orientations of an object, one can be produced from the other b a single rotation about an arbitrar ais. Computer Animation: Algorithms and Techniques 53

26 ! BA A 2 Y θ Α Α 2 φ θ 2 B A A 2 φ cos A A 2 A A 2 X A k R B ( k φ)a Z θ k ( k) θ + k θ 2 FIGURE 26. Interpolating ais-angle representations. 54 Computer Animation: Algorithms and Techniques

27 . [ s, v ] [ s 2, v 2 ] [ s s 2 v v 2, s v 2 + s 2 v + v v 2 ] (EQ 20) [ 0, v ] [ 0, v 2 ] [ 0, v v 2 ] iff v v 2 0 (EQ 2) ( q ) 2 [ s, v] q where q s (EQ 22) q q q ( q ) (EQ 23) v' Rot() v q v q (EQ 24) Rot q ( Rot p () v ) q ( p v p ) q (( pq) v ( pq )) Rot pq () v (EQ 25) Rot ( Rot() v ) q ( q v q ) q v (EQ 26) q Rot θ, (,, ) [ cos( θ 2), sin( θ 2) (,, ) ] (EQ 27) q Rot θ, (,, ) [ cos( θ 2), sin( ( θ) 2) ( (,, ) )] [ cos( θ 2), sin( θ 2) ( (,, ) )] [ cos( θ 2), sin( θ 2),, ] Rot θ, (,, ) q (EQ 28) 27. Computer Animation: Algorithms and Techniques 55

28 56 Computer Animation: Algorithms and Techniques