Quaternion Interpolation

Transcription

1 Quaternion Interpolation

2 3D Rotation Repreentation (reiew) Rotation Matrix orthornormal column/row bad for interpolation Fixed Angle rotate about global axe bad for interpolation, gimbal lock Euler Angle rotate about local axe ame problem a fixed angle ILE5030 Computer Animation and Special Effect

3 3D Rotation Repreentation (reiew) Axi angle rotate about A by, (A x,a y,a z,) good interpolation, no gimbal lock bad for compounding rotation Quaternion imilar to axi angle but in different form q=[,] good for compounding rotation ILE5030 Computer Animation and Special Effect 3

4 ILE5030 Computer Animation and Special Effect 4 Quaternion Math (reiew) Addition Multiplication Multiplication i aociatie but not commutatie q and q repreent the ame orientation,,,,,, q q q q 3 3 ) ( ) ( q q q q q q

5 Quaternion Rotation (reiew) To rotate a ector uing quaternion Repreent the ector a [0, ] Repreent the rotation a a quaternion q ' Rot q and cq ha the ame rotation effect to c i a calar q ( ) q q ILE5030 Computer Animation and Special Effect 5

6 Viualizing Rotation View rotation a point lying on an n-d phere -angle rotation unit circle in D pace -angle rotation unit phere in 3D pace Interpolating rotation mean moing on n-d phere How about 3-angle rotation (quaternion)? ILE5030 Computer Animation and Special Effect 6

7 Quaternion Interpolation A quaternion i a point on a 4D unit phere Unit quaternion: q=(,x,y,z), q = Interpolating rotation mean moing on 4D phere ILE5030 Computer Animation and Special Effect 7

8 Linear Interpolation Linear interpolation generate unequal pacing of point after projecting to circle ILE5030 Computer Animation and Special Effect 8

9 Spherical Linear Interpolation (lerp) Want equal increment along arc connecting two quaternion on the pherical urface in( u ) inu lerp( q, q, u) q q in in Normalize to regain unit quaternion q q ILE5030 Computer Animation and Special Effect 9

10 Slerp Recall that q and q repreent ame rotation Slerp can go the LONG way! Hae to go the hort way q q 0 q q q ILE5030 Computer Animation and Special Effect 0

11 Ueful Analogie Euclidean Space Poition Linear interpolation 4D Spherical Space Orientation Spherical linear interpolation ILE5030 Computer Animation and Special Effect

12 What if there are multiple egment? A linear interpolation in Euclidean pace, we can hae firt order dicontinuity Need a cubic cure interpolation to maintain firt order continuity ILE5030 Computer Animation and Special Effect

13 Bezier Interpolation on 4D Sphere Hae to perform interpolation on 4D phere Contruct Bezier cure by iteratiely applying lerp ILE5030 Computer Animation and Special Effect 3

14 Bezier Interpolation in Euclidean Space Colinearity of the control 3-6 point 3 0 P(u) = [u3 u u ] at either ide of an endpoint guarantee -3 the 3 3t 0order pcontinuity P (0) = 3(p-p0), P ()=3(p3-p) p0 p p0 p p3 p p3 ILE5030 Computer Animation and Special Effect 4

15 Bezier Interpolation in Euclidean Space Automatically generate control point p n- p n a n p n+ b n+ p n+ a n+ ILE5030 Computer Animation and Special Effect 5

16 Bezier Interpolation on 4D phere Automatically generating interior (pherical) control point q n- a n b n q n q n- q n+ Double the arc double Biect the pan q' Bi ect( q' n, qn ) q' n n ( qn, qn ) ( qn q n) qn q n q q n n ILE5030 Computer Animation and Special Effect 6

17 De Cateljau Contruction of Bezier Cure Contructing Bezier cure by multiple linear interpolation u=/3 P(/3) ILE5030 Computer Animation and Special Effect 7

18 De Cateljau Contruction on 4D Sphere p lerp(q n,a n, 3) p lerp(a n,b n, 3) u=/3 a n p b n p 3 lerp(b n,q n, 3) p lerp(p, p, 3) p P(/3) p 3 p 3 lerp(p, p 3, 3) p lerp(p, p 3, 3) q n q n ILE5030 Computer Animation and Special Effect 8

19 Bezier Interpolation in Euclidean Space Automatically generate control point b n p n- p n a n p n+ p n+ a n+ ILE5030 Computer Animation and Special Effect 9