1. Animating with Quaternions
Given quaternion representations, interpolate through them
Cubic, interpolating
Use operations efficient with quaternion representation
Halfway between 2 quaternions
Spherical Linear interpolation

2. Review of Quaternion Math
(cos(q/2),sin(q/2)*A)
[s1,v1] + [s2,v2] = [s1+s2,v1+v2]
[s1,v1] * [s2,v2] = [s1*s2-v1.v2,s1*v2+s2*v1+v1Xv2]
||q|| = sqrt(s*s + x*x + y*y + z*z)
q-1 = [-s,v]/||q||2

3. Rotation Matrix of a Unit Quaternion
[ w,x,y,z]
1-2y2-2z2
2xy-2wz
2xz+2wy
1-2x2-2y2
2xz-2wy
2yz+2wx
1-2x2-2z2
2xy+2wz
2yz-2wx

4. Review of Quaternion Rotations
Rotq(v) = v’ = q * [0,v] * q-1
Rotq(v) = Rotk*q(v)
Rotqp(v) = Rotq(Rotp(v) )

5. spherical linear interpolation
Quaternions as points on a 4D sphere
Unit quaternion: q=(s,x,y,z), ||q|| = 1
Want equal increment along arc connecting two quaternions on surface of sphere
spherical linear interpolation

6. Linearly interpolate between quaternions
q=(sq,xq,yq,zq), p=(sp,xp,yp,zp) q p
Note: Simply adding quaternions gives mid-rotation
(because a scalar multiple of a quaternion represents the same rotation, you don’t need to divide by 2 to get the ‘average’ rotation)

7. Interpolation Scalar: K = (1-a) A + a B Euler Angles:
(linear interpolation: lerp)
Euler Angles:
[x,y,z] = (1- a)[xa,ya,za] + a[xb,yb,zb]
Quaternions?:
Linear interpolation of quaternion values would give unequal rotation increments - need to slerp
(spherical linear interpolation)

8. Quaternion : Power Operator
[ cos(q/2), sin(q /2) * A] a =
[cos(aq/2), sin(aq /2) * A]
q a rotates to q orientation as a goes from 0 to 1
Remember: q and -q represent same orientation
q a and (-q) a give different rotation sequences
0< q < p gives short squence; p < q <2 p gives long one
Since first term in q is cos(q/2), q gives short sequence if first term is positive

9. Quaternions as points on a 4D sphere
Remember: Rot[s,v] == Rot[-s,-v]
Use quaternion representations that are ‘close’ to each other:
Given quaternion p, should you interpolate to q or -q?
p and q are close if their 4D dot product is greater than zero.

10. Interpolation of rotations
K = (1-a) A + a B
Scalar:
(lerp)
Euler Angles:
[x,y,z] = (1-a)[xa,ya,za] + a[xb,yb,zb]
Quaternions:
q = (qbqa-1) a qa
(slerp)
q = slerp(qa,qb, a)

11. Review of Bezier Interpolation
Using Catmull-Romm interp of interior control points p0 p1 p2 p3
P(u) = [u3 u2 u 1]
p0 p1 p2 p3 -1 1 3 -6 -3

12. Bezier Interpolation of Quaternions

13. Interpolation Background
De Casteljau construction of Bezier curve
u=1/3
P(1/3)

14. Interpolation Background
Easy to compute quaternion half-way point
q1/2 = q0 + q01

15. Quaternion - cubic interpolationpn
pn-1
pn+1
pn+2 an 1
an+1
bn+1

16. Quaternion Interpolation
Automatically generating interior (spherical) control points
Q n
P n
A n
P n+1
Bisect the span
P n-1
Double the arc

17. Quaternion Interpolation
Q n
Double(q1,q2) = q2q1-1q2
P n
Q n =Double(P n-1, P n)
P n-1

18. Construct Midpoint q1 + q2 = [s1,x1,y1,z1] + [s2,x2,y2,z2] Q n P n A n
Bisect(p,q) = (p+q)/||p+q||
A n =Bisect(Q n, P n+1)

19. Spherical Linear Interpolation

20. Repeated mid-point interpolation
For u = 1/4

21. Quaternion Interpolation
Linearly interpolating between 4 keys
Cubic Bezier interpolation among same 4 keys