animating with quaternions

Rick Parent - CIS681

Animating with Quaternions

Given quaternion representations, interpolate through them

Use operations efficient with quaternion representation

Halfway between 2 quaternions

Spherical Linear interpolation

Cubic, interpolating


Review of Quaternion Math

(cos(/2),sin(/2)*A)

[s1,v1] * [s2,v2] = [s1*s2-v1.v2,s1*v2+s2*v1+v1Xv2]

q-1 = [-s,v]/||q||2

||q|| = sqrt(s*s + x*x + y*y + z*z)

[s1,v1] + [s2,v2] = [s1+s2,v1+v2]


Rotation Matrix of a Unit Quaternion

1-2y2-2z2 2xy-2wz 2xz+2wy

1-2x2-2y22xz-2wy 2yz+2wx

1-2x2-2z22xy+2wz 2yz-2wx

[ w,x,y,z]


Review of Quaternion Rotations

Rotq(v) = v’ = q * [0,v] * q-1 Rotq(v) = Rotk*q(v)

Rotqp(v) = Rotq(Rotp(v) )


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


Linearly interpolate between quaternions

q=(sq,xq,yq,zq), p=(sp,xp,yp,zp)

q p

r() (1 )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)

Rotr(

1

2)(v) Rotqp (v)


Interpolation

Euler Angles: [x,y,z] = (1- )[xa,ya,za] + [xb,yb,zb]

K = (1-) A + BScalar: (linear interpolation: lerp)

Quaternions?: Linear interpolation of quaternion values would give unequal rotation increments - need to slerp (spherical linear interpolation)


Quaternion : Power Operator

[ cos(/2), sin( /2) * A] [cos(/2), sin( /2) * A] =

q rotates to q orientation as goes from 0 to 1

Remember: q and -q represent same orientation

q and (-q) give different rotation sequences0< < gives short squence; < <2 gives long one

Since first term in q is cos(/2), q gives short sequence if first term is positive


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.


Interpolation of rotations

Euler Angles: [x,y,z] = (1-a)[xa,ya,za] + a[xb,yb,zb]

K = (1-) A + BScalar: (lerp)

Quaternions: q = (qbqa-1) qa (slerp)

q = slerp(qa,qb, )


Review of Bezier Interpolation

P(u) = [u3 u2 u 1]

p0 p1 p2 p3

-1 1

1

3

3

33 3

-6

-3

-3 00

0 00

p0

p1

p2 p3

Using Catmull-Romm interp of interior control points


Bezier Interpolation of Quaternions


Interpolation Background

De Casteljau construction of Bezier curve

u=1/3

P(1/3)


Interpolation Background

Easy to compute quaternion half-way point

u=1/2

P(1/2)

q1/2 = q0 + q01


Quaternion - cubic interpolation

pn

pn-1

pn+1

pn+2

an+1

an

bn+1

1


Quaternion Interpolation

Automatically generating interior (spherical) control points

P n-1

P n

P n+1

Q n

A n

Double the arc

Bisect the span



P n-1

P n

Q n Double(q1,q2) = q2q1-1q2

Q n =Double(P n-1, P n)


Construct Midpoint

P n-1

P n

P n+1

Q n

A n Bisect(p,q) = (p+q)/||p+q||

A n =Bisect(Q n, P n+1)

q1 + q2 = [s1,x1,y1,z1] + [s2,x2,y2,z2]


Spherical Linear Interpolation

p1 slerp(qn ,an, 13)

p2 slerp(an ,bn1, 13)

p3 slerp(bn1,qn1, 13)

p12 slerp(p1, p2, 13)

p23 slerp(p2, p3, 13)

p slerp(p12, p23, 13)


Repeated mid-point interpolation

For u = 1/4

temp slerp(qn ,an, 12) qn an

p1 slerp(qn ,temp, 12) qn temp

temp slerp(an ,bn1, 12) an bn1

p2 slerp(qn , temp, 12) qn temp

temp slerp(bn1,qn1, 12) bn1 qn1

p3 slerp(bn1,temp, 12) bn1 temp

temp slerp(p1, p2, 12) p1 p2

p12 slerp(p1, temp, 12) p1 temp


p23 slerp(p2,temp, 12) p2 temp


p23 slerp(p12,temp, 12) p12 temp



Linearly interpolating between 4 keys

Cubic Bezier interpolation among same 4 keys

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animating with quaternions

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