Animating Rotations
and Using Quaternions
What We’ll Talk About
•
•
•
•
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Animating Translation
Animating 2D Rotation
Euler Angle representation
3D Angle problems
Quaternions
Animating with quaternions
Translation
2D Rotation
3D Rotation
Euler
Problems
Quaternions
Q. Animation
Review: Translation
P=X
Y
Z
1
1
1
T=
P’ = TP =
tx
ty
1 tz
x + tx
y + ty
z + tz
1
1
Animation: interpolate over tx,ty,tz in T
Move from (10,20,30) to (10,50,40), in time = 0 to 10
∆x = (50-20)/10 ∆y = (40-30)/10
1
1
10
20
1 30
1
1
10
23
1 31
1
Time = 0
1
1
1
10
26
1 32
1
Time = 1
Time = 2
1
…
1
10
50
1 40
1
Time = 10
Review: 2D Rotation
R=
cos Ө -sin Ө
sin Ө cos Ө
P’ = RP
2D Rotation
3D Rotation
Euler
Problems
Quaternions
Q. Animation
Ө
1
Animating 2D Rotations
cos 0 -sin 0
sin 0 cos 0
time
Rotation
0
0
3
90
cos 90 -sin 90
sin 90 cos 90
cos 60 -sin 60
sin 60 cos 60
cos 30 -sin 30
sin 30 cos 30
1
Number of frames: 3
∆R = (90-0)/3 = 30
1
1
1
Review: 3D Rotations
1
Rx =
3D Rotation
Euler
Problems
Quaternions
Q. Animation
cos Ө -sin Ө
sin Ө cos Ө
1
cos Ө
Ry =
sin Ө
1
-sin Ө
cos Ө
1
Rz =
cos Ө -sin Ө
sin Ө cos Ө
1
1
Orientation
specified by a
combination of
rotations in a
predetermined
order: RxRyRz
Euler Angles
Euler
Quaternions
Q. Animation
• Euler’s Theorem: any orientation can be
expressed as a single rotation about an axis
• Can lead to Gimbal lock
• Gives you the basic idea
• Euler representation is good basis for calculating
quaternions, which work well
• Ideas:
– Orientation represented by an angle and an (x,y,z)
vector, e.g. (A1, Ө1)
– Axes represent local coordinate system, not global
• Rotation order is reverse of global, i.e. RzRyRx
Euler Angles
Euler
Quaternions
Q. Animation
• Determine axis of rotation from 1st line a
to second line b: cross product a x b
• Determine angle between lines
dot product = |a| |b| cos Ø
Ø = acos ( a b/ (|a||b|) )
note: normalized angle
Ө1
• Animation with k = 0..1 :
Ø
Ө2
axisk = rotate(k Ø ) a
anglek = (1-k)Ө1 +kӨ2
Can you do it?
• Line 1: P1 = (0,1,0) P2 = (1,0,1)
• Line 2: Q1 = (1,1,1) Q2 = (3,3,3)
• Cross product:
x
y
z
1
-1
1
2
2
2
(-4,0,-4)
• acos a•b/|a||b|
acos(2/(√3 * 2√3) = acos(1/3) = 70.5 degrees
Motivation
• We would like to represent 3D
rotations about an arbitrary axis
• We would like to be able to
apply a series of arbitrary
rotations and have it actually
work..
– Direct interpolation of matrices
leads to nonsense
– Gimbal lock occurs when the
axes of two of the three gimbals
needed to compensate for
rotations in 3D space are driven
to the same direction, e.g.
(0,90,0)
Problems
Euler
Quaternions
Q. Animation
What is a quaternion?
Quaternions
Q. Animation
• Alternative to Euler angles for specifying
orientation
• 4-tuple: use 3 numbers for axis of rotation
+ 1 for angle of rotation
• Let q be a quaternion:
q = s,v
= s, vx, vy, vz
= s + vxi + vyj + Vzk
Quaternion algebra
Quaternions
Q. Animation
Quaternions are
like complex
numbers, with one
normal component
and 3 imaginary
components, i,j,k.
A. i2 = j2 = k2 = -1 = i-j-k
B. ij = k = -ji
jk = i = -kj
ki = j = -ik
(a+bi)(c+di) =
C. q1 + q2 = (s1 + s2, v1 + v2)
(ac-bd) +(cb-ad)i
D. q1*q2 = q3
if s1 = s2 = 0, then q3 = (v1•v2 ,v1 x v2)
general: q3 = (s1s2- v1•v2, s1v2 +s2v1 + v1x v2 )
E. q1•q2 = (s1s2 + v1•v2)
Not all Quaternions
Represent Rotations
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•
•
•
•
Quaternions
Q. Animation
Only unit-length quaternions are rotations
|| q || = sqrt( s2 + v v) = 1
q = 1/||q|| [s,v]
q-1 = 1/||q||2 [s,-v]
The inverse rotation is rotation by the
same amount by the negative axis
• Unit quaternion defined by
q = (cos Ө/2, sin Ө/2[x,y,z])
Suppose I had an Euler
Rotation…
q = (cos Ө/2, sin Ө/2[x,y,z]) where Ө
and (x,y,z) are the Euler angle and axis
respectively
2. Normalize using quaternion
normalization rules: q
1.
Note:
q = -q = (-s,-v)
= cos (-Ө/2), sin(-Ө/2)[-x,-y,-z]
= cos (Ө/2), -sin(Ө/2)[-x,-y,-z]
=( cos (Ө/2), sin(Ө/2)[x,y,z] )
Quaternions
Q. Animation
Convert Q to
matrix form
M = 1-2y2-2z2 2xy + 2sz 2xy -2sy
2xy + 2sz 1-2x2-2z2 2yz – 2sx
2xz – 2sy 2yz – 2sx 1-2x2-2y2
Quaternions
Q. Animation
Rotating a point
p by q
• Rq(p) = q p q
where q = [s,-v]
(conjugate of q)
where p = [0,(px, py, pz)]
Rq(p) = [s,v][0,p][s,-v]
=[0,(s2-v v)p + 2v(v p) +2s(v x p)]
Quaternions
Q. Animation
Consecutive Rotations
•
•
•
•
R2(R1(p))
q2(q1 p q1)q2 = (q2 q1) p (q2 q1)
=R21(p)
Very convenient…
Note about conjugates
• q = [s,-v]
• q-1= 1/|q|2 q (normalize!!)
• For unit quaternions,
q-1= q
Animating Rotations Using
Quaternions
•
A.
How to find intermediate rotations?
Linear interpolation:
lerp( q1, q2, u) = q1(1-u) + uq2
• Advantage: easier to find q’ vs. animating rotation
matrix
• Problem: will speed up in the midst of rotation
q1
q: point on a 4D
unit sphere
Ө
q2
Animating Rotations Using
Quaternions
B. Spherical Interpolation
slerp( q1, q2, u) = [sin(1-u) Ө/sin Ө ] q1 + [sin uӨ/sin Ө ] q2
Where q1•q2 = cos Ө (note: use smaller Ө )
Why?
Spherical Trigonometry
Plane trigonometry
a
=
b
=
c
Same formula, but a,b,c are
arc lengths
sin
sin ß
sin
b
a
ß
b
a
ß
c
c