Animating Rotations and Using Quaternions

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Animating Rotations and Using Quaternions
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What Well Talk About

• Animating Translation
• Animating 2D Rotation
• Euler Angle representation
• 3D Angle problems
• Quaternions
• Animating with quaternions

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Review Translation
Translation2D Rotation3D RotationEulerProblems
QuaternionsQ. Animation
P X Y Z 1
P TP
x txy tyz tz 1
1 tx 1 ty 1tz
1
T
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 10 1 20 130
1
1 10 1 23 131
1
1 10 1 50 140
1
1 10 1 26 132
1

Time 0
Time 1
Time 2
Time 10
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Review 2D Rotation
2D Rotation3D RotationEulerProblemsQuaternions
Q. Animation
cos ? -sin ? sin ? cos ?
1
P RP
?
R
Animating 2D Rotations
Number of frames 3?R (90-0)/3 30
cos 90 -sin 90 sin 90 cos 90
1
cos 60 -sin 60 sin 60 cos 60
1
cos 0 -sin 0 sin 0 cos 0
1
cos 30 -sin 30 sin 30 cos 30
1
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Review 3D Rotations
3D RotationEulerProblemsQuaternionsQ.
Animation
1 cos ? -sin ? sin ? cos ?
1
Rx
Orientation specified by a combination of
rotations in a predetermined order RxRyRz
cos ? sin ? 1 -sin ?
cos ? 1

Ry
cos ? -sin ? sin ? cos ?
1 1

Rz
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Euler Angles
EulerQuaternionsQ. Animation

• Eulers 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, ?1)
• Axes represent local coordinate system, not
  global
• Rotation order is reverse of global, i.e. RzRyRx

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Euler Angles
EulerQuaternionsQ. Animation

• Determine axis of rotation from 1st line a to
  second line b cross product a x b
• Determine angle between linesdot product a
  b cos ØØ acos ( a b/ (ab) ) note
  normalized angle
• Animation with k 0..1 axisk rotate(k Ø ) a
  anglek (1-k)?1 k?2

?1
Ø
?2
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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(-4,0,-4)
• acos ab/ab acos(2/(v3 2v3) acos(1/3)
  70.5 degrees

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Motivation
ProblemsEulerQuaternionsQ. Animation

• 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)

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What is a quaternion?
QuaternionsQ. 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

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Quaternion algebra
QuaternionsQ. Animation
Quaternions are like complex numbers, with one
normal component and 3 imaginary components,
i,j,k. (abi)(cdi) (ac-bd) (cb-ad)i

• i2 j2 k2 -1 i-j-k
• ij k -ji jk i -kj ki j -ik
• q1 q2 (s1 s2, v1 v2)
• q1q2 q3if s1 s2 0, then q3 (v1v2 ,v1 x
  v2)general q3 (s1s2- v1v2, s1v2 s2v1 v1x
  v2 )
• q1q2 (s1s2 v1v2)

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Not all Quaternions Represent Rotations
QuaternionsQ. Animation

• Only unit-length quaternions are rotations
• q sqrt( s2 v v) 1
• q 1/q s,v
• q-1 1/q2 s,-v
• The inverse rotation is rotation by the same
  amount by the negative axis
• Unit quaternion defined by
• q (cos ?/2, sin ?/2x,y,z)

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Suppose I had an Euler Rotation

• q (cos ?/2, sin ?/2x,y,z) where ? and
  (x,y,z) are the Euler angle and axis respectively
• Normalize using quaternion normalization rules q

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Note
QuaternionsQ. Animation

• q -q (-s,-v) cos (-?/2),
  sin(-?/2)-x,-y,-z cos (?/2),
  -sin(?/2)-x,-y,-z ( cos (?/2),
  sin(?/2)x,y,z )

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Convert Q to matrix form
QuaternionsQ. Animation

• M 1-2y2-2z2 2xy 2sz 2xy -2sy 2xy
  2sz 1-2x2-2z2 2yz 2sx
• 2xz 2sy 2yz 2sx 1-2x2-2y2

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Rotating a point p by q
QuaternionsQ. Animation

• Rq(p) q p qwhere q s,-v (conjugate of
  q)where p 0,(px, py, pz)Rq(p)
  s,v0,ps,-v 0,(s2-v v)p 2v(v p) 2s(v
  x p)

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Consecutive Rotations

• R2(R1(p))
• q2(q1 p q1)q2 (q2 q1) p (q2 q1)
• R21(p)
• Very convenient

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Note about conjugates

• q s,-v
• q-1 1/q2 q (normalize!!)
• For unit quaternions, q-1 q

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Animating Rotations Using Quaternions

• How to find intermediate rotations?
• Linear interpolationlerp( 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
?
q2
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Animating Rotations Using Quaternions

• Spherical Interpolation
• slerp( q1, q2, u) sin(1-u) ?/sin ? q1 sin
  u?/sin ? q2
• Where q1q2 cos ? (note use smaller ? )
• Why?
• Plane trigonometry
• a b c
• sin sin ß sin

Spherical Trigonometry Same formula, but a,b,c
arearc lengths
b
a
ß
b
a
ß
c
c