Quaternions and Interpolation of Rotations

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Quaternions and Interpolation of Rotations

• Ladislav Kavan

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Rotations in R2 and R3

• Recall a rotation in R3 is a linear
  transformation, whose matrix

is orthonormal and det(M) 1. Similarly, a
rotation in R2 is a linear transformation with
matrix
which is orthonormal and det(M') 1.
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Complex Numbers

• C ... the set of complex numbers
• A complex number z?C can be written as z a
  ib,
• where "i" is the imaginary unit, satisfying i2
  -1
• z a - ib is the complex conjugate of z
• z ?(zz) ?(a2 b2) is the complex norm of
  z
• z is unit complex number if z1
• Every z?C can be written as z zei?, because
  of
• Euler's identity
• ei? cos(?) isin(?)

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Complex Numbers and 2D Rotation

• Rotation of (x1, x2)T ? R2 by angle ? can be
  written using a 2?2 matrix as

• Define
• x?C such that x x1 ix2
• y?C such that y y1 iy2
• Then we can write the equation above as
• yei?x
• Why? Because of Euler's identity.

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Complex Numbers and 2D Rotation

• Observe ei? is a unit complex number, because
• ei?2 cos2(?) sin2(?) 1
• We have a unique correspondence between 2D
  rotations and unit complex numbers
• Unit complex numbers form a unit circle in the
  complex plane
• Identity is simply 1?C
• Inverse rotation is complex conjugate,
• because e-i? (ei?)
• What about 3D rotations?

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Quaternions

• generalization of complex numbers
• discovered by W.R.Hamilton

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Quaternions

• Three imaginary units i, j, k with properties
• i2 -1, j2 -1, k2 -1
• ij -ji k, jk -kj i, ki -ik j
• Exercise derive this from i2 j2 k2 ijk
  -1
• Quaternion is a four-tuple q w xi yj zk
• 1 0i 0j 0k is a quaternion unit (1?Q)
• addition of quaternions, scalar multiplication
  component-wise
• compact notation q w r, where r xi yj zk
  (w is called the real part and r the vector one)
• q can be interpreted as a R4 vector, w as a real
  number and r as a R3 vector

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Quaternion Multiplication

• Multiplication is what makes it an algebra. If
• q0 w0 x0i y0j z0k w0 r0
• q1 w1 x1i y1j z1k w1 r1
• then the quaternion multiplication is defined as
• q0 q1 w0 w1 - ltr0, r1gt w0 ? r1 w1 ? r0 r0
  ? r1
• (when interpreting r0 and r1 as R3 vectors)
• Quaternion multiplication Q ? Q ? Q
• associative
• non-commutative
• Dot product of quaternions also possible Q ? Q ?
  R
• ltq0, q1gt w0 w1 x0 x1 y0 y1 z0 z1

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Quaternion Operations

• Let q w xi yj zk w r
• We define
• quaternion conjugate q w - xi - yj - zk w -
  r
• quaternion norm q ?(qq) ?(w2x2y2z2)
• unit quaternion has q 1
• Properties ?p,q?Q
• p q pq
• (p q) q p
• if p is unit pp pp 1

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Quaternion Operations

• Euler's identity generalizes to quaternions
• er? cos(?) rsin(?)
• Each q?Q can be written as qqer? (for some
  ??R)
• Especially, for a unit quaternion qer?
• Power of a unit quaternion q can be therefore
  defined as
• qt ert? (cos(t?) rsin(t?))
• Warning er?es? ? er?s? in general, because
  quaternions are not commutative

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Unit Quaternions and 3D Rotations

• A rotation in R3 unit axis a?R3 and angle ??R
• Let x (x1, x2, x3)T ? R3, and v ix1 jx2
  jx3 ? Q
• Define unit quaternion q cos(?/2) a sin(?/2)
• Then
• v' q v q
• is a quaternion with zero real part (like v).
• Denote v' ix1' jx2' jx3' and x' (x1',
  x2', x3')T
• Fact x' is a rotation of x about axis a with
  angle ?
• A 3D rotation can be represented by a unit
  quaternion
• Is this correspondence unique?

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Unit Quaternions and 3D Rotations

• No, quaternions q and -q represent the same
  rotation
• (since -q v -q q v q)
• Lets rotate v' by another unit quaternion p
• v'' p v' p p q v q p (p q) v (p q)
• Especially, if p q
• v'' (qq) v (qq) v
• unit quaternion multiplication corresponds to
  composition of rotations
• conjugation of unit quaternion corresponds to
  inverse rotation
• 1 is the identity (zero angle rotation)

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Quaternion to Rotation Matrix

• Let q w xi yj zk ? Q, q 1.
• Rotation by q corresponds to multiplication by
  matrix

• Conversion from matrix to quaternion is also
  possible (but a bit tricky because of numerical
  issues, see Shoemake 1985)
• composition of rotations is faster in quaternions
• transformation of vector is faster by matrix

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Interpolation of Rotations

• Consider two rotations R0, R1
• The task is to define a rotation R(t) ?t?0,1,
  s.t.
• R(0) R0
• R(1) R1
• ?t?0,1 R(t) is a rotation
• The angular velocity of an object transformed by
  R(t) is constant
• First idea interpolate rotation matrices
• Problem the resulting matrix might not be a
  rotation (the orthonormality might be violated)

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Interpolation of 2D Rotations

• Let 2D rotations R0, R1 are represented by unit
  complex numbers z0, z1
• Consider interpolation

Fulfills conditions 1,2,3 R(t) is a unit complex
number Violates condition 4
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Interpolation of 2D Rotations

• Express the complex numbers z0, z1 as
• z0 ei?
• z1 ei?
• Consider interpolation R(t) ei(1-t)? it?
  z0(z0 z1)t
• Fulfills conditions 1,2,3 and 4 the angle of
  rotation is interpolated linearly
• How to generalize to 3D? Using quaternions!
  (but be careful quaternions do not commute
  like complex numbers)

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Interpolation of 3D Rotations

• Let p wp rp, q wq rq are unit
  quaternions.
• Then the interpolation
• R(t) p(pq)t
• is called a Spherical Linear Interpolation
  (SLERP), and
• fulfills conditions 1,2,3 and 4.
• Catch -q represents the same rotation as q
  what happens if we employ -q instead of q?
• A scalar part of pq is wpwq ltrp,rqgt ltp,qgt
• Since p and q are unit, ltp,qgtcos(?), ??0,?)
• Observe 2? is the angle of rotation pq

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Spherical Linear Interpolation

• If we apply -q instead of q, we get
• -ltp,qgt -cos(?) cos(? - ?)
• The angle of rotation -pq is thus 2? - 2?
• Example for ? 60 degrees

interpolating pq
interpolating -pq
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Spherical Linear Interpolation
The same on a quaternion sphere (a 2D slice below)
p
pq interpolation
-pq interpolation
q
0
-q

• Conclusion If we want the shortest-path
  interpolation, we must ensure ltp,qgt ? 0
• (by using -q instead of q)

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Spherical Linear Interpolation

• Unit quaternions p, q are points on unit sphere
  in R4.
• SLERP interpolates along the shortest arc on the
  unit sphere in R4. This gives formula

• where ? is again the angle inclined by p and q.
• This formula is
• more efficient for computation
• equivalent to R(t) p(pq)t

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Spherical Linear Interpolation

• Input
• unit quaternions p, q,
• interpolation parametr t?lt0,1gt
• The algorithm
• c_sig ltp,qgt
• if c_sig lt 0 then q -q c_sig -c_sig
• sig acos(c_sig)
• res sin((1-t)sig)
• res sin(tsig)
• res / sin(sig)
• return res

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Comparison

• Quaternions have 4 elements
• Better than matrices (9 elements)
• However, 3D rotation has only 3 DoFs
• 3 scalars are sufficient (Euler angles)
• Any 3D rotation M can be expressed as
• M Rx(?)Ry(?)Rz(?),
• where Rx, Ry, Rz are rotations about coordinate
  axis
• Why dont use Euler angles instead of quaternions?

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Gimbal Lock

• M Rx(?)Ry(?)Rz(?)
• If ? 90 degrees, Rx does the same rotation as
  Rz
• One DoF lost
• Not good for interpolation