¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

3.F Quaternions
A close cousin to the Rodrigues vector is the quaternion. An excellent description of
quaternions, their history and properties can be found in Altmann’s book. They have also
been much used in robotics for describing rotations. It is defined as a four component
vector in relation to the axis-angle representation as follows, where [uvw] are the
components of the unit vector representing the rotation axis, and θ is the rotation angle.
q = q(q1,q2,q3,q4) = q( u sinθ/2, v sinθ/2, w sinθ/2, cosθ/2) (3.F.1)
Note that many authors put the fourth component in the first position, i.e.
q = ( cosθ/2, u. sinθ/2, v sinθ/2, w sinθ/2). This set of components was obtained by
Rodrigues prior to Hamilton’s invention of quaternions and their algebra. Some authors
refer to the Euler-Rodrigues parameters for rotations in the notation (λ,Λ) where λ is
equivalent to q4 and Λ is equivalent to the vector (q1,q2,q3). The particular form of the
quaternion that we are interested in has a unit norm (√{q1 2 +q2 2 +q3 2 + q4 2 }=1) but
quaternions in general may have arbitrary “length”. Yet another notation writes the unit
quaternion q is an ordered set of four real numbers of the following form,
3
2
q = (q0 ;q) = (q0 ;q1 ;q2 ;q3 ), satisfying ∑ qi =1. (3.F.2)
The quaternion q = { q0
, −q
} is associated with the inverse rotation,
€
3.F.1 Conversions
i= 0
−1
g .
If one has the rotation in (orthogonal) matrix form (g or a), it can be converted to a
quaternion via the following formulae.
ε ijk Δg jk
qi = ±
4 1+ tr Δg
⎛ q ⎞ ⎡
1 sinθ
⎜ ⎟ 2
[a(2,3) − a(3,2)]/ norm
⎤ ⎡
⎢
⎥ ±[Δg(2,3) − Δg(3,2)]/2 1+ tr( Δg)
⎤
⎢
⎥
⎜ q2⎟ ⎢ sinθ
⎜ ⎟ =
2
[a(1,3) − a(3,1)]/ norm ⎥
⎢
⎥
⎢ ±[Δg(3,1) − Δg(1,3)]/2 1+ tr( Δg)
⎥
⎜ q €
=
3⎟
⎢ sinθ
⎜
⎝ q
⎟
2
[a(1,2) − a(2,1)]/ norm ⎥ ⎢
⎥ (3.F.3)
±[Δg(1,2) − Δg(2,1)]/2 1+ tr Δg
⎢
⎥ ⎢
( ) ⎥
4⎠
⎣ ⎢ cosθ
2 ⎦ ⎥
⎣
⎢ ± 1+ tr( Δg)
/2 ⎦
⎥
( ) (3.F.3)
norm = [ a(2,3) − a(3,2) ]
€
2 + [ a(1,3) − a(3,1) ] 2 + [ a(1,2) − a(2,1) ] 2 (3.F.4)
cos θ 2 = 1 2
1+ tr( a)
(3.F.5)
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