Rotation Operator (vector Space)

Quaternions

Quaternions are defined similar to with the difference that the half angle is used instead of the full angle .

This means that the first 3 components are components of a vector defined from

$q_1\ \hat{f_1}\ +\ q_2\ \hat{f_2}\ +\ \ q_3\ \hat{f_1}\ =\ \sin \frac{\theta}{2}\quad \hat{e_3}=\frac{\sin \frac{\theta}{2}}{\sin\theta}\quad \bar E$

and that the fourth component is the scalar

$q_4=\cos \frac{\theta}{2}$

As the angle defined from the canonical form is in the interval

one would normally have that . But a "dual" representation of a rotation with quaternions is used, i.e.

and

are two alternative representations of one and the same rotation.

The entities are defined from the quaternions by

Using quaternions the matrix of the rotation operator is

$\begin{bmatrix} 2({q_1}^2+{q_4}^2)-1 &2({q_1}{q_2}-{q_3}{q_4}) &2({q_1}{q_3}+{q_2}{q_4}) \\ 2({q_1}{q_2}+{q_3}{q_4}) &2({q_2}^2+{q_4}^2)-1 &2({q_2}{q_3}-{q_1}{q_4}) \\ 2({q_1}{q_3}-{q_2}{q_4}) &2({q_2}{q_3}+{q_1}{q_4}) &2({q_3}^2+{q_4}^2)-1 \\ \end{bmatrix}$

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Rotation Operator (vector Space) - Quaternions