Rotation Operator (vector Space) - Numerical Example

Numerical Example

Consider the reorientation corresponding to the Euler angles \alpha=10^\circ \quad \beta=20^\circ \quad \gamma=30^\circ \quad relative a given base vector system

Corresponding matrix relative to this base vector system is (see Euler angles#Matrix orientation)

\begin{bmatrix} 0.771281 & -0.633718 & 0.059391 \\ 0.613092 & 0.714610 & -0.336824 \\ 0.171010 & 0.296198 & 0.939693 \end{bmatrix}

and the quaternion is

(0.171010,\ -0.030154,\ 0.336824,\ 0.925417)

The canonical form of this operator

\begin{bmatrix} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{bmatrix}

with is obtained with

The quaternion relative to this new system is then

(0,\ 0,\ 0.378951,\ 0.925417) = (0,\ 0,\ \sin\frac{\theta}{2},\ \cos\frac{\theta}{2})

Instead of making the three Euler rotations

the same orientation can be reached with one single rotation of size around

Read more about this topic:  Rotation Operator (vector Space)

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