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)

Famous quotes containing the word numerical:

    “The moment a mere numerical superiority by either states or voters in this country proceeds to ignore the needs and desires of the minority, and for their own selfish purpose or advancement, hamper or oppress that minority, or debar them in any way from equal privileges and equal rights—that moment will mark the failure of our constitutional system.”
    —Franklin D. Roosevelt (1882–1945)

    “There is a genius of a nation, which is not to be found in the numerical citizens, but which characterizes the society.”
    —Ralph Waldo Emerson (1803–1882)