Euler Axis (Axis-Angle)¶
The Euler axis representation describes rotations using a rotation axis and angle.
Overview¶
Also known as axis-angle representation, this describes any rotation as a single rotation about a unit vector (axis) by a specified angle.
Mathematical Representation¶
An Euler axis rotation is specified by:
- Unit vector (axis): \(\hat{n} = [n_x, n_y, n_z]\) where \(|\hat{n}| = 1\)
- Rotation angle: \(\theta\) (in radians)
Together: \([\theta, n_x, n_y, n_z]\) (4 parameters)
Rodrigues' Rotation Formula¶
Any vector \(\vec{v}\) can be rotated about axis \(\hat{n}\) by angle \(\theta\) using:
\[\vec{v}_{rot} = \vec{v}\cos\theta + (\hat{n} \times \vec{v})\sin\theta + \hat{n}(\hat{n} \cdot \vec{v})(1-\cos\theta)\]
Advantages¶
- Intuitive: Natural geometric interpretation
- Minimal representation: Efficient for single rotations
- Useful for visualization: Easy to show rotation axis
Disadvantages¶
- Composition complexity: Combining rotations is not straightforward
- Singularity at zero rotation: Axis becomes undefined
- Interpolation: Non-linear
Applications¶
Best used for:
- Visualizing rotation axes
- Specifying rotations geometrically
- Converting from/to other representations