Module: tfg.geometry.transformation.axis_angle

This module implements axis-angle functionalities.

The axis-angle representation is defined as $\theta\mathbf{a}$, where $\mathbf{a}$ is a unit vector indicating the direction of rotation and $\theta$ is a scalar controlling the angle of rotation. It is important to note that the axis-angle does not perform rotation by itself, but that it can be used to rotate any given vector $\mathbf{v} \in {\mathbb{R}^3}$ into a vector $\mathbf{v}'$ using the Rodrigues' rotation formula:

$$\mathbf{v}'=\mathbf{v}\cos(\theta)+(\mathbf{a}\times\mathbf{v})\sin(\theta) +\mathbf{a}(\mathbf{a}\cdot\mathbf{v})(1-\cos(\theta)).$$

More details about the axis-angle formalism can be found on this page.

Note: Some of the functions defined in the module expect a normalized axis $\mathbf{a} = [x, y, z]^T$ as inputs where $x^2 + y^2 + z^2 = 1$.

Functions

from_euler(...): Converts Euler angles to an axis-angle representation.

from_euler_with_small_angles_approximation(...): Converts small Euler angles to an axis-angle representation.

from_quaternion(...): Converts a quaternion to an axis-angle representation.

from_rotation_matrix(...): Converts a rotation matrix to an axis-angle representation.

inverse(...): Computes the axis-angle that is the inverse of the input axis-angle.

is_normalized(...): Determines if the axis-angle is normalized or not.

rotate(...): Rotates a 3d point using an axis-angle by applying the Rodrigues' formula.