pytransform3d.rotations.q_prod_vector

pytransform3d.rotations.q_prod_vector(q, v)

Apply rotation represented by a quaternion to a vector.

We use Hamilton’s quaternion multiplication.

To apply the rotation defined by a unit quaternion \(\boldsymbol{q} \in S^3\) to a vector \(\boldsymbol{v} \in \mathbb{R}^3\), we first represent the vector as a quaternion: we set the scalar part to 0 and the vector part is exactly the original vector \(\left(\begin{array}{c}0\\\boldsymbol{v}\end{array}\right) \in \mathbb{R}^4\). Then we left-multiply the quaternion and right-multiply its conjugate

\[\begin{split}\left(\begin{array}{c}0\\\boldsymbol{w}\end{array}\right) = \boldsymbol{q} \cdot \left(\begin{array}{c}0\\\boldsymbol{v}\end{array}\right) \cdot \boldsymbol{q}^*.\end{split}\]

The vector part \(\boldsymbol{w}\) of the resulting quaternion is the rotated vector.

Parameters:

qarray-like, shape (4,)

Unit quaternion to represent rotation: (w, x, y, z)

varray-like, shape (3,)

3d vector

Returns:

warray, shape (3,)

3d vector

See also

rotor_apply

The same operation with a different name.

concatenate_quaternions

Hamilton’s quaternion multiplication.