pytransform3d.transformations.dq_prod_vector

pytransform3d.transformations.dq_prod_vector(dq, v)

Apply transform represented by a dual quaternion to a vector.

To apply the transformation defined by a unit dual quaternion \(\boldsymbol{q}\) to a point \(\boldsymbol{v} \in \mathbb{R}^3\), we first represent the vector as a dual quaternion: we set the real part to (1, 0, 0, 0) and the dual part is a pure quaternion with the scalar part 0 and the vector as its vector part \(\left(\begin{array}{c}0\\\boldsymbol{v}\end{array}\right) \in \mathbb{R}^4\). Then we left-multiply the dual quaternion and right-multiply its dual quaternion conjugate

\[\begin{split}\left(\begin{array}{c}1\\0\\0\\0\\0\\\boldsymbol{w}\end{array}\right) = \boldsymbol{q} \cdot \left(\begin{array}{c}1\\0\\0\\0\\0\\\boldsymbol{v}\end{array}\right) \cdot \boldsymbol{q}^*.\end{split}\]

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

Parameters:

dqarray-like, shape (8,)

Unit dual quaternion to represent transform: (pw, px, py, pz, qw, qx, qy, qz)

varray-like, shape (3,)

3d vector

Returns:

warray, shape (3,)

3d vector