conjugate

Quaternion.conjugate

Conjugate of quaternion

A quaternion, whose form is \(\mathbf{q} = \begin{pmatrix}q_w & q_x & q_y & q_z\end{pmatrix}\), has a conjugate of the form \(\mathbf{q}^* = \begin{pmatrix}q_w & -q_x & -q_y & -q_z\end{pmatrix}\).

A product of the quaternion with its conjugate yields:

\[\begin{split}\mathbf{q}\mathbf{q}^* = \begin{bmatrix}q_w^2 + q_x^2 + q_y^2 + q_z^2\\ \mathbf{0}_v \end{bmatrix}\end{split}\]

A versor (normalized quaternion) multiplied with its own conjugate gives the identity quaternion back.

\[\mathbf{q}\mathbf{q}^* = \begin{bmatrix}1 & 0 & 0 & 0 \end{bmatrix}\]

Returns:

q* – Conjugated quaternion.

Return type:

numpy.array

Examples

>>> q = Quaternion([0.603297, 0.749259, 0.176548, 0.20850])
>>> q.conjugate
array([0.603297, -0.749259, -0.176548, -0.20850 ])