Axis angle to quaternion
Let \(\langle\hat{a},\alpha\rangle\) be an axis angle representing a counter-clockwise rotation around the axis \(\hat{a}\) by \(\alpha\) degrees. The quaternion \(q\) representing the same transformation is defined as
\[\begin{align*} q_x &= \hat{a}_x \sin\left(\alpha/2\right)\\ q_y &= \hat{a}_y \sin\left(\alpha/2\right)\\ q_z &= \hat{a}_z \sin\left(\alpha/2\right)\\ q_w &= \cos\left(\alpha/2\right) \end{align*}\]