Convert dual quaterion to rotation/translation

• Given a non-unit dual quaternion as input \(\hat q'=q'_0 + \epsilon q'_\epsilon\)
• • How to compute the components of \(\hat q' / \|\hat q'\| = \hat q = q_0 + \epsilon q_\epsilon \) ?

• Express the parameterization in rotation-translation:
• • \(\displaystyle \hat q = q_0 + \frac{\epsilon}{2} q_t\,q_0\;\), \(\;\;q_t=(t_x,t_y,t_z,0)\)

• First, normalize the non dual component: \(\hat q' / \|q_0'\|\)
• • \(\Rightarrow\)
    \(q_0=q_0' / \|q_0'\| \)

• Second, enforce the parameterization of the dual component: \(\frac{1}{2}q_t\,q_0 = q_\epsilon' / \|q'_0\|\)
• • \(\Rightarrow\)
    \( q_t = 2 \, q'_\epsilon \, q_0^\star / \|q'_0\|\)