Dual number and dual quaternion

• Dual number \(a=a_0+\epsilon a_\epsilon\)
• • Dual element \(\epsilon\) is nilpotent : \(\epsilon \neq 0\), \(\epsilon^2 = 0\)
  • \(\epsilon\) commonly used to model infinitesimal quantity (ex. automatic differentiation)

• Dual quaternion \(\hat q\) = Generalization of quaternion to dual numbers
• \(\hat q = q_0 + \epsilon \, q_{\epsilon}\)
• • - \(q_0\): pure rotation component
  • - \(q_{\epsilon}\): encodes translation component (dual part)

• Given a unit quaternion \(q_0\), and a translation \(t=(t_x,t_y,t_z)\), the associated unit dual quaternion is
• • \(\displaystyle \hat q = q_0 + \frac{\epsilon}{2}\,q_t\,q_0\)
    \(\;\;q_t = (t_x,t_y,t_z,0)\)

• Unit dual quaternion (\(\|\hat q\|=1\)) describe the set of rigid transformation as screw motion.