Axis-angle representation with dual quaternion

• Similarily to quaternion: "angle/axis" correspondance
• • - \(\displaystyle \hat q=\cos(\hat \theta/2)+ \hat n\, \sin(\hat \theta/2) \)

• • - \(\hat \theta=\theta_0 + \epsilon \theta_{\epsilon}\)
  • • - \(\theta_0\): angle of rotation
    • - \(\theta_{\epsilon}=t\cdot n_0\;\): quantity of translation along \(n_0\)

• • - \(\hat n=n_0+\epsilon\,n_{\epsilon}\)
  • • - \(n_0\): axis of rotation
    • - \(n_{\epsilon}=\frac{1}{2}\left( (n_0 \times t)\, cotan(\theta_0/2)+t\right) \times n_0\): called the moment of the rotation axis.