Relation between quaternion and rotation

Consider

- a quaternion \(q=(s,w)\) of unit norm \(\|q\|=1\).
- a vector \(v=(v_x,v_y,v_z)\) assimilated to the pure quaternion \(q_v=(v,0)=(v_x,v_y,v_z,0)\).

Then

- \(q_{v'}=\mathcal{R}_q(v)=q\,q_v\,q^{\star}\) is a pure quaternion \(q_{v'}=(v_x',v_y',v_z',0)\)
- And \(v'=(v_x',v_y',v_z')\) is the rotation of vector \(v\) around the axis \(n=s/\|s\|\), with angle \(2\; acos(w)\).

Demonstration

\(\mathcal{R}_q(v)=(s,w)\;(v,0)\;(-s,w)= \; ... \; =((w ^ 2-s ^ 2)\, v + 2 (s \cdot v)\,s + 2 w\,(s\times v),0)\)

As \(\|q\|=1\), we can write \(q=(s,w)=(n\sin(\phi),\cos(\phi))\), where \(\|n\|=1\)

Then \(\displaystyle \mathcal{R}_q(v) = (\underbrace{(\cos ^ 2(\phi)-\sin ^ 2(\phi))}_{\cos(2\phi)}\,v + \underbrace{2 \sin ^ 2 (\phi)}_{1-\cos (2 \phi)} (n \cdot v)\,n + \underbrace{2\, \cos(\phi) \sin(\phi)}_{\sin(2\phi)}\;n\times v\,,\;0)\)

\(\Rightarrow\) Rodrigues formula for axis \(n\) and angle \(2 \phi\).

The unit quaternion \(q=(n\,\sin(\theta/2),\cos(\theta/2))\) represents the rotation of angle \(\theta\) around the axis \(n\).