Quaternion double cover

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Parent: Quaternion index

Source: Solà 2017

$$ \mathbf{q} = \left[ \begin{array}{c} q_w\ \mathbf{q}_v \end{array} \right] = \left[ \begin{array}{c} \cos\frac{\phi}{2} \ \mathbf{u} \sin\frac{\phi}{2} \end{array} \right] $$ where $\phi$ is the angle rotated by $\mathbf{q}$ on objects in the 3D space $\mathbb{R}^3$.

quaternion-double-cover

Recap

  • $\theta$ is the angle in quaternion space (s. unit quaternions )
  • $\phi$ as the angle in 3D space $\mathbb{R}^3$

Therefore, the angle is halved in quaternion space. $$\theta = \phi / 2$$