Axis-Angle

Rodrigues' rotation formula

We denote a vector, \(\vec{v}'\) from a vector, \(\vec{v}\) rotated with a given axis of rotation \(\vec{k}\) by an angle \(\theta\).

\[ \vec{v}'=\vec{v}\cos\theta+(\vec{k} \times \vec{v})\sin\theta + \vec{k}(\vec{k} \cdot \vec{v})(1-\cos\theta)=(I+\sin\theta(\vec{k}^\times)+(1-\cos\theta)(\vec{k}^\times)^2)\vec{v} \]
\[ ((\vec{k}^\times)^2 = \vec{k}\vec{k}^T - I) \]

And this is exactly same context of ACTIVE(ALIBI) rotation matrix.

\[ R_{active}=I+\sin\theta(\vec{k}^\times)+(1-\cos\theta)(\vec{k}^\times)^2 \]

As a PASSIVE(ALIAS) rotation matrix, you need to transpose the ACTIVE rotation matrix.

\[ R_{passive}=I-\sin\theta(\vec{k}^\times)+(1-\cos\theta)(\vec{k}^\times)^2 \]

Derive Axis-Angle from Rotation Matrix

\(\theta=\cos^{-1}(\frac{tr(R)-1}{2}) + 2\pi m\)

\(\vec{k}^{\times}=\frac{R-R^T}{2\sin\theta}\) (ACTIVE)

\(\vec{k}^{\times}=\frac{R^T-R}{2\sin\theta}\) (PASSIVE)