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Axis and Angel of Rotation

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A rotation $ Q$ in $ \mathbb{R}^3$ has an axis of rotation, i.e. $ Q$ fixes an unit vector $ u$, and corresponds to a plane rotation by an angle $ \vartheta$ in the plane orthogonal to $ u$.

With respect to an orthonormal right-handed coordinate system $ u,v,w$, the matrix representation of $ Q$ is given by:

$\displaystyle \tilde Q = \left(\begin{array}{ccc} 1&0&0 \\ 0&\cos\vartheta & -\sin\vartheta \\ 0&\sin\vartheta & \cos\vartheta \\ \end{array}\right) \,. $

For the angle of rotation

$\displaystyle \cos\vartheta = \frac{1}{2}\left(\operatorname{Spur} Q - 1\right) $

holds.
(Authors: Höllig/Reble/Höfert)

Example:

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  automatically generated 1/12/2007