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Mathematik-Online lexicon:

Axis and Angel of Rotation

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With the rotation-matrix

$\displaystyle Q = \frac{1}{2} \left( \begin{array}{ccc} 1 & -\sqrt{2} & 1 \\ \s... ...& 0 & -\sqrt{2} \\ 1 & \sqrt{2} & 1 \end{array} \right) \in \mathbb{R}^{3,3} $

you get

$\displaystyle Q^{\operatorname t}Q = \frac{1}{4} \left( \begin{array}{ccc} 4 & ... ...\\ 0 & 0 & 4 \end{array} \right) = E \Rightarrow Q^{\operatorname t}= Q^{-1} $

and

$\displaystyle \operatorname{det} Q = \operatorname{det} \frac{1}{2} \left( \beg... ...\ \sqrt{2} & 0 & -\sqrt{2} \\ 1 & \sqrt{2} & 1 \end{array} \right) = + 1\,. $

Compute the rotationaxis, the eigenvector $ v$ to the eigenvalue $ \lambda = 1$:

$\displaystyle 2(Q-1E) = \left( \begin{array}{ccc} -1 & -\sqrt{2} & 1 \\ \sqrt{2} & -2 & -\sqrt{2} \\ 1 & \sqrt{2} & -1 \end{array} \right) $

With 2nd row $ - \sqrt{2} \cdot$ 1st row and 3rd row $ +$ 1st row you get

$\displaystyle \left( \begin{array}{ccc} -1 & -\sqrt{2} & 1 \\ 0 & -4 & 0 \\ 0 & 0 & 0 \end{array} \right) $

and so the eigenvector $ v = \frac{1}{\sqrt{2}} \left( \begin{array}{c} 1 \\ 0 \\ 1 \end{array} \right)$.

To compute $ \varphi$ you compute the angle between an unit vector $ n$ with $ n \bot v$ and the unit vector $ x = Qn$.

$\displaystyle n = \left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right)\,, x... ...{1}{2} \left( \begin{array}{c} -\sqrt{2} \\ 0 \\ \sqrt{2} \end{array} \right) $

$\displaystyle cos \varphi = n \cdot x = 0 \Rightarrow \varphi = \pm \frac{\pi}{2}\,. $

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  automatisch erstellt am 12.  1. 2007