Rotation Formula

$\begin{figure}\begin{center}\BoxedEPSF{Rotation_Formula.epsf scaled 1200}\end{center}\end{figure}$

A formula which relates the Vector ${\bf r}'$ to the Angle $\Phi$ in the above figure (Goldstein 1980). Referring to the figure,

$\begin{eqnarray*} {\bf r}' &=& \overrightarrow{ON}+\overrightarrow{NV}+\overrigh... ...ot{\bf r})(1-\cos \Phi ) +({\bf r}\times \hat {\bf n})\sin \Phi. \end{eqnarray*}$

The Angle $\Phi$ and unit normal $\hat{\bf n}$ may also be expressed as Euler Angles. In terms of the Euler Parameters,

$\begin{displaymath} {\bf r}' = {\bf r}({e_0}^2-{e_1}^2-{e_2}^2-{e_3}^2)+2{\bf e}({\bf e}\cdot{\bf r}) +2({\bf r}\times \hat {\bf n})\sin\Phi. \end{displaymath}$

See also Euler Angles, Euler Parameters

References

Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.