next up previous
Next: Velocity Transformations Up: Appendix Previous: Appendix

Eulerian Rotation

Figure: Eulerian Rotation $E(\Omega,\Theta,\Phi)$(after Madelung (1964)): the transformation between system $S(X,Y,Z)$ and system $S'(X',Y',Z')$ can be expressed by the three right-handed principal rotations: 1. $< \Omega, Z>$ around the $Z$-axis towards the ascending node $\ascnode$, 2. $< \vartheta, X>$ around the ascending node axis, 3. $<\phi, Z>$ around the $Z'$-axis towards the $+X'$-axis.
\begin{figure}\begin{center}
\epsfig{file=eulerian.eps,width=10cm}\end{center}\end{figure}

In this paper we describe transformations between cartesian coordinate systems in Euclidean space. Let system S be defined by the orthonormal right-handed basis vectors $X,Y,Z$ and system $S'$ by the orthonormal right-handed basis vectors $X',Y',Z'$ with a common origin $O$. The position of system $S'$ in system $S$ is then defined by the angular coordinates of its pole ( $Z'=(\theta,\Psi=\Omega-90^\circ$)) and the prime meridian angle $\phi$ (see Fig.4) which is the angular distance between prime meridian $X'$ and ascending node $\ascnode$. The Eulerian transformation matrix from $S$ to $S'$ is then defined by (Madelung, 1964):

  $\textstyle E(\Omega,\theta,\phi) =$ $\displaystyle \left( \begin{array}{ccc}
\par
\cos\phi \cos\Omega-\sin\phi \sin\...
...n\Omega \sin\theta & -\cos\Omega \sin\theta & \cos\theta\\
\end{array} \right)$ (39)

Such that a vector $\mathbf{v}$ given in $S$ has coordinates $\mathbf{v'} = E*\mathbf{v}$ in $S'$. This corresponds to three principal rotations:
\begin{displaymath}E = R_3(\phi)*R_1(\theta)*R_3(\Omega) = <\phi, Z>*< \theta, X>*< \Omega, Z> \end{displaymath} (40)

in the notation of Hapgood (1992) where '*' denotes matrix multiplication. The three principal rotations are on the other hand given by
\begin{displaymath}R_1(\zeta)=<\zeta,X> = E(0,\zeta,0) \hspace{0.5cm}
R_2(\zeta...
...90^{\circ}) \hspace{0.5cm} R_3(\zeta)=<\zeta,Z> = E(0,0,\zeta).\end{displaymath} (41)

Note that all rotation matrices are orthogonal, s.t. $E^{-1} = E^T$ and transformations between all systems defined in this paper can easily be calculated by a series of matrix multiplications.


next up previous
Next: Velocity Transformations Up: Appendix Previous: Appendix
Markus Fraenz 2017-03-13