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.

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):

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

$$E = R_3(\phi)*R_1(\theta)*R_3(\Omega) = <\phi, Z>*< \theta, X>*< \Omega, Z>$$ (40)

in the notation of Hapgood (1992) where '*' denotes matrix multiplication. The three principal rotations are on the other hand given by

$$R_1(\zeta)=<\zeta,X> = E(0,\zeta,0) \hspace{0.5cm} R_2(\zeta... ...90^{\circ}) \hspace{0.5cm} R_3(\zeta)=<\zeta,Z> = E(0,0,\zeta).$$ (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.