The transformations between the seven geocentric coordinate systems:

- geocentric equatorial
inertial for epoch J2000.0 (GEI
_{2000}) - geocentric equatorial inertial for epoch-of-date (GEI)
- geographic (GEO)
- geocentric solar ecliptic (GSE)
- geocentric solar magnetospheric (GSM)
- solar magnetic (SM)
- geomagnetic (MAG)

can be broken down into six fundamental transformations as shown in the figure below. The symbol beside each arrow refers to the matrix for the transformation associated with the arrow (and the direction of the arrow indicates the sense of the transformation). These symbols and the matrices are specified in more detail below.

The **P** and **T**_{n} matrices are specified here using
the notation of
Hapgood (1992):

- The
*GEI to GEO*transformation is given by the matrix**T**_{1}= <*theta*,Z>, where the rotation angle*theta*is the Greenwich mean sidereal time. This transformation is a rotation in the plane of the Earth's equator from the First Point of Aries to the Greenwich meridian. - The
*GEI to GSE*transformation is given by the matrix**T**_{2}= <*lambda*_{O},Z>*<*epsilon*,X>, where the rotation angle*lambda*_{O}is the Sun's ecliptic longitude and the angle*epsilon*is the obliquity of the ecliptic. This transformation is a rotation from the Earth's equator to the plane of the ecliptic followed by a rotation in the plane of the ecliptic from the First Point of Aries to the Earth-Sun direction. - The
*GSE to GSM*transformation is given by the matrix**T**_{3}= <*- psi*,X>, where the rotation angle*psi*is the the GSE-GSM angle. This transformation is a rotation in the GSE YZ plane from the GSE Z axis to the GSM Z axis. - The
*GSM to SM*transformation is given by the matrix**T**_{4}= <*- mu*,Y>, where the rotation angle*mu*is the dipole tilt. This transformation is a rotation in the GSM XZ plane from the GSM Z axis to the geomagnetic dipole axis. - The
*GEO to MAG*transformation is given by the matrix**T**_{5}= <*lat*-90,Y>*<*long*,Z>, where the rotation angle*lat*is the latitude and angle*long*is the longitude of the geomagnetic pole (as defined by the axis of the dipole component of the geomagnetic field). This transformation is a rotation in the plane of the Earth's equator from the Greenwich meridian to the meridian containing the dipole axis, followed by a rotation in that meridian from the rotation axis to the dipole axis. - The
*GEI*transformation is given by the matrix_{2000}to GEI**P**= <*-z*,Z>*<_{A}*theta*,Y>*<_{A}*-zeta*,Z>, where the rotation angles_{A}*z*,_{A}*theta*and_{A}*zeta*are the precession angles. This transformation is a precession correction as described by Hapgood (1995)._{A}

The full set of tranformation matrices between the various geocentric
coordinate systems can be obtained by multiplication of the matrices for these
fundamental transformations, **P** and **T**_{n}, as shown in
the table below.

From | |||||||

To | GEI_{2000} | GEI | GEO | GSE | GSM | SM | MAG |

GEI_{2000} |
1 | P^{-1} | P^{-1}T_{1}^{-1} | P^{-1}T_{2}^{-1} | P^{-1}T_{2}^{-1}T_{3}^{-1} |
P^{-1}T_{2}^{-1}T_{3}^{-1}T_{4}^{-1} |
P^{-1}T_{1}^{-1}T_{5}^{-1} |

GEI | P | 1 |
T_{1}^{-1} | T_{2}^{-1} |
T_{2}^{-1}T_{3}^{-1} |
T_{2}^{-1}T_{3}^{-1}T_{4}^{-1} |
T_{1}^{-1}T_{5}^{-1} |

GEO | T_{1}P |
T_{1} | 1 | T_{1}T_{2}^{-1} | T_{1}T_{2}^{-1}T_{3}^{-1} |
T_{1}T_{2}^{-1}T_{3}^{-1}T_{4}^{-1} |
T_{5}^{-1} |

GSE | T_{2}P | T_{2} | T_{2}T_{1}^{-1} |
1 | T_{3}^{-1} |
T_{3}^{-1}T_{4}^{-1} |
T_{2}T_{1}^{-1}T_{5}^{-1} |

GSM | T_{3}T_{2}P |
T_{3}T_{2} | T_{3}T_{2}T_{1}^{-1} |
T_{3} | 1 | T_{4}^{-1} | T_{3}T_{2}T_{1}^{-1}T_{5}^{-1} |

SM | T_{4}T_{3}T_{2}P |
T_{4}T_{3}T_{2} |
T_{4}T_{3}T_{2}T_{1}^{-1} |
T_{4}T_{3} | T_{4} | 1 | T_{4}T_{3}T_{2}T_{1}^{-1}T_{5}^{-1} |

MAG | T_{5}T_{1}P |
T_{5}T_{1} | T_{5} | T_{5}T_{1}T_{2}^{-1} |
T_{5}T_{1}T_{2}^{-1}T_{3}^{-1} |
T_{5}T_{1}T_{2}^{-1}T_{3}^{-1}T_{4}^{-1} |
1 |

Last updated 4 June 1997 by Mike Hapgood (Email: M.Hapgood@rl.ac.uk)