# Coordinate transformations between geocentric systems

## Overview

The transformations between the seven geocentric coordinate systems:

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 fundamental transformations

The P and Tn matrices are specified here using the notation of Hapgood (1992):

1. The GEI to GEO transformation is given by the matrix T1 = <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.
2. The GEI to GSE transformation is given by the matrix T2 = <lambdaO,Z>*<epsilon,X>, where the rotation angle lambdaO 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.
3. The GSE to GSM transformation is given by the matrix T3 = <- 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.
4. The GSM to SM transformation is given by the matrix T4 = <- 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.
5. The GEO to MAG transformation is given by the matrix T5 = <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.
6. The GEI2000 to GEI transformation is given by the matrix P = <-zA,Z>*<thetaA,Y>*<-zetaA,Z>, where the rotation angles zA, thetaA and zetaA are the precession angles. This transformation is a precession correction as described by Hapgood (1995).

## Index of all tranformations

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 Tn, as shown in the table below.

 From To GEI2000 GEI GEO GSE GSM SM MAG GEI2000 1 P-1 P-1T1-1 P-1T2-1 P-1T2-1T3-1 P-1T2-1T3-1T4-1 P-1T1-1T5-1 GEI P 1 T1-1 T2-1 T2-1T3-1 T2-1T3-1T4-1 T1-1T5-1 GEO T1P T1 1 T1T2-1 T1T2-1T3-1 T1T2-1T3-1T4-1 T5-1 GSE T2P T2 T2T1-1 1 T3-1 T3-1T4-1 T2T1-1T5-1 GSM T3T2P T3T2 T3T2T1-1 T3 1 T4-1 T3T2T1-1T5-1 SM T4T3T2P T4T3T2 T4T3T2T1-1 T4T3 T4 1 T4T3T2T1-1T5-1 MAG T5T1P T5T1 T5 T5T1T2-1 T5T1T2-1T3-1 T5T1T2-1T3-1T4-1 1

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