Coordinate transformations between geocentric systems
The transformations between the seven
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
- 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.
- 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.
- 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
- 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.
- The GEO to MAG transformation is given by the matrix T5
= <lat-90,Y>*<long,Z>, where the rotation angle lat
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 GEI2000 to GEI transformation is given by the
matrix P = <-zA,Z>*<thetaA,Y>*<-zetaA,Z>,
where the rotation angles zA, thetaA and
the precession angles.
This transformation is a precession correction as described by
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.
Last updated 4 June 1997 by Mike