## LOCAL SYSTEMS

**Description:
**

The many systems established by various countries throughout the world for positioning on and mapping of the Earth’s surface were established with an astrogeodetic origin to the datum. In the UK the origin of the datum for the Ordnance Survey is at Herstmonceux in Sussex. The origin of the datum for the European system is at Potsdam and the origin of the datum for North America is at Meades Ranch in Kansas. At each of these origins the relationship between the geoid and ellipsoid is defined in terms of the tangential planes between the respective surfaces. Such systems use reference ellipsoids which most closely fit the geoid in their area of coverage and are defined by the following eight parameters:

- The size and shape of the ellipsoid, defined by the semi-major axis a, and one other chosen from the semi-minor axis b or the flattening f or the eccentricity e (two parameters).

- The minor axis of the ellipsoid is orientated parallel to the mean spin axis of the Earth as defined by IERS (two parameters).

- The geodetic, latitude and longitude are defined as being the same as the astronomical, i.e. gravitational latitude and longitude and therefore with respect to the geoid, then the tangential planes must be parallel and the deviations of the vertical at the origin are zero. In this case the vertical to the geoid and the normal to the ellipsoid are coincident. If the tangential planes are further defined as being coincident then the separation between the geoid and ellipsoid surface, N, is also zero. This constitutes the definition of four parameters; two for the deviations of the vertical, East–West and North–South, one for the separation and one for the implicit definition of the zero direction of longitude.

Satellite datums or datums that are required for global use may be defined as follows.

- The Earth’s mass centre is the origin of the coordinate system, e.g. if a Cartesian coordinate system is being used then at the Earth’s mass centre, X = Y = Z = 0 (three parameters).

- The orientation of one of the coordinate axes is defined in inertial space, in practical terms that means with respect to the ‘fixed’ stars (two parameters).

- The orientation of the zero direction in the plane at right angles to the defined direction, e.g. the direction in which X is always 0 (one parameter).

- The shape and size of the reference ellipsoid as above (two parameters).

It follows that all properly defined geodetic systems will have their axes parallel and can be related to each other by simple translations in X, Y and Z.

The goodness of fit can be indicated by an examination of values for the deviation of the vertical (ξ ) and geoid–ellipsoid separation (N), as indicated in Figure Consider a meridian section through the geoid–ellipsoid (Figure , it is obvious that the north–south component of the deviation of the vertical.

**A meridianal section**