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An Earth ellipsoid is a mathematical figure approximating the shape of the Earth, used as a reference frame for computations in geodesy, astronomy and the geosciences. Various different ellipsoids have been used as approximations.
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It is an ellipsoid of revolution, whose short (polar) axis (connecting the two flattest spots called geographical north and south poles) is approximately aligned with the rotation axis of the Earth. The ellipsoid is defined by the equatorial axis a and the polar axis b; their difference is about 21 km or 0.335 percent. Additional parameters are the mass function J2, the correspondent gravity formula, and the rotation period (usually 86164 seconds).
Many methods exist for determination of the axes of an Earth ellipsoid, ranging from meridian arcs up to modern satellite geodesy or the analysis and interconnection of continental geodetic networks. Amongst the different set of data used in national surveys are several of special importance: the Bessel ellipsoid of 1841, the international Hayford ellipsoid of 1924, and (for GPS positioning) the WGS84 ellipsoid.
Historical method of determining the ellipsoid
High precision land surveys can be used to determine the distance between two places at nearly the same longitude by measuring a base line and a chain of triangles. (Suitable stations for the end points are rarely at the same longitude). The distance Δ along the meridian from one end point to a point at the same latitude as the second end point is then calculated by trigonometry. The surface distance Δ is reduced to Δ', the corresponding distance at mean sea level. The intermediate distances to points on the meridian at the same latitudes as other stations of the survey may also be calculated.
The geographic latitudes of both end points, φs (standpoint) and φf (forepoint) and possibly at other points are determined by astrogeodesy, observing the zenith distances of sufficient numbers of stars. If latitudes are measured at end points only, the radius of curvature at the midpoint of the meridian arc can be calculated from R = Δ'/(|φs-φf|). A second meridian arc will allow the derivation of two parameters required to specify a reference ellipsoid. Longer arcs with intermediate latitude determinations can completely determine the ellipsoid. In practice multiple arc measurements are used to determine the ellipsoid parameters by the method of least squares. The parameters determined are usually the semi-major axis,
Geodesy no longer uses simple meridian arcs, but complex networks with hundreds of fixed points linked by the methods of satellite geodesy.
Historical Earth ellipsoids
The reference ellipsoid models listed below have had utility in geodetic work and many are still in use. The older ellipsoids are named for the individual who derived them and the year of development is given. In 1887 the English surveyor Colonel Alexander Ross Clarke CB FRS RE was awarded the Gold Medal of the Royal Society for his work in determining the figure of the Earth. The international ellipsoid was developed by John Fillmore Hayford in 1910 and adopted by the International Union of Geodesy and Geophysics (IUGG) in 1924, which recommended it for international use.
At the 1967 meeting of the IUGG held in Lucerne, Switzerland, the ellipsoid called GRS-67 (Geodetic Reference System 1967) in the listing was recommended for adoption. The new ellipsoid was not recommended to replace the International Ellipsoid (1924), but was advocated for use where a greater degree of accuracy is required. It became a part of the GRS-67 which was approved and adopted at the 1971 meeting of the IUGG held in Moscow. It is used in Australia for the Australian Geodetic Datum and in South America for the South American Datum 1969.
The GRS-80 (Geodetic Reference System 1980) as approved and adopted by the IUGG at its Canberra, Australia meeting of 1979 is based on the equatorial radius (semi-major axis of Earth ellipsoid)
An ellipsoidal model describes only the ellipsoid's geometry and a normal gravity field formula to go with it. Commonly an ellipsoidal model is part of a more encompassing geodetic datum. For example, the older ED-50 (European Datum 1950) is based on the Hayford or International Ellipsoid. WGS-84 is peculiar in that the same name is used for both the complete geodetic reference system and its component ellipsoidal model. Nevertheless, the two concepts—ellipsoidal model and geodetic reference system—remain distinct.
Note that the same ellipsoid may be known by different names. It is best to mention the defining constants for unambiguous identification.