Figure of the Earth
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The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earth's size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and water areas. This is, in fact, the surface on which actual Earth measurements are made. It is not suitable, however, for exact mathematical computations because the formulas which would be required to take the irregularities into account would necessitate a prohibitive amount of computations. The topographic surface is generally the concern of topographers and hydrographers.
The Pythagorean spherical concept offers a simple surface which is mathematically easy to deal with. Many astronomical and navigational computations use it as a surface representing the Earth. While the sphere is a close approximation of the true figure of the Earth and satisfactory for many purposes, to the geodesists interested in the measurement of long distancesspannig continents and oceansa more exact figure is necessary. The idea of a flat Earth, however, is still acceptable for surveys of small areas. Planetable surveys are made for relatively small areas and no account is taken of the curvature of the Earth. A survey of a city would likely be computed as though the Earth were a plane surface the size of the city. For such small areas, exact positions can be determined relative to each other without considering the size and shape of the total Earth.
Contents 
Ellipsoid of revolution
Since the Earth is in fact flattened slightly at the poles and bulges somewhat at the equator, the geometrical figure used in geodesy to most nearly approximate the shape of the Earth is an ellipsoid of revolution. The ellipsoid of revolution is the figure which would be obtained by rotating an ellipse about its shorter axis. An ellipsoid of revolution describing the figure of the Earth is called a reference ellipsoid.
An ellipsoid of revolution is uniquely defined by specifying two dimensions. Geodesists, by convention, use the semimajor axis and flattening. The size is represented by the radius at the equator  the semimajor axis  and designated by the letter <math>a<math>. The shape of the ellipsoid is given by the flattening, <math>f<math>, which indicates how closely the ellipsoid approaches a spherical shape. The difference between the reference ellipsoid representing the Earth and a sphere is very small, only one part in 300 approximately.
Note that for a flattened ellipsoid, the polar radius of curvature is larger that the equatorial one, even though the Earth's surface is closer to the Earth's centre at the poles than at the equator. This circumstance has formed the basis for attempts to determine the flattening of the mean Earth ellipsoid by socalled grade measurements.
Historical Earth ellipsoids
The ellipsoids 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 mathematician Col 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 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 GRS67 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 Geodetic Reference System 1967 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.
Name  Equatorial radius  Inverse flattening  Where used 
Australian National  6378,160  298.25  Australia 
Krassowsky (1940)  6,378,245m  298.3  Russia 
International (1924)  6,378,388  297  Europe 
Clarke (1880)  6,378,249.145  293.465  France, Africa 
Clarke (1866)  6,378,206.4  294.9786982  North America 
Bessel (1841)  6,377,397.155  299.1528128  Europe, Japan 
Airy (1830)  6,377,563.396  299.3249646  Great Britain 
Everest (1830)  6,377,276.345  300.8017  India 
WGS66 (1966)  6,378,145  298.25  USA/DoD 
GRS67 (1967)  6,378,160  298.247167427  
South American (1969)  6378160  298.25  South America 
WGS72 (1972)  6,378,135  298.26  USA/DoD 
GRS80 (1979)  6,378,137  298.257222101  
WGS84  6,378,137  298.2572235630 
Note that GRS80 (Geodetic Reference System 1980) as approved and adopted by the IUGG at its Canberra, Australia meeting of 1979 is originally defined based on the equatorial radius (semimajor axis of Earth ellipsoid) <math>a<math>, total mass <math>GM<math>, dynamic form factor <math>J_2<math> and angular velocity of rotation <math>\omega<math>, making the inverse flattening <math>1/f<math> a derived quantity. The minute difference in <math>1/f<math> seen between GRS80 and WGS84 was produced by inaccurate numerical evaluation from the defining constants...
Note also that some of the above ellipsoid models are actually geodetic datums: e.g., while GRS80 defines only the geometric shape of its ellipsoid and a normal gravity field formula to go with it, WGS84 defines a complete geodetic reference system realized in the terrain. Similarly, the older ED50 (European Datum 1950) is based on the Hayford or International Ellipsoid.
More complicated figures
The possibility that the Earth's equator is an ellipse rather than a circle and therefore that the ellipsoid is triaxial has been a matter of scientific controversy for many years. Modern technological developments have furnished new and rapid methods for data collection and since the launching of the first Russian sputnik, orbital data has been used to investigate the theory of ellipticity.
A second theory, more complicated than triaxiality, proposed that observed longperiodic orbital variations of the first Earth satellites indicate an additional depression at the south pole accompanied by a bulge of the same degree at the north pole. It is also contended that the northern middle latitudes were slightly flattened and the southern middle latitudes bulged in a similar amount. This concept suggested a slightly pearshaped Earth and was the subject of much public discussion. Modern geodesy tends to retain the ellipsoid of revolution and treat triaxiality and pear shape as a part of the geoid figure: they are represented by the spherical harmonic coefficients <math>C_{22},S_{22}<math> and <math>C_{30}<math>, respectively, corresponding to degree and order numbers 2,2 for the triaxiality and 3,0 for the pear shape.
Geoid
It was stated earlier that measurements are made on the apparent or topographic surface of the Earth and it has just been explained that computations are performed on an ellipsoid. One other surface is involved in geodetic measurement: the geoid. In geodetic surveying, the computation of the geodetic coordinates of points is commonly performed on a reference ellipsoid closely approximating the size and shape of the Earth in the area of the survey. The actual measurements made on the surface of the Earth with certain instruments are however referred to the geoid. The ellipsoid is a mathematically defined regular surface with specific dimensions. The geoid, on the other hand, coincides with that surface to which the oceans would conform over the entire Earth if free to adjust to the combined effect of the Earth's mass attraction (gravitation) and the centrifugal force of the Earth's rotation, i.e., gravity. As a result of the uneven distribution of the Earth's mass, the geoidal surface is irregular and, since the ellipsoid is a regular surface, the separations between the two, referred to as geoid undulations, geoid heights, or geoid separations, will be irregular as well.
The geoid is a surface along which the gravity potential is everywhere equal and to which the direction of gravity is always perpendicular. The latter is particularly important because optical instruments containing levelling devices are commonly used to make geodetic measurements. When properly adjusted, the vertical axis of the instrument coincides with the direction of gravity and is, therefore, perpendicular to the geoid. The angle between the plumb line which is perpendicular to the geoid (sometimes called "the vertical") and the perpendicular to the ellipsoid (sometimes called "the ellipsoidal normal") is defined as the deflection of the vertical. It has two components: an eastwest and a northsouth component.
Correlations to Geophysics and Geology
Earth rotation and Earth's interior
Dertermining the exact figure of the Earth is no only a geodetic operation or a task of geometry, but is alo correlated to geophysics. Without any idea of the Earth's interior we can state a "constant density" of 5,5 g/cm³  and by theoretical reasons (see Leonhard Euler, A. Wangerin etc.) such a body rorating like the Earth would have an obliquity of 1:230.
In fact the measured flattening is 1:298,25  which is more similar to a sphere and a heavy argument that the Earth's core is very compact. Therefore the density must be a function of the depth, reaching from about 2,7 g/cm³ at the surface (rock density of granite, limestone etc., see regional geology) up to approx. 15 within the inner core. The modern seismology yields a value of 16 g/cm³ (iron or hydrogen) at the center of the earth.
Global and regional Gravity Field
Another implication to the physical exploration of the Earth's interior is the gravity field which can be measured very exactly at the surface and by satellites. If the true vertical does not corresponde to the theoretical one (in fact the deflection amounts from 2" to 50") because the topography and all geological masses are slightly disturbing the gravity field. Therefore the gross stucture of the earth's crust and mantle can be determined by geodeticgeophysical models of the subsurface.
See also
Literature
 Guy Bomford, Geodesy, Oxford 1962 and 1880.
 G.Bomford, Determination of the European geoid by means of vertical deflections. Rpt of Comm.14, IUGG 10th Gen.Ass., Rome 1954.
 K.Ledersteger/ G.Gerstbach, Die horizontale Isostasie / Das isostatische Geoid 31. Ordnung. Geowissenschaftliche Mitteilungen Band 5, TU Wien 1975.
 H.Moritz/ B.Hofmann, Physical Geodesy, Edition Springer, Wien & New York 2005.