Figure of the Earth

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 distances-spanning continents and oceans-a more exact figure is necessary. The idea of a flat Earth, however, is still acceptable for surveys of small areas. Plane-table 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.

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 . The shape of the ellipsoid is given by the flattening, , 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 so-called
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 GRS-67 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.

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 (semi-major axis of
Earth ellipsoid) , total mass , dynamic
form factor and angular velocity of rotation
, making the inverse flattening a
derived quantity. The minute difference in 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 and
, 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 east-west and a north-south component.