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. It has been suggested that geodetic system be merged into this article or section. ... Earth is the third planet in the Solar system. ...

The Pythagorean concept of a spherical Earth 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. Closer approximations range from modelling the shape of the entire Earth as an oblate spheroid, to the use of spherical harmonics or local approximations in terms of local reference ellipsoids. 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. The Pythagoreans were an Hellenic organization of astronomers, musicians, mathematicians, and philosophers; who believed that all things are, essentially, numeric. ... An oblate spheroid is ellipsoid having a shorter axis and two equal longer axes. ... In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ... In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the true figure of the Earth or other planetary body. ... The notion of a flat Earth refers to the idea that the inhabited surface of Earth is flat, rather than curved (see Spherical 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. In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the true figure of the Earth or other planetary body. ...

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 a. The shape of the ellipsoid is given by the flattening, f, 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. The flattening, ellipticity, or oblateness of an oblate spheroid is the relative difference between its equatorial radius a and its polar radius b: The flattening of the Earth is 1:298. ...

For such a flattened ellipsoid, the polar radius of curvature is larger than the equatorial

(a² / b),

even though the Earth's surface is closer to the Earth's centre at the poles than at the equator. Conversely, the equator's vertical radius of curvature is smaller than the polar

(b² / a).

This circumstance has formed the basis for attempts to determine the flattening of the mean Earth ellipsoid by so-called grade measurements. Grade measurement is the geodetic determination of the local radius of curvature of the figure of the Earth by determining the difference in astronomical latitude between two locations on the same meridian, the metric distance between which is known. ...

Historical Earth ellipsoids

The 19 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 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 John Fillmore Hayford in 1910 and adopted by the International Union of Geodesy and Geophysics (IUGG) in 1924, which recommended it for international use. John Fillmore Hayford (May 19, 1868 - March 10, 1925) was eminent United States geodesist. ...

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. It has been suggested that this article or section be merged with Datum. ...

The 20th, "General Purpose", model is not recognized in any specific datum/reference system but, rather, is an extrapolative radial refinement relative to the most recent, formally recognized models (keeping in mind Earth's maximun elevation is about 8.85 km above sea level and the maximum depression is about 11.52 km below it, providing a potential 20.37 km deviation from the prescribed radius at a given point—almost as much difference as between the equatorial and polar radii!):

The GRS-80 (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) a, total mass GM, dynamic form factor J₂ and angular velocity of rotation ω, making the inverse flattening 1 / f a derived quantity. The minute difference in 1 / f seen between GRS-80 and WGS-84 was produced by inaccurate numerical evaluation from the defining constants... Pierre Louis Maupertuis, here wearing lapmudes or a fur coat from his Lapland expedition. ... Photograph of Everest Colonel Sir George Everest (July 4, 1790 - December 1, 1866) was a British surveyor and geographer, and Surveyor-General of India from 1830 to 1843. ... George Biddell Airy Sir George Biddell Airy (July 27, 1801–January 2, 1892) was British Astronomer Royal from 1835 to 1881. ... Friedrich Wilhelm Bessel (July 22, 1784 – March 17, 1846) was a German mathematician, astronomer, and systematizer of the Bessel functions (which, despite their name, were discovered by Daniel Bernoulli). ... Friedrich Robert Helmert (* July 31, 1843 in Freiberg, Saxonia; † June 15, 1917 in Potsdam) was a celebrated German geodesist and an important writer on the theory of errors. ... John Fillmore Hayford (May 19, 1868 - March 10, 1925) was eminent United States geodesist. ... The World Geodetic System defines a fixed global reference frame for the Earth, for use in geodesy and navigation. ... The World Geodetic System defines a fixed global reference frame for the Earth, for use in geodesy and navigation. ... Definition GRS 80, or Geodetic Reference System 1980, is a geodetic reference system consisting of a global reference ellipsoid and a gravity field model. ... The World Geodetic System defines a fixed global reference frame for the Earth, for use in geodesy and navigation. ... The International Earth Rotation Service (IERS) is the body responsible for maintaining global time and reference frame standards, notably through its Earth Orientation Paramater (EOP) and International Celestial Reference System (ICRS) groups. ...

Some of the above ellipsoid models are actually geodetic datums: e.g., while GRS-80 defines only the geometric shape of its ellipsoid and a normal gravity field formula to go with it, WGS-84 defines a complete geodetic reference system realized in the terrain. Similarly, the older ED-50 (European Datum 1950) is based on the Hayford or International Ellipsoid. A Geodetic Datum is an adjustment to position of origin and shape of the planet, used by cartographers and satellite navigation systems to translate positions indicated on their products to their real position on earth. ... ED 50 (European Datum 1950) is a geodetic datum which was defined after World War II for the international connection of geodetic networks. ...

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 long periodic 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 C₂₂,S₂₂ and C₃₀, respectively, corresponding to degree and order numbers 2,2 for the triaxiality and 3,0 for the pear shape. The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ...

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 GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ... In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the true figure of the Earth or other planetary body. ... The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ... In physics, gravitation or gravity is the tendency of objects with mass to accelerate toward each other. ... Gravity is a force of attraction that acts between bodies that have mass. ...

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. The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ... A plumb line is a reference line guided by a string or cord weighted at the end with a large weight known as a plumb bob. ... The vertical deflections (deflections of the plumb line, astro-geodetic deflections) are important parameters of the local gravity field. ...

Correlations to geophysics and geology

Earth rotation and Earth's interior

Dertermining the exact figure of the Earth is not only a geodetic operation or a task of geometry, but is also related to geophysics. Without any idea of the Earth's interior, we can state a "constant density" of 5.5 g/cm³ and, according to theoretical arguments (see Leonhard Euler, A. Wangerin, etc.), such a body rotating like the Earth would have an flattening of 1:230. This article or section should include material from Erdmessung. ... Table of Geometry, from the 1728 Cyclopaedia. ... Geophysics, the study of the earth by quantitative physical methods, especially by seismic reflection and refraction, geodesy, gravity, magnetic, electrical, electromagnetic, and radioactivity methods. ... Leonhard was the first to use the term function to describe an expression involving various arguments; i. ... The flattening, ellipticity, or oblateness of an oblate spheroid is the relative difference between its equatorial radius a and its polar radius b: The flattening of the Earth is 1:298. ...

In fact the measured flattening is 1:298.25, which is more similar to a sphere and a strong 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 approximately 15 within the inner core. Modern seismology yields a value of 16 g/cm³ (iron or hydrogen) at the center of the earth. Earth, also known as the Earth or Terra, is the third planet outward from the Sun. ... Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ... Quarrying granite for the Mormon Temple, Utah Territory. ... Geology (from Greek γη- (ge-, the earth) and λογος (logos, word, reason)) is the science and study of the Earth, its composition, structure, physical properties, history and the processes that shape it. ... Seismology (from the Greek seismos = earthquake and logos = word) is the scientific study of earthquakes and the movement of waves through the Earth. ... General Name, Symbol, Number iron, Fe, 26 Chemical series transition metals Group, Period, Block 8, 4, d Appearance lustrous metallic with a grayish tinge Atomic mass 55. ...

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. The true vertical does not correspond 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 structure of the earth's crust and mantle can be determined by geodetic-geophysical models of the subsurface. The gravity field is the field of force, caused by the gravitation of the Earth, and influenced by the Earth rotation, the atmosphere and by geological bodies. ... A satellite is any object that orbits another object (which is known as its primary). ... An object is in a vertical position when it is aligned in an up-down direction, perpendicular to the horizon. ... [[ Deflection happens when an object hits a plane surface In physics In physics deflection is the event where an object collides and bounces against a plane surface. ... Surface of the Earth Topography, a term in geography, has come to refer to the lay of the land, or the physiogeographic characteristics of land in terms of elevation, slope, and orientation. ... Earth cutaway from core to exosphere. ...

See also

• Earth radius, flattening, Geosciences
• Eratosthenes, Pierre Bouguer, Friedrich Robert Helmert

The radius of Earth (or any planet) is the distance from its center to a point on its surface at mean sea level. ... The flattening, ellipticity, or oblateness of an oblate spheroid is the relative difference between its equatorial radius a and its polar radius b: The flattening of the Earth is 1:298. ... Earth science (also known as geoscience or the geosciences), is an all-embracing term for the sciences related to the planet Earth. ... Eratosthenes (Ερατοσθένης) Eratosthenes (Ερατοσθένης) (276 BC - 194 BC) was a Hellenistic mathematician, geographer and astronomer. ... Pierre Bouguer (February 16, 1698 – August 15, 1758) was a French mathematician. ... Friedrich Robert Helmert (* July 31, 1843 in Freiberg, Saxonia; † June 15, 1917 in Potsdam) was a celebrated German geodesist and an important writer on the theory of errors. ...

External links

• Reference Ellipsoids (PCI Geomatics)
• Reference Ellipsoids (ScanEx)

Literature

• Guy Bomford, Geodesy, Oxford 1962 and 1880.
• Guy Bomford, Determination of the European geoid by means of vertical deflections. Rpt of Comm. 14, IUGG 10th Gen. Ass., Rome 1954.
• Karl Ledersteger and Gottfried Gerstbach, Die horizontale Isostasie / Das isostatische Geoid 31. Ordnung. Geowissenschaftliche Mitteilungen Band 5, TU Wien 1975.
• Helmut Moritz and Bernhard Hofmann, Physical Geodesy. Springer, Wien & New York 2005.