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An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a
reference frame In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
for computations in
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
,
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, and the
geoscience Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four sphe ...
s. Various different
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as th ...
s have been used as approximations. It is a
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...
(an ellipsoid of
revolution In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ...
) whose minor axis (shorter diameter), which connects the geographical
North Pole The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Ma ...
and
South Pole The South Pole, also known as the Geographic South Pole, Terrestrial South Pole or 90th Parallel South, is one of the two points where Earth's axis of rotation intersects its surface. It is the southernmost point on Earth and lies antipod ...
, is approximately aligned with the Earth's axis of rotation. The ellipsoid is defined by the ''equatorial axis'' (''a'') and the ''polar axis'' (''b''); their radial difference is slightly more than 21 km, or 0.335% of ''a'' (which is not quite 6,400 km). 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 The Bessel ellipsoid (or Bessel 1841) is an important reference ellipsoid of geodesy. It is currently used by several countries for their national geodetic surveys, but will be replaced in the next decades by modern ellipsoids of satellite geode ...
of 1841, the international Hayford ellipsoid of 1924, and (for GPS positioning) the
WGS84 The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also desc ...
ellipsoid.


Types

There are two types of ellipsoid: mean and reference. A data set which describes the global
average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
of the Earth's surface curvature is called the ''mean Earth Ellipsoid''. It refers to a theoretical coherence between the geographic latitude and the meridional curvature of the
geoid The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
. The latter is close to the
mean sea level There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ...
, and therefore an ideal Earth ellipsoid has the same
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
as the geoid. While the mean Earth ellipsoid is the ideal basis of global geodesy, for
regional In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and the interaction of humanity and t ...
networks a so-called ''reference ellipsoid'' may be the better choice. When geodetic measurements have to be computed on a mathematical reference surface, this surface should have a similar curvature as the regional geoid; otherwise, reduction of the measurements will get small distortions. This is the reason for the "long life" of former reference ellipsoids like the Hayford or the
Bessel ellipsoid The Bessel ellipsoid (or Bessel 1841) is an important reference ellipsoid of geodesy. It is currently used by several countries for their national geodetic surveys, but will be replaced in the next decades by modern ellipsoids of satellite geode ...
, despite the fact that their main axes deviate by several hundred meters from the modern values. Another reason is a judicial one: the
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
s of millions of boundary stones should remain fixed for a long period. If their reference surface changes, the coordinates themselves also change. However, for international networks, GPS positioning, or
astronautics Astronautics (or cosmonautics) is the theory and practice of travel beyond Earth's atmosphere into outer space. Spaceflight is one of its main applications and space science its overarching field. The term ''astronautics'' (originally ''astron ...
, these regional reasons are less relevant. As knowledge of the Earth's figure is increasingly accurate, the International Geoscientific Union IUGG usually adapts the axes of the Earth ellipsoid to the best available data.


Reference ellipsoid

In
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
, a reference ellipsoid is a mathematically defined surface that approximates the
geoid The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
, which is the truer, imperfect
figure of the Earth Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A sphere is a well-known historical approxim ...
, or other planetary body, as opposed to a perfect, smooth, and unaltered sphere, which factors in the undulations of the bodies'
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
due to variations in the composition and density of the interior, as well as the subsequent
flattening Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution ( spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening ...
caused by the
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
from the rotation of these massive objects (for planetary bodies that do rotate). Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
,
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
, and
elevation The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § ...
are defined. In the context of standardization and geographic applications, a ''geodesic reference ellipsoid'' is the mathematical model used as foundation by spatial reference system or
geodetic datum A geodetic datum or geodetic system (also: geodetic reference datum, geodetic reference system, or geodetic reference frame) is a global datum reference or reference frame for precisely representing the position of locations on Earth or other pla ...
definitions.


Ellipsoid parameters

In 1687
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
published the Principia in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of a flattened ("oblate")
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as th ...
of revolution, generated by an
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
rotated around its minor diameter; a shape which he termed an
oblate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ci ...
. In geophysics,
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
, and related areas, the word 'ellipsoid' is understood to mean 'oblate ellipsoid of revolution', and the older term 'oblate spheroid' is hardly used.Torge, W (2001) Geodesy (3rd edition), published by de Gruyter, For bodies that cannot be well approximated by an ellipsoid of revolution a triaxial (or scalene) ellipsoid is used. The shape of an ellipsoid of revolution is determined by the shape parameters of that
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
. The
semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lon ...
of the ellipse, , becomes the equatorial radius of the ellipsoid: the semi-minor axis of the ellipse, , becomes the distance from the centre to either pole. These two lengths completely specify the shape of the ellipsoid. In geodesy publications, however, it is common to specify the semi-major axis (equatorial radius) and the
flattening Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution ( spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening ...
, defined as: : f=\frac. That is, is the amount of flattening at each pole, relative to the radius at the equator. This is often expressed as a fraction 1/; then being the "inverse flattening". A great many other ellipse parameters are used in
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
but they can all be related to one or two of the set , and . A great many ellipsoids have been used to model the Earth in the past, with different assumed values of and as well as different assumed positions of the center and different axis orientations relative to the solid Earth. Starting in the late twentieth century, improved measurements of satellite orbits and star positions have provided extremely accurate determinations of the earth's center of mass and of its axis of revolution; and those parameters have been adopted also for all modern reference ellipsoids. The ellipsoid WGS-84, widely used for mapping and
satellite navigation A satellite navigation or satnav system is a system that uses satellites to provide autonomous geo-spatial positioning. It allows satellite navigation devices to determine their location ( longitude, latitude, and altitude/ elevation) to hig ...
has close to 1/300 (more precisely, 1/298.257223563, by definition), corresponding to a difference of the major and minor semi-axes of approximately (more precisely, 21.3846857548205 km). For comparison, Earth's
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
is even less elliptical, with a flattening of less than 1/825, while
Jupiter Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousand ...
is visibly oblate at about 1/15 and one of Saturn's triaxial moons, Telesto, is highly flattened, with between 1/3 and 1/2 (meaning that the polar diameter is between 50% and 67% of the equatorial.


Determination

Arc measurement Arc measurement, sometimes degree measurement (german: Gradmessung), is the astrogeodetic technique of determining of the radius of Earth – more specifically, the local Earth radius of curvature of the figure of the Earth – by relating the la ...
is the historical method of determining the ellipsoid. Two meridian arc measurements will allow the derivation of two parameters required to specify a
reference ellipsoid An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
. For example, if the measurements were hypothetically performed exactly over the equator plane and either geographical pole, the radii of curvature so obtained would be related to the equatorial radius and the polar radius, respectively ''a'' and ''b'' (see: Earth polar and equatorial radius of curvature). Then, the
flattening Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution ( spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening ...
would readily follow from its definition: :f=(a-b)/a. For two arc measurements each at arbitrary average latitudes \varphi_i, i=1,\,2, the solution starts from an initial approximation for the equatorial radius a_0 and for the flattening f_0. The theoretical
Earth's meridional radius of curvature Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, denot ...
M_0(\varphi_i) can be calculated at the latitude of each arc measurement as: :M_0(\varphi_i)=\frac where e_0^2 = 2f_0-f_0^2. Then discrepancies between empirical and theoretical values of the radius of curvature can be formed as \delta M_i=M_i-M_0(\varphi_i). Finally, corrections for the initial equatorial radius \delta a and the flattening \delta f can be solved by means of a
system of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in t ...
formulated via
linearization In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linea ...
of M: :\delta M_i \approx \delta a (\partial M / \partial a) + \delta f (\partial M / \partial f) where the partial derivatives are: :\partial M / \partial a \approx 1 :\partial M / \partial f \approx -2 a_0 (1-1.5 \sin^2\varphi_i) Longer arcs with multiple intermediate-latitude determinations can completely determine the ellipsoid that best fits the surveyed region. In practice, multiple arc measurements are used to determine the ellipsoid parameters by the method of least squares adjustment. The parameters determined are usually the semi-major axis, a, and any of the semi-minor axis, b,
flattening Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution ( spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening ...
, or eccentricity. Regional-scale systematic effects observed in the radius of curvature measurements reflect the geoid undulation and the deflection of the vertical, as explored in astrogeodetic leveling.
Gravimetry Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest. Units of measurement G ...
is another technique for determining Earth's flattening, as per
Clairaut's theorem Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise ...
. Modern
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
no longer uses simple meridian arcs or ground triangulation networks, but the methods of satellite geodesy, especially
satellite gravimetry Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest. Units of measurement Gr ...
.


Geodetic coordinates


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 A geodetic datum or geodetic system (also: geodetic reference datum, geodetic reference system, or geodetic reference frame) is a global datum reference or reference frame for precisely representing the position of locations on Earth or other plan ...
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 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) a, total mass GM, dynamic form factor J_2 and angular velocity of rotation \omega, making the inverse flattening 1/f a derived quantity. The minute difference in 1/f seen between GRS-80 and WGS-84 results from an unintentional truncation in the latter's defining constants: while the WGS-84 was designed to adhere closely to the GRS-80, incidentally the WGS-84 derived flattening turned out to be slightly different than the GRS-80 flattening because the normalized second degree zonal harmonic gravitational coefficient, that was derived from the GRS-80 value for J_2, was truncated to eight significant digits in the normalization process.NIMA Technical Report TR8350.2, "Department of Defense World Geodetic System 1984, Its Definition and Relationships With Local Geodetic Systems", Third Edition, 4 July 199

/ref> An ellipsoidal model describes only the ellipsoid's geometry and a
normal gravity In geodesy and geophysics, theoretical gravity or normal gravity is an approximation of the true gravity on Earth's surface by means of a mathematical model representing Earth. The most common model of a smoothed Earth is a rotating Earth ellips ...
field formula to go with it. Commonly an ellipsoidal model is part of a more encompassing
geodetic datum A geodetic datum or geodetic system (also: geodetic reference datum, geodetic reference system, or geodetic reference frame) is a global datum reference or reference frame for precisely representing the position of locations on Earth or other pla ...
. 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.


See also

*
Equatorial bulge An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere. On E ...
*
Earth radius of curvature Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, den ...
*
Geodetic datum A geodetic datum or geodetic system (also: geodetic reference datum, geodetic reference system, or geodetic reference frame) is a global datum reference or reference frame for precisely representing the position of locations on Earth or other pla ...
* Great ellipse *
Meridian arc In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is to ...
*
Normal gravity In geodesy and geophysics, theoretical gravity or normal gravity is an approximation of the true gravity on Earth's surface by means of a mathematical model representing Earth. The most common model of a smoothed Earth is a rotating Earth ellips ...
*
Planetary coordinate system A planetary coordinate system is a generalization of the geographic coordinate system and the geocentric coordinate system for planets other than Earth. Similar coordinate systems are defined for other solid celestial bodies, such as in the ''selen ...
*
History of geodesy The history of geodesy deals with the historical development of measurements and representations of the Earth. The corresponding scientific discipline, '' geodesy'' ( /dʒiːˈɒdɪsi/), began in pre-scientific antiquity and blossomed during th ...
* Planetary ellipsoid


References


Bibliography

* P. K. Seidelmann (Chair), et al. (2005), “Report Of The IAU/IAG Working Group On Cartographic Coordinates And Rotational Elements: 2003,” ''Celestial Mechanics and Dynamical Astronomy'', 91, pp. 203–215. **Web address: https://astrogeology.usgs.gov/Projects/WGCCRE * ''OpenGIS Implementation Specification for Geographic information - Simple feature access - Part 1: Common architecture'', Annex B.4. 2005-11-30 **Web address: http://www.opengeospatial.org


External links


Geographic coordinate system
( SPENVIS help page)
Coordinate Systems, Frames and Datums
{{DEFAULTSORT:Earth Ellipsoid Geodesy Earth sciences