Reference ellipsoid

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Flattened sphere OblateSpheroid.PNG
Flattened sphere

In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, which is the truer, imperfect figure of the Earth, or other planetary body, as opposed to a perfect, smooth, and unaltered sphere, which factors in the undulations of the bodies' gravity due to variations in the composition and density of the interior, as well as the subsequent flattening caused by the centrifugal force 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, longitude, and elevation are defined.

Contents

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 definitions.

Ellipsoid parameters

In 1687 Isaac Newton 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 of revolution, generated by an ellipse rotated around its minor diameter; a shape which he termed an oblate spheroid. [1] [2]

In geophysics, geodesy, and related areas, the word 'ellipsoid' is understood to mean 'oblate ellipsoid of revolution', and the older term 'oblate spheroid' is hardly used. [3] [4] 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. The semi-major axis of the ellipse, a, becomes the equatorial radius of the ellipsoid: the semi-minor axis of the ellipse, b, 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) a and the flattening f, defined as:

That is, f is the amount of flattening at each pole, relative to the radius at the equator. This is often expressed as a fraction 1/m; m = 1/f then being the "inverse flattening". A great many other ellipse parameters are used in geodesy but they can all be related to one or two of the set a, b and f.

A great many ellipsoids have been used to model the Earth in the past, with different assumed values of a and b 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 has f close to 1/300 (more precisely, 1/298.257223563, by definition), corresponding to a difference of the major and minor semi-axes of approximately 21 km (13 miles) (more precisely, 21.3846857548205 km). For comparison, Earth's Moon is even less elliptical, with a flattening of less than 1/825, while Jupiter is visibly oblate at about 1/15 and one of Saturn's triaxial moons, Telesto, is highly flattened, with f between 1/3 and 1/2 (meaning that the polar diameter is between 50% and 67% of the equatorial.

Geodetic coordinates

Geodetic coordinates P(F,l,h) Geodetic coordinates.svg
Geodetic coordinates P(ɸ,λ,h)

Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude (north/south) ϕ, longitude (east/west) λ, and ellipsoidal height h (also known as geodetic height [5] ).

The triad is also known as Earth ellipsoidal coordinates [6] (not to be confused with ellipsoidal-harmonic coordinates ).

Historical Earth ellipsoids

Equatorial (a), polar (b) and mean Earth radii as defined in the 1984 World Geodetic System revision (not to scale) WGS84 mean Earth radius.svg
Equatorial (a), polar (b) and mean Earth radii as defined in the 1984 World Geodetic System revision (not to scale)

Currently the most common reference ellipsoid used, and that used in the context of the Global Positioning System, is the one defined by WGS 84.

Traditional reference ellipsoids or geodetic datums are defined regionally and therefore non-geocentric, e.g., ED50. Modern geodetic datums are established with the aid of GPS and will therefore be geocentric, e.g., WGS 84.

See also

Notes

  1. Heine, George (September 2013). "Euler and the Flattening of the Earth". Math Horizons. 21 (1): 25–29. doi:10.4169/mathhorizons.21.1.25. S2CID   126412032.
  2. Choi, Charles Q. (12 April 2007). "Strange but True: Earth Is Not Round". Scientific American. Retrieved 4 May 2021.
  3. Torge, W (2001) Geodesy (3rd edition), published by de Gruyter, ISBN   3-11-017072-8
  4. Snyder, John P. (1993). Flattening the Earth: Two Thousand Years of Map Projections. University of Chicago Press. p. 82. ISBN   0-226-76747-7.
  5. National Geodetic Survey (U.S.).; National Geodetic Survey (U.S.) (1986). Geodetic Glossary. NOAA technical publications. U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Ocean Service, Charting and Geodetic Services. p. 107. Retrieved 2021-10-24.
  6. Awange, J.L.; Grafarend, E.W.; Paláncz, B.; Zaletnyik, P. (2010). Algebraic Geodesy and Geoinformatics. Springer Berlin Heidelberg. p. 156. ISBN   978-3-642-12124-1 . Retrieved 2021-10-24.

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Geographic coordinate system System to specify locations on Earth

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Spheroid Surface formed by rotating an ellipse

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Ellipsoid Quadric surface that looks like a deformed sphere

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.

Earth radius Distance from the Earth surface to a point near its center

Earth radius 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 6,378 km (3,963 mi) to a minimum of nearly 6,357 km (3,950 mi).

World Geodetic System Geodetic reference system

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Figure of the Earth Size and shape used to model the Earth for geodesy

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Geodetic datum Reference frame for measuring location

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Flattening Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution

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 is f and its definition in terms of the semi-axes of the resulting ellipse or ellipsoid is

Spatial reference system System to specify locations on Earth

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Earth-centered, Earth-fixed coordinate system 3-D coordinate system centered on the Earth

The Earth-centered, Earth-fixed coordinate system, also known as the Geocentric coordinate system, is a cartesian spatial reference system that represents locations in the vicinity of the Earth as X, Y, and Z measurements from its center of mass. Its most common use is in tracking the orbits of satellites and in satellite navigation systems for measuring locations on the surface of the Earth, but it is also used in applications such as tracking crustal motion.

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.

Earth 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.

Vertical datum Reference surface for vertical positions

In geodesy, surveying, hydrography and navigation, vertical datum or altimetric datum, is a reference surface for vertical positions, such as the elevations of Earth-bound features and altitudes of satellite orbits and in aviation. In planetary science, vertical datums are also known as zero-elevation surface or zero-level reference.

Geodetic coordinates

Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude (north/south) ϕ, longitude (east/west) λ, and ellipsoidal heighth. The triad is also known as Earth ellipsoidal coordinates.

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 selenographic coordinates for the Moon. The coordinate systems for almost all of the solid bodies in the Solar System were established by Merton E. Davies of the Rand Corporation, including Mercury, Venus, Mars, the four Galilean moons of Jupiter, and Triton, the largest moon of Neptune.

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