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Encyclopedia > Reference ellipsoid

In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. 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. It has been suggested that geodetic system be merged into this article or section. ... The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ... The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earths size and shape is to be defined. ... A geodetic network is a network of triangles which are measured exactly by techniques of terrestrial surveying or by satellite geodesy. ... Latitude, usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ... Longitude, sometimes denoted by the Greek letter Î» (lambda),[1][2] describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ... Basic Definition In geography, the elevation of a geographic location is its height above mean sea level (or some other fixed point). ...

Mathematically, a reference ellipsoid is usually an oblate (flattened) spheroid with two different axes: An equatorial radius (the semi-major axis $a,!$), and a polar radius (the semi-minor axis $b,!$). More rarely, a scalene ellipsoid with three axes (triaxial——$a_x,,a_y,,b,!$) is used, usually for modeling the smaller, irregularly shaped moons and asteroids. The polar axis here is the same as the rotational axis, and is not the magnetic or orbital pole. The geometric center of the ellipsoid is placed at the center of mass of the body being modeled, and not the barycenter in a multi-body system. An oblate spheroid is ellipsoid having a shorter axis and two equal longer axes. ... In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ... World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses IlhÃ©u das Rolas, in SÃ£o TomÃ© and PrÃ­ncipe. ... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ... For other uses of the word pole, see Pole (disambiguation). ... In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ... 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... 253 Mathilde, a C-type asteroid. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

In working with elliptic geometry, several parameters are commonly utilized, all of which are trigonometric functions of an ellipse's angular eccentricity, $o!varepsilon,!$: All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other... In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and eccentricity. ...

$o!varepsilon=arccosleft(frac{b}{a}right)=2arctanleft(sqrt{frac{a-b}{a+b}}right);,!$

Due to rotational forces, the equatorial radius is usually larger than the polar radius. This ellipticity or flattening, $f,!$, determines how close to a true sphere an oblate spheroid is, and is defined as 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. ... A sphere is a perfectly symmetrical geometrical object. ...

$f=operatorname{ver}(o!varepsilon)=2sinleft(frac{o!varepsilon}{2}right)^2=1-cos(o!varepsilon)=frac{a-b}{a}.,!$

For Earth, $f,!$ is around 1/300, and is very gradually decreasing over geologic time scales. 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 nearly 1/3 to 1/2! Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Apparent magnitude: up to -12. ... Adjectives: Jovian Atmosphere Surface pressure: 20â€“200 kPa[4] (cloud layer) Composition: ~86% Molecular hydrogen ~13% Helium 0. ... Adjectives: Saturnian Atmosphere Surface pressure: 140 kPa Composition: >93% hydrogen >5% helium 0. ... Telesto is a figure in Greek mythology. ...

Such flattening is related to the eccentricity, $e,!$, of the cross-sectional ellipse by (This page refers to eccentricity in mathematics. ...

$e^2=f(2-f)=sin(o!varepsilon)^2=frac{a^2-b^2}{a^2}.,!$

It is traditional when defining a reference ellipsoid to specify the semi-major equatorial radius $a,!$ (usually in meters) and the inverse of the flattening ratio $1/f,!$. The semi-minor polar radius is then easily derived. The metre, or meter (symbol: m) is the SI base unit of length. ...

Coordinates

A primary use of reference ellipsoids is to serve as a basis for a coordinate system of latitude (north/south), longitude (east/west), and elevation (height). For this purpose it is necessary to identify a zero meridian, which for Earth is usually the Prime Meridian. For other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the crater Airy-0. It is possible for many different coordinate systems to be defined upon the same reference ellipsoid. Latitude, usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ... Longitude, sometimes denoted by the Greek letter Î» (lambda),[1][2] describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ... Basic Definition In geography, the elevation of a geographic location is its height above mean sea level (or some other fixed point). ... On the earth, a meridian is a north-south line between the North Pole and the South Pole. ... Location of the Prime Meridian Prime Meridian in Greenwich The Prime Meridian, also known as the International Meridian or Greenwich Meridian, is the meridian (line of longitude) passing through the Royal Greenwich Observatory, Greenwich, England â€” it is the meridian at which longitude is 0 degrees. ... Airy-0 is a crater on Mars whose location defines the position of the prime meridian of that planet. ...

The longitude measures the rotational angle between the zero meridian and the measured point. By convention for the Earth, Moon, and Sun it is expressed as degrees ranging from −180° to +180° For other bodies a range of 0° to 360° is used. An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ...

The latitude measures how close to the poles or equator a point is along a meridian, and is represented as angle from −90° to +90°, where 0° is the equator. The common or geographic latitude is the angle between the equatorial plane and a line that is normal to the reference ellipsoid. Depending on the flattening, it may be slightly different from the geocentric latitude, which is the angle between the equatorial plane and a line from the center of the ellipsoid. For non-Earth bodies the terms planetographic and planetocentric are used instead. A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

The coordinates of a geodetic point are customarily stated as geodetic latitude and longitude, i.e., the direction in space of the geodetic normal containing the point, and the height h of the point over the reference ellipsoid. If these coordinates, i.e., latitude $phi,!$, longitude $lambda,!$ and height h, are given, one can compute the geocentric rectangular coordinates of the point as follows:

$X_t=[N+h]cos(phi)cos(lambda);,!$
$Y_t=[N+h]cos(phi)sin(lambda);,!$
$Z_t=[cos(o!varepsilon)^2N+h]sin(phi);,!$

where

$N=N(phi)=frac{a}{sqrt{1-(sin(phi)sin(o!varepsilon))^2}},!$

is the radius of curvature in the prime vertical. Curvature is the amount by which a geometric object deviates from being flat. ... In astronomy and astrology, the prime vertical is the vertical circle passing east and west through the zenith, and intersecting the horizon in its east and west points. ...

In contrast, extracting $phi,!$, $lambda,!$ and h from the rectangular coordinates usually requires iteration: It has been suggested that this article or section be merged with Guess value. ...

Letting $phi_c=arctan(sec(o!varepsilon)^2tan(psi_t));,!$,

` $phi_p=phi_c:;phi_c=arctan!left(frac{qquad;;a^2Z_tquad,+frac{1}{4}[N(phi_p)sin(phi_p)]^3sin(2o!varepsilon)^2}{!!!!!a^2sqrt{X_t^2+Y_t^2},-[N(phi_p)cos(phi_p)]^3sin(o!varepsilon)^2}right);,!$ Repeat until $phi_c=phi_p,!$: $phi=phi_c.,!$ `

Or, introducing the geocentric, $psi,!$, and parametric, or reduced, $beta,!$, latitudes: Latitude, usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ...

$psi_t=arctanleft(frac{Z_t}{sqrt{X_t^2+Y_t^2}}right);$ and $beta_c=arctan(sec(o!varepsilon)tan(psi_t));,!$,
` $phi_p=phi_c:;phi_c=arctan!left(frac{qquad,Z_tqquad+bsin(beta_c)^3tan(o!varepsilon)^2}{sqrt{X_t^2+Y_t^2};-acos(beta_c)^3sin(o!varepsilon)^2}right);,!$ $beta_p=beta_c:;beta_c=arctanleft(cos(o!varepsilon)tan(phi_c)right);;,!$ Repeat until $phi_c=phi_p,!$ and $beta_c=beta_p,!$: $phi=phi_c;quadbeta=beta_c;quadpsi=arctan(cos(o!varepsilon)tan(beta)).,!$ `
Once $phi,!$ is determined, then h can be isolated:
$h=sec(phi){color{white}dot{{color{black}sqrt{X_t^2+Y_t^2}}}}-N;=;csc(phi)Z_t-cos(o!varepsilon)^2N,,!$
${}_{color{white}8.}=cos(phi){color{white}dot{{color{black}sqrt{X_t^2+Y_t^2}}}},+,sin(phi)left[Z_t+sin(o!varepsilon)^2Nsin(phi)right]-N.,!$

Common reference ellipsoids for the Earth

Currently the most common reference ellipsoid used, and that used in the context of the Global Positioning System, is WGS 84. WGS 84 is the 1984 revision of the World Geodetic System. ...

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. It has been suggested that this article or section be merged with Geodetic system. ... ED 50 (European Datum 1950) is a geodetic datum which was defined after World War II for the international connection of geodetic networks. ... Over fifty GPS satellites such as this NAVSTAR have been launched since 1978. ...

The following table lists some of the most common ellipsoids:

Name Equatorial axis (m) Polar axis (m) Inverse flattening,
$1/f,!$
Clarke 1866 6 378 206.4 6 356 583.8 294.978 698 2
International 1924 6 378 388 6 356 911.9 297.0
GRS 1980 6 378 137 6 356 752.3141 298.257 222 101
WGS 1984 6 378 137 6 356 752.3142 298.257 223 563
Sphere (6371 km) 6 371 000 6 371 000 0

See Figure of the Earth for a more complete historical list. The metre or meter is a measure of length. ... Definition GRS 80, or Geodetic Reference System 1980, is a geodetic reference system consisting of a global reference ellipsoid and a gravity field model. ... WGS 84 is the 1984 revision of the World Geodetic System. ... The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earths size and shape is to be defined. ...

Ellipsoids for non-Earth bodies

Reference ellipsoids are also useful for geodetic mapping of other planetary bodies including planets, their satellites, asteroids and comet nuclei. Some well observed bodies such as the Moon and Mars now have quite precise reference ellipsoids. Apparent magnitude: up to -12. ... Adjectives: Martian Atmosphere Surface pressure: 0. ...

For rigid-surface nearly-spherical bodies, which includes all the rocky planets and many moons, ellipsoids are defined in terms of the axis of rotation and the mean surface height excluding any atmosphere. Mars is actually egg shaped, where its north and south polar radii differ by approximately 6 km, however this difference is small enough that the average polar radius is used to define its ellipsoid. The Earth's Moon is effectively spherical, having no bulge at its equator. Where possible a fixed observable surface feature is used when defining a reference meridian. In geometry, an oval or ovoid (from Latin ovum, egg) is any curve resembling an egg or an ellipse. ... A kilometer (Commonwealth spelling: kilometre), symbol: km is a unit of length in the metric system equal to 1,000 metres (from the Greek words Ï‡Î¯Î»Î¹Î± (khilia) = thousand and Î¼Î­Ï„ÏÎ¿ (metro) = count/measure). ...

For gaseous planets like Jupiter, an effective surface for an ellipsoid is chosen as the equal-pressure boundary of one bar. Since they have no permanent observable features the choices of prime meridians are made according to mathematical rules. Adjectives: Jovian Atmosphere Surface pressure: 20â€“200 kPa[4] (cloud layer) Composition: ~86% Molecular hydrogen ~13% Helium 0. ... The bar (symbol bar) and the millibar (symbol mbar, also mb) are units of pressure. ...

Small moons, asteroids, and comet nuclei frequently have irregular shapes. For some of these, such as Jupiter's Io, a scalene (triaxial) ellipsoid is a better fit than the oblate spheroid. For highly irregular bodies the concept of a reference ellipsoid may have no useful value, so sometimes a spherical reference is used instead and points identified by planetocentric latitude and longitude. Even that can be problematic for non-convex bodies, such as Eros, in that latitude and longitude don't always uniquely identify a single surface location. Atmospheric characteristics Atmospheric pressure trace Sulfur dioxide 90% Io (eye-oe, IPA: , Greek á¿™ÏŽ) is the innermost of the four Galilean moons of Jupiter. ... Look up Convex set in Wiktionary, the free dictionary. ... The asteroid 433 Eros (eer-os) was named after the Greek god of love Eros. ...

Since the Earth, like all planets, is not a perfect sphere, the radius of Earth can refer to various values. ... The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earths size and shape is to be defined. ... The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ...

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