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Geodetic coordinates are a type of
curvilinear In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inv ...
orthogonal coordinate system In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
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 ...
based on 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 ...
''. They include geodetic latitude (north/south) , ''
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 ...
'' (east/west) , and ellipsoidal height (also known as geodetic height). The triad is also known as Earth ellipsoidal coordinates (not to be confused with '' ellipsoidal-harmonic coordinates'').


Definitions

Longitude measures the rotational
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
between the zero meridian and the measured point. By convention for the Earth, Moon and Sun, it is expressed in degrees ranging from −180° to +180°. For other bodies a range of 0° to 360° is used. For this purpose, it is necessary to identify a ''zero meridian'', which for Earth is usually the
Prime Meridian A prime meridian is an arbitrary meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian in a 360°-system) form a great ...
. For other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the crater
Airy-0 Airy-0 is a crater inside the larger Airy Crater on Mars, whose location defined the position of the prime meridian of that planet. It is about 0.5 km (0.3 mile) across and lies within the dark region Sinus Meridiani, one of the early ...
. It is possible for many different coordinate systems to be defined upon the same reference ellipsoid. Geodetic latitude measures how close to the poles or equator a point is along a meridian, and is represented as an angle from −90° to +90°, where 0° is the equator. The ''geodetic 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 latitude 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 '' se ...
'' and '' planetocentric latitude'' are used instead. Ellipsoidal height (or ellipsoidal
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
), also known as geodetic height (or geodetic altitude), is the distance between the point of interest and the ellipsoid surface, evaluated along the ellipsoidal normal vector; it is defined as a signed distance such that points inside the ellipsoid have negative height.


Geodetic vs. geocentric coordinates

Geodetic latitude and '' geocentric latitude'' have different definitions. Geodetic latitude is defined as the angle between the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can also ...
ial plane and the
surface normal In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
at a point on the ellipsoid, whereas geocentric latitude is defined as the angle between the equatorial plane and a radial line connecting the centre of the ellipsoid to a point on the surface (see figure). When used without qualification, the term latitude refers to geodetic latitude. For example, the latitude used in
geographic coordinates The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various ...
is geodetic latitude. The standard notation for geodetic latitude is . There is no standard notation for geocentric latitude; examples include , , . Similarly, geodetic altitude is defined as the height above the ellipsoid surface, normal to the ellipsoid; whereas '' geocentric altitude'' is defined as the distance to the reference ellipsoid along a radial line to the geocenter. When used without qualification, as in aviation, the term
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
refers to geodetic altitude (possibly with further refinements, such as in
orthometric height The orthometric height is the vertical distance ''H'' along the plumb line from a point of interest to a reference surface known as the ''geoid'', the vertical datum that approximates mean sea level. Orthometric height is one of the scientific fo ...
s). Geocentric altitude is typically used in
orbital mechanics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of ...
(see orbital altitude). If the impact of Earth's
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 ...
is not significant for a given application (e.g.,
interplanetary spaceflight Interplanetary spaceflight or interplanetary travel is the crewed or uncrewed travel between stars and planets, usually within a single planetary system. In practice, spaceflights of this type are confined to travel between the planets of the ...
), the
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 ...
may be simplified as a
spherical Earth Spherical Earth or Earth's curvature refers to the approximation of figure of the Earth as a sphere. The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of Greek philosophers. ...
, in which case the geocentric and geodetic latitudes equal and the latitude-dependent geocentric radius simplifies to a global mean Earth's radius (see also:
spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
).


Conversion

Given geodetic coordinates, one can compute the '' geocentric Cartesian coordinates'' of the point as follows: :\begin X &= \big( N + h\big)\cos\cos \\ Y &= \big( N + h\big)\cos\sin \\ Z &= \left( \frac N + h\right)\sin \end where and are the equatorial radius (
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 ...
) and the polar radius (
semi-minor 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 lo ...
), respectively. is the '' prime vertical radius of curvature'', function of latitude : :N = \frac, In contrast, extracting , and from the rectangular coordinates usually requires
iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
as and are mutually involved through :A guide to coordinate systems in Great Britain. This is available as a pdf document at [] Appendices B1, B2Osborne, P (2008)
The Mercator Projections
Section 5.4
:\lambda = \operatorname(Y,X). :h=\frac - N, :\phi = \arctan\left( (Z / p)/(1 - e^2 N / (N + h)) \right). where p = \sqrt. More sophisticated methods are
available In reliability engineering, the term availability has the following meanings: * The degree to which a system, subsystem or equipment is in a specified operable and committable state at the start of a mission, when the mission is called for at a ...
.


See also

* Local geodetic coordinates *
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 ...
*
Geodesics on an ellipsoid The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an ''oblate ellipsoid'', a slightly flattened sphere. A ''geodes ...
*
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 ...


References

{{reflist Geodesy Orthogonal coordinate systems