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A position representation is the parameters used to express a position relative to a reference. When representing positions relative to the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
, it is often most convenient to represent ''
vertical position Vertical position or vertical location, also known as vertical level or simply level, is a position along a vertical direction above or below a given vertical datum (reference level). Vertical distance or vertical separation is the distance betw ...
'' (height or depth) separately, and to use some other parameters to represent horizontal position. There are also several applications where only the horizontal position is of interest, this might e.g. be the case for ships and ground vehicles/cars. It is a type of
geographic coordinate system 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 ...
. There are several options for horizontal position representations, each with different properties which makes them appropriate for different applications.
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 UTM are common horizontal position representations. The horizontal position has two
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, and thus two parameters are sufficient to uniquely describe such a position. However, similarly to the use of
Euler angles The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> Th ...
as a formalism for representing rotations, using only the minimum number of parameters gives singularities, and thus three parameters are required for the horizontal position to avoid this.


Latitude and longitude

The most common horizontal position representation is
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 ...
and
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 ...
. The parameters are intuitive and well known, and are thus suited for communicating a position to humans, e.g. using a position plot. However, latitude and longitude should be used with care in mathematical expressions (including calculations in computer programs). The main reason is the singularities at the
Poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in ...
, which makes longitude undefined at these points. Also near the poles the latitude/longitude grid is highly non-linear, and several errors may occur in calculations that are sufficiently accurate on other locations. Another problematic area is the meridian at ± 180° longitude, where the longitude has a discontinuity, and hence specific program code must often be written to handle this. An example of the consequences of omitting such code is the crash of the navigation systems of twelve
F-22 Raptor The Lockheed Martin F-22 Raptor is an American single-seat, twin-engine, all-weather stealth tactical fighter aircraft developed for the United States Air Force (USAF). As the result of the USAF's Advanced Tactical Fighter (ATF) program, th ...
fighter aircraft while crossing this meridian.


''n''-vector

''n''-vector is a three parameter non-singular horizontal position representation that can replace latitude and longitude. Geometrically, it is a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
which is normal to the
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 ...
. The vector is decomposed in an Earth centered earth fixed
coordinate system 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 sig ...
. It behaves the same at all Earth positions, and it holds the mathematical one-to-one property. The vector formulation makes it possible to use standard 3D vector algebra, and thus ''n''-vector is well-suited for mathematical calculations, e.g. adding, subtracting, interpolating and averaging positions. Using three parameters, ''n''-vector is inconvenient for communicating a position directly to humans and before showing a position plot, a conversion to latitude/longitude might be needed.


Local flat Earth assumption

When carrying out several calculations within a limited area, a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
might be defined with the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
at a specified Earth-fixed position. The origin is often selected at the surface of the reference ellipsoid, with the ''z''-axis in the vertical direction. Hence (three dimensional) position vectors relative to this coordinate frame will have two horizontal and one vertical parameter. The axes are typically selected as North-East-Down or East-North-Up, and thus this system can be viewed as a
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 the meridians and parallels. For small areas a local coordinate system can be convenient for relative positioning, but with increasing (horizontal) distances, errors will increase and repositioning of the
tangent point In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
may be required. The alignment along the north and east directions is not possible at the
Poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in ...
, and near the Poles these directions might have significant errors (here the linearization is valid only in a very small area).


Grid reference system

Instead of one local Cartesian grid, that needs to be repositioned as the position of interest moves, a fixed set of
map projections In cartography, map projection is the term used to describe a broad set of Transformation (function) , transformations employed to represent the two-dimensional curved Surface (mathematics), surface of a globe on a Plane (mathematics), plane. In ...
covering the Earth can be defined.


UTM

UTM is one such system, dividing the Earth into 60 longitude zones (and with UPS covering the
Polar regions The polar regions, also called the frigid zones or polar zones, of Earth are the regions of the planet that surround its geographical poles (the North and South Poles), lying within the polar circles. These high latitudes are dominated by floa ...
). UTM is widely used, and the coordinates approximately corresponds to meters north and east. However, as a set of map-projections it has inherent
distortions In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal ...
, and thus most calculations based on UTM will not be exact. The crossing of zones gives additional complexity.


Comparison

When deciding which parameters to use for representing position in a specific application, there are several properties that should be considered. The following table gives a summary of what to consider.


See also

* Rotation formalisms in three dimensions *
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) , ''longitude'' (east/west) , and ellipsoidal height (also known as geo ...
*
Geodetic 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 pla ...
* Plane coordinates


References

{{Use dmy dates, date=July 2011 Navigation Geodesy Geographic coordinate systems Geographic position