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 surfa ...

, 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. 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 pol ...

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 lette ...

and UTM are common horizontal position representations. The horizontal position has two degrees of freedom, and thus two parameters are sufficient to uniquely describe such a position. However, similarly to the use of Euler angles 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 pol ...

and

. 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 C ...

, 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

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fighter aircraft while crossing this meridian.

''n''-vector

''n''-vector is a three parameter

non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...

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. The vector is decomposed in an Earth centered earth fixed coordinate system. 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 In mathematics, vector algebra may mean: * Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space. * The algebraic operations in vector calculus, namely the specific additional stru ...

, 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 might be defined with the

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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, lineari ...

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 may be required. The alignment along the north and east directions is not possible at the

, 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 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, 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.

References

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