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What are coordinates?

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IN GEOGRAPHY

A geographic coordinate system enables every location on the Earth to be specified in three coordinates, using mainly a spherical coordinate system.

The Earth is not a sphere, but an irregular shape approximating an ellipsoid; the challenge is to define a coordinate system that can accurately state each topographical feature as an unambiguous set of numbers.

Positions are determined with latitude and longitude.

IN MATHEMATICS

A coordinate is a number that determines the location of a point along some line or curve. A list of two, three, or more coordinates can be used to determine the location of a point on a surface, volume, or higher-dimensional domain.

For example, the longitude is a coordinate which determines the position of a point along the Earth's equator, and latitude is another coordinate that defines a poisition along a meridian. The pair of coordinates consising of a latitude and a longitude determines a point on the surface of the Earth.

A systematic method of assigning such a coordinate list to each point in the domain is called a coordinate system. There is an infinitude of coordinate systems that one could define for any domain, and many that are used in specific contexts. The usual latitude-longitude coordinate system, for example, is widely used in geography and astronomy. The Cartesian coordinate system for the Euclidean plane and Euclidean space is the basis of analytic geometry.

Coordinates are typically named after the system used to assign them; thus one says "the Cartesian coordinates of a point p" to mean "the coordinates assigned to p by a Cartesian coordinate system".

In mathematics and its applications, a coordinate (or co-ordinate) system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. This concept is part of the theory of manifolds.[1] "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring. For complicated spaces, it is often not possible to provide one consistent coordinate system for the entire space. In this case, a collection of coordinate systems, called graphs, are put together to form an atlas covering the whole space. A simple example (which motivates the terminology) is the surface of the earth.

Although a specific coordinate system is useful for numerical calculations in a given space, the space itself is considered to exist independently of any particular choice of coordinates. From this point of view, a coordinate on a space is simply a function from the space (or a subset of the space) to the scalars. When the space has additional structure, one restricts attention to the functions which are compatible with this structure. Examples include:

   * Continuous functions on topological space;
   * Smooth functions on smooth manifolds;
   * Measurable functions on measure spaces;
   * Rational functions on algebraic varieties;
   * Linear functionals on vector spaces.

The coordinates on a space transform naturally (by pullback) under the group of automorphisms of the space, and the set of all coordinates is a commutative ring called the coordinate ring of the space.

In informal usage, coordinate systems can have singularities: these are points where one or more of the coordinates is not well-defined. For example, the origin in the polar coordinate system (r,θ) on the plane is singular, because although the radial coordinate has a well-defined value (r = 0) at the origin, θ can be any angle, and so is not a well-defined function at the origin.

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