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In Mathematics and its applications, a coordinate system is a system for assigning an ''n''- Tuple of Number s or Scalars to each Point in an ''n''- Dimension al space. "Scalars" in many cases means Real Number s, but, depending on context, can mean Complex Number s 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 '''charts''', 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:
The coordinates on a space transform naturally (by Pullback ) under the Group of Automorphism s 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 Polar Coordinates (''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. EXAMPLES The prototypical example of a coordinate system is the Cartesian coordinate system, which describes a point ''P'' in the Euclidean Space Rn by an N-tuple P of real numbers r These numbers ''r''1, ..., ''rn'' are called the ''coordinates'' of the point ''P''. If a subset ''S ''of a Euclidean space is mapped Continuously onto another topological space, this defines coordinates in the image of S. That can be called a parametrization of the image, since it assigns numbers to points. That correspondence is unique only if the mapping is Bijective . The system of assigning Longitude and Latitude to geographical locations is a coordinate system. In this case the ''parametrization'' fails to be unique at the north and south poles. Defining a coordinate system based on another one In Geometry and Kinematics , coordinate systems are used not only to describe the (linear) position of points, but also to describe the Angular Position of axes, planes, and Rigid Bodies . In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the body, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation Matrix , which includes, in its three columns, the Cartesian Coordinates of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three Unit Vectors aligned with those axes. TRANSFORMATIONS A coordinate transformation is a conversion from one system to another, to describe the same space. With every Bijection from the space to itself two coordinate transformations can be associated:
For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to -3, so that the coordinate of each point becomes 3 more. SYSTEMS COMMONLY USED Some coordinate systems are the following:
ASTRONOMICAL SYSTEMS Coordinate systems on the sphere are particularly important in astronomy: see Astronomical Coordinate Systems . LESS COMMON COORDINATE SYSTEMS The following coordinate systems have special uses. They all have the properties of being Orthogonal Coordinate Systems , that is the Coordinate Surfaces meet at right angles.
GEOGRAPHICAL SYSTEMS Geography and Cartography utilize various Geographic Coordinate System s to map positions on the 3-dimensional globe to a 2-dimensional document. The coordinate system. SEE ALSO EXTERNAL LINKS |
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