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In Mathematics as applied to Geometry , Physics or Engineering , a coordinate system is a system for assigning a Tuple of Number s to each Point in an ''n''- Dimension al space. "Numbers" in many cases means Real Number s, but, depending on context, can mean Complex Number s or elements of some other Field . If the space or Manifold is curved, it may not be possible to provide one consistent coordinate system for the entire space. In this case, a set of coordinate systems, called '''charts''', are put together to form an Atlas covering the whole space. When the space has some additional Algebraic Structure , then the co-ordinates will also transform under Rings or Group s; a particularly famous example in this case are the Lie Group s. Although any 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. By convention the origin of the coordinate system in Cartesian coordinates is the point (0, 0, ..., 0), which may be assigned to any given point of Euclidean Space . In Physics , a Scalar is a Physical Quantity which assumes a single value which is a "real" quantity independent of the coordinate system. In this sense coordinates are not scalars (although, of course, a scalar field can be defined which for one particular coordinate system corresponds to a particular coordinate). In some coordinate systems some points are associated with multiple tuples of coordinates, e.g. the origin in Polar Coordinates : ''r'' = 0 but θ can be any angle. EXAMPLES An example of a coordinate system is to describe 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. 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, 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. If the bijection is an Involution , e.g. a Reflection , then the two associated coordinate transformations are the same, e.g., in 1D, ''x'' becomes 7-''x''. Examples of bijections include the invertible Affine Transformation s. Of these, the Similarity Transformations preserve distance ratios, hence magnitude ratios, and angles, so that e.g. decomposition of a vector into perpendicular components is preserved. In the case that vector quantities are considered in relation to position and displacement, as in Vector Field s, a similarity transformation of space is normally accompanied by a corresponding ''linear'' transformation of the other vector quantities, to preserve angles between e.g. a force and a displacement, hence preserve e.g. dot products up to scaling. The transformation is linear because, as opposed to position, most vector quantities have a natural origin, e.g. zero force. However, velocity translation preserves the laws of motion, because an Inertial Frame Of Reference is preserved. (But if there is e.g. air-resistance, a velocity translation will affect tacitly assumed stationarity of air.) In diagrams showing vectors of multiple physical dimensions, e.g. forces and displacements, scaling of one kind of vectors does not affect relevant properties: a force and a displacement having the same length in a diagram has no particular significance. SINGULARITIES Some choices of coordinate systems may lead to paradoxes, for example, close to a Black Hole , but can be understood by changing the choice of coordinate system. At an actual Mathematical Singularity the coordinate system breaks down. SYSTEMS COMMONLY USED Some coordinate systems are the following:
ASTRONOMICAL SYSTEMS
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