co-ordinates (See coordinate)
adj : of
equal importance, rank, or degree n : a number that identifies a
position relative to an axis [syn: co-ordinate]
2 bring into common action, movement, or condition; "coordinate the painters, masons, and plumbers"; "coordinate his actions with that of his colleagues"; "coordinate our efforts"
3 be co-ordinated; "These activities co-ordinate well"
4 bring (components or parts) into proper or desirable coordination correlation; "align the wheels of my car"; "ordinate similar parts" [syn: align, ordinate] [also: co-ordinating, co-ordinates, co-ordinated, co-ordinate]
- third-person singular of coordinate
In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. "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 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:
- Continuous functions on topological spaces;
- Smooth functions on smooth manifolds;
- Measurable functions on measure spaces;
- Rational functions on algebraic varieties;
- Linear functionals on vector spaces.
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.
ExamplesThe 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 = (r1, ..., rn)
- r1, ..., rn.
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 oneIn 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 node, 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.
TransformationsA 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:
- such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
- such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)
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 usedSome coordinate systems are the following:
- The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for three-dimensional flat space, uses three numbers representing distances.
- Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
- The polar
- Circular coordinate system (commonly referred to as the polar coordinate system) represents a point in the plane by an angle and a distance from the origin.
- Cylindrical coordinate system represents a point in space by an angle, a distance from the origin and a height.
- Spherical coordinate system represents a point in space with two angles and a distance from the origin.
- Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates.
- Generalized coordinates are used in the Lagrangian treatment of mechanics.
- Canonical coordinates are used in the Hamiltonian treatment of mechanics.
- Intrinsic coordinates describe a point upon a curve by the length of the curve to that point and the angle the tangent to that point makes with the x-axis.
- Parallel coordinates visualise a point in n-dimensional space as a polyline connecting points on n vertical lines.
A list of common coordinate systemsThe following coordinate systems all have the properties of being orthogonal coordinate systems, that is the coordinate surfaces meet at right angles.
Geographical systemsGeography and cartography utilize various geographic coordinate systems to map positions on the 3-dimensional globe to a 2-dimensional document.
The Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) coordinate systems both use a metric-based cartesian grid laid out on a conformally projected surface to locate positions on the surface of the Earth. The UTM system is not a single map projection but a series of map projections, one for each of sixty zones. The UPS system is used for the polar regions, which are not covered by the UTM system.
During medieval times, the stereographic coordinate system was used for navigation purposes. The stereographic coordinate system was superseded by the latitude-longitude system, and more recently, the Global Positioning System.
Although no longer used in navigation, the stereographic coordinate system is still used in modern times to describe crystallographic orientations in the field of materials science.
Astronomical systemsCoordinate systems on the sphere are particularly important in astronomy: see astronomical coordinate systems.
coordinates in Afrikaans: Koördinaatstelsel
coordinates in Asturian: Coordenada
coordinates in Bulgarian: Координата
coordinates in Catalan: Coordenada
coordinates in Czech: Soustava souřadnic
coordinates in Danish: Koordinatsystem
coordinates in German: Koordinatensystem
coordinates in Modern Greek (1453-): Σύστημα συντεταγμένων
coordinates in Spanish: Sistema de coordenadas
coordinates in Esperanto: Koordinatsistemo
coordinates in French: Système de coordonnées
coordinates in Scottish Gaelic: Siostaman cho-chomharran
coordinates in Galician: Sistema de coordenadas
coordinates in Korean: 좌표계
coordinates in Italian: Sistema di riferimento
coordinates in Hebrew: קואורדינטות
coordinates in Luxembourgish: Koordinate-system
coordinates in Lithuanian: Koordinačių sistema
coordinates in Dutch: Coördinaat
coordinates in Japanese: 座標
coordinates in Norwegian: Koordinatsystem
coordinates in Polish: Układ współrzędnych
coordinates in Portuguese: Sistema de coordenadas
coordinates in Russian: Система координат
coordinates in Slovenian: Koordinatni sistem
coordinates in Albanian: Sistemi koordinativ
coordinates in Slovak: Sústava súradníc
coordinates in Finnish: Koordinaatisto
coordinates in Swedish: Koordinatsystem
coordinates in Tamil: பகுமுறை வடிவவியல்
coordinates in Turkish: Küresel koordinat sistemi
coordinates in Chinese: 坐標系統
coordinates in Ukrainian: Системи координат
Cartesian coordinates, abscissa, altitude, azimuth, boundaries, bounds, bourns, circumference, circumscription, compass, confines, cylindrical coordinates, declination, edges, equator coordinates, fringes, latitude, limitations, limits, longitude, marches, metes, metes and bounds, ordinate, outlines, outskirts, pale, parameters, perimeter, periphery, polar coordinates, right ascension, skirts, verges