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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property 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. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how 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 Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, measurable functions are Well-behaved functions between measurable spaces. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

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. The notion of pullback in Mathematics is a fundamental one It refers to two different but related processes precomposition and fiber-product In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety

In informal usage, coordinate systems can have singularities: these are points where one or more of the coordinates is not well-defined. In Mathematics, the term well-defined is used to specify that a certain concept or object (a function, a property, a relation, etc 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. In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by

Examples

The Cartesian coordinate system in the plane.

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 = (r₁, . In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple . . , r_n)

of real numbers

r₁, . . . , r_n.

These numbers r₁, . . . , r_n are called the coordinates linear polynomials 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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output 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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

The system of assigning longitude and latitude to geographical locations is a coordinate system. Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the 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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Kinematics ( Greek κινειν, kinein, to move is a branch of Classical mechanics which describes the motion of objects without This article deals with orientation of reference axes or frames In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected 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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane 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. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length

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:

• 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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

Systems commonly used

Some 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. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane
• Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves. Curvilinear coordinates are a Coordinate system for the Euclidean space based on some transformation that converts the standard Cartesian coordinate system to a coordinate
• The polar coordinate systems:
  • 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. In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by
  • Cylindrical coordinate system represents a point in space by an angle, a distance from the origin and a height. The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually
  • Spherical coordinate system represents a point in space with two angles and a distance from the origin. In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial
• Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates. In Geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century are a way to assign six Homogenous coordinates to each line In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations
• Generalized coordinates are used in the Lagrangian treatment of mechanics. By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system
• Canonical coordinates are used in the Hamiltonian treatment of mechanics. In Mathematics and Classical mechanics, canonical coordinates are particular sets of coordinates on the Phase space, or equivalently on the Cotangent Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.
• 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. Parallel coordinates is a common way of visualizing High-dimensional Geometry and analyzing Multivariate data. In Mathematics, an n -dimensional space is a Topological space whose Dimension is n (where n is a fixed Natural

A list of common coordinate systems

The following coordinate systems all have the properties of being orthogonal coordinate systems, that is the coordinate surfaces meet at right angles. In Mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 q 2. The coordinate surfaces of a three dimensional Coordinate system are the Surfaces on which a particular coordinate of the system is constant while the coordinate

• Orthogonal coordinates
• Two dimensional orthogonal coordinate systems

• Three dimensional orthogonal coordinate systems

Geographical systems

Geography and cartography utilize various geographic coordinate systems to map positions on the 3-dimensional globe to a 2-dimensional document. In Mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 q 2. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by Parabolic coordinates are a two-dimensional orthogonal Coordinate system in which the Coordinate lines are Confocal Parabolas A three-dimensional two-center bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal Coordinate system. In Mathematics, biangular coordinates are a Coordinate system for the plane where A and B are two fixed points and the position In Mathematics, two-center bipolar coordinates is a coordinate system based on two coordinates which give distances from two fixed centers C1 and C2 In Mathematics, hyperbolic coordinates are a useful method of locating points in Quadrant I of the Cartesian plane \{(x y \:\ x > 0\ Elliptic coordinates are a two-dimensional orthogonal Coordinate system in which the Coordinate lines are confocal Ellipses and Hyperbolae In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial Parabolic coordinates are a two-dimensional orthogonal Coordinate system in which the Coordinate lines are Confocal Parabolas A three-dimensional Parabolic cylindrical coordinates are a three-dimensional orthogonal Coordinate system that results from projecting the two-dimensional parabolic coordinate Paraboloidal coordinates are a three-dimensional orthogonal Coordinate system (\lambda \mu \nu that generalizes the two-dimensional parabolic Oblate spheroidal coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional elliptic coordinate system Prolate spheroidal coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional elliptic coordinate system Ellipsoidal coordinates are a three-dimensional orthogonal Coordinate system (\lambda \mu \nu that generalizes the two-dimensional elliptic coordinate Elliptic cylindrical coordinates are a three-dimensional orthogonal Coordinate system that results from projecting the two-dimensional elliptic coordinate Toroidal coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional bipolar coordinate system Bispherical coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional bipolar coordinate system Bipolar cylindrical coordinates are a three-dimensional orthogonal Coordinate system that results from projecting the two-dimensional bipolar coordinate Conical coordinates are a three-dimensional orthogonal Coordinate system consisting of concentric spheres (described by their radius r and by two Geography (from Greek γεωγραφία - geografia) is the study of the Earth and its lands features inhabitants and phenomena A geographic coordinate system enables every location on the Earth to be specified in three coordinates using mainly a spherical coordinate system.

The Global Positioning System uses the WGS84 coordinate system. Basic concept of GPS operation A GPS receiver calculates its position by carefully timing the signals sent by the constellation of GPS Satellites high above the Earth The World Geodetic System defines a reference frame for the earth for use in Geodesy and Navigation.

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 Universal Transverse Mercator ( UTM) Coordinate system is a grid-based method of specifying locations on the surface of the Earth The Universal Polar Stereographic (UPS coordinate system is used in conjunction with the Universal Transverse Mercator (UTM coordinate system to locate positions on the surface 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 systems

Coordinate systems on the sphere are particularly important in astronomy: see astronomical coordinate systems. Astronomical coordinate systems are Coordinate systems used in astronomy to describe the location of objects in the sky and in the universe

See also

• active and passive transformation
• frame of reference
• Galilean transformation

External links

• Hexagonal Coordinate System
• Coordinates of a point Interactive tool to explore coordinates of a point