Coordinate system

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See coordinates (mathematics) for a more elementary introduction to this topic.

In mathematics and applications, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. "Numbers" in many cases means real numbers, but, depending on context, can mean complex numbers 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 coordinates will also transform under rings or groups; a particularly famous example in this case are the Lie groups.

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 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.

Contents

1 Examples
2 Transformations
3 Systems commonly used
4 Astronomical systems
5 See also

Examples

An example of a coordinate system is to describe a point P in the Euclidean space Rⁿ by an n-tuple

P = (r₁, ..., r_n)

of real numbers

r₁, ..., r_n.

These numbers r₁, ..., r_n 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:

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 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.
Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
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.
* 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.
** Geographic coordinate system
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.

Astronomical systems

Celestial coordinate system
* Horizontal coordinate system
* Equatorial coordinate system - based on Earth rotation
* Ecliptic coordinate system - based on Solar System rotation
* Galactic coordinate system - based on Milky Way rotation
extragalactic coordinate systems
* supergalactic coordinate system - based on plane of local supercluster of galaxies
* comoving coordinates - valid to particle horizon

Coordinate system

Examples

Transformations

Systems commonly used

Astronomical systems

See also