Coordinate system

In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of 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 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 cobbled together to form an atlas for the space.

When the space has some additional algebraic structure, then the co-ordinates will also transform under rings or groupss; 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. Other coordinate systems do not, however, have a clear notion of origin. For example polar coordinates (r,θ) assign the point (x,y) = (0,0) the value r = 0 but θ 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 = (r₁,...,r_n)

r₁,...,r_n.

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.

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: