A coordinate system is a system for assigning an 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 members of any of many other rings or ring-like algebraic structures.
Although any specific coordinate system is useful for numerical calculations in a given space, the space itself is independent of any particular choice of coordinates. Some choices of coordinate systems lead to paradoxes, for example, close to a black hole, but can be understood by changing the choice of coordinate system.
A coordinate transformation is a conversion from one system to another, to describe the same space.
An example of a coordinate system is to describe a point P in the Euclidean space Rn by an n-tuple P=(r1,...,rn) of real numbers r1,...,rn.
These numbers r1,...,rn are called the coordinates of the point P.
If a subset S of an Euclidean space is mapped continuously onto another topological space, this defines coordinates in the image of S.
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.
- For any finite-dimensional vector space and any basis, the coefficients of the basis vectors can be used as coordinates. Changing the basis is a coordinate transformation, a linear transformation that can be summarized by a matrix, and is computationally the same as a mapping of points to other points keeping the bases the same: e.g. in 2D:
- a clockwise rotation is a mapping of points to other points which changes the coordinates the same as keeping the points in place but rotating the coordinate axes anti-clockwise.
- an expansion by a factor two in the direction of one basis vector is a mapping of points to other points which changes the coordinates the same as keeping the points in place but halving the magnitude of that basis vector (in both cases the corresponding coordinate is doubled).
- a mapping of points to other points which distorts a rectangle to a parallellogram changes the coordinates the same as keeping the points in place but changing the basis vectors from being two sides of that parallellogram to perpendicular ones, two sides of that rectangle.
- The polar coordinate systems
- Celestial coordinate system
- extragalactic coordinate systems
- Binary coordinate system