The simplest example of a coordinate system is the identification of points on a line with real numbers using the *number line*. In this system, an arbitrary point

*O* (the

*origin*) is chosen on a given line. The coordinate of a point

*P* is defined as the signed distance from

*O* to

*P*, where the signed distance is the distance taken as positive or negative depending on which side of the line

*P* lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.

^{[4]}
### Cartesian coordinate system

The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines.

In three dimensions, three mutually orthogonal planes are chosen and the three coordinates of a point are the signed distances to each of the planes.^{[5]} This can be generalized to create *n* coordinates for any point in *n*-dimensional Euclidean space.

Depending on the direction and order of the coordinate axes, the three-dimensional system may be a right-handed or a left-handed system. This is one of many coordinate systems.

### Polar coordinate system

Another common coordinate syst

In three dimensions, three mutually orthogonal planes are chosen and the three coordinates of a point are the signed distances to each of the planes.^{[5]} This can be generalized to create *n* coordinates for any point in *n*-dimensional Euclidean space.

Depending on the direction and order of the coordinate axes, the three-dimensional system may be a right-handed or a left-handed system. This is one of many coordinate systems.

### Polar coordinate system

Another comm

Depending on the direction and order of the coordinate axes, the three-dimensional system may be a right-handed or a left-handed system. This is one of many coordinate systems.

Another common coordinate system for the plane is the *polar coordinate system*.^{[6]} A point is chosen as the *pole* and a ray from this point is taken as the *polar axis*. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is *r* for given number *r*. For a given pair of coordinates (*r*, θ) there is a single point, but any point is represented by many pairs of coordinates. For example, (*r*, θ), (*r*, θ+2π) and (−*r*, θ+π) are all polar coordinates for the same point. The pole is represented by (0, θ) for any value of θ.

### Cylindrical and spherical coordinate systems

Main articles:

[7] Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (

*r*,

*z*) to polar coordinates (

*ρ*,

*φ*) giving a triple (

*ρ*,

*θ*,

*φ*).

^{[8]}
### Homogeneous coordinate system

A point in the plane may be represented in *homogeneous coordinates* by a triple (*x*, *y*, *z*) where *x*/*z* and *y*/*z* are the Cartesian coordinates of the

A point in the plane may be represented in *homogeneous coordinates* by a triple (*x*, *y*, *z*) where *x*/*z* and *y*/*z* are the Cartesian coordinates of the point.^{[9]} This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the use of infinity. In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values.

### Other commonly used systems

Some other common coordinate systems are the following: