Coordinates (mathematics)

Coordinates are numbers which describe the location of points in a plane or in space. For example, the height above sea level is a coordinate which is useful for describing points near the surface of the earth. A coordinate system, in a plane or in space, is a systematic method of assigning a pair or a triple of numbers to each point in the plane or in space (respectively) which describe its position uniquely. For example, the triple consisting of latitude, longitude and altitude (height above sea level) define a coordinate system near to the surface of the earth.

Coordinates may be defined in more general contexts. For example, if one is not interested in height, then latitude and longitude form a coordinate system on the surface of the earth, which is (approximately) a sphere. Coordinates such as these are also important in astronomy for describing the location of objects in the (night) sky: see Celestial coordinate systems for further examples. For simplicity, however, this article will restrict attention to coordinate systems in a plane and in space.

Cartesian coordinates

In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represented by a pair of numbers $(x,\; y)$.

• $x$ is the signed distance from the y-axis to the point P, and
• $y$ is the signed distance from the x-axis to the point P.

In the three-dimensional Cartesian coordinate system, a point P in the xyz-space is represented by a triple of numbers $(x,\; y,\; z)$.

• $x$ is the signed distance from the yz-plane to the point P,
• $y$ is the signed distance from the xz-plane to the point P, and
• $z$ is the signed distance from the xy-plane to the point P.

Polar coordinates

The polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles. They are the most common systems of curvilinear coordinates.

The term polar coordinates often refers to circular coordinates (two-dimensional). Other commonly used polar coordinates are cylindrical coordinates and spherical coordinates (both three-dimensional).

Circular coordinates

The circular coordinate system, commonly referred to as the polar coordinate system, is a two-dimensional polar coordinate system, defined by an origin, O, and a ray (or semi-infinite line) L leading from this point. L is also called the polar axis. In terms of the Cartesian coordinate system, one usually picks O to be the origin (0,0) and L to be the positive x-axis (the right half of the x-axis).

In the circular coordinate system, a point P is represented by a pair (r, θ). Using terms of the Cartesian coordinate system,

• $0leq\{r\}$ (radius) is the distance from the origin to the point P, and
• (azimuth) is the angle between the positive x-axis and the line from the origin to the point P.

Possible coordinate transformations from one circular coordinate system to another include:

• change of zero direction (such as making north the zero direction)
• changing from the angle increasing counterclockwise to increasing clockwise or conversely (as in a compass)
• change of scale

and combinations. More generally, transformations of the corresponding Cartesian coordinates can be translated into transformations from one circular coordinate system to another by basically transforming to Cartesian coordinates, transforming those, and transforming back to circular coordinates. This is e.g. needed for:

• change of origin
• change of scale in one direction

A minor change is changing the range to e.g.

Circular coordinates can be convenient in situations where only the distance, or only the direction to a fixed point matters, rotations about a point, etc. (by taking the special point as the origin).

A complex number can be viewed as a point or a position vector on a plane, the so-called complex plane or Argand diagram. Here the circular coordinates are r = |z|, called the absolute value or modulus of z, and φ = arg(z), called the complex argument of z. These coordinates (mod-arg form) are especially convenient for complex multiplication and powers.

Cylindrical coordinates

The cylindrical coordinate system is a three-dimensional polar coordinate system. In the cylindrical coordinate system, a point P is represented by a triple (r, θ, h). Using terms of the Cartesian coordinate system,

• $0leq\{r\}$ (radius) is the distance between the z-axis and the point P,
• (azimuth or longitude) is the angle between the positive x-axis and the line from the origin to the point P projected onto the xy-plane, and
• $h$ (height) is the signed distance from xy-plane to the point P.

Note: some sources use $z$ for $h$; there is no "right" or "wrong" convention, but it is necessary to be aware of the convention being used.

Cylindrical coordinates involve some redundancy; θ loses its significance if r = 0.

Cylindrical coordinates are useful in analyzing systems that are symmetrical about an axis. For example the infinitely long cylinder that has the Cartesian equation $x^2+y^2=c^2$ has the very simple equation $r=c$ in cylindrical coordinates.

Spherical coordinates

The spherical coordinate system is a three-dimensional polar coordinate system. In this coordinate system, a point $P$ is represented by a triple (ρ,θ,φ). Using terms of the Cartesian coordinate system,

• $0leqrho$ (radius) is the distance between the point $P$ and the origin,
• $0leqphileq\; 180^circ$ (zenith, colatitude or polar angle) is the angle between the $z$-axis and the line from the origin to the point P, and
• $0leqthetaleq\; 360^circ$ (azimuth or longitude) is the angle between the positive $x$-axis and the line from the origin to the point P projected onto the $xy$-plane.

There are different conventions for the exact letters used for the angles (for example, physics sources typically use $phi$ for the longitude and $theta$ for the colatitude).

The concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates.

Transformations between coordinate systems

Because there are many different possible coordinate systems for describing points in the plane or in space, it is important to understand how they are related. Such relations are described by coordinate transformations which give formulae for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (x,y) and polar coordinates (r,θ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x = r cos θ and y = r sin θ.

See also

• coordinate rotation
• coordinate system
• curvilinear coordinates
• nabla in cylindrical and spherical coordinates
• parabolic coordinates
• Vector (geometric)
• vector fields in cylindrical and spherical coordinates

Spherical coordinates

• celestial coordinate system
• Euler angles
• gimbal lock
• spherical harmonic
• yaw, pitch and roll

External links

• Frank Wattenberg has made some attractive animations illustrating spherical and cylindrical coordinate systems.
• http://www.physics.oregonstate.edu/bridge/papers/spherical.pdf is a description of the different conventions in use for naming components of spherical coordinates, along with a proposal for standardizing this.
• http://mathworld.wolfram.com/SphericalCoordinates.html is a very thorough review of all possible partial derivatives etc.