- 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 andlongitude (height above sea level) define a coordinate system near to the surface of the earth.altitude 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 inastronomy for describing the location of objects in the (night) sky: seeCelestial coordinate system s 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 system s in which a point is identified by a distance from some fixed feature in space and one or moresubtended angle s. They are the most common systems ofcurvilinear 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 thepolar 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 theCartesian 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

* $0leq\; heta<360^circ$ (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 scaleand 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 directionA minor change is changing the range $0leq\; heta<360^circ$ to e.g. $-180^circ<\; hetaleq180^circ$

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. 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 (Argand diagram 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 theCartesian coordinate system ,

* $0leq\{r\}$ (radius ) is the distance between the "z"-axis and the point P,

* $0leq\; heta<360^circ$ (azimuth orlongitude ) 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.

**pherical coordinates**The

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 thespherical coordinate system Cartesian coordinate system ,

* $0leq\; ho$ (radius ) is the distance between the point $P$ and the origin,

* $0leqphileq\; 180^circ$ (zenith ,colatitude orpolar angle ) is the angle between the $z$-axis and the line from the origin to the point P, and

* $0leq\; hetaleq\; 360^circ$ (azimuth orlongitude ) 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 $heta$ 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".**ee also***

coordinate rotation

*coordinate system

*curvilinear coordinates

*nabla in cylindrical and spherical coordinates

*parabolic coordinates

*Vector (geometric)

*vector fields in cylindrical and spherical coordinates **pherical coordinates***

celestial coordinate system

*Euler angles

*gimbal lock

*spherical harmonic

*yaw, pitch and roll **External links*** Frank Wattenberg has made some attractive animations illustrating [

*http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/spherical/body.htm spherical*] and [*http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/cylindrical/body.htm 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.

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