This article describes some of the common coordinate systems that appear in elementary mathematics. For advanced topics, please refer to coordinate system. For more background, see Cartesian coordinate system.
The coordinates of a point are the components of a tuple of numbers used to represent the location of the point in the plane or space. A coordinate system is a plane or space where the origin and axes are defined so that coordinates can be measured.
Cartesian coordinates
In the twodimensional Cartesian coordinate system, a point P in the xyplane is represent by a tuple of two components (x,y).
 x is the signed distance from the yaxis to the point P, and
 y is the signed distance from the xaxis to the point P.
In the threedimensional Cartesian coordinate system, a point P in the xyzspace is represent by a tuple of three components (x,y,z).
 x is the signed distance from the yzplane to the point P,
 y is the signed distance from the xzplane to the point P, and
 z is the signed distance from the xyplane to the point P.
For advanced topics, please refer to Cartesian coordinate system.
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.
The term polar coordinates often refers to circular coordinates (twodimensional). Other commonly used polar coordinates are cylindrical coordinates and spherical coordinates (both threedimensional).
Circular coordinates
The circular coordinate system, often referred to simply as the polar coordinate system, is a twodimensional polar coordinate system, defined by an origin, O, and a semiinfinite 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 xaxis (the right half of the xaxis).
In the circular coordinate system, a point P is represented by a tuple of two components (r,θ). Using terms of the Cartesian coordinate system,

(radius) is the distance from the origin to the point P, and

(azimuth) is the angle between the positive xaxis and the line from the origin to the point P.
Cylindrical coordinates
The cylindrical coordinate system is a threedimensional polar coordinate system.
In the cylindrical coordinate system, a point P is represented by a tuple of three components (r,θ,h). Using terms of the Cartesian coordinate system,

(radius) is the distance between the zaxis and the point P,

(azimuth or longitude) is the angle between the positive xaxis and the line from the origin to the point P projected onto the xyplane, and
 h (height) is the signed distance from xyplane 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 threedimensional polar coordinate system.
In the spherical coordinate system, a point P is represented by a tuple of three components (ρ,φ,θ). Using terms of the Cartesian coordinate system,

(radius) is the distance between the point P and the origin,

(colatitude or polar angle) is the angle between the zaxis and the line from the origin to the point P, and

(azimuth or longitude) is the angle between the positive xaxis and the line from the origin to the point P projected onto the xyplane.
NB: The above convention is the standard used by American mathematicians and American calculus textbooks. However, most physicists, engineers, and nonAmerican mathematicians interchange the symbols φ and θ above, using φ to denote the azimuth and θ the colatitude. One should be very careful to note which convention is being used by a particular author. It should be noted that, regardless of how one labels the coordinates, one argument against the conventional American mathematical definition is the fact that it produces a lefthanded coordinate system, rather than the usual convention of a righthanded coordinate system.
The spherical coordinate system also involves some redundancy; φ loses its significance if ρ = 0, and θ loses its significance if ρ = 0 or φ = 0 or .
To construct a point from its spherical coordinates: from the origin, go ρ along the positive zaxis, rotate φ about yaxis toward the direction of the positive xaxis, and rotate θ about the zaxis toward the direction of the positive yaxis.
Spherical coordinates are useful in analyzing systems that are symmetrical about a point; a sphere that has the Cartesian equation x^{2} + y^{2} + z^{2} = c^{2} has the very simple equation ρ = c in spherical coordinates.
Spherical coordinates are the natural coordinates for physical situations where there is spherical symmetry. In such a situation, one can describe waves using spherical harmonics. Another application is ergonomic design, where ρ is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out.
The concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates.
See also: Celestial coordinate system
Conversion between coordinate systems
Cartesian and circular
where u_{0} is the Heaviside step function with u_{0}(0) = 0 and sgn is the signum function. Here the u_{0} and sgn functions are being used as "logical" switches which are used as shorthand substitutes for several if ... then statements. Some computer languages include a bivariate arctangent function atan2(y,x) which finds the value for θ in the correct quadrant given x and y.
Cartesian and cylindrical
Cartesian and spherical
Cylindrical and spherical
See also
 For spherical coordinates:
 Credit to original articles:
External links
 Frank Wattenberg has made some nice 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.
Last updated: 05072005 17:49:24
Last updated: 05132005 07:56:04