Editing Analytic geometry (section)

==Coordinates==
{{Main|Coordinate system}}
[[File:Cartesian-coordinate-system.svg|thumb|right|250px|Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.]]

In analytic geometry, the [[Euclidean plane|plane]] is given a coordinate system, by which every [[point (geometry)|point]] has a pair of [[real number]] coordinates. Similarly, [[Euclidean space]] is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following:<ref name=stewart>[[James Stewart (mathematician)|Stewart, James]] (2008). ''Calculus: Early Transcendentals'', 6th ed., Brooks Cole Cengage Learning. {{ISBN|978-0-495-01166-8}}</ref>

===Cartesian coordinates (in a plane or space)===
{{main|Cartesian coordinate system}}
The most common coordinate system to use is the [[Cartesian coordinate system]], where each point has an ''x''-coordinate representing its horizontal position, and a ''y''-coordinate representing its vertical position.  These are typically written as an [[ordered pair]] (''x'',&nbsp;''y'').  This system can also be used for three-dimensional geometry, where every point in [[Euclidean space]] is represented by an [[Tuple|ordered triple]] of coordinates (''x'',&nbsp;''y'',&nbsp;''z'').

===Polar coordinates (in a plane)===
{{main|Polar coordinate system}}
In [[polar coordinates]], every point of the plane is represented by its distance ''r'' from the origin and its [[angle]] ''θ'', with ''θ'' normally measured counterclockwise from the positive ''x''-axis. Using this notation, points are typically written as an ordered pair (''r'', ''θ'').  One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae: <math display="block">x = r\, \cos\theta,\, y = r\, \sin\theta; \, r = \sqrt{x^2+y^2},\, \theta = \arctan(y/x).</math> This system may be generalized to three-dimensional space through the use of [[Cylindrical coordinates|cylindrical]] or [[Spherical coordinates|spherical]] coordinates.

===Cylindrical coordinates (in a space)===
{{main|Cylindrical coordinate system}}
In [[cylindrical coordinates]], every point of space is represented by its height ''z'', its [[radius]] ''r'' from the ''z''-axis and the [[angle]] ''θ'' its projection on the ''xy''-plane makes with respect to the horizontal axis.

===Spherical coordinates (in a space)===
{{main|Spherical coordinate system}}
In spherical coordinates, every point in space is represented by its distance ''ρ'' from the origin, the [[angle]] ''θ'' its projection on the ''xy''-plane makes with respect to the horizontal axis, and the angle ''φ'' that it makes with respect to the ''z''-axis. The names of the angles are often reversed in physics.<ref name=stewart/>