Editing Polar coordinate system (section)

{{Short description|Coordinates comprising a distance and an angle}}
[[Image:Examples of Polar Coordinates.svg|thumb|Points in the polar coordinate system with pole ''O'' and polar axis ''L''. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,{{nbsp}}60°). In blue, the point (4,{{nbsp}}210°).]]

In [[mathematics]], the '''polar coordinate system''' specifies a given [[point (mathematics)|point]] in a [[plane (mathematics)|plane]] by using a distance and an angle as its two [[coordinate system|coordinates]]. These are
*the point's distance from a reference point called the ''pole'', and
*the point's direction from the pole relative to the direction of the ''polar axis'', a [[ray (geometry)|ray]] drawn from the pole.

The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''[[azimuth]]''.<ref name="brown">{{Cite book |last=Brown |first=Richard G. |url=https://archive.org/details/advancedmathemat00rich_0 |title=Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis |publisher=McDougal Littell |year=1997 |isbn=0-395-77114-5 |editor-last=Andrew M. Gleason |location=Evanston, Illinois}}</ref> The pole is analogous to the origin in a [[Cartesian coordinate system]].

Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as [[spiral]]s. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates.

The polar coordinate system is extended to three dimensions in two ways: the [[cylindrical coordinate system]] adds a second distance coordinate, and the [[spherical coordinate system]] adds a second angular coordinate.

[[Grégoire de Saint-Vincent]] and [[Bonaventura Cavalieri]] independently introduced the system's concepts in the mid-17th century, though the actual term ''polar coordinates'' has been attributed to [[Gregorio Fontana]] in the 18th century. The initial motivation for introducing the polar system was the study of [[circular motion|circular]] and [[orbital motion]].