Editing Bipolar coordinates (section)

{{short description|2-dimensional orthogonal coordinate system based on Apollonian circles}}
{{distinguish|Two-center bipolar coordinates|Biangular coordinates}}
[[File:Iso1.svg|thumb|right|350px|Bipolar coordinate system]]

'''Bipolar coordinates''' are a two-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] based on the [[Apollonian circles]].<ref name=bip>Eric W. Weisstein, '''Concise Encyclopedia of Mathematics CD-ROM''', ''Bipolar Coordinates'', CD-ROM edition 1.0, May 20, 1999<!-- Bot generated title --> {{Cite web |url=http://bbs.sachina.pku.edu.cn/Stat/Math_World/math/b/b233.htm |title=Bipolar Coordinates |access-date=December 9, 2006 |archive-date=December 12, 2007 |archive-url=https://web.archive.org/web/20071212005309/http://bbs.sachina.pku.edu.cn/Stat/Math_World/math/b/b233.htm |url-status=dead }}</ref> There is also a third system, based on two poles ([[biangular coordinates]]).  

The term "bipolar" is further used on occasion to describe other curves having two singular points (foci), such as [[ellipse]]s, [[hyperbola]]s, and [[Cassini oval]]s.  However, the term ''bipolar coordinates'' is reserved for the coordinates described here, and never used for systems associated with those other curves, such as [[elliptic coordinates]].

[[File:Bipolar_coordinates.svg|thumb|right|350px|Geometric interpretation of the bipolar coordinates. The angle σ is formed by the two foci and the point '''P''', whereas ''τ'' is the logarithm of the ratio of distances to the foci. The corresponding circles of constant ''σ'' and ''τ'' are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal.]]