Bipolar coordinates

Not to be confused with Two-center bipolar coordinates or Biangular coordinates.
Bipolar coordinate system

Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles.[1] 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 ellipses, hyperbolas, and Cassini ovals. 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.

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.
  1. ^ Eric W. Weisstein, Concise Encyclopedia of Mathematics CD-ROM, Bipolar Coordinates, CD-ROM edition 1.0, May 20, 1999 "Bipolar Coordinates". Archived from the original on December 12, 2007. Retrieved December 9, 2006.