thumb|right|350px|Bipolar coordinate system
Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. There are also other systems, based on two poles (biangular coordinates, two-center bipolar 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.
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.
Definition
The system is based on two foci F<sub>1</sub> and F<sub>2</sub>. Referring to the figure at right, the σ-coordinate of a point P equals the angle F<sub>1</sub> P F<sub>2</sub>, and the τ-coordinate equals the natural logarithm of the ratio of the distances d<sub>1</sub> and d<sub>2</sub>:
:<math>
\tau = \ln \frac{d_1}{d_2}.
</math>
If, in the Cartesian system, the foci are taken to lie at (−a, 0) and (a, 0), the coordinates of the point P are
:<math>
x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}, \qquad y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma}.
</math>
The coordinate τ ranges from <math>-\infty</math> (for points close to F<sub>1</sub>) to <math>\infty</math> (for points close to F<sub>2</sub>). The coordinate σ is only defined modulo 2π, and is best taken to range from −π to π, by taking it as the negative of the acute angle F<sub>1</sub> P F<sub>2</sub> if P is in the lower half plane.
Proof that coordinate system is orthogonal
The equations for x and y can be combined to give
:<math>
x + i y = a i \cot\left( \frac{\sigma + i \tau}{2}\right)
</math>
- Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.