Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. 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.

The system is based on two foci F₁ and F₂. Referring to the figure at right, the σ-coordinate of a point P equals the angle F₁ P F₂, and the τ-coordinate equals the natural logarithm of the ratio of the distances d₁ and d₂:

If, in the Cartesian system, the foci are taken to lie at (−a, 0) and (a, 0), the coordinates of the point P are

The coordinate τ ranges from ${\displaystyle -\infty }$ (for points close to F₁) to ${\displaystyle \infty }$ (for points close to F₂). 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 F₁ P F₂ if P is in the lower half plane.

The equations for x and y can be combined to give

or

This equation shows that σ and τ are the real and imaginary parts of an analytic function of x+iy (with logarithmic branch points at the foci), which in turn proves (by appeal to the general theory of conformal mapping) (the Cauchy-Riemann equations) that these particular curves of σ and τ intersect at right angles, i.e., it is an orthogonal coordinate system.