# Two-center bipolar coordinates

Not to be confused with bipolar coordinates. ‹See Tfd›
Two-center bipolar coordinates.

In mathematics, two-center bipolar coordinates is a coordinate system, based on two coordinates which give distances from two fixed centers, $c_1$ and $c_2$.[1] This system is very useful in some[which?] scientific applications(e.g. To calculate the electric field of a dipole on a plane).[2][3]

## Cartesian coordinates

Cartesian coordinates and polar coordinates.

The transformation to Cartesian coordinates $(x,\ y)$ from two-center bipolar coordinates $(r_1,\ r_2)$ is

$x = \frac{r_1^2-r_2^2}{4a}$
$y = \pm \frac{1}{4a}\sqrt{16a^2r_1^2-(r_1^2-r_2^2+4a^2)^2}$

where the centers of this coordinate system are at $(+a,\ 0)$ and $(-a,\ 0)$.[1]

## Polar coordinates

The transformation to polar coordinates from two-center bipolar coordinates is

$r = \sqrt{\frac{r_1^2+r_2^2-2a^2}{2}}$
$\theta = \arctan \left( \frac{\sqrt{8a^2(r_1^2+r_2^2 - 2a^2)-(r_1^2 - r_2^2)^2}}{r_1^2 - r_2^2}\right)\,\!$

where $2 a$ is the distance between the poles (coordinate system centers).