Two-center bipolar coordinates

Not to be confused with bipolar coordinates.

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 $c_{2}$. This system is very useful in some scientific applications.

Cartesian coordinates

Cartesian coordinates and polar coordinates.

The transformation to Cartesian coordinates $(x,\ y)$ from two-center bipolar coordinates $(r_{1},\ r_{2})$ is

$x={\frac {r_{1}^{2}-r_{2}^{2}}{4a}}$

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

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

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

See also

• Biangular coordinates