Two-center bipolar coordinates

For related concepts, see 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 scientific applications (e.g. calculating the electric field of a dipole on a plane).^[2][3]

Transformation to Cartesian coordinates

Cartesian coordinates and polar coordinates.

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

$x = \frac{r_2^2-r_1^2}{4a}$

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

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

Transformation to polar coordinates

When x>0 the transformation to polar coordinates from two-center bipolar coordinates is

$r = \sqrt{\frac{r_1^2+r_2^2-2a^2}{2}}$

$\theta = \arctan \left(\frac{\sqrt{r_1^4-8a^2r_1^2-2r_1^2r_2^2-(4a^2-r_2^2)^2}}{r_2^2-r_1^2} \right)\,\!$

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

See also

• Biangular coordinates
• Lemniscate of Bernoulli
• Oval of Cassini
• Cartesian oval
• Ellipse

References

1. Weisstein, Eric W., "Bipolar coordinates", MathWorld.
2. R. Price, The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods.
3. The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.