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

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

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

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

$r={\sqrt {{\frac {r_{1}^{2}+r_{2}^{2}-2a^{2}}{2}}}}$

$\theta =\arctan \left({\frac {{\sqrt {r_{1}^{4}-8a^{2}r_{1}^{2}-2r_{1}^{2}r_{2}^{2}-(4a^{2}-r_{2}^{2})^{2}}}}{r_{2}^{2}-r_{1}^{2}}}\right)\,\!$

where $2a$ 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.