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, ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$.[1] This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane).[2][3]

For related concepts, see Bipolar coordinates.

Two-center bipolar coordinates.

Transformation to Cartesian coordinates

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

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

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

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

Transformation to polar coordinates

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

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

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

where ${\displaystyle 2a}$ 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.