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Two-center bipolar coordinates

For related concepts, see 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 $c_{2}$ .^[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

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

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