Two-center bipolar coordinates

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In mathematics, two-center bipolar coordinates is a coordinate system, based on two coordinates which give distances from two fixed centers^[1]. This system is very useful in some scientific applications ^[2]^[3]

[edit] Cartesian coordinates

To Cartesian coordinates from two-center bipolar coordinates^[1]

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

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

[edit] Polar coordinates

To polar coordinates from two-center bipolar coordinates

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

$\theta = \arctan \left[ \sqrt{\frac{8c^2(r_1^2+r_2^2-2c^2)}{r_1^2-r_2^2}-1}\right]$

Where 2c is the distance between the poles.

[edit] References

^ ^a ^b Weisstein, Eric W., Bipolar coordinates at MathWorld.
^ R. Price, "[http://www.physics.utah.edu/~rprice/AREA51DOCS/paperIIa.pdf The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods.]"
^ [http://arxiv.org/PS_cache/gr-qc/pdf/0502/0502034.pdf The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method]