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

  1. ^ a b Eric W. Weisstein, Bipolar coordinates at 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.


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