Two-center bipolar coordinates

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Two-center 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, C1 and C2 [1] This system is very useful in some scientific applications[2][3] It should not be confused with so-called bipolar coordinates.

[edit] Cartesian coordinates

Cartesian coordinates and polar coordinates.
Cartesian coordinates and polar coordinates.

Transformation to Cartesian coordinates (x,\ y) from two-center bipolar coordinates (r_1,\ r_2)[1]

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

where the centers of this coordinate system are at (+a, 0) and (-a, 0).

[edit] Polar coordinates

To polar coordinates from two-center bipolar coordinates

r = \sqrt{\frac{r_1^2+r_2^2-2a^2}{2}}
\theta = \arctan \left[ \frac{\sqrt{8a^2(r_1^2+r_2^2 - 2a^2)-(r_1^2 - r_2^2)^2}}{r_1^2 - r_2^2}\right]\,\!

Where 2a is the distance between the poles (coordinate system centers).

[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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