Two-center bipolar coordinates

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Not to be confused with bipolar coordinates. ‹See Tfd›
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[which?] scientific applications(e.g. To calculate the electric field of a dipole on a plane).[2][3]

Cartesian coordinates[edit]

Cartesian coordinates and polar coordinates.

The transformation to Cartesian coordinates (x,\ y) from two-center bipolar coordinates (r_1,\ r_2) is

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).[1]

Polar coordinates[edit]

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

See also[edit]


  1. ^ a b 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.