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 mathworld|urlname=BipolarCoordinates|title=Bipolar coordinates] This system is very useful in some scientific applications [ [http://www.physics.utah.edu/~rprice/AREA51DOCS/paperIIa.pdf R. Price, The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods.] ] [ [http://arxiv.org/abs/gr-qc/0502034v1 The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.] ] It should not be confused with so-called bipolar coordinates.

Cartesian coordinates

Transformation to Cartesian coordinates $(x,\; y)$ from two-center bipolar coordinates $(r\_1,\; r\_2)$:$x\; =\; frac\{r\_1^2-r\_2^2\}\{4a\}$

:$y\; =\; pm\; frac\{1\}\{4a\}sqrt\{16a^2r\_1^2-(r\_1^2-r\_2^2+4a^2)^2\}$where the centers of this coordinate system are at (+"a", 0) and (-"a", 0).

Polar coordinates

To polar coordinates from two-center bipolar coordinates:$r\; =\; sqrt\{frac\{r\_1^2+r\_2^2-2a^2\}\{2$

:$heta\; =\; arctan\; left\; [\; frac\{sqrt\{8a^2(r\_1^2+r\_2^2\; -\; 2a^2)-(r\_1^2\; -\; r\_2^2)^2\{r\_1^2\; -\; r\_2^2\}\; ight]\; ,!$ Where 2"a" is the distance between the poles (coordinate system centers).

References