Two-center bipolar coordinates

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For related concepts, see 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, and .[1] This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane).[2][3]

Transformation to Cartesian coordinates[edit]

Cartesian coordinates and polar coordinates.

The transformation to Cartesian coordinates from two-center bipolar coordinates is

where the centers of this coordinate system are at and .[1]

Transformation to polar coordinates[edit]

When x>0 the transformation to polar coordinates from two-center bipolar coordinates is

where 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.