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Two-center bipolar coordinates

From Wikipedia, the free encyclopedia
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

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When the centers are at and , the transformation to Cartesian coordinates from two-center bipolar coordinates is

[1]

Transformation to polar coordinates

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

where is the distance between the poles (coordinate system centers).

Applications

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Polar plotters use two-center bipolar coordinates to describe the drawing paths required to draw a target image.

See also

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References

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  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.