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Bipolar Coordinates

Bipolar coordinates are a two-dimensional system of coordinates. There are two commonly defined types of bipolar coordinates, the first of which is defined by

where u in [0,2pi), v in (-infty,infty). The following identities show that curves of constant u and v are circles in xy-space.

x^2+(y-acotu)^2=a^2csc^2u

(3)

(x-acothv)^2+y^2=a^2csch^2v.

(4)

The scale factors are

The Laplacian is

del ^2=((coshv-cosu)^2)/(a^2)((partial^2)/(partialu^2)+(partial^2)/(partialv^2)).

(7)

Laplace's equation is separable.

Two-center bipolar coordinates are two coordinates giving the distances from two fixed centers r_1 and r_2, sometimes denoted r and r^'. For two-center bipolar coordinates with centers at (+/-c,0),

Combining (8) and (9) gives

r_1^2-r_2^2=4cx.

(10)

Solving for Cartesian coordinates x and y gives

Solving for polar coordinates gives

See also

Bipolar Cylindrical Coordinates, Polar Coordinates

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References

Lockwood, E. H. "Bipolar Coordinates." Ch. 25 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 186-190, 1967.

Referenced on Wolfram|Alpha

Bipolar Coordinates

Cite this as:

Weisstein, Eric W. "Bipolar Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BipolarCoordinates.html

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