## Bipolar Coordinates

Bipolar coordinates are a 2-D system of coordinates. There are two commonly defined types of bipolar coordinates, the first of which is defined by

 (1) (2)

where , . The following identities show that curves of constant and are Circles in -space.
 (3)

 (4)

The Scale Factors are
 (5) (6)

The Laplacian is
 (7)

Laplace's Equation is separable.

Two-center bipolar coordinates are two coordinates giving the distances from two fixed centers and , sometimes denoted and . For two-center bipolar coordinates with centers at ,

 (8) (9)

Combining (8) and (9) gives
 (10)

Solving for Cartesian Coordinates and gives
 (11) (12)

Solving for Polar Coordinates gives
 (13) (14)

References

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

© 1996-9 Eric W. Weisstein
1999-05-26