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) |

(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) |

(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

1999-05-26