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

			(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)