Question

Please explain in the most simple way possible what Bipolar coordinates are and what each coordinate represent in them compared to cartesian coordinates. A reference book would be great too.

Answer 1

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 , .

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 ,

			(3)
			(4)

Combining (3) and (4) gives

(5)

Solving for Cartesian coordinates and gives

			(6)

Reference:

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