# Bipolar coordinates

The numbers and which are connected with the Cartesian orthogonal coordinates and by the formulas

where . The coordinate lines are two families of circles with poles and and the (half-c)ircles orthogonal with these .

Figure: b016470a

The Lamé coefficients are:

The Laplace operator is:

Bipolar coordinates in space (bispherical coordinates) are the numbers and , which are connected with the orthogonal Cartesian coordinates and by the formulas:

where . The coordinate surfaces are spheres (), the surfaces obtained by rotation of arcs of circles () and half-planes passing through the -axis. The system of bipolar coordinates in space is formed by rotating the system of bipolar coordinates on the plane around the -axis.

The Lamé coefficients are:

The Laplace operator is:

#### References

[1] | E. Madelung, "Die mathematischen Hilfsmittel des Physikers" , Springer (1957) |

#### Comments

#### References

[a1] | O. Veblen, J.H.C. Whitehead, "The foundations of differential geometry" , Cambridge Univ. Press (1932) |

**How to Cite This Entry:**

Bipolar coordinates. D.D. Sokolov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Bipolar_coordinates&oldid=11655