Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci in bipolar coordinates-axis, the axis of rotation) in the bispherical coordinate system.

Definition

is

coordinate equals the natural logarithm to the foci

.

Coordinate surfaces

correspond to intersecting tori of different radii

are non-intersecting spheres of different radii

plane.

Inverse formulae

The formulae for the inverse transformation are:

Scale factors

are equal

whereas the azimuthal scale factor equals

Thus, the infinitesimal volume element equals

and the Laplacian is given by

by substituting the scale factors into the general formulae found in orthogonal coordinates.

Applications

The classic applications of bispherical coordinates are in solving partial differential equations, e.g., Laplace's equation, for which bispherical coordinates allow a separation of variables. However, the Helmholtz equation is not separable in bispherical coordinates. A typical example would be the electric field surrounding two conducting spheres of different radii.

References

Bibliography

External links

• MathWorld description of bispherical coordinates