Bispherical coordinates

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.

in bipolar coordinates remain points (on the

-axis, the axis of rotation) in the bispherical coordinate system.

The most common definition of bispherical coordinates

( τ , σ , ϕ )

coordinate of a point

equals the angle

coordinate equals the natural logarithm of the ratio of the distances

to the foci The coordinates ranges are -∞ <

Surfaces of constant

correspond to intersecting tori of different radii that all pass through the foci but are not concentric.

The surfaces of constant

are non-intersecting spheres of different radii that surround the foci.

tori are centered in the

The formulae for the inverse transformation are: where

The scale factors for the bispherical coordinates

are equal whereas the azimuthal scale factor equals Thus, the infinitesimal volume element equals and the Laplacian is given by Other differential operators such as

can be expressed in the coordinates

( σ , τ )

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

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.

Illustration of bispherical coordinates, which are obtained by rotating a two-dimensional bipolar coordinate system about the axis joining its two foci. The foci are located at distance 1 from the vertical z -axis. The red self-intersecting torus is the σ=45° isosurface, the blue sphere is the τ=0.5 isosurface, and the yellow half-plane is the φ=60° isosurface. The green half-plane marks the x - z plane, from which φ is measured. The black point is located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0.841, -1.456, 1.239).