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 and in bipolar coordinates remain points (on the -axis, the axis of rotation) in the bispherical coordinate system.
The most common definition of bispherical coordinates is
where the coordinate of a point equals the angle and the coordinate equals the natural logarithm of the ratio of the distances and to the foci
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. The centers of the constant- spheres lie along the -axis, whereas the constant- tori are centered in the plane.
The formulae for the inverse transformation are:
The scale factors for the bispherical coordinates and 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 and can be expressed in the coordinates 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.
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