Bispherical coordinates are a threedimensional orthogonal coordinate system that results from rotating the twodimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci F_{1} and F_{2} in bipolar coordinates remain points (on the zaxis, the axis of rotation) in the bispherical coordinate system.
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The most common definition of bispherical coordinates (σ,τ,φ) is
where the σ coordinate of a point P equals the angle F_{1}PF_{2} and the τ coordinate equals the natural logarithm of the ratio of the distances d_{1} and d_{2} to the foci
Surfaces of constant σ correspond to intersecting tori of different radii
that all pass through the foci but are not concentric. The surfaces of constant τ are nonintersecting spheres of different radii
that surround the foci. The centers of the constantτ spheres lie along the zaxis, whereas the constantσ tori are centered in the xy plane.
The scale factors for the bispherical coordinates σ and τ 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 (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.
