Bispherical coordinates are a three-dimensionalorthogonalcoordinate system that results from rotating the two-dimensionalbipolar coordinate system about the axis that connects the two foci. Thus, the twofoci${\displaystyle F_1}$ and${\displaystyle F_2}$ inbipolar coordinates remain points (on the${\displaystyle z}$-axis, the axis of rotation) in the bispherical coordinate system.

The most common definition of bispherical coordinates${\displaystyle (\tau ,\sigma ,\varphi )}$ is

where the${\displaystyle \sigma }$ coordinate of a point${\displaystyle P}$ equals the angle${\displaystyle F_{1}P{F}_2}$ and the${\displaystyle \tau }$ coordinate equals thenatural logarithm of the ratio of the distances${\displaystyle d_1}$ and${\displaystyle d_2}$ to the foci

${\displaystyle \tau =\ln \frac{d_1}{d_2}}$

The coordinates ranges are −∞ <${\displaystyle \tau }$ < ∞, 0 ≤${\displaystyle \sigma }$ ≤${\displaystyle \pi }$ and 0 ≤${\displaystyle \varphi }$ ≤ 2${\displaystyle \pi }$.

Surfaces of constant${\displaystyle \sigma }$ correspond to intersecting tori of different radii

${\displaystyle z^2+{\left(\sqrt{x^2+y^2}-a\cot \sigma \right)}^2=\frac{a^2}{{\sin }^2\sigma }}$

that all pass through the foci but are not concentric. The surfaces of constant${\displaystyle \tau }$ are non-intersecting spheres of different radii

${\displaystyle \left(x^2+y^2\right)+{\left(z-a\coth \tau \right)}^2=\frac{a^2}{{\sinh }^2\tau }}$

that surround the foci. The centers of the constant-${\displaystyle \tau }$ spheres lie along the${\displaystyle z}$-axis, whereas the constant-${\displaystyle \sigma }$ tori are centered in the${\displaystyle xy}$ plane.

The formulae for the inverse transformation are:

where$R=\sqrt{x^2+y^2+z^2}$ and$Q=\sqrt{{\left(R^2+a^2\right)}^2-{\left(2az\right)}^2}.$

The scale factors for the bispherical coordinates${\displaystyle \sigma }$ and${\displaystyle \tau }$ are equal

${\displaystyle h_{\sigma }=h_{\tau }=\frac{a}{\cosh \tau -\cos \sigma }}$

whereas the azimuthal scale factor equals

${\displaystyle h_{\varphi }=\frac{a\sin \sigma }{\cosh \tau -\cos \sigma }}$

Thus, the infinitesimal volume element equals

${\displaystyle dV=\frac{a^3\sin \sigma }{{\left(\cosh \tau -\cos \sigma \right)}^3}d\sigma d\tau d\varphi }$

and the Laplacian is given by

Other differential operators such as${\displaystyle \mathrm{\nabla }\cdot \mathbf{F}}$ and${\displaystyle \mathrm{\nabla }\times \mathbf{F}}$ can be expressed in the coordinates${\displaystyle (\sigma ,\tau )}$ by substituting the scale factors into the general formulae found inorthogonal coordinates.

The classic applications of bispherical coordinates are in solvingpartial differential equations, e.g.,Laplace's equation, for which bispherical coordinates allow aseparation of variables. However, theHelmholtz equation is not separable in bispherical coordinates. A typical example would be theelectric field surrounding two conducting spheres of different radii.

• Morse PM, Feshbach H (1953).Methods of Theoretical Physics, Parts I and II. New York: McGraw-Hill. pp. 665–666,1298–1301.
• Korn GA,Korn TM (1961).Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 182.LCCN 59014456.
• Zwillinger D (1992).Handbook of Integration. Boston, MA: Jones and Bartlett. p. 113.ISBN 0-86720-293-9.
• Moon PH, Spencer DE (1988). "Bispherical Coordinates (η, θ, ψ)".Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer Verlag. pp. 110–112 (Section IV, E4Rx).ISBN 0-387-02732-7.

• MathWorld description of bispherical coordinates

• Three-dimensional coordinate systems
• Orthogonal coordinate systems

• Articles with short description
• Short description is different from Wikidata