Bispherical coordinates

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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).

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 F_{1} and F_{2} in bipolar coordinates remain points (on the z-axis, the axis of rotation) in the bispherical coordinate system.


The most common definition of bispherical coordinates (\sigma, \tau, \phi) is

x = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \cos \phi

y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \sin \phi

z = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}

where the \sigma coordinate of a point P equals the angle F_{1} P F_{2} and the \tau coordinate equals the natural logarithm of the ratio of the distances d_{1} and d_{2} to the foci

\tau = \ln \frac{d_{1}}{d_{2}}

Coordinate surfaces[edit]

Surfaces of constant \sigma correspond to intersecting tori of different radii

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 \tau are non-intersecting spheres of different radii

\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-\tau spheres lie along the z-axis, whereas the constant-\sigma tori are centered in the xy plane.

Inverse formulae[edit]

The formulae for the inverse transformation are:

\sigma = \arccos((R^2-a^2)/Q)
\tau = \operatorname{arsinh}(2 a z/Q)
\phi = \operatorname{atan}(y/x)

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

Scale factors[edit]

The scale factors for the bispherical coordinates \sigma and \tau are equal

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

whereas the azimuthal scale factor equals

h_\phi = \frac{a \sin \sigma}{\cosh \tau - \cos\sigma}

Thus, the infinitesimal volume element equals

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

and the Laplacian is given by

\nabla^2 \Phi =
\frac{\left( \cosh \tau - \cos\sigma \right)^3}{a^2 \sin \sigma} 
& \left[
\frac{\partial}{\partial \sigma}
\left( \frac{\sin \sigma}{\cosh \tau - \cos\sigma}
\frac{\partial \Phi}{\partial \sigma}
\right) \right. \\[8pt]
&{} \quad + \left.
\sin \sigma \frac{\partial}{\partial \tau}
\left( \frac{1}{\cosh \tau - \cos\sigma}
\frac{\partial \Phi}{\partial \tau}
\right) + 
\frac{1}{\sin \sigma \left( \cosh \tau - \cos\sigma \right)}
\frac{\partial^2 \Phi}{\partial \phi^2}

Other differential operators such as \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (\sigma, \tau) 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.



  • Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. pp. 665–666. 
  • 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. 

External links[edit]