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-interecting 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).
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-interecting 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 F1 and F2 in bipolar coordinates remain points (on the z-axis, the axis of rotation) in the bispherical coordinate system.

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[edit] Definition

The most common definition of bispherical coordinates (σ,τ,φ) 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 σ coordinate of a point P equals the angle F1PF2 and the τ coordinate equals the natural logarithm of the ratio of the distances d1 and d2 to the foci

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

[edit] Coordinate surfaces

Surfaces of constant σ 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 τ 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-τ spheres lie along the z-axis, whereas the constant-σ tori are centered in the xy plane.

[edit] Inverse formulae

[edit] Scale factors

The scale factors for the bispherical coordinates σ and τ 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) + \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}} \right]

Other differential operators such as \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (σ,τ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

[edit] Applications

The classic applications of bispherical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which bispherical coordinates allow a separation of variables. A typical example would be the electric field surrounding two conducting spheres of different radii.

[edit] See also

  • Three dimensional orthogonal coordinate systems


[edit] References

[edit] Bibliography

  • 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 59-14456. 
  • 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. 

[edit] External links

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