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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).

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.

Contents

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}}
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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.

Inverse formulae

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.

Applications

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.

See also

  • Three dimensional orthogonal coordinate systems

References

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. ed.). New York: Springer Verlag. pp. 110–112 (Section IV, E4Rx). ISBN 0-387-02732-7.  

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