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.

Definition[edit]

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


\begin{align}
\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}
\right]
\end{align}

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.

Applications[edit]

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.

References[edit]

Bibliography[edit]

  • 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 5914456 Check |lccn= value (help). 
  • 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]