Bispherical coordinates

Bispherical coordinates

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

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

:x = a frac{sin sigma}{cosh au - cos sigma} cos phi

:y = a frac{sin sigma}{cosh au - cos sigma} sin phi

:z = a frac{sinh au}{cosh au - cos sigma}

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

: au = ln frac{d_{1{d_{2

Coordinate surfaces

Surfaces of constant sigma correspond to intersecting tori of different radii

:z^{2} +left( sqrt{x^{2} + y^{2 - a cot sigma ight)^{2} = frac{a^{2{sin^{2} sigma}

that all pass through the foci but are not concentric. The surfaces of constant au are non-intersecting spheres of different radii

:left( x^{2} + y^{2} ight) +left( z - a coth au ight)^{2} = frac{a^{2{sinh^{2} au}

that surround the foci. The centers of the constant- au spheres lie along the z-axis, whereas the constant-sigma tori are centered in the xy plane.

Inverse formulae

cale factors

The scale factors for the bispherical coordinates sigma and au are equal

:h_{sigma} = h_{ au} = frac{a}{cosh au - cossigma}

whereas the azimuthal scale factor equals

:h_{phi} = frac{a sin sigma}{cosh au - cossigma}

Thus, the infinitesimal volume element equals

:dV = frac{a^{3}sin sigma}{left( cosh au - cossigma ight)^{3 dsigma d au dphi

and the Laplacian is given by

: abla^{2} Phi =frac{left( cosh au - cossigma ight)^{3{a^{2}sin sigma} left [ frac{partial}{partial sigma}left( frac{sin sigma}{cosh au - cossigma}frac{partial Phi}{partial sigma} ight) + sin sigma frac{partial}{partial au}left( frac{1}{cosh au - cossigma}frac{partial Phi}{partial au} ight) + frac{1}{sin sigma left( cosh au - cossigma ight)}frac{partial^{2} Phi}{partial phi^{2 ight]

Other differential operators such as abla cdot mathbf{F} and abla imes mathbf{F} can be expressed in the coordinates (sigma, au) 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 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.

See also

References

Bibliography

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* | pages = p. 182

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External links

* [http://mathworld.wolfram.com/BisphericalCoordinates.html MathWorld description of bispherical coordinates]


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