{{Short description|Three-dimensional orthogonal coordinate system}} [[Image:Bispherical coordinates.png|thumb|right|350px|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 <math>F_{1}</math> and <math>F_{2}</math> in bipolar coordinates remain points (on the <math>z</math>-axis, the axis of rotation) in the bispherical coordinate system.
==Definition== The most common definition of bispherical coordinates <math>(\tau, \sigma, \phi)</math> is
:<math>\begin{align} 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}, \end{align}</math>
where the <math>\sigma</math> coordinate of a point <math>P</math> equals the angle <math>F_{1} P F_{2}</math> and the <math>\tau</math> coordinate equals the natural logarithm of the ratio of the distances <math>d_{1}</math> and <math>d_{2}</math> to the foci
:<math> \tau = \ln \frac{d_{1}}{d_{2}} </math>
The coordinates ranges are −∞ < <math>\tau</math> < ∞, 0 ≤ <math>\sigma</math> ≤ <math>\pi</math> and 0 ≤ <math>\phi</math> ≤ 2<math>\pi</math>.
===Coordinate surfaces=== Surfaces of constant <math>\sigma</math> correspond to intersecting tori of different radii
:<math> z^{2} + \left( \sqrt{x^2 + y^2} - a \cot \sigma \right)^2 = \frac{a^2}{\sin^2 \sigma} </math>
that all pass through the foci but are not concentric. The surfaces of constant <math>\tau</math> are non-intersecting spheres of different radii
:<math> \left( x^2 + y^2 \right) + \left( z - a \coth \tau \right)^2 = \frac{a^2}{\sinh^2 \tau} </math>
that surround the foci. The centers of the constant-<math>\tau</math> spheres lie along the <math>z</math>-axis, whereas the constant-<math>\sigma</math> tori are centered in the <math>xy</math> plane.
===Inverse formulae===
The formulae for the inverse transformation are:
:<math>\begin{align} \sigma &= \arccos\left(\dfrac{R^2-a^2}{Q}\right), \\ \tau &= \operatorname{arsinh}\left(\dfrac{2az}{Q}\right), \\ \phi &= \arctan\left(\dfrac{y}{x}\right), \end{align}</math>
where <math display="inline">R = \sqrt{x^2 + y^2 + z^2}</math> and <math display="inline">Q = \sqrt{\left(R^2 + a^2\right)^2 - \left(2 a z\right)^2}.</math>
===Scale factors===
The scale factors for the bispherical coordinates <math>\sigma</math> and <math>\tau</math> are equal
:<math> h_\sigma = h_\tau = \frac{a}{\cosh \tau - \cos\sigma} </math>
whereas the azimuthal scale factor equals
:<math> h_\phi = \frac{a \sin \sigma}{\cosh \tau - \cos\sigma} </math>
Thus, the infinitesimal volume element equals
:<math> dV = \frac{a^3 \sin \sigma}{\left( \cosh \tau - \cos\sigma \right)^3} \, d\sigma \, d\tau \, d\phi </math>
and the Laplacian is given by
:<math> \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} </math>
Other differential operators such as <math>\nabla \cdot \mathbf{F}</math> and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau)</math> 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.
==References== {{reflist}}
==Bibliography== *{{cite book | author = Morse PM, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Parts I and II | publisher = McGraw-Hill | location = New York | pages = 665–666, 1298–1301 }} *{{cite book | author = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | page = 182 | lccn = 59014456}} *{{cite book | author = Zwillinger D | year = 1992 | title = Handbook of Integration | publisher = Jones and Bartlett | location = Boston, MA | isbn = 0-86720-293-9 | page = 113}} *{{cite book | author = Moon PH, Spencer DE | year = 1988 | chapter = Bispherical Coordinates (η, θ, ψ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer Verlag | location = New York | isbn = 0-387-02732-7 | pages = 110–112 (Section IV, E4Rx)}}
==External links== *[http://mathworld.wolfram.com/BisphericalCoordinates.html MathWorld description of bispherical coordinates]
{{Orthogonal coordinate systems}}
Category:Three-dimensional coordinate systems Category:Orthogonal coordinate systems