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| [[Image:Bispherical coordinates.png|thumb|right|350px|Illustration of bispherical coordinates, which are obtained by rotating a two-dimensional [[bipolar coordinates|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).]]
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| '''Bispherical coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] that results from rotating the two-dimensional [[bipolar coordinates|bipolar coordinate system]] about the axis that connects the two foci. Thus, the two [[Focus (geometry)|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.
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| ==Definition==
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| The most common definition of bispherical coordinates <math>(\sigma, \tau, \phi)</math> is
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| :<math>
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| x = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \cos \phi
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| </math>
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| :<math>
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| y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \sin \phi
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| </math>
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| :<math>
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| z = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}
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| </math>
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| 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
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| :<math>
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| \tau = \ln \frac{d_{1}}{d_{2}}
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| </math>
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| ===Coordinate surfaces===
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| Surfaces of constant <math>\sigma</math> correspond to intersecting tori of different radii
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| :<math>
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| z^{2} +
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| \left( \sqrt{x^2 + y^2} - a \cot \sigma \right)^2 = \frac{a^2}{\sin^2 \sigma}
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| </math>
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| that all pass through the foci but are not concentric. The surfaces of constant <math>\tau</math> are non-intersecting spheres of different radii
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| :<math>
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| \left( x^2 + y^2 \right) +
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| \left( z - a \coth \tau \right)^2 = \frac{a^2}{\sinh^2 \tau}
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| </math>
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| 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.
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| ===Inverse formulae===
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| The formulae for the inverse transformation are:
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| :<math>\sigma = \arccos((R^2-a^2)/Q)</math>
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| :<math>\tau = \operatorname{arsinh}(2 a z/Q)</math>
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| :<math>\phi = \operatorname{atan}(y/x) </math>
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| where <math>R=\sqrt{x^2+y^2+z^2}</math> and <math>Q=\sqrt{(R^2+a^2)^2-(2 a z)^2}.</math>
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| ===Scale factors===
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| The scale factors for the bispherical coordinates <math>\sigma</math> and <math>\tau</math> are equal
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| :<math>
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| h_\sigma = h_\tau = \frac{a}{\cosh \tau - \cos\sigma}
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| </math>
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| whereas the azimuthal scale factor equals
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| :<math>
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| h_\phi = \frac{a \sin \sigma}{\cosh \tau - \cos\sigma}
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| </math>
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| Thus, the infinitesimal volume element equals
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| :<math>
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| dV = \frac{a^3 \sin \sigma}{\left( \cosh \tau - \cos\sigma \right)^3} \, d\sigma \, d\tau \, d\phi
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| </math>
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| and the Laplacian is given by
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| :<math>
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| \begin{align}
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| \nabla^2 \Phi =
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| \frac{\left( \cosh \tau - \cos\sigma \right)^3}{a^2 \sin \sigma}
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| & \left[
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| \frac{\partial}{\partial \sigma}
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| \left( \frac{\sin \sigma}{\cosh \tau - \cos\sigma}
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| \frac{\partial \Phi}{\partial \sigma}
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| \right) \right. \\[8pt]
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| &{} \quad + \left.
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| \sin \sigma \frac{\partial}{\partial \tau}
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| \left( \frac{1}{\cosh \tau - \cos\sigma}
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| \frac{\partial \Phi}{\partial \tau}
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| \right) +
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| \frac{1}{\sin \sigma \left( \cosh \tau - \cos\sigma \right)}
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| \frac{\partial^2 \Phi}{\partial \phi^2}
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| \right]
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| \end{align}
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| </math>
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| 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]].
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| ==Applications==
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| The classic applications of bispherical coordinates are in solving [[partial differential equations]],
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| e.g., [[Laplace's equation]], for which bispherical coordinates allow a
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| [[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.
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| ==References==
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| {{Empty section|date=July 2010}}
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| ==Bibliography==
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| *{{cite book | author = Morse PM, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | pages = 665–666}}
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| *{{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 = 5914456}}
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| *{{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}}
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| *{{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)}}
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| ==External links==
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| *[http://mathworld.wolfram.com/BisphericalCoordinates.html MathWorld description of bispherical coordinates]
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| {{Orthogonal coordinate systems}}
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| [[Category:Coordinate systems]]
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She is known by the name of Myrtle Shryock. Hiring is her day job now and she will not alter it anytime soon. California is our beginning location. What I love performing is to gather badges but I've been taking on new things recently.
My web site; at home std test