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| {{ distinguish|Ecliptic coordinate system}}
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| [[Image:Elliptical coordinates grid.svg|thumb|right|352px|Elliptic coordinate system]]
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| In [[geometry]], the '''elliptic coordinate system''' is a two-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] in which
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| the [[Coordinate system#Coordinate line|coordinate lines]] are [[confocal]] [[ellipse]]s and [[hyperbola]]e. The two [[Focus (geometry)|foci]]
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| <math>F_{1}</math> and <math>F_{2}</math> are generally taken to be fixed at <math>-a</math> and
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| <math>+a</math>, respectively, on the <math>x</math>-axis of the [[Cartesian coordinate system]].
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| ==Basic definition==
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| The most common definition of elliptic coordinates <math>(\mu, \nu)</math> is
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| :<math>
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| x = a \ \cosh \mu \ \cos \nu
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| </math>
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| :<math>
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| y = a \ \sinh \mu \ \sin \nu
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| </math>
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| where <math>\mu</math> is a nonnegative real number and <math>\nu \in [0, 2\pi].</math>
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| On the [[complex plane]], an equivalent relationship is
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| :<math>
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| x + iy = a \ \cosh(\mu + i\nu)
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| </math>
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| These definitions correspond to ellipses and hyperbolae. The trigonometric identity
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| :<math>
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| \frac{x^{2}}{a^{2} \cosh^{2} \mu} + \frac{y^{2}}{a^{2} \sinh^{2} \mu} = \cos^{2} \nu + \sin^{2} \nu = 1
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| </math>
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| shows that curves of constant <math>\mu</math> form [[ellipse]]s, whereas the hyperbolic trigonometric identity
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| :<math>
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| \frac{x^{2}}{a^{2} \cos^{2} \nu} - \frac{y^{2}}{a^{2} \sin^{2} \nu} = \cosh^{2} \mu - \sinh^{2} \mu = 1
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| </math>
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| shows that curves of constant <math>\nu</math> form [[hyperbola]]e.
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| ==Scale factors==
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| In an [[orthogonal coordinate system]] the lengths of the basis vectors are known as scale factors. The scale factors for the elliptic coordinates <math>(\mu, \nu)</math> are equal to
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| :<math>
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| h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu} = a\sqrt{\cosh^{2}\mu - \cos^{2}\nu}.
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| </math>
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| Using the ''double argument identities'' for [[Hyperbolic_functions#Identities|hyperbolic functions]] and [[Trigonometric_function#Identities|trigonometric functions]], the scale factors can be equivalently expressed as
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| :<math>
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| h_{\mu} = h_{\nu} = a\sqrt{\frac{1}{2} (\cosh2\mu - \cos2\nu}).
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| </math>
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| Consequently, an infinitesimal element of area equals
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| :<math>
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| dA = h_{\mu} h_{\nu} d\mu d\nu
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| = a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu
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| = a^{2} \left( \cosh^{2}\mu - \cos^{2}\nu \right) d\mu d\nu
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| = \frac{a^{2}}{2} \left( \cosh 2 \mu - \cos 2\nu \right) d\mu d\nu
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| </math>
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| and the Laplacian reads
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| :<math>
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| \nabla^{2} \Phi
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| = \frac{1}{a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right)}
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| \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right)
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| = \frac{1}{a^{2} \left( \cosh^{2}\mu - \cos^{2}\nu \right)}
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| \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right)
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| = \frac{2}{a^{2} \left( \cosh 2 \mu - \cos 2 \nu \right)}
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| \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right).
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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>(\mu, \nu)</math> by substituting
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| the scale factors into the general formulae found in [[orthogonal coordinates]].
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| ==Alternative definition==
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| An alternative and geometrically intuitive set of elliptic coordinates <math>(\sigma, \tau)</math> are sometimes used,
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| where <math>\sigma = \cosh \mu</math> and <math>\tau = \cos \nu</math>. Hence, the curves of constant <math>\sigma</math> are ellipses, whereas the curves of constant <math>\tau</math> are hyperbolae. The coordinate <math>\tau</math> must belong to the interval [-1, 1], whereas the <math>\sigma</math>
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| coordinate must be greater than or equal to one.
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|
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| The coordinates <math>(\sigma, \tau)</math> have a simple relation to the distances to the foci <math>F_{1}</math> and <math>F_{2}</math>. For any point in the plane, the ''sum'' <math>d_{1}+d_{2}</math> of its distances to the foci equals <math>2a\sigma</math>, whereas their ''difference'' <math>d_{1}-d_{2}</math> equals <math>2a\tau</math>.
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| Thus, the distance to <math>F_{1}</math> is <math>a(\sigma+\tau)</math>, whereas the distance to <math>F_{2}</math> is <math>a(\sigma-\tau)</math>. (Recall that <math>F_{1}</math> and <math>F_{2}</math> are located at <math>x=-a</math> and <math>x=+a</math>, respectively.)
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| A drawback of these coordinates is that the points with [[Cartesian coordinates]] (x,y) and (x,-y) have the same coordinates <math>(\sigma, \tau)</math>, so the conversion to Cartesian coordinates is not a function, but a [[multivalued function|multifunction]].
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| :<math> | |
| x = a \left. \sigma \right. \tau
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| </math>
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| :<math>
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| y^{2} = a^{2} \left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right).
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| </math>
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| ==Alternative scale factors==
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| The scale factors for the alternative elliptic coordinates <math>(\sigma, \tau)</math> are
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| :<math>
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| h_{\sigma} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{\sigma^{2} - 1}}
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| </math>
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| :<math>
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| h_{\tau} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{1 - \tau^{2}}}.
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| </math>
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| Hence, the infinitesimal area element becomes
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| :<math>
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| dA = a^{2} \frac{\sigma^{2} - \tau^{2}}{\sqrt{\left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)}} d\sigma d\tau
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| </math>
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| and the Laplacian equals
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| :<math>
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| \nabla^{2} \Phi =
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| \frac{1}{a^{2} \left( \sigma^{2} - \tau^{2} \right) }
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| \left[
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| \sqrt{\sigma^{2} - 1} \frac{\partial}{\partial \sigma}
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| \left( \sqrt{\sigma^{2} - 1} \frac{\partial \Phi}{\partial \sigma} \right) +
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| \sqrt{1 - \tau^{2}} \frac{\partial}{\partial \tau}
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| \left( \sqrt{1 - \tau^{2}} \frac{\partial \Phi}{\partial \tau} \right)
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| \right].
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| </math>
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| Other differential operators such as <math>\nabla \cdot \mathbf{F}</math>
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| and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau)</math> by substituting
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| the scale factors into the general formulae
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| found in [[orthogonal coordinates]].
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| ==Extrapolation to higher dimensions==
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| Elliptic coordinates form the basis for several sets of three-dimensional [[orthogonal coordinates]].
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| The [[elliptic cylindrical coordinates]] are produced by projecting in the <math>z</math>-direction.
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| The [[prolate spheroidal coordinates]] are produced by rotating the elliptic coordinates about the <math>x</math>-axis, i.e., the axis connecting the foci, whereas the [[oblate spheroidal coordinates]] are produced by rotating the elliptic coordinates about the <math>y</math>-axis, i.e., the axis separating the foci.
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| ==Applications==
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| The classic applications of elliptic coordinates are in solving [[partial differential equations]],
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| e.g., [[Laplace's equation]] or the [[Helmholtz equation]], for which elliptic coordinates are a natural description of a system thus allowing a [[separation of variables]] in the [[partial differential equations]]. Some traditional examples are solving systems such as electrons orbiting a molecule or planetary orbits that have an elliptical shape.
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| The geometric properties of elliptic coordinates can also be useful. A typical example might involve
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| an integration over all pairs of vectors <math>\mathbf{p}</math> and <math>\mathbf{q}</math>
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| that sum to a fixed vector <math>\mathbf{r} = \mathbf{p} + \mathbf{q}</math>, where the integrand
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| was a function of the vector lengths <math>\left| \mathbf{p} \right|</math> and <math>\left| \mathbf{q} \right|</math>. (In such a case, one would position <math>\mathbf{r}</math> between the two foci and aligned with the <math>x</math>-axis, i.e., <math>\mathbf{r} = 2a \mathbf{\hat{x}}</math>.) For concreteness, <math>\mathbf{r}</math>, <math>\mathbf{p}</math> and <math>\mathbf{q}</math> could represent the [[momentum|momenta]] of a particle and its decomposition products, respectively, and the integrand might involve the kinetic energies of the products (which are proportional to the squared lengths of the momenta).
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| ==See also==
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| *[[Curvilinear coordinates]]
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| *[[Generalized coordinates]]
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| *[[Mean motion]]
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| ==References==
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| * {{springer|title=Elliptic coordinates|id=p/e035440}}
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| * Korn GA and Korn TM. (1961) ''Mathematical Handbook for Scientists and Engineers'', McGraw-Hill.
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| * Weisstein, Eric W. "Elliptic Cylindrical Coordinates." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html
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| {{Orthogonal coordinate systems}}
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| [[Category:Coordinate systems]]
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