Elliptic coordinate system

2013-11-28T11:50:06Z

90.84.144.236: /* Basic definition */

[[File:Elliptic cylindrical coordinates.png|thumb|right|350px|[[Coordinate system#Coordinate surface|Coordinate surfaces]] of elliptic cylindrical coordinates. The yellow sheet is the prism of a half-hyperbola corresponding to ν=-45°, whereas the red tube is an elliptical prism corresponding to μ=1. The blue sheet corresponds to ''z''=1. The three surfaces intersect at the point '''P''' (shown as a black sphere) with [[Cartesian coordinate system|Cartesian coordinates]] roughly (2.182, -1.661, 1.0). The foci of the ellipse and hyperbola lie at ''x'' = ±2.0.]]

'''Elliptic cylindrical coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] that results from projecting the two-dimensional [[elliptic coordinates|elliptic coordinate system]] in the
perpendicular <math>z</math>-direction. Hence, the [[Coordinate system#Coordinate surface|coordinate surfaces]] are [[prism (geometry)|prisms]] of confocal [[ellipse]]s and [[hyperbola]]e. The two [[Focus (geometry)|foci]]
<math>F_{1}</math> and <math>F_{2}</math> are generally taken to be fixed at <math>-a</math> and
<math>+a</math>, respectively, on the <math>x</math>-axis of the [[Cartesian coordinate system]].

==Basic definition==

The most common definition of elliptic cylindrical coordinates <math>(\mu, \nu, z)</math> is

:<math>
x = a \ \cosh \mu \ \cos \nu
</math>

:<math>
y = a \ \sinh \mu \ \sin \nu
</math>

:<math>
z = z
\!</math>

where <math>\mu</math> is a nonnegative real number and <math>\nu \in [0, 2\pi)</math>.

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

:<math>
\frac{x^{2}}{a^{2} \cosh^{2} \mu} + \frac{y^{2}}{a^{2} \sinh^{2} \mu} = \cos^{2} \nu + \sin^{2} \nu = 1
</math>

shows that curves of constant <math>\mu</math> form [[ellipse]]s, whereas the hyperbolic trigonometric identity

:<math>
\frac{x^{2}}{a^{2} \cos^{2} \nu} - \frac{y^{2}}{a^{2} \sin^{2} \nu} = \cosh^{2} \mu - \sinh^{2} \mu = 1
</math>

shows that curves of constant <math>\nu</math> form [[hyperbola]]e.

==Scale factors==

The scale factors for the elliptic cylindrical coordinates <math>\mu</math> and <math>\nu</math> are equal

:<math>
h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu}
</math>

whereas the remaining scale factor <math>h_{z}=1</math>.
Consequently, an infinitesimal volume element equals

:<math>
dV = a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu dz
</math>

and the Laplacian equals

:<math>
\nabla^{2} \Phi = \frac{1}{a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right)} \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right) + \frac{\partial^{2} \Phi}{\partial z^{2}}
</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>(\mu, \nu, z)</math> by substituting
the scale factors into the general formulae found in [[orthogonal coordinates]].

==Alternative definition==

An alternative and geometrically intuitive set of elliptic coordinates <math>(\sigma, \tau, z)</math> are sometimes used, 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>
coordinate must be greater than or equal to one.

The coordinates <math>(\sigma, \tau, z)</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 (x,y) 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>.
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.)

A drawback of these coordinates is that they do not have a 1-to-1 transformation to the [[Cartesian coordinates]]
:<math>
x = a\sigma\tau \!
</math>

:<math>
y^{2} = a^{2} \left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right)
</math>

==Alternative scale factors==

The scale factors for the alternative elliptic coordinates <math>(\sigma, \tau, z)</math> are

:<math>
h_{\sigma} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{\sigma^{2} - 1}}
</math>

:<math>
h_{\tau} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{1 - \tau^{2}}}
</math>

and, of course, <math>h_{z}=1</math>. Hence, the infinitesimal volume element becomes

:<math>
dV = a^{2} \frac{\sigma^{2} - \tau^{2}}{\sqrt{\left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)}} d\sigma d\tau dz
</math>

and the Laplacian equals

:<math>
\nabla^{2} \Phi =
\frac{1}{a^{2} \left( \sigma^{2} - \tau^{2} \right) }
\left[
\sqrt{\sigma^{2} - 1} \frac{\partial}{\partial \sigma}
\left( \sqrt{\sigma^{2} - 1} \frac{\partial \Phi}{\partial \sigma} \right) +
\sqrt{1 - \tau^{2}} \frac{\partial}{\partial \tau}
\left( \sqrt{1 - \tau^{2}} \frac{\partial \Phi}{\partial \tau} \right)
\right] +
\frac{\partial^{2} \Phi}{\partial z^{2}}
</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 elliptic cylindrical coordinates are in solving [[partial differential equations]],
e.g., [[Laplace's equation]] or the [[Helmholtz equation]], for which elliptic cylindrical coordinates allow a
[[separation of variables]]. A typical example would be the [[electric field]] surrounding a
flat conducting plate of width <math>2a</math>.

The three-dimensional [[wave equation]], when expressed in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the [[Mathieu differential equation]]s.

The geometric properties of elliptic coordinates can also be useful. A typical example might involve
an integration over all pairs of vectors <math>\mathbf{p}</math> and <math>\mathbf{q}</math>
that sum to a fixed vector <math>\mathbf{r} = \mathbf{p} + \mathbf{q}</math>, where the integrand
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).

==Bibliography==
*{{cite book | author = [[Philip M. Morse|Morse PM]], [[Herman Feshbach|Feshbach H]] | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | isbn = 0-07-043316-X | page = 657 | lccn = 5211515}}
*{{cite book | author = [[Henry Margenau|Margenau H]], Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | publisher = D. van Nostrand | location = New York | pages = 182–183 | lccn = 5510911 }}
*{{cite book | author = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | id = ASIN B0000CKZX7 | page = 179 | lccn = 5914456}}
*{{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | page = 97 | lccn = 6725285}}
*{{cite book | author = Zwillinger D | year = 1992 | title = Handbook of Integration | publisher = Jones and Bartlett | location = Boston, MA | isbn = 0-86720-293-9 | page = 114}} Same as Morse & Feshbach (1953), substituting ''u''<sub>''k''</sub> for ξ<sub>''k''</sub>.
*{{cite book | author = Moon P, Spencer DE | year = 1988 | chapter = Elliptic-Cylinder Coordinates (η, ψ, z) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer-Verlag | location = New York | pages = 17–20 (Table 1.03) | isbn = 978-0-387-18430-2}}

==External links==
*[http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html MathWorld description of elliptic cylindrical coordinates]

{{Orthogonal coordinate systems}}

[[Category:Coordinate systems]]

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Elliptic coordinate system