Elliptic coordinate system

{{Short description|2D coordinate system whose coordinate lines are confocal ellipses and hyperbolae}} {{ distinguish|Ecliptic coordinate system}} thumb|right|352px|Elliptic coordinate system In geometry, the '''elliptic coordinate system''' is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two 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 coordinates <math>(\mu, \nu)</math> is

:<math>\begin{align} x &= a \ \cosh \mu \ \cos \nu \\ y &= a \ \sinh \mu \ \sin \nu \end{align}</math>

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

On the complex plane, an equivalent relationship is

:<math>x + iy = a \ \cosh(\mu + i\nu)</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 ellipses, 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 hyperbolae.

===Scale factors===

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

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

Using the ''double argument identities'' for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as

:<math>h_{\mu} = h_{\nu} = a\sqrt{\frac{1}{2} (\cosh2\mu - \cos2\nu)}.</math>

Consequently, an infinitesimal element of area equals

:<math>\begin{align} dA &= h_{\mu} h_{\nu} d\mu d\nu \\ &= a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu \\ &= a^{2} \left( \cosh^{2}\mu - \cos^{2}\nu \right) d\mu d\nu \\ &= \frac{a^{2}}{2} \left( \cosh 2 \mu - \cos 2\nu \right) d\mu d\nu \end{align}</math>

and the Laplacian reads

:<math>\begin{align} \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{1}{a^{2} \left( \cosh^{2}\mu - \cos^{2}\nu \right)} \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right) \\ &= \frac{2}{a^{2} \left( \cosh 2 \mu - \cos 2 \nu \right)} \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{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>(\mu, \nu)</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)</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)</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>. 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 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 multifunction.

:<math> x = a \left. \sigma \right. \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)</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>

Hence, the infinitesimal area element becomes

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

==Extrapolation to higher dimensions==

Elliptic coordinates form the basis for several sets of three-dimensional orthogonal coordinates: #The elliptic cylindrical coordinates are produced by projecting in the <math>z</math>-direction. #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. #Ellipsoidal coordinates are a formal extension of elliptic coordinates into 3-dimensions, which is based on confocal ellipsoids, hyperboloids of one and two sheets.

Note that (ellipsoidal) Geographic coordinate system is a different concept from above.

==Applications==

The classic applications of elliptic coordinates are in solving partial differential equations, 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.

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 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).

==References== * {{springer|title=Elliptic coordinates|id=p/e035440}} * Korn GA and Korn TM. (1961) ''Mathematical Handbook for Scientists and Engineers'', McGraw-Hill. * Weisstein, Eric W. "Elliptic Cylindrical Coordinates." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html

Category:Two-dimensional coordinate systems Category:Ellipses