Elliptic coordinate system - en.LinkFang.org

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 \({\displaystyle F_{1}}\) and \({\displaystyle F_{2}}\) are generally taken to be fixed at \({\displaystyle -a}\) and \({\displaystyle +a}\), respectively, on the \({\displaystyle x}\)-axis of the Cartesian coordinate system.


Basic definition

The most common definition of elliptic coordinates \({\displaystyle (\mu ,\nu )}\) is

\({\displaystyle x=a\ \cosh \mu \ \cos \nu }\)
\({\displaystyle y=a\ \sinh \mu \ \sin \nu }\)

where \({\displaystyle \mu }\) is a nonnegative real number and \({\displaystyle \nu \in [0,2\pi ].}\)

On the complex plane, an equivalent relationship is

\({\displaystyle x+iy=a\ \cosh(\mu +i\nu )}\)

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

\({\displaystyle {\frac {x^{2}}{a^{2}\cosh ^{2}\mu }}+{\frac {y^{2}}{a^{2}\sinh ^{2}\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1}\)

shows that curves of constant \({\displaystyle \mu }\) form ellipses, whereas the hyperbolic trigonometric identity

\({\displaystyle {\frac {x^{2}}{a^{2}\cos ^{2}\nu }}-{\frac {y^{2}}{a^{2}\sin ^{2}\nu }}=\cosh ^{2}\mu -\sinh ^{2}\mu =1}\)

shows that curves of constant \({\displaystyle \nu }\) 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 \({\displaystyle (\mu ,\nu )}\) are equal to

\({\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}=a{\sqrt {\cosh ^{2}\mu -\cos ^{2}\nu }}.}\)

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

\({\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {{\frac {1}{2}}(\cosh 2\mu -\cos 2\nu )}}.}\)

Consequently, an infinitesimal element of area equals

\({\displaystyle dA=h_{\mu }h_{\nu }d\mu d\nu =a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)d\mu d\nu =a^{2}\left(\cosh ^{2}\mu -\cos ^{2}\nu \right)d\mu d\nu ={\frac {a^{2}}{2}}\left(\cosh 2\mu -\cos 2\nu \right)d\mu d\nu }\)

and the Laplacian reads

\({\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)={\frac {1}{a^{2}\left(\cosh ^{2}\mu -\cos ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)={\frac {2}{a^{2}\left(\cosh 2\mu -\cos 2\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right).}\)

Other differential operators such as \({\displaystyle \nabla \cdot \mathbf {F} }\) and \({\displaystyle \nabla \times \mathbf {F} }\) can be expressed in the coordinates \({\displaystyle (\mu ,\nu )}\) by substituting the scale factors into the general formulae found in orthogonal coordinates.

Alternative definition

An alternative and geometrically intuitive set of elliptic coordinates \({\displaystyle (\sigma ,\tau )}\) are sometimes used, where \({\displaystyle \sigma =\cosh \mu }\) and \({\displaystyle \tau =\cos \nu }\). Hence, the curves of constant \({\displaystyle \sigma }\) are ellipses, whereas the curves of constant \({\displaystyle \tau }\) are hyperbolae. The coordinate \({\displaystyle \tau }\) must belong to the interval [-1, 1], whereas the \({\displaystyle \sigma }\) coordinate must be greater than or equal to one.

The coordinates \({\displaystyle (\sigma ,\tau )}\) have a simple relation to the distances to the foci \({\displaystyle F_{1}}\) and \({\displaystyle F_{2}}\). For any point in the plane, the sum \({\displaystyle d_{1}+d_{2}}\) of its distances to the foci equals \({\displaystyle 2a\sigma }\), whereas their difference \({\displaystyle d_{1}-d_{2}}\) equals \({\displaystyle 2a\tau }\). Thus, the distance to \({\displaystyle F_{1}}\) is \({\displaystyle a(\sigma +\tau )}\), whereas the distance to \({\displaystyle F_{2}}\) is \({\displaystyle a(\sigma -\tau )}\). (Recall that \({\displaystyle F_{1}}\) and \({\displaystyle F_{2}}\) are located at \({\displaystyle x=-a}\) and \({\displaystyle x=+a}\), respectively.)

A drawback of these coordinates is that the points with Cartesian coordinates (x,y) and (x,-y) have the same coordinates \({\displaystyle (\sigma ,\tau )}\), so the conversion to Cartesian coordinates is not a function, but a multifunction.

\({\displaystyle x=a\left.\sigma \right.\tau }\)
\({\displaystyle y^{2}=a^{2}\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right).}\)

Alternative scale factors

The scale factors for the alternative elliptic coordinates \({\displaystyle (\sigma ,\tau )}\) are

\({\displaystyle h_{\sigma }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{\sigma ^{2}-1}}}}\)
\({\displaystyle h_{\tau }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{1-\tau ^{2}}}}.}\)

Hence, the infinitesimal area element becomes

\({\displaystyle dA=a^{2}{\frac {\sigma ^{2}-\tau ^{2}}{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}}d\sigma d\tau }\)

and the Laplacian equals

\({\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left[{\sqrt {\sigma ^{2}-1}}{\frac {\partial }{\partial \sigma }}\left({\sqrt {\sigma ^{2}-1}}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\sqrt {1-\tau ^{2}}}{\frac {\partial }{\partial \tau }}\left({\sqrt {1-\tau ^{2}}}{\frac {\partial \Phi }{\partial \tau }}\right)\right].}\)

Other differential operators such as \({\displaystyle \nabla \cdot \mathbf {F} }\) and \({\displaystyle \nabla \times \mathbf {F} }\) can be expressed in the coordinates \({\displaystyle (\sigma ,\tau )}\) 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 \({\displaystyle z}\)-direction. The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the \({\displaystyle x}\)-axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the \({\displaystyle y}\)-axis, i.e., the axis separating the foci.


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 \({\displaystyle \mathbf {p} }\) and \({\displaystyle \mathbf {q} }\) that sum to a fixed vector \({\displaystyle \mathbf {r} =\mathbf {p} +\mathbf {q} }\), where the integrand was a function of the vector lengths \({\displaystyle \left|\mathbf {p} \right|}\) and \({\displaystyle \left|\mathbf {q} \right|}\). (In such a case, one would position \({\displaystyle \mathbf {r} }\) between the two foci and aligned with the \({\displaystyle x}\)-axis, i.e., \({\displaystyle \mathbf {r} =2a\mathbf {\hat {x}} }\).) For concreteness, \({\displaystyle \mathbf {r} }\), \({\displaystyle \mathbf {p} }\) and \({\displaystyle \mathbf {q} }\) 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).

See also


Categories: Two-dimensional coordinate systems

Information as of: 09.07.2020 10:20:11 CEST

Source: Wikipedia (Authors [History])    License : CC-by-sa-3.0

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