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

Basic definition

The most common definition of elliptic coordinates $(\mu ,\nu )$ is

{\begin{aligned}x&=a\ \cosh \mu \ \cos \nu \\y&=a\ \sinh \mu \ \sin \nu \end{aligned}}

where $\mu$ is a nonnegative real number and $\nu \in [0,2\pi ].$

On the complex plane, an equivalent relationship is

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

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

{\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 $\mu$ form ellipses, whereas the hyperbolic trigonometric identity

{\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 $\nu$ 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 $(\mu ,\nu )$ are equal to

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

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

Consequently, an infinitesimal element of area equals

{\begin{aligned}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{aligned}}

and the Laplacian reads

{\begin{aligned}\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{aligned}}

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

Alternative definition

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

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

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

x=a\left.\sigma \right.\tau

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 $(\sigma ,\tau )$ are

h_{\sigma }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{\sigma ^{2}-1}}}

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

Hence, the infinitesimal area element becomes

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

\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 $\nabla \cdot \mathbf {F}$ and $\nabla \times \mathbf {F}$ can be expressed in the coordinates $(\sigma ,\tau )$ 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 $z$ -direction.
The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the $x$ -axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the $y$ -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 $\mathbf {p}$ and $\mathbf {q}$ that sum to a fixed vector $\mathbf {r} =\mathbf {p} +\mathbf {q}$ , where the integrand was a function of the vector lengths $\left|\mathbf {p} \right|$ and $\left|\mathbf {q} \right|$ . (In such a case, one would position $\mathbf {r}$ between the two foci and aligned with the $x$ -axis, i.e., $\mathbf {r} =2a\mathbf {\hat {x}}$ .) For concreteness, $\mathbf {r}$ , $\mathbf {p}$ and $\mathbf {q}$ 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).

References

"Elliptic coordinates", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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

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Orthogonal coordinate systems
Two dimensional	Cartesian Polar (Log-polar) Parabolic Bipolar Elliptic
Three dimensional	Cartesian Cylindrical Spherical Parabolic Paraboloidal Oblate spheroidal Prolate spheroidal Ellipsoidal Elliptic cylindrical Toroidal Bispherical Bipolar cylindrical Conical 6-sphere