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

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

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

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

and the Laplacian reads

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

Other differential operators such as ${\displaystyle \nabla \cdot \mathbf {F} }$ 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:

1. The elliptic cylindrical coordinates are produced by projecting in the ${\displaystyle z}$-direction.
2. 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.
3. 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 ${\displaystyle \mathbf {p} }$ 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).