Elliptic coordinate system

Basic definition

The most common definition of elliptic coordinates $(μ, ν)$ is

\begin{matrix} x & = a \cosh μ \cos ν \\ y & = a \sinh μ \sin ν \end{matrix}

where $μ$ is a nonnegative real number and $ν \in [0, 2 π] .$

On the complex plane, an equivalent relationship is

x + i y = a \cosh (μ + i ν)

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

\frac{x^{2}}{a^{2} \cosh^{2} μ} + \frac{y^{2}}{a^{2} \sinh^{2} μ} = \cos^{2} ν + \sin^{2} ν = 1

shows that curves of constant $μ$ form ellipses, whereas the hyperbolic trigonometric identity

\frac{x^{2}}{a^{2} \cos^{2} ν} - \frac{y^{2}}{a^{2} \sin^{2} ν} = \cosh^{2} μ - \sinh^{2} μ = 1

shows that curves of constant $ν$ 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 $(μ, ν)$ are equal to

h_{μ} = h_{ν} = a \sqrt{\sinh^{2} μ + \sin^{2} ν} = a \sqrt{\cosh^{2} μ - \cos^{2} ν} .

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

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

Consequently, an infinitesimal element of area equals

\begin{matrix} d A & = h_{μ} h_{ν} d μ d ν \\ = a^{2} (\sinh^{2} μ + \sin^{2} ν) d μ d ν \\ = a^{2} (\cosh^{2} μ - \cos^{2} ν) d μ d ν \\ = \frac{a^{2}}{2} (\cosh 2 μ - \cos 2 ν) d μ d ν \end{matrix}

and the Laplacian reads

\begin{matrix} \nabla^{2} Φ & = \frac{1}{a^{2} (\sinh^{2} μ + \sin^{2} ν)} (\frac{\partial^{2} Φ}{\partial μ^{2}} + \frac{\partial^{2} Φ}{\partial ν^{2}}) \\ = \frac{1}{a^{2} (\cosh^{2} μ - \cos^{2} ν)} (\frac{\partial^{2} Φ}{\partial μ^{2}} + \frac{\partial^{2} Φ}{\partial ν^{2}}) \\ = \frac{2}{a^{2} (\cosh 2 μ - \cos 2 ν)} (\frac{\partial^{2} Φ}{\partial μ^{2}} + \frac{\partial^{2} Φ}{\partial ν^{2}}) \end{matrix}

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

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Alternative definition

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

The coordinates $(σ, τ)$ 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 $2 a σ$ , whereas their difference $d_{1} - d_{2}$ equals $2 a τ$ . Thus, the distance to $F_{1}$ is $a (σ + τ)$ , whereas the distance to $F_{2}$ is $a (σ - τ)$ . (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 $(σ, τ)$ , so the conversion to Cartesian coordinates is not a function, but a multifunction.

x = a σ τ

y^{2} = a^{2} (σ^{2} - 1) (1 - τ^{2}) .

Alternative scale factors

The scale factors for the alternative elliptic coordinates $(σ, τ)$ are

h_{σ} = a \sqrt{\frac{σ^{2} - τ^{2}}{σ^{2} - 1}}

h_{τ} = a \sqrt{\frac{σ^{2} - τ^{2}}{1 - τ^{2}}} .

Hence, the infinitesimal area element becomes

d A = a^{2} \frac{σ^{2} - τ^{2}}{\sqrt{(σ^{2} - 1) (1 - τ^{2})}} d σ d τ

and the Laplacian equals

\nabla^{2} Φ = \frac{1}{a^{2} (σ^{2} - τ^{2})} [\sqrt{σ^{2} - 1} \frac{\partial}{\partial σ} (\sqrt{σ^{2} - 1} \frac{\partial Φ}{\partial σ}) + \sqrt{1 - τ^{2}} \frac{\partial}{\partial τ} (\sqrt{1 - τ^{2}} \frac{\partial Φ}{\partial τ})] .

Other differential operators such as $\nabla \cdot F$ and $\nabla \times F$ can be expressed in the coordinates $(σ, τ)$ 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 $p$ and $q$ that sum to a fixed vector $r = p + q$ , where the integrand was a function of the vector lengths $| p |$ and $| q |$ . (In such a case, one would position $r$ between the two foci and aligned with the $x$ -axis, i.e., $r = 2 a \hat{x}$ .) For concreteness, $r$ , $p$ and $q$ 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).