elliptic coordinates

Not to be confused with Ecliptic coordinate system.

Elliptic coordinate system

Elliptic coordinates are 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 $(μ,ν)$ is

$x = a \ \cosh \mu \ \cos \nu$

$y = a \ \sinh \mu \ \sin \nu$

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

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 $μ$ 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 $ν$ form hyperbolae.

Scale factors

The scale factors for the elliptic coordinates $(μ,ν)$ are equal

$h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu}$

To simplify the computation of the scale factors, double angle identities can be used to express them equivalently as

$h_{\mu} = h_{\nu} = a\sqrt{\frac{1}{2} (\cosh2\mu - \cos2\nu})$

Consequently, an infinitesimal element of area equals

$dA = a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu$

and the Laplacian equals

$\nabla^{2} \Phi = \frac{1}{a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right)} \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right)$

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

Alternative definition

An alternative and geometrically intuitive set of elliptic coordinates $(σ,τ)$ are sometimes used, where $σ = coshμ$ and $τ = cosν$ . 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 they do not have a 1-to-1 transformation to the Cartesian coordinates

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

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

Alternative scale factors

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

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 allow a separation of variables. A typical example would be the electric field surrounding a flat conducting plate of width $2 a$ .

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).