# Elliptic coordinates: Wikis

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# Encyclopedia

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 F1 and F2 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 $\nu \in [0, 2\pi).$

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 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 F1 and F2. For any point in the plane, the sum d1 + d2 of its distances to the foci equals 2aσ, whereas their difference d1d2 equals 2aτ. Thus, the distance to F1 is a(σ + τ), whereas the distance to F2 is a(σ − τ). (Recall that F1 and F2 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 2a.

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).

• Three dimensional orthogonal coordinate systems

## References

• Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.