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

Basic definition

The most common definition of elliptic coordinates <math>(\mu, \nu)</math> is

<math>

x = a \ \cosh \mu \ \cos \nu</math>

<math>

y = a \ \sinh \mu \ \sin \nu</math>

where <math>\mu</math> is a nonnegative real number and <math>\nu \in [0, 2\pi)</math>.

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

<math>

\frac{x^{2}}{a^{2} \cosh^{2} \mu} + \frac{y^{2}}{a^{2} \sinh^{2} \mu} = \cos^{2} \nu + \sin^{2} \nu = 1</math>

shows that curves of constant <math>\mu</math> form ellipses, whereas the hyperbolic trigonometric identity

<math>

\frac{x^{2}}{a^{2} \cos^{2} \nu} - \frac{y^{2}}{a^{2} \sin^{2} \nu} = \cosh^{2} \mu - \sinh^{2} \mu = 1</math>

shows that curves of constant <math>\nu</math> form hyperbolae.

Scale factors

The scale factors for the elliptic coordinates <math>(\mu, \nu)</math> are equal

<math>

h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu}</math>

Consequently, an infinitesimal element of area equals

<math>

dA = a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu</math>

and the Laplacian equals

<math>

\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)</math>

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

Alternative definition

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

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

A drawback of these coordinates is that they do not have a 1-to-1 transformation to the Cartesian coordinates

<math>

x = a\sigma\tau</math>

<math>

y^{2} = a^{2} \left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right)</math>

Alternative scale factors

The scale factors for the alternative elliptic coordinates <math>(\sigma, \tau)</math> are

<math>

h_{\sigma} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{\sigma^{2} - 1}}</math>

<math>

h_{\tau} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{1 - \tau^{2}}}</math>

Hence, the infinitesimal area element becomes

<math>

dA = a^{2} \frac{\sigma^{2} - \tau^{2}}{\sqrt{\left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)}} d\sigma d\tau</math>

and the Laplacian equals

<math>

\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]</math>

Other differential operators such as <math>\nabla \cdot \mathbf{F}</math> and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau)</math> 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 <math>z</math>-direction.The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the <math>x</math>-axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the <math>y</math>-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 <math>2a</math>.

The geometric properties of elliptic coordinates can also be useful. A typical example might involve an integration over all pairs of vectors <math>\mathbf{p}</math> and <math>\mathbf{q}</math> that sum to a fixed vector <math>\mathbf{r} = \mathbf{p} + \mathbf{q}</math>, where the integrand was a function of the vector lengths <math>\left| \mathbf{p} \right|</math> and <math>\left| \mathbf{q} \right|</math>. (In such a case, one would position <math>\mathbf{r}</math> between the two foci and aligned with the <math>x</math>-axis, i.e., <math>\mathbf{r} = 2a \mathbf{\hat{x}}</math>.) For concreteness, <math>\mathbf{r}</math>, <math>\mathbf{p}</math> and <math>\mathbf{q}</math> 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

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

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