Elliptic coordinate system

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Not to be confused with Ecliptic coordinate system. ‹See Tfd›
Elliptic coordinate system

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 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[edit]

The most common definition of elliptic coordinates (\mu, \nu) is


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

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

where \mu 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 \mu 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 \nu form hyperbolae.

Scale factors[edit]

In an orthogonal coordinate system the lengths of the basis vectors are known as scale factors. The scale factors for the elliptic coordinates (\mu, \nu) are equal to


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


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

Consequently, an infinitesimal element of area equals


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

and the Laplacian reads


\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).

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

Alternative definition[edit]

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

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


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

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

Alternative scale factors[edit]

The scale factors for the alternative elliptic coordinates (\sigma, \tau) 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 (\sigma, \tau) by substituting the scale factors into the general formulae found in orthogonal coordinates.

Extrapolation to higher dimensions[edit]

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[edit]

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 \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).

See also[edit]

References[edit]