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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.

Contents

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).

See also

  • Three dimensional orthogonal coordinate systems

References

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