Elliptic coordinates

Elliptic coordinates

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 (mu, u) is

:x = a cosh mu cos u

:y = a sinh mu sin u

where mu is a nonnegative real number and u in [0, 2pi).

On the complex plane, an equivalent relationship is

:x + iy = a cosh(mu + i u)

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} u + sin^{2} u = 1

shows that curves of constant mu form ellipses, whereas the hyperbolic trigonometric identity

:frac{x^{2{a^{2} cos^{2} u} - frac{y^{2{a^{2} sin^{2} u} = cosh^{2} mu - sinh^{2} mu = 1

shows that curves of constant u form hyperbolae.

cale factors

The scale factors for the elliptic coordinates (mu, u) are equal

:h_{mu} = h_{ u} = asqrt{sinh^{2}mu + sin^{2} u}.

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

:h_{mu} = h_{ u} = asqrt{frac{1}{2} (cosh2mu - cos2 u}).

Consequently, an infinitesimal element of area equals

:dA = a^{2} left( sinh^{2}mu + sin^{2} u ight) dmu d u

and the Laplacian equals

: abla^{2} Phi = frac{1}{a^{2} left( sinh^{2}mu + sin^{2} u ight)} left( frac{partial^{2} Phi}{partial mu^{2 + frac{partial^{2} Phi}{partial u^{2 ight).

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

Alternative definition

An alternative and geometrically intuitive set of elliptic coordinates (sigma, au) are sometimes used, where sigma = cosh mu and au = cos u. Hence, the curves of constant sigma are ellipses, whereas the curves of constant au are hyperbolae. The coordinate au must belong to the interval [-1, 1] , whereas the sigma coordinate must be greater than or equal to one. The coordinates (sigma, au) 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 2asigma, whereas their "difference" d_{1}-d_{2} equals 2a au.Thus, the distance to F_{1} is a(sigma+ au), whereas the distance to F_{2} is a(sigma- au). (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 ight. au

:y^{2} = a^{2} left( sigma^{2} - 1 ight) left(1 - au^{2} ight).

Alternative scale factors

The scale factors for the alternative elliptic coordinates (sigma, au) are

:h_{sigma} = asqrt{frac{sigma^{2} - au^{2{sigma^{2} - 1

:h_{ au} = asqrt{frac{sigma^{2} - au^{2{1 - au^{2}.

Hence, the infinitesimal area element becomes

:dA = a^{2} frac{sigma^{2} - au^{2{sqrt{left( sigma^{2} - 1 ight) left( 1 - au^{2} ight) dsigma d au

and the Laplacian equals

: abla^{2} Phi = frac{1}{a^{2} left( sigma^{2} - au^{2} ight) }left [sqrt{sigma^{2} - 1} frac{partial}{partial sigma} left( sqrt{sigma^{2} - 1} frac{partial Phi}{partial sigma} ight) + sqrt{1 - au^{2 frac{partial}{partial au} left( sqrt{1 - au^{2 frac{partial Phi}{partial au} ight) ight] .

Other differential operators such as abla cdot mathbf{F} and abla imes mathbf{F} can be expressed in the coordinates (sigma, au) 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} ight| and left| mathbf{q} ight|. (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).

ee also

*Curvilinear coordinates
*Generalized coordinates

References

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


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