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 $F_$ and $F_$ 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 $(\mu, \nu)$ is

:$\begin x &= a \ \cosh \mu \ \cos \nu \\ y &= a \ \sinh \mu \ \sin \nu \end$

where $\mu$ is a nonnegative real number and \nu \in , 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

:$\frac + \frac = \cos^ \nu + \sin^ \nu = 1$

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

:$\frac - \frac = \cosh^ \mu - \sinh^ \mu = 1$

shows that curves of constant $\nu$ form hyperbolae.

Scale factors

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

:$h_ = h_ = a\sqrt = a\sqrt.$

Using the ''double argument identities'' for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as

:$h_ = h_ = a\sqrt.$

Consequently, an infinitesimal element of area equals

:$\begin dA &= h_ h_ d\mu d\nu \\ &= a^ \left( \sinh^\mu + \sin^\nu \right) d\mu d\nu \\ &= a^ \left( \cosh^\mu - \cos^\nu \right) d\mu d\nu \\ &= \frac \left( \cosh 2 \mu - \cos 2\nu \right) d\mu d\nu \end$

and the Laplacian reads

:$\begin \nabla^ \Phi &= \frac \left( \frac + \frac \right) \\ &= \frac \left( \frac + \frac \right) \\ &= \frac \left( \frac + \frac \right) \end$

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

Alternative definition

An alternative and geometrically intuitive set of elliptic coordinates $(\sigma, \tau)$ 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 $F_$ and $F_$. For any point in the plane, the ''sum'' $d_+d_$ of its distances to the foci equals $2a\sigma$, whereas their ''difference'' $d_-d_$ equals $2a\tau$.
Thus, the distance to $F_$ is $a(\sigma+\tau)$, whereas the distance to $F_$ is $a(\sigma-\tau)$. (Recall that $F_$ and $F_$ 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.

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

:$y^ = a^ \left( \sigma^ - 1 \right) \left(1 - \tau^ \right).$

Alternative scale factors

The scale factors for the alternative elliptic coordinates $(\sigma, \tau)$ are

:$h_ = a\sqrt$

:$h_ = a\sqrt.$

Hence, the infinitesimal area element becomes

:$dA = a^ \frac d\sigma d\tau$

and the Laplacian equals

:
\nabla^ \Phi =
\frac
\left \sqrt \frac
\left( \sqrt \frac \right) +
\sqrt \frac
\left( \sqrt \frac \right)
\right

Other differential operators such as $\nabla \cdot \mathbf$
and $\nabla \times \mathbf$ 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

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.
# Ellipsoidal coordinates are a formal extension of elliptic coordinates into 3-dimensions, which is based on confocal ellipsoids, hyperboloids of one and two sheets.

Note that (ellipsoidal) Geographic coordinate system is a different concept from above.

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 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 $\mathbf$ and $\mathbf$
that sum to a fixed vector $\mathbf = \mathbf + \mathbf$, where the integrand
was a function of the vector lengths $\left, \mathbf \$ and $\left, \mathbf \$. (In such a case, one would position $\mathbf$ between the two foci and aligned with the $x$-axis, i.e., $\mathbf = 2a \mathbf$.) For concreteness, $\mathbf$, $\mathbf$ and $\mathbf$ 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

*Curvilinear coordinates
* Ellipsoidal coordinates
*Generalized coordinates
*Bipolar coordinates

References

*
* Korn GA and Korn TM. (1961) ''Mathematical Handbook for Scientists and Engineers'', McGraw-Hill.
* Weisstein, Eric W. "Elliptic Cylindrical Coordinates." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html

{{Orthogonal coordinate systems

Two-dimensional coordinate systems
Ellipses