Elliptic coordinate system

Elliptic coordinate system

In geometry, the elliptic(al) 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

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

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

${\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {{\frac {1}{2}}(\cosh 2\mu -\cos 2\nu )}}.}$

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

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

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

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 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

• Curvilinear coordinates
• Generalized coordinates

References

• Hazewinkel, Michiel, ed. (2001), "Elliptic coordinates", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• 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