Elliptic coordinate system

Short description: 2D coordinate system whose coordinate lines are confocal ellipses and hyperbolae

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 [math]\displaystyle{ F_{1} }[/math] and [math]\displaystyle{ F_{2} }[/math] are generally taken to be fixed at [math]\displaystyle{ -a }[/math] and [math]\displaystyle{ +a }[/math], respectively, on the [math]\displaystyle{ x }[/math]-axis of the Cartesian coordinate system.

Basic definition

The most common definition of elliptic coordinates [math]\displaystyle{ (\mu, \nu) }[/math] is

[math]\displaystyle{ \begin{align} x &= a \ \cosh \mu \ \cos \nu \\ y &= a \ \sinh \mu \ \sin \nu \end{align} }[/math]

where [math]\displaystyle{ \mu }[/math] is a nonnegative real number and [math]\displaystyle{ \nu \in [0, 2\pi]. }[/math]

On the complex plane, an equivalent relationship is

[math]\displaystyle{ x + iy = a \ \cosh(\mu + i\nu) }[/math]

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

[math]\displaystyle{ \frac{x^{2}}{a^{2} \cosh^{2} \mu} + \frac{y^{2}}{a^{2} \sinh^{2} \mu} = \cos^{2} \nu + \sin^{2} \nu = 1 }[/math]

shows that curves of constant [math]\displaystyle{ \mu }[/math] form ellipses, whereas the hyperbolic trigonometric identity

[math]\displaystyle{ \frac{x^{2}}{a^{2} \cos^{2} \nu} - \frac{y^{2}}{a^{2} \sin^{2} \nu} = \cosh^{2} \mu - \sinh^{2} \mu = 1 }[/math]

shows that curves of constant [math]\displaystyle{ \nu }[/math] 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 [math]\displaystyle{ (\mu, \nu) }[/math] are equal to

[math]\displaystyle{ h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu} = a\sqrt{\cosh^{2}\mu - \cos^{2}\nu}. }[/math]

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

[math]\displaystyle{ h_{\mu} = h_{\nu} = a\sqrt{\frac{1}{2} (\cosh2\mu - \cos2\nu)}. }[/math]

Consequently, an infinitesimal element of area equals

[math]\displaystyle{ \begin{align} 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 \end{align} }[/math]

and the Laplacian reads

[math]\displaystyle{ \begin{align} \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) \end{align} }[/math]

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

Alternative definition

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

The coordinates [math]\displaystyle{ (\sigma, \tau) }[/math] have a simple relation to the distances to the foci [math]\displaystyle{ F_{1} }[/math] and [math]\displaystyle{ F_{2} }[/math]. For any point in the plane, the sum [math]\displaystyle{ d_{1}+d_{2} }[/math] of its distances to the foci equals [math]\displaystyle{ 2a\sigma }[/math], whereas their difference [math]\displaystyle{ d_{1}-d_{2} }[/math] equals [math]\displaystyle{ 2a\tau }[/math]. Thus, the distance to [math]\displaystyle{ F_{1} }[/math] is [math]\displaystyle{ a(\sigma+\tau) }[/math], whereas the distance to [math]\displaystyle{ F_{2} }[/math] is [math]\displaystyle{ a(\sigma-\tau) }[/math]. (Recall that [math]\displaystyle{ F_{1} }[/math] and [math]\displaystyle{ F_{2} }[/math] are located at [math]\displaystyle{ x=-a }[/math] and [math]\displaystyle{ x=+a }[/math], respectively.)

A drawback of these coordinates is that the points with Cartesian coordinates (x,y) and (x,-y) have the same coordinates [math]\displaystyle{ (\sigma, \tau) }[/math], so the conversion to Cartesian coordinates is not a function, but a multifunction.

[math]\displaystyle{ x = a \left. \sigma \right. \tau }[/math]

[math]\displaystyle{ y^{2} = a^{2} \left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right). }[/math]

Alternative scale factors

The scale factors for the alternative elliptic coordinates [math]\displaystyle{ (\sigma, \tau) }[/math] are

[math]\displaystyle{ h_{\sigma} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{\sigma^{2} - 1}} }[/math]

[math]\displaystyle{ h_{\tau} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{1 - \tau^{2}}}. }[/math]

Hence, the infinitesimal area element becomes

[math]\displaystyle{ dA = a^{2} \frac{\sigma^{2} - \tau^{2}}{\sqrt{\left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)}} d\sigma d\tau }[/math]

and the Laplacian equals

[math]\displaystyle{ \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]. }[/math]

Other differential operators such as [math]\displaystyle{ \nabla \cdot \mathbf{F} }[/math] and [math]\displaystyle{ \nabla \times \mathbf{F} }[/math] can be expressed in the coordinates [math]\displaystyle{ (\sigma, \tau) }[/math] 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:

1. The elliptic cylindrical coordinates are produced by projecting in the [math]\displaystyle{ z }[/math]-direction.
2. The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the [math]\displaystyle{ x }[/math]-axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the [math]\displaystyle{ y }[/math]-axis, i.e., the axis separating the foci.
3. 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 [math]\displaystyle{ \mathbf{p} }[/math] and [math]\displaystyle{ \mathbf{q} }[/math] that sum to a fixed vector [math]\displaystyle{ \mathbf{r} = \mathbf{p} + \mathbf{q} }[/math], where the integrand was a function of the vector lengths [math]\displaystyle{ \left| \mathbf{p} \right| }[/math] and [math]\displaystyle{ \left| \mathbf{q} \right| }[/math]. (In such a case, one would position [math]\displaystyle{ \mathbf{r} }[/math] between the two foci and aligned with the [math]\displaystyle{ x }[/math]-axis, i.e., [math]\displaystyle{ \mathbf{r} = 2a \mathbf{\hat{x}} }[/math].) For concreteness, [math]\displaystyle{ \mathbf{r} }[/math], [math]\displaystyle{ \mathbf{p} }[/math] and [math]\displaystyle{ \mathbf{q} }[/math] 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

• Hazewinkel, Michiel, ed. (2001), "Elliptic coordinates", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/e035440 
• 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