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 $$F_{1}$$ 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


 * $$\begin{align}

x &= a \ \cosh \mu \ \cos \nu \\ y &= a \ \sinh \mu \ \sin \nu \end{align}$$

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


 * $$h_{\mu} = h_{\nu} = a\sqrt{\frac{1}{2} (\cosh2\mu - \cos2\nu)}.$$

Consequently, an infinitesimal element of area equals


 * $$\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}$$

and the Laplacian reads


 * $$\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}$$

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:
 * 1) The elliptic cylindrical coordinates are produced by projecting in the $$z$$-direction.
 * 2) 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.
 * 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 $$\mathbf{p}$$ 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).