Elliptic coordinate system

Not to be confused with Ecliptic coordinate system.

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

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

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

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

Consequently, an infinitesimal element of area equals

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

and the Laplacian reads

${\begin{aligned}\nabla ^{2}\Phi &={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\\&={\frac {1}{a^{2}\left(\cosh ^{2}\mu -\cos ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\\&={\frac {2}{a^{2}\left(\cosh 2\mu -\cos 2\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)\end{aligned}}$

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).

See also

• Curvilinear coordinates
• Ellipsoidal coordinates
• Generalized coordinates
• Bipolar coordinates

References

• "Elliptic coordinates", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• 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