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# 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

$$x=a\ \cosh \mu \ \cos \nu$$
$$y=a\ \sinh \mu \ \sin \nu$$

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

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

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

Categories: Two-dimensional coordinate systems

Information as of: 09.07.2020 10:20:11 CEST

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