Tijdeman's theorem

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

$dA=h_{\mu }{h}_{\nu }d\mu d\nu =a^2({\sinh }^2\mu +{\sin }^2\nu )d\mu d\nu =a^2({\cosh }^2\mu -{\cos }^2\nu )d\mu d\nu =\frac{a^2}{2}(\cosh 2\mu -\cos 2\nu )d\mu d\nu$

and the Laplacian reads

${\mathrm{\nabla }}^2\Phi =\frac{1}{a^2({\sinh }^2\mu +{\sin }^2\nu )}(\frac{{\partial }^2\Phi }{\partial {\mu }^2}+\frac{{\partial }^2\Phi }{\partial {\nu }^2})=\frac{1}{a^2({\cosh }^2\mu -{\cos }^2\nu )}(\frac{{\partial }^2\Phi }{\partial {\mu }^2}+\frac{{\partial }^2\Phi }{\partial {\nu }^2})=\frac{2}{a^2(\cosh 2\mu -\cos 2\nu )}(\frac{{\partial }^2\Phi }{\partial {\mu }^2}+\frac{{\partial }^2\Phi }{\partial {\nu }^2}).$

Other differential operators such as $\mathrm{\nabla }\cdot 𝐅$ and $\mathrm{\nabla }\times 𝐅$ 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.

$x=a\sigma \tau$

$y^2=a^2({\sigma }^2-1)(1-{\tau }^2).$

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

$dA=a^2\frac{{\sigma }^2-{\tau }^2}{\sqrt{({\sigma }^2-1)(1-{\tau }^2)}}d\sigma d\tau$

and the Laplacian equals

${\mathrm{\nabla }}^2\Phi =\frac{1}{a^2({\sigma }^2-{\tau }^2)}[\sqrt{{\sigma }^2-1}\frac{\partial }{\partial \sigma }\left(\sqrt{{\sigma }^2-1}\frac{\partial \Phi }{\partial \sigma }\right)+\sqrt{1-{\tau }^2}\frac{\partial }{\partial \tau }\left(\sqrt{1-{\tau }^2}\frac{\partial \Phi }{\partial \tau }\right)].$

Other differential operators such as $\mathrm{\nabla }\cdot 𝐅$ and $\mathrm{\nabla }\times 𝐅$ 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 $𝐩$ and $𝐪$ that sum to a fixed vector $𝐫=𝐩+𝐪$, where the integrand was a function of the vector lengths $\left|𝐩\right|$ and $\left|𝐪\right|$. (In such a case, one would position $𝐫$ between the two foci and aligned with the $x$-axis, i.e., $𝐫=2a\widehat{𝐱}$.) For concreteness, $𝐫$, $𝐩$ and $𝐪$ 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
• Mean motion

References

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• 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