elliptic cylindrical coordinates

The most common definition of elliptic cylindrical coordinates $(\backslash mu,\; \backslash nu,\; z)$ is

:$$

x = a \ \cosh \mu \ \cos \nu

:$$

y = a \ \sinh \mu \ \sin \nu

:$$

z = z

where $\backslash mu$ is a nonnegative real number and $\backslash nu\; \backslash in\; [0,\; 2\backslash pi]$.

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 $\backslash mu$ 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 $\backslash nu$ form hyperbolae.

The scale factors for the elliptic cylindrical coordinates $\backslash mu$ and $\backslash nu$ are equal

:$$

h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu}

whereas the remaining scale factor $h\_\{z\}=1$.

Consequently, an infinitesimal volume element equals

:$$

dV = a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu dz

and the Laplacian equals

:$$

\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{\partial^{2} \Phi}{\partial z^{2}}

Other differential operators such as $\backslash nabla\; \backslash cdot\; \backslash mathbf\{F\}$ and $\backslash nabla\; \backslash times\; \backslash mathbf\{F\}$ can be expressed in the coordinates $(\backslash mu,\; \backslash nu,\; z)$ by substituting

the scale factors into the general formulae found in orthogonal coordinates.

An alternative and geometrically intuitive set of elliptic coordinates $(\backslash sigma,\; \backslash tau,\; z)$ are sometimes used, where $\backslash sigma\; =\; \backslash cosh\; \backslash mu$ and $\backslash tau\; =\; \backslash cos\; \backslash nu$. Hence, the curves of constant $\backslash sigma$ are ellipses, whereas the curves of constant $\backslash tau$ are hyperbolae. The coordinate $\backslash tau$ must belong to the interval {{math|[−1, 1]}}, whereas the $\backslash sigma$

coordinate must be greater than or equal to one.

The coordinates $(\backslash sigma,\; \backslash tau,\; z)$ have a simple relation to the distances to the foci $F\_\{1\}$ and $F\_\{2\}$. For any point in the (x,y) plane, the sum $d\_\{1\}+d\_\{2\}$ of its distances to the foci equals $2a\backslash sigma$, whereas their difference $d\_\{1\}-d\_\{2\}$ equals $2a\backslash tau$.

Thus, the distance to $F\_\{1\}$ is $a(\backslash sigma+\backslash tau)$, whereas the distance to $F\_\{2\}$ is $a(\backslash sigma-\backslash tau)$. (Recall that $F\_\{1\}$ and $F\_\{2\}$ are located at $x=-a$ and $x=+a$, respectively.)

A drawback of these coordinates is that they do not have a 1-to-1 transformation to the Cartesian coordinates

:$$

x = a\sigma\tau

:$$

y^{2} = a^{2} \left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right)

The scale factors for the alternative elliptic coordinates $(\backslash sigma,\; \backslash tau,\; z)$ 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}}}

and, of course, $h\_\{z\}=1$. Hence, the infinitesimal volume element becomes

:$$

dV = a^{2} \frac{\sigma^{2} - \tau^{2}}{\sqrt{\left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)}} d\sigma d\tau dz

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] +

\frac{\partial^{2} \Phi}{\partial z^{2}}

Other differential operators such as $\backslash nabla\; \backslash cdot\; \backslash mathbf\{F\}$

and $\backslash nabla\; \backslash times\; \backslash mathbf\{F\}$ can be expressed in the coordinates $(\backslash sigma,\; \backslash tau)$ by substituting

the scale factors into the general formulae

found in orthogonal coordinates.

The classic applications of elliptic cylindrical coordinates are in solving partial differential equations,

e.g., Laplace's equation or the Helmholtz equation, for which elliptic cylindrical coordinates allow a

separation of variables. A typical example would be the electric field surrounding a

flat conducting plate of width $2a$.

The three-dimensional wave equation, when expressed in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equations.

The geometric properties of elliptic coordinates can also be useful. A typical example might involve

an integration over all pairs of vectors $\backslash mathbf\{p\}$ and $\backslash mathbf\{q\}$

that sum to a fixed vector $\backslash mathbf\{r\}\; =\; \backslash mathbf\{p\}\; +\; \backslash mathbf\{q\}$, where the integrand

was a function of the vector lengths $\backslash left|\; \backslash mathbf\{p\}\; \backslash right|$ and $\backslash left|\; \backslash mathbf\{q\}\; \backslash right|$. (In such a case, one would position $\backslash mathbf\{r\}$ between the two foci and aligned with the $x$-axis, i.e., $\backslash mathbf\{r\}\; =\; 2a\; \backslash mathbf\{\backslash hat\{x\}\}$.) For concreteness, $\backslash mathbf\{r\}$, $\backslash mathbf\{p\}$ and $\backslash mathbf\{q\}$ 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).

• {{cite book | author = Morse PM, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | isbn = 0-07-043316-X | page = 657 | lccn = 52011515}}
• {{cite book | author = Margenau H, Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | url = https://archive.org/details/mathematicsofphy0002marg| url-access = registration| publisher = D. van Nostrand | location = New York | pages = [https://archive.org/details/mathematicsofphy0002marg/page/182 182]–183 | lccn = 55010911 }}
• {{cite book | author = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | id = ASIN B0000CKZX7 | page = 179 | lccn = 59014456}}
• {{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | page = 97 | lccn = 67025285}}
• {{cite book | author = Zwillinger D | year = 1992 | title = Handbook of Integration | publisher = Jones and Bartlett | location = Boston, MA | isbn = 0-86720-293-9 | page = 114}} Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
• {{cite book | author = Moon P, Spencer DE | year = 1988 | chapter = Elliptic-Cylinder Coordinates (η, ψ, z) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer-Verlag | location = New York | pages = 17–20 (Table 1.03) | isbn = 978-0-387-18430-2}}

• [http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html MathWorld description of elliptic cylindrical coordinates]

{{Orthogonal coordinate systems}}

Category:Three-dimensional coordinate systems

Category:Orthogonal coordinate systems