Prolate spheroidal coordinates

File:ProlateSpheroidCoord.png

The most common definition of prolate spheroidal coordinates $(\backslash mu,\; \backslash nu,\; \backslash varphi)$ is

:$$

x = a \sinh \mu \sin \nu \cos \varphi

:$$

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

:$$

z = a \cosh \mu \cos \nu

where $\backslash mu$ is a nonnegative real number and $\backslash nu\; \backslash in\; [0,\; \backslash pi]$. The azimuthal angle $\backslash varphi$ belongs to the interval $[0,\; 2\backslash pi]$.

The trigonometric identity

:$$

\frac{z^2}{a^2 \cosh^2 \mu} + \frac{x^2 + y^2}{a^2 \sinh^2 \mu} = \cos^2 \nu + \sin^2 \nu = 1

shows that surfaces of constant $\backslash mu$ form prolate spheroids, since they are ellipses rotated about the axis

joining their foci. Similarly, the hyperbolic trigonometric identity

:$$

\frac{z^2}{a^2 \cos^2 \nu} - \frac{x^2 + y^2}{a^2 \sin^2 \nu} = \cosh^2 \mu - \sinh^2 \mu = 1

shows that surfaces of constant $\backslash nu$ form

hyperboloids of revolution.

The distances from the foci located at $(x,\; y,\; z)\; =\; (0,\; 0,\; \backslash pm\; a)$ are

:$$

r_\pm = \sqrt{x^2 + y^2 + (z \mp a)^2} = a(\cosh \mu \mp \cos \nu).

The scale factors for the elliptic coordinates $(\backslash mu,\; \backslash nu)$ are equal

:$$

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

whereas the azimuthal scale factor is

:$$

h_\varphi = a \sinh\mu \sin\nu,

resulting in a metric of

:$$

\begin{align}

ds^2 &= h_\mu^2 d\mu^2 + h_\nu^2 d\nu^2 + h_\varphi^2 d\varphi^2 \\

&= a^2 \left[ (\sinh^2\mu + \sin^2\nu) d\mu^2 + (\sinh^2\mu + \sin^2\nu) d\nu^2 + (\sinh^2\mu \sin^2\nu) d\varphi^2 \right].

\end{align}

Consequently, an infinitesimal volume element equals

:$$

dV = a^3 \sinh\mu \sin\nu ( \sinh^2 \mu + \sin^2 \nu) \, d\mu \, d\nu \, d\varphi

and the Laplacian can be written

:$$

\begin{align}

\nabla^2 \Phi = {} & \frac{1}{a^2 (\sinh^2 \mu + \sin^2 \nu)}

\left[ \frac{\partial^2 \Phi}{\partial \mu^2} +

\frac{\partial^2 \Phi}{\partial \nu^2} +

\coth \mu \frac{\partial \Phi}{\partial \mu} +

\cot \nu \frac{\partial \Phi}{\partial \nu}

\right] \\[6pt]

& {} + \frac{1}{a^2 \sinh^2 \mu \sin^2\nu}

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

\end{align}

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 $(\backslash mu,\; \backslash nu,\; \backslash varphi)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

File:Prolate spheroidal coordinates degenerate.png; this is illustrated here with two black spheres, one on each sheet of the hyperboloid and located at (x, y, ±z). However, neither of the definitions presented here are degenerate.]]

An alternative and geometrically intuitive set of prolate spheroidal coordinates $(\backslash sigma,\; \backslash tau,\; \backslash phi)$ 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 prolate spheroids, whereas the curves of constant $\backslash tau$ are hyperboloids of revolution. The coordinate $\backslash tau$ belongs to the interval [−1, 1], whereas the $\backslash sigma$ coordinate must be greater than or equal to one.

The coordinates $\backslash sigma$ and $\backslash tau$ 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 $2a\backslash sigma$, 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 $z=-a$ and $z=+a$, respectively.) This gives the following expressions for $\backslash sigma$, $\backslash tau$, and $\backslash varphi$:

:$$

\sigma = \frac 1 {2a} \left(\sqrt{x^2+y^2+(z+a)^2}+\sqrt{x^2+y^2+(z-a)^2}\right)

:$$

\tau = \frac 1 {2a} \left(\sqrt{x^2+y^2+(z+a)^2}-\sqrt{x^2+y^2+(z-a)^2}\right)

:$$

\varphi = \arctan\left(\frac y x \right)

Unlike the analogous oblate spheroidal coordinates, the prolate spheroid coordinates (σ, τ, φ) are not degenerate; in other words, there is a unique, reversible correspondence between them and the Cartesian coordinates

:$$

x = a \sqrt{(\sigma^2 - 1) (1 - \tau^2)} \cos \varphi

:$$

y = a \sqrt{(\sigma^2 - 1) (1 - \tau^2)} \sin \varphi

:$$

z = a\ \sigma\ \tau

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

while the azimuthal scale factor is now

:$$

h_\varphi = a \sqrt{\left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)}

Hence, the infinitesimal volume element becomes

:$$

dV = a^3 (\sigma^2 - \tau^2) \, d\sigma \, d\tau \, d\varphi

and the Laplacian equals

:$$

\begin{align}

\nabla^2 \Phi = {} &

\frac{1}{a^2 (\sigma^2 - \tau^2)}

\left\{ \frac{\partial}{\partial \sigma} \left[

\left( \sigma^2 - 1 \right) \frac{\partial \Phi}{\partial \sigma}

\right] + \frac{\partial}{\partial \tau} \left[

(1 - \tau^2) \frac{\partial \Phi}{\partial \tau}

\right]

\right\} \\

& {} + \frac{1}{a^2 (\sigma^2 - 1) (1 - \tau^2)}

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

\end{align}

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.

As is the case with spherical coordinates, Laplace's equation may be solved by the method of separation of variables to yield solutions in the form of prolate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant prolate spheroidal coordinate (See Smythe, 1968).

{{reflist}}

• {{cite book |vauthors=Morse PM, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | page = 661}} Uses ξ₁ = a cosh μ, ξ₂ = sin ν, and ξ₃ = cos φ.
• {{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 | last = Smythe | first = WR| title = Static and Dynamic Electricity |edition = 3rd | publisher = McGraw-Hill | location = New York | year = 1968}}
• {{cite book |vauthors=Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | page = 97 | lccn = 67025285}} Uses coordinates ξ = cosh μ, η = sin ν, and φ.

• {{cite book |vauthors=Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers |url=https://archive.org/details/mathematicalhand0000korn |url-access=registration | publisher = McGraw-Hill | location = New York | page = [https://archive.org/details/mathematicalhand0000korn/page/177 177] | lccn = 59014456}} Korn and Korn use the (μ, ν, φ) coordinates, but also introduce the degenerate (σ, τ, φ) coordinates.
• {{cite book |vauthors=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/180 180]–182 | lccn = 55010911 }} Similar to Korn and Korn (1961), but uses colatitude θ = 90° - ν instead of latitude ν.
• {{cite book |vauthors=Moon PH, Spencer DE | year = 1988 | chapter = Prolate Spheroidal Coordinates (η, θ, ψ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer Verlag | location = New York | isbn = 0-387-02732-7 | pages = 28–30 (Table 1.06)}} Moon and Spencer use the colatitude convention θ = 90° − ν, and rename φ as ψ.

• {{cite book |vauthors=Landau LD, Lifshitz EM, Pitaevskii LP | year = 1984 | title = Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) | edition = 2nd | publisher = Pergamon Press | location = New York | isbn = 978-0-7506-2634-7 | pages = 19–29 }} Treats the prolate spheroidal coordinates as a limiting case of the general ellipsoidal coordinates. Uses (ξ, η, ζ) coordinates that have the units of distance squared.

• [http://mathworld.wolfram.com/ProlateSpheroidalCoordinates.html MathWorld description of prolate spheroidal coordinates]

{{Orthogonal coordinate systems}}

Category:Three-dimensional coordinate systems

Category:Orthogonal coordinate systems