Prolate spheroidal coordinates

Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the axis on which the foci are located. Rotation about the other axis produces the oblate spheroidal coordinates.

This coordinate system can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the "z"-axis. One example is solving for the wavefunction of an electron moving in the electromagnetic field of two positively charged nuclei, as in the hydrogen ion, H₂⁺. Another example is solving for the electric field generated by two small electrode tips. Other limiting cases include fields generated by a line segment (μ=0) or a line with a missing segment (ν=0).

Definition

The most common definition of prolate spheroidal coordinates $(mu,\; u,\; phi)$ is

:$x\; =\; a\; sinh\; mu\; sin\; u\; cos\; phi$

:$y\; =\; a\; sinh\; mu\; sin\; u\; sin\; phi$

:$z\; =\; a\; cosh\; mu\; cos\; u$

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

The trigonometric identity

:$frac\{z^\{2\{a^\{2\}\; cosh^\{2\}\; mu\}\; +\; frac\{x^\{2\}\; +\; y^\{2\{a^\{2\}\; sinh^\{2\}\; mu\}\; =\; cos^\{2\}\; u\; +\; sin^\{2\}\; u\; =\; 1$

shows that surfaces of constant $mu$ form prolate spheroids, since they are ellipses rotated about the axis joining their foci. Similarly, the hyperbolic trigonometric identity

:$frac\{z^\{2\{a^\{2\}\; cos^\{2\}\; u\}\; -\; frac\{x^\{2\}\; +\; y^\{2\{a^\{2\}\; sin^\{2\}\; u\}\; =\; cosh^\{2\}\; mu\; -\; sinh^\{2\}\; mu\; =\; 1$

shows that surfaces of constant $u$ form
hyperboloids of revolution.

cale factors

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

:$h\_\{mu\}\; =\; h\_\{\; u\}\; =\; asqrt\{sinh^\{2\}mu\; +\; sin^\{2\}\; u\}$

whereas the azimuthal scale factor equals

:$h\_\{phi\}\; =\; a\; sinhmu\; sin\; u$

Consequently, an infinitesimal volume element equals

:$dV\; =\; a^\{3\}\; sinhmu\; sin\; u\; left(\; sinh^\{2\}mu\; +\; sin^\{2\}\; u\; ight)\; dmu\; d\; u\; dphi$

and the Laplacian can be written

:$abla^\{2\}\; Phi\; =\; frac\{1\}\{a^\{2\}\; left(\; sinh^\{2\}mu\; +\; sin^\{2\}\; u\; ight)\}\; left\; [frac\{partial^\{2\}\; Phi\}\{partial\; mu^\{2\; +\; frac\{partial^\{2\}\; Phi\}\{partial\; u^\{2\; +\; coth\; mu\; frac\{partial\; Phi\}\{partial\; mu\}\; +\; cot\; u\; frac\{partial\; Phi\}\{partial\; u\}\; ight]\; +\; frac\{1\}\{a^\{2\}\; sinh^\{2\}mu\; sin^\{2\}\; u\}frac\{partial^\{2\}\; Phi\}\{partial\; phi^\{2$

Other differential operators such as $abla\; cdot\; mathbf\{F\}$ and $abla\; imes\; mathbf\{F\}$ can be expressed in the coordinates $(mu,\; u,\; phi)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

Alternative definition

An alternative and geometrically intuitive set of prolate spheroidal coordinates $(sigma,\; au,\; phi)$ are sometimes used, where $sigma\; =\; cosh\; mu$ and $au\; =\; cos\; u$. Hence, the curves of constant $sigma$ are prolate spheroids, whereas the curves of constant $au$ are hyperboloids of revolution. The coordinate $au$ belongs to the interval [-1, 1] , whereas the $sigma$ coordinate must be greater than or equal to one. The coordinates $sigma$ and $au$ 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 $2asigma$, whereas their "difference" $d\_\{1\}-d\_\{2\}$ equals $2a\; au$. Thus, the distance to $F\_\{1\}$ is $a(sigma+\; au)$, whereas the distance to $F\_\{2\}$ is $a(sigma-\; au)$. (Recall that $F\_\{1\}$ and $F\_\{2\}$ are located at $z=-a$ and $z=+a$, respectively.)

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 (in technical language, a 1-to-1 mapping) between them and the Cartesian coordinates

:$x\; =\; a\; sqrt\{left(\; sigma^\{2\}\; -\; 1\; ight)\; left(1\; -\; au^\{2\}\; ight)\}\; cos\; phi$

:$y\; =\; a\; sqrt\{left(\; sigma^\{2\}\; -\; 1\; ight)\; left(1\; -\; au^\{2\}\; ight)\}\; sin\; phi$

:$z\; =\; a\; sigma\; au$

Alternative scale factors

The scale factors for the alternative elliptic coordinates $(sigma,\; au,\; phi)$ are

:$h\_\{sigma\}\; =\; asqrt\{frac\{sigma^\{2\}\; -\; au^\{2\{sigma^\{2\}\; -\; 1$

:$h\_\{\; au\}\; =\; asqrt\{frac\{sigma^\{2\}\; -\; au^\{2\{1\; -\; au^\{2\}$

while the azimuthal scale factor is now

:$h\_\{phi\}\; =\; a\; sqrt\{left(\; sigma^\{2\}\; -\; 1\; ight)\; left(\; 1\; -\; au^\{2\}\; ight)\}$

Hence, the infinitesimal volume element becomes

:$dV\; =\; a^\{3\}\; left(\; sigma^\{2\}\; -\; au^\{2\}\; ight)\; dsigma\; d\; au\; dphi$

and the Laplacian equals

:$abla^\{2\}\; Phi\; =\; frac\{1\}\{a^\{2\}\; left(\; sigma^\{2\}\; -\; au^\{2\}\; ight)\}left\{frac\{partial\}\{partial\; sigma\}\; left\; [\; left(\; sigma^\{2\}\; -\; 1\; ight)\; frac\{partial\; Phi\}\{partial\; sigma\}\; ight]\; +\; frac\{partial\}\{partial\; au\}\; left\; [\; left(\; 1\; -\; au^\{2\}\; ight)\; frac\{partial\; Phi\}\{partial\; au\}\; ight]\; ight\}+\; frac\{1\}\{a^\{2\}\; left(\; sigma^\{2\}\; -\; 1\; ight)\; left(\; 1\; -\; au^\{2\}\; ight)\}frac\{partial^\{2\}\; Phi\}\{partial\; phi^\{2$

Other differential operators such as $abla\; cdot\; mathbf\{F\}$ and $abla\; imes\; mathbf\{F\}$ can be expressed in the coordinates $(sigma,\; au)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

As is the case with spherical coordinates, Laplaces 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).

See also

References

Bibliography

No angles convention

* Uses ξ₁ = a cosh μ, ξ₂ = sin ν, and ξ₃ = cos φ.

* Same as Morse & Feshbach (1953), substituting "u"_"k" for ξ_"k".

*

* | pages = p. 97 Uses coordinates ξ = cosh μ, η = sin ν, and φ.

Angle convention

* | pages = p. 177 Korn and Korn use the (μ, ν, φ) coordinates, but also introduce the degenerate (σ, τ, φ) coordinates.

*| pages = pp. 180–182 Similar to Korn and Korn (1961), but uses colatitude θ = 90° - ν instead of latitude ν.

* Moon and Spencer use the colatitude convention θ = 90° - ν, and re-name φ as ψ.

Unusual convention

* 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.

External links

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