Prolate spheroidal coordinates

From Wikipedia, the free encyclopedia

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.

1 Basic definition
2 Scale factors
3 Alternative definition
4 Alternative scale factors
5 References

[edit] Basic definition

The most common definition of prolate spheroidal coordinates $(μ,ν,φ)$ is

$x = a \ \sinh \mu \ \sin \nu \ \cos \phi$

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

$z = a \ \cosh \mu \ \cos \nu$

where $μ$ is a nonnegative real number and $\nu \in [0, 2\pi)$ . The azimuthal angle $φ$ also belongs to the interval $[0,2π)$ .

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 $μ$ 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 $ν$ form hyperboloids of revolution.

[edit] Scale factors

The scale factors for the elliptic coordinates $(μ,ν)$ are equal

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

whereas the azimuthal scale factor equals

$h_{\phi} = a \sinh\mu \ \sin\nu$

Consequently, an infinitesimal volume element equals

$dV = a^{3} \sinh\mu \ \sin\nu \ \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu d\phi$

and the Laplacian can be written

$\nabla^{2} \Phi = \frac{1}{a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right)} \left[ \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} + \coth \mu \frac{\partial \Phi}{\partial \mu} + \cot \nu \frac{\partial \Phi}{\partial \nu} \right] + \frac{1}{a^{2} \sinh^{2}\mu \sin^{2}\nu} \frac{\partial^{2} \Phi}{\partial \phi^{2}}$

Other differential operators such as $\nabla \cdot \mathbf{F}$ and $\nabla \times \mathbf{F}$ can be expressed in the coordinates $(μ,ν)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

[edit] Alternative definition

An alternative and geometrically intuitive set of prolate spheroidal coordinates $(σ,τ,φ)$ are sometimes used, where $σ = coshμ$ and $τ = cosν$ . Hence, the curves of constant $σ$ are prolate spheroids, whereas the curves of constant $τ$ are hyperboloids of revolution. The coordinate $τ$ belongs to the interval [-1, 1], whereas the $σ$ coordinate must be greater than or equal to one.

The coordinates $σ$ and $τ$ 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 $2 a σ$ , whereas their difference $d 1 - d 2$ equals $2 a τ$ . Thus, the distance to $F 1$ is $a (σ + τ)$ , whereas the distance to $F 2$ is $a (σ - τ)$ . (Recall that $F 1$ and $F 2$ are located at $x = - a$ and $x = + a$ , respectively.)

Unlike the similar elliptic coordinates, prolate spheroid coordinates do have a 1-to-1 transformation to the Cartesian coordinates

$x = a \sqrt{\left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right)} \cos \phi$

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

z = a στ

[edit] Alternative scale factors

The scale factors for the alternative elliptic coordinates $(σ,τ,φ)$ 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

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

Hence, the infinitesimal volume element becomes

$dV = a^{3} \left( \sigma^{2} - \tau^{2} \right) d\sigma d\tau d\phi$

and the Laplacian equals

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

Other differential operators such as $\nabla \cdot \mathbf{F}$ and $\nabla \times \mathbf{F}$ can be expressed in the coordinates $(σ,τ)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.