Prolate spheroidal coordinates

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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 is particularly useful when the boundary conditions of a differential equation are defined on a prolate spheroid or a hyperboloid of revolution (e.g. two electrode tips) or, in the degenerate cases, a line segment (μ=0) or a line with a missing segment (ν=0).

[edit] Basic definition

The most common definition of prolate spheroidal coordinates <math>(\mu, \nu, \phi)</math> is

<math>

x = a \ \sinh \mu \ \sin \nu \ \cos \phi </math>

<math>

y = a \ \sinh \mu \ \sin \nu \ \sin \phi </math>

<math>

z = a \ \cosh \mu \ \cos \nu </math>

where <math>\mu</math> is a nonnegative real number and <math>\nu \in [0, \pi]</math>. The azimuthal angle <math>\phi</math> belongs to the interval <math>[0, 2\pi)</math>.

The trigonometric identity

<math>

\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 </math>

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

<math>

\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 </math>

shows that surfaces of constant <math>\nu</math> form hyperboloids of revolution.

[edit] Scale factors

The scale factors for the elliptic coordinates <math>(\mu, \nu)</math> are equal

<math>

h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu} </math>

whereas the azimuthal scale factor equals

<math>

h_{\phi} = a \sinh\mu \ \sin\nu </math>

Consequently, an infinitesimal volume element equals

<math>

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

and the Laplacian can be written

<math>

\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}} </math>

Other differential operators such as <math>\nabla \cdot \mathbf{F}</math> and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\mu, \nu)</math> 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 <math>(\sigma, \tau, \phi)</math> are sometimes used, where <math>\sigma = \cosh \mu</math> and <math>\tau = \cos \nu</math>. Hence, the curves of constant <math>\sigma</math> are prolate spheroids, whereas the curves of constant <math>\tau</math> are hyperboloids of revolution. The coordinate <math>\tau</math> belongs to the interval [-1, 1], whereas the <math>\sigma</math> coordinate must be greater than or equal to one.

The coordinates <math>\sigma</math> and <math>\tau</math> have a simple relation to the distances to the foci <math>F_{1}</math> and <math>F_{2}</math>. For any point in the plane, the sum <math>d_{1}+d_{2}</math> of its distances to the foci equals <math>2a\sigma</math>, whereas their difference <math>d_{1}-d_{2}</math> equals <math>2a\tau</math>. Thus, the distance to <math>F_{1}</math> is <math>a(\sigma+\tau)</math>, whereas the distance to <math>F_{2}</math> is <math>a(\sigma-\tau)</math>. (Recall that <math>F_{1}</math> and <math>F_{2}</math> are located at <math>x=-a</math> and <math>x=+a</math>, respectively.)

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

<math>

x = a \sqrt{\left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right)} \cos \phi </math>

<math>

y = a \sqrt{\left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right)} \sin \phi </math>

<math>

z = a\ \sigma\ \tau </math>

[edit] Alternative scale factors

The scale factors for the alternative elliptic coordinates <math>(\sigma, \tau, \phi)</math> are

<math>

h_{\sigma} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{\sigma^{2} - 1}} </math>

<math>

h_{\tau} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{1 - \tau^{2}}} </math>

while the azimuthal scale factor is now

<math>

h_{\phi} = a \sqrt{\left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)} </math>

Hence, the infinitesimal volume element becomes

<math>

dV = a^{3} \left( \sigma^{2} - \tau^{2} \right) d\sigma d\tau d\phi </math>

and the Laplacian equals

<math>

\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}} </math>

Other differential operators such as <math>\nabla \cdot \mathbf{F}</math> and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau)</math> 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).

[edit] References

• Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.
• Smythe, William R. (1968). Static and Dynamic Electricity, 3rd ed., New York: McGraw-Hill.