Oblate spheroidal coordinates

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Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius <math>a</math> in the <math>xy</math> plane. (Rotation about the other axis produces the prolate spheroidal coordinates.)

This coordinate system is particularly useful when the boundary conditions of a differential equation are defined on an oblate spheroid or a hyperboloid of revolution (e.g. a nozzle) or, in the degenerate cases, a circular disc (μ=0) or on a plane with a circular hole (ν=0).

[edit] Basic definition

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

<math>

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

<math>

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

<math>

z = a \ \sinh \mu \ \sin \nu </math>

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

The trigonometric identity

<math>

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

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

<math>

\frac{x^{2} + y^{2}}{a^{2} \cos^{2} \nu} - \frac{z^{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 coordinates <math>\mu</math> and <math>\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 \cosh\mu \ \cos\nu </math>

Consequently, an infinitesimal volume element equals

<math>

dV = a^{3} \cosh\mu \ \cos\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{1}{\cosh \mu} \frac{\partial}{\partial \mu} \left( \cosh \mu \frac{\partial \Phi}{\partial \mu} \right) + \frac{1}{\cos \nu} \frac{\partial}{\partial \nu} \left( \cos \nu \frac{\partial \Phi}{\partial \nu} \right) \right] + \frac{1}{a^{2} \left( \cosh^{2}\mu\cos^{2}\nu \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>(\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 oblate 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 focal ring. For any point, the sum <math>d_{1}+d_{2}</math> of its distances to the focal ring equals <math>2a\sigma</math>, whereas their difference <math>d_{1}-d_{2}</math> equals <math>2a\tau</math>. Thus, the "far" distance to the focal ring is <math>a(\sigma+\tau)</math>, whereas the "near" distance is <math>a(\sigma-\tau)</math>.

Unfortunately, oblate spheroid coordinates do not have a 1-to-1 transformation to the Cartesian coordinates

<math>

x = a\sigma\tau \cos \phi </math>

<math>

y = a\sigma\tau \sin \phi </math>

<math>

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

[edit] Alternative scale factors

The scale factors for the alternative oblate spheroidal 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>

whereas the azimuthal scale factor is <math>h_{\phi} = a \sigma \tau</math>.

Hence, the infinitesimal volume element can be written

<math>

dV = a^{3} \sigma \tau \frac{\sigma^{2} + \tau^{2}}{\sqrt{\left( \sigma^{2} + 1 \right) \left( 1 - \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 oblate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant oblate 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.