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.)
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[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 [0, 2\pi)</math>. The azimuthal angle <math>\phi</math> also 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 = \sinh \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.
[edit] References
- Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.
