Paraboloidal coordinates
Paraboloidal coordinates are a three-dimensional orthogonal coordinate system <math>(\lambda, \mu, \nu)</math> that generalizes the two-dimensional parabolic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the paraboloidal coordinate system is not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
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Basic formulae
The Cartesian coordinates <math>(x, y, z)</math> can be produced from the ellipsoidal coordinates <math>( \lambda, \mu, \nu )</math> by the equations
- <math>
x^{2} = \frac{\left( A - \lambda \right) \left( A - \mu \right) \left( A - \nu \right)}{B - A}</math>
- <math>
y^{2} = \frac{\left( B - \lambda \right) \left( B - \mu \right) \left( B - \nu \right)}{A - B}</math>
- <math>
z = \frac{1}{2} \left( A + B - \lambda - \mu -\nu \right)</math>
where the following limits apply to the coordinates
- <math>
\lambda < B < \mu < A < \nu</math>
Consequently, surfaces of constant <math>\lambda</math> are elliptic paraboloids
- <math>
\frac{x^{2}}{\lambda - A} + \frac{y^{2}}{\lambda - B} = 2z + \lambda</math>
and surfaces of constant <math>\nu</math> are likewise
- <math>
\frac{x^{2}}{\nu - A} + \frac{y^{2}}{\nu - B} = 2z + \nu</math>
whereas surfaces of constant <math>\mu</math> are hyperbolic paraboloids
- <math>
\frac{x^{2}}{\mu - A} + \frac{y^{2}}{\mu - B} = 2z + \mu </math>
Scale factors
The scale factors for the paraboloidal coordinates <math>(\lambda, \mu, \nu )</math> are
- <math>
h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right)}{ \left( A - \lambda \right) \left( B - \lambda \right)}}</math>
- <math>
h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \nu - \mu \right) \left( \lambda - \mu \right)}{ \left( A - \mu \right) \left( B - \mu \right)}}</math>
- <math>
h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \nu \right) \left( \mu - \nu \right)}{ \left( A - \nu \right) \left( B - \nu \right)}}</math>
Hence, the infinitesimal volume element equals
- <math>
dV = \frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right) \left( \nu - \mu\right)}{8\sqrt{\left( A - \lambda \right) \left( B - \lambda \right) \left( A - \mu \right) \left( \mu - B \right) \left( \nu - A \right) \left( \nu - B \right) }} \ d\lambda d\mu d\nu</math>
Differential operators such as <math>\nabla \cdot \mathbf{F}</math> and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\lambda, \mu, \nu)</math> by substituting the scale factors into the general formulae found in orthogonal coordinates.
References
- Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.
Categories
Coordinate systems