Paraboloidal coordinates

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Paraboloidal coordinates are a three-dimensional orthogonal coordinate system $(λ,μ,ν)$ 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.

[edit] Basic formulae

The Cartesian coordinates $(x, y, z)$ can be produced from the ellipsoidal coordinates $(λ,μ,ν)$ by the equations

$x^{2} = \frac{\left( A - \lambda \right) \left( A - \mu \right) \left( A - \nu \right)}{B - A}$

$y^{2} = \frac{\left( B - \lambda \right) \left( B - \mu \right) \left( B - \nu \right)}{A - B}$

$z = \frac{1}{2} \left( A + B - \lambda - \mu -\nu \right)$

where the following limits apply to the coordinates

λ < B < μ < A < ν

Consequently, surfaces of constant $λ$ are elliptic paraboloids

$\frac{x^{2}}{\lambda - A} + \frac{y^{2}}{\lambda - B} = 2z + \lambda$

and surfaces of constant $ν$ are likewise

$\frac{x^{2}}{\nu - A} + \frac{y^{2}}{\nu - B} = 2z + \nu$

whereas surfaces of constant $μ$ are hyperbolic paraboloids

$\frac{x^{2}}{\mu - A} + \frac{y^{2}}{\mu - B} = 2z + \mu$

[edit] Scale factors

The scale factors for the paraboloidal coordinates $(λ,μ,ν)$ are

$h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right)}{ \left( A - \lambda \right) \left( B - \lambda \right)}}$

$h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \nu - \mu \right) \left( \lambda - \mu \right)}{ \left( A - \mu \right) \left( B - \mu \right)}}$

$h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \nu \right) \left( \mu - \nu \right)}{ \left( A - \nu \right) \left( B - \nu \right)}}$

Hence, the infinitesimal volume element equals

$dV = \frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right) \left( \nu - \mu\right)}{8\sqrt{\left( A - \lambda \right) \left( B - \lambda \right) \left( A - \mu \right) \left( \mu - B \right) \left( \nu - A \right) \left( \nu - B \right) }} \ d\lambda d\mu d\nu$

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