Bispherical coordinates
Paraboloidal coordinates are a three-dimensional orthogonal coordinate system that generalizes the two-dimensional parabolic coordinate system. Similar to the related ellipsoidal coordinates, the paraboloidal coordinate system has orthogonal quadratic coordinate surfaces that are not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
Basic formulae
The Cartesian coordinates can be produced from the ellipsoidal coordinates by the equations
where the following limits apply to the coordinates
Consequently, surfaces of constant are elliptic paraboloids
and surfaces of constant are likewise
whereas surfaces of constant are hyperbolic paraboloids
Scale factors
The scale factors for the paraboloidal coordinates are
Hence, the infinitesimal volume element equals
Differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.
References
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Bibliography
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