Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas.

Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges.

Two-dimensional parabolic coordinates

are defined by the equations, in terms of Cartesian coordinates:

form confocal parabolae

form confocal parabolae

). The foci of all these parabolae are located at the origin.

can be converted to parabolic coordinates by:

Two-dimensional scale factors

are equal

Hence, the infinitesimal element of area is

and the Laplacian equals

by substituting the scale factors into the general formulae found in orthogonal coordinates.

Three-dimensional parabolic coordinates

The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional orthogonal coordinates. The parabolic cylindrical coordinates-direction. Rotation about the symmetry axis of the parabolae produces a set of confocal paraboloids, the coordinate system of tridimensional parabolic coordinates. Expressed in terms of cartesian coordinates:

is defined

form confocal paraboloids

form confocal paraboloids

). The foci of all these paraboloids are located at the origin.

The Riemannian metric tensor associated with this coordinate system is

Three-dimensional scale factors

The three dimensional scale factors are:

are the same as in the two-dimensional case. The infinitesimal volume element is then

and the Laplacian is given by

by substituting the scale factors into the general formulae found in orthogonal coordinates.

See also

• Parabolic cylindrical coordinates

• Orthogonal coordinate system

• Curvilinear coordinates

Bibliography

• Same as Morse & Feshbach (1953), substituting uk for ξk.

External links

• MathWorld description of parabolic coordinates