Parabolic coordinates

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.

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

form confocal parabolae that open upwards (i.e., towards

form confocal parabolae that open downwards (i.e., towards

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

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

The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional orthogonal coordinates.

The parabolic cylindrical coordinates are produced by projecting in the

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: where the parabolae are now aligned with the

form confocal paraboloids that open upwards (i.e., towards

form confocal paraboloids that open downwards (i.e., towards

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

The Riemannian metric tensor associated with this coordinate system is The three dimensional scale factors are: It is seen that the scale factors

The infinitesimal volume element is then and the Laplacian is given by Other differential operators such as

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

In green, confocal parabolae opening upwards, In red, confocal parabolae opening downwards,
Coordinate surfaces of the three-dimensional parabolic coordinates. The red paraboloid corresponds to τ=2, the blue paraboloid corresponds to σ=1, and the yellow half-plane corresponds to φ=-60°. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.0, -1.732, 1.5).