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

Two-dimensional parabolic coordinates are defined by the equations, in terms of Cartesian coordinates: The curves of constant \sigma form confocal parabolae that open upwards (i.e., towards +y), whereas the curves of constant \tau form confocal parabolae that open downwards (i.e., towards -y). The foci of all these parabolae are located at the origin. The Cartesian coordinates x and y can be converted to parabolic coordinates by:

Two-dimensional scale factors

The scale factors for the parabolic coordinates are equal Hence, the infinitesimal element of area is and the Laplacian equals Other differential operators such as and can be expressed in the coordinates 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 are produced by projecting in the z-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: where the parabolae are now aligned with the z-axis, about which the rotation was carried out. Hence, the azimuthal angle \varphi is defined The surfaces of constant \sigma form confocal paraboloids that open upwards (i.e., towards +z) whereas the surfaces of constant \tau form confocal paraboloids that open downwards (i.e., towards -z). 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: It is seen that the scale factors h_{\sigma} and h_{\tau} are the same as in the two-dimensional case. The infinitesimal volume element is then and the Laplacian is given by Other differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.