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 ${\displaystyle (\sigma ,\tau )}$ are defined by the equations, in terms of Cartesian coordinates:

The curves of constant ${\displaystyle \sigma }$ form confocal parabolae

that open upwards (i.e., towards ${\displaystyle +y}$), whereas the curves of constant ${\displaystyle \tau }$ form confocal parabolae

that open downwards (i.e., towards ${\displaystyle -y}$). The foci of all these parabolae are located at the origin.

The Cartesian coordinates ${\displaystyle x}$ and ${\displaystyle y}$ can be converted to parabolic coordinates by:

The scale factors for the parabolic coordinates ${\displaystyle (\sigma ,\tau )}$ are equal