# Parabolic coordinates

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 $(\sigma ,\tau )$ are defined by the equations, in terms of Cartesian coordinates:
    $x=\sigma \tau$
    $y={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)$
The curves of constant $\sigma$ form confocal parabolae
    $2y={\frac {x^{2}}{\sigma ^{2}}}-\sigma ^{2}$
that open upwards (i.e., towards $+y$), whereas the curves of constant $\tau$ form confocal parabolae
    $2y=-{\frac {x^{2}}{\tau ^{2}}}+\tau ^{2}$
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:
    $\sigma =\operatorname {sign} (x){\sqrt {{\sqrt {x^{2}+y^{2}}}-y}}$
    $\tau ={\sqrt {{\sqrt {x^{2}+y^{2}}}+y}}$

## Two-dimensional scale factors

The scale factors for the parabolic coordinates $(\sigma ,\tau )$ are equal
    $h_{\sigma }=h_{\tau }={\sqrt {\sigma ^{2}+\tau ^{2}}}$
Hence, the infinitesimal element of area is
    $dA=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau$
and the Laplacian equals
    $\nabla ^{2}\Phi ={\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}\Phi }{\partial \sigma ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \tau ^{2}}}\right)$
Other differential operators such as $\nabla \cdot \mathbf {F}$ and $\nabla \times \mathbf {F}$ can be expressed in the coordinates $(\sigma ,\tau )$ 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:
    $x=\sigma \tau \cos \varphi$
    $y=\sigma \tau \sin \varphi$
    $z={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)$
where the parabolae are now aligned with the $z$-axis, about which the rotation was carried out. Hence, the azimuthal angle $\varphi$ is defined
    $\tan \varphi ={\frac {y}{x}}$
The surfaces of constant $\sigma$ form confocal paraboloids
    $2z={\frac {x^{2}+y^{2}}{\sigma ^{2}}}-\sigma ^{2}$
that open upwards (i.e., towards $+z$) whereas the surfaces of constant $\tau$ form confocal paraboloids
    $2z=-{\frac {x^{2}+y^{2}}{\tau ^{2}}}+\tau ^{2}$
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
    $g_{ij}={\begin{bmatrix}\sigma ^{2}+\tau ^{2}&0&0\\0&\sigma ^{2}+\tau ^{2}&0\\0&0&\sigma ^{2}\tau ^{2}\end{bmatrix}}$

## Three-dimensional scale factors

The three dimensional scale factors are:
    $h_{\sigma }={\sqrt {\sigma ^{2}+\tau ^{2}}}$
    $h_{\tau }={\sqrt {\sigma ^{2}+\tau ^{2}}}$
    $h_{\varphi }=\sigma \tau$
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
    $dV=h_{\sigma }h_{\tau }h_{\varphi }\,d\sigma \,d\tau \,d\varphi =\sigma \tau \left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau \,d\varphi$
and the Laplacian is given by
    $\nabla ^{2}\Phi ={\frac {1}{\sigma ^{2}+\tau ^{2}}}\left[{\frac {1}{\sigma }}{\frac {\partial }{\partial \sigma }}\left(\sigma {\frac {\partial \Phi }{\partial \sigma }}\right)+{\frac {1}{\tau }}{\frac {\partial }{\partial \tau }}\left(\tau {\frac {\partial \Phi }{\partial \tau }}\right)\right]+{\frac {1}{\sigma ^{2}\tau ^{2}}}{\frac {\partial ^{2}\Phi }{\partial \varphi ^{2}}}$
Other differential operators such as $\nabla \cdot \mathbf {F}$ and $\nabla \times \mathbf {F}$ can be expressed in the coordinates $(\sigma ,\tau ,\phi )$ by substituting the scale factors into the general formulae found in orthogonal coordinates.