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.