Parabolic coordinates

In green, confocal parabolae opening upwards, $2y=\frac{x^2}{{\sigma }^2}-{\sigma }^2$ In red, confocal parabolae opening downwards, $2y=-\frac{x^2}{{\tau }^2}+{\tau }^2$

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:

$x=\sigma \tau$

$y=\frac{1}{2}({\tau }^2-{\sigma }^2)$

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:

$\sigma =\mathrm{sign}(x)\sqrt{\sqrt{x^2+y^2}-y}$

$\tau =\sqrt{\sqrt{x^2+y^2}+y}$

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

$dA=({\sigma }^2+{\tau }^2)d\sigma d\tau$

and the Laplacian equals

${\mathrm{\nabla }}^2\mathrm{\Phi }=\frac{1}{{\sigma }^2+{\tau }^2}(\frac{{\partial }^2\mathrm{\Phi }}{\partial {\sigma }^2}+\frac{{\partial }^2\mathrm{\Phi }}{\partial {\tau }^2})$

Other differential operators such as $\mathrm{\nabla }\cdot 𝐅$ and $\mathrm{\nabla }\times 𝐅$ 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

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).

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:

$x=\sigma \tau \cos \phi$

$y=\sigma \tau \sin \phi$

$z=\frac{1}{2}({\tau }^2-{\sigma }^2)$

where the parabolae are now aligned with the $z$-axis, about which the rotation was carried out. Hence, the azimuthal angle $\phi$ is defined

$\tan \phi =\frac{y}{x}$

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

$g_{ij}=\left[\begin{array}{ccc}{\sigma }^2+{\tau }^2& 0& 0\\ 0& {\sigma }^2+{\tau }^2& 0\\ 0& 0& {\sigma }^2{\tau }^2\end{array}\right]$

Three-dimensional scale factors

The three dimensional scale factors are:

$h_{\sigma }=\sqrt{{\sigma }^2+{\tau }^2}$

$h_{\tau }=\sqrt{{\sigma }^2+{\tau }^2}$

$h_{\phi }=\sigma \tau$

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

$dV=h_{\sigma }{h}_{\tau }{h}_{\phi }d\sigma d\tau d\phi =\sigma \tau ({\sigma }^2+{\tau }^2)d\sigma d\tau d\phi$

and the Laplacian is given by

${\mathrm{\nabla }}^2\mathrm{\Phi }=\frac{1}{{\sigma }^2+{\tau }^2}[\frac{1}{\sigma }\frac{\partial }{\partial \sigma }\left(\sigma \frac{\partial \mathrm{\Phi }}{\partial \sigma }\right)+\frac{1}{\tau }\frac{\partial }{\partial \tau }\left(\tau \frac{\partial \mathrm{\Phi }}{\partial \tau }\right)]+\frac{1}{{\sigma }^2{\tau }^2}\frac{{\partial }^2\mathrm{\Phi }}{\partial {\phi }^2}$

Other differential operators such as $\mathrm{\nabla }\cdot 𝐅$ and $\mathrm{\nabla }\times 𝐅$ can be expressed in the coordinates $(\sigma ,\tau ,\varphi )$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

See also

• Parabolic cylindrical coordinates
• Orthogonal coordinate system
• Curvilinear coordinates

Bibliography

• Template:Cite book
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• Template:Cite book Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
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External links

• Template:Springer
• MathWorld description of parabolic coordinates