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

${\displaystyle x=\sigma \tau }$

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

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

${\displaystyle 2y={\frac {x^{2}}{\sigma ^{2}}}-\sigma ^{2}}$

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

${\displaystyle 2y=-{\frac {x^{2}}{\tau ^{2}}}+\tau ^{2}}$

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:

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

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

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

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

Hence, the infinitesimal element of area is

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

and the Laplacian equals

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

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

${\displaystyle x=\sigma \tau \cos \varphi }$

${\displaystyle y=\sigma \tau \sin \varphi }$

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

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

${\displaystyle \tan \varphi ={\frac {y}{x}}}$

The surfaces of constant ${\displaystyle \sigma }$ form confocal paraboloids

${\displaystyle 2z={\frac {x^{2}+y^{2}}{\sigma ^{2}}}-\sigma ^{2}}$

that open upwards (i.e., towards ${\displaystyle +z}$) whereas the surfaces of constant ${\displaystyle \tau }$ form confocal paraboloids

${\displaystyle 2z=-{\frac {x^{2}+y^{2}}{\tau ^{2}}}+\tau ^{2}}$

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

The Riemannian metric tensor associated with this coordinate system is

${\displaystyle g_{ij}={\begin{bmatrix}\sigma ^{2}+\tau ^{2}&0&0\\0&\sigma ^{2}+\tau ^{2}&0\\0&0&\sigma ^{2}\tau ^{2}\end{bmatrix}}}$

The three dimensional scale factors are:

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

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

${\displaystyle h_{\varphi }=\sigma \tau }$

It is seen that the scale factors ${\displaystyle h_{\sigma }}$ and ${\displaystyle h_{\tau }}$ are the same as in the two-dimensional case. The infinitesimal volume element is then

${\displaystyle dV=h_{\sigma }h_{\tau }h_{\varphi }\,d\sigma \,d\tau \,d\varphi =\sigma \tau \left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau \,d\varphi }$

and the Laplacian is given by

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

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

• Parabolic cylindrical coordinates
• Orthogonal coordinate system
• Curvilinear coordinates

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• Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 114. ISBN 0-86720-293-9. Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
• Moon P, Spencer DE (1988). "Parabolic Coordinates (μ, ν, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 34–36 (Table 1.08). ISBN 978-0-387-18430-2.

• "Parabolic coordinates", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• MathWorld description of parabolic coordinates