Parabolic coordinates

File:Parabolic coords.svg

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

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 <math>(\sigma, \tau)</math> are defined by the equations, in terms of Cartesian coordinates:

<math>

x = \sigma \tau </math>

<math>

y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) </math>

The curves of constant <math>\sigma</math> form confocal parabolae

<math>

2y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2} </math>

that open upwards (i.e., towards <math>+y</math>), whereas the curves of constant <math>\tau</math> form confocal parabolae

<math>

2y = -\frac{x^{2}}{\tau^{2}} + \tau^{2} </math>

that open downwards (i.e., towards <math>-y</math>). The foci of all these parabolae are located at the origin.

The Cartesian coordinates <math>x</math> and <math>y</math> can be converted to parabolic coordinates by:

<math>

\sigma = \operatorname{sign}(x)\sqrt{\sqrt{x^{2} +y^{2}}-y} </math>

<math>

\tau = \sqrt{\sqrt{x^{2} +y^{2}}+y} </math>

Two-dimensional scale factors

The scale factors for the parabolic coordinates <math>(\sigma, \tau)</math> are equal

<math>

h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}} </math>

Hence, the infinitesimal element of area is

<math>

dA = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau </math>

and the Laplacian equals

<math>

\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) </math>

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

Three-dimensional parabolic coordinates

File:Parabolic coordinates 3D.png

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 <math>z</math>-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:

<math>

x = \sigma \tau \cos \varphi </math>

<math>

y = \sigma \tau \sin \varphi </math>

<math>

z = \frac{1}{2} \left(\tau^{2} - \sigma^{2} \right) </math>

where the parabolae are now aligned with the <math>z</math>-axis, about which the rotation was carried out. Hence, the azimuthal angle <math>\varphi</math> is defined

<math>

\tan \varphi = \frac{y}{x} </math>

The surfaces of constant <math>\sigma</math> form confocal paraboloids

<math>

2z = \frac{x^{2} + y^{2}}{\sigma^{2}} - \sigma^{2} </math>

that open upwards (i.e., towards <math>+z</math>) whereas the surfaces of constant <math>\tau</math> form confocal paraboloids

<math>

2z = -\frac{x^{2} + y^{2}}{\tau^{2}} + \tau^{2} </math>

that open downwards (i.e., towards <math>-z</math>). The foci of all these paraboloids are located at the origin.

The Riemannian metric tensor associated with this coordinate system is

<math> g_{ij} = \begin{bmatrix} \sigma^2+\tau^2 & 0 & 0\\0 & \sigma^2+\tau^2 & 0\\0 & 0 & \sigma^2\tau^2 \end{bmatrix} </math>

Three-dimensional scale factors

The three dimensional scale factors are:

<math>h_{\sigma} = \sqrt{\sigma^2+\tau^2}</math>

<math>h_{\tau} = \sqrt{\sigma^2+\tau^2}</math>

<math>h_{\varphi} = \sigma\tau</math>

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

<math>

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 </math>

and the Laplacian is given by

<math>

\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} </math>

Other differential operators such as <math>\nabla \cdot \mathbf{F}</math> and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau, \phi)</math> 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