thumb|right|384px|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
thumb|right|300px|[[Coordinate system#Coordinate surface|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
- Same as Morse & Feshbach (1953), substituting u<sub>k</sub> for ξ<sub>k</sub>.
External links
- MathWorld description of parabolic coordinates