Editing Parabolic coordinates (section)

==Three-dimensional parabolic coordinates==

[[Image:Parabolic coordinates 3D.png|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 coordinate system|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 manifold|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>