Parabolic coordinates

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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.

1 Two-dimensional parabolic coordinates
2 Two-dimensional scale factors
3 Three-dimensional parabolic coordinates
4 Three-dimensional scale factors
5 An alternative formulation
6 See also
7 References

[edit] Two-dimensional parabolic coordinates

Two-dimensional parabolic coordinates <math>(\sigma, \tau)</math> are defined by the equations

<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.

[edit] 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.

[edit] Three-dimensional parabolic 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 <math>z</math>-direction. Rotation about the symmetry axis of the parabolae produces a set of confocal paraboloids, forming a coordinate system that is also known as "parabolic coordinates"

<math>

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

<math>

y = \sigma \tau \sin \phi </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>\phi</math> is defined

<math>

\tan \phi = \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.

[edit] Three-dimensional scale factors

The scale factors <math>h_{\sigma}</math> and <math>h_{\tau}</math> are the same as in the two-dimensional case. The scale factor for the azimuthal angle <math>\phi</math> equals

<math>

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

Hence, the infinitesimal volume element is

<math>

dV = \sigma\tau \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau d\phi </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) + \left( \frac{1}{\sigma^{2}} + \frac{1}{\tau^{2}} \right) \frac{\partial^{2} \Phi}{\partial \phi^{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, \phi)</math> by substituting the scale factors into the general formulae found in orthogonal coordinates.

[edit] An alternative formulation

Conversion from Cartesian to parabolic coordinates is effected by means of the following equations:

<math> \phi = \arctan {y \over x}. </math>

<math>

\begin{vmatrix}d\eta\\d\xi\\d\phi\end{vmatrix} = \begin{vmatrix}

   \frac{x}{\sqrt{x^2+y^2+z^2}}

& \frac{y}{\sqrt{x^2+y^2+z^2}} &-1+\frac{z}{\sqrt{x^2+y^2+z^2}}\\

   \frac{x}{\sqrt{x^2+y^2+z^2}}

& \frac{y}{\sqrt{x^2+y^2+z^2}} &1 +\frac{z}{\sqrt{x^2+y^2+z^2}}\\ \frac{-y}{x^2+y^2}&\frac{x}{x^2+y^2}&0 \end{vmatrix} \cdot \begin{vmatrix}dx\\dy\\dz\end{vmatrix} </math>

If φ=0 then a cross-section is obtained; the coordinates become confined to the x-z plane:

If η=c (a constant), then

<math> \left. z \right|_{\eta = c} = {x^2 \over 2 c} - {c \over 2}. </math>

This is a parabola whose focus is at the origin for any value of c. The parabola's axis of symmetry is vertical and the concavity faces upwards.

If ξ=c then

<math> \left. z \right|_{\xi = c} = {c \over 2} - {x^2 \over 2 c}. </math>

This is a parabola whose focus is at the origin for any value of c. Its axis of symmetry is vertical and the concavity faces downwards.

Now consider any upward parabola η=c and any downward parabola ξ=b. It is desired to find their intersection:

regroup,

factor out the x,

<math> x^2 \left( {b + c \over 2 b c} \right) = {b + c \over 2}, </math>

cancel out common factors from both sides,

take the square root,

x is the geometric mean of b and c. The abscissa of the intersection has been found. Find the ordinate. Plug in the value of x into the equation of the upward parabola:

then plug in the value of x into the equation of the downward parabola:

z_c = z_b, as should be. Therefore the point of intersection is

<math> P : \left( \sqrt{b c}, {b - c \over 2} \right). </math>

Draw a pair of tangents through point P, each one tangent to each parabola. The tangential line through point P to the upward parabola has slope:

The tangent through point P to the downward parabola has slope:

The products of the two slopes is

The product of the slopes is negative one, therefore the slopes are perpendicular. This is true for any pair of parabolas with concavities in opposite directions.

Such a pair of parabolas intersect at two points, but when φ is restricted to zero, it actually confines the other coordinates η and ξ to move in a half-plane with x>0, because x<0 corresponds to φ=π.

Thus a pair of coordinates η and ξ specify a unique point on the half-plane. Then letting φ range from 0 to 2π the half-plane revolves with the point (around the z-axis as its hinge): the parabolas form paraboloids. A pair of opposing paraboloids specifies a circle, and a value of φ specifies a half-plane which cuts the circle of intersection at a unique point. The point's Cartesian coordinates are [Menzel, p. 139]:

<math> z = \begin{matrix}\frac{1}{2}\end{matrix} ( \xi - \eta ). </math>

<math>

\begin{vmatrix}dx\\dy\\dz\end{vmatrix} = \begin{vmatrix}

\frac{1}{2}\sqrt{\frac{\xi}{\eta}}\cos\phi

&\frac{1}{2}\sqrt{\frac{\eta}{\xi}}\cos\phi &-\sqrt{\xi\eta}\sin\phi\\

\frac{1}{2}\sqrt{\frac{\xi}{\eta}}\sin\phi

&\frac{1}{2}\sqrt{\frac{\eta}{\xi}}\sin\phi &\sqrt{\xi\eta}\cos\phi\\ -\frac{1}{2}&\frac{1}{2}&0 \end{vmatrix} \cdot \begin{vmatrix}d\eta\\d\xi\\d\phi\end{vmatrix} </math>