Definitions

# 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

Two-dimensional parabolic coordinates $\left(sigma, tau\right)$ are defined by the equations


x = sigma tau,


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

The curves of constant $sigma$ form confocal parabolae


2y = frac{x^{2}}{sigma^{2}} - sigma^{2}

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


2y = -frac{x^{2}}{tau^{2}} + tau^{2}

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

## Two-dimensional scale factors

The scale factors for the parabolic coordinates $\left(sigma, tau\right)$ are equal


h_{sigma} = h_{tau} = sqrt{sigma^{2} + tau^{2}}

Hence, the infinitesimal element of area is


dA = left(sigma^{2} + tau^{2} right) dsigma dtau

and the Laplacian equals


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 $nabla cdot mathbf\left\{F\right\}$ and $nabla times mathbf\left\{F\right\}$ can be expressed in the coordinates $\left(sigma, tau\right)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

## 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 $z$-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"


x = sigma tau cos phi


y = sigma tau sin phi


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

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


tan phi = frac{y}{x}

The surfaces of constant $sigma$ form confocal paraboloids


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

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


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

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

## Three-dimensional scale factors

The three dimensional scale factors are:

$h_\left\{sigma\right\} = sqrt\left\{sigma^2+tau^2\right\}$
$h_\left\{tau\right\} = sqrt\left\{sigma^2+tau^2\right\}$
$h_\left\{phi\right\} = sigmatau,$

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


dV = h_sigma h_tau h_phi = sigmatau left(sigma^{2} + tau^{2} right),dsigma,dtau,dphi

and the Laplacian is given by


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^2tau^2}frac{partial^2 Phi}{partial phi^2}

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

## An alternative formulation

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

$xi = sqrt\left\{sqrt\left\{ x^2 + y^2 + z^2 \right\} + z\right\},$
$eta = sqrt\left\{sqrt\left\{ x^2 + y^2 + z^2 \right\} - z\right\},$
$phi = arctan \left\{y over x\right\}.$


begin{vmatrix}detadxidphiend{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}dxdydzend{vmatrix}

$etage 0,quadxige 0$

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

$eta = -z + sqrt\left\{ x^2 + z^2\right\},$
$xi = z + sqrt\left\{ x^2 + z^2\right\}.$

If η=c (a constant), then

$left. z right|_\left\{eta = c\right\} = \left\{x^2 over 2 c\right\} - \left\{c over 2\right\}.$
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

$left. z right|_\left\{xi = c\right\} = \left\{c over 2\right\} - \left\{x^2 over 2 c\right\}.$
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:

$\left\{x^2 over 2 c\right\} - \left\{c over 2\right\} = \left\{b over 2\right\} - \left\{x^2 over 2 b\right\},$
regroup,
$\left\{x^2 over 2 c\right\} + \left\{x^2 over 2 b\right\} = \left\{b over 2\right\} + \left\{c over 2\right\},$
factor out the x,
$x^2 left\left(\left\{b + c over 2 b c\right\} right\right) = \left\{b + c over 2\right\},$
cancel out common factors from both sides,
$x^2 = b c, ,$
take the square root,
$x = sqrt\left\{b c\right\}.$
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:
$z_c = \left\{b c over 2 c\right\} - \left\{c over 2\right\} = \left\{b - c over 2\right\},$
then plug in the value of x into the equation of the downward parabola:
$z_b = \left\{b over 2\right\} - \left\{b c over 2 b\right\} = \left\{b - c over 2\right\}.$
zc = zb, as should be. Therefore the point of intersection is
$P : left\left(sqrt\left\{b c\right\}, \left\{b - c over 2\right\} right\right).$

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:

$\left\{d z_c over d x\right\} = \left\{x over c\right\} = \left\{ sqrt\left\{ b c\right\} over c\right\} = sqrt\left\{ b over c\right\} = s_c.$
The tangent through point P to the downward parabola has slope:
$\left\{d z_b over d x\right\} = - \left\{x over b\right\} = \left\{ - sqrt\left\{ b c \right\} over b\right\} = - sqrt\left\{ \left\{c over b\right\} \right\} = s_b.$

The products of the two slopes is

$s_c s_b = - sqrt\left\{ \left\{b over c\right\}\right\} sqrt\left\{ \left\{c over b\right\}\right\} = -1.$
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]:

$x = sqrt\left\{xi eta\right\} cos phi,$
$y = sqrt\left\{xi eta\right\} sin phi,$
$z = begin\left\{matrix\right\}frac\left\{1\right\}\left\{2\right\}end\left\{matrix\right\} \left(xi - eta \right).$


begin{vmatrix}dxdydzend{vmatrix} = begin{vmatrix} frac{1}{2}sqrt{frac{xi}{eta}}cosphi &frac{1}{2}sqrt{frac{eta}{xi}}cosphi &-sqrt{xieta}sinphi frac{1}{2}sqrt{frac{xi}{eta}}sinphi &frac{1}{2}sqrt{frac{eta}{xi}}sinphi &sqrt{xieta}cosphi -frac{1}{2}&frac{1}{2}&0 end{vmatrix} cdot begin{vmatrix}detadxidphiend{vmatrix}

## Bibliography

• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. ISBN 0-07-043316-X,
• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand.
• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill.
• Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag.
• Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. Same as Morse & Feshbach (1953), substituting uk for ξk.
• Moon P, Spencer DE (1988). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions. corrected 2nd ed., 3rd print ed., New York: Springer-Verlag.