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.

Contents

[edit] Two-dimensional parabolic coordinates

Two-dimensional parabolic coordinates (σ,τ) are defined by the equations

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

The curves of constant σ form confocal parabolae

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

that open upwards (i.e., towards + y), whereas the curves of constant τ 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.

[edit] Two-dimensional scale factors

The scale factors for the parabolic coordinates (σ,τ) 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) d\sigma d\tau

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{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (σ,τ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

[edit] Three-dimensional parabolic coordinates

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).
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 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 = στcosφ
y = στsinφ
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 φ is defined

\tan \phi = \frac{y}{x}

The surfaces of constant σ 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 τ 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.

[edit] Three-dimensional scale factors

The three dimensional scale factors are:

h_{\sigma} = \sqrt{\sigma^2+\tau^2}
h_{\tau} = \sqrt{\sigma^2+\tau^2}
h_{\phi} = \sigma\tau\,

It is seen that The scale factors hσ and hτ are the same as in the two-dimensional case. The infinitesimal volume element is then

dV = h_\sigma h_\tau h_\phi = \sigma\tau \left( \sigma^{2} + \tau^{2} \right)\,d\sigma\,d\tau\,d\phi

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

Other differential operators such as \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (σ,τ,φ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

[edit] An alternative formulation

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

\eta = - z + \sqrt{ x^2 + y^2 + z^2 },
\xi = z + \sqrt{ x^2 + y^2 + z^2 },
\phi = \arctan {y \over x}.
\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}
\eta\ge 0,\quad\xi\ge 0

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

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

If η=c (a constant), then

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

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|_{\xi = c} = {c \over 2} - {x^2 \over 2 c}.

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:

{x^2 \over 2 c} - {c \over 2} = {b \over 2} - {x^2 \over 2 b},

regroup,

{x^2 \over 2 c} + {x^2 \over 2 b} = {b \over 2} + {c \over 2},

factor out the x,

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

cancel out common factors from both sides,

x^2 = b c, \,

take the square root,

x = \sqrt{b c}.

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 = {b c \over 2 c} - {c \over 2} = {b - c \over 2},

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

z_b = {b \over 2} - {b c \over 2 b} = {b - c \over 2}.

zc = zb, as should be. Therefore the point of intersection is

P : \left( \sqrt{b c}, {b - c \over 2} \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:

{d z_c \over d x} = {x \over c} = { \sqrt{ b c} \over c} = \sqrt{ b \over c} = s_c.

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

{d z_b \over d x} = - {x \over b} = { - \sqrt{ b c } \over b} = - \sqrt{ {c \over b} } = s_b.

The products of the two slopes is

s_c s_b = - \sqrt{ {b \over c}} \sqrt{ {c \over b}} = -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{\xi \eta} \cos \phi,
y = \sqrt{\xi \eta} \sin \phi,
z = \begin{matrix}\frac{1}{2}\end{matrix} ( \xi - \eta ).
\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}

[edit] See also

  • Three dimensional orthogonal coordinate systems


[edit] Bibliography

  • Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand, pp. 185–186. LCCN 55-10911. 
  • Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, p. 180. LCCN 59-14456, ASIN B0000CKZX7. 
  • Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag, p. 96. LCCN 67-25285. 
  • Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett, p. 114. ISBN 0-86720-293-9.  Same as Morse & Feshbach (1953), substituting uk for ξk.
  • Moon P, Spencer DE (1988). "Parabolic Coordinates (μ, ν, ψ)", Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, corrected 2nd ed., 3rd print ed., New York: Springer-Verlag, pp. 34–36 (Table 1.08). ISBN 978-0387184302. 

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