Parabolic coordinates

Short description: Two-dimensional orthogonal coordinate system

In green, confocal parabolae opening upwards, 2y=x2σ2−σ2 In red, confocal parabolae opening downwards, 2y=−x2τ2+τ2

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

* 1 Two-dimensional parabolic coordinates
* 2 Two-dimensional scale factors
* 3 Three-dimensional parabolic coordinates
* 4 Three-dimensional scale factors
* 5 See also
* 6 Bibliography
* 7 External links

## Two-dimensional parabolic coordinates

Two-dimensional parabolic coordinates (σ,τ) are defined by the equations, in terms of Cartesian coordinates:

x=στ

y=12(τ2−σ2)

The curves of constant σ form confocal parabolae

2y=x2σ2−σ2

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

2y=−x2τ2+τ2

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

The Cartesian coordinates x and y can be converted to parabolic coordinates by:

σ=sign⁡(x)x2+y2−y

τ=x2+y2+y

## Two-dimensional scale factors

The scale factors for the parabolic coordinates (σ,τ) are equal

hσ=hτ=σ2+τ2

Hence, the infinitesimal element of area is

dA=(σ2+τ2)dσdτ

and the Laplacian equals

∇2Φ=1σ2+τ2(∂2Φ∂σ2+∂2Φ∂τ2)

Other differential operators such as ∇⋅𝐅 and ∇×𝐅 can be expressed in the coordinates (σ,τ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

## 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).

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, the coordinate system of tridimensional parabolic coordinates. Expressed in terms of cartesian coordinates:

x=στcos⁡φ

y=στsin⁡φ

z=12(τ2−σ2)

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⁡φ=yx

The surfaces of constant σ form confocal paraboloids

2z=x2+y2σ2−σ2

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

2z=−x2+y2τ2+τ2

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

The Riemannian metric tensor associated with this coordinate system is

gij=[σ2+τ2000σ2+τ2000σ2τ2]

## Three-dimensional scale factors

The three dimensional scale factors are:

hσ=σ2+τ2
hτ=σ2+τ2
hφ=στ

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=hσhτhφdσdτdφ=στ(σ2+τ2)dσdτdφ

and the Laplacian is given by

∇2Φ=1σ2+τ2[1σ∂∂σ(σ∂Φ∂σ)+1τ∂∂τ(τ∂Φ∂τ)]+1σ2τ2∂2Φ∂φ2

Other differential operators such as ∇⋅𝐅 and ∇×𝐅 can be expressed in the coordinates (σ,τ,ϕ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

## See also

* Parabolic cylindrical coordinates
* Orthogonal coordinate system
* Curvilinear coordinates

## Bibliography

* Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. pp. 660. ISBN 0-07-043316-X.
* Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 185–186. https://archive.org/details/mathematicsofphy0002marg.
* Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. pp. 180. ASIN B0000CKZX7.
* Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 96.
* Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. pp. 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 978-0-387-18430-2.

## External links

* Hazewinkel, Michiel, ed. (2001), "Parabolic coordinates", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/p071170
* MathWorld description of parabolic coordinates

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* t
* e

Orthogonal coordinate systems

Two dimensional|

* Cartesian
* Polar
* Parabolic
* Bipolar
* Elliptic

Three dimensional|

* Cartesian coordinate system
* Cylindrical coordinate system
* Spherical coordinate system
* Parabolic cylindrical coordinates
* Paraboloidal coordinates
* Oblate spheroidal coordinates
* Prolate spheroidal coordinates
* Ellipsoidal coordinates
* Elliptic cylindrical coordinates
* Toroidal coordinates
* Bispherical coordinates
* Bipolar cylindrical coordinates
* Conical coordinates
* 6-sphere coordinates
* Flat-ring cyclide coordinates
* Flat-disk cyclide coordinates
* Bi-cyclide coordinates
* Cap-cyclide coordinates
* Concave bi-sinusoidal single-centered coordinates
* Concave bi-sinusoidal double-centered coordinates
* Convex inverted-sinusoidal spherically aligned coordinates
* Quasi-random-intersection cartesian coordinates

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