Parabolic coordinates


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 (sigma, au) are defined by the equations

:x = sigma au,

:y = frac{1}{2} left( au^{2} - sigma^{2} ight)

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 au form confocal parabolae

:2y = -frac{x^{2{ au^{2 + au^{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 (sigma, au) are equal

:h_{sigma} = h_{ au} = sqrt{sigma^{2} + au^{2

Hence, the infinitesimal element of area is

:dA = left( sigma^{2} + au^{2} ight) dsigma d au

and the Laplacian equals

: abla^{2} Phi = frac{1}{sigma^{2} + au^{2 left( frac{partial^{2} Phi}{partial sigma^{2 + frac{partial^{2} Phi}{partial au^{2 ight)

Other differential operators such as abla cdot mathbf{F} and abla imes mathbf{F} can be expressed in the coordinates (sigma, au) 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 au cos phi

:y = sigma au sin phi

:z = frac{1}{2} left( au^{2} - sigma^{2} ight)

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

: an 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 au form confocal paraboloids

:2z = -frac{x^{2} + y^{2{ au^{2 + au^{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_{sigma} = sqrt{sigma^2+ au^2}:h_{ au} = sqrt{sigma^2+ au^2}:h_{phi} = sigma au,

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

:dV = h_sigma h_ au h_phi = sigma au left( sigma^{2} + au^{2} ight),dsigma,d au,dphi

and the Laplacian is given by

: abla^2 Phi = frac{1}{sigma^{2} + au^{2 left [frac{1}{sigma} frac{partial}{partial sigma} left( sigma frac{partial Phi}{partial sigma} ight) +frac{1}{ au} frac{partial}{partial au} left( au frac{partial Phi}{partial au} ight) ight] +frac{1}{sigma^2 au^2}frac{partial^2 Phi}{partial phi^2}

Other differential operators such as abla cdot mathbf{F} and abla imes mathbf{F} can be expressed in the coordinates (sigma, au, phi) 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{sqrt{ x^2 + y^2 + z^2 } + z}, : eta = sqrt{sqrt{ x^2 + y^2 + z^2 } - z}, : phi = arctan {y over x}.

:egin{vmatrix}deta\dxi\dphiend{vmatrix}=egin{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}&0end{vmatrix}cdotegin{vmatrix}dx\dy\dzend{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{ x^2 + z^2}, : xi = z + sqrt{ x^2 + z^2}.

If η="c" (a constant), then: left. z ight|_{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 ight|_{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} ight) = {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} ight).

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 = egin{matrix}frac{1}{2}end{matrix} ( xi - eta ).

:egin{vmatrix}dx\dy\dzend{vmatrix}=egin{vmatrix} frac{1}{2}sqrt{frac{xi}{etacosphi&frac{1}{2}sqrt{frac{eta}{xicosphi&-sqrt{xieta}sinphi\ frac{1}{2}sqrt{frac{xi}{etasinphi&frac{1}{2}sqrt{frac{eta}{xisinphi&sqrt{xieta}cosphi\-frac{1}{2}&frac{1}{2}&0end{vmatrix}cdotegin{vmatrix}deta\dxi\dphiend{vmatrix}

ee also

Bibliography

* | pages = p. 660

* | pages = pp. 185&ndash;186

*, ASIN B0000CKZX7 | pages = p. 180

* | pages = p. 96

* Same as Morse & Feshbach (1953), substituting "u""k" for ξ"k".

*

External links

* [http://mathworld.wolfram.com/ParabolicCoordinates.html MathWorld description of parabolic coordinates]


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • parabolic coordinates — parabolinės koordinatės statusas T sritis fizika atitikmenys: angl. parabolic coordinates vok. parabolische Koordinaten, f rus. параболические координаты, f pranc. coordonnées paraboliques, f …   Fizikos terminų žodynas

  • Parabolic cylindrical coordinates — are a three dimensional orthogonal coordinate system that results from projecting the two dimensional parabolic coordinate system in theperpendicular z direction. Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic… …   Wikipedia

  • Coordinates (mathematics) — Coordinates are numbers which describe the location of points in a plane or in space. For example, the height above sea level is a coordinate which is useful for describing points near the surface of the earth. A coordinate system, in a plane or… …   Wikipedia

  • Parabolic cylinder function — In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation:frac{d^2f}{dz^2} + left(az^2+bz+c ight)f=0.This equation is found, for example, when the technique of separation of variables …   Wikipedia

  • Action-angle coordinates — In classical mechanics, action angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving …   Wikipedia

  • Paraboloidal coordinates — are a three dimensional orthogonal coordinate system (λ,μ,ν) that generalizes the two dimensional parabolic coordinate system. Similar to the related ellipsoidal coordinates, the paraboloidal coordinate system has orthogonal quadratic coordinate… …   Wikipedia

  • Curvilinear coordinates — Curvilinear, affine, and Cartesian coordinates in two dimensional space Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian… …   Wikipedia

  • Orthogonal coordinates — In mathematics, orthogonal coordinates are defined as a set of d coordinates q = (q1, q2, ..., qd) in which the coordinate surfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate surface for a particular… …   Wikipedia

  • Oblate spheroidal coordinates — Figure 1: Coordinate isosurfaces for a point P (shown as a black sphere) in oblate spheroidal coordinates (μ, ν, φ). The z axis is vertical, and the foci are at ±2. The red oblate spheroid (flattened sphere) corresponds to μ=1, whereas the blue… …   Wikipedia

  • Ellipsoidal coordinates — are a three dimensional orthogonal coordinate system (λ,μ,ν) that generalizes the two dimensional elliptic coordinate system. Unlike most three dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.