- Parabolic coordinates
Parabolic coordinates are a two-dimensional orthogonal
coordinate system in which thecoordinate line s areconfocal parabola s. 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 thepotential 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 . Theparabolic 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". Theabscissa of the intersection has been found. Find theordinate . 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
paraboloid s. 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. 185–186
*, 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]
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