NationMaster - Encyclopedia: 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. In mathematics, orthogonal coordinates are defined as a set of coordinates that have no off-diagonal terms in their metric tensor, i. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... Confocal means having the same foci. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...

Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges. The Stark effect is the splitting of a spectral line into several components in the presence of an electric field. ... Potential theory may be defined as the study of harmonic functions. ...

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

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.

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

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

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 $nabla cdot mathbf{F}$ 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. In mathematics, orthogonal coordinates are defined as a set of coordinates that have no off-diagonal terms in their metric tensor, i. ...

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, forming a coordinate system that is also known as "parabolic coordinates" Fig. ... In mathematics, orthogonal coordinates are defined as a set of coordinates that have no off-diagonal terms in their metric tensor, i. ...

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.

Three-dimensional scale factors

The three dimensional scale factors are:

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

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

$h_{phi} = sigmatau,$

It is seen that The scale factors $h σ$ and $h τ$ 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 $nabla cdot mathbf{F}$ 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. In mathematics, orthogonal coordinates are defined as a set of coordinates that have no off-diagonal terms in their metric tensor, i. ...

An alternative formulation

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

$eta = - z + sqrt{ x^2 + y^2 + z^2 },$

$&# 0;= z + sqrt{ x^2 + y^2 + z^2 },$

$phi = arctan {y over x}.$

$begin{vmatrix}detad&# 0;dphiend{vmatrix} = begin{vmatrix} frac{x}{sqrt{x^2+y^2+z^2}} & frac{y}{sqrt{x^2+y^2+z^2}} &-1+frac{z}{sqrt{x^2+y^2+z^2}} frac{x}{sqrt{x^2+y^2+z^2}} & frac{y}{sqrt{x^2+y^2+z^2}} &1 +frac{z}{sqrt{x^2+y^2+z^2}} frac{-y}{x^2+y^2}&frac{x}{x^2+y^2}&0 end{vmatrix} cdot begin{vmatrix}dxdydzend{vmatrix}$

$etage 0,quad&# 0;ge 0$

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

$eta = -z + sqrt{ x^2 + z^2},$

$&# 0;= z + sqrt{ x^2 + z^2}.$

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. A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane...

If ξ=c then

$left. z right|_{&# 0;= c} = {c over 2} - {x^2 over 2 c}.$

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: The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... Abscissa means the x coordinate on an (x, y) graph; the input of a mathematical function against which the output is plotted. ... Ordinate means the y coordinate on an (x, y) graph; the plotted output of a mathematical function. ...

$z_c = {b c over 2 c} - {c over 2} = {b - c over 2},$

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

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

z_c = z_b, 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]: Paraboloid of revolution Hyperbolic paraboloid In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic paraboloid). ...

$x = sqrt{&# 0;eta} cos phi,$

$y = sqrt{&# 0;eta} sin phi,$

$z = begin{matrix}frac{1}{2}end{matrix} ( &# 0;- eta ).$

$begin{vmatrix}dxdydzend{vmatrix} = begin{vmatrix} frac{1}{2}sqrt{frac{&# 0;{eta}}cosphi &frac{1}{2}sqrt{frac{eta}{&# 0;}cosphi &-sqrt{&# 0;eta}sinphi frac{1}{2}sqrt{frac{&# 0;{eta}}sinphi &frac{1}{2}sqrt{frac{eta}{&# 0;}sinphi &sqrt{&# 0;eta}cosphi -frac{1}{2}&frac{1}{2}&0 end{vmatrix} cdot begin{vmatrix}detad&# 0;dphiend{vmatrix}$

Bibliography

Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 660. ISBN 0-07-043316-X, LCCN 52-11515.

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 u_k for ξ_k.

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.

Full name : Philip McCord Morse Founding ORSA President (1952) B.S. Physics, 1926, Case Institute; Ph. ... Institute Professor Emeritus Herman Feshbach of Cambridge, a renowned nuclear physicist and champion of equal opportunity at MIT and around the world, died December 22 2000 of congestive heart failure at Youville Hospital in Cambridge. ... The Library of Congress Control Number or LCCN is a serially based system of numbering books in the Library of Congress in the United States. ... Henry Margenau (1901 - February 8, 1997) was a German-U.S. physicist, historian and philosopher of science, and Christian writer. ... The Library of Congress Control Number or LCCN is a serially based system of numbering books in the Library of Congress in the United States. ... The Library of Congress Control Number or LCCN is a serially based system of numbering books in the Library of Congress in the United States. ... The Library of Congress Control Number or LCCN is a serially based system of numbering books in the Library of Congress in the United States. ...

External links

MathWorld description of parabolic coordinates

Categories: Coordinate systems

Results from FactBites:

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