Parabolic cylindrical coordinates: Difference between revisions

Revision as of 04:24, 1 March 2013

Coordinate surfaces of the conical coordinates. The constants b and c were chosen as 1 and 2, respectively. The red sphere represents r=2, the blue elliptic cone aligned with the vertical z-axis represents μ=cosh(1) and the yellow elliptic cone aligned with the (green) x-axis corresponds to ν² = 2/3. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.26, -0.78, 1.34). The elliptic cones intersect the sphere in taco-shaped curves.

Conical coordinates are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius $r$ ) and by two families of perpendicular cones, aligned along the z- and x-axes, respectively.

Basic definitions

The conical coordinates $(r, μ, ν)$ are defined by

x = \frac{r μ ν}{b c}

y = \frac{r}{b} \sqrt{\frac{(μ^{2} - b^{2}) (ν^{2} - b^{2})}{(b^{2} - c^{2})}}

z = \frac{r}{c} \sqrt{\frac{(μ^{2} - c^{2}) (ν^{2} - c^{2})}{(c^{2} - b^{2})}}

with the following limitations on the coordinates

ν^{2} < c^{2} < μ^{2} < b^{2}

Surfaces of constant $r$ are spheres of that radius centered on the origin

x^{2} + y^{2} + z^{2} = r^{2}

whereas surfaces of constant $μ$ and $ν$ are mutually perpendicular cones

\frac{x^{2}}{μ^{2}} + \frac{y^{2}}{μ^{2} - b^{2}} + \frac{z^{2}}{μ^{2} - c^{2}} = 0

\frac{x^{2}}{ν^{2}} + \frac{y^{2}}{ν^{2} - b^{2}} + \frac{z^{2}}{ν^{2} - c^{2}} = 0

In this coordinate system, both Laplace's equation and the Helmholtz equation are separable.

Scale factors

The scale factor for the radius $r$ is one ( $h_{r} = 1$ ), as in spherical coordinates. The scale factors for the two conical coordinates are

h_{μ} = r \sqrt{\frac{μ^{2} - ν^{2}}{(b^{2} - μ^{2}) (μ^{2} - c^{2})}}

h_{ν} = r \sqrt{\frac{μ^{2} - ν^{2}}{(b^{2} - ν^{2}) (c^{2} - ν^{2})}}

References

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Bibliography

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

MathWorld description of conical coordinates

Template:Orthogonal coordinate systems

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+[[File:Conical coordinates.png|thumb|380px|right|[[Coordinate system#Coordinate surface|Coordinate surfaces]] of the conical coordinates.  The constants ''b'' and ''c'' were chosen as 1 and 2, respectively.  The red sphere represents ''r''=2, the blue elliptic cone aligned with the vertical ''z''-axis represents μ=cosh(1) and the yellow elliptic cone aligned with the (green) ''x''-axis corresponds to ν<sup>2</sup> = 2/3.  The three surfaces intersect at the point '''P''' (shown as a black sphere) with [[Cartesian coordinate system|Cartesian coordinates]] roughly (1.26, -0.78, 1.34).  The elliptic cones intersect the sphere in taco-shaped curves.]]
+'''Conical coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] consisting of
+concentric spheres (described by their radius <math>r</math>) and by two families of perpendicular cones, aligned along the ''z''- and ''x''-axes, respectively.
+==Basic definitions==
+The conical coordinates <math>(r, \mu, \nu)</math> are defined by
+:<math>
+x = \frac{r\mu\nu}{bc}
+</math>
+:<math>
+y = \frac{r}{b} \sqrt{\frac{\left( \mu^{2} - b^{2} \right) \left( \nu^{2} - b^{2} \right)}{\left( b^{2} - c^{2} \right)} }
+</math>
+:<math>
+z = \frac{r}{c} \sqrt{\frac{\left( \mu^{2} - c^{2} \right) \left( \nu^{2} - c^{2} \right)}{\left( c^{2} - b^{2} \right)} }
+</math>
+with the following limitations on the coordinates
+:<math>
+\nu^{2} < c^{2} < \mu^{2} < b^{2}
+</math>
+Surfaces of constant <math>r</math> are spheres of that radius centered on the origin
+:<math>
+x^{2} + y^{2} + z^{2} = r^{2}
+</math>
+whereas surfaces of constant <math>\mu</math> and <math>\nu</math> are mutually perpendicular cones
+:<math>
+\frac{x^{2}}{\mu^{2}} + \frac{y^{2}}{\mu^{2} - b^{2}} + \frac{z^{2}}{\mu^{2} - c^{2}} = 0
+</math>
+:<math>
+\frac{x^{2}}{\nu^{2}} + \frac{y^{2}}{\nu^{2} - b^{2}} + \frac{z^{2}}{\nu^{2} - c^{2}} = 0
+</math>
+In this coordinate system, both [[Laplace's equation]] and the [[Helmholtz equation]] are separable.
+==Scale factors==
+The scale factor for the radius <math>r</math> is one (<math>h_{r} = 1</math>), as in [[spherical coordinates]].  The scale factors for the two conical coordinates are
+:<math>
+h_{\mu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \mu^{2} \right) \left( \mu^{2} - c^{2} \right)}}
+</math>
+:<math>
+h_{\nu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \nu^{2} \right) \left( c^{2} - \nu^{2} \right)}}
+</math>
+==References==
+{{reflist}}
+==Bibliography==
+*{{cite book | author = [[Philip M. Morse|Morse PM]], [[Herman Feshbach|Feshbach H]] | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | isbn = 0-07-043316-X|lccn=52011515 | page = 659}}
+*{{cite book | author = [[Henry Margenau|Margenau H]], Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | publisher = D. van Nostrand | location = New York | pages = 183&ndash;184 | lccn = 5510911 }}
+*{{cite book | author = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | id = ASIN B0000CKZX7 | page = 179 | lccn = 5914456}}
+*{{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | pages = 991&ndash;100 | lccn = 6725285}}
+*{{cite book | author = Arfken G | year = 1970 | title = Mathematical Methods for Physicists | edition = 2nd | publisher = Academic Press | location = Orlando, FL | pages = 118–119 | id = ASIN B000MBRNX4}}
+*{{cite book | author = Moon P, Spencer DE | year = 1988 | chapter = Conical Coordinates (r, θ, λ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer-Verlag | location = New York | pages = 37–40 (Table 1.09) | isbn = 978-0-387-18430-2}}
+==External links==
+*[http://mathworld.wolfram.com/ConicalCoordinates.html MathWorld description of conical coordinates]
+{{Orthogonal coordinate systems}}
+[[Category:Coordinate systems]]

Parabolic cylindrical coordinates: Difference between revisions

Revision as of 04:24, 1 March 2013

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Basic definitions

Scale factors

References

Bibliography

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Parabolic cylindrical coordinates: Difference between revisions

Revision as of 04:24, 1 March 2013

Basic definitions

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References

Bibliography

External links

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