conical coordinates
File:Conical coordinates.png of the conical coordinates. The constants {{math|b}} and {{math|c}} were chosen as 1 and 2, respectively. The red sphere represents {{math|1=r = 2}}, the blue elliptic cone aligned with the vertical {{math|z}}-axis represents μ=cosh(1) and the yellow elliptic cone aligned with the (green) {{mvar|x}}-axis corresponds to {{math|1=ν2 = 2/3}}. The three surfaces intersect at the point {{math|P}} (shown as a black sphere) with Cartesian coordinates roughly (1.26, −0.78, 1.34). The elliptic cones intersect the sphere in spherical conics.]]
Conical coordinates, sometimes called sphero-conal or sphero-conical coordinates, are a three-dimensional orthogonal coordinate system consisting of
concentric spheres (described by their radius {{mvar|r}}) and by two families of perpendicular elliptic cones, aligned along the {{math|z}}- and {{math|x}}-axes, respectively. The intersection between one of the cones and the sphere forms a spherical conic.
Basic definitions
The conical coordinates are defined by
:
x = \frac{r\mu\nu}{bc}
:
y = \frac{r}{b} \sqrt{\frac{\left( \mu^{2} - b^{2} \right) \left( \nu^{2} - b^{2} \right)}{\left( b^{2} - c^{2} \right)} }
:
z = \frac{r}{c} \sqrt{\frac{\left( \mu^{2} - c^{2} \right) \left( \nu^{2} - c^{2} \right)}{\left( c^{2} - b^{2} \right)} }
with the following limitations on the coordinates
:
\nu^{2} < c^{2} < \mu^{2} < b^{2}.
Surfaces of constant {{mvar|r}} are spheres of that radius centered on the origin
:
x^{2} + y^{2} + z^{2} = r^{2},
whereas surfaces of constant and are mutually perpendicular cones
:
\frac{x^{2}}{\mu^{2}} + \frac{y^{2}}{\mu^{2} - b^{2}} + \frac{z^{2}}{\mu^{2} - c^{2}} = 0
and
:
\frac{x^{2}}{\nu^{2}} + \frac{y^{2}}{\nu^{2} - b^{2}} + \frac{z^{2}}{\nu^{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 {{mvar|r}} is one ({{math|1=hr = 1}}), as in spherical coordinates. The scale factors for the two conical coordinates are
:
h_{\mu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \mu^{2} \right) \left( \mu^{2} - c^{2} \right)}}
and
:
h_{\nu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \nu^{2} \right) \left( c^{2} - \nu^{2} \right)}}.
References
{{reflist}}
Bibliography
- {{cite book | author = Morse PM, 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 = Margenau H, Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | url = https://archive.org/details/mathematicsofphy0002marg| url-access = registration| publisher = D. van Nostrand | location = New York | pages = [https://archive.org/details/mathematicsofphy0002marg/page/183 183]–184 | lccn = 55010911 }}
- {{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 = 59014456}}
- {{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | pages = 991–100 | lccn = 67025285}}
- {{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}}