Английская Википедия:Conical coordinates

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Файл:Conical coordinates.png

Coordinate surfaces of the conical coordinates. The constants Шаблон:Math and Шаблон:Math were chosen as 1 and 2, respectively. The red sphere represents Шаблон:Math, the blue elliptic cone aligned with the vertical Шаблон:Math-axis represents μ=cosh(1) and the yellow elliptic cone aligned with the (green) Шаблон:Mvar-axis corresponds to Шаблон:Math. The three surfaces intersect at the point Шаблон:Math (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 Шаблон:Mvar) and by two families of perpendicular elliptic cones, aligned along the Шаблон:Math- and Шаблон:Math-axes, respectively. The intersection between one of the cones and the sphere forms a spherical conic.

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 Шаблон:Mvar 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> and

<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 Шаблон:Mvar is one (Шаблон: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>

and

<math>

h_{\nu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \nu^{2} \right) \left( c^{2} - \nu^{2} \right)}}. </math>

Bibliography

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book

External links

• MathWorld description of conical coordinates