Conical coordinates

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Conical coordinates are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius <math>r</math>) and by two families of perpendicular cones.

[edit] 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.

[edit] 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>