Conical coordinates

Basic definitions

The conical coordinates (r, \mu, \nu) 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}

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 \mu and \nu 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 r is one (), 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

Bibliography