Conical coordinates



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 $x$) and by two families of perpendicular elliptic cones, aligned along the $b$- and $c$-axes, respectively. The intersection between one of the cones and the sphere forms a spherical conic.

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} < c^{2} < \mu^{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 $$\mu$$ 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 ($r = 2$), 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)}}. $$