Conical coordinates

Basic definitions

The conical coordinates ${\displaystyle (r,\mu ,\nu )}$ are defined by

${\displaystyle x=\frac{r\mu \nu }{bc}}$

${\displaystyle y=\frac{r}{b}\sqrt{\frac{\left({\mu }^2-b^2\right)\left({\nu }^2-b^2\right)}{\left(b^2-c^2\right)}}}$

${\displaystyle z=\frac{r}{c}\sqrt{\frac{\left({\mu }^2-c^2\right)\left({\nu }^2-c^2\right)}{\left(c^2-b^2\right)}}}$

with the following limitations on the coordinates

${\displaystyle {\nu }^2<c^2<{\mu }^2<b^2.}$

Surfaces of constant r are spheres of that radius centered on the origin

${\displaystyle x^2+y^2+z^2=r^2,}$

whereas surfaces of constant ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are mutually perpendicular cones

${\displaystyle \frac{x^2}{{\mu }^2}+\frac{y^2}{{\mu }^2-b^2}+\frac{z^2}{{\mu }^2-c^2}=0}$

and

${\displaystyle \frac{x^2}{{\nu }^2}+\frac{y^2}{{\nu }^2-b^2}+\frac{z^2}{{\nu }^2-c^2}=0.}$

In this coordinate system, both Laplace's equation and the Helmholtz equation are separable.