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.

The scale factor for the radius r is one (h_r = 1), as in spherical coordinates. The scale factors for the two conical coordinates are

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

and

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

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• MathWorld description of conical coordinates