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

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