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.
Basic definitions
The conical coordinates ( r , μ , ν ) {\displaystyle (r,\mu ,\nu )} are defined by
x = r μ ν b c {\displaystyle x={\frac {r\mu \nu }{bc}}} y = r b ( μ 2 − b 2 ) ( ν 2 − b 2 ) ( b 2 − c 2 ) {\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)}}}} z = r c ( μ 2 − c 2 ) ( ν 2 − c 2 ) ( c 2 − b 2 ) {\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
ν 2 < c 2 < μ 2 < b 2 . {\displaystyle \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 , {\displaystyle x^{2}+y^{2}+z^{2}=r^{2},}whereas surfaces of constant μ {\displaystyle \mu } and ν {\displaystyle \nu } are mutually perpendicular cones
x 2 μ 2 + y 2 μ 2 − b 2 + z 2 μ 2 − c 2 = 0 {\displaystyle {\frac {x^{2}}{\mu ^{2}}}+{\frac {y^{2}}{\mu ^{2}-b^{2}}}+{\frac {z^{2}}{\mu ^{2}-c^{2}}}=0}and
x 2 ν 2 + y 2 ν 2 − b 2 + z 2 ν 2 − c 2 = 0. {\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.
Scale factors
The scale factor for the radius r is one (hr = 1), as in spherical coordinates. The scale factors for the two conical coordinates are
h μ = r μ 2 − ν 2 ( b 2 − μ 2 ) ( μ 2 − c 2 ) {\displaystyle h_{\mu }=r{\sqrt {\frac {\mu ^{2}-\nu ^{2}}{\left(b^{2}-\mu ^{2}\right)\left(\mu ^{2}-c^{2}\right)}}}}and
h ν = r μ 2 − ν 2 ( b 2 − ν 2 ) ( c 2 − ν 2 ) . {\displaystyle h_{\nu }=r{\sqrt {\frac {\mu ^{2}-\nu ^{2}}{\left(b^{2}-\nu ^{2}\right)\left(c^{2}-\nu ^{2}\right)}}}.}Bibliography
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