Coordinate surfaces of the conical coordinates. The constants
b and
c were chosen as 1 and 2, respectively. The red sphere represents
r = 2, the blue elliptic cone aligned with the vertical
zaxis represents μ=cosh(1) and the yellow elliptic cone aligned with the (green)
xaxis corresponds to
ν^{2} = 2/3. The three surfaces intersect at the point
P (shown as a black sphere) with
Cartesian coordinates roughly (1.26, 0.78, 1.34). The elliptic cones intersect the sphere in
spherical conics.
Conical coordinates, sometimes called spheroconal or spheroconical coordinates, are a threedimensional 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 xaxes, 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 (h_{r} = 1), 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
 $h_{\nu }=r{\sqrt {\frac {\mu ^{2}\nu ^{2}}{\left(b^{2}\nu ^{2}\right)\left(c^{2}\nu ^{2}\right)}}}.$


Two dimensional  

Three dimensional  
