# Conical coordinates

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 z-axis represents μ=cosh(1) and the yellow elliptic cone aligned with the (green) x-axis corresponds to ν2 = 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 taco-shaped curves.

Conical coordinates are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius $r$) and by two families of perpendicular cones, aligned along the z- and x-axes, respectively.

## 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

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

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

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

## Bibliography

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• Moon P, Spencer DE (1988). "Conical Coordinates (r, θ, λ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 37–40 (Table 1.09). ISBN 978-0-387-18430-2.