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 ) and by two families of perpendicular elliptic cones, aligned along the - and -axes, 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

:$x = \frac$

:$y = \frac \sqrt$

:$z = \frac \sqrt$

with the following limitations on the coordinates

:$\nu^ < c^ < \mu^ < b^.$

Surfaces of constant are spheres of that radius centered on the origin

:$x^ + y^ + z^ = r^,$

whereas surfaces of constant $\mu$ and $\nu$ are mutually perpendicular cones

:$\frac + \frac + \frac = 0$
and
:$\frac + \frac + \frac = 0.$

In this coordinate system, both Laplace's equation and the Helmholtz equation are separable.

Scale factors

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

:$h_ = r \sqrt$

and
:$h_ = r \sqrt.$

References

Bibliography

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External links

MathWorld description of conical coordinates

{{Orthogonal coordinate systems

Three-dimensional coordinate systems
Orthogonal coordinate systems