Conical coordinates

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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.

[edit] Basic definitions

The conical coordinates (r,μ,ν) are defined by

with the following limitations on the coordinates

ν² < c² < μ² < b²

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

x² + y² + z² = r²

whereas surfaces of constant μ and ν are mutually perpendicular cones

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

[edit] 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

[edit] References

• Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.