Conical coordinates

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 are defined by

with the following limitations on the coordinates

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

whereas surfaces of constant and are mutually perpendicular cones


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


Light cone conical coordinates

An alternative set of (non-orthogonal) conical coordinates have been derived[1]

where are spherical polar coordinates. The corresponding inverse relations are

The infinitesimal Euclidean distance between two points in these coordinates

and are orthogonal coordinates on the surface of the cone given by . If the path between any two points is constrained to this surface, then the geodesic distance between any two points

and is


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