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.
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
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
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
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