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.
The conical coordinates are defined by
with the following limitations on the coordinates
Surfaces of constant are spheres of that radius centered on the origin
whereas surfaces of constant and are mutually perpendicular cones
and
In this coordinate system, both Laplace's equation and the Helmholtz equation are separable.
The scale factor for the radius is one (), as in spherical coordinates. The scale factors for the two conical coordinates are
and