Example 17.10.13 [Polar coordinates in space] The standard polar coordinates in space (§7.1) are obtained by rotating the plane polar coordinates around the x-axis. From (17.10.48) and (17.10.56) we have
and from (17.10.57) hX = hY = eξ, hZ = eξ cos η. The substitution r = eξ reduces these expressions to the usual form.
Rotation of the plane bi-polar coordinate system around the x-axis generates bi-spherical coordinates in which the coordinate surfaces are spheres, spindles and half-planes. Rotation around the y-axis generates toroidal coordinates in which the coordinate surfaces are tori and intersecting spheres.