Some authors define stereographic projection from the north pole (0, 0, 1) onto the plane , which is tangent to the unit sphere at the south pole (0, 0, −1). This can be described as a composition of a projection onto the equatorial plane described above, and a homothety from it to the polar plane. The homothety scales the image by a factor of 2 (a ratio of a diameter to a radius of the sphere), hence the values  and  produced by this projection are exactly twice those produced by the equatorial projection described in the preceding section. For example, this projection sends the equator to the circle of radius 2 centered at the origin. While the equatorial projection produces no infinitesimal area distortion along the equator, this pole-tangent projection instead produces no infinitesimal area distortion at the south pole.
Other authors use a sphere of radius  and the plane . In this case the formulae become
In general, one can define a stereographic projection from any point  on the sphere onto any plane  such that
* is perpendicular to the diameter through , and
* does not contain .
As long as  meets these conditions, then for any point  other than  the line through  and  meets  in exactly one point , which is defined to be the stereographic projection of P onto E.