Denote the stereographic projection performed with as the north pole to the equatorial plane of . Clearly when is the north pole , i.e. , then is the usual stereographic projection.

As we have already known that, for arbitrary , the image of is

Here the point is being understood as a point in by adding zero in the last coordinate. For the inverse map, it is not hard to see that

The purpose of this entry is to compute the Jacobian of the, for example, by comparing the ratio of volumes.

First pick two arbitrary points and denote and . The Euclidean distance between and is