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