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

Note that

As already computed in this entry, we know that

Therefore,

Finally, we have

Thus,

thanks to . Again, from this entry, we obtain

In other words, we have just shown that the Euclidean distance has been scaled by

Clearly this scaling factor has limit as . To find the limit, it suffices to find the limit of

We rewrite this as follows

Hence, we have to find the limit of

as . If we write and , then by sending step by step, it is not hard to verify that

as . Hence,

where and denote the first and coordinates of , respectively.

Hence by taking implies that tangent vectors at are scaled by a (conformal) factor . Therefore, the standard metric on is expressed in terms of by

Thus the Jacobian of at the point is .

**Corollary**. If is the North pole , then is the zero vector. Consequently,

**Application**. As a consequence of the above finding, for the conformal transformation given by

the Jacobian of its inverse is

Clearly,

and

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