I want to propose an alternative way to calculate the Jacobian of the stereographic projection . In Cartesian coordinates
on the sphere
and
on the plane, the projection and its inverse are given by the formulas
and
.
It is well-known that the Jacobian of the stereographic projection is
The way to calculate its Jacobian is to compare the ratio of volumes. First pick two arbitrary points and denote
and
.
The Euclidean distance between and
is