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

Note that

and

Therefore,

In other words, we have just shown that the Euclidean distance has been scaled by . Therefore, the standard metric on induced from fulfills

Hence the standard volume element on satisfies

Thus the Jacobian of the stereographic projection at the point is the coefficient .

See also:

- Stereographic projection, 5
- Stereographic projection, 4
- Stereographic projection, 3
- Stereographic projection, 2
- Stereographic projection

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