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# The spherical metric via stereographic projection.

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Find the induced metric of the sphere S^2 in stereographic projection coordinates (M, N), where

M = - x/(z-1), N = -y/(z-1),

with x^2+y^2+z^2=1.

##### Solution Summary

This solution provides a detailed explanation of how to compute the metric induced on the plane by stereographic projection of the sphere into the plane. Step by step calculations are given.

##### Solution Preview

The solution to this problem requires some tedious calculations involving partial derivatives and algebra. I will skip some details, but I will present enough material so that you can fill in the details yourself.

SOLUTION:

First let me change M to u and N to v, so that we have the equations

(1) u = x/(1-z), v = y/(1-z), x^2 + y^2 + z^2 = 1.

Our first task is to write x, y, z in terms of u and v. To do this, we calculate

(2) u^2 + v^2 = (x^2 + y^2) / ...

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