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.
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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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