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## Main Question or Discussion Point

In any book on differentiable manifolds, the stereographic projection map P from the n-Sphere to the (n-1)-plane is discussed as part of an example of how one might cover a sphere with an atlas. This is usually followed by a comment such as "it is obvious" or "it can be shown" that the inverse projection P^{-1} is given by such and such. Now, it is not at all "obvious" to me algebraically how this can be accomplished. The inverse function theorem of couse can guarantee that the inverse projection exists but, being an existential theorem, provides no means for finding the inverse. How does one approach a problem such as this? Apparently, the geometry of the situation can be used to derive the inverse but I'm lousy at geometry so I'm not sure how to approach it from this angle (no pun intended) either. Thank you for any suggestions.