A stereographic manipulation (part 2)

Algebra Level pending

First, do have a look at the first part.

Continuing where we left off, we found our two stereographic projections, \(f\) and \(f'\) which cover the whole of our circle \( S^1 \) We can think of \(f\) as a mappings of the form : \[f_{1} : U_1 \rightarrow \overline{U}_1\] where \(U_1 \subseteq S^1 \backslash (0,1) \subseteq \mathbb{R}\) and \(\overline{U}_1 \subseteq \mathbb{R}\)

and similarly for \(f'\) : \[f_{2} : U_2 \rightarrow \overline{U}_2\] where \(U_2 \subseteq S^1 \backslash (0,-1) \subseteq \mathbb{R}\) and \(\overline{U}_2 \subseteq \mathbb{R}\)

First, convince yourself that these two functions are bijections!

Now, let \( 0 \neq \overline{x} \in \overline{U}_2 \). How does \( \overline{x} \) look like in \(\overline{U}_1\),i.e. to what element in \(\overline{U}_1\) does \(\overline{x}\) corresponds to?

Hint : Find the point \((x,y) \in S^1 \backslash \{ (0,1) ; (0,-1) \}\) such that \(f_2(x,y) = \overline{x} \) and then project this point in \(\overline{U}_1\) via the function \(f_1 \)


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