# 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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