A stereographic manipulation

Algebra Level 3

Let \( S^1 = \{ (x,y) \in \mathbb{R} \backslash x^2+y^2 = 1 \} \). Consider the stereographic projection :

Let \( \mathcal{L} \) be the line passing through the point \( N=(0,1) \in S^1 \) and some point \( P=(x,y) \in S^1 \) such that \( (x,y) \neq (0,1) \). Then \( \mathcal{L} \) crosses the x-axis at a point \( P'=(\overline{x},0) \). This defines a mapping : \[f : S^1 \backslash \{ (0,1) \} \rightarrow \mathbb{R} \] \[ (x,y) \mapsto \overline{x}=f(x,y) \]

Now, do the same, but this time, with the line \( \mathcal{L'} \) passing through the point \( (0,-1) \in S^1 \). Call the new function it induces by \( f' \).

What is \( (f+f')(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} ) \)?


Credit : Google Image.
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