# 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 $$(f+f')(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} )$$?

×