1. ## Stereographic projection

Hl! I have problems with this exercises

Let $\displaystyle \varepsilon$ the Stereographic projection of $\displaystyle \mathbb{S}^1- \{i \}$, that is

$\displaystyle \varepsilon : \mathbb{S}^1 - \{i \} \longrightarrow{\mathbb{R}}$

where

$\displaystyle \varepsilon(z)=\displaystyle\frac{RE(z)}{1-Im(z)}$

Find a formula for $\displaystyle \varepsilon^{-1} :\mathbb{R}\rightarrow{\mathbb{S}^1}- \{i \}$

I tried to take the line that passes through i and a point on the real axis to see where the circle intersects but I can not find the formula

Thanks

2. ## Re: Stereographic projection

The stereographic projection maps

$\displaystyle i \to \infty$

$\displaystyle -i\to 0$

$\displaystyle 1\to 1$

Therefore its inverse sends

$\displaystyle \infty \to i$

$\displaystyle 0 \to -i$

$\displaystyle 1 \to 1$

it's easy to write down the corresponding Moebius transformation

3. ## Re: Stereographic projection

Originally Posted by Idea
The stereographic projection maps

$\displaystyle i \to \infty$

$\displaystyle -i\to 0$

$\displaystyle 1\to 1$

Therefore its inverse sends

$\displaystyle \infty \to i$

$\displaystyle 0 \to -i$

$\displaystyle 1 \to 1$

it's easy to write down the corresponding Moebius transformation
You can make the first correspondence , please

4. ## Re: Stereographic projection

$\displaystyle \varepsilon ^{-1}(u)=\frac{i u+1}{u+i}$

$\displaystyle u\in \mathbb{R}$

5. ## Re: Stereographic projection

Originally Posted by Idea
$\displaystyle \varepsilon ^{-1}(u)=\frac{i u+1}{u+i}$

$\displaystyle u\in \mathbb{R}$
Thanks

but, why the domain is $\displaystyle \mathbb{R}$and the codomain is$\displaystyle \mathbb{S}^1- \{i \}$ ?