# Stereographic projection

#### cristianoceli

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

#### 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

1 person

#### cristianoceli

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

#### Idea

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

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

1 person

#### cristianoceli

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

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

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

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