TITLE: Showing diffeomorphism between $S^1 \subset \mathbb{R}^2$ and $\mathbb{RP}^1$
QUESTION [6 upvotes]: I am trying to construct a diffeomorphism between $S^1 = \{x^2 + y^2 = 1; x,y \in \mathbb{R}\}$ with subspace topology and $\mathbb{R P}^1 = \{[x,y]: x,y \in \mathbb{R}; x \vee y \not = 0  \}$ with quotient topology and I am a little stuck.
I have shown that both are smooth manifolds, and I used stereographic projection for $S^1$, but now I am runing into trouble when I give the homeomorphism between $S^1$ and $\mathbb{RP}^1$ as the map that takes a line in $\mathbb{RP}^1$ to the point in $S^1$ that you get when letting the parallel line go through the respective pole used in the stereographic projection. 
If I use the south and north poles I get a potential homeomorphism, but I cannot capture the horizontal line in my image, but when I pick say north and east then my map is not well defined as I get different lines for the same point in $S^1$. Can somebody give me a hint how to make this construction work, or is it better to move to a different representation of $S^1$ ?

REPLY [9 votes]: Consider the stereographic atlas $\{(U_\alpha, \varphi_\alpha), (U_\beta, \varphi_\beta)\}$  where $U_\alpha=\mathbb S^1-\{(0, 1)\}$ and $U_\beta=\mathbb S^1-\{(0, -1)\}$. Here $\varphi_\alpha$ is given by $$(x, y)\mapsto u=\frac{x}{1-y},$$ whereas $\varphi_\beta$ is given by $$(x, y)\mapsto v=\frac{x}{1+y}.$$ I don't know if I have given the right names for the south and north projection, but you decide later..Now consider the atlas $\{(U_1, \varphi_1), (U_2, \varphi_2)\}$ for $\mathbb R\mathbb P^1$ given by $\varphi_1:U_1\rightarrow \mathbb R-\{0\}$, $$[(x, y)]\mapsto \frac{y}{x}$$ and $\varphi_2:U_2\rightarrow \mathbb R-\{0\}$ is given by, $$[(x, y)]\mapsto \frac{x}{y}.$$ Define $\psi:\mathbb R\mathbb P^1\rightarrow \mathbb S^1$ setting $$\psi(u)=\left\{\begin{array}{ccc}\varphi_\alpha^{-1}\circ \varphi_1&\textrm{if}&u\in U_1\\ \varphi_\beta^{-1}\circ \varphi_2&\textrm{if}&u\in U_2\end{array}\right..$$ In this case $\psi$ is defined in $U_1\cup U_2=\mathbb R\mathbb P^1$, but for it to be well defined you must check that in $U_1\cap U_2$, $$\varphi_\alpha^{-1}\circ \varphi_1=\varphi_\beta^{-1}\circ \varphi_2.$$ This will be your diffeomorphism..