I know how to construct the charts. I will have: $$\phi_i(x_1,x_2) : U_i \to \mathbb{R}$$ where $Y_i \subset \mathbb{RP}^1$ defined by: $U_i :=\{(x_1,x_2) : x_i > 0\},$
$\phi_1(x_1,x_2) = \frac{x_2}{x_1},$ for example.
Let $S^1$ the unit circle with the stereographic projection:
$\mathcal{A} = \{(U_N,\phi_N), (U_S,\phi_S)\},$ defined as follows:
$$U_N := \{(x,y) \in \mathbb{R}^2\cap S^1\}$$ such that $ y \neq 1.$
$$U_S := \{(x,y) \in \mathbb{R}^2\cap S^1\}$$ such that $ y \neq -1.$
$$\phi_N(x,y) := \frac{x}{1-y},$$ $$\phi_S(x,y) := \frac{x}{1+y}.$$
How can I construct a diffeomorphism between these two manifolds using these charts? I don't know how to define it.. I don't know how to choose the functions defined on every open set and what open set to choose.
Thanks!