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It seems that there is a standard technique for an idea similar to the stereographic projection. I don't know how can I use it. For example here in this exercise, how can I use it ? I'm really sorry for being so stupid....

Define a birational map from an irreducible quadric hypersurface $X \subset P^3$ to $P^2$, by analogy with the stereographic projection, and find the open sets $U \subset X$, $V \subset P^2$, that are isomorphic.

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View $\mathbb{P}^2$ as a hyperplane $H \subset \mathbb{P}^3$ and choose a point $P_0$ that lies on $X$ but is not on $H$. Consider the rational map $X \dashrightarrow H$ that sends $P \in X$ to the point on $H$ that intersects the line through $P$ and $P_0$. You need to check: (1) this is actually a rational map; (2) this map is invertible as a rational map (the inverse has a similarly concrete geometric description); (3) what are subsets of $X$, resp. $H$, where the map, resp. its inverse, is not well-defined.

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