TITLE: Understanding the connection between the projective space and the affine plane
QUESTION [5 upvotes]: Suppose we have a point $P=[x,y,z]\in \mathbb P^2$. 
Then at least one of the coordinates is not zero. Suppose $z\neq 0$. So we have write $P$ as $[x/z,y/z,1]$ and this point belongs to $(x/z,y/z)$ in the affine plane. 
But the every point on the line at infinity is of the form $[x,y,0]$, right? What are the images of this points?
Vice versa a point $Q=(a,b)$ in the affine plane belongs to the projective point $[a,b,1]$. Which points will be send to points on the line at infinity ?

REPLY [9 votes]: I want to add to the good answer of Ashwin a representation of the projective plane that I consider very nice and intuitive  that is given embedding the projective plane $\mathbb{P}^2$ in $\mathbb{R}^3$.
The figure illustrates the situation.
The points on a plane $z=k$ have coordinates $P=(a,b,k)$ and correspond to points of $\mathbb{P}^2$ as $(a,b,k) \rightarrow (a/k,b/k,1)$ (usually we chose $k=1$ ), but we can identify these points also with the straight lines from the origin that contain these points.
Now, for any such line, we have a point at infinity $(a,b,0)$ that correspond to a line in the plane $z=0$ passing through the point $(a,b,0)$ that gives the ''direction'' at infinity, in the sense that this line is the ''limit position'' of the line passing through $(a,b,k)$ when $k \rightarrow 0$.