We are working in the projective space $RP^2$. For a general element $x \in RP^2$ we use the notation $x := [(x_0, x_1, x_2)].$ In $RP^2$ we typically choose the line $x_2 = 0$ to be the 'line at infinity'. Then all points in $RP^2$, which are not at infinity, satisfy $x_2 \not = 0$. We can then rewrite all these points in the form $A := [(y_0, y_1, 1)]$, by linearity of the coset. By this identification we can consider the projective point $A$ to be the point $(y_0, y_1)$ in the affine space.
Now suppose that I choose a different line to be the line at infinity. Let $l_1 \subset RP^2$ be given by the condition that $x_0 + x_1 = 0$. In this case, I am very confused on how I can construct affine points and how we can find coordinates for them.
Any help would be greatly appreciated.
We can simply construct a projective transformation which maps the line $l_1 <-> x_0 + x_1 = 0$ to the line $ l_2 <-> x_2 = 0$, call such a transformation $\phi$. When the line at infinity is $l_2$, we can easily define a bijection between projective points not contained in $l_2$ and affine points, call this bijection $f$. We now find that the bijection $f \circ \phi$ which will give affine coordinates to each projective point in $RP^2$ which is not contained in $l_1$.