# Turning affine planes into projective planes

How can we show that an affine plane of order $n$ can always be turned into a projective plane of order $n$?

Say I start with an affine plane, and split it into $n+1$ parallel classes, add a point $\infty_i$ to each class $C_i$, extend every line in $C_i$ to contain $\infty_i$, and finally connect all the "$\infty_i$" points into a line. I think I am on the correct track with this construction. Could someone use this start to perhaps complete the question?

• Looks like a good start to me (+1)! Isn't the rest case-by-case verification of the axioms of a projective plane? – Jyrki Lahtonen Jul 18 '12 at 6:26
• Yeah, but I am having trouble with the clarity of those cases. hence why I put the question here. :) – Xuan Huang Jul 18 '12 at 7:03
• Two lines in your projective plane are either both affine lines extended by a point or the other is the line at infinity. Treat those cases separately. The affine case splits further according to whether the two lines are parallel or not. – Jyrki Lahtonen Jul 18 '12 at 7:22
• I still can't put it together. How disappointing. – Xuan Huang Jul 18 '12 at 15:38
• Would you be able to show me these cases? – Xuan Huang Jul 18 '12 at 21:53

As an example let us show that any two lines, $L_1$ and $L_2$, of your projective plane intersect at exactly one point.
If one of them, say $L_1$, consists of the points at infinity only, then $L_2$ intersects the affine plane along a line $L_2'$. Therefore $L_2$ contains exactly one infinite point, namely the one associated with all the affine lines parallel to $L_2'$. Therefore $L_1\cap L_2$ is a singleton.
If both $L_1$ and $L_2$ intersect the affine plane (along the lines $L_1'$ and $L_2'$), then there are two possibilities. If $L_1'$ and $L_2'$ are not parallel, then they intersect at a finite point. The same holds then for $L_1$ and $L_2$, as the infinite points are distinct (due to them not being parallel). OTOH, if $L_1'$ and $L_2$' are parallel, then they don't have any common affine points. But then the same point at infinity is tagged on them to get the lines $L_1$ and $L_2$, so the projective lines intersect at an infinite point.