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?

  • 2
    $\begingroup$ Looks like a good start to me (+1)! Isn't the rest case-by-case verification of the axioms of a projective plane? $\endgroup$ – Jyrki Lahtonen Jul 18 '12 at 6:26
  • $\begingroup$ Yeah, but I am having trouble with the clarity of those cases. hence why I put the question here. :) $\endgroup$ – Xuan Huang Jul 18 '12 at 7:03
  • $\begingroup$ 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. $\endgroup$ – Jyrki Lahtonen Jul 18 '12 at 7:22
  • $\begingroup$ I still can't put it together. How disappointing. $\endgroup$ – Xuan Huang Jul 18 '12 at 15:38
  • $\begingroup$ Would you be able to show me these cases? $\endgroup$ – 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.

The other axioms of a projective plane are verified in the same way. For example, you show that there is a line through any pair of points by splitting it into separate cases according to whether neither, one, or both of the points are infinite or affine.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.