
Re: a question on the real projective plane
Posted:
Jan 28, 2014 5:18 PM


Arturo has given you plenty of substantial theory; but you may also like some more elementary hints.
In article <lc3vgo$vu0$1@dontemail.me>, David Bernier <david250@videotron.ca> wrote:
> I like the view that the real projective plane has as points > the lines through (0, 0, 0) in R^3 .
Any direction vector for such a line gives homogeneous coordinates (x,y,z) for a projective point. Nonzero scalar multiples (kx,ky,kz) do just as well. Thus, for example, the point (1,2,3) has alternative names (2,4,6) etc. Also each line has homogeneous coordinates [l,m,n], with the property that point (x,y,z) lies on line [l,m,n] iff lx + my + nz = 0.
> So, there is a line in the real projective plane > called the line at infinity, which is > an addition to the affine plane....
Only if you choose it so. You can switch between the real projective and affine planes by removing or inserting one line. For example, if you delete the line [1,1,1], having equation x + y + z = 0, then you get barycentric coordinates in the affine plane. A more popular choice is to delete the line [0,0,1], having equation z = 0. Then any point (x,y,z) not on that line has Cartesian coordinates (x/z, y/z).
> .... > In the real projective plane, the line x = 0 meets the "line at > infinity" at a point, say Q. I think the two lines meet at > a single point. Obviously, Q is a point at infinity.
The line [1,0,0], having equation x = 0, meets the line at infinity [0,0,1] where both x and z are zero, so your point Q is (0,1,0), or (0,6,0) or whatever.
> > Thinking of finite projective planes from finite fields, it > seems to me that points at infinity are not special , they're just > like the others.
That's true of _all_ projective planes. None of them has special points at infinity unless you _choose_ a line to be the line at infinity (and delete it to get an affine plane, finite or not).
> .... > So does there exist a bijection: > > f: RP^2 > RP^2 (real projective plane) > > mapping points to points bijectively, lines to lines bijectively > that preserves the incidence relation, and > that maps (0, 0) to a point on the line at infinity?
Such bijections are called collineations. For the real projective plane, just think about multiplying each (x,y,z) on the right by an invertible square matrix. In your example, the point (0,0) (Cartesian) is (0,0,1) (homogeneous), so you're asking for a matrix which maps (0,0,1) to (something,something,0). Use any 3x3 invertible matrix A whose entry A_33 = 0.
HTH
Ken Pledger.

