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Topic: a question on the real projective plane
Replies: 9   Last Post: Jan 30, 2014 5:59 PM

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 Ken.Pledger@vuw.ac.nz Posts: 1,374 Registered: 12/3/04
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@dont-email.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. Non-zero 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.