Index

We have now seen Desargues' Theorem for the Euclidean plane as well as for the affine plane. Now we will look at Desargues' Theorem for the Projective Plane. Recall that prior versions of Desargues' Theorem were split into two parts. For the projective plane, these two parts reduce to one.

The Desarguesian planes that we have seen so far satisfy axioms that deal with parallel lines. But as we have seen, the parallel postulate fails be satisfied in any projective plane. For example, a pencil of parallel lines in the regular Euclidean plane meet at a unique point at infinity in the real projective plane, and all such points at infinity (one for each pencil of parallel lines) lie along a line at infinity. With this in mind, let's now look at how Desargue's Theorem applies to the real projective plane.

Recall that in an affine plane we start with distinct points x, y, z, x', y' and z' (we're changing the element labels from the way they were originally introduced for reasons that will be apparent shortly) where x and y and z are noncollinear with l (x, y) || l (x',y') || l (z,z') and l (x, y) || l (x',y') || and l (x, z) || l (x',z'). In the projective plane we will require instead that (as in Part II), because after we adjoin a line at infinity to our affine plane to form the projective plane, the pencil of parallel lines l (x, y) || l (x',y') || l (z,z') will meet at a point at infinity.

Here is an example of this:

Created with Cinderella

The reason that we are able to write Desargues' Theorem in the projective plane in just one statement is because (as we saw with the affine plane) Theorem I deals with the case where the lines l (x,x'), l (y,y') and l (z,z') are parallel and Theorem II deals with the case when they intersect at a given point. In the projective plane all lines intersect.

Notice also that a projective plane will be Desarguesian if and only if it satisfies the converse of Desargue's Theorem. This is because the duality principle holds for any projective plane!

In addition, we can rewrite Pappus' Theorem for a projective plane after adding the line at infinity to the corresponding (derived) affine plane.

Here is an example of this.

Created with Cinderella

We now propose the following theorem.

For a proof of this see (Batten, pg. 59).

Now let's look at a few more implications of Desargues' Theorem for the projective plane. It is true that any set of 10 points that satisfies the properties described above will be a Desarguesian configuration, but an exciting example of this can occur from a (c,l) - collineation. Now it is obvious why we have labeled the points as we have. We can see that f(x) = x', f(y) = y' and f(z) = z' are mappings under a collineation such that c is the center. We can see that in the figure shown above of the Desarguesian configuration, our points x, y and z are not in . (This is obvious as they are certainly distinct from c, and recall that for our axis l, p = f(p) = p' for any point p on l ). In this case rather, l is the line that contains u, v and w, as all three of these are fixed points (Recall that we call a central collineation where the center does not lie on the axis a homology.

It is easy to see why this is not true in the case of the Fano plane. Any non-identity (c,l)-collineation must be an elation and it is impossible to then choose x, x', y, y', z, z' as would be necessary (see the solution to Problem 25). The rest of the proof is simple and can be seen from the figure of the Desargeusian configuration above.

Here is an interesting theorem about Desarguesian projective planes:

A proof of this is in (Bennett, pg. 193)

We now come to a very exciting conclusion about Desarguesian projective planes.

References: Batten, Bennett, Beutelspacher & Rosenbaum, Polster