Problem 26: Desargues' Theorem in the Affine Plane

Index

We have seen the proof of Desargues' Theorem for the Euclidean plane, but we haven't yet seen that the Desarguesian configuration exists beyond that. In fact it does, and we will see how important it is in proving many interesting theorems about both projective and affine planes.

Let's begin by looking at Desargue's Theorem in the coordinate affine plane. Recall that we defined the coordinate affine plane to be the affine plane created over F2 ( F is a field, possibly finite) . To show that Desargues' Theorem does indeed hold in F2 we need to prove the following theorems concerning parallelism.

Theorem 1. Let u, v, z, and w be elements of F2. Then ( u, v, w, z ) is a parallelogram if and only if z = u - v + w.

Let's first look at a picture of this. Its easy to see that this must be true.

Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

Now let's show a quick proof. There exists a line (DC above) with equation ax + by = c (for some scalars a, b, c in F, a and b not both zero) that contains the points u = (u1, u2) and v = (v 1, v2) in F2 such that the point w = (w1, w2) is not on that line. We can see that both w and u - v + w satisfy the equation ax + by = w1a + w2b while the points u and v do not. It is certainly clear that w = (w1, w2) will satisfy this equation: let's see that u - v + w does as well.

( u1 - v1 + w1) a + (u2 - v 2+ w2) = u1a + u2b - u1a - u2b + w1a + w2b = c - c + w1a + w2b = w1a + w2b c.

Thus, the line through u and v is parallel to the line through w and u -v + w. Similarly, the line through u and w is parallel to the line through v and u - v + w. Thus (u, v, w, u - v + w ) is a parallelogram. In addition, by the axioms of an affine plane we know that u - v + w will be the unique poing such that (u, v, w, u - v + w) forms a parallelogram.

Theorem 2. If z, u and v are distinct collinear points in F2 then there is an element t in F, such that v = u + t (u - z).

This figure is a quick illustration of this theorem.

Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

The addition of the point (or vector, which is the same thing) t(u - z) to the point u will just be another point that lies along the line collinear with u and z. So for any point v, it will just be some multiple of u - z added to u.

Here is the third theorem that will be needed later on:

Theorem 3. If l (u,v) || l (z, w) in F2, then u - v is a nonzero multiple of w - z. Conversely, if u, v and z are noncollinear, and u - v is a nonzero multiple of w - z, then l (u , v) || l (z, w) .

This is a quick proof given the 2 theorems we've already seen . Let x be the unique point in F2 such that ( u, v, z, x ) is a parallelogram. It follows then by Theorem 1 that x = u - v + z. By theorem 2 we know that x = z + t (w - z) and for some nonzero scalar t as x is certainly distinct from z. Combining these two results, we are left with u - v + z = z + t ( w- z ). So u - v = t ( w - z ). Thus, u - v is obviously a non - zero multiple of w - z. Try proving the converse for yourself.

Ok, now that we have seen these three Theorems, we propose the following:

Theorem 4. For any field F, the affine plane over F is Desarguesian, meaning that Desargues' Theorem is true.

This is very exciting and we will see the important implications of this a bit later.

Certainly other affine planes that are not created via coordinization can be Desarguesian. We call all such affine planes Desarguesian affine planes.

From theorem 4 it is easy to see that the real coordinate plane is Desarguesian because R is a field. In addition it can easily be shown that the rational coordinate plane is Desarguesian as well (See Bennett, pg. 52).

So, we now have seen several examples of Desarguesian affine planes. Does there exist an affine plane that is not Desarguesian? In 1902, F.R. Moulton produced an example (the Moulton plane) of an affine plane in which Desargues Theorems do not hold. Here is a figure of the plane.

In this plane the point set P is the set of ordered pairs of real numbers. The line set L is the set of all Euclidean horizontal lines, vertical lines and lines with negative slope. In addition, "broken" Euclidean lines are in L - meaning lines that have positive slope m above the x axis and slope 2m below the x axis, where m is a fixed constant.

PROBLEM 26: (a) Using Theorems 1 - 3 given above, prove the result of Theorem 4. (That is, show that Parts I and II of Desargues' Theorem are true in the affine plane over the field F)

(b) First show that the Moulton plane is an affine plane. Then show that it fails to satisfy Desargues Thm. (If this kind of thing excites you, you should start reading Part II - Geometries on Surfaces, of Poster.)

Solutions

References: Batten, Bennett, Beutelspacher & Rosenbaum, Polster