The first and the second axioms for affine planes are really just the first and fifth of Euclid's postulates! So the Euclidean plane is an affine plane (but not visa-versa: in the subject of finite geometry we are only interested in properties of incidence). Recall from Problem 18 that we can derive an affine plane from a projective plane by removing a line and the points with which it is incident. If we remove a line from the real projective plane (the projective plane associated with the field of real numbers) we are left with the real affine plane.

Let's look further into the real affine plane:

The real coordinate plane (R²) is the affine plane in which we define our points and lines as follows: our set of points P := {( x, y ) such that x and y are both real numbers } and our set of lines L :={( x ,y ): ax + by = c, where a and b are not both 0 }. Now a, b and c must be real numbers where . It is easy to see that these sets P and L define the point and line set of the Euclidean plane. We know that axioms 1 and 2 must hold in either the rational coordinate plane Q² or the real coordinate plane R², as any two points are on one and only one line. In addition, in the rational or real coordinate planes, given a point p not on a line l, there is then a unique line that contains p that is parallel to l. Axiom 3 is trivial.

Thus, we can see that there also must exist a rational affine plane such that our point set P is really just the rational coordinate plane Q², i.e., the points of the form P := {( x, y ) such that x and y are rational numbers}. In addition, our lines are of the form: L := {( x, y ): ax + by = c} where a, b and c are rational numbers and .

These are both examples of infinite affine planes, but as we've already seen in previous problems, finite affine planes certainly exist. We have already seen an example of the smallest affine plane (recall the affine plane embedded into the Fano plane). It has four points and six lines. You can check for yourself that all of the axioms of an affine plane hold for this case:

Notice that if we let a = ( 0,0 ), b = ( 1,0 ), c = ( 0,1 ), and d = ( 1,1 ), this example of an affine plane can be viewed as the affine coordinate plane over Z₂. This is similar to Problem 21, where we learned how to build a model of a projective plane from any given field. We can also use a similar method to create affine planes.

If we have a field F, then we can define a point set (actually a 2-dimensional vector space over F) P as F², i.e., P:= {( x, y ) such that x and y are elements of F }. In addition, our line set will be of the form: L:={( x ,y ): ax +by = c such that a, b, c are in F, and a and b are not both 0}. Thus, our point and line sets create the coordinate affine plane with the obvious incidence relation.

References: Batten, Bennett, Beutelspacher & Rosenbaum, Polster