Incidence Geometry

We are now ready to begin our study of geometries in earnest. We will study neutral geometry, based on the axioms of Hilbert. This means that we will study all that we can, almost, without the introduction of a Parallel Postulate of any sort. At the appropriate time we will add a parallel postulate and see where we will be led.

For an ease of notation, let denote the statement that the point B lies between the point A and the point C.

Definition: Let be any line and let A and B be any points that do not lie on . If A=B or if the segment AB contains no point lying on , we say that A and B are on the same side of .   If if and if AB contains a point of , we say that A and B are on opposite sides of .  

Let us quickly review the incidence axioms.

Incidence Axiom 1: For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q.

Incidence Axiom 2: For every line there exists at least two distinct points incident with .

Incidence Axiom 3: There exist three distinct points with the property that no line is incident with all three of them.

This does not seem like much, but already we can prove several easy properties that any set satisfying these three axioms must have.

Theorem 5.1:  If and m are distinct lines that are not parallel, then and m have a unique point in common.

Let's be brave and give a formal and an informal proof of this theorem. Having done that, I think that you will see how an informal proof is really a rigorous proof, just not a formal proof.

Proof: First, the formal proof. We shall break the statement into its three constituent parts.

P: .

Q: .

R: and m have a unique point in common.

We are to prove then that

An informal proof of this result follows much the same line, but is easier to read.

Proof: Since and m are not parallel and since , they must have at least one point in common. Assume that they have more than one point in common. They then have at least two points in common. Axiom I-1 says that two points determine a unique line, so , which is contrary to the hypothesis. Thus, and m have a unique point in common.

Definition: Two or more lines are concurrent if they intersect in one common point. 

Definition: Two or more points are collinear if they are all incident with the same line. 

We have four other results to mention.

Theorem 5.2:  For every line there is at least one point not incident with it.

Theorem 5.3:  every point there is at least one line not incident with it.

Theorem 5.4:  every point there exist at least two lines incident with it.

Theorem 5.5:  exist three distinct lines which are not concurrent.

To introduce you to the concept of a model for geometry, let us look at a simple example of some mathematical object which satisfies the three axioms of incidence, based on our interpretation of the undefined concepts.

Example: Consider the set . We shall interpret a point to be a singleton subset of U. Thus, , and are points. We shall interpret a line to be a doubleton subset of U. The lines are then , , and . We shall agree that a point is incident with a line if it is a subset of the line. Now, before we continue, we must verify that each of the Incidence Axioms is valid in this particular example.

Axiom I-1: If X and Y are any of the points of this geometry, then is the unique line which contains them, for there are only three possible lines.

Axiom I-2: If is any line in this geometry, then and are two distinct points incident with it.

Axiom I-3: The points , and are three distinct points which are not collinear.

Thus this is a model of a geometry which satisfies the Incidence Axioms. Such a geometry is called an incidence geometry. There are a number of different ways of visualizing this geometry.

Example: Again, consider the set . We shall interpret a point to be a doubleton subset of U. The points are then , , and . We shall interpret a line to be a singleton subset of U. Thus, , and are lines. We shall agree that a point is incident with a line if it contains the line as a subset. Now, we must verify that each of the Incidence Axioms is valid in this example.

Axiom I-1: If X and Y are any of the points of this geometry, then is the unique line which contains them. There are three possible points and each must contain two elements from U. Thus, the intersection in the set U will be nonempty.

Axiom I-2: If is any line in this geometry, then and are two distinct points incident with it.

Axiom I-3: The points , and are three distinct points which are not collinear, since their intersection is empty.

Thus this, too, is a model of an incidence geometry. It is sometimes referred to as the dual geometry to the previous example.

Note that since the three incidence axioms hold for each of these two examples, the five theorems must also hold. One further item to note about these geometries--there are no parallel lines. Given a line and a point not on that line in each of these two examples, there is no line through that point parallel to the given line. We say that these two models exhibit the elliptic parallel property. This property is not inherent in the incidence axioms, but is inherent in the examples. There are other examples of incidence geometries which do not exhibit the elliptic parallel property.

This implies that we cannot prove the Euclidean Parallel Postulate based only on the incidence axioms. In fact we cannot prove that parallel lines even exist, based solely on the incidence axioms. Furthermore, we cannot prove that they do not exist.

Example: If we take and take the same interpretation for point and line as in Example 1, then we will have an incidence geometry which exhibits the Euclidean parallel property--unique parallels.

Example: If we take and take the same interpretation for point and line as in Example 1, then we will have an incidence geometry which exhibits the hyperbolic parallel property--multiple parallels.

Definition: We say that two models for incidence geometry are isomorphic if there is a one-to-one correspondence between the points of the models, , and the lines of the models , which preserves the incidence relations; i.e., P' is incident with if and only if P is incident with .

Note that we will not be able to have a model with the elliptic parallel property isomorphic to a model with the hyperbolic or Euclidean parallel property, for incidence would not be preserved.