Incidence Geometry

September 22, 2009

c 2009 Charles Delman What is Incidence Geometry?

O Incidence geometry is geometry involving only points, lines, and the incidence relation. It ignores the relations of betweenness and congruence.

O Incidence geometry based on the three neutral incidence axioms might be termed neutral incidence geometry.

O These axioms might be modified or expanded to obtain special types of incidence geometry.

c 2009 Charles Delman 1 Definitions Recall the following definitions based on the relation of incidence:

O A set of points is collinear if there is a common line incident with all of them. A set of points that is not collinear is called noncollinear.

O A set of lines is concurrent if there is a common point incident with all of them.

O Two lines are parallel if there is no point incident with both of them. (That is, they do not intersect; in this case, the word “intersect” is given a meaning distinct from its set-theoretic one, since lines are not necessarily considered to be sets of points.) More generally, a set of lines is parallel if, for any two distinct lines in the set, there is no point incident with both of them.

c 2009 Charles Delman 2 The Axioms of Neutral Incidence Geometry Recall the three neutral incidence axioms:

O Axiom I-1: For every point P and for every point Q that is distinct from P , there is a unique line l incident with P and Q. (That is, there is a line incident with both P and Q, and if lines l and m are each incident with both P and Q, then l = m.)

O Axiom I-2: For every line l there exist (at least) two points incident with l.

O Axiom I-3: There exist three distinct noncollinear points.

c 2009 Charles Delman 3 Recall also the following notation:

←→ O The unique line incident with points P and Q is denoted by PQ.

c 2009 Charles Delman 4 Propositions of Neutral Incidence Geometry It is instructive and useful to establish some theorems that can be proven using only these three axioms. In fact, for the following five propositions, we will not even need to use Axiom I-2! Of course, these theorems remain true (and proven) when more axioms are added. The propositions are numbered in accordance with the textbook.

O Proposition 2.1 If l and m are distinct lines that are not parallel, then l and m have a unique point in common. To get a feel for this proposition, let us consider an equivalent statement, its contrapositive.

O Proposition 2.1 If l and m do not have a unique point in common, then l and m are parallel or l = m.

c 2009 Charles Delman 5 A Proof of Proposition 2.1 Proof. [This proof is based on the first statement: If l and m are distinct lines that are not parallel, then l and m have a unique point in common.] Let l be a line, and let m 6= l be a line that is not parallel to l. [Here we assume the conditions of the theorem. Note that l can be any line, but then restrictions depending on l apply to our choice of m.] Claim 1. Lines l and m have a point in common. The claim follows immediately from the definition of parallel. Claim 2. Lines l and m do not have two distinct points in common. Let P be a point they have in common. [Choosing such a point P is justified by Claim 1.] By way of contradiction, suppose Q lies on both l and m and Q 6= P . [Note that this is really an existence statement: there exists a point Q lying on both l and m such that Q 6= P . Alternatively, we could have let Q be a point on both l and m (which we know exists -

c 2009 Charles Delman 6 the set of such points is not empty, by Claim 1) and supposed only that Q 6= P . Then we would be proving (still by contradiction) the equivalent universal statement that, for all points Q lying on both l and m, Q = P . In general, (6 ∃a ∈ A)p(a) is logically equivalent to (∀a ∈ A)¬p(a).] Since Q 6= P , it follows by the uniqueness conclusion of Axiom I-1 that l = m. This contradicts our initial assumption that l 6= m; therefore, we conclude there is no such point Q, so P is unique.

c 2009 Charles Delman 7 To summarize, without the commentary, we have: Proof. Let l be a line, and let m 6= l be a line that is not parallel to l. Claim 1. Lines l and m have a point in common. The claim follows immediately from the definition of parallel. Claim 2. Lines l and m do not have two distinct points in common. Let P be a point they have in common. By way of contradiction, suppose Q lies on both l and m and Q 6= P . Since Q 6= P , it follows by the uniqueness conclusion of Axiom I-1 that l = m. This contradicts our initial assumption that l 6= m; therefore, we conclude there is no such point Q, so P is unique.

c 2009 Charles Delman 8 Another proof of Proposition 2.1 Proof. [This proof is based on the second statement: If l and m do not have a unique point in common, then l and m are parallel or l = m.] Assume it is not the case that l and m have a unique point in common. Then either they have no point in common or there exist two distinct points that they have in common. Case 1: l and m have no point in common. Then, by definition, they are parallel. Case 2: There are two distinct points common to l and m. Then by the uniqueness conclusion of Axiom I-1, l = m.

c 2009 Charles Delman 9 O Proposition 2.2 There exist three distinct lines that are not concurrent.

O Proposition 2.3 For every line, there is at least one point not incident with it.

O Proposition 2.4 For every point, there is at least one line not incident with it.

O Proposition 2.5 For every point, there are at least two lines incident with it.

c 2009 Charles Delman 10 Parallelism Properties

O Now that we have seen some models of incidence geometry, we see that different properties with respect to parallel lines may apply. The same property need not even apply to all the lines and points of a given model. However, geometries with uniform properties are generally of greater interest, and the following three possibilities are most important. o If l is a line and P is a point not lying on l, then there is a unique line through P that is parallel to l. (Euclidean Parallel Property) o If l is a line and P is a point not lying on l, then there exist two distinct lines through P that are parallel to l. (Hyperbolic Parallel Property) o Any two distinct lines are incident with a common point. (That is, if l is a line and P is a point not lying on l, then there are no lines through P that are parallel to l. (Elliptic Parallel Property)

c 2009 Charles Delman 11 Affine and Projective Geometry

O Incidence geometry with the additional assumption of the Euclidean Parallel Property is called affine geometry.

O A model of affine geometry is called an affine plane.

O There is a very important construction, inspired by visual perspective, that adds points to an affine plane in order that all lines intersect. The resulting new model is called the projective completion of the affine plane. (The name “projective” comes from the fact that rays of light project from points in space to our eyes.)

c 2009 Charles Delman 12 The Plane in Perspective When a horizontal plane is viewed in perspective, parallel lines appear to meet at a point infinitely far away. All the points where parallel lines can meet appear to lie on a line, which we call the horizon.

c 2009 Charles Delman 13 This is because the rays of light from the two lines to a fixed point, representing an eye, form two intersecting planes. (The rays from behind your head would cause the lines to continue beyond the horizon if you could see them.)

c 2009 Charles Delman 14 If you imagine the rays of light projecting onto a sphere with your eye as a point at the center (the visual sphere), the two lines would project to great circles intersecting at a pair of antipodal points.

By analyzing the geometry of projected light, artists and mathematicians during the renaissance learned to produce realistic depictions of scenes on a planar surface, as if the surface of the painting or drawing were a window. Their analysis gave rise to projective geometry.

c 2009 Charles Delman 15 The Projective Completion of an Affine Plane O To create the projective completion of an affine plane, we define two lines of the affine plane to be equivalent if they are parallel or equal (the same). It is obvious that this relation is reflexive and symmetric. That it is transitive requires a bit of thought.

O We define a point of the projective completion to be either a point of the original affine plane or an equivalence class of lines. Each of the new points is called a point at infinity. We also introduce a new line, l∞, the line at infinity or horizon, which is incident with each of the points at infinity (but not any of the original points of the affine plane). A line of the original affine plane passes though a point on the horizon if it is in the equivalence class that defines that point.

c 2009 Charles Delman 16 In summary: O Let A denote the original affine plane. Let A∗ denote its projective completion. o If l is a line of A, then l is incident with [l].([l] denotes the equivalence class of l under the relation just defined.) No other points at infinity lie on l. o If l is a line of A, then the point [l] lies on l∞. No other points lie on l∞. o All the original incidence relations of A apply in A∗.

O It turns out that if we forget about how the line and points at infinity were created, they do not have any features that distinguish them from any other lines or points, respectively.

O Example: The projective completion of the four-point affine plane is the seven-point projective plane.

c 2009 Charles Delman 17 Projective Geometry O It is easy to check that A∗ satisfies the following four axioms. (Note that you must check several cases, depending on whether or not each point or line lies at infinity.) o (Axiom I-1) If P and Q are distinct points, there is a unique line incident with both of them. o (Axiom I-2+) Every line is incident with at least three distinct points. o (Axiom I-3) There exist three noncollinear points. o (Elliptic Parallel Property) Any two distinct lines intersect.

O These are the axioms of projective geometry. A model of projective geometry is called a projective plane.

O Projective geometry has many applications, from abstract algebra to computer graphics.

c 2009 Charles Delman 18