Projective Geometry - Lecture Notes Chapter 1

1.1 Foundations

 Def: A geometry is a pair G = (, I), where is a set and I a relation on that is symmetric and reflexive, i.e.
  1. If (x,y) I then (y,x) I.
  2. (x,x) I for all x .
Examples:
  1. 3-dimensional Euclidean geometry
  2. Cube graph
  3. Rooted tree
Def: A subgeometry of G = (, I) is G' = (', I') where ' is a subset of and I' is the restriction of I to '.

Example:
In Euclidean plane, the set of points interior to the unit circle.

Def: Let G = (, I) be a geometry. A flag of G is a set of elements of that are mutually incident. A flag is called maximal if there is no element x \ such that {x} is also a flag.

Examples:

  1. Tetrahedral graph
  2. Flag
Def: A geometry G = (, I) has rank r if one can partition into r sets such that each maximal flag of G intersects each of these sets in exactly one element. In particular, each maximal flag has exactly r elements.

Examples:

  1. R3 has rank 3.
  2. Tetrahedral graph has rank 3.
  3. A rooted tree is not a rank geometry.
  4. A vector space of "dimension" r is a geometry of rank r with subspaces of dimension i as the partition.
Any geometry of rank r gives a subgeometry of rank r-1 by removing a type. Thus, any rank r geometry can be considered as a rank 2 geometry.

Lemma 1.1.1: Let G be a geometry of rank r. Then no two distinct elements of the same type are incident.

A rank 2 geometry is often called an incidence structure. The types are called points and blocks. One writes G = (P,B,I).

1.2 The axioms of projective geometry

Let G be a rank 2 geometry. We will call the blocks of G lines.

Axiom 1: For any two distinct points P and Q, ! line incident with both, denoted by PQ.
Lemma 1.2.1: Two distinct lines are incident with at most one point.

Axiom 2: (Veblen-Young) If A, B, C and D are 4 points such that AB intersects CD, then AC intersects BD.
This axiom was formerly known as Pasch's Axiom, but that is really a misnomer.

Axiom 3: Any line is incident with at least 3 points.
So much for graphs!

Def: A projective space is a rank 2 geometry that satisfies axioms 1, 2 and 3.

A projective space is nondegenerate if it also satisfies:
Axiom 4: There exist at least 2 lines.
Def: A projective plane is a projective space in which each two lines have at least one common point. This condition is a stronger version of Axiom 2 and will be referred to as Axiom 2'.

Example: The Fano Plane.

The Principle of Duality

Def: Let A be a proposition concerning geometries of rank 2, whose elements we shall call points and blocks. We obtain the proposition A dual to A by interchanging the words "point" and "block".

Def: Let G be a rank 2 geometry with point set 1, block set 2, and incidence set I. The geometry G dual to G has point set 2, block set 1 and two elements of G are incident if and only if they are incident in G.

Note that the definition implies that (G) = G.

Theorem 1.2.2: (Principle of Duality). Let be a class of rank 2 geometries. Suppose that has the property: if contains the geometry G, then it also contains the dual geometry G. Then the following assertion is true:

if A is a proposition that is true for all G in , then A is also true for all G in .

Lemma 1.2.3: Any projective plane P also satisfies the propositions that are dual to axioms 1,2',3 and 4.

Theorem 1.2.4: (Principle of Duality for Projective Planes). If a proposition A is true for all projective planes then the dual proposition A also holds for all projective planes.

Warning: P and P need not be isomorphic.