Projective Geometry - Lecture Notes Chapter 1

1.6 Affine Geometries

Def: Let P be a projective space of dimension d > 1 and a hyperplane. Define A = P\ as follows:
  1. The points of A are the points of P\.
  2. The lines of A are the lines of P not in .
  3. The t-dimensional subspaces of A are the t-dimensional subspaces of P which are not contained in .
  4. Incidence is inherited.
The rank 2 geometry of points and lines of A is called an affine space of dimension d. An affine plane is an affine space of dimension 2. The set of all subspaces of A is called an affine geometry.

For fixed t (1td-1), the rank 2 geometry consisting of the points and t-dimensional subspaces of A is denoted by At (thus affine space is the geometry A1).

Subspaces of an affine space are called flats.

is called the hyperplane at infinity and its points are called points at infinity (sometimes improper points).

P is called the projective closure of A.

Examples:

1. affine plane of order 2

2. affine plane of order 3

3. Consider the example of section 1.4. Let be the plane 0001T. The points of this plane (points at infinity) are those with last coordinate 0. The points of the affine space P\ are thus: 0001, 0011, 0101, 0111, 1001, 1011, 1101, and 1111. The projective plane 1000T becomes the affine plane whose points are : 0001, 0011, 0101 and 0111. The line labeled a of the projective space becomes the affine line whose points are 0001 and 1001.

Def: Let G = (,,I) be a rank 2 geometry. A parallelism of G is an equivalence relation || on the block set satisfying Playfair's axiom (i.e., given a point P and a block B not containing P, there is a unique block B' containing P with B'||B.)

Theorem 1.6.1: For t {1,...,d-1}, At has a parallelism.

The parallelism of the above theorem is called the natural parallelism.

For the natural parallelism, any two distinct parallel t-flats span a (t+1)-flat. Two subspaces of arbitrary dimension are parallel if one is parallel to a subspace of the other. Thus, if is the hyperplane at infinity of P, then subspaces U and W are parallel if U W, or vice-versa. In particular, a line g is parallel to a hyperplane H if g is parallel to some line h of H, meaning that gh.

Lemma 1.6.2: Let A = P\ where P is a d-dimensional projective space.

  1. Each line not parallel to a hyperplane H meets H in precisely one point of A.
  2. If d = 2, any two non-parallel lines intersect in a point of A.
Corollary 1.6.3: Any affine plane A satisfies:
  1. Through any two distinct points there passes exactly one line.
  2. Playfair's axiom.
  3. There exist 3 points not on a common line.
Theorem 1.6.4: Let S = (,,I) be a rank 2 geometry satisfying 1), 2) and 3) above. Then S is an affine plane.

Theorem 1.6.5: Let A be a finite d-dimensional affine space of order q. Then

  1. There exists an integer q2 such that any line of A is incident with exactly q points.
  2. If U is a t-flat, then |U| = qt.
Corollary 1.6.6: The total number of points in an affine plane of order q is q2.

Theorem 1.6.7: If S is a rank 2 geometry satisfying:

  1. 2 points determine a unique line,
  2. there are q2 points in total, and
  3. each line has exactly q points.
Then S is an affine plane.