Projective Geometry I - Lecture Notes

3.1 The Axiom P5 of Desargues

P5: Two triangles that are centrally perspective are axially perspective.

Theorem 3.1: P5 holds in the real projective plane.

Def: A configuration is a set of points and lines such that two distinct points lie on at most one line.

Example: Desargues configuration.

3.2 Moulton's Example

The Moulton plane.

3.3 Axioms for Projective Space

Def: A projective 3-space is a set whose elements are called points, together with certain subsets called lines and certain other subsets called planes such that:

S1: Two distinct points determine a unique line.
S2: Three noncollinear points lie on a unique plane.
S3: A line meets a plane in at least one point.
S4: Two planes have at least a line in common.
S5: There exist 4 noncoplanar points, no three of which are collinear.
S6: Every line has at least three points.

Example: Real Projective 3-space.

Theorem 3.6: P5 holds in any projective 3-space, where we do not assume that all points necessarily lie in a plane. In particular, P5 holds for any plane that lies in a projective 3- space.

Rmk: There are competing definitions for projective 3-space. In some of them, Thm 3.6 is false (Hartvigson).

3.4 Principle of Duality

Proposition 3.7: Let \pi be a projective plane. Let \pi* be the set of lines of \pi, and define a line of \pi* to be a pencil of lines in \pi. Then \pi* is a projective plane. Furthermore, if \pi satisfies P5 then so does \pi*.

\pi* is called the dual projective plane of \pi.

Converse of Desargues' axiom.

Principle of Duality

Proposition 3.11: If a projective plane contains a line with n+1 points, then all lines have n+1 points and there are n² + n + 1 points in the plane.

Proposition 3.12: If a projective plane contains a point with a pencil of n+1 lines, then all points have pencils of n+1 lines and there are n² + n + 1 lines in the plane.

Remark: The dual of the dual of a plane is the original plane. The dual of a plane need not be isomorphic to the original plane, but this is true for the real projective plane.