Projective Geometry I - Lecture Notes Week 2

2.1 Completion of the Affine Plane

Definitions: Ideal points, line at infinity, completion of A.

Def: A projective plane P is a set, whose elements are called points, and a set of subsets, called lines, satisfying:

P3' : There exist four points, no three of which are collinear.

(Show that P3' is equivalent to P3 and P4, assuming P1 and P2)

Prop 2.2: The completion of an affine plane A is a projective plane.

Example: the completion of the real affine plane is the real projective plane.

Example: the completion of AG(2,2) is PG(2,2).

Example: In R3, points are lines through the origin and lines are planes through the origin.

Example: Let V be a 3-dimensional vector space over some scalar field. Points are 1- dimensional subspaces of V and lines are 2-dimensional subspaces of V. If the scalars are the reals, this is just the previous example.

The relationship between affine planes and projective planes.

Comment on the author's use of the word "reasonable".

2.2 Homogeneous Coordinates for the Real Projective Plane

Homogeneous coordinates.

Def: Two planes are isomorphic if there is a bijection taking collinear points to collinear points. Isomorphism, Automorphism.

Prop 2.9: The projective plane defined by homogeneous coordinates of real numbers is isomorphic to the completion of the Euclidean affine plane.

In the real projective plane, P2(R), removal of any line gives the Euclidean plane. Show how the various conics are unified by this viewpoint.

Sphere Model of the real projective plane.

Topological representaton as a Mobius strip with a disk attached to its boundary.