## 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:

- P1: Two distinct points of P lie on one and only one line.
- P2: Two distinct lines meet in precisely one point.
- P3: There exist three noncollinear points.
- P4: Every line contains at least three points.

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 **R**^{3}, 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, **P**^{2}(**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.