## Projective Geometry I - Lecture Notes Week 1

Some comments on the text.
** 1.1 Affine Planes**
**Definitions**: *Affine plane, collinear, parallel*
**Example**: the real affine plane (Euclidean Plane,** R**^{2}).

**Prop 1.5:** Parallelism is an equivalence relation.

*Note how axiom A2 is used here.*

**Prop 1.6**: Two distinct lines have at most one point in common.

**Prop 1.7**: An affine plane has at least 4 points.

**Example**: AG(2,2)

**Def**: *pencil of lines* (pencil of parallel lines)

**1.2 Transformations of the Affine Plane**
Relabelling of AG(2,2) or a change of coordinates in **R**^{2} are examples of relabelling.
**Def**: *automorphism* of an affine plane.(Transformation, symmetry)

**Convention**: primes used to denote images under an automorphism.

different symbols indicate distinct elements.

**Observation**: Automorphisms satisfy the two "algebraic conditions":

- (P {\cup} Q)' = P' {\cup} Q'
- (l {\cdot} m)' = l' {\cdot} m'

**Prop 1.14**: An automorphism transforms parallel lines to parallel lines.
**Prop 1.15**: The set of automorphisms, Aut **A**, is closed under composition and inverses.

**Def**: A *dilatation* of an affine plane is an automorphism for which the image of a line is
parallel to the original line.

**Examples**: Stretching and translations in A^{2}(**R**) are dilatations.

**Prop 1.18**: The set of dilatations, *Dil ***A**, is closed under composition and inverses.

**Prop 1.19**: A dilatation which leaves two distinct points fixed is the identity.

**Cor 1.20**: A dilatation is determined by the images of two points.

**Def**: A dilatation with no fixed points, or the identity is called a *translation*.

A dilatation with one fixed point is called a *central dilatation*.

**Prop 1.22**: For any non-identity translation, for any two points P, Q we have PP' || QQ'.

**Prop 1.23**: The translations form a subset *Tran ***A** of Dil **A** which is closed under composition
and inverses. Furthermore, for any t in Tran **A** and s in Dil **A**, sts^{-1} is in Tran **A**.

Comment on algebraic notation.