## Projective Geometry I - Lecture Notes Week 1

1.1 Affine Planes
Definitions: Affine plane, collinear, parallel

Example: the real affine plane (Euclidean Plane, R2).

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 R2 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 A2(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.