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.
Def: pencil of lines (pencil of parallel lines)
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":
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.