PROJECTIVE GEOMETRY COURSE

§ 6: *PM as a base for geometries with a finer structure*

We start from the real projective plane of the straight lines and planes through *O*. A *projective transformation* is a bijection from this projective plane onto itself that preserves the
incidence of points and lines and the cross ratio.

Let *GP* be the group of projective transformations, and *LR ^{3}* the group of regular linear transformations of ℜ

Let

Now take an arbitrary line out of *P ^{2}* and call it the line at infinity. We ourselves take the line

The projective transformations that map

In this § we further identify each affine transformation with a regular linear transformation of ℜ

; hence, such an affine transformation is the product of: first an orthogonal projection onto {x_{3}=0}, then a linear transformation in {x_{3}=0}, and finally a translation with
translation vector (a,b,1).

A *similarity transformation* is an affine transformation that works in {x_{3}=0} as a product of a rotation or reflection and a positive scalar (the similarity factor).
So the matrix of a similarity transformation has the form

(with similarity factor ρ).

The similarity transformations form the similarity group *GS*.

Finally we define a *congruence transformation* as a similarity transformation with similarity factor 1. The congruence transformations form the congruence group *GC*.

Thus we find *GC*⊂*GS*⊂*GA*⊂*GP*.

For *X*∈{*P*, *A*, *S*, *C*} we define a *X*-concept as a concept that is invariant under the
transformations of *GX*.

So {*P*-concepts}⊂{*A*-concepts}⊂{*S*-concepts}⊂{*C*-concepts}.

For example, points, lines and incidence are *P*-concepts, parallel is an *A*-concept, angle a *S*-concept, and distance a *C*-concept.

We define the *X*-geometry as the geometry of the *X*-concepts. The *C*-geometry is the good old euclidian geometry.

*O22* Define the concept "parallel" in *A*-geometry, and show this is not a *P*-concept.

*O23* Prove that "angle" is a *S*-concept, and "distance" a *C*-concept.

*O24* Give for each of the following concepts the letters *X* so that that concept is a *X*-concept: quadrilateral, parallellogram, rhomb, square.

*O25* We call two figures *X*-equal if there is an element of *GX* that maps one of them into the other. Are any two conics *A*-equal?

*O26* Suppose we have three proper points *A*, *B* and *C* on a line *l*, and the point at infinity *L*_{∞} on *l*.

Define the following *A*-concepts using cross ratio (We count ∞/∞=1; a/∞=0, ∞/a=∞ for a≠∞,0 (resp.); *XY*=-*YX*; etc.)

i) "*X* lies between *A* and *B* on *l* ";

ii) "*AB* is three times as great as *BC* ";

iii) "*B* is the centre of *AC* ".