Fundamental theorem of the projective geometry

The fundamental theorem of the projective geometry is stated as follows:

In projective Geometry of the plan, the fundamental theorem is a base of powerful demonstration, which practically allows all to show. But it is a little frustrating insofar as it is too powerful for certain small properties, it does not make it possible to really include/understand why such small property is true. It is as if one took a power hammer to crush a fly.

All depends on the Axiome S which one decided to use. In this precise case, the axioms are:

axioms of the plan projective-all-short:

a projective plan is a unit
It is a whole of points
Certains subsets are called lines.

axioms of the projective plan of incidence:

There exist at least 2 points in the plan.
Each line has at least 3 points.
For two distinct points there exists one and only one line which is incidental for them.
Two distinct lines have one and only one common point.
For any line there exists at least a nonincidental point on this line.

More the axiom of the fundamental projective plan:

If a unidimensional projective transformation has 3 distinct fixed points, then it is the transformation identity.

The fundamental theorem of the plane projective geometry is stated as follows:

There exists one and only one projective transformation of a rectilinear division into itself or another rectilinear division, transforming 3 points distinct from the first into three points distinct from the second.

The Vocabulaire is strange and is explained by the gropings in the discovery of the increasingly moderate concepts.

unidimensional Configuration (in: one-dimensional-form): (2 cases) together of incidental points on the same line, also called rectilinear division, or together incidental lines at the same point, also called beam of right-hand sides (in: pencil off lines).
projective Transformation, projectivity (in: projectivity): composition of prospects.
Perspective: (4 cases) that is to say a Bijection between a rectilinear division and a beam such as a point and its right-hand side-image are incidental (in: elementary correspondence); that is to say the opposite bijection; that is to say combination of both in an order or the other (in: perspectivity).
unidimensional projective Transformation: it is a pleonastic expression to recall that this projective transformation works only on unidimensional beings of the plan or space of higher size.
ref.: Atlas of mathematics, Fritz Reinhardt and Heinrich Soeder, 1974, Pochothèque in French 1997.
ref. for English: Coxeter, 1987. English adopted the vocabulary of Poncelet, the French that of Chasles.

One should in any rigor call it the fundamental theorem of the projective plane geometry but the use decided some differently.

It is necessary to show the existence and unicity.

Unicity . Let us admit that there exists possible a first projective transformation T1 which transforms HAS B C in A3, B3 C3. In does there exist the different one? That is to say a second projective transformation T2 which with the same property. As the TP are bijective, one can consider the reverse of T2 (=T2⁻¹). Let us consider now the transformer made up P=T2⁻¹ (T1 ()) It transforms has in has, B out of B and C out of C. It has 3 distinct fixed points, then according to the axiom above it is the transformation identity; P=I.

However, by definition of T2⁻¹, I=T2⁻¹ (T2 ()). We can transform this equation:

T2⁻¹ (T1 ()) = T2⁻¹ (T2 ()). Let us combine on the left with T2

T2 = T2. Let us apply the associativeness of this law of composition interns

T2 (T2⁻¹) = T2 (T2⁻¹). Remplaçons T2 (T2⁻¹ by I

I = I and, I being the neutral element,

T1 () = T2 (), which shows the unicity of a possible projective transformation T1 which transforms HAS B C in A3, B3 C3.

Foot-note. For the notation of the internal law of composition of the transformers, one has the choice between T2 O T1 which decides " Round T2 T1" and " means; one applies T1 then T2" and the other notation by parenthesizing-multi-stage T2 (T1 (… etc = " T2 of T1 of… etc". This last notation is adopted here.

Existence.

does there Exist at least a transformer projective unidimensional which transforms ABC into A3B3C3?

Here the existence is clarified only for the case where 4 couples are made distinct points, namely B3, B; C3, C; A3, has and C3, A. Are Delta the line B-B3, Z=Delta inter C-C3, W=Delta inter A-A3, b=Delta inter A-C3. Initially let us specify that these lines and these points always exist because of the axioms of the projective plan of incidence pointed out supra. The auxiliary points has and C in fact are confused with has and C3. One considers 2 transformer projective unidimensional:

Tz of center Z which transforms has, B, C in has, B, C.

Tw of center W which transforms has, B, C in A3, B3, C3.

let us form the made up transformation U=Tw (Tz ()).

U transforms HAS B C in A3 B3 C3. The existence is thus shown.

Comment on the axiomatic aspect: this demonstration of unicity can appear disappointing insofar as it rises mechanically from the axiom of the 3 fixed points. It is to only move back for better jumping, but one can nothing there, it is the axiomatic step. If one wants to go up further in the axioms, should for example be shown the following theorem: " If a unidimensional projective transformation has 3 distinct fixed points, then it is the transformation identité." With this intention, one must start from a system of axioms plus upstream.

A reference: (in:) Projective Geometry, H.S.M. Coxeter, Springer, 1987,1998. Chap 2.1.Axioms

Projective

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