3.1. group of collineations of an affine plane 3.2. the

3.1. GROUP OF COLLINEATIONS OF AN AFFINE PLANE Definition

= 

{ψ : 

3.1.1.

Let

it

be

}

→  | where ψ - is bijection

=(,,)

an

affine

plane

and

set of bijections to set points  on yourself.

Collineation of affine plane  called a bijection ψ ∈ , such that

∀ ∈ , ψ (  ) ∈ 

(1)

Otherwise, a collineation of the affine plane  is a bijection of set  on yourself, that preserves lines. It is known that the set of bijections to a set over itself is a group on rassociated with the binary action  of composition in it, which is known as total group or symmetric groups [1], [31], [32], [38], [63], [65], [66],[67],[70], [71]. In a collineation ψ of an affine plans, image ψ (P) to a point P to plans often mark briefly P '. PROPOSITION 3.1.1. Every bijections of set to points  on yourself to affine plane =(, , ) is a his collineation. CONSEQUENCE. Collineation set Kol of the affine plane =(, , ) forms symmetrical groups about the composition  , namely (Kol,  ) is symmetric group. It is clear that identical bijections id :  →  is a collineation of the affine plane

=(, ,

), we call identical collineations of . In this collineation, every point of  passes itself, as well as every line  passes itself. Definition 3.1.2. An point P of the affine plan  called fix point his associated with a collineation δ , if coincides with the image itself δ (P), briefly when P= δ ( P ) . According to this definition, we have this PROPOSITION 3.1.1. Every point of the affine plane, is a fix point related to his identical collineation.

3.2. THE DILATATION GROUP OF THE AFFINE PLANE Definition 3.2.1. [15], [28] Dilatation of an affine plane =(, , ) called a its collineation δ such that

∀P ≠ Q ∈ , δ ( PQ ) || PQ

(2)

According to axiom A1 of affine plane definitions, the line that passes through two different points P, Q we have written PQ. From the fact that dilatations δ is bijection, worth implication

P ≠ Q ⇔ δ ( P ) ≠ δ (Q ) ,

(3)

therefore line δ ( PQ ) also written δ ( P )δ (Q ) and propositions (2), in these circumstances, takes the view

∀P ≠ Q ∈ , δ ( P )δ (Q ) || PQ

(2')

It is clear that identical collineations id  of an affine plan =(, , ) It is an dilatation of his, who called his identical dilation. However, by being bijection an dilatation δ of an affine plane, the inverse his δ −1 is also bijection. It's a dilatation his affine plan the bijections δ −1 ? By (3), easily shown that is true this PROPOSITION 3.2.1. Inverse bijections δ −1 of a dilatations δ of an affine plan is also an dilatation of that plan. Proff. coming soon PROPOSITION 3.2.2. Composition of two dilatations of a affine plan is agin an extension of his. Proff. coming soon Let it be Zgj = { δ ∈ Kol δ is dilatation of } the dilatation set of affine plane =(, , ). As such it is the subset of collineations Kol . Propositions 3.2.2 indicates that (Zgj ,  ) is a substructure of the symmetric group (Kol ,  ) of collineations of the affine plan . Propositions 3.2.1 indicates that this substructure is a subgroup of the group (Kol ,  ), [2], [33], [35], [36], [37], [38], [64], [68], [70], [71]. Is obtained in that way this THEOREM 3.2.1. Set Zgj of dilatations a affine plan  forms a group with respect to composition  . Proff. coming soon Definition 3.2.2. Let it be  an dilatation of affine plane  = (, , ), P his one point. Lines that passes by P and  P  called trace of points P regarding dilatations  . Proff. coming soon THEOREM 3.2.2. For an point P to an affine plane =(, , ), not fixed related to an his dilatation δ , is true propositions

∀Q ∈  - {P}, Q ∈ P δ ( P ) ⇒ δ (Q ) ∈ P δ ( P ) . Proff. coming soon

THEOREM 3.2.3.[28] If two different assigned points P, Q of an affine plane =(, , ), are defined their image P' = δ(P) and Q' = δ(Q) by an his dilatations δ ≠ id , then image

R' = δ(R) of an other points R ∈  − {P,Q} determined as follows: P′

R ∉ PQ ⇒ δ ( R ) =  RP ∩

QRQ′ , S′

P′ R ∈ PQ ⇒ ∃S ∈  , S ∉ PQ, δ ( R ) =  RP ∩  RS .

Proff. coming soon CONSEQUENCE 1. If the tracks P δ ( P ) , Q δ (Q ) the two points P, Q of an affine plane are expected, then their cutting points P δ ( P ) ∩ Q δ (Q ) is a fixed point related to his dilatations δ . Proff. coming soon CONSEQUENCE 2. If an point Q of affine plans is to trace P δ (P) to an his point P, then the its image δ (Q) locates at the a trace. Otherwise, in an affine plan, line, which is an a trace of his points by an dilatation, is a trace for every other points of it Proff. coming soon CONSEQUENCE 3. Two dilatations δ 1 ≠ id and δ 2 ≠ id of an affine plane =(,,), are equal if and only if, when two points P ≠ Q ∈  are simultaneously true equations

( )

( )

( )

( )

= δ1 P δ= P and δ 1 Q δ2 Q 2

(6)

Otherwise, an dilatation δ ≠ id of an affine plane is completely determined by giving his image according to two different points of the plans. Proff. coming soon THEOREM 3.2.4. For every dilatation δ ≠ id of an affine plane =(, , ) exists in the plane least two not fixed points about what dilatation. Proff. coming soon THEOREM 3.2.5. If an affine plane =(, , ) has two fixed points about an dilatationthen he dilatation is identical dilatation id  of his. Proff. coming soon CONSEQUENCE. For every dilatation δ ≠ id to an affine plan =(, , ) if in the plan has an fixed point with respect to that dilatation, then it is only. Proff. coming soon

THEOREM 3.2.6. For every dilatation δ ≠ id to an affine plan =(, , ), which has a fixed point V associated with it dilatation, is true propositions

( )

∀P ∈  ,V ∈ Pδ P ,

(8)

otherwise, all the tracks regarding with dilatation δ cross the point V. Proff. coming soon THEOREM 3.2.7. An affine plane =(,,) has not fix point related to an dilatation

δ ≠ id then and only then, when all the tracks Pδ ( P ) for all P ∈  are parallel between

themselves. Proff. coming soon THEOREMS we can give also this wording: For every dilatation δ ≠ id of an affine plane =(,,), worth propositions “has not fixed point by δ ”

⇔

∀ P, Q

∈

, P δ (P)||Q δ (Q).

(9)

Proff. coming soon The last two theorems summarized in this PROPOSITION 3.2.3. In an affine plane related to dilatation δ ≠ id all traces P δ (P) for all P

∈ , or cross the by a single point, or are parallel between themselves.

3. TRANSLATIONS GROUPS OF AN AFFINE PLANE By Propositions 3.2.3, in an affine plane all traces related an dilatation of his or cross the by a single point, or are parallel between themselves. This fact leads us to this Definition 3.3.1. [15], [14], [28], Translation of an affine plans  = (, , ), called identical dilatation id his and every other of its dilatation, about which heaffine plane has not fixed points. If  is an translation different from identical translation id , then, by Theorems 3.2.7, all traces related to  form the a set of parallel lines. According to Propositions 1.2.4, at every point P ∈  pass at least three lines out , among which only one is its a traceof translations  . Because || parallelism relation on  It is an equivalence relation, according Propositionit 1.2.1, then  =/|| is an a cleavage of  in the equivalence classes by parallelism [1], [33], [35], [37], [40]. Each class has representative an line that passes from of random point P.

Definition 3.3.2. For one translation σ ≠ id , equivalence classes of the cleavage π =  / ||, which contained tracks by σ of points of the plan  = (, , ) called the direction of his translations σ and marked π σ . So, for σ ≠ id , direction π σ represented by single the trace by σ every point P ∈  . for translation id , We say that there are undefined direction. Otherwise we say that has the same the direction with every other translations  of the plane  , namely true accept the propositions: For every translation  of the plane ,  id =   . 

(10)

Subject of review at this point would be the set of translations of the affine plane =(,,): Zhv = { σ ∈ Zgj σ is translation of } . Let it be  : Zhv  Zhv an scalene application of Zhv on yourself. For every translation id id  , its image α (σ ) is again an translation, that can be α (σ ) =  or α (σ )   , So there is a certain direction or indefinitely. The first equation, in the case where σ = id , takes the view α (

id )= id , and the second α ( id )  id , that it is not possible to  is application. To avoid this, yet accept that for every application  : Zhv  Zhv , is true equalization

α ( id )= id .

(10')

THEOREM 3.3.1: If a certain point P of an affine plane  = (, , ), Its image is

( ) − {P } determined as follows:

( )

determined P ' = σ P according to an his translations σ ≠ id , then image Q ' = σ Q a other point Q ∈ 

P' Q  PP '  (Q)   QPP '   PQ S P' S' Q  PP '  S  PP ', S'   PP   PS dhe (Q)  PP '   SQ '

(11) (11')

CONSEQUENCE. Two translations σ 1 ≠ id , σ 2 ≠ id of an affine plane  = (, , ), are equal only when for a point P ∈  is true equalization

σ 1 (P ) = σ 2 (P )

(12)

Otherwise, one translations σ ≠ id of an affine plane is completely determined by giving her the likeness to an point according to the plan. PROPOSITION 3.3.1. The inverse translation σ −1 of an translation σ to an affine plane is also an translation in the affine plane.

CONSEQUENCE. For every translation σ to an affine plane  = (, , ), σ and σ −1 have the same direction. PROPOSITION 3.3.2. The composition of the two translations in affine plane is agin a translation of his. PROPOSITION 3.3.3. If translations σ 1 and σ 2 have the same direction with translation σ to a affine plane  = (, , ), then and composition σ 2  σ 1 has the same the direction, otherwise ∀σ 1, σ 2 , σ ∈ Zhv , π σ = πσ = π σ ⇒ π σ σ = πσ . (13) 1

2

2

1

THEOREM 3.3.2. Set Zhv of translations to an affine plane  form a group about the

(

)

composition , which is a sub-group of the group Zgj  ,  to dilatations of affine plans .

(

) ,  ) of him plane.

THEOREM 3.3.3. Group Zhv ,  of translations to the affine plane  is normal sub- group

(

of the group of dilatations Zgj 

CONSEQUENCE 1. For every dilatations δ ∈ Zgj  and for every translations σ ∈ Zhv of affine plane  = (, , ), translations σ and δ −1  σ  δ of his have the same direction. According to the understanding of the normal sub-group [1], [31], [32], [33], [36], [37], [64], [67], from this Theorem also it shows that there is true the implication

(

)

∀ δ , σ ∈ Zgj  × Zhv , δ  σ = σ  δ. Because δ ∈ Zhv ⇒ δ ∈ Zgj  , from that comes true the implication

(

)

σ  δ. ∀ δ , σ ∈ Zhv × Zhv , δ  σ =

(14)

This indicates that is true this

(

)

CONSEQUENCE 2. The translations group Zhv ,  of an affine plane  is (abelian) commutative. By definition of an Abelian Groups [2], [31], [32], [33], [36], [37], [64], [66], [67], [68], this means that besides propositions (14), are true even these propositions: ∀σ 1, σ 2 , σ 3 ∈ ZhvA,

(σ

1

 σ2 )  σ3 = σ 1  (σ 2  σ 3 )

∀σ ∈ Zhv , σ = id  id = σ σ ∀σ ∈ Zhv , ∃σ −1 ∈ Zhv σ  σ −1 = id 

(14''') v

(14')

(14'')