Affine maps and some theorems in plane geometry

DEPARTMENT OF MATHEMATICS NGUYEN THI THU AFFINE MAPS AND THREE THEOREMS IN PLANE GEOMETRY GRADUATION THESIS Major: Geometry Hanoi - 2019... DEPARTMENT OF MATHEMATICS NGUYEN THI THU

DEPARTMENT OF MATHEMATICS

NGUYEN THI THU

AFFINE MAPS AND THREE THEOREMS

IN PLANE GEOMETRY

GRADUATION THESIS

Major: Geometry

Hanoi - 2019

DEPARTMENT OF MATHEMATICS

NGUYEN THI THU

AFFINE MAPS AND THREE THEOREMS

I would like to express my gratitudes to the teachers of the ment of Mathematics, Hanoi Pedagogical University 2, the teachers

Depart-in the geometry group as well as the teachers Depart-involved The lecturershave imparted valuable knowledge and facilitated for me to completethe course and the thesis

In particular, I would like to express my deep respect and gratitude

to Assoc.Prof.Nguyen Nang Tam, who has direct guidance, help mecomplete this thesis

Due to time, capacity and conditions are limited, so the thesis cannot avoid errors Then, I look forward to receiving valuable commentsfrom teachers and friends

Hanoi, May - 2019Student

Nguyen Thi Thu

I assure that the data and the results of this thesis are true and notidentical to other topic I also assure that all the help for this thesishas been acknowledged and that the results presented in the thesishas been identified clearly.

Hanoi, May - 2019Student

Nguyen Thi Thu

Acknowledgements 1

1.1 Affine space 3

1.1.1 Definition of affine space 3

1.1.2 Examples 4

1.1.3 Some properties 5

1.2 Affine maps 7

1.2.1 Definition 7

1.2.2 Examples 8

1.2.3 The properties of affine maps 8

1.2.4 Images and pre-image of flats by affine mapping 12 1.2.5 The single ratio and affine map 12

1.2.6 The parallel projections 14

1.3 Affine transformations 15

1.3.1 Affine Isomorphism 15

1.3.2 Affine transformation 16

1.3.3 Affine mapping in affine coodinates 19

1.3.4 Homological affine mapping 20

1.3.5 Homologicalgline affine mapping 22

1.3.6 Involutory affine 24

1.4 Exercises 26

2 THREE THEOREMS IN PLANE GEOMETRY 32 2.1 The single ratio in plane 32

2.2 Thales’ theorem 33

2.3 Desargues’ theorem 35

2.3.1 Ceva’s theorem 35

2.3.2 Menelaus’ theorem 36

2.3.3 Desargues’ theorem 36

2.4 Pappus’ theorem 40

2.5 Exercises 42

1 Rationale

Pure affine geometry, in the sense that there are no distance, angles,perpendicular Instead of, it is the notions of subtraction of points toproduce a vector During the process, I learned about affine geometryand found that it has many applications and played an important role

in mathematics With the desire to study more deeply in geometryand to be guided by the supervisor, I chose the topic Affine maps andsome theorems in plane geometry for the graduation thesis

2 Aims of the study

The purpose of this thesis reseaches into affine mappings and threetheorems of affine geometry in the plane

3 The object and scope of the study

The affine mappings and three theorems of affine geometry in theplane

The references relate to affine geometry

4 Research methods

To research the textbooks, references,document relate to affine etry

geom-5 Main contents

The thesis consists of 2 chapters

Chapter 1: Affine maps

1.1 Affine spaces

1.1.1 Definition of affine space

Definition 1.1.1 (see,[1- pp.1,2]) Let E be a K -vector space (where

K is a field) An affine space over E is a set A together with a map

A× E → A(M, v) 7→ M + v = Nsuch that:

(1) M +−→

0 = M for all M ∈ A, where −→

0 is the identity element ofE;

(2) M + (−→v + −→w ) = (M + −→v ) + w for all M ∈

A and −→v , −→w ∈ E;and

(3) given M, N ∈ A, there exists a unique v ∈ E such that M + −→v =

N

Remark 1.1.2 (See,[1- p.2]) Note that if M ∈ A and −→v ∈ E , thenotation M +−→v means only the image of the pair (M, −→v ) via the abovemap A× E → A Hence, the four signs + appearing in condition 2have different meanings: three of them represent the above map, andthe other, ordinary vector addition in the vector space E

Remark 1.1.3 (see,[1- page 2]) The unique vector determined by thepoints M and N is denoted by −−→

M N Hence, we have the fundamentalrelation M +−−→

M N = N Remark 1.1.4 The dimension of an affine space A is defined to bethe dimension of its associated vector space E We shall write

dimA = dimE

1.1.2 Examples

Example 1.1.5 (see [1-page 3]) The standard example of an affinespace is given by A = E, that is, the points of this affine space arethe elements of the vector space The action is

A× E → A(M, −→v ) 7→ M + −→vwhere the sum is ordinary vector addition

Example 1.1.6 In Euclid space 2-dimension E2 and 3-dimension E3are affine space that associated with vector space (free) 2-dimension,3-dimesion.

M N = −−→

N M(5) Given M ∈ A, −→v ∈ E, there is a unique N ∈

M N = −−→

M P implies N = P (8) −−→

(2) Let us assume M + −→u = N + −→u , and subtract −→u We obtain(M + −→u ) − −→u = (N + −→u ) − −→u Hence, M + (−→u − −→u ) =

Remark 1.2.2 Since all of the linear maps −→

f M are equal for anaffinity f , i.e they do not depend on the point M , we shall denotethis map simply by −→

M N is an arbitrary vector, we have, for every point M ∈ A1and for every vector −→v ∈ E

1,

f (M + −→v ) = f (M ) +−→f (−→v ).

Theorem 1.2.3 A map between two affine spaces f : A1 → A2 is

an affine mapping if and only if there exists a linear map −→

f : E1 →

E2 between the corresponding K-vector spaces such that f (M + v) =

f (M ) +−→

f (v) for all M ∈ A1 and v ∈ E1

Proof If f is an afine mapping, we take −→

f = −→

f M, for some M , and

we are done

Conversely, if there exists a linear map −→

f with this property, since

N = M +−−→

M N , for all M, N ∈ A1, we have f (N ) = f (M ) +−→

f (−−→

M N ),and hence, −→

1.2.2 Examples

Example 1.2.4 The ”consatnts mapping” that maps f to apoint isaffine; the associated linear mapping is the zero mapping

Example 1.2.5 The mapping f = id : A → A, f (M ) = M , for all

M ∈ A is the affine mapping with −→

f = id, that is −→

f (v) = v, for all

v ∈ E

1.2.3 The properties of affine maps

In this section A1 and A2 are affine spaces over the K-vector spaces

E1 and E2, respectively

(a) Let φ : E1 → E2 be a linear map and suppose given two points

M ∈ A1 and N ∈ A2 Then there exists a unique affinity

f :A1 →A2 such that f (M ) = N and −→

f = φ

Proof We will prove the uniqueness.Assume that there exists

an affine mapping g : A1 → A2 which has the associated linearmapping −→g

(b) The composition g ◦ f of two affine mapping f : A1 → A2 ,

g :A2 → A3 is an afine mapping The associated linear mapping

is the composition of the associated linear mapping, i.e,

−−→

g ◦ f = −→g ◦−→f

Indeed, for any M, N ∈A we have:

(c) Let M0, M1, , Mn be affinely indepenent points in an afine space

A1 Let M00, M10, , Mn0 be points in an affine space A2 Then,there exists an affine mapping f : A1 → A2 such that f (Mi) =

Mi0, i = 0, 1, , n.If n = dimA1 + 1, then this affine is unique.Proof Since the points M0, M1, , Mn are affinely indepen-dent, the vectors −−−−→

M0M1, ,−−−−→

M0Mn are linearly independent Weknow that there exists a linear map φ : E1 → E2 such thatφ(−−−→

(d) Let f : A1 → A2 be an affine mapping with associated linearmap −→

f Then f is injective if and only if −→

f is injective and f issurjective if and only if −→

f is surjective

Proof Let us assume that f is injective, and suppose−→

f (−→v ) = −→0 Let −→v = −−→M N We have 0 = −→f (−−→M N ) = −−−−−−−→f (M )f (N ), and hence

f (M ) = f (N ) This implies M = N and −→v = −−→M N = −→0 , that

is, −→

f is injective

Assume now that −→

f is injective and suppose f (M ) = f (N ).Then −→

is, M = N , and f is injective

Assume now that f is surjective and let −→v ∈ E

2 Let −→v = −−→M Nand choose M, N such that f (M ) = M0 and f (N ) = N0 Then

M0N0 = −→v , and hence −→f is surjective.

Assume now that −→

f is surjective and let N ∈ A2 Take anypoint M ∈ A1 and a vector v ∈ E1 such that −→

f (−→v ) = −−→M N Then f (M + −→v ) = f (M ) +−→f (−→v ) = f (M ) + −−−−−→f (M )N = N , andhence f is surjective.2 In particular, f is bijective if and only if

−

→

f is bijective

1.2.4 Images and pre-image of flats by affine mapping

Definition 1.2.6 Let f :A1 → A2 be an affine map with associatedlinear map −→

1.2.5 The single ratio and affine map

Definition 1.2.7 Let P, Q, R ∈ Abe distinct colinear points and thenumber λ ∈ K such that −→

Inversely, let f : A → A be a mapping that turns three collinearpoints into three collinear points and does not change the single ratio

of them Let I be fixed point such that I ∈ A and I0 = f (I)

For any M ∈A , M0 = f (M ), we construct:

We will prove that −→

f is associated linear mapping of f Now, we will show that −→

f (λ−→x ) = λ−→f (−→x ).

If λ = 0, λ = 1, −→x = 0, obviously We will consider λ 6= 0, λ 6=

1, −→x 6= 0 Let −IM = −→ →x ,−→IN = λ−→x then M, I, N are collinear and[N M I] = λ Therefore, N0, M0, I0 are collinear and [N0M0I0] = λ or

IP = 12 (−→x + −→y ), then M, N, P are collinear and [M N P ] = −1.Therefore, M0, N0, P0 are collinear and [M0N0P0] = −1 or −−→

1.2.6 The parallel projections

Definition 1.2.10 Let A be an affine space over K-vector space E,

α be an m − f lat in A directed by −→α , −→β be a vector subspace of Esuch that −→α ⊕−→β = E.

We define: f : A → α is follows: for M ∈ A: βM-flat passing through

M directed by −→

β Then α ∩ βM = M0 Let f (M ) = M0 Then themap f is called the parallel projection on m-flat α among −→

β Remark 1.2.11 i) For any M ∈ α : f (M ) = M

ii) The parallel projection is an affine map

Indeed, consider the mapping

Hence, −→

f is linear affine map

Take M, N ∈ A and f (M ) = M0, f (N ) = N0, we have:

f is linear map associated with f Therefore f is affine map.2

f : E1 → E2 is also linear isomorphism

Indeed, take I ∈ A and put f (I) = I0 then for any M of A1 wehave −−→

b) A1 is an afine isomorphism with A2 if and only if E1 is alsoisomorphism E2.

c) A1 is an affine isomorphim A2 if and only if they have the samedimension

d) Let f : A →A0 be an afine isomorphim Then its inverse

f−1 : A2 →A1 is also an afine isomorphim with associated linearmapping −→

f −1

1.3.2 Affine transformation

Definition 1.3.2 An affine isomorphism f : A → A is called anaffinity from an affine space A into itself

Definition 1.3.3 (Fixed point): A point M ∈ A is a fixed point of

an affinity f : A →A if and only if f (M ) = M

Theorem 1.3.4 Let A0, A1, , An ∈ A and A01, A02, , A0n ∈ A be,respectively, affinely independent points Then, there exists a uniqueaffinity f : A→ A such that f (Ai) = A0i, i = 0, 1, , n

Theorem 1.3.5 ( Affine Group)(see, [1-page 54]) The set of all tive affinities from an affine space A into itself is a group with respect

bijec-to composition of maps, called the affine group or group of affinities,and is denoted GA

Theorem 1.3.7 An affinity f is a translation if and only if there is

f = id.For this reason we shall denote translations by T− →v and we will say that

T− →v is the translation by vector −→v , where −→v is called the translationvector

Remark 1.3.8 (see,[1-page 67]) The set of all translations of an affinespace A is a group with respect to composition of maps The identityelement is translation by the vector −→

0 The group properties followfrom the equalities T− →u ◦T−→

v =T− →v −→u +−→v (the composition of translations is atranslation) and T− →v1=T

−−→v (the inverse of a translation is a translation).This group, denoted by T, is a subgroup of the group of affinities GA

N = M + 1−λ1 −−−−−→

M f (M )Moreover, it is clear that N is the unique fixed point To see this let

us assume that there are two different fixed points N, N0 Then

Remark 1.3.11 (see, [1-page 69]) The set of all homotheties of anaffine spaceAis not a group with respect to composition of maps Theidentity translation is not a homothety Even if we add the identity tothe set of all homotheties, we still dont have a group, since the com-position of homotheties with different centers and inverse similituderatios is a translation.

1.3.3 Affine mapping in affine coodinates

Let f : An →An be affine map and ,

(O0, ε0) be affine frames, where (O, ε), ε = {−→e

1, , −→e

n}, ε0 = {−→

e01, ,−→

e0n}For any point X ∈ An and the coordinate of point X is (x1, , xn)

X0 = f (x) and the coordinate of point X0 is (x01, , x0n)

O0 = f (O) and the coordinate of point O’ is (b1, , bn)

1.3.4 Homological affine mapping

Definition 1.3.12 Let A be an affine space over the K-vector space

E, and m-fat α directed by −→α , −→β is subspace of E such that −→α ⊕−→β =

E Consider the mapping

f :A → A

M 7→ M0

For any M ∈ A, and M1 is intersection point of α with the flat

passing through M and directed by −→

β such that −−−→

M1M0 = λ−−−→

M1M Then, we said to be a homological affine mapping which has basis α,directed by −→

1 : −→

An = −→α ⊕−→β → −→α , −→p

2 : −→

An = −→α ⊕−→β → −→β

Proof Indeed, For any M, N ∈ A : f (M ) = M0, f (N ) = N0

Since f is a homological affine map, so −−−→

i i = m + 1, m + 2, , n

Therefore, the coordinate expression of homological affine map is:

0 0 0 · · · 1 0 · · · 0

0 0 0 · · · 0 λ · · · 0 · · · · · ·

1.3.5 Homologicalgline affine mapping

Definition 1.3.14 In affine space An, let α be hypersurfaces, and −→

β

be vector space 1-dimension such that −→

β ⊂ −→α The affine transformation f : An → An fix every points of α and forany M ∈ An such that f (M ) = M0 then −−−→

M M0 ∈ −→β The affinetransformation f is said to be homologicalgline affine mapping withbasis α, directed by −→

β Theorem 1.3.15 Let α be hypersurface and two points N, N0 ∈ α/

2, , −−→en−1−→

e0n} is basis of −→A where −→

e0n = −−→

ON0.Therefore, there is one and only one affine transformation f :A → Asuch that f (O) = O, ϕ(−→e

Assume that M ∈An has coordinate (xi), and the coordinate of f (M )

is (x0i) By the form coordinate of affine map, we have: x0 = Ax,or

Proof Assume that f : A →A is an involution, then f2 = IdALet −→

f : E1 → E1 is linear involution of f , then −→

f2 = IdE

1.4 Exercises

Exercise 1.4.1 Prove that Affine mapping f is the parallel projection

if and only if f2 = f

Solution

Let f is the parallel projection, we wil show that f2 = f

Consider the mapping f : A →A, for any M ∈ A, f (M ) = M0

A = −→α +−→β Thus,E = −→α ⊕−→βLet f be an affine mapping, I ∈ A and f (I) = I0 Put α = (I0, −→α )Consider the mapping f : A →A

M 7→ f (M ) = M0

We will show that f is the parallel projection which have basis α, anddirected by −→

M M0 ∈ kerϕ = −→β , that mean M0 ∈ (M,−→β ) (*)

B0 = (0, 0, 0), B1 = (0, 1, 0), B2 = (2, 0, 1), B3 = (1, 0, 1)are affinelyindependents

a) Prove that systems A0, A1, A2, A3 and B0, B1, B2, B3 are affinelyindependent

b) Find coordinate expression of the affine transformation f : A3 →

A3 such that f (Ai) = Bi, i = 0, 1, 2, 3

c) Finf fixed points of f

d) Find coordinate expression of f in the affine frame {A0, A1, A2, A3}

Solution

1 −1 1

0 −1 −1

Thus, A0, A1, A2, A3 are affinely independent

Similarly, B0, B1, B2,3 are affinely independent

So, f has equation

x2 = 1

7, x3 =

37

Thus, The fixed point M (87,17,37)

THREE THEOREMS IN PLANE GEOMETRY

In Chapter, we will present the single ratio in plane, three theorem

in plane : Thales’ theorem, Desargues’ theorem and Pappus’ theoremand give some exercise to apply three theorem The content is writtenbase on references [2], [3], [6]

2.1 The single ratio in plane

Definition 2.1.1 InA2 Let P,Q,R be three distinct collinear points.Then we have a number λ ∈ R such that −→

RP = λ−→

RQ The number λ

is called the single ratio of triple point p,Q,R and denoted by

λ = [P, Q, R] = (P, Q, R)

Assume that R has character different from 2 If [P, Q, R] = −1 then

R is called midpoint of pair point (P,Q)

Notation that the hypothesis P 6= Q then λ 6= 1

THREE THEOREMS IN PLANE GEOMETRY< /h2>

In Chapter, we will present the single ratio in plane, three theorem

in plane : Thales’ theorem,... and Pappus’ theoremand give some exercise to apply three theorem The content is writtenbase on references [2], [3], [6]

2.1 The single ratio in plane< /h3>

Definition 2.1.1 In< h3>A2... A1, A2, A3 are affinely independent

Similarly, B0, B1, B2,3 are affinely independent