Introduction

Affine geometry is a form of geometry featuring the unique parallel line property

where the notion of angle is undefined and lengths cannot be compared in different directions

(that is, Euclid's third and fourth postulates are meaningless). Affine geometry is derived

from ordered geometry and satisfying nine POSTULATEs in ordered geometry.

In this geometry the single parallel line, concern Postulat Playfair, “ through a known

point, which is not passing through known line, can be made only a parallel line to it”, hold

important role. Since a circle is not included in this geometry and angel is never measured,

then it means this geometry has basic postulate 1, 2, and 5 from Euclid

History

1. Leonhard Euler

Leonhard Euler (15 April 1707 – 18 September

1783) was a pioneering Swiss mathematician and physicist.

He made important discoveries in fields as diverse as

infinitesimal calculus and graph theory. He is also

introduced much of the modern mathematical terminology

and notation, particularly for mathematical analysis, such as

the notion of a mathematical function. He is also renowned

for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his

adult life in St. Petersburg, Russia, and in Berlin, Prussia

At the age of thirteen he enrolled at the University of Basel, and in 1723, received his

Master of Philosophy with a dissertation that compared the philosophies of Descartes and

Newton. In 1726, Euler completed a dissertation on the propagation of sound with the title

De Sono. In 1727, he entered the Paris Academy Prize Problemcompetition, where the

problem that year was to find the best way to place the masts on a ship.

And then Euler introduced Affine Geometry in 1748 with term affine (in Latin, means

“related”) in his book Introduction in analysis Infinitorum (volume 2, chapter XVIII). His

first idea for this kind of geometry, for each point (x,y) is mapped to new point (ax + cy + e,

bx + dy + f), then it would be developed for circles, angels, and distances in affine geometry.

2. August Mobius

August Ferdinand

Möbius (November 17, 1790 –

September 26, 1868) was a

German mathematician and

theoretical astronomer

Möbius studied at the University of Leipzig,

where he switched from law to mathematics, physics,

and astronomy. He joined Gauss at the observatory in

Göttingen, and later Pfaff in Halle. Throughout his academic career, he bore the title of

astronomer and taught mechanics. He also wrote a Textbook of Statics (1837) and studied

systems of lenses. His most famous contributions, however, are in the field of pure

mathematics. While Möbius was not the inventor of the one-sided so-called Möbius strip,

which is actually a discovery of Johann Benedict Listing, he did introduce the notion of

orientability, which allowed him to put a minus sign in front of lengths, areas, and

volumes. Furthermore, the well-known strip which carries his name is not the only one-

sided surface that he considered; he described a whole class of polyhedra with this

property, which he called extraordinary. They all have volume zero, and violate Euler's

polyhedral formula. The smallest has 10 triangular faces, 15 edges, and 6 vertices. The

notion of the dual polyhedron is also due to Möbius.

August Mobius (1827) wrote affine geometry in his book Der Barycentrische

Calcul (chapter 3). The algebraic tools developed by august Mobius in his book include a

formula for the cross-ratio, general solution for various fundamental problems such as

determining a conic section passing through given point, and the abstract formulation of

the duality principle and the algebraic characterization of affine transformation.

Möbius was the first to introduce homogeneous coordinates into projective

geometry. Homogeneous coordinates have a range of applications, including computer

graphics and 3D computer vision, where they allow affine transformations

3. Felix Klein

Klein was born in Düsseldorf, to Prussian parents; his father

was a Prussian government official's secretary stationed in the

Rhine Province. He attended the Gymnasium in Düsseldorf,

then studied mathematics and physics at the University of

Bonn, 1865–1866, intending to become a physicist. At that

time, Julius Plücker held Bonn's chair of mathematics and

experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest

was geometry. Klein received his doctorate, supervised by Plücker, from the University of

Bonn in 1868

In 1871, while at Göttingen, Klein made major discoveries in geometry. He published

two papers On the So-called Non-Euclidean Geometry showing that Euclidean and non-

Euclidean geometries could be considered special cases of a projective surface with a

specific conic section adjoined.

Next, Felix Klein also give definition about geometry as follows: a geometry defined

by a group of transformations, if definition and the prevailed theorem for the property of

the shape is “invariant” (not change) by those transformation of G, but it is not “invariant”

by other group transformation that containing G.

In 1872, he introduced Erlangen Program. This program was set out in Klein's

inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on

the occasion. The Program proposed a unified approach to geometry that became the

accepted view. Klein showed how the essential properties of a given geometry could be

represented by the group of transformations that preserve those properties

Development of Affine Geometry

1. Edwin B. Wilson and Gilbert N. Lewis (1912) developed affine geometri to express

special relativity theory

2. Herman Weyl (1918) refer to affine geometry to support his article “Space, Time, and

Matter” . he use it in addition and subtraction of vector

3. Graciela Birman and Katsumi Nomizu (1984) in their article Trigonometry in Lorenzian

Geometry describe “affine plane”

SUBJECT MATTER

1. POSTULATEs and theorems in Affine Geometry

The basic of Affine geometry is Ordered Geometry. Affine plane is known as special

condition of ordered plane. The fundamental definition is same those are point and

intermediacy. Affine geometry is derived from ordered geometry by adding two extra

POSTULATEs.

POSTULATE 1 There are at least two points.

POSTULATE 2 If A and B are two distinct points, there is at least one point C for which

[ABC].

POSTULATE 3 If [ABC], then A and C are distinct: A ≠ C.

POSTULATE 4 If [ABC], then [CBA] but not [BCA].

POSTULATE 5 If C and D are distinct points on the line AB, then A is on the line CD.

POSTULATE 6 If AB is a line, there is a point C not on this line.

POSTULATE 7 If ABC is a triangle and [BCD] and [CEA] then there is, on the line DE, a

point F for which [AFB].

POSTULATE 8 All points are in one plane.

POSTULATE 9 For every partition of all the points on a line into two non-empty sets, such that

no point of either lies between two points of the other, there is a point of one set which lies

between every other point of that set and every point of the other set.

POSTULATE 10 For any point A and any line r, not through A, there is at most one line

through A, in the plane Ar, which does not meet r.

POSTULATE 11 If A, A', B, B', C, C, O are seven distinct points, such that AA', BB', CC are

three distinct lines through O, and if the line AB is parallel to A'B', and BC to B'C, then also CA

is parallel to C'A'.

.

The affine POSTULATE of parallelism tell us that, for any point A and any line r, there is

exactly one line through A, in the plane Ar, which does not meet r. Hence the two rays from A

parallel to r are always collinear, any two lines in a plane that do not meet are parallel, and

parallelism is an equivalence relation. The last remark comprises three properties:

a. Parallelism is reflexive. (Each line is parallel to itself.)

b. Parallelism is symmetric. (If p is parallel to r, then r is parallel to p.)

c. Parallelism is transitive. (If p and q are parallel to r, then p is parallel to q)

POSTULATE 11 is probably familiar to most readers either as an affine form of Desargues's

theorem

THEOREM 1.1

If ABC and A'B'C’ are two triangles with distinct vertices, so placed that the line BC is parallel

to B'C’, CA to C'A', and AB to A'B', then the three lines AA', BB', CC’ are either concurrent or

parallel

Proof :

If the three lines AA', BB', CC’ are not all parallel, some two of them must meet. The notation

being symmetrical, we may suppose that these two are AA' and BB', meeting in O, as in

Picture6b. Let OC meet B'C’ in C1. By POSTULATE 11, applied to AA', BB', C1, the line ACis

parallel to A' C1 as well as to A'C’. By POSTULATE 10, C1 lies on A'C’ as well as on B'C’.

Since A'B'C’ is a triangle, C1 coincides with C’. Thus, if AA', BB', CC’ are not parallel, they are

concurrent

THEOREM 1.2

If A, A', B, B', C, C’ are six distinct points on three distinct parallel lines AA', BB', CC’, so

placed that the line AB is parallel to A'B', and BC to B'C’, then also CA is parallel to C'A'.

Proof.

C’

A

B’

C

A’

B

1. Given six distinct points A, A’, B, B’, C, and C’ on three distinct parallel lines

(AA’//BB’//CC’).

2. AB//A’B’ and BC//B’C’

3. Through A’ draw A’C’ parallel to AC to meet B’C’ in C1 (Line construction)

4. AB//A’B’ and BC//B’C’ and AC//A’C1

5. AA’//BB’//CC1

6. CC1// CC’

7. CC’ and CC1 meet in C

8. Statement (7) contradicts with POSTULATE “For point C and line AA’, not through C, there

is at most one line through C, which does not meet AA’”

9. C’ and C1 are coincides

2. Affine Transformation

DEFINITION 2.1

Four non-collinear points A, B, C, D are said to form a parallelogram ABCD if the line AB is

parallel to DC, and BC to AD. Its vertices are the four points; its sides are the four segments AB,

BC, CD, DA, and its diagonals are the two segments AC and BD. Since B and D are on opposite

sides of AC, the diagonals meet in a point called the center.

C1

C’

A

B’

C

A’

BC1

C’

A

B’

C

A’

B

a. Dilatasion

DEFINITION 2.2 Dilatation to be a transformation which transforms each line into a parallel

line.

THEOREM 2.3

Two given segments, AB and A'B', on parallel lines, determine a unique dilatation AB →A'B'.

Proof:

1. Construct any segment which is not parallel or coincide to AB and A’B’, let the segment is

CP.

2. To find the image of C, construct a line from A to C then construct a line through A’ which

is parallel to AC.

3. Construct a line from B to C then construct a line through B’ which is parallel to BC.

4. Let the intersection of line through A’ which is parallel to AC and line through B’ which is

parallel to BC is C’.

5. C’ is the image of C.

6. Then by the same way, we can find the image of P as well.

7. Let the image of P is P’.

8. Based on the theorem 1, then lines AA’, BB’, CC’, and PP’ are either concurrent or parallel.

9. If AA’, BB’, CC’, and PP’ are concurrent, then CP is parallel to C’P’.

10. If AA’, BB’, CC’, and PP’ are parallel, then CP is parallel to C’P’.

11. CP and C'P' are parallel, so that the transformation is indeed a dilatation.

DEFINITION 2.3

Invers of dilatation AB→ A’B’ is dilatation A’B’→ AB.

DEFINITION 2.4

The product of 2 dilatations is a dilatation which is followed by the other dilatation.

Then the result of the product of 2 dilatation AB→ A’B’ dan A’B'→ A’’B’’ is dilatation

A’B'→ A’’B’’.

The result of the product of dilatation by its invers is identity ABAB

The line which join pairs of corresponding points are invariant lines.The all these lines

are either concurrent or parallel. If the lines are concurrent, their intersection O is an

invariant point, and we have a central dilatation, and the point is unique.

b. Translation

DEFINITION 2.5

If, on the other hand, the invariant lines are parallel, there is no invariant point, and we have a

translationABA'B', where not only is AB parallel to A'B' but also AA' is parallel to BB'. If these

two parallel lines are distinct, AA'B'B is a parallelogram. If not, we can use auxiliary

parallelograms AA'C'C and C'CBB' (or AA'D'D and D'DBB').

Two applications theorem 2 suffice to prove that, when A, B, A' are given, B' is independent of

the choice of C (or D). Hence:

THEOREM 2.4 Any two points A and A' determine a unique translation A A'.

Proof.

1. Given 2 any points A and A’

2. Draw AA’ (Line construction)

3. Let B be an any point that doesn’t lie on AA’

4. From B, draw a line parallel to AA’ (Line construction)

5. From A’, draw a line parallel to AB (Line construction)

6. Suppose B’ is a common point of (4) and (5)

7. AA’ // BB’ and AB // A’B’

8. ABB’A’ is a parallelogram

9. Invariant lines are parallel

10. A→B is a translation.

THEOREM 2.5 The dilatation AB→A'B' transforms every point between A and B into a point

between A' and B'.

Proof

There are 2 cases for proofing this theorem, such as:

a. The invariant lines are parallel

A’

A

A’

A B

A’

A B

A’

A

B

B

A’

A

If the lines AB and A’B’ are distinct, and A’B’ is an image of AB. So that, there are

invariant lines a and b passing through AA’ and BB’ respectively. Construct line c which

is parallel to AA’ and BB’. Hence, there is point C that is the intersection of segment AB

in line c, then we get [ACB]. Point C’ is the image of C. Therefore, we obtain [A’C’B’].

b. The Iinvariant lines are concurrent

If the lines AB and A’B’ are distinct such that AA’ and AB’ has an intersection in point O

which is not placed between A and A’, and also O is not placed between B and B’. A’B’ is

an image of AB. So that, there are invariant lines a and b passing through AA’ and BB’

respectively. Construct line c which divides angle AOB. Hence, there is point C that is the

intersection of segment AB in line c, then we get [ACB]. Point C’ is the image of C.

Therefore, we obtain [A’C’B’].

THEOREM 2.6 The product of two translations AB and B C is the translation A C

Proof

A’C’ B’

AC B

bca

If the product of two translation is not translation, then surely there is invariant point O,

O is moved to O’ by the first translation A B.

Since O is invariant point then O is moved to O’ by the first translation BC.

O’Ois invers of OO’

The product of two translations has a invariant point, if one of the translation is the invers

of the others and the result of product is identity.

The product of two translation is translation, in which dilatation that there is no invariant

point.

The result of two translationsatisfy commutative property.

let two translation don’t allow parallel line. By constructing parallelogram ABCD, seem that

AB is same with DC and BC is same with AD,

AC = (AB)(BC) = (AD)(DC) = (BC)(AB) (proven)

If two translations allow at same line, let the translations are T and X. Let Y translation is a

translation which does not allow parallel line with the translations, then X and Y are

commutative, and also X and TY.

T(XY) = T(YX) = (TY)X= X(TY)

(TX)Y = (XT)Y, such that TX = XT (proven)

c. Half-turn

DEFINITION 2.6

Any two distinct points, A and B, are interchanged by a unique dilatation ABBA, or, more

concisely, AB which we call a half-turn. (Of course, AB is the same as BA.)

If C is any point outside the line AB, the half-turn transforms C into the point D in which

the line through B parallel to AC meets the line through A parallel to BC

.

Therefore ADBC is a parallelogram, and the same half-turn can be expressed as C↔D. The

invariant lines AB and CD, being the diagonals of the parallelogram, intersect in a point O, which

is the invariant point of the half-turn. It follows that any segment AB has a midpoint which can

be denned to be the invariant point of the half-turn A↔B, we have proved that the center of a

parallelogram is the midpoint of each diagonal, that is, that the two diagonals "bisect" each other.

To see how the half-turn transforms an arbitrary point T on AB, we merely have to join

this point to C (or D) and then draw a parallel line through D (or C) and there is T’ on AB.

By considering their effect on an arbitrary point B, we may express any two half-turns as

A↔ B and B↔C. If their product has an invariant point O, a half-turn A↔B transform O to O’.

Hence, A↔B is same as O↔O'.Then O’ is transformed to O by B↔C, so B↔C is same as O↔

O'. In this case A↔B = B↔C there is invariant point, while every other case, there is no

invariant point. Hence

THEOREMA 2.7

The product of two half-turns A↔B and B↔C is the translation A↔C.

Proof

If A↔B is not same as B↔C, then (A↔B)(B↔C) does not have invariant point. So, it will be a

translation. If ABCD is parallelogram, then A↔B is same as C↔D and A↔D is same as C↔B.

This relation also satisfy when the parallelogram change to be line with four distinct points

which the position is symmetrical ordered.

THEOREMA 2.8

The half-turns A↔B and C↔D are equal if and only if the translations A↔D and C↔B are

equal.

Proof

a. If A↔B = C↔D then A↔D = C↔B

A↔D = (A↔B) (B↔D)

= (C↔D) (B↔D)

= (C↔D) (D↔B)

= C↔D

b. If A↔D = C↔B then A↔B = C↔D

A↔B = (A↔D) (D↔B)

= (C↔B) (D↔B)

= (C↔B) (B↔D)

= C↔D

If C and D is , and called C’, then C’ is the mid point of AB if only if translation A→C’

and C’→B is equal.

Then we can make parallelogram AC’A’B’ and BC’B’A’ and there is triangle ABC with

A’, B’, and C’ are the mid point of the side. Then we get a theorem as

APPLICATION

1. THE APPLICATION OF AFFINE TRANSFORMATION IN OBJECT DESIGN

Affine transformation can be applied by architects to design a building before they really build

the building. They can construct the design of building by using computer programming such as

Visual Basic. Both of them applying theory of affine transformation to minimize the cost of

building program.

The first step (before doing transformation), we should determine the angle point of the object.

3D transformation uses three coordinate point, namely x-axis, y-axis, and z-axis. Coordinate of

x-axis indicates the width, coordinate of y-axis means length, and z-axis shows the height or

depth. Next, joint the point with its pairs with edge. Finally, we will transform the result object in

computer programming.

Usually, computer programming use the basic of Affine geometry, such as

The transformation from P(Px,Py) to Q(Qx,Qy) by using formula

Qx = aPx + cPy + trx

Qy = bPx + dPy + try, or frequently the program use the combination of translation, scaling, and

rotataion by using matrix.

On 3D Transformation we have the matrix are

For Translation

For Scaling

For rotation

Rotation of x-axis

Rotation of z-axis

Rotation of y-axis

The example of algorithm of Visual Basic

2. THE APPLICATION OF AFFINE TRANSFORMATION IN JUMP WORKBENCH

SOFTWARE

Affine transformation is a useful operation that can be applied in Jump Workbench software

features by changing the coordinate system, changing of units of measurements, and referencing

scanned paper. When a map that is either paper or digital image is not printed or exported in the

same coordinate system as the vector data. Then the map is needed to be registered (place, rotate,

scale) and transformed. Affine transformation can do the process of changing the map features

with sufficient accuracy.

3. APPLIACTION OF AFFINE TRANSFORMATION IN BATIK MOTIF

Affine transformation also can be implemented in fashion model especially in batik

motif. Nowadays many batik company use fractal code to produce batik with variety motif.

The main characteristic of fractal becomes basic of fractal code, that is having resemblance

with it self. And furthermore Jacquin introduced an automatic scheme in coding image which

is known as Partitioned Iterated Function System (PIFS). PIFS concept is dividing image

into range block which is not overlapping to each other.

BIBLIOGRAPHY

Budiarto, M. T dan Masriyah . 2010. Sistem Geometri (Edisi Revisi). UNESA: University Press