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Affine Geometry

A. 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 Axioms in ordered geometry.

Affine geometry can be developed in terms of the geometry of vectors, with or without the notion of coordinates. An affine space is distinguished from a vector space of the same dimension by 'forgetting' the origin 0 (sometimes known as free vectors).

1. HISTORY

There are some mathematicians that gave contribution and influence for the development of affine geometry.

a. 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 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.

He is considered to be the preeminent mathematician of the 18th century, and one of the greatest mathematicians to have ever lived. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes.

Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother. 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. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly

Picture 1Leonhard Euler

discovered his new pupil's incredible talent for mathematics. Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was pursuing an (ultimately unsuccessful) attempt to obtain a position at the University of Basel. In 1727, he entered the Paris Academy Prize Problem competition, where the problem that year was to find the best way to place the masts on a ship. He won second place, losing only to Pierre Bouguer—a man now known as "the father of naval architecture". Euler subsequently won this coveted annual prize twelve times in his career.

Euler worked in almost all areas of mathematics: geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes Euler's name is associated with a large number of topics.

Euler is the only mathematician to have two numbers named after him: the immensely important Euler's Number in calculus, e, approximately equal to 2.71828, and the Euler-Mascheroni Constant γ (gamma) sometimes referred to as just "Euler's constant", approximately equal to 0.57721. It is not known whether γ is rational or irrational.

In 1748, Euler introduced the term affine (Latin affinis, "related") in his book Introductio in analysin infinitorum (chapter XVII). He was the first mathematician who studied affine geometry. 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.

b. August Ferdinand Möbius

August Ferdinand Möbius (November 17, 1790 – September 26, 1868) was a German mathematician and theoretical astronomer.

August Möbius was born in Schulpforta, Saxony-Anhalt, and was descended on his mother's side from religious reformer Martin Luther. He studied mathematics under Carl Friedrich Gauss and Johann Pfaff. Möbius died in Leipzig in 1868 at the age of 77.

He introduced homogeneous coordinates for the first time, and he has been best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict Listing around the same time. The Möbius configuration, formed by two mutually inscribed tetrahedra, is also named after him. Many mathematical concepts are named after him, including the Möbius transformations, important in projective geometry and the Möbius transform of number

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Picture 2August Ferdinand Möbius

theory. His interest in number theory led to the important Möbius function μ(n) and the Möbius inversion formula. In Euclidean geometry, he systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.

In 1827, August Möbius wrote on affine geometry in his Der barycentrische Calcul (chapter 3). The algebraic tool developed by August Möbius in Der barycentrische Calcul included a formula for the cross-ratio, general solutions for various fundamental problems, such as determining a conic section passing through given points, and the abstract formulation of the duality principle and the algebraic characterization of affine transformations.

c. Felix Klein

After Felix Klein's Erlangen program, affine geometry was recognized as a generalization of Euclidean geometry. The name “Affine Geometry” came from Erlangen Program.

Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician, known for his work in group theory, complex analysis, non-Euclidean geometry, and on the connections between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day.

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 Gymnasi-um 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. This had the remarkable corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was, putting Euclidean and non-Euclidean geometries on the same footing, and ending all controversy surrounding non-Euclidean geometry. Cayley never accepted Klein's argument, believing it to be circular.

Klein's synthesis of geometry as the study of the properties of a space that is invariant under a given group of transformations, known as the Erlangen Program (1872), profoundly influenced the evolution of mathematics. 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 (and remains) the accepted view. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that

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Picture 3Christian Felix Klein

preserve those properties. Thus the Program's definition of geometry encompassed both Euclidean and non-Euclidean geometry.

Today the significance of Klein's contributions to geometry is more than evident, but not because those contributions are now seen as strange or wrong. On the contrary, those contributions have become so much a part of our present mathematical thinking that it is hard for us to appreciate their novelty, and the way in which they were not immediately accepted by all his contemporaries.

Klein accepted a chair at the University of Göttingen in 1886. From then until his 1913 retirement, he sought to re-establish Göttingen as the world's leading mathematics research center. Yet he never managed to transfer from Leipzig to Göttingen his own role as the leader of a school of geometry. At Göttingen, he taught a variety of courses, mainly on the interface between mathematics and physics, such as mechanics and potential theory.

Around 1900, Klein began to take an interest in mathematical instruction in schools. In 1905, he played a decisive role in formulating a plan recommending that the rudiments of differential and integral calculus and the function concept be taught in secondary schools. This recommendation was gradually implemented in many countries around the world. In 1908, Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Congress of Mathematicians. Under his guidance, the German branch of the Commission published many volumes on the teaching of mathematics at all levels in Germany.

The London Mathematical Society awarded Klein its De Morgan Medal in 1893. He was elected a member of the Royal Society in 1885, and was awarded its Copley medal in 1912. He retired the following year due to ill health, but continued to teach mathematics at his home for some years more. He died in Göttingen in 1925.

2. DEVELOPMENT OF AFFINE GEOMETRY

In 1912, Edwin B. Wilson and Gilbert N. Lewis developed an affine geometry to express the special theory of relativity.

In 1918, Hermann Weyl referred to affine geometry for his text Space, Time, Matter. He uses affine geometry to introduce vector addition and subtraction at the earliest stages of his development of mathematical physics. Later, E. T. Whittaker wrote:

Weyl's geometry is interesting historically as having been the first of the affine geometries to be worked out in detail: it is based on a special type of parallel transport [...using] worldlines of light-signals in four-dimensional space-time. A short element of one of these world-lines may be called a nulvector; then the parallel transport in question is such that it carries any null-vector at one point into the position of a null-vector at a neighboring point.

In 1984, "the affine plane associated to the Lorentzian vector space L2" was described by Graciela Birman and Katsumi Nomizu in an article entitled "Trigonometry in Lorentzian geometry.

B. SUBJECT MATTER

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1. AXIOMS FOR AFFINE GEOMETRY

In this axiomatic treatment, we regard the real affine plane as a special case of the ordered plane. Accordingly, the primitive concepts are point and intermediacy, satisfying nine Axioms in ordered geometry. Hence, affine geometry is derived from ordered geometry by adding two extra axioms:

AXIOM 1 There are at least two points.

AXIOM 2 If A and B are two distinct points, there is at least one point C for which [ABC].

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

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

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

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

AXIOM 7 If ABC is a triangle and [BCD] and [CEA] then there is, on the line DE, a point F for which [AFB].

AXIOM 8 All points are in one plane.

AXIOM 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.

AXIOM 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.

AXIOM 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'.

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Picture 4

Picture 5

B A’

C

B’

A

C’

The affine axiom 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)

Axiom 11 is probably familiar to most readers either as an affine form of Desargues's theorem. We shall see that it implies:

THEOREM 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 Picture 6b. Let OC meet B'C’ in C1. By Axiom 11, applied to AA', BB', C1, the line AC is parallel to A' C1 as well as to A'C’. By Axiom 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.

Roughly speaking, Axiom 11 is the converse of one half of Theorem 1. The converse of the other half is

THEOREM 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.

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Picture 6a Picture 6b

Picture 7

1. Given six distinct points A, A’, B, B’, C, and C’ on three distinct parallel lines (AA’//BB’//CC’ ).( Premise)

2. AB//A’B’ and BC//B’C’ ( Premise)3. Through A’ draw A’C’ parallel to AC to meet B’C’ in C1 (Line construction)4. AB//A’B’ dan BC//B’C’ dan AC//A’C1 (Corollary 3)5. AA’//BB’//CC1 (Theorem 1 and corollary 1)6. CC1// CC’ (Corollary 1 and 5)7. CC’ and CC1 meet in C (Corollary 6)8. Statement (7) contradicts with axiom 10

(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 (Corollary 8)

2. AFFINE TRANSFORMATIONS

a. DILATASION

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.

DEFINITION 1 Dilatation to be a transformation which transforms each line into a parallel line.

THEOREM 3 Two given segments, AB and A'B', on parallel lines, determine a unique dilatation AB A'B'.

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B A’

C

B’

A

C’

C1 B A’

C

B’

A

C’

C1

Picture 8a Picture 8b

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’ (axiom 11).10. If AA’, BB’, CC’, and PP’ are parallel, then CP is parallel to C’P’ (theorem 2).11. CP and C'P' are parallel, so that the transformation is indeed a dilatation

b. TRASLATION

DEFINITION 2 The product of two dilatations, AB A'B' and A'B' A"B", is the dilatation AB A"B". In particular, the product of a dilatation with its inverse is the identity, AB AB. Thus all the dilatations together form a (continuous) group.

The lines PP' which join pairs of corresponding points are invariant lines. The all these lines are either concurrent or parallel. If the lines PP' are concurrent, their intersection O is an invariant point, and we have a central dilatation

OA OA'

(where A' lies on the line OA). The invariant point O is unique; for, if O and Ox were two such, the dilatation would be OO1 —> OO1, which is the identity.

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Picture 9a Picture 9b

If, on the other hand, the lines PP' are parallel, there is no invariant point, and we have a translation AB A'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') as in Picture 9. 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 4 Any two points A and A' determine a unique translation A A'.

Proof.

1. Given 2 any points A and A’ (Premise)

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) (Corollary 4 and 5)

7. AA’ // BB’ and AB // A’B’ (Corollary 4 and 5)

8. ABB’A’ is a parallelogram (Corollary 7)

9. Invariant lines are parallel (Corollary 8)

10. A→B is a translation (Corollary 9)

THEOREM 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. If invarian lines are parallel

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A

A’

Picture 10

A

A’

A

A’

B A

A’

B A

A’

B

B’

Picture 11

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. If invarian lines are concurent

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 6 The product of two translations A B and B C is the translation A C

Proof.

Suppose A C is a dilatation, so that it will have an invariant point, namely O.

Since A C is the result of A B and B C , then translation A B caused O O’ and translation B C caused O‘ O

Therefore, O‘ O is the inverse of O‘ O then the result of translation is an identity (O O). We can conclude that A C has no invariant point, so that A C is a translation.

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Picture 13

Picture 12

a c b

BCA

B’C’A’

ac b

BCA

B’C’A’

DEFINITION 3 Any two distinct points, A and B, are interchanged by a unique dilatation AB BA, or, more concisely, A B which we call a half-turn. (Of course, A B is the same as B A.)

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 (Picture 13). Therefore ADBC is a parallelogram, and the same half-turn can be expressed as CD. 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 on AB, we merely have to join this point to C (or D) and then draw a parallel line through D (or C).

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, each of them must be expressible in the form O O', that is, they must coincide. In every other case, there is no invariant point. Hence

THEOREM 7 The product of two half-turns AB and B C is the translation A C.

Proof. We have seen (Picture 14) that, If A is any point outside the line PQ, then there is half turn A B. From B outside line QR there is also half turn from B to C. Based on theorem 4 any two points A and C determine a unique translation A C.

Suppose A C is a dilatation, so that it will have an invariant point, namely O. Since A C is the result of A B and B C , then translation A B caused O O’ and translation B C caused O‘ O.

Therefore, O‘ O is the inverse of O‘ O then the result of translation is an identity (O O). We can conclude that A C has no invariant point, so that A C is a translation.

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Picture 14

Picture 15

A

B

C

Q

P

R

THEOREM 8 The half-turns A B and C D are equal if and only if the translations A D and C B are equal.

In fact, the relation implies

and, conversely, the relation implies

THEOREM 9 If the three diagonals of a hexagon (not necessarily convex) all have the same midpoint, any two opposite sides are parallel

Proof. Since the three diagonals have the same midpoint implies

Its mean DE parallel to BA

Its mean DC parallel to FA

So EF parallel to CB.

THEOREM 10 The line joining the midpoints of two sides of a triangle is parallel to the third side, and the line through the midpoint of one side parallel to another passes through the midpoint of the third.

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Picture 16

A B

C

DE

F A

B C

D

EF

Proof. Two figures are said to be homothetic if they are related by a dilatation, congruent if they are related by a translation or a half-turn. In particular, a directed segment AB is congruent to its "opposite" segment BA by the half-turn A B. Thus, in Picture 15, the four small triangles

AC'B', C'BA', B'A'C, A'B'C

Are all congruent, and each of them is homothetic to the large triangle ABC.

C. APPLICATIONS AND RELATIONSHIPS

1. THE APPLICATION OF AFFINE GEOMETRY IN IMAGE PROCESSING

An affine transformation is the composition of translation, rotation, scaling, and shear. Affine transformation is applied in computer imaging process. An image could have perspective irregularities due to the position of the camera. The example of the image below can help the explanation of affine transformation in imaging process. The principle used this process is to correct a range of perspective distortions by transforming measurements from the ideal coordinates to those actually used.

Look at the pictures below.

In picture 17(a), the part face shown is in fronto-parallel plane (no depth difference among parts of the thing in the image). The circular hole is shown as a circle

In picture 17(b), the circular hole is shown as an ellipse in the image. If in picture 17(a), the distance between parts is not shown and there is no difference in depth; however, in picture 17(b) the distance and depth among parts is clearly imaged.

The transformation of the face part shown in these pictures is approximated by a planar affine transformation.

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Picture 16

Picture 17a Picture 17b

Another example of affine transformation is shown in the pictures below.These transformations map each point in 3D space to a potentially different point in 3D space.

We can build up many types of transformation by using a combination of transformation including translation, shear, scaling, reflection, and rotation.

One of the most important is a combination of a translation and a rotation, this is important for mechanics because it can represent all possible movements of a solid body.

Original object the result of scaling the result of translation

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.

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Picture 18a Picture 18cPicture 18b

3. THE APPLICATION OF AFFINE TRANSFORMATION IN GEOREFERENCE MAP FEATURES

Affine transformation can be applied by architects to design a building before they really build the building. They can construct manual miniature or by using computer. Both of them applying theory of affine transformation including translation, dilatation, rotation, scaling.

Nowadays, architects tend to use computer to construct 3D model of a building since it is more efficient than the manual. They use some programs in a computer, for example visual basic. Then, they can transform an object to a new position (translation, scaling, or rotation to the object) by using matrices. The following matrices are the example of matrices to translate and scale an object.

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Picture 20

Picture 21

4. THE APLICATION 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. Pertition scheme used is rectangular

REFERENCES

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Matrix for translation 3D object Matrix for scaling 3D object

Coxeter. 1969. Introduction to Geometry. Canada: John Wiley & Sons, Inc.

http://en.wikipedia.org/wiki/Affine_geometry. Accessed on Tuesday, March 19, 2013 at 07:00

http://en.wikipedia.org/wiki/August_Ferdinand_M%C3%B6bius. Accessed on Tuesday, March 19, 2013 at 07:30

http://en.wikipedia.org/wiki/Felix_Klein. Accessed on Tuesday, March 19, 2013 at 07:30

http://en.wikipedia.org/wiki/Leonhard_Euler. Accessed on Tuesday, March 19, 2013 at 07:30

http://homepages.inf.ed.ac.uk/rbf/HIPR2/affine.htm. Accessed on Tuesday, March 26, 2013 at 14:21

http://uqu.edu.sa/files2/tiny_mce/plugins/filemanager/files/4282164/Affine%20geometry.pdf. Accessed on Tuesday, March 19, 2013 at 07:00

http://www.euclideanspace.com/maths/geometry/affine/. Accessed on Saturday, may 11, 2013 at 20:00

http://www.quantdec.com/GIS/affine.htm. Accessed on Tuesday, March 26, 2013 at 14: 41

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