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Home , Complete quadrangle, Concurrent lines, Pencil (mathematics), Perspectivity, Synthetic geometry

Degree project

Projective

Author: Wu Wei Supervisor: Hans Frisk Examiner: Per Anders Svensson Course Code: 2MA41E Subject: Mathematics Level: Bachelor

Department Of Mathematics

Wei Wu

June 13, 2019

Abstract Projective geometry is a branch of mathematics which is founda- tionally based on an . In this thesis, six for two-dimensional projective geometry are chosen to build the structure for proving some further results like Pappus’ and Pascal’s . This work is mainly in synthetic geometry. Contents

1 Introduction 5 1.1 Informal description of projective geometry ...... 5 1.2 What is projective geometry? ...... 6 1.3 History ...... 6 1.4 What you will find in this thesis ...... 7

2 The axiomatic system and 8 2.1 First five axioms for the projective ...... 9 2.2 Duality ...... 11 2.3 An infinite model for ...... 13

3 Harmonic sets 15 3.1 Construction of the fourth of a harmonic set ...... 16 3.2 Construction of the fourth of a harmonic set ...... 19

4 Perspectivities and Projectivities 21 4.1 Perspectivities ...... 22 4.2 Projectivities ...... 23 4.3 Construction of a projectivity between pencils of points . . . 26

5 Point and line conics 31

6 Conclusion 36

2 List of Figures

1 Vanishing line BC and points B,C...... 5 2 Finite model with 13 points...... 8 3 A ...... 9 4 The points E, F , G from a ABCD...... 10 5 Illustration of Desargues’ in 3D . . . 11 6 The diagonal lines EF , BC, AD of a complete abcd...... 13 7 4ABC and 4A0B0C0 are perspective from the line l. Points AC · A0C0 = R, BC · B0C0 = Q, AB · A0B0 = P all lie on line l. 14 8 A projective plane model in 3D...... 14 9 Harmonic set H(AB, CD)...... 15 10 Five collinear points form a quadrangular set (AA0)(BB)(CD). 16 11 Six collinear points form a quadrangular set (AA0)(BB0)(CD). 16 12 Unique point D for harmonic set H(AB, CD)...... 17 13 H(AB, CD) ⇔ H(CD, AB)...... 18 14 H(ab, cd)...... 19 15 of lines...... 21 16 Pencil of points...... 21 17 Perspectivity between pencils of points...... 22 18 Perspectivity between pencils of lines...... 23 19 Perspectivity between a pencil of points and a pencil of lines. 23 20 A perspectivity between pencils of points...... 24 21 A projectivity between pencils of points...... 24 22 ABC ∧ A00B00C00...... 24 23 abc ∧ a00b00c00...... 24 24 abc ∧ A00B00C00...... 25 25 Axis of projectivity h for ABC ∧ A0B0C0...... 27 26 A P1P2P3P4P5P6...... 28 27 ABC ∧ BCA ...... 29 28 points C,F ...... 29 29 Only one invariant point C...... 30 30 Five points U, U 0, A, B and C generate a point conic. . . . . 32 31 Lines from S and R are projectively related...... 32 32 The Pascal’s line is outside of the ...... 33 33 Construction of a ...... 34 34 Construction of a ...... 34 35 Five lines u, u0, a, b, and c generate a line conic...... 34 36 The Brianchon’s point B is inside of the ellipse...... 35 37 The Brianchon’s point B is outside of the ellipse...... 35

3 List of Tables

1 Point and line conic...... 31

4 1 Introduction

1.1 Informal description of projective geometry

Projective geometry is a subject which originates from visual arts: using figures to record the by observation. The transformation that maps objects onto the plane is an example of a projective transformation. We have a lot of projective transformations in our daily life. One example is that if we see a from above, it is a ; but if we observe it from the side, it will be a .

We are all familiar with that has one character- istic of pairs of lines: lines never meet. However, an exception is discovered by our eyes. The railway should be a pair of parallel lines but when we see the end of the railway it looks like they will meet somewhere in the end. If the parallel lines will meet at infinity the space is transformed into a new type of geometric object, the projective space. Projective space can thus be defined as an extension of in which two lines always meet in an infinite point. In some way, Euclidean geometry is about the object itself while projective geometry is about investigating the object under observation.

Take as an example the sides of a (see figure 1). Point B is the infinite point of lines AB and DB. We call those infinite points the vanishing points and the line which goes through the vanishing points is the vanishing line [4].

Figure 1: Vanishing line BC and points B,C.

5 1.2 What is projective geometry?

Let us now turn to projective geometry as a branch of mathematics. The plane geometry of ’s Elements is the same as the geometry of lines and : the tools are the ruler (the straight-edge or unmarked ruler) and the compass. A Danish geometer Georg Mohr (1640-1697) and an Italian Lorenzo Mascheroni (1750-1800) have independently discovered that in the geometrical constructions nothing is lost by only using the compass. For example for four points A, B, C, and D. We can find the intersection point of lines AB and CD only by using compass although the actual process is quite complicated.

However, what’s going on if we only use rulers in geometry? It looks unacceptable since we can not even complete Euclid’s first proposition (To construct an equilateral triangle on a given finite straight line) by only using a ruler. Can we only use a ruler to develop a geometry which is not including circles, distance, , betweenness and parallelism? The answer is yes; this is projective geometry. It has not so much structure but anyhow full of beauty. Here, the basic concepts of projective geometry are listed here [2].

• P oint, line and incident are taken as undefined terms

• Collinear: Any number of points that are incident with the same line are said to be collinear.

• Concurrent: Any number of lines incident with a point are said to be concurrent.

1.3 History

As previously stated in section 1.1, projective geometry is a sub- ject which originates from visual and it begins with work of an architect. In 1425, an Italian architect Brunelleschi started to discuss the geometri- cal theory of perspective which was summarized in a treatise by Alberti a few years later. Although Menaechmus, Euclid, and Apol- lonius studied conics in the fourth and third centuries B.C., the earliest projective theorems were discovered by in the third century A.D. The French mathematician, J. V. Poncelet (1788-1867), was the first to prove such theorems by purely projective reasoning. The Ger- man astronomer Johann Kepler (1571-1630) and the French architect Girard Desargues (1591-1661) introduced however the concept of a point at infinity much earlier.

6 Poncelet could then construct a projective space from ordinary space by introducing a line at infinity consisting of all the points at infinity. In 1871, provided an algebraic foundation for projective geometry in terms of ””. This means projective geometry could be analyzed with coordinates. However, in this thesis, we will not focus on analytic projective geometry [1].

1.4 What you will find in this thesis

In this thesis, we select Judith N. Cederberg’s book as our main material and take its six axioms. All definitions and theorems are built on the six axioms. Also, this thesis will using GeoGebra for visualization to help us understand the projective geometry. The paper will first introduce axiomatic system then find a particular set of points and lines, next talk about the relationships between the particular set of points and lines, finally describe the constructions which on the projective plane.

In section 2 the axioms are introduced and in section 3 a special set of four points, the harmonic set, is constructed. This relation between four points is invariant under the projective transformations which is the topic of section 4. Finally, in section 5 the point and line conics are studied.

7 2 The axiomatic system and duality

The statements that are accepted without proofs are known as ax- ioms. Other statements which can be proved by using the axioms are called theorems.

Mathematically, an axiomatic system is a collection of axioms, and one or all of the axioms can be used to prove theorems logically. A mathematical theory consists of an axiomatic system and theorems which are derived by the axiomatic system. In this thesis, we take the six axioms from our key material [1] to develop the axiomatic system for the projective plane.

In this thesis only models with an infinite number of points will be considered. Finite models have also been studied [1], they use 1-3 below but have another fourth axiom. Our first four axioms guarantee that there are at least four points on each line and in total 13 point (see figure 2). The infinite model is presented at the end of this section.

If one includes axioms including the third , like in [2], it is possible to prove our Axiom 5 (Desargues’ theorem). This thesis is only about 2D and then we have to take it as an axiom, see however figure 5 in page 11 for a visualization in 3D.

Figure 2: Finite model with 13 points.

In this axiomatic system, the undefined concepts are point and line and incident. Points are said to be collinear if they are incident with the same line and lines are said to be concurrent if they intersect at the same point. The first five axioms are presented below. The sixth axiom will be given in section 4.

8 2.1 First five axioms for the projective plane

Axiom 1 Any two distinct points are incident with exactly one line.

Axiom 2 Any two distinct lines are incident with at least one point.

Notice that although Axiom 1 is the same as in Euclidean geometry, Axiom 2 is not. In projective geometry there are no parallel lines. Therefore we have to introduce the concept of an ideal point which is the point at infin- ity added in projective geometry as the assumed intersection of two parallel lines. Also, Axiom 1 and 2 do not show the existence of either points or lines.

Axiom 3 There exist at least four points, no three of which are collinear.

Axiom 3 shows us that points and lines exist in the projective plane so we can consider a set of three noncollinear points and a set of four points no three collinear, the triangle and the quadrangle.

Definition 2.1 A set of three noncollinear points forms a triangle (4ABC), and the three points determine three lines. The points A, B, C are called vertices and the lines a, b, c are called sides of the triangle. (figure 3)

Figure 3: A triangle.

Definition 2.2 A set of four points A, B, C, D, no three collinear forms a (complete) quad- rangle, and the four points determine six lines AB and CD, AC and BD,

9 AD and BC. The points are called vertices and the lines are called sides of the quadrangle. The points E, F , G at which pairs of opposite sides AB and CD, AC and BD, AD and BC intersect are called diagonal points of the quadrangle. (figure 4)

Figure 4: The diagonal points E, F , G from a complete quadrangle ABCD.

We see in figure 4 that the three diagonal points from a complete quad- rangle form a triangle and this observation is taken as the fourth axiom.

Axiom 4 The three diagonal points of a complete quadrangle are never collinear.

Axiom 1-4 describe the essential properties of the projective plane, all about the points and lines themselves. The next two axioms are more about the relation between a set of points and lines. In this section we only intro- duce the first five axioms; we put the sixth axiom in section 5.

Axiom 5 is about two relationships between pairs of . Before we show the axiom, the properties of the relation will be given as a definition.

Definition 2.3 Triangles 4ABC and 4A0B0C0 are said to be perspective from a point if the three lines joining corresponding vertices, AA0, BB0 and CC0 are concurrent. The triangles are said to be perspective from a line if the three points of intersection of corresponding sides, AB · A0B0, AC · A0C0, and BC · B0C0 are collinear. (figure 7, page 14)

10 The corresponding axiom is given below.

Axiom 5 (Desargues’ theorem) If two triangles are perspective from a point then they are perspective from a line.

In plane projective geometry we have to set the Desargues’ theorem as an axiom since this statement does not hold for some that also satisfy Axiom 1-4 [2].

For a visualization of Desargues’ theorem in 3D, see figure 5: A trian- gular OABC, points A0,B0 and C0 are on the sides of the triangular pyramid. 4ABC and 4A0B0C0 are perspective from the point O. We can observe that point D, E and F lie on the line which is the same as the meeting line of plane ABC and plane A0B0C0. Note that the lines A0B0 and AB must cross each other since they lie in the plane OAB.

Figure 5: Illustration of Desargues’ theorem in 3D projective space.

2.2 Duality

The dual of a statement is replacing each occurrence of the word ”line” by the word ”point” and vice verse. Axioms 1 and 2 are nearly dual statements.

If in an axiomatic system, the dual of any axiom can be proven as a theorem, it satisfies the duality principle. To ensure that the axiom system

11 in this thesis satisfies the duality principle, we must prove the dual of each axiom. Some of the proofs will be given below.

Theorem 2.1 (Dual of Axiom 1) Any two distinct lines are incident with exactly one point.

Proof. By Axiom 2, any two distinct lines are incident with at least one point, assume there are two points incident with the two distinct lines. But this assumption is a violation of Axiom 1. Hence this assumption is invalid. Therefore any two distinct lines are incident with exactly one point.

Here we introduce the concept of a quadrilateral which is the dual of a quadrangle, and figure 6 in page 13 will help us understand the concept more directly.

Definition 2.4 A (complete) quadrilateral is a set of four lines a, b, c and d, no three concur- rent, and the four lines determine six points. The points are called vertices and the lines are called sides of the quadrilateral. If a, b, c, d are four lines of the quadrilateral, a · b and c · d, a · c and b · d, and a · d and b · c are said to be pairs of opposite vertices. The lines joining pairs of opposite vertices are called diagonal lines.

We see in figure 6 that the diagonal lines are not concurrent and this can be proven.

Theorem 2.2 (Dual of Axiom 4) The three diagonal lines of a complete quadrilateral are never concurrent.

Proof. Let abcd be an arbitrary complete quadrilateral. See figure 6 in page 13, let points A = a · c, B = a · d, C = c · b, D = b · d, E = a · b, and F = c · d. Then the diagonal lines are EF , BC, AD and assume they are concurrent. The diagonal points of quadrangle ABCD are E,F,G. Point G is on lines BC and AD, these two lines cross on EF (the three lines where assumed to be concurrent), so point G is on line EF . This is a contradiction since according to axiom 4 the three diagonal points are not collinear. Therefore the diagonal lines EF , BC, AG are never concurrent.

12 Figure 6: The diagonal lines EF , BC, AD of a complete quadrilateral abcd.

Finally the dual of the Desargues’ theorem can be proved.

Theorem 2.3 (Dual of Axiom 5) If two triangles are perspective from a line then they are perspective from a point.

Proof. See figure 7 in page 14 to follow the proof. Assume 4ABC and 4A0B0C0 are perspective from a line l, let AC · A0C0 = R, BC · B0C0 = Q, AB · A0B0 = P . To prove the two triangles are perspective form a point, we need to show lines AA0,BB0, and CC0 are concurrent. Let O = AA0 · BB0, then we have a look at 4RAA0 and 4QBB0. It is easy to see lines AB, A0B0 and RQ are concurrent and intersect at the point P . So we can say 4RAA0 and 4QBB0 are perspective from a point. By using Axiom 5 we should get 4RAA0 and 4QBB0 are perspective from a line. Hence we get points AR · BQ = C, A0R · B0Q = C0, AA0 · BB0 = O and they are collinear. So 4ABC and 4A0B0C0 are indeed perspective from the point O.

2.3 An infinite model for projective plane

For a visualisation of the duality we need a model for the projective plane, see figure 8 in page 14. A point P in the plane α is represented by a

13 Figure 7: 4ABC and 4A0B0C0 are perspective from the line l. Points AC · A0C0 = R, BC · B0C0 = Q, AB · A0B0 = P all lie on line l. ray a through the origin O. The ray a is the normal vector to a plane through O which intersects the α-plane along a line l. To obtain the projective plane we add to α-plane the ideal points, represented by lines in the xz-plane. The ideal line corresponds then to the xz-plane. To each point in the projective plane corresponds a line and vice versa.

Figure 8: A projective plane model in 3D.

So far this section have focused on the axiomatic system, the following section will discuss a set of important points.

14 3 Harmonic sets

This section is going to introduce a particular set of four collinear points (and a dual set of four concurrent lines) which is defined in terms of a quadrangle. This four points have a special relation and it turns out that this relation remains after a so called projective transformation, which will be the subject of section 4.

Definition 3.1 Four collinear points, A, B, C, D are said to form a harmonic set H(AB, CD) if there is a complete quadrangle EF GH in which two opposite sides FE and GH pass through point A, two other opposite sides EH and FG pass through B, while the remaining side EG pass through point C and FH pass though point D, respectively. C is called the harmonic conjugate of D (or D is the harmonic conjugate of C) with respect to A and B. (figure 9)

Figure 9: Harmonic set H(AB, CD).

See figure 9, following the definition, with the quadrangle EF GH we can have another harmonic set H(BA, CD) since points A and B both are the intersection of two opposite sides of a quadrangle. Also, we have H(BA, DC) from the same quadrangle since the two points C and D are both the points of the remaining side.

So the quadrangle have at least four related harmonic sets which mean: H(AB, CD) ⇔ H(BA, CD) ⇔ H(AB, DC) ⇔ H(BA, DC) Note: The symbol ⇔ means assume ... then we have ....

We have six sides of the quadrangle so normally we get six intersec- tions with an arbitrary line in the plane. If the line goes through one

15 diagonal point we get five intersections (figure 10). We use the symbol (AA0)(BB0)(CD) to denote the six intersection points A, A0,B,B0,C,D. In this case, we call it a quadrangular set (figure 11). If the line goes through two diagonal points it is a harmonic set and they are particular cases of quadrangular sets. H(AB, CD) can also be written as (AA)(BB)(CD).

Figure 10: Five collinear points form a quadrangular set (AA0)(BB)(CD).

Figure 11: Six collinear points form a quadrangular set (AA0)(BB0)(CD).

3.1 Construction of the fourth point of a harmonic set

Focusing on the quadrangle AEBH (see figure 12 in page 17) we can find the three diagonal points F , G and D and by Axiom 4 they are three distinct points which are never collinear. Points A and F are collinear, points B and

16 G are collinear and points C, G, F are collinear, hence point D is distinct from point A, B and C. Is point D unique if we have points A, B and C? The answer is yes and we set the uniqueness of point D as a theorem.

Figure 12: Unique point D for harmonic set H(AB, CD).

Theorem 3.1 If A.B and C are three distinct, collinear points, then D, the harmonic con- jugate of C with respect to A and B, is unique.

Proof. Let EF GH and E0F 0G0H0 be two quadrangles which have the same A, B, C then find their conjugate points D and D0 (see figure 12). This means E0F 0 · G0H0 = A, F 0H0 · E0G0 = B, and F 0G0 · AB = C. Let E0H0 · AB = D0. We need to show D = D0. To do this, we need Axiom 5 and its dual. By the dual of Axiom 5 4EFG and 4E0F 0G0 are perspective from a point since they are perspective from a line AB. Also we can say EE0, FF 0 and GG0 are concurrent. Similarly 4F GH and 4F 0G0H0 are perspective from AB, hence FF 0, HH0 and GG0 are concurrent. Therefore EE0, FF 0, HH0 and GG0 are all concurrent. By Axiom 5, 4EHG and 4E0H0G0 are per- spective from a line AB since they are perspective from a point. We get the following relationship EH · E0H0, EG · E0G0=B, and HG · H0G0=A are collinear. Since EH · AB=D and E0H0 · AB=D0, we get D=D0.

17 If the point D is the harmonic conjugate to C with respect to A and B it turns out that the point A is conjugate to B with respect to C and D. This is the replaceability of a harmonic set.

Theorem 3.2 H(AB, CD) ⇔ H(CD, AB).

Proof. Assume H(AB, CD), then there is a quadrangle EF GH. Set n=AB, we get EG·n=C, FH ·n=D, EF ·GH=A and EH ·FG=B. Let DG·FC=S and GE · FH=T . Then consider the quadrangle T GSF . See figure 13. The question is now if ST goes through point A? Think about 4THE and 4SGF , we can find points TE · SF =C, TH · SG=D and HE · GF =B all lie on line AB, so they are perspective from a line; by the dual of Axiom 5, they should perspective from a point. We can find that the point is A. It means A is incident with TS. Therefore H(CD, AB).

Figure 13: H(AB, CD) ⇔ H(CD, AB).

Summarize it then we get a corollary.

Corollary 3.2.1 H(AB, CD) ⇔ H(AB, DC) ⇔ H(BA, CD) ⇔ H(BA, DC) ⇔ H(CD, AB) ⇔ H(CD,BA) ⇔ H(DC, AB) ⇔ H(DC, AB)

18 3.2 Construction of the fourth line of a harmonic set

As we said in Section 2, the dual of a theorem is also a theorem. We give here the definition of the dual of fourth point of a harmonic set. The proofs of the dual theorems will be similar to the previous proofs.

Definition 3.2 See figure 14. Four concurrent lines, a, b, c, d are said to form a harmonic set H(ab, cd) if there is a complete quadrilateral in which two opposite vertices A and B lie on a, two other opposite vertices C and E lie on b, while the remaining two vertices D and F lie on lines c and d, respectively. The quadrilateral AD, AE, BC, BD forms H(ab, cd).

Figure 14: H(ab, cd).

Then we would also have the uniqueness and replaceability of H(ab, cd). The two theorems are presented below, and the proofs are similar with the previous theorems.

Theorem 3.3 If a, b and c are three distinct concurrent lines, then d, the harmonic con- jugate of c with respect to a and b, is unique.

Theorem 3.4 H(ab, cd) ⇔ H(cd, ab)

From the theorems in this section we got H(AB, CD) ⇔ H(CD, AB) and H(ab, cd) ⇔ H(cd, ab). Are there any relation H(AB, CD) ⇔ H(ab, cd)?

19 Yes if we have a special relation, a so called perspectivity, between the lines and the points. This is the subject of the next section.

20 4 Perspectivities and Projectivities

In this section we will talk about mappings from a set of points to a set of points, a set of lines to a set of lines, a set of points to a set of lines and vice versa. We learn how to use points and lines constructions to obtain correspondences synthetically.

There are two relationships to describe mappings. Before we learn the relationships, we need to know two basic concepts. They are presented as definitions with figures.

Definition 4.1 The set of all lines through a point P is called a pencil of lines with center P.(figure 15)

Definition 4.2 The set of all points on a line p is called a pencil of points with axis p.(figure 16)

Figure 15: Pencil of lines. Figure 16: Pencil of points.

With the two basic definitions, one relationship of the mappings be- tween pencils (the pencil of points, the pencil of lines and the pencil of points and lines) is known as a perspectivity. We show the definition of perspectivity in the following and present some corresponding elementary mappings by figures.

00 00 We use the symbol Z to denote this mapping relationship which is defined as perspectivity related.

21 4.1 Perspectivities

Definition 4.3 A one-to-one mapping between two pencils is called a perspectivity.

Definition 4.3.1 A one-to-one mapping between two pencils of points with axis p and p0 is called a perspectivity if each line joining the point A on p with the corre- sponding point A0 on p0 is incident with a fixed point O. O is called the center O 0 of the perspectivity. We can use A ∧ A to denote this perspectivity.(figure 17)

Figure 17: Perspectivity between pencils of points.

Definition 4.3.2 A one-to-one mapping between two pencils of lines with centers P and P 0 is called a perspectivity if each point of intersection of the corresponding lines a on P and a0 on P 0 lies on a fixed line o. o is called the axis of the per- o 0 spectivity. We can use a ∧ a to denote this perspectivity. (figure 18, page 23)

Definition 4.3.3 A one-to-one mapping between a pencil of points with axis p and a pencil of lines with center P is called a perspectivity if each point A on p is incident with the corresponding line a0 on P . 0 0 We can use A Z a or a Z A to denote the perspectivity. (figure 19, page 23)

0 Notice definition 4.3.3 includes two types of perspectivity, one is A Z a , 0 0 another is a Z A. We use A Z a to denote the perspectivity with a pencil of

22 Figure 18: Perspectivity between pencils of lines.

Figure 19: Perspectivity between a pencil of points and a pencil of lines.

0 points first, and then a pencil of lines; a Z A denotes the perspectivity with a pencil of lines first and a pencil of points second. Since perspectivities are one-to-one mappings, the inverses are also perspectivities.

4.2 Projectivities

Here we introduce another relationship of the mappings between the pencils.

Definition 4.4 A one-to-one mapping between the elements of two pencils is called a pro- jectivity if it consists of a finite product of perspectivities.

When a projectivity exists between two pencils, the pencils are said to be projectively related, we use the symbol 00∧00 to denote it.

In figure 20 and figure 21 in page 24 we show the mappings in 3D as an illustration of the difference between a perspectivity and a projectivity. So perspectivity is a special case of projectivity.

23 Figure 20: A perspectivity be- Figure 21: A projectivity between tween pencils of points. pencils of points.

Next we present figures of a projectivity between the pencils of points (figure 22), the pencils of lines (figure 23) and a pencil of points and a pencil of lines (figure 24 in page 25).

Figure 22: ABC ∧ A00B00C00.

Figure 23: abc ∧ a00b00c00.

If a projectivity maps either a pencil of points or a pencil of lines onto itself, it is called a projectivity on the pencil. If a point (line) goes to the same point (line), we say it is an invariant point (line). If the mapping relates each point to itself, it is the identity mapping.

24 Figure 24: abc ∧ A00B00C00.

After the introduction of projectivity, we can give the final axiom which describes an important property of projectivities on pencils. We formulate it for points, but it can also be formulated for a pencil of lines.

Axiom 6 If a projectivity leaves invariant each of three distinct points on a line, it leaves invariant every point on the line.

With help of the Axiom 6, we can prove the fundamental theorem in projective geometry.

Theorem 4.1 (fundamental theorem) A projectivity between two pencils is uniquely determined by three pairs of corresponding elements.

Proof. For the construction of the projectivity, see section 4.3. The uniqueness follows from Axiom 6 in the following way. Assume there exists a mapping T which we denote ABC ∧ A0B0C0 and another mapping S also denoted ABC ∧ A0B0C0. Then TS−1 maps ABC ∧ ABC. Axiom 6 tells us it is the identity, TS−1 = I, so mapping S = T . The proofs for the other two projectivities a done in the same way.

Corollary 4.1.1 If in a projectivity between two distinct pencils an element corresponds to itself, then the projectivity is a perspectivity (i.e., the mapping requires only one perspectivity).

25 Proof. Take two distinct lines p and p0 with a common point P where points A and B lie on p and points A0 and B0 lie on p0. If P is an invariant point we have ABP ∧ A0B0P . We can take O = AB0 · A0B as the center of the perspectivity. The proofs for the other two projectivities a done in the same way.

After we have introduced the sixth axiom and the fundamental theorem, we now turn to constructions of projectivities.

4.3 Construction of a projectivity between pencils of points

Here we present a theorem which is about constructing the images under a projectivity between pencils of points. This theorem is also almost showing the Pappus’ theorem. We present the theorem with a definition and a figure.

Definition 4.5 If A and A0, B and B0 are pairs of corresponding points, the cross joins of those pairs are the lines AB0 and BA0.

The corresponding theorem is shown below.

Theorem 4.2 A projectivity between two distinct pencils of points determines a unique line called the axis of projectivity, which contains the intersections of the cross joints of all pairs of corresponding points.

Proof. Check figure 25 in page 27 to follow the proof. Consider two distinct pencils of points with axes p and p0. Assume ABC ∧ A0B0C0. Set point P = p · p0 which is none of the six points. Setting A0 as the center, we have a perspectivity between points A, B, C and lines A0A, A0B,A0C; then change the center to A, we have another perspectivity between points A0,B0,C0 and lines AA0, AB0, AC0. Hence we have a projectivity between lines A0A, A0B,A0C and lines AA0, AB0, AC0. According to fundamental theorem (theorem 4.1) and corollary 4.1.1, there is a perspectivity between lines AA0, AB0, AC0 and lines A0A, A0B,A0C with

26 a axis of the perspectivity. Let point F = AB0 · BA0 and G = AC0 · CA0 then we have the axis h which is determined by points F and G. Thus, by axis h we can find the mapping of another distinct point D. Let E = A0D ·h then the mapping of point D is D0 = AE ·p0. Next we are going to prove that the line is unique. To prove that the line h is unique, we need to show that the line h is independent of the choices for the centers of pencils. Set R = p · h and 0 A0 A Q = p · h, the image of P is Q since P ∧ Q and Q ∧ Q; the pre image of A0 A P is R since R ∧ R and R ∧ P . According to theorem 4.1, the image is unique. Then we can say points R and Q are independent of the choices for the centers of pencils, and the two points determine the axis h. Therefore, points R and Q determine a unique line h.

Figure 25: Axis of projectivity h for ABC ∧ A0B0C0.

From theorem 4.2, we can prove one of the most well-known theorem which is Pappus’s theorem. However, first we need a new definition.

Definition 4.6

A hexagon is given by a set of six distinct points called vertices, they are P1,

P2, P3, P4, P5 and P6. Lines P1P2, P2P3, P3P4, P4P5, P5P6 and P6P1 are called the sides of the hexagon P1P2P3P4P5P6. Points P1 and P4, P2 and

P5, P3 and P6 are pairs of opposite vertices. Lines P1P2 and P4P5, P2P3 and P5P6, P3P4 and P6P1 are pairs of opposite sides. The three points A, B and C which are the intersection of opposite sides are called diagonal points. (figure 26, page 28)

27 Figure 26: A hexagon P1P2P3P4P5P6.

Theorem 4.3 (Pappus’s theorem) If alternate vertices of a hexagon lie on two lines, then the diagonal points are collinear.

The proof of Pappus’s theorem is similar to the proof of theorem 4.2 which refer to the hexagon AB0CA0BC0A.

Axiom 6 and the fundamental theorem also show that a projective trans- formation of a line into itself can not have more than two invariant points without being the identity. The elliptic projectivity have no invariant point, the parabolic projectivity have one invariant point and the hyperbolic pro- jectivity have two invariant points. We will show these three constructions now. Question 1 How is the elliptic projectivity looking like? Answer 1 For an elliptic projectivity three perspectivities are needed. An example, ABC ∧ BCA, is given in figure 27 in page 29. First we do the perspectivity O 0 0 0 ABC ∧ A B C , then we put B = A1, C = B1 and A = C1. The second 0 0 0 perspectivity has A1 as the center and maps the points A ,B ,C on the line h. Finally we take A0 as the center and maps the line h to line AC. Question 2 How do we construct projectivities with one or two invariant points? Answer 2 Assume we have a quadrangular set (AD)(BE)(FC), see figure 28 in page

28 Figure 27: ABC ∧ BCA

29, there exists a quadrangle QRSP such that A = p · SP , B = p · SQ, C = p · SR. D = p · QR, E = p · RP , F = p · QP where p = AB. The P Q projectivity AEC ∧ SRC ∧ BDC leaves C invariant. Let G = SR · PQ. It is P Q clear that point F is also an invariant point since F ∧ G ∧ F .

The projectivity AECF Z BDCF is hyperbolic if C and F are distinct.

Figure 28: Invariant points C,F .

The projectivity AEC Z BDC is parabolic if C and F are coincident, see figure 29 in page 30.

29 Figure 29: Only one invariant point C.

30 5 Point and line conics

The following part of this thesis describes the construction of conics in greater detail by using a different style, compared to previous sections, no rigorous style with definitions, theorems and proofs anymore. A table will present the concepts and the point and line constructions will be exhibited by figures [3]. According to the axiomatic system: two points determine a line, three points determine a triangle and four points determine a quadrangle. Five points determine a point conic by a projectivity of pencils of lines. What can be clearly seen in table 1 is the characteristics of a point conic and its dual.

Point conic Line conic

A point conic is the set of points of A line conic is the set of lines intersection of corresponding lines joining corresponding points of two of two projectively, but not per- projectively, but not perspectively, spectively, related pencils of lines related pencils of points with dis- with distinct centers U and U 0. tinct axes u and u0. These lines These points and the lines p and and the points P and Q and the q and the center F determine the line f determine the projectivity. projectivity. A to a point conic is a line A point of contact of a line conic that has exactly one point in com- is a point that lies on exactly one mon with the point conic. line of the line conic.

Table 1: Point and line conic.

The figure 30 in page 32 shows that a point conic is determined by five points. Points U and U 0 are taken as the centers for the pencil of lines, take also three other points A, B and C in the plane, no three of the five chosen points can be collinear. We can now construct the two lines p = AB and q = AC; and the point F = UB · U 0C. The two lines p and q and the point F are now used for the projectivity of the pencils in the following way. A ray from U will meet the line q at a point K, this point is mapped by a perspectivity with F to a point L on p. The final perspectivity gives the line U 0L. The intersection UK · U 0L gives a point X on the point conic.

31 Figure 30: Five points U, U 0, A, B and C generate a point conic.

Using this construction we realize that the five points U, U 0, A, B, and C also belong to the point conic. For example, sending a ray from point U towards point U 0 will intersect with the corresponding line at point U 0. The name point conic reminds us about the well known conics, the ellipse, the parabola and the hyperbola and it turns out it is exactly these three that can be generated by this construction. To show this, coordinates must be used. From now on we often drop the name point conic and simplify say conic.

It is obvious from the construction in figure 30 that the three points K,F,L lie on a line. This observation was first made by (1623- 1662) and he formulated Pascal’s theorem. It tells us that the diagonal points of a hexagon which is inscribed in a point conic are collinear. The theorem is almost directly seen from the figure. The line which is determined by three diagonal points is called Pascal’s line.

Figure 31: Lines from S and R are projectively related.

For full understanding of Pascal’s theorem let us take three new points, S, Y and R on the point conic constructed with centers U and U 0 above. The three points U, U 0 and X are the same as in figure 30. Let us now

32 investigate if the diagonal points W , F , Z of the hexagon UXU 0SYRU are collinear?

Let M = UR ·SY and N = U 0S ·YR, if we set U as the center, we have a perspectivity between points S, W, M, Y and lines US,UX,UR,UY . Then change the center to U 0, we have a projectivity between lines US,UX,UR,UY and lines U 0S, U 0X,U 0R,U 0Y . Next we set line YR as the axis and we have a perspectivity between lines U 0S, U 0X,U 0R,U 0Y and points N,Z,R,Y . Hence we get a projectivity between points S, W, M, Y and points N,Z,R,Y . Since point Y goes to itself, Y is an invariant point. According to corol- lary 4.1.1, SWM ∧ NZR is a perspectivity! Then we can define the center F = SN · MR. Therefore the diagonal points W, F, Z lie also in this case on a straight line. This shows that there is nothing special with centers U and U 0, we can equally well use S and R as centers and we still have a projectivity between the pencils of lines.

Sometimes the Pascal’s line can lie outside the conic as in figure 32.

Figure 32: The Pascal’s line is outside of the ellipse.

As indicated previously, the parabola and the hyperbola can also be generated by the same construction as in figure 30. Figure 33 and figure 34 in page 34 are two illustrations for parabola and hyperbola, line DFE is the Pascal’s line.

After the introduction of the point conic construction, the dual con- struction will be presented. Figure 35 in page 34 reveals that lines u and u0 are taken as the lines for the pencil of points. Take also three other lines a, b and c in the plane, no three of the five chosen lines can be concurrent. We can now construct the two points P = a·c, Q = a·b and line f is determined by points b · u0 and c · u. The two points P and Q and the line f are now used for the projectivity of the pencils in the following way. Take a point A on u0, the ray from point P through A will meet the line f at a point F ,

33 Figure 33: Construction of a Figure 34: Construction of a hy- parabola. perbola. the intersection point of QF and u gives a point B on the line u. In this way we get a new line x = AB of the line conic. From this construction we can realize that the joins between opposite vertices of the hexagon u0xucabu0 goes through a point F in Figure 35. This point is called Brianchon’s point.

Figure 35: Five lines u, u0, a, b, and c generate a line conic.

In figure 36 and figure 37 in page 35, a hexagon 1234561 is circumscribed around a , those lines connecting opposite vertices (12 and 45, 23 and 56, 34 and 61) meet in a single point (Brianchon’s point). This observation was first made by Charles Julien Brianchon (1783-1864) and he formulated it as a theorem: Brianchon’s theorem.

34 Figure 36: The Brianchon’s point Figure 37: The Brianchon’s point B is inside of the ellipse. B is outside of the ellipse.

35 6 Conclusion

Overall, this study explained two famous theorems in the axiomatic system, Pappus’ theorem and Pascal’s theorem. Also a lot of points and lines constructions were shown. The aim of the present research was to understand the basics of projective geometry. A limitation in this thesis is that only 2D is considered and focus on the constructions alone without coordinates.

The study of projective geometry could now go in another three direc- tions: finite geometry, analytical geometry (with coordinates) or computer graphics. And one modern application of projection is pictures dis- played on a screen.

References

[1] Judith N. Cederberg, 2001, A Course in Modern Geometries, Springer.

[2] H.S.M, Coxeter, 1991, Projective Geometry, Springer.

[3] Bengt Ulin, 2000, Projective Geometry, Ekelunds f¨orlag.

[4] Dan Pedoe, 1976, Geometry and the Visual Arts, Dover edition.

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