Degree project Projective Geometry Author: Wu Wei Supervisor: Hans Frisk Examiner: Per Anders Svensson Course Code: 2MA41E Subject: Mathematics Level: Bachelor Department Of Mathematics Projective Geometry Wei Wu June 13, 2019 Abstract Projective geometry is a branch of mathematics which is founda- tionally based on an axiomatic system. In this thesis, six axioms for two-dimensional projective geometry are chosen to build the structure for proving some further results like Pappus' and Pascal's theorems. 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 duality 8 2.1 First five axioms for the projective plane . 9 2.2 Duality . 11 2.3 An infinite model for projective plane . 13 3 Harmonic sets 15 3.1 Construction of the fourth point of a harmonic set . 16 3.2 Construction of the fourth line 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 triangle. 9 4 The diagonal points E, F , G from a complete quadrangle ABCD. .............................. 10 5 Illustration of Desargues' theorem in 3D projective space. 11 6 The diagonal lines EF , BC, AD of a complete quadrilateral 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 Pencil 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 hexagon P1P2P3P4P5P6..................... 28 27 ABC ^ BCA ........................... 29 28 Invariant 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 ellipse. 33 33 Construction of a parabola. 34 34 Construction of a hyperbola. 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 shape 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 cylinder from above, it is a circle; but if we observe it from the side, it will be a rectangle. We are all familiar with Euclidean geometry that has one character- istic of pairs of lines: parallel 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 Euclidean space 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 cube (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 Euclid's Elements is the same as the geometry of lines and circles: 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, angles, 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, Archimedes and Apol- lonius studied conics in the fourth and third centuries B.C., the earliest projective theorems were discovered by Pappus of Alexandria 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, Felix Klein provided an algebraic foundation for projective geometry in terms of "homogeneous coordinates". 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 Axiom 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 dimension, 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.