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A Perspective on Conics in the Real Projective Plane

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A Perspective on Conics in the Real Projective Plane A Perspective on Conics in the Real Projective Plane by Joseph D. Brown III (Under the direction of Theodore Shifrin) Abstract In this paper we will introduce the reader to projective geometry and what it means for two figures to be projectively equivalent. This leads to a discussion on the projective generation of conics. Special attention is given to the relationship between the projective group and conics, as well as the concept of projective duality. Index words: Projective Group, Perspectivities, Projective Plane, Conics A Perspective on Conics in the Real Projective Plane by Joseph D. Brown III B.S., The University of Georgia, 2006 B.S.Ed., The University of Georgia, 2006 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Master of Arts Athens, Georgia 2009 c 2009 Joseph D. Brown III All Rights Reserved A Perspective on Conics in the Real Projective Plane by Joseph D. Brown III Approved: Major Professor: Theodore Shifrin Committee: Edward Azoff Malcolm Adams Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia May 2009 Dedication To my parents David and Susan, my life partner and best friend Justin, my friends and teachers. iv Acknowledgments I am indebted to my advisor, Ted Shifrin, for his support, guidance, and friendship over the years. In particular, I would like to thank him for introducing me to the beautiful subject of projective geometry. His contributions have been invaluable to my studies and growth as a scholar. I would like to thank the other members of my committee: Ed Azoff for his encouragement throughout my graduate school career; Malcolm Adams for advising me early on, and for his contributions to this research project. Thanks to Justin for continuing to encourage me to work harder, Jen for always being willing to listen and help, and Jon for keeping me sane. I would also like to recognize Uncle Keith for providing me with a place to escape and relax whenever I needed it. v Table of Contents Page Acknowledgments . v List of Figures . vii Chapter 1 Introduction . 1 2 Preliminaries . 3 2.1 Introduction to projective n-space . 3 2.2 Perspectivities . 19 2.3 The theorems of Desargues and Pappus . 25 2.4 Projective duality . 31 3 Conics in the projective plane . 38 3.1 Projective generation of the conic . 39 3.2 The theorems of Pascal and Brianchon . 44 3.3 Quadrics and embedding R2 conics in P2 ............ 47 3.4 The dual conic C∗ ......................... 54 3.5 Classification of conics . 58 Bibliography . 61 vi List of Figures 2.1 Any line passing through the origin is identified by a second point (a, b) lying on the line. 4 2.2 P1 is homeomorphic to the unit circle. 5 2.3 Stereographic projection . 6 2.4 Antipodal points of a sphere are identified . 8 2.5 D with antipodal points on the boundary identified. 9 2.6 P2 is a M¨obius strip with a disk attached to its boundary . 10 2.7 Plane in R3 spanned by p and q ......................... 11 1 2.8 Parallel lines intersect at the line at infinity P∞. 15 2.9 The projection π with center P .......................... 19 2.10 The perspectivity f : ` → m with center P .................... 20 2.11 Cross-ratio . 22 2.12 Euclidean perspective of the cross-ratio . 23 2.13 Composition of perspectivities . 24 2.14 Special case of Desargues’ theorem . 26 2.15 Desargues’ theorem . 27 2.16 Special case of Pappus’ theorem . 28 2.17 Pappus’ theorem . 29 2.18 Dually, points correspond to lines and a line is a pencil of lines through a point. 32 2.19 Concurrent lines translate dually to collinear points. 33 2.20 Example from page 13 with dual. 34 vii 2.21 Pappus’ dual theorem . 36 2 2∗ 3.1 The pencils λP and λQ in P and in P ..................... 39 3.2 The intersection point X[s,t] of `[s,t] and m[s,t]. ................. 40 3.3 Conic interpretation of cross-ratio . 44 3.4 Pascal’s Theorem . 45 3.5 Brianchon’s Theorem . 46 2 2 2 3.6 Compare 6x1x2 + 8x2 = 1 with 9y1 − y2 =1................... 49 1 3.7 The intersection of E and P∞ is empty . 53 1 3.8 The intersection of P and P∞ is [0, 0, 1] . 54 1 3.9 The intersection of H and P∞ is {[0, a, b], [0, −a, b]} . 55 3.10 The dual conic C∗. ................................ 56 3.11 Comparing the conic C with its dual C∗. .................... 58 3.12 The dual conic C∗ ................................. 59 3.13 The dual of the composition of perspectivities. 60 viii Chapter 1 Introduction In the eighteenth century a German philosopher by the name of Immanuel Kant stated that geometry is the study of the properties of the physical space that we live in. However, after Kant’s death in 1804, Klein, Gauss, Lobachevski, and Bolyai proved that he could not be more wrong with the development of non-Euclidean geometries. In particular, Klein redefined geometry to consist of sets of objects and relations among them. Then, the relation-preserving mappings form a group of automorphisms. In this paper we will start by giving an introduction of a special type of non-Euclidean geometry called projective geometry. In this geometry, we study projections and properties that are invariant under a projection. We shall start our investigation of projective geometry with an algebraic approach. In particular, we will be working in P1 and P2. In the treatment of projective geometry we include some classic theorems as well as the concept of projective duality. The theorem which says any projective transformation is the composition of at most two perspectivities was motivated by exploration in Geometer’s Sketchpad. I used Geometer’s Sketchpad to prove this theorem by construction. The main reference for this chapter is Abstract Algebra: A Geometric Approach by Ted Shifrin. In the third chapter we direct the focus to projective conics and show the projective generation of conics. We describe the relationship between conics and projective transforma- tions. Along with reading and understanding the proofs, I did great deal of exploration on conics in Geometer’s Sketchpad. Geometer’s Sketchpad files are available upon request by sending me an email at [email protected]. We close this chapter with a treatment of 1 the dual conic and the question of classifying conics based on the geometry of the projective transformation. Main references for this chapter are Abstract Algebra: A Geometric Approach by Ted Shifrin, and Geometry: A Comprehensive Course by Dan Pedoe. 2 Chapter 2 Preliminaries 2.1 Introduction to projective n-space Definition 2.1.1. We define projective n-space, denoted Pn, to be the set of all lines through the origin in Rn+1. That is, if x, y ∈ Rn+1 − {0} then we say that x ≡ y if y = cx for some c ∈ R − {0}. Then Pn is the set of all ≡ equivalence classes. The topological structure of Pn is the quotient topology coming from (Rn+1 − {0})/ ≡. We check that ≡ is an equivalence relation. 1. x = 1 · x, so x ≡ x. Thus, ≡ is reflexive. 2. Suppose x ≡ y. Then there is some c ∈ R − {0} such that x = cy. So (1/c)x = y, with 1/c ∈ R − {0}. Hence y ≡ x and so ≡ is symmetric. 3. Suppose x ≡ y and y ≡ z. Then there are c, d ∈ R − {0} such that x = cy and y = dz. So x = c(dz) = (cd)z, with cd ∈ R − {0}. Therefore, x ≡ z and so ≡ is transitive. In the case of n = 1, we consider the set of all lines that pass through the origin in R2. Every line that passes through the origin can be identified by another point (a, b) lying on the line. Therefore, we describe any element of P1 by an equivalence class [(a, b)] or [a, b] for short. Notice that for a 6= 0, b/a is the slope of the line in R2. See Figure 2.1. Example 2.1.1. Consider the line ` ⊂ R2 given by the equation 4x + 2y = 0. The slope of this line is −4/2 = −2, and so this corresponds to the element [2, −4] ≡ [1, −2] ∈ P1. 3 Figure 2.1: Any line passing through the origin is identified by a second point (a, b) lying on the line. Example 2.1.2. Consider the line m ⊂ R2 given by the equation x = 0. This line has undefined or infinite slope and (0, 1) is a point on it. Therefore the line m is the element [0, 1] ∈ P1. We refer to this as the point at infinity. The slope of any line through the origin is either a real number or infinite. So we have the one-to-one correspondence between P1 and R1 ∪ {∞}. We remark here that often it will be convenient to think of P1 as R ∪ {∞}. Now we will briefly discuss the topology of P1. 1 (x0, x1) 2 For each element [x0, x1] ∈ , consider the unit vector u = ∈ . We have a P p 2 2 R x0 + x1 1 2 2 two-to-one correspondence between points on the unit circle S = {(x0, x1): x0 + x1 = 1} and P1, because vectors u and −u represent the same point in P1. Since each point in the lower semicircle is identified with a point in the upper semicircle, then we only consider the upper semicircle and its endpoints. Note, we still have these endpoints identified with each 4 Figure 2.2: P1 is homeomorphic to the unit circle.
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