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Affine and Projective Geometry Affine and Projective Geometry Adrien Deloro Şirince ’17 Summer School Contents Table of Contents ii List of Lectures iii Introduction1 Week I: Affine and Projective Planes2 I.1 Affine planes ............................. 3 I.1.1 The Definition ........................ 3 I.1.2 Membership as Incidence .................. 3 I.1.3 The Affine Plane over a Skew-field............. 4 I.2 Projective planes........................... 7 I.2.1 The need for Projective planes ............... 7 I.2.2 The Definition ........................ 8 I.2.3 From Affine to Projective, and Back............ 9 I.3 Arguesian Planes........................... 12 I.3.1 Desargues’s Projective Theorem .............. 12 I.3.2 Free Planes and Non-Arguesian Planes........... 15 I.3.3 Pappus’ Theorem....................... 17 I.3.4 Pappus Implies Desargues.................. 19 I.4 Coordinatising Arguesian Planes Explicitly ............ 21 I.4.1 The Frame .......................... 22 I.4.2 Addition............................ 23 I.4.3 Multiplication......................... 26 I.5 Duality ................................ 28 I.5.1 One Finitary Aspect..................... 28 I.5.2 Duality ............................ 30 I.5.3 The Dual Plane........................ 31 I.5.4 Dualising Desargues and Pappus.............. 32 Week II: Group-Theoretic Aspects 34 II.1 Arguesianity and the Group of Dilations.............. 34 II.1.1 Dilations and Translations.................. 34 II.1.2 The (p,p)-Desargues Property................ 37 II.1.3 The (c,p)-Desargues Property................ 39 II.2 A Vector Space from an Affine Plane................ 40 II.2.1 A Ring ............................ 40 II.2.2 A Skew-Field......................... 41 II.2.3 Dimensionality........................ 42 i II.3 An Application: SO(3,R) ...................... 44 II.3.1 The Geometry and Non-Degeneracy Axioms . 44 II.3.2 The First Axiom....................... 45 II.3.3 The Second Axiom...................... 46 II.3.4 The Third and Fourth Axioms ............... 47 II.4 Projectivities ............................. 48 II.4.1 Projectivities......................... 48 II.4.2 Group-theoretic Interpretation of Pappus’ Theorem . 49 ii List of Lectures Week I Lecture 1 (Introduction; Affine Planes).................. 2 Lecture 2 (Projective Planes)....................... 7 Lecture 3 (Desargues’ Theorem) ..................... 11 Lecture 4 (Pappus’ Theorem)....................... 17 Lecture 5 (Coordinates).......................... 21 Lecture 6 (Duality)............................. 28 Week II Lecture 7 (Coordinatising like adult people (1))............. 34 Lecture 8 (Coordinatising like adult people (2))............. 40 Lecture 9 (The Geometry of SO3(R))................... 43 Lecture 10 (Projectivities)......................... 48 iii Introduction If parallel lines meet at infinity, then all trains for the infinite are running to a disaster. Kryszanowski Prerequisites In order to follow this class, one needs to know: • high-school geometry: in the plane, using coordinates, vectors, equations of lines; • vector spaces, linear maps, matrices; • fields: essentially the definition; • groups (subgroups, normal subgroups, . ) and group actions (stabilisers, orbits, . ) will play a prominent role in week 2. Recommended Reading [1] Artin, Emil. Geometric Algebra. Only Chapter 2. [2] Cameron, Peter. Projective Planes and Spaces. Mostly Chapters 1, 2, 3. Notes available online on the author’s webpage. [3] Hartshorne, Robin. Foundations of Projective Geometry. Technically out of print but on the internet one finds a modern typeset. [4] Hilbert, David. Foundations of Geometry. Only Chapter 5 on Desargues’ Theorem. 1 Chapter I: Affine and Projective Planes Lecture 1 (Introduction; Affine Planes) The Quartet and the Orchestra. We begin with an analogy. Beethoven’s Ninth Symphony (op. 125) uses an extensive orchestra and a choir. After com- pletion Beethoven quickly returned to composing string quartets (op. 130–132 and 135), which are no less praised by amateurs than the symphony. The dis- crepancy of means and effects between the monumentality of the massive orches- tration on the one hand, and the elegant economy of just four instruments on the other, is significant. None is “better” music than the other: the symphony targets universality, whereas the quartets aim at spirituality and purity. And mathematics is like music. There are various layers of mathematical structures, from elementary aspects to the whole orchestra of modern mathe- matics. And none is better for the true mathematician will enjoy both. For instance when studying the real plane R2 we have very different notions and tools: • points and lines; • coordinates and vectors; • distances; • angles; the dot product; • the underlying complex multiplication. In this class we shall however restrict ourselves to the most basic layer where one has only points and lines. Painting from Nature. This leads to the notion of an abstract affine plane. It is just a collection of points and lines with the possibility for some points to be on some lines or not. But of course for this to be interesting one must start with intuition, or physical observation in our small part of the universe (a sheet of paper can do), in order to find axioms which describe the behaviour. 2 I.1. Affine planes I.1.1. The Definition Definition (affine plane). An affine plane A is given by an incidence geometry, i.e.: • a set of points P; • a set of lines L; • a relation I (for “incidence”) on P × L, which says when a point is on a line or not; subject to the following axioms AP1, AP2, AP3 listed hereafter (we shall have to pause for auxiliary definitions). 2 AP1. ∀(a, b) ∈ P a 6= b → (∃!` ∈ L aI` ∧ bI`). (The meaning is clear: given any two distinct points, there is a unique line incident to both. It is convenient to call this line (ab).) The second axiom requires a definition. Definition (parallel). Two lines `, m are parallel if either ` = m or there is no point incident to both: ` k m iff (` = m) ∨ (∀a ∈ P¬(aI` ∧ aIm)) AP2. ∀(a, `) ∈ P × L ∃!m ∈ L (aIm) ∧ (` k m). (Intuitive meaning: through any point there is a unique line parallel to a given line. It is convenient to call this line `a.) Definition (collinear). Points a1, . , an are collinear if there is a line ` such that each akI`. AP3. There exist three non-collinear points. Axioms AP1, AP2, AP3 define affine planes. Notice that we speak only about points and lines; words such as angle, distance, or even between are banned from our vocabulary. Exercise. AP3 is added to prevent degenerate configurations, the most ex- treme of which being P = ∅ = L. List all configurations satisfying AP1 and AP2 but not AP3. I.1.2. Membership as Incidence Notice that we are using an abstract relation I; however we are used to thinking that a point belongs to a line, in the set-theoretic sense. Indeed, any not-too- small affine plane can be replaced by a similar one where the incidence relation is the usual membership. We need to give a precise meaning to “similar”. 3 Definition (isomorphism). Two incidence geometries (P1, L1,I1) and (P2, L2,I2) are isomorphic if there are bijections f : P1 → P2 and g : L1 → L2 such that: ∀(a, `) ∈ P1 × L1 aI1`b ⇔ f(a)I2g(`) Proposition. Let A = (P, L,I) be an affine plane where each line is incident to at least two points. Then there is an affine plane A0 = (P0, L0, ∈) (where ∈ is the usual membership relation) which is isomorphic to A. Proof. The idea is to keep the same points, and write new lines as sets of points. To each ` ∈ L associate S` = {a ∈ P : aI`}, a subset of P. Now 0 consider L = {S` : ` ∈ L}, a family of subsets of P. Finally let f be the 0 identity function on P and g : L → L map ` to S`. We claim that (P, L,I) and (P0, L0, ∈) are isomorphic via f and g. Clearly f is a bijection; g is clearly onto but it also is one-to-one. For if S` = Sm then since there are at least two distinct points a, b ∈ P incident to `, one has a, b ∈ S` = Sm, so a and b are also incident to m. But by AP1, one must have ` = m. So g is injective as well. Now for any (a, `) ∈ P × L, one has aI` iff a ∈ S`, which is the definition of an isomorphism. From now on we shall leave aside the case where there are lines with only one (or even zero) point; such lines should be removed from L and simply forgotten. So we are free to consider that ∈ will always be the incidence relation. As a consequence, instead of always saying that a is incident to ` we can use the common phrases: “a is on `”, “a belongs to `”, “` contains a”. I.1.3. The Affine Plane over a Skew-field We wrote the definition of an affine plane by painting from nature. It is time to check that the usual plane R2 is indeed an affine plane in our abstract sense. Proposition. R2 with the usual points and lines is an affine plane. Proof. Points are pairs of real numbers (x, y) ∈ R2; lines are sets of the form −→ `p,−→v = {p + t v : t ∈ R} 2 −→ −→ where p ∈ R and v 6= 0 . Notice that one always has p ∈ `p,−→v . We must check axioms AP1, AP2, AP3; we shall actually give the proof as it will reveal something of interest. −→ −→ Claim 1. Two lines `1 = ` −→ and `2 = ` −→ are equal iff v1 and v2 are −−→p1,v1 p2,v2 proportional vectors, and p1p2 is a multiple of those. Verification. First suppose `1 = `2. Since p2 ∈ `1 there is t ∈ R with p2 = −→ −−→ −→ −→ p1 + tv1. So p1p2 = tv1 is a multiple of v1. But also p1 ∈ `2 so there is s ∈ R −→ with p1 = p2 + sv2; now −→ −→ −→ p1 = p2 + sv2 = p1 + tv1 + sv2 −→ which simplifies into t−→v + s−→v = 0 . Since we know that neither −→v nor −→v is −→ 1 2 1 2 0 , we deduce that they are proportional. 4 Conversely we shall prove one inclusion, and the other will follow by sym- −→ metry.
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