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Projective Geometry Lecture Notes

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Projective Geometry Lecture Notes Thomas Baird March 26, 2014 Contents 1 Introduction 2 2 Vector Spaces and Projective Spaces 4 2.1 Vector spaces and their duals . 4 2.1.1 Fields . 4 2.1.2 Vector spaces and subspaces . 5 2.1.3 Matrices . 7 2.1.4 Dual vector spaces . 7 2.2 Projective spaces and homogeneous coordinates . 8 2.2.1 Visualizing projective space . 8 2.2.2 Homogeneous coordinates . 13 2.3 Linear subspaces . 13 2.3.1 Two points determine a line . 14 2.3.2 Two planar lines intersect at a point . 14 2.4 Projective transformations and the Erlangen Program . 15 2.4.1 Erlangen Program . 16 2.4.2 Projective versus linear . 17 2.4.3 Examples of projective transformations . 18 2.4.4 Direct sums . 19 2.4.5 General position . 20 2.4.6 The Cross-Ratio . 22 2.5 Classical Theorems . 23 2.5.1 Desargues' Theorem . 23 2.5.2 Pappus' Theorem . 24 2.6 Duality . 26 3 Quadrics and Conics 28 3.1 Affine algebraic sets . 28 3.2 Projective algebraic sets . 30 3.3 Bilinear and quadratic forms . 31 3.3.1 Quadratic forms . 33 3.3.2 Change of basis . 33 1 3.3.3 Digression on the Hessian . 36 3.4 Quadrics and conics . 37 3.5 Parametrization of the conic . 40 3.5.1 Rational parametrization of the circle . 42 3.6 Polars . 44 3.7 Linear subspaces of quadrics and ruled surfaces . 46 3.8 Pencils of quadrics and degeneration . 47 4 Exterior Algebras 52 4.1 Multilinear algebra . 52 4.2 The exterior algebra . 55 4.3 The Grassmanian and the Pl¨ucker embedding . 59 4.4 Decomposability of 2-vectors . 59 4.5 The Klein Quadric . 61 5 Extra details 64 6 The hyperbolic plane 65 7 Ideas for Projects 68 1 Introduction Projective geometry has its origins in Renaissance Italy, in the development of perspective in painting: the problem of capturing a 3-dimensional image on a 2-dimensional canvas. It is a familiar fact that objects appear smaller as the they get farther away, and that the apparent angle between straight lines depends on the vantage point of the observer. The more familiar Euclidean geometry is not well equipped to make sense of this, because in Euclidean geometry length and angle are well-defined, measurable quantities independent of the observer. Projective geometry provides a better framework for understanding how shapes change as perspective varies. The projective geometry most relevant to painting is called the real projective plane, and is denoted RP 2 or P (R3). Definition 1. The real projective plane, RP 2 = P (R3) is the set of 1-dimensional sub- spaces of R3. This definition is best motivated by a picture. Imagine an observer sitting at the origin in R3 looking out into 3-dimensional space. The 1-dimensional subspaces of R3 can be understood as lines of sight. If we now situate a (Euclidean) plane P that doesn't contain the origin, then each point in P determines unique sight line. Objects in (subsets of) R3 can now be \projected" onto the plane P , enabling us to transform a 3-dimensional scene into a 2-dimensional image. Since 1-dimensional lines translate into points on the plane, we call the 1-dimensional lines projective points. Of course, not every projective point corresponds to a point in P , because some 1- dimensional subspaces are parallel to P . Such points are called points at infinity. To 2 Figure 1: A 2D representation of a cube Figure 2: Projection onto a plane 3 motivate this terminology, consider a family of projective points that rotate from pro- jective points that intersect P to one that is parallel. The projection onto P becomes a family of points that diverges to infinity and then disappears. But as projective points they converge to a point at infinity. It is important to note that a projective point is only at infinity with respect to some Euclidean plane P . One of the characteristic features of projective geometry is that every distinct pair of projective lines in the projective plane intersect. This runs contrary to the parallel postulate in Euclidean geometry, which says that lines in the plane intersect except when they are parallel. We will see that two lines that appear parallel in a Euclidean plane will intersect at a point at infinity when they are considered as projective lines. As a general rule, theorems about intersections between geometric sets are easier to prove, and require fewer exceptions when considered in projective geometry. According to Dieudonn´e's History of Algebraic Geometry, projective geometry was among the most popular fields of mathematical research in the late 19th century. Pro- jective geometry today is a fundamental part of algebraic geometry, arguably the richest and deepest field in mathematics, of which we will gain a glimpse in this course. 2 Vector Spaces and Projective Spaces 2.1 Vector spaces and their duals Projective geometry can be understood as the application of linear algebra to classical geometry. We begin with a review of basics of linear algebra. 2.1.1 Fields The rational numbers Q, the real numbers R and the complex numbers C are examples of fields. A field is a set F equipped with two binary operations F × F ! F called addition and multiplication, denoted + and · respectively, satisfying the following axioms for all a; b; c 2 R. 1. (commutativity) a + b = b + a, and ab = ba. 2. (associativity)(a + b) + c = a + (b + c) and a(bc) = (ab)c 3. (identities) There exists 0; 1 2 F such that a + 0 = a and a · 1 = a 4. (distribution) a(b + c) = ab + ac and (a + b)c = ac + bc 5. (additive inverse) There exists −a 2 F such that a + (−a) = 0 1 1 6. (multiplicative inverse) If a 6= 0, there exists a 2 F such that a · a = 1. Two other well know operations are best defined in terms of the above properties. For example a − b = a + (−b) 4 and if b 6= 0 then 1 a=b = a · : b We usually illustrate the ideas in this course using the fields R and C, but it is worth remarking that most of the results are independent of the base field F . A perhaps unfamiliar example that is fun to consider is the field of two elements Z=2. This is the field containing two elements Z=2 = f0; 1g subject to addition and multiplication rules 0 + 0 = 0; 0 + 1 = 1 + 0 = 1; 1 + 1 = 0 0 · 0 = 0; 0 · 1 = 1 · 0 = 0; 1 · 1 = 1 and arises as modular arithmetic with respect to 2. 2.1.2 Vector spaces and subspaces Let F denote a field. A vector space V over F is set equipped with two operations: • Addition: V × V ! V ,(v; w) 7! v + w. • Scalar multiplication: F × V ! V ,(λ, v) 7! λv satisfying the following list of axioms for all u; v; w 2 V and λ, µ 2 F . • (u + v) + w = u + (v + w) (additive associativity) • u + v = v + u (additive commutativity) • There exists a vector 0 2 V called the zero vector such that, 0 + v = v for all v 2 V (additive identity) • For every v 2 V there exists −v 2 V such that v + −v = 0 (additive inverses) • (λ + µ)v = λv + µv (distributivity) • λ(u + v) = λu + λv (distributivity) • λ(µv) = (λµ)v (associativity of scalar multiplication) • 1v = v ( identity of scalar multiplication) I will leave it as an exercise to prove that 0v = 0 and that −1v = −v. Example 1. The set V = Rn of n-tuples of real numbers is a vector space over F = R in the usual way. Similarly, F n is a vector space over F . The vector spaces we consider in this course will always be isomorphic to one of these examples. Given two vector spaces V; W over F , a map φ : V ! W is called linear if for all v1; v2 2 V and λ1; λ2 2 F , we have φ(λ1v1 + λ2v2) = λ1φ(v1) + λ2φ(v2): 5 Example 2. A linear map φ from Rm to Rn can be uniquely represented by an n × m matrix of real numbers. In this representation, vectors in Rm and Rn are represented by column vectors, and φ is defined using matrix multiplication. Similarly, a linear map from Cm to Cn can be uniquely represented by an n × m matrix of complex numbers. We say that a linear map φ : V ! W is an isomorphism if φ is a one-to-one and onto. In this case, the inverse map φ−1 is also linear, hence is also a isomorphism. Two vector spaces are called isomorphic if there exists an isomorphism between them. A vector space V over F is called n-dimensional if it is isomorphic to F n. V is called finite dimensional if it is n-dimensional for some n 2 f0; 1; 2; ::::g. Otherwise we say that V is infinite dimensional. A vector subspace of a vector space V is a subset U ⊂ V containing zero which is closed under addition and scalar multiplication. This makes U into a vector space in it's own right, and the inclusion map U,! V in linear. Given any linear map φ : V ! W , we can define two vector subspaces: the image φ(V ) := fφ(v)jv 2 V g ⊆ W and the kernel ker(φ) := fv 2 V jφ(v) = 0g ⊆ V: The dimension of φ(V ) is called the rank of φ, and the dimension of ker(φ) is called the nullity of φ.
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