PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin [email protected] 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles in the plane, Pythagoras’ Theorem, or something more analytic like three-dimensional geometry using dot products and vector products. In either scenario this is usually called Euclidean geometry and it involves notions like distance, length, angles, areas and so forth. So what’s wrong with it? Why do we need something different? Here are a few reasons: • Projective geometry started life over 500 years ago in the study of perspective drawing: the distance between two points on the artist’s canvas does not rep- resent the true distance between the objects they represent so that Euclidean distance is not the right concept. The techniques of projective geometry, in particular homogeneous coordinates, provide the technical underpinning for perspective drawing and in particular for the modern version of the Renaissance artist, who produces the computer graphics we see every day on the web. • Even in Euclidean geometry, not all questions are best attacked by using dis- tancesandangles. Problemsaboutintersectionsoflinesandplanes, forexample are not really metric. Centuries ago, projective geometry used to be called “de- 2 scriptive geometry” and this imparts some of the flavour of the subject. This doesn’t mean it is any less quantitative though, as we shall see. • The Euclidean space of two or three dimensions in which we usually envisage geometry taking place has some failings. In some respects it is incomplete and asymmetric, and projective geometry can counteract that. For example, we know that through any two points in the plane there passes a unique straight line. But we can’t say that any two straight lines in the plane intersect in a unique point, because we have to deal with parallel lines. Projective geometry evens things out – it adds to the Euclidean plane extra points at infinity, where parallel lines intersect. With these new points incorporated, a lot of geometrical objects become more unified. The different types of conic sections – ellipses, hyperbolas and parabolas – all become the same when we throw in the extra points. • It may be that we are only interested in the points of good old R2 and R3 but there are always other spaces related to these which don’t have the structure of a vector space – the space of lines for example. We need to have a geometrical and analytical approach to these. In the real world, it is necessary to deal with such spaces. The CT scanners used in hospitals essentially convert a series of readings from a subset of the space of straight lines in R3 into a density distribution. At a simpler level, an optical device maps incoming light rays (oriented lines) to outgoing ones, so how it operates is determined by a map from the space of straight lines to itself. 3 Projective geometry provides the means to describe analytically these auxiliary spaces of lines. In a sense, the basic mathematics you will need for projective geometry is something you have already been exposed to from your linear algebra courses. Projective ge- ometry is essentially a geometric realization of linear algebra, and its study can also help to make you understand basic concepts there better. The difference between the points of a vector space and those of its dual is less apparent than the difference between a point and a line in the plane, for example. When it comes to describing the space of lines in three-space, however, we shall need some additional linear algebra called exterior algebra which is essential anyway for other subjects such as differential geometry in higher dimensions and in general relativity. At this level, then, you will need to recall the basic properties of : • vector spaces, subspaces, sums and intersections • linear transformations • dual spaces After we have seen the essential features of projective geometry we shall step back and ask the question “What is geometry?” One answer given many years ago by Felix Klein was the rather abstract but highly influential statement: “Geometry is the study of invariants under the action of a group of transformations”. With this point of view both Euclidean geometry and projective geometry come under one roof. But more than that, non-Euclidean geometries such as spherical or hyperbolic geometry can be treated in the same way and we finish these lectures with what was historically a driving force for the study of new types of geometry — Euclid’s axioms and the parallel postulate. 2 Projective spaces 2.1 Basic definitions Definition 1 Let V be a vector space. The projective space P(V) of V is the set of 1-dimensional vector subspaces of V. Definition 2 If the vector space V has dimension n+1, then P(V) is a projective space of dimension n. A 1-dimensional projective space is called a projective line, and a 2-dimensional one a projective plane. 4 For most of the course, the field F of scalars for our vector spaces will be either the real numbers R or complex numbers C. Our intuition is best served by thinking of the real case. So the projective space of Rn+1 is the set of lines through the origin. Each such line intersects the unit n-sphere Sn = {x ∈ Rn+1 : P x2 = 1} in two i i points ±u, so from this point of view P(Rn+1) is Sn with antipodal points identified. Since each line intersects the lower hemisphere, we could equally remove the upper hemisphere and then identify opposite points on the equatorial sphere. When n = 1 this is just identifying the end points of a semicircle which gives a circle, but when n = 2 it becomes more difficult to visualize: If this were a course on topology, this would be a useful starting point for looking at some exotic topological spaces, but it is less so for a geometry course. Still, it does explain why we should think of P(Rn+1) as n-dimensional, and so we shall write it as Pn(R) to make this more plain. A better approach for our purposes is the notion of a representative vector for a point of P(V). Any 1-dimensional subspace of V is the set of multiples of a non-zero vector v ∈ V. We then say that v is a representative vector for the point [v] ∈ P(V). Clearly if λ 6= 0 then λv is another representative vector so [λv] = [v]. Now suppose we choose a basis {v ,...,v } for V. The vector v can be written 0 n n X v = x v i i i=0 and the n+1-tuple (x ,...,x ) provides the coordinates of v ∈ V. If v 6= 0 we write 0 n the corresponding point [v] ∈ P(V) as [v] = [x ,x ,...,x ] and these are known as 0 1 n homogeneous coordinates for a point in P(V). Again, for λ 6= 0 [λx ,λx ,...,λx ] = [x ,x ,...,x ]. 0 1 n 0 1 n Homogeneous coordinates give us another point of view of projective space. Let U ⊂ P(V) be the subset of points with homogeneous coordinates [x ,x ,...,x ] 0 0 1 n 5 such that x 6= 0. (Since if λ 6= 0, x 6= 0 if and only if λx 6= 0, so this is a 0 0 0 well-defined subset, independent of the choice of (x ,...,x )). Then, in U , 0 n 0 [x ,x ,...,x ] = [x ,x (x /x ),...,x (x /x )] = [1,(x /x ),...,(x /x )]. 0 1 n 0 0 1 0 0 n 0 1 0 n 0 Thus we can uniquely represent any point in U by one of the form [1,y ,...,y ], so 0 1 n U ∼= Fn. 0 The points we have missed out are those for which x = 0, but these are the 1- 0 dimensional subspaces of the n-dimensional vector subspace spanned by v ,...,v , 1 n which is a projective space of one lower dimension. So, when F = R, instead of thinking of Pn(R) as Sn with opposite points identified, we can write Pn(R) = Rn ∪Pn−1(R). A large chunk of real projective n-space is thus our familiar Rn. Example: The simplest example of this is the case n = 1. Since a one-dimensional projective space is a single point (if dimV = 1, V is the only 1-dimensional subspace) the projective line P1(F) = F ∪pt. Since [x ,x ] maps to x /x ∈ F we usually call 0 1 1 0 thisextrapoint[0,1]thepoint∞. WhenF = C, thecomplexnumbers, theprojective line is what is called in complex analysis the extended complex plane C∪{∞}. Having said that, there are many different copies of Fn inside Pn(F), for we could have chosen x instead of x , or coordinates with respect to a totally different basis. i 0 Projective space should normally be thought of as a homogeneous object, without any distinguished copy of Fn inside. 2.2 Linear subspaces Definition 3 AlinearsubspaceoftheprojectivespaceP(V)isthesetof1-dimensional vector subspaces of a vector subspace U ⊆ V. Note that a linear subspace is a projective space in its own right, the projective space P(U). Recall that a 1-dimensional projective space is called a projective line. We have the following two propositions which show that projective lines behave nicely: 6 Proposition 1 Through any two distinct points in a projective space there passes a unique projective line. Proof: Let P(V) be the projective space and x,y ∈ P(V) distinct points. Let u,v be representative vectors. Then u,v are linearly independent for otherwise u = λv and x = [u] = [λv] = [v] = y. Let U ⊆ V be the 2-dimensional vector space spanned by u and v, then P(U) ⊂ P(V) is a line containing x and y. Suppose P(U0) is another such line, then u ∈ U0 and v ∈ U0 and so the space spanned by u,v (namely U) is a subspace of U0. But U and U0 are 2-dimensional so U = U0 and the line is thus unique. 2 Proposition 2 In a projective plane, two distinct projective lines intersect in a unique point. Proof: Let the projective plane be P(V) where dimV = 3. Two lines are defined by P(U ),P(U ) where U ,U are distinct 2-dimensional subspaces of V. Now from 1 2 1 2 elementary linear algebra dimV ≥ dim(U +U ) = dimU +dimU −dim(U ∩U ) 1 2 1 2 1 2 so that 3 ≥ 2+2−dim(U ∩U ) 1 2 and dim(U ∩U ) ≥ 1. 1 2 But since U and U are 2-dimensional, 1 2 dim(U ∩U ) ≤ 2 1 2 with equality if and only if U = U . As the lines are distinct, equality doesn’t occur 1 2 and so we have the 1-dimensional vector subspace U ∩U ⊂ V 1 2 which is the required point of intersection in P(V). 2 7 Remark: Themodelofprojectivespaceasthespherewithoppositepointsidentified illustrates this proposition, for a projective line in P2(R) is defines by a 2-dimensional subspace of R3, which intersects the unit sphere in a great circle. Two great circles intersect in two antipodal points. When we identify opposite points, we just get one intersection. Insteadofthesphericalpicture, let’sconsiderinsteadthelinkbetweenprojectivelines and ordinary lines in the plane, using the decomposition P2(R) = R2 ∪P1(R). Here we see that the real projective plane is the union of R2 with a projective line P1(R). Recall that this line is given in homogeneous coordinates by x = 0, so 0 it corresponds to the 2-dimensional space spanned by (0,1,0) and (0,0,1). Any 2- dimensional subspace of R3 is defined by a single equation a x +a x +a x = 0 0 0 1 1 2 2 and if a and a are not both zero, this intersects U ∼= R2 (the points where x 6= 0) 1 2 0 0 where 0 = a +a (x /x )+a (x /x ) = a +a y +a y 0 1 1 0 2 2 0 0 1 1 2 2 which is an ordinary straight line in R2 with coordinates y ,y . The projective line 1 2 has one extra point on it, where x = 0, i.e. the point [0,a ,−a ]. Conversely, any 0 2 1 straight line in R2 extends uniquely to a projective line in P2(R). Two lines in R2 are parallel if they are of the form a +a y +a y = 0, b +a y +a y = 0 0 1 1 2 2 0 1 1 2 2 8 but then the added point to make them projective lines is the same one: [0,a ,−a ], 2 1 so the two lines meet at a single point on the “line at infinity” P1(R). 2.3 Projective transformations If V,W are vector spaces and T : V → W is a linear transformation, then a vector subspace U ⊆ V gets mapped to a vector subspace T(U) ⊆ W. If T has a non- zero kernel, T(U) may have dimension less than that of U, but if kerT = 0 then dimT(U) = dimU. In particular, if U is one-dimensional, so is T(U) and so T gives a well-defined map τ : P(V) → P(W). Definition 4 A projective transformation from P(V) to P(W) is the map τ defined by an invertible linear transformation T : V → W. Note that if λ 6= 0, then λT and T define the same linear transformation since [(λT)(v)] = [λ(T(v))] = [T(v)]. The converse is also true: suppose T and T0 define the same projective transformation τ. Take a basis {v ,...,v } for V, then since 0 n τ([v ]) = [T0(v )] = [T(v )] i i i we have T0(v ) = λ T(v ) i i i for some non-zero scalars λ and also i n n X X T0( v ) = λT( v ) i i i=0 i=0 for some non-zero λ. But then n n n n X X X X λT(v ) = λT( v ) = T0( v ) = λ T(v ). i i i i i i=0 i=0 i=0 i=0 Since T is invertible, T(v ) are linearly independent, so this implies λ = λ. Then i i T0(v ) = λT(v ) for all basis vectors and hence for all vectors and so i i T0 = λT. 9 Example: You are, in fact, already familiar with one class of projective transfor- mations – M¨obius transformations of the extended complex plane. These are just projective transformations of the complex projective line P1(C) to itself. We de- scribe points in P1(C) by homogeneous coordinates [z ,z ], and then a projective 0 1 transformation τ is given by τ([z ,z ]) = ([az +bz ,cz +dz ]) 0 1 0 1 0 1 where ad−bc 6= 0. This corresponds to the invertible linear transformation (cid:18) (cid:19) a b T = . c d ItisconvenienttowriteP1(C) = C∪{∞}wherethepoint∞isnowthe1-dimensional space z = 0. Then if z 6= 0, [z ,z ] = [z,1] and 1 1 0 1 τ([z,1]) = [az +b,cz +d] and if cz +d 6= 0 we can write az +b τ([z,1]) = [ ,1] cz +d which is the usual form of a M¨obius transformation, i.e. az +b z 7→ . cz +d The advantage of projective geometry is that the point ∞ = [1,0] plays no special role. If cz +d = 0 we can still write τ([z,1]) = [az +b,cz +d] = [az +b,0] = [1,0] and if z = ∞ (i.e. [z ,z ] = [1,0]) then we have 0 1 τ([1,0]) = [a,c]. Example: If we view the real projective plane P2(R) in the same way, we get some less familiar transformations. Write P2(R) = R2 ∪P1(R) where the projective line at infinity is x = 0. A linear transformation T : R3 → R3 can then be written as 0 the matrix   d b b 1 2 T = c1 a11 a12 c a a 2 21 22 10