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4 Lines and Cross Ratios 4 Lines and cross ratios At this stage of this monograph we enter a significant didactic problem. There are three concepts which are intimately related and which unfold their full power only if they play together. These concepts are performing calculations with geometric objects, determinants and determinant algebra and geometric incidence theorems. The reader may excuse that in a beginners book that makes only little assumptions on the pre-knowledge one has to introduce these concepts one after the other. Thereby we will sacrifice some mathe- matical beauty for clearness of exposition. Still we highly recommend to read the following chapters (at least) twice. In order to get an impression of the interplay of the different concepts. This and the next section is devoted to the relations of RP2 to calcula- tions in the underlying field R. For this we will first find methods to relate points in a projective plane to the coordinates over R. Then we will show that elementary operations like addition and multiplication can be mimicked in a purely geometric fashion. Finally we will use these facts to derive in- teresting statements about the structure of projective planes and projective transformations. 4.1 Coordinates on a line Assume that two distinct points [p] and [q] in R are given. How can we describe the set of all points on the line throughP these two points. It is clear that we can implicitly describe them by first calculating the homogeneous coordinates of the line through p and q and then selecting all points that are incident to this line. However, there is also a very direct and explicit way of describing these points, as the following theorem shows: Lemma 4.1. Let [p] and [q] be two distinct points in R. The set of all points on the line through these points is given by P [λ p + µ q] λ, µ R with λ or µ being non-zero . { · · ∈ } ! ! 74 4 Lines and cross ratios Proof. The proof is an exercise in elementary linear algebra. For λ, µ R (with λ or µ being non-zero) let r = λ p + µ q be a representative∈ of a point. We have to show that this point is· on the· line through [p] and [q]. In other words we must prove that λ p + µ q, p q = 0. This is an immediate consequence of the arithmetic rules" · for the· scalar× $ and vector product. We have: λ p + µ q, p q = λ p, p q + µ q, p q " · · × $ = λ" p,· p ×q +$ µ"q, p· q× $ = λ" 0+×µ $0 " × $ =0· · The first two equations hold by multilinearity of the scalar product. The third equation comes from the fact that p, p q and q, p q are always zero. Conversely, assume that [r] is a" point× on$ the" line× spanned$ by [p] and [q]. This means that there is a vector l R3 with ∈ l, p = l, q = l, r =0. " $ " $ " $ The points [p] and [q] are distinct, thus p and q are linearly independent. Consider the matrix M with row vectors p, q, r. This matrix cannot have full rank since the product M l is the zero vector. Thus r must lie in the span of p and q. Since r itself is not· the zero vector we have a representation of the form r = λ p + µ q with λ or µ being non-zero. · · %& The last proof is simply an algebraic version of the geometric fact that we consider a line as the linear span of two distinct points on it. In the form r = λ p+µ q we can simultaneously multiply both parameters λ and µ by the same factor· ·α and still obtain the same point [r]. If one of the two parameters is non-zero we can normalize this parameter to 1. Using this fact we can express almost all points on the line through [p] and [q] by the expression λ p + q; λ R. The only point we miss is [p] itself. Similarly, we obtain all· points except∈ of [q] by the expression p + µ q. Let us interpret these relations within the framework of concrete coordinates· of points in the standard embedding of the Euclidean plane. For this we set o = (0, 0, 1) (the corresponding point [o] represents the origin of the coordinate system of R2 embedded on the z = 1 plane) and x∞ = (1, 0, 0) (the corresponding point [x∞] represents the infinite point in direction of the x-axis). The points represented by vectors λ x∞ + o =(λ, 0, 1) are the finite points on the line joining [o] and [x∞] (this is· the embedded x-axis). Each such point (λ, 0, 1)is bijectively associated to a real parameter λ R. It is important∈ to notice that this way of assigning real numbers to points in the projective plane is heavily dependent on the choice of the reference points. It will be our next aim to reconstruct this relation of real parameters in a purely projective setup. 4.2 The real projective line 75 4.2 The real projective line y ( 1, 1) (1, 1) − (2, 1) x (1, 0) Fig. 4.1. Homogeneous coordinates on the real projective line. The last section focused on viewing a single line in R from the projective viewpoint. In the expression λ p + µ q the parametersP (λ, µ) themselves can be considered as homogeneous· coordinates· on the 1-dimensional projective line spanned by p and q. In this section we want to step back from our consid- erations of the projective plane and study the situation of a (self-contained) projective line. We will do this in analogy to the homogeneous setup for the projective plane. For the moment, we again restrict ourselves to the real case. A real (Euclidean) line is a 1-dimensional object that could be isomor- phically associated to the real numbers R. Each point on the line uniquely corresponds to exactly one real number. Increasing this real number further and further we will move the corresponding point further and further out. Decreasing the parameter will move the point further and further out in the opposite direction. In a projective setup we will compactify this situation by adding one point at infinity on this line. If we increase or decrease the real parameter we will in the limit process reach this unique infinite point. Algebraically we can model this process again by introducing homogeneous coordinates. A finite point with parameter λ on the line will be represented by a two-dimensional vector (λ, 1) (or any non-zero multiple of this vector). The unique infinite point corresponds to the vector (1, 0) (or any non-zero multiple R2−{(0,0)} of this vector). Formally we can describe this space as R−{0} . The picture above gives an impression of the situation. The original line is now embedded on the y = 1 line. Each two dimensional vector represents a one dimensional subspace of R2. For finite points the intersection of this subspace with the line gives the corresponding point on the line. The infinite point is represented by any vector on the x-axis. (The reader should notice that this setup is com- pletely analogous to the setup for the real projective plane that we described in Section 3.1). Topologically a projective line has the shape of a circle — a one dimensional road on which we come back to the start-point if we travel long enough in one direction. We call this space RP1. In the projective plane RP2 we can consider any line as an isomorphic copy of RP1. 76 4 Lines and cross ratios In analogy to the real projective plane we define a projective transfor- mation by the multiplication of the homogeneous coordinate vector with a matrix. This time it must be a 2 2 matrix × x ab x y '→ cd · y " # " # " # If we consider our points (λ, 1) represented by a real parameter λ, then matrix multiplication induces the following action on the parameter λ: a λ + b λ · . '→ c λ + d · A point gets mapped to infinity if the denominator of the above ratio vanishes. An argument completely analogous to the proof of Theorem 3.4 shows: Theorem 4.1. Let [a], [b], [c] RP1 be three points of which no two are co- incident and let [a#], [b#], [c#] ∈RP1 be another three points of which no two are coincident, then there exists∈ a 2 2 matrix M such that [M a] = [a#], [M b] = [b#] and [M c] = [c#]. × · · · In other words the image of three points uniquely determines a projective transformation. The projective transformations arise in a natural way if we represent points on the line with respect to two different sets of reference vectors, as the following lemma shows. Lemma 4.2. Let # be the line spanned by two points [p] and [q] in RP2. Let [a] and [b] be two other distinct points on #. Consider the vector λp + µq (that represents a point on #). This vector can also be written as αa + βb for certain α,β. The parameters (α,β) can be expressed in (λ, µ) by a linear transformation, that only depends on a, b, p and q. Proof. Theorem 4.1 ensures that the point represented by λp + µq can also be expressed in the form αa + βb. Since p is in the span of a and b it can be written as p = αpa + βpb. Similarly, q can be written as q = αqa + βqb.
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