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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, and algebra and geometric 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 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

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 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 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 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 75 4.2 The

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 . 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 — 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. Thus the expression λp + µq can be written as λ(αpa + βpb)+µ(αqa + βqb). Thus we have αa + βb =(αpλ + αqµ)a +(βpλ + βqµ)b Since a and b are linearly independent we have α α α λ = p q . β βp βq · µ " # " # " # %& Finally, we want to describe how perspectivities from one line to another induce projective maps on the coordinates of the lines. For this let # and ## be two lines and let o be a point not incident to either of them. Furthermore assume that [a] and [b] are points on # and that [a#] and [b#] are the corresponding projected images in a projection through o from # to ##. With these settings we obtain: 4.3 Cross ratios (a first encounter) 77

! O b b p a p!

a!

Fig. 4.2. Projective scales under projections.

Lemma 4.3. There exists a number τ R such that the image of a point αa + βb under the projection is α#a# + β#∈b#, with (α#,β#) = (ατ,β). Proof. One way to geometrically express the desired result is to say that the line (αa + βb) (α#a# + β#b#) is incident to o. This happens if and only if the scalar product× of o and this line is zero. In this case we have

0= (αa + βb) (α#a# + β#b#),o = "(αa α#a#×) + (βb β#b#) +$ (αa β#b#) + (βb α#a#),o = αα" # (×a a#),o + ββ× # (b b#),o ×+ αβ# (a b#×),o + βα$# (b a#),o = αβ#"(a × b#),o$+ βα# "(b ×a#),o$ " × $ " × $ " × $ " × $ The first and second equality is just expanding the cross product by distribu- tivity. The third equality holds since o is on a a# and o is on b b#. The last line being zero can also be written as × ×

! ! α $(a×b ),o& α ( ! )= ! . β · − $(b×a ),o& β ! $(a×b ),o& Setting τ = ! gives the desired claim. − $(b×a ),o& %&

4.3 Cross ratios (a first encounter)

In the previous sections we have seen that many geometric magnitudes (among them seemingly natural magnitudes like distances or ratios of distances) do not remain invariant under projective transformations. Cross ratios are the simplest magnitudes that stay invariant under pro- jective transformations. Cross ratios will play an important role throughout each of the following chapters. Before we introduce cross ratios, we will set up a little notation that will help to abbreviate many of the formulas we will have to consider from now a b on. If a = 1 and b = 1 are two dimensional vectors we will use a b " 2# " 2# 78 4 Lines and cross ratios

a b [a, b] := det 1 1 a b " 2 2# as an abbreviation of the determinant of the 2 2 matrix formed by these two vectors. We will also use ×

a1 b1 c1 [a, b, c] := det a b c  2 2 2 a3 b3 c3   as an abbreviation of a 3 3-determinant if a, b, c are three-dimensional vec- tors. The reader should× be careful not to confuse these brackets with the notion we use for equivalence classes. A cross ratio is assigned to an ordered quadruple of points on a line. We first restrict ourselves to the case of calculating the cross ratio for four points in RP1. Later on we will define the cross ratio for four arbitrary points on a line in the projective plane RP2. We first define the cross-ratio on the level of homogeneous coordinates and then prove that the cross ratio is actually only depending on the projective points represented by these coordinates. Definition 4.1. Let a, b, c, d be four non-zero vectors in R2. The cross ratio (a, b; c, d) is the following magnitude: [a, c][b, d] (a, b; c, d) := [a, d][b, c] We will now show that the value of the cross ratio does not change under various transformations. Lemma 4.4. For any real non-zero parameters λ ,λ ,λ ,λ R we have a b c d ∈ (a, b; c, d) = (λaa,λbb; λcc,λdd). Proof. Since [p, q] represents a determinant with columns p and q we have [λpp,λqq]=λpλq[p, q]. Applying this to the definitions of cross ratios we get: [λ a,λ c][λ b,λ d] λ λ λ λ [a, c][b, d] [a, c][b, d] a c b d = a b c d = . [λaa,λdd][λbb,λcc] λaλbλcλd[a, d][b, c] [a, d][b, c] Canceling all λs is feasible, since they were assumed to be non-zero. The equality of the leftmost and the rightmost term is exactly the claim. %& This lemma proves that it makes sense to speak of the cross ratio

([a], [b]; [c], [d]) of four points on in RP2 since the concrete choices of the representatives are irrelevant for the value of the cross ratio. Cross ratios are also invariant under projective transformations. We obtain: 4.4 Elementary properties of the cross ratio 79

Lemma 4.5. Let M be a 2 2 matrix with non-vanishing determinant and let a, b, c, d be four vectors in R2×, then we have (a, b; c, d) = (M a, M b; M c, M d). · · · · Proof. We have [M p, M q] = det(M) [p.q]. This gives: · · · [M a, M c][M b, M d] det(M)2[a, c][b, d] [a, c][b, d] · · · · = = . [M a, M d][M b, M c] det(M)2[a, d][b, c] [a, d][b, c] · · · · Canceling all determinants is feasible, since M was assumed to be invertible. The equality of the leftmost and the rightmost term is exactly the claim. %& Taking the last two lemmas together proofs a remarkable robustness of the cross ratio. Not only it is independent of the vectors representing the points. It is even invariant under projective transformations. This in turn has the consequence that if we have a line in RP2 with two points p and q such that the points on the line are represented by λp + µq the cross ratio of four points on this line can be calculated by using the parameters (λ, µ) as one- dimensional homogeneous coordinates. The value of this the cross ratio is well defined according to Lemma 4.2, Lemma 4.2 and Lemma 4.5. Thus we have encountered our first genuine projective measure: the cross ratio. From now on we will only very rarely have to distinguish between a point [p] in and its representation in homogeneous coordinates. The reason is that whenever we want to link projective entities to measures we can do this via cross ratios. If no confusion can arise we will from now on identify a point [p] with the homogeneous coordinate vector p representing it. Whenever we speak of the point p we mean the [p] and if we speak of the vector p we mean the element of Rd representing it.

4.4 Elementary properties of the cross ratio

In this section we will collect a few elementary facts that are useful whenever one calculates with cross ratios.

4.4.1 Cross ratios and the line of real numbers

Readers already familiar with cross ratios may have noticed that our approach to cross ratios is not the one taken most often by text books. Usually cross ra- tios are introduced by expressions concerning the oriented distances of points on a line. For reference we will briefly also present this approach. For this let # be any line and let a, b, c, d be four points on this line. We assume that # is equipped with an orientation (a preferred direction) and we denote by a, b the directed (Euclidean) distance from a to b (this means that a, b =| | b, a ). If # represents the line of real numbers each point a | | −| | corresponds to a number xa R and we can simply set a, b = xb xa. Now the cross ratio is usually defined∈ as | | − 80 4 Lines and cross ratios a, c b, c (a, b; c, d)= | | | | . a, d b, d | |(| | (In the german literature the cross ratio is called Doppelverh¨altnis – a “ratio of ratios”.) It is easy to see that this definition agrees with our setup for all a b finite points a, b, c, d. We can introduce homogeneous coordinates , , 1 1 " # " # c d and for the points. The determinant then becomes 1 1 " # " # ab det = a b = a, b . 11 − −| | " # An easy calculation shows the identity of both setups. Compared to this ap- proach via oriented lengths the approach taken in the last section has the ad- vantage that it also treats infinite points correctly. Sometimes the form above provides a nice shortcut when calculating the cross ratio for finite points. For some positions of the input values the cross ratio becomes infinite. This happens whenever either a and d coincide or when b and c coincide. It will later on turn out useful not to consider this as an unpleasant special case. We simply can consider the results of the cross ratio themselves as points on a projectively closed line. The infinite value is then nothing else as a representation of the infinite point. If we assume in this interpretation that three of the entries (say a, b and c) are distinct and fixed then the map

d (a, b; c, d) '→ itself becomes a projective transformation. If one wants to calculate with infi- nite numbers the following rules will be consistent with all operations through- out this book: 1/ = 0; 1/0= ; 1 + = . ∞ ∞ ∞ ∞

4.4.2 Permutations of cross ratios

The cross ratio is not independent of the order of the entries. However, if we know the cross ratio (a, b; c, d)=λ we can reconstruct the cross ratio for any permutation of a, b, c and d. We obtain Theorem 4.2. Let a, b, c, d be four points on a projective line with cross ratio (a, b; c, d)=λ. then we have (i) (a, b; c, d) = (b, a; d, c) = (c, d; a, b) = (d, c; b, a) (ii) (a, b; d, c) = 1/λ (iii) (a, c; b, d) = 1 λ (iv) The values for− the remaining permutations are a consequence of these three rules. 4.4 Elementary properties of the cross ratio 81

Proof. Statement (i) is clear from Definition 4.1 and the anti-commutativity of the determinant. Statement (ii) is obvious from the definition since it just exchanges numerator and denominator. Statement (iii) requires a little elementary calculations. The expression we want to prove is

(a, c; b, d) = 1 (a, b; c, d) − On the determinant level this reads [a, b][c, d] [a, c][b, d] =1 [a, d][c, b] − [a, d][b, c]

Multiplying by [a, d][b, c] this translates to

[a, b][c, d] [a, c][b, d] + [a, d][b, c] = 0. − Since the cross ratio is invariant under projective transformations we may assume that all points are finite and we can represent them by numbers λa, λb, λc and λd. The Determinants then become differences and our expression reads

(λ λ )(λ λ ) (λ λ )(λ λ ) + (λ λ )(λ λ ) = 0. a − b c − d − a − c b − d a − d b − c Expanding all terms we get

(λ λ + λ λ λ λ λ λ ) a c b d − a d − b c (λaλb + λcλd λaλd λcλb) −+(λ λ + λ λ − λ λ − λ λ ) = 0. a b d c − a c − d b This is obviously true since all summands cancel. Finally, it is obvious that we can generate all possible permutations of points by application of the three rules. The first rule allows to bring any letter to the front position. The second and third equation describes two specific transpositions from which all permutations of the last three letters can be generated. %& Remark 4.1. If (a, b; c, d)=λ, the six values that of the cross ratio for permu- tations of these points are: 1 1 λ 1 λ λ, , 1 λ, , , − . λ − 1 λ 1 λ λ − − In particular these six functions form a group isomorphic to S3. 82 4 Lines and cross ratios

o d! d c! c b a b!

a!

Fig. 4.3. Cross ratios under projections. We have: (a, b; c, d) = (a!,b!; c!,d!)

4.4.3 Cross ratios and perspectivities

We now want to demonstrate how cross ratios stay invariant under geometric projections. This is an immediate corollary of the fact that projections induce a projective transformation (Lemma 4.3) and the invariance of the cross ratio under projective transformations. We get:

Corollary 4.1. Let o be a point and let # and ## be two lines not passing through o. If four points a, b, c, d on # are projected by the eye point o to four points a#,b#,c#,d# on ##, then the cross ratios satisfy: (a, b; c, d) = (a#,b#; c#,d#).

This corollary justifies another concept. We can assign to any quadruple of lines that pass through one point o a cross ratio. We can assign this cross ratio in the following way. We cut the four lines by an arbitrary line #. The four points of intersection define a cross ratio. The last theorem shows that the value of this cross ratio is independent from the specific choice of #. Thus we can call it the cross ratio of the lines. This fact is nothing else but a consequence of the fact that in every concept must have a reasonable dual. So, if one can assign a cross ratio to four points on a line one must also be able to assign a cross ratio to four lines through a point.

4.4.4 Cross ratios in RP2

Sometimes it is very inconvenient to calculate cross ratios of four points on a line in the real projective plane RP2 by first introducing a projective scale on the line. However there is a possibility to calculate the cross ratio much more directly by using quotients of 3 3 determinants. × Lemma 4.6. Let a, b, c, d be four collinear points in the projective plane RP2 and let o be a point not on this line. Then one can calculate the cross ratio via: [o, a, c][o, b, d] (a, b; c, d)= . [o, a, d][o, b, c] 4.4 Elementary properties of the cross ratio 83

Proof. Similar to the proof of Lemma 4.4 and 4.5 it is easy to see that the value oft his expression does not depend on the specific choice of the representing vectors, and that it is invariant under projective transformations. Hence we may assume w.l.o.g. that we have

1 0 0 a = 0 ,b= 1 ,o= 0 . 0 0 1       Under these assumptions the points c and d have coordinates.

c1 d1 c = c2 ,d= d2 .  0   0      All determinants reduce to 2 2 determinants and the theorem follows im- mediately. × %&