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 mathematical 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 calculations 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 interesting 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 PR are given. How can we
describe the set of all points on the line through 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 PR . The set of all points
on the line through these points is given by
!
{[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.
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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)
(1, 0)
(2, 1)
x
Fig. 4.1. Homogeneous coordinates on the real projective line.
The last section focused on viewing a single line in PR from the projective
viewpoint. In the expression λ · p + µ · q the parameters (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 considerations 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 isomorphically 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
2
−{(0,0)}
. The picture
of this vector). Formally we can describe this space as R R−{0}
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 completely 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 transformation 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
.
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:
λ '→
Theorem 4.1. Let [a], [b], [c] ∈ RP1 be three points of which no two are coincident 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 = αp a + βp b. Similarly, q can be written as q = αq a + βq b. Thus
the expression λp + µq can be written as λ(αp a + βp b) + µ(αq a + βq b). Thus
we have
αa + βb = (αp λ + αq µ)a + (βp λ + βq µ)b
Since a and b are linearly independent we have
" # "
# " #
α
αp αq
λ
=
·
.
β
βp βq
µ
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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 projection 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)
b!
b
O
77
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 distributivity. 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
β
!
),o&
· (− $(a×b
$(b×a! ),o& ) =
α!
β! .
!
),o&
Setting τ = − $(a×b
$(b×a! ),o& gives the desired claim.
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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 projective 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
" #
a1
b1
are two dimensional vectors we will use
on. If a =
and b =
a2
b2
78
4 Lines and cross ratios
"
#
a b
[a, b] := det 1 1
a2 b 2
as an abbreviation of the determinant of the 2 × 2 matrix formed by these two
vectors. We will also use


a 1 b 1 c1
[a, b, c] := deta2 b2 c2 
a 3 b 3 c3
as an abbreviation of a 3 × 3-determinant if a, b, c are three-dimensional vectors. 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, b; c, d) :=
[a, c][b, 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 λa , λb , λc , λd ∈ R we have
(a, b; c, d) = (λa a, λb b; λc c, λd d).
Proof. Since [p, q] represents a determinant with columns p and q we have
[λp p, λq q] = λp λq [p, q]. Applying this to the definitions of cross ratios we get:
[λa a, λc c][λb b, λd d]
λa λb λc λd [a, c][b, d]
[a, c][b, d]
=
=
.
[λa a, λd d][λb b, λc c]
λa λb λc λd [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.
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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 onedimensional 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 projective space 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 equivalence class [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 ratios 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, b; c, d) =
|a, c| ( |b, c|
.
|a, d| |b, d|
(In the german literature the cross ratio is called Doppelverhältnis – 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 approach via oriented lengths the approach taken in the last section has the advantage 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 infinite numbers the following rules will be consistent with all operations throughout 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
(λa − λb )(λc − λd ) − (λa − λc )(λb − λd ) + (λa − λd )(λb − λc ) = 0.
Expanding all terms we get
(λa λc + λb λd − λa λd − λb λc )
−(λa λb + λc λd − λa λd − λc λb )
+(λa λb + λd λc − λa λc − λd λb ) =
0.
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.
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Remark 4.1. If (a, b; c, d) = λ, the six values that of the cross ratio for permutations 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 projective geometry 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 immediately.
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