** Next:** Harmonic quadruples and von
** Up:** The Real Projective Line
** Previous:** Basic definitions

In the previous section we saw that the action of
on
the projective line is transitive. In fact, we can say much more
than that:

**Theorem 2.1** (First fundamental theorem)
If

and

are two triples of distinct points on
the projective line, there exits a unique projective transformation
that sends

to

,

to

and

to

.

The proof hinges on the following exercise:

**Exercise 2.1** (10)
Let

, and

be three distinct lines throught the origin
in

. Show that it is possible to find a basis

of

such that

lies on

,

lies on

and

lies on

. Moreover, show that such a
basis is uniquely defined up to multiples.

**Exercise 2.2** (05)
Prove the first fundamental theorem.

**Exercise 2.4**
Explain how the following figure constitutes a third proof of the
first fundamental theorem (uniqueness excepted).

Having seen that any triple of distinct points can be mapped onto any
other triple of distict points by a *unique* projective transformation,
we cannot expect the same to hold for quadruples of points.

**Definition 2.1**
Let

be a quadruple of points on the projective
line with

, and

different from each other,
and let

be the unique projective transformation taking

, and

respectively onto the lines of
slope

, and

.
The

*cross-ratio*
is defined
as the point

.

**Proposition 2.1**
Show that if

is a projective transformation and

is a quadruple of points on the projective
line with

, and

different from each other,
then the cross-ratio

equals

.

**Exercise 2.5** (10)
Prove the above result.

**Exercise 2.6** (05)
By indentifying points in the projective line with the slopes of the
corresponding lines, verify that the cross-ratio of four numbers

can be written as:

**Exercise 2.7** (05)
Use the preceding exercise to give a purely computational proof of the
invariance of the cross-ratio.

**Exercise 2.8** (05)
A projective transformation sends 0 to

,

to

, and

to 0.
Where does it send

?

We are now ready to state an important characterization of projective
transformations of the real projective line.

**Theorem 2.2** (Second fundamental theorem)
A map from the projective line to itself is a projective transformation
if and only if it preserves cross-ratios.

The proof is given in the following exercises.

**Exercise 2.9** (05)
Show that a map from the projective line to itself that preserves
cross-ratios and that fixes the lines with slope

, and

is the identity map.

**Exercise 2.10** (05)
Prove theorem

2.2.

** Next:** Harmonic quadruples and von
** Up:** The Real Projective Line
** Previous:** Basic definitions
Juan Carlos Alvarez
2000-10-27