Next: Bibliography
Up: The Real Projective Line
Previous: The fundamental theorems
In the nineteenth century, some geometers where unsatisfied with
the characterization of projective transformations given by
the second fundamental theorem on the grounds that it was not
sufficiently geometric. In this section we describe the characterization
they came up with.
Definition 3.1
A quadruple of points on the projective line is said to be a
harmonic quadruple if its crossratio equals
.
Theorem 3.1
In the following figure the quadruple
is harmonic.
Exercise 3.1 (05)
Find in the figure two perspectives such that their composition
interchanges
and
while fixing both
and
.
Exercise 3.2 (05)
Use that perspectives preserve crossratios to deduce the theorem
from the previous exercise.
Theorem 3.2 (von Staudt's theorem)
A map from the projective line to itself is projective if and
only if it sends harmonic quadruples onto harmonic quadruples.
The proof is given in the form of exercises.
Exercise 3.3 (20)
Let
be a function satisfying the following properties:
Show that
for all
.
Exercise 3.4 (00)
Show that if
and
are any real numbers, then

.

Exercise 3.5 (15)
Let
be a map from the real projective line to itself that preserves
harmonic quadruples and fixes the lines with slopes
and
.
If
is the function such that
is the
slope of the image under
of the line of slope
, show that
satisfies all hypotheses in exercise
3.3 and is, therefore,
equal to the identity. Use this to prove von Staudt's theorem.
Next: Bibliography
Up: The Real Projective Line
Previous: The fundamental theorems
Juan Carlos Alvarez
20001027