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** Up:** The Real Projective Plane
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As explained at the end of chapter one, an affine transformation on
is a pair consisting of an invertible
matrix and a vector . If is a point in
, the action
of on is .

**Exercise 2.1** (10)
Let

be an affine transformation on

and let

be the

matrix

Show that upon identifying

with

via the chart

the projective tranformation defined by

equals the affine transformation
defined by

.

**Exercise 2.2** (15)
Use the preceding exercise to justify the following claim:

*the group of
affine transformations on
is the subgroup of projective
transformations that fix the hyperplane at infinity.*
The notion of *parallelism* is typical of affine geometry. It can
be easily interpreted in projective geometry if we fix the hyperplane at
infinity.

**Exercise 2.3** (05)
Use figure 2 to show that two staight lines on the plane

are
parallel if and only if the corresponding projective lines intersect at
infinity.

Figure 2.

**Exercise 2.4** (00)
Show that two affine subspaces in

are parallel if and only if all
the points in their intersection lie on the hyperplane at infinity.

Part of the philosophy of projective geometry is to think of
as a subset of
. Equivalently, we may say that
is a *compactification* of
. This point of view
is helpful in the solution of many geometric problems. Here is a
pretty example:

**Exercise 2.5** (25)
A

*convex set* in

is one that contains every line segment
joining any two of its points. For example a disc is convex, but a croissant is
not. Show that any unbounded convex subset on the plane contains a half
line.

Likewise, some theorems of projective geometry are easier to prove if
we reduce them to theorems in affine geometry. The classical examples
are the theorems of Pappus and Desargues, which we permit ourselves to
write in the form of experiments.

**Note.**
In the geometric constructions that follow, we will denote points by
uppercase letters and lines by lowercase letters. The symbol
denotes the line passing through the points and , while
denotes the point of intersection of the lines and
.

**Theorem 2.1** (Pappus)
Draw two lines on the projective plane and three points on each line.
Denote the points on the first line by

, and the points
on the second line by

. Draw the lines that join
points denoted by different letters (i.e., we do not draw the lines

, or

). The points

,
and

are collinear.

Figure 3.

The proof is given in the following two exercises, the first of which is to
prove the affine version of the theorem.

**Exercise 2.6** (10)
Draw two lines on the plane and draw points

on the
first line, and points

on the second line such that

is parallel to

, and

is parallel to

. Show
that

is necessarily parallel to

.

Figure 4.

Now comes the reduction of the projective result to the affine
result.

**Exercise 2.7** (10)
Let

be any projective transformation that sends the points

and

to infinity. Apply

to the
points and lines in the configuration of theorem

2.1 and verify
that the new configuration is like the one of the previous exercise.
Use this to prove Pappus' theorem.

**Theorem 2.2** (Desargues)
Draw the lines

, and

on the projective plane, and draw points

,

, and

. The points

, and

are collinear
if and only if the lines

, and

are concurrent.

Figure 5.

**Exercise 2.8** (20)
Find and prove an affine version of Desargues' theorem. Reduce
Desargues' theorem to its affine version by using a well-chosen
projective transformation.

** Next:** The fundamental theorems
** Up:** The Real Projective Plane
** Previous:** Basic concepts
Juan Carlos Alvarez
2001-01-22