Graphics Reference
In-Depth Information
a
Moebius
strip
a
disk
a
a
disk with antipodal boundary
points identified
disk glued to Moebius strip
along their boundary
(a)
(b)
Figure 3.9.
Visualizing the projective plane.
open disk topologically, but that is not a useful way of looking at it if one wants to
use its vector space structure.
It should be pointed out that there is actually nothing special about ideal
points even though their name would suggest otherwise. From an intrinsic point
of view, every point of
P
n
looks like every other point of
P
n
. The reader needs to
understand what really happened here. Projective spaces are
abstract
sets (sets of
equivalence classes) and we decided to coordinatize their points, that is, we decided
to associate a tuple of numbers with each point. Although our choice of coordinates
is a natural one, it is only one out of infinitely many ways that one can assign coor-
dinates to the points of these abstract sets. In the case of
P
2
, our chosen coordinates
imbedded
R
2
in
P
2
in a special way. Which points end up being called ideal points
is totally a function of how
R
2
is imbedded. Their special nature is purely an artifact
of the construction. For example, one can think of a sphere as the Euclidean plane
with one extra point “at infinity” added. This would seem to make this one point
special, but it is only special because of the particular representation. A sphere,
thought of in the abstract, has all points and their neighborhoods looking the same.
We shall return to the issue of coordinates in the next section and Sections 8.13 and
10.3.
•
Notation.
If
is the line in the plane defined by equation (3.14), let
denote the
ideal point [-b,a,0].
•
It follows from the discussion in the previous section that the association of
with the family of lines parallel to
sets up a one-to-one correspondence between
ideal points and families of parallel lines in the plane.
A
line
in
P
2
is any subset
L
of
P
2
of the form
Definition.
L
=
[
{
]
}
(
)
π
(
)
XYZ aX bY cZ
,,
++=
0
for fixed
a,b,c
000
,, .
Points of
P
2
that lie on a line are said to be
collinear
.
The tuple (a,b,c) that defines the line
L
above is not unique because any non-
zero multiple defines the same line. On the other hand, it is easy to show that the
