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(2) Every point of
L
other than
P
1
can be expressed
uniquely
in the form
[s
p
1
+
p
2
]. By allowing s to equal • and using the convention that [s
p
1
+
p
2
] =
P
1
when
s =•, we shall say in the future that
every
point of
L
can be so expressed. Similarly,
every point of
L
can be expressed uniquely in the form [
p
1
+ t
p
2
], where the point for
t =•is identified with
P
2
.
Proof.
The fact that every point
Q
= [
q
] of
L
can be written in the form [s
p
1
+ t
p
2
]
follows form the fact that the vector
q
must be a linear combination of the vectors
p
1
and
p
2
because the determinant (3.17) is zero. Showing that
P
1
,
P
2
, and
P
are distinct
points if st π 0 is straightforward. Applying Theorem 3.4.1.1 to those three points
implies that they lie on
L
because the fact that the bottom row is a linear combina-
tion of the top two rows means that the determinant in the theorem is zero. This basi-
cally proves (1). The existence part of (2) follows easily from (1). The uniqueness part
of (2) is easily checked.
Note that identifying [s
p
1
+
p
2
] with
P
1
when s =•is reasonable because
1
1
È
Í
Ê
Ë
ˆ
¯
˘
˙
=+
È
Í
˘
[
]
=
˙
Æ
[]
s
pp
+
s
pp
+
pp p
as
s
Æ•.
1
2
1
2
1
2
1
s
s
The same argument justifies identifying [
p
1
+ t
p
2
] with
P
2
when t =•.
Theorem 3.4.1.4(2) can be interpreted as saying that every line in
P
2
can be para-
meterized by the extended reals
R
*. In the first parameterization we have in effect
made
P
1
the new ideal point.
P
2
is the new ideal point for the second. The parame-
terizations are not unique because they depend on the choice the two distinct points
with respect to which they are defined. More importantly, the next example shows
that it also depends on the
representatives p
1
and
p
2
we have chosen.
3.4.1.5. Example.
Consider the line
L
defined by the points
P
1
= [1,3,2] and
P
2
= [-1,0,4]. By Theorem 3.4.1.4(2),
{
[
]
}
»-
{
[
]
}
L
=-
s
13 2
,,
s
s
+
4
s
Œ
R
10 4
,, .
The point
Q
= [0,3,6] on
L
would be assigned the parameter s = 1. On the other hand,
P
1
= [2,6,4] and choosing the representative (2,6,4) for
P
1
would have represented
L
as
{
[
]
}
»-
{
[
]
}
2164 4
s
-
,,
s
s
+
s
Œ
R
104
,, ,
and assigned the parameter 1/2 to
Q
.
Next, we would like to define the cross-ratio for four points on a line in
P
2
. One
approach would be to take advantage of our earlier definition of the cross-ratio for
points in
R
2
. The only complication is that one of the points might be an ideal point
and one would have to give special definitions in those cases. It would be nice to give
a more intrinsic definition. Although a metric-free definition was given by C.G. von
Staudt (
Beiträge zur Geometrie der Lage
, 1847), the modern approach is based on
metric considerations. We shall use a coordinate-based approach using homogeneous
coordinates. In the end, one would of course want to check that all definitions agree.
