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3.4.1.8. Corollary.
Using the notation in Theorem 3.4.1.4, if we require the repre-
sentatives
p
1
and
p
2
to always add up to a representative of a fixed point, then the
parameterization described in part (2) of that theorem will depend
only
on the points
P
1
and
P
2
but not on their representatives.
Consider the projective line
P
1
and the three special points [1,0], [0,1], and [1,1]
that correspond to the points •, 0, and 1. Note how the coordinates of the standard
representatives of the first two points add up to the coordinates of the representative
of the third. We are ready to coordinatize points in projective space.
Definition.
Let
L
be a line in
P
2
and let
I
,
O
, and
U
be three distinct points on
L
.
Choose representations
I
= [
i
],
O
= [
o
], and
U
= [
u
] for the points so that
i
+
o
=
u
.
The map
1
Æ
j :
PL
defined by
(
[
]
)
=+
[
]
j XY
,
X
i
Y
o
is called the
standard parameterization of
L
with respect to the points
I
,
O
, and
U
. Using
the standard identification of
P
1
and
R
* we shall also describe the map j with the
formulas
()
=+
[
]
()
=
j
x
x
io
and
j
,
3.4.1.9. Theorem.
The standard parameterization of a line
L
with respect to three
of its points is a one-to-one and onto map that depends only on the points and not
on their representatives.
Proof.
The theorem follows from Corollary 3.4.1.8.
Definition.
Let j be the standard parameterization of a line
L
in
P
2
with respect to
points
I
,
O
, and
U
. If
P
Œ
L
and if j
-1
(
P
) = [X,Y], then (X,Y) will be called the
homo-
geneous coordinates
of
P
with respect to the
coordinate system
defined by
I
,
O
, and
U
.
Let x = X/Y or • depending on whether Y π 0 or Y = 0. The number x will be called
the (
extended real
or
affine
)
coordinate
of
P
with respect to the given coordinate system.
The points [1,0], [0,1], and [1,1] define the
standard coordinate system
for
P
1
and the
coordinates with respect to it are called the
standard coordinates
.
Let
1
Æ
j
k
:
PL
(3.22a)
be the standard parameterization of a line
L
with respect to the coordinate system
defined by points
I
k
,
O
k
, and
U
k
on
L
, k = 1,2. Express
I
k
,
O
k
, and
U
k
in the form
I
k
= [
i
k
],
O
k
= [
o
k
], and
U
k
= [
u
k
] with
i
k
+
o
k
=
u
k
. By Theorem 3.4.1.4 there are con-
stants a, b, c, and d, so that
