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terization of
L
with respect to the points
A
= [
a
],
D
= [
d
], and
C
= [
c
] with
a
+
d
=
c
.
Then j(1,0) =
A
, j(0,1) =
D
, j(1,1) =
C
, and the right-hand side of equation (3.24) eval-
uates to b
2
/b
1
. Now, j(b
1
,b
2
) =
B
= [
a
+ k
d
] =j(1,k), which implies that k = b
2
/b
1
. In
the notation of Definition 1, the cross-ratio (
AD
,
BC
) was defined to be k/k¢, but k¢=1
here, so that the theorem is proved.
For yet another derivation and explanation of the projective invariance of the
cross-ratio see [Blin98].
We now move on to parameterizing points in the projective plane. What we did
for a projective line will generalize in a natural way. Three points were enough to
determine a coordinate system for
R
2
. We need a fourth for
P
2
.
Definition.
Let
I
,
J
,
O
, and
U
be four points of
P
2
no three of which are collinear.
Choose representations
I
= [
i
],
J
= [
j
],
O
= [
o
], and
U
= [
u
] for the points so that
i
+
j
+
o
=
u
. The map
2
2
j :
PP
Æ
defined by
(
[
]
)
=++
[
]
j XYZ
,,
X
i
Y
j
Z
o
is called the
standard parameterization of
P
2
with respect to the coordinate system
I
,
J
,
O
, and
U
. Using the standard inclusion of
R
2
in
P
2
we shall also describe the map j
with the formulas
(
)
=++
[
]
(
[
]
)
=+
[
]
j
xy
,
x
i
y
j
o
and
j
xy
, ,
0
x
i
y
j
,
3.4.1.14. Theorem.
The standard parameterization of
P
2
with respect to four 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 the natural analogs of Theorem 3.4.1.4, Lemma
3.4.1.7, and Corollary 3.4.1.8, whose proofs are left as exercises for the reader.
Definition.
Let j be the standard parameterization of
P
2
with respect to points
I
,
J
,
O
, and
U
. If
P
Œ
P
2
and if j
-1
(
P
) = [X,Y,Z], then (X,Y,Z) will be called the
homoge-
neous coordinates
of
P
with respect to the
coordinate system
defined by
I
,
J
,
O
, and
U
.
For those points
P
for which Z π 0, let x = X/Z and y = Y/Z and call the pair (x,y) the
(
affine
)
coordinates
of
P
with respect to the given coordinate system. The points [1,0,0],
[0,1,0], [0,0,1], and [1,1,1] define the
standard coordinate system
for
P
2
and the coor-
dinates with respect to it are called the
standard coordinates
.
3.4.1.15. Example.
Suppose the standard coordinates for a point
P
in
P
2
are
(1,2) = [1,2,1]. What are the coordinates of
P
with respect to the coordinate system
defined by
I
= (-3,0) = [-3,0,1],
J
= (0,-2) = [0,-2,1],
O
= (-1,-1) = [-1,-1,1], and
U
= (-2,-1) = [-2,-1,1]?
