Graphics Reference
In-Depth Information
Solution.
First of all, we must find representatives
i
and
o
for
I
and
O
, respectively,
so that
i
+
o
= (7,1). Since the equation
()
+
()
=
()
a
21
,
b
31
,
71
,
has solution a =-4 and b = 5, we can let
i
= (-8,-4) and
o
= (15,5). Next, the standard
parameterization j of
P
1
with respect to the given
I
,
O
, and
U
is defined by
-+
-+
8 5
45
x
x
()
=--
[
(
)
+
(
)
]
=-
[
]
=
j xx
8
,
4
15 5
,
8
x
+
15
,
-
4
x
+
5
.
Since j(5/6) = 5, it follows that 5/6 are the coordinates of
P
in the new coordinate
system. Alternatively, we could follow the proof of Theorem 3.4.1.10 and solve the
system of equations
()
=--
(
)
+
(
)
1 0
,
a
8
,
4
c
15 5
,
()
=--
(
)
+
(
)
0 1
,
b
8
,
4
d
15 5
,
for a, b, c, and d to get a = 1/4, b =-3/4, c = 1/5, and d =-2/5. Then
1
4
1
5
Ê
ˆ
1
2
3
5
5
6
Á
Á
˜
˜
=
Ê
Ë
ˆ
¯
()
--
51
´
3
4
2
5
Ë
¯
which leads to the same answer.
We are ready for the third definition of the cross-ratio (
AD
,
BC
) of four distinct
points
A
,
B
,
C
, and
D
on a line
L
in
P
2
.
Definition 3.
Let j be the standard parameterization of
L
with respect to a coordi-
nate system
I
,
O
, and
U
. If j([a
1
,a
2
]) =
A
, j([b
1
,b
2
]) =
B
, j([c
1
,c
2
]) =
C
, and j([d
1
,d
2
])
=
D
, then define
aa
bb
dd
cc
dd
bb
1
2
1
2
1
2
1
2
(
)
=
AD BC
,
.
(3.24)
aa
cc
1
2
1
2
1
2
1
2
3.4.1.13. Theorem.
The definition of the cross-ratio (
AD
,
BC
) of four distinct points
A
,
B
,
C
, and
D
on a line
L
in the projective plane via equation (3.24) is well defined
and agrees with those in equations (3.19) and (3.20).
Proof.
To show that the definition is independent of the coordinate system one only
needs to show that the right-hand side of equation (3.24) is unchanged under a trans-
formation of the form in equation (3.23). A straightforward computation does that.
Next, it suffices to show agreement with Definition 1. Let j be the standard parame-
