Graphics Reference
In-Depth Information
Figure 3.6.
Computing the cross-ratios from
x-coordinates.
y
L
D
C
B
A
x
A¢
B¢
C¢
D¢
are distinct collinear points that have distinct x-coordinates, then one can compute
their cross-ratio from their x-coordinate values. In other words, if
A
¢(x
1
,0),
B
¢(x
2
,0),
C
¢(x
3
,0), and
D
¢(x
4
,0) are the projections of the points on the x-axis, then
xx
xx
-
-
xx
xx
-
-
xx
xx
-
-
xx
xx
-
-
2
1
3
1
2
1
4
3
(
)
=¢
(
)
=
AD BC
,
A D B C
¢
,
¢
¢
=
.
4
2
4
3
4
2
3
1
Similarly, one can use the y-coordinates if those are distinct.
3.2.2. Example.
To compute the cross-ratio of the points
A
(2,0),
B
(0,-1),
C
(6,2),
and
D
(8,3) on the line
L
defined by the equation x - 2y - 2 = 0, assuming that
L
is
oriented to the right.
Solution.
By definition of the cross-ratio, we have that
||
AB
BD
||
||
CD
AC
||
5
25
◊
-
5
45
1
8
(
)
=
AD BC
,
◊
=
=-
.
||
||
||
||
Since the x-coordinates 2, 0, 6, and 8 of the points are distinct, an easier way to
compute the cross-ratio is to use these values and formula (3.1):
02
80
-
-
86
62
-
-
1
8
◊
=-
The answer is the same.
We point out another interesting geometric consequence of Theorem 3.2.1. Since
the cross-ratio is preserved by a perspectivity, it follows that the view of four collinear
points is completely determined once one knows the position of three of those points
in the view. For example, consider a railroad track that consists of two rails and
equally spaced ties or a ladder with equally spaced rungs. What are the possible per-
spective views of this track or ladder? Well, the position and relative spacing of any
three of these ties or rungs in the view can be quite arbitrary except for some minor
