Graphics Reference
In-Depth Information
The figures above pointed out aspects of some specific perspectivities. What are
the abstract invariant properties that characterize such maps in general? Earlier we
mentioned some geometric properties that perspectivities between lines do not pre-
serve. Is there anything that all perspectivities preserve? Yes, there is, and it is called
the cross-ratio.
Definition.
Let
A
,
B
,
C
, and
D
be distinct collinear points on an
oriented
line
L
.
The
cross-ratio
in which
B
and
C
divide
A
and
D
, denoted by (
AD
,
BC
), is defined to
be the following quotient of ratios of division:
(
)
AD B
AD C
,
,
||
AB
BD
||
||
AC
CD
||
||
AB
BD
||
||
CD
AC
||
(
)
=
AD BC
,
=
=
◊
||
,
(
)
||
||
||
||
||
||
||
where || || denotes the
signed
distance between points of the
oriented
line
L
.
Although developed by the ancient Greeks, the modern development of the cross-
ratio is due to A.F. Moebius (
Der Barycentrische Calcul
, 1827) and independently to
M. Chasles (various publications from 1829-1865). The term “cross-ratio” was coined
by W.K. Clifford in 1878.
Because the ratios of division are independent of the orientation of the line, so is
the cross-ratio. The cross-ratio can be (and is often) defined for points
A
,
B
,
C
, and
D
, where only
three
of those four points are distinct if one defines it to be • in the
duplicate point case. We shall not do so here.
To explain the somewhat mysterious concept of cross-ratio, we first look at the
case of four numbers a, b, c, and d. See Figure 3.4. By definition
ba
db
-
-
ca
dc
-
-
ba
db
-
-
dc
ca
-
-
(
)
=
ad bc
,
=
.
(3.1)
Figure 3.4(a) shows the intervals involved in the formula. Figure 3.4(b) shows some
values of (ad,bc), where we fixed a, c, and d and let b vary. We see that the cross-ratio
changes from (d - c)/(c - a) to 0 as b increases from -• to a. It then increases from
0 to • as b increases from a to d. The cross-ratio decreases from (d - c)/(c - a) to -•
as b decreases from +• to d. In general, if one fixes three distinct points
A
,
C
, and
D
on a line
L
, then the function
value of (ad,bc) for different b
b - a
d - b
d- c
d- c
c- a
0
1
c- a
•
c- a
d- c
(
)
- •
- •
+ •
a
c
d
a
b
c
d
(a)
(b)
Figure 3.4.
The cross-ratio for four numbers.
