Graphics Reference
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Figure 3.10.
Defining when a point
is between two points.
B
U
Bo
0
A
u
1
A
O
(a)
(b)
Similarly, one shows that replacing Q by Q ¢ will preserve formula (3.20). Let Q ¢=[ q ¢],
q ¢= p + t q . Express the points A , B , C , and D in terms of P and Q ¢ and relate the new
expressions with the old ones. The details are left to the reader.
Finally, to show that Definition 1 and 2 agree, choose P = A and Q = D . In this
case, a = 0 and d =•. These values imply that the quotient in (3.20) is just b/c, the
same value that Definition 1 asserts. This proves the theorem.
We shall give a third definition of the cross-ratio shortly. First, we study the
problem of coordinatizing the points of the projective line and plane. Let us start with
lines. The standard way that one assigns coordinates (real numbers) to a line in
Euclidean space (thought of as an abstract set of vectors without coordinates) is to
decide on a unit of distance, pick a start point o (the “zero”), and pick a direction for
the line that defines which half with respect to o will get positive numbers and which
will get negative numbers. We can accomplish this by picking two points: the start
point o and another point u (the “+1”) that is a unit distance form o . See Figure
3.10(a). Each point p of the line is then assigned the number t, where p - o = t( u -
o ). Note how o and u get assigned the numbers 0 and 1, respectively. What is differ-
ent about a line in P 2 is that it is topologically a circle. Picking two points on a circle
does not orient it. In Figure 3.10(b), which of the points A or B is “between” O and
U ? Figure 3.10(a) shows that this is not a problem for a line in R 2 . For a circle or pro-
jective line we have to pick three points, but it is convenient to pick them in a special
way. This will also restore the dependency on representations of points that we lost
in Theorem 3.4.1.4. The next lemma is the basic property we need.
3.4.1.7. Lemma. Let I = [ i 1 ] = [ i 2 ], O = [ o 1 ] = [ o 2 ], and U = [ u 1 ] = [ u 2 ] be three
distinct points on a line in P 2 . If
i
+=
o u
,
j
=
12
, ,
(3.21)
j
j
j
then i 1 = c i 2 , o 1 = c o 2 , and u 1 = c u 2 , for some c π 0.
Proof.
Let i 1 = a i 2 , o 1 = b o 2 , and u 1 = c u 2 . Equations (3.21) imply that
(
) =+.
ab c c
i
+==+
ou i
o i
cc
o
2
2
2
2
2
2
2
Since the vectors i 2 and o 2 are linearly independent, we must have a = b = c.
The importance of Lemma 3.4.1.7 is the following:
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