PROJECTIVE TRANSFORMATIONS
7
. A and
r x', we
k —a.
Dy h and
umn, we
— a') 9^-0,
, a linear
3 x of a
the corre-
mation is
act points
the other
if A'=A,
= x. Thus
1 transfor
me is inter-
raight line
linate of a
fine V, the
is a pro-
respective
of x such
£3', X4' of
2J
x' determine four distinct points A', B', C', D' of l'. For,
if i^j,
x ' ^aXj+P _ A (Xj — Xj)
1 1 yXi-\-8 7 Xj-\-8 (7X1+5) (7x^+5) ’
{A 'B'C'D') = -f- x \~ x \ = = (a BCD)
X3 —X2 X4 — X2 X3—X2 X4 — X2
since, if U denotes 7x1+5,
r =iirJwHiiJi£) =i -
If A'¿¿A, project the points +', B', C', D' from any con
venient vertex V on to any line AB\ through + and distinct
v'
from l, obtaining the points A\=A, B1, C1, D\ of Fig. 2. Let
V be the intersection of BB1 with CC1 and let VD\ meet l at
P. Then
(.ABCP) = UxBiCxDx) = {A'B'C'D') = {ABCD).
From the first and last we have P=D, as proved above.
Holding xi, X2, X3 fixed, but allowing X4 to vary, we obtain
two projective ranges on l and l'. If A'=A, we use 1' itself
as ABi and see that the ranges are perspective.