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.