§3] 
HOMOGENEOUS COORDINATES 
9 
Since A'P+PB' =A'B', we may replace A'P by A'B'—PB' 
and get 
PB' = 
krj-A'B' 
£+kri 
Let A have the coordinates £', rj', referred to A', B'. Then 
,_ PR , ky'-A'B'-W-ttfk-A'B' 
PA=PB'-AB' = PB'~ 
Similarly, if B has the coordinates £1, 771, referred to A', B', 
P£^(vh — £vi)k-A'B' 
Hence, by division, 
X r{r}£' — £7/) r _ —c(£l+kl]i) 
y s(t/£i — £771) S £-pkr]' 
Since we are concerned only with the ratio of x to y, we may 
set 
x=-rt] r^T], y = 5771^—5^17/. 
Since the location of A and B with reference to A' and B' 
is at our choice, as also the constant c (and hence r and 5), 
the values of ry and — r£' are at our choice, likewise srji and 
—s£ 1. There is, however, the restriction A ¿¿B, whence v 
Thus a change of reference points and constant multiplier c 
gives rise to a linear transformation 
T : 
X=a£+/3r], y = t£+<5t7, 
& 
Ô 
*0, 
of coordinates, and conversely every such transformation can 
be interpreted as the formulas for a change of reference points 
and constant multiplier. 
4. Examples of Invariants. The linear functions 
l = ax+by, L=Ax-\-By 
become, under the preceding linear transformation T, 
a(a£+/37/) +6(7^+577) =a' £+b'r], A' ^-\-B'rj, 
where 
o! — cux-\-by } b r = a$ J rbb, A =Aoc-\-By, B — Afi-\-Bb.