ONE-DIMENSIONAL PROJECTIVE TRANSFORMATIONS. 3 
4. Invariant Points. When the points of a line are shifted 
into new positions by a projective transformation T, does it 
ever happen that one or more of the points are unaltered in 
position? To answer this question, we reason as follows: The 
coordinate of a point «, which remains at rest or unaltered in 
position, 7. e., which is transformed into itself, must satisfy 
EEE 
cx +d * 
Clearing of fraction, we see that the coordinates of all such 
points satisfy the quadratic equation 
ca’ + (d—a)« —b=0; (3) 
whence we conclude that a projective transformation 7 
leaves unaltered two points on the line, and their coordi- 
nates are given by the roots of equation (3). These two 
points are generally distinct, but for special values of a, 
b, c, d, they may coincide. They are called the invariant 
points of the transformation. Two transformations will 
not generally have the same invariant points; but, as we 
shall learn, an unlimited number of transformations may 
have one or both invariant points in common. 
There is one particular transformation that leaves every 
point of the line invariant. . If b=c=0 and d=a in 
equation (1), we get x,=«a. This shows that x always 
equals x, or that every point on the line is transformed 
into itself. This transformation is called the identical pro- 
jective transformation. 
THEOREM 1. A projective transformation of the points on a line 
leaves invariant either two distinct points, two coincident points, or 
all points on the line. 
the equation 
5. Characteristic Equation of T. Let T be given in the 
homogeneous form as follows: 
px, = an by, ; 
Oh CoCr °) 
We indicate another way of finding the invariant points of 
T. Set «,=«x and y,=y in the above equations and trans- 
pose; thus we get 
