2 ONE-DIMENSIONAL PROJECTIVE TRANSFORMATIONS. 
$1. General Properties of One-Dimensional 
Projective Transformations. 
2. Analytic Definition. A projective transformation in one 
dimension is defined analytically by the equation 
Ne eas (1) 
or, in homogeneous coordinates by the pair of equations 
px, = ax by, f 
py, = cx dy. ) 
In these equations the coefficients, a, b, c, d, are constants, and 
the variables are x, x, and «, y, x, y, Both constants and 
variables are to be regarded as complex numbers, unless oth- 
erwise expressly stated. 
The determinant ¢ a is called the determinant of the 
transformation T ; it is assumed, for the present at least, 
that this determinant does not vanish. 
A projective transformation of the points on a line should 
be looked upon as an operation which, when applied to a finite 
set of points or to the range of all points on the line, has the 
effect of rearranging and redistributing the points of the set 
or range so as to form a new set ora new range. ‘The sets or 
ranges of points which are related to one another by a project- 
ive transformation are said to be projective. 
3. A Transformation and its Inverse. The transformation 
T expressed by equation (1) transforms the point x into «,, 
where x is any point on the line. Equation (1) may be solved 
for x, giving us i <a. = (2) 
The transformation expressed by this equation is called the 
inverse of 7 and is symbolized by 7-7. YT‘ transforms a 
point x, into v. The two transformations T and T™ are so 
related to each other that if T transforms a point P into P,, 
T~‘ transforms P, back to P. 
