44 THEORY OF PROJECTION. 
the points of each pair. The intersection of these lines is the 
required center. All other pairs of corresponding points are 
obtained by drawing the rays of the projecting pencil. But 
in assuming that we have a perspective projection we assume 
at the same time that O is a self-corresponding point on the 
two lines. Thus we see, as before, that we choose three 
points on/ and their three corresponding points onl’, and 
thereby the projection is completely determined. 
THEOREM 28. The projection of one range upon another is com- 
pletely determined by three points on one range, and their three 
corresponding points on the other. 
53. Projective Transformation. In Fig. 2 if l’ be revolved 
about O until it coincides with /, any point P’ on l’ will be 
brought to some point P, on 1, so that OP’=OP,. The two 
ranges of points are then considered as existing on the same 
line /. The operation of projecting by means of the conic K 
a range of points on / into a new range on /’ and then by revo- 
lution about O bringing the new range back to / will be called 
a projective transformation. The effect of a projective 
transformation is to shift the points of a line into new posi- 
tions so that there is a projective relation between the old 
and new positions of the points. 
THEOREM 29. Given two lines, / and U’, intersecting at O; a pro- 
jective transformation of the points on] is completely determined 
by means of a conic K touching both / and I’. 
54. Analytic Representation of a Projective Transforma- 
tion. We now proceed to consider the analytical aspect of a 
projective transformation of the points ona line. To this end 
we shall make use of the theorem that the cross-ratio of any 
four points (A BCX) is equal to that of their four corres- 
ponding points (A’B’C’X’). Taking O as the origin, let the 
distances to the four points A, B, C, and X be a, b, ¢, and x; 
and let the distances to the four corresponding points be a, b,, 
c, and 2z,. Since the cross-ratios of these two sets of points 
are equal we have 
