Projective Geometry - Cross Ratio

Projections do not keep distances constant, nor do they enlarge or shrink them in an obvious way. In Figure 2, for instance, D'C' is a little smaller than CD, but A'B' is much larger than AB. There is, however, a rather obscure constancy about a projection's effect on distance. It is known as the "cross ratio." If A, B, C, and D are points in order on a line and if they are projected through a point P into points A', B', C', and D' on another line, then the two expressions and are equal.

Cross rations play an important part in many of projective geometry's theorems.