Cross-ratio - Projective Geometry

Projective Geometry

Cross-ratio is a projective invariant in the sense that it is preserved by the projective transformations of a projective line. In particular, if four points lie on a straight line L in R2 then their cross-ratio is a well-defined quantity, because any choice of the origin and even of the scale on the line will yield the same value of the cross-ratio. Furthermore, let {L_i, 1 ≤ i ≤ 4}, be four distinct lines in the plane passing through the same point Q. Then any line L not passing through Q intersects these lines in four distinct points P_i (if L is parallel to L_i then the corresponding intersection point is "at infinity"). It turns out that the cross-ratio of these points (taken in a fixed order) does not depend on the choice of a line L, and hence it is an invariant of the 4-tuple of lines {L_i}. This can be understood as follows: if L and L′ are two lines not passing through Q then the perspective transformation from L to L′ with the center Q is a projective transformation that takes the quadruple {P_i} of points on L into the quadruple {P_i′} of points on L′. Therefore, the invariance of the cross-ratio under projective automorphisms of the line implies (in fact, is equivalent to) the independence of the cross-ratio of the four collinear points {P_i} on the lines {L_i} from the choice of the line that contains them.