Projective Lines, Cross-Ratios, Homographies

Abstract

To any four distinct points (a _i ) _{i = 1,2,3,4} on a projective line (considered by itself or inside a projective space), we associate a scalar, denoted by [a _i ] = [a _l, a ₂, a ₃, a ₄], and called the cross-ratio of these four points. For points on an affine line D, the cross-ratio is defined to be the same as on the completion $\tilde D = D \cup {\infty _D}$ (cf. 5.A).