Chapter 13 - Cross ratio

Abstract

This chapter presents the properties of the unoriented cross ratio in classical geometry. A measure that is invariant under arbitrary projective maps must be more complicated than Euclidean distance. One can easily show that any function that depends on only two or three points and is invariant under arbitrary projective maps must assume only a finite set of values, and therefore is not a useful measure. In classical projective geometry, the simplest real-valued invariant is the cross ratio of four collinear points. The cross ratio of four collinear points is invariant under arbitrary projective maps. The cross ratio has a number of symmetry properties which follow directly from the defining formulas. For one thing, the cross ratio does not change if one swap the first pair of arguments with the second pair, or reverse the order of both pairs simultaneously. Swapping the innermost (or outermost) two arguments is numerically equivalent to computing one minus the original ratio, and moving to the antipodal range. Reversing only the first or last pair has the effect of exchanging the two coordinates of the cross ratio. If the cross-ratio is viewed as a (two-sided) real number, this operation is equivalent to taking its reciprocal.