Chapter Twelve - The Cross Ratio: November–December 1998

Abstract

This chapter deals with the cross ratio in geometry. Geometry is the study of properties of a figure that stay the same as the figure undergoes some transformation. For example, in Euclidean geometry, the allowable transformations are rotations and translations. Properties that stay constant include distances and angles. For projective geometry such as homogeneous coordinates, the transformations include perspective projections. The cross ratio is as much a property of the four mutually intersecting lines p, q, r, and s as it is of the four collinear points A, B, C, and D. Any line m that is thrown across the lines will generate four points with the same cross ratio. Each of these line collections will have the same cross ratio. The value found after intersection is constant no matter where line m is placed. It is also constant if the whole diagram undergoes a homogeneous transformation. One can project any of these figures prospectively and also get an unchanged cross ratio. From this, one can also see the necessity of this arrangement of ratio of ratios in constructing a quantity that remains homogeneously meaningful. So, even though perspective transformations do not preserve distances, or ratios of distances, they do preserve these ratios of ratios of distances.