The Cross-Ratio

The cross-ratio of four points A, B, C and D on a line l in projective space is the ratio of the ratios of the mixed distances |AC|:|AD| / |BD|:|BC| , or, written in bracket notation, ([AC][BD]):([AD][BC]).

In the following construction you can see a very important property of the cross-ratio: It is invariant under projective transformation, thus if the four points are the intersections of four concurrent lines with a fifth projection line, the cross-ratio is constant. Move the line by dragging the points marked move.

Enable Java for an interactive figure created with Cinderella

This gives rise to the natural definition of cross-ratios of four concurrent lines (by the cross-ratio of any four points obtained by intersecting with a fifth line) and also the cross ratio of four points not necessarily collinear with respect to a center of projection.

About the figure: It has been created with Cinderella. Since Cinderella does not do algebraic calculations out of the box, the necessary divisions of the lengths are simulated using von-Staudt-constructions. Von-Staudt-constructions have been used by (guess who) von-Staudt to show the field structure of point-line-construction without refering to coordinates. The points |AC|, |AD|,... are at distance |AC|, |AD|,... from the point 0, the two intermediate results |AC|:|AD| and |BD|:|BC| are at the appropriate distances from 0, and the the fat point labelled with the cross-ratio is at cross-ratio-distance from 0. While moving the intersection line, the cross-ratio-point does not move, which shows that its position (or distance from 0 is invariant under projective transformations of the points A, B, C and D.

Ulli Kortenkamp, June 8, 2000. Many thanks to Seth Teller for asking for a method to display the cross-ratio with Cinderella.