the cross-ratio
Take a photograph of four fence-posts standing evenly along a straight road. In the photo the posts are no longer evenly spaced — perspective squashes the far ones together — so 'distance' and 'ratio of distances' are not preserved. Astonishingly, though, one particular combination of the four positions comes out identical in the photo and in reality. That magic combination is the cross-ratio, and it is the single most important number in projective geometry: the fundamental quantity that perspective and projection leave untouched.
For four collinear points A, B, C, D, the cross-ratio is the 'ratio of two ratios'. Using signed lengths along the line, define (A, B; C, D) = (AC / BC) / (AD / BD), that is (AC . BD) / (BC . AD), where AC means the signed length from A to C, and so on. Equivalently, if the points have coordinates a, b, c, d on the line, (A, B; C, D) = ((c - a)(d - b)) / ((c - b)(d - a)). The key theorem: any projective transformation (any perspectivity or sequence of them — exactly the operations a camera or a shadow performs) leaves this number unchanged, even though it scrambles all the individual distances. There is a dual version for four concurrent lines: the cross-ratio of four lines through a point equals the cross-ratio of the four points where any transversal cuts them.
Why it matters: the cross-ratio is THE projective invariant — almost every quantity preserved by projection can be expressed through it, and it is the tool that lets projective geometry measure anything at all. It is also the engine behind harmonic conjugates (the special value -1) and the projective treatment of conics. Two honest cautions. First, the value depends on the order of the four points: permuting them produces six possible values (the famous set including a value v and 1/v, 1 - v, and so on), so always state the order. Second, it is genuinely four points' worth of information — there is no projective invariant of just three points on a line, because a projective map can send any three collinear points to any other three.
Four collinear points with coordinates a = 0, b = 1, c = 2, d = 3. Cross-ratio (A, B; C, D) = ((c - a)(d - b)) / ((c - b)(d - a)) = ((2 - 0)(3 - 1)) / ((2 - 1)(3 - 0)) = (2 . 2) / (1 . 3) = 4/3. Photograph this row from any angle: the four images, however unevenly spaced, still give cross-ratio 4/3.
Distances change under projection; this one ratio-of-ratios does not — the fundamental projective invariant.
The value depends on the order of the four points: the six orderings give up to six values v, 1/v, 1 - v, 1/(1 - v), (v - 1)/v, v/(v - 1). So 'the cross-ratio is 4/3' is only meaningful once the labelling A, B; C, D is fixed. There is no analogous projective invariant for three points — you need four.