The cross-ratio

Four gold points A, B, C and D on the ground line are seen from the eye O and drawn on the ice-blue picture line as A′, B′, C′ and D′. Change the tilt of the picture line first: the points on it crowd together at one end and spread apart at the other, and the first row in the corner, A′B′ : B′C′, changes with them; lengths are not kept, and neither are their ratios. The more the picture line slants against the ground line, the more this ratio moves when you drag the eye. Dragging the small ice-blue point only slides the picture line without turning it: every length scales by the same factor, and this ratio stays put. The two rows below are cross-ratios: take the ratios AC : BC and AD : BD, and divide the first by the second. The cross-ratio (A, B; C, D) on the ground line and (A′, B′; C′, D′) on the picture line always agree. Drag the picture line behind the eye and they still do: a line of sight is a whole line, and A′ and the others fall on the part that runs back past the eye. As the picture line turns nearly parallel to a line of sight, that point flies off the picture towards infinity, and the two cross-ratios still agree; exactly parallel, or with the picture line through the eye, the corner shows a dash. Nor is the cross-ratio one fixed number: drag “Where C sits on the ground line” and both cross-ratios change together, and stay equal. It is fixed by the four points alone, wherever they are seen from.