Two on a Line
We collapse the red and blue lines into one purple line – but do not merge their intervals (when distinct elements become one, they are said to become degenerate).
These we keep apart for now on the now-merged,
purple line: but
wherever they go, and no matter how they appear,
they remain projectively indistinguishable from each other.
They can be separate, and they can be overlapped—and they can be neither: they can ‘just touch’ ends, so that adjacent end-points of separated intervals merge, and become degenerate. When this happens, the red and blue intervals share an end and the intervals become “linked”, or, “daisy-chained”. Try it!
We are now but a step away from forming a sequence of joined-up, projectively-indistinguishable intervals, on a line—which is the closest we will ever get, by strictly projective means, to forming a regular ruler.