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## 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
the two perspectivities must become co-planar. In fact, all the
participating elements fall into a single plane.

Our red and blue intervals remain projectively identical, or equivalent, to the black interval, by their perspectivities, and hence to each other, by the projectivity.

We said we'd keep the intervals apart, but now that we are here, try dragging point S, or point P, along its line.  You will see the corresponding interval shift along the purple line:

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.