Imaginary Circling Measure (3)

Propagating the Overlap

The first (blue) point (index 0) of the measure controlled by directrices D1 and D2 is here shown being overlapped
by the n'th (blue-turned-conveniently-red) point (index n-1) of the same (*) cycle,
which completes that cycle, and
starts the next.

So point n-1 of any cycle is point 0 of the following cycle.

Accordingly, this measure has a finite stepping-rate of n steps per cycle (s.c-1).

 Among other elements, you may drag point H to adjust the overlap. Note that when the overlap is degenerate (when the last point of the cycle is incident with the first), the cycling is exact, and the measure is stationary. When the overlap is an interval (when the last point of the cycle is not incident with the first, and in fact steps "beyond" it), the cycling is inexact, and the measure seems to shift, or “precess”. We project, and so “measure”, the overlap back onto the original conic, using the dashed black line from point n-1,to project point S on to the conic as yellow point F. This allows us, using F and D1, to “attach” the geometrically constant overlap (the black bars) to every blue point of the first cycle, and so to mark the corresponding points of the following cycle.

In this fashion the overlap is propagated through cycles.

for, after n steps, when a new cycle commences,
the overlaps are themselves identically overlapped.

If, during a pass, “old” points were replaced by “new”, somewhat as frames of a “movie” are replaced,
each of the old points would seem to move to its new position,
and the entire measure would appear to shift uniformly along.

* Caveat: There is no actual guarantee that just these red positions depicted here, and said to be overtaking or overlapping the blue, are in fact the continuation round the conic of the points of the same, originally-blue, cycle. They could be the points of another distribution, one also controlled by the directrices D1 and D2. The overlap shown is a necessary, but not sufficient condition for such continuity. For our present purpose - namely, illustration - we assume the continuity.

The difficulty just noted is fundamental →.

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