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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).
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Among other elements, you may drag point H to adjust the overlap. Note that when the overlap is a point, the cycling is exact, and the measure is stationary. When the overlap is an interval, the cycling is inexact, and the measure seems to shift, or “precess”. |
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We project, and so “measure”, the overlap
This allows us, using F and D1, to “attach” the geometrically constant overlap (the black bars) |
Here we use the fact that intervals produced by exactly the same projective transformation are geometrically indistinguishable, which is to say that they are the same as, or ‘equal’ to, each other—or, equivalently, that they have the same projective ‘size’. To see this exemplified, drag either larger blue point round the conic below. Wherever either blue bar is, you should be able to lay one exactly on the other, because they are the same ‘size’. By dragging the larger white point, you can “attach” the black bar to to either end of either blue bar—which is the method described on the left . |
The corollary is that intervals that project their end-points, via a single point on the conic, onto the same pair of points on that same conic, are geometrically identical, or, have the same projective ‘size’. |
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 →.