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Imaginary Circling Measure (2)

The disposition of the points of an imaginary circling measure,
on one of the two zero-stepping conics,
is graphed versus transform step, below.

  • Adjust the conic by moving the smaller yellow points
  • Move the whole measure using the large yellow point
  • Adjust the scaling and position of the yellow graph lines by moving white points
  • Move the measure-line using its orange point
Please enable Java for an interactive construction (with Cinderella).
The yellow graph-lines are obtained from a special case of transformation of a line into itself, namely, the “SRM”, the one having the invariant double points lying degenerately together, and at infinity. See the white points and lines. So the graph-lines are constructed from first principles, and are exact—not measured-out, and inexact.

If you drag either of the directrices, D1 or D2, on the conic round the conic in either sense,
you will see the measure either expand or collapse, depending.

If it expands, you will observe that the measure at some stage overlaps itself.
In other words, as we have already seen, this measure cycles,
with a finite number of steps per cycle.

You will see that the overlay can, but need not be, “exact”,
meaning that the points of a given cycle can, but need not,
exactly fall on the corresponding points of previous or subsequent cycles.

If the overlay is exact, then the
whole measure, though repeating,
will appear to be stationary,
like a standing wave.

If the overlay is not exact, then points of a given cycle
must fall alongside corresponding points of the preceding (or of the subsequent) cycle,
and exactly this same shift of all the points
must occur per cycle, cumulatively
so the entire measure
will show a phase-shift:
it will seem to precess,
like a travelling wave.

The graph of the disposition untangles this overlaying behaviour
by setting the repeating cycles out, side by side.

This measure cycles forever,
locally * only (i.e., without cycling through )
on localised, circular or elliptical conics,
but, rather like tan θ,
both locally and through ∞ on the straight line,
and the other, non-localised, non-circular, non-elliptical conics.

The Measure on one of the two single-stepping conics
is graphed below, versus transform step,
to display the wave-like, oscillatory character
of an imaginary cycling measure.

Please enable Java for an interactive construction (with Cinderella).

This graph of a measure and of a wave in motion is actually equivalent to the cycling of an inexact measure. Corresponding points of cycles would move along, and round in exactly the manner seen here.

But you are required to imagine all the points of the measure
being over-laid, per shift of any one point to its new position.

That is to say, a shift of any one point
is accomplished by the prior shifting of all the others:
it is a “knock-on” effect, like a Mexican wave.

It is indeed very like the propagation of a pulse
round a material ring (more properly called a torus):
a portion of the ring, receiving a “push” from behind,
moves forward a little, and so imparts a push
to the next portion—and so on.

The push clearly circulates,
but the pushed portions of a perfectly elastic ring would “spring back” completely, and leave the ring as a whole undisplaced.

If the whole ring is to move, it cannot be perfecly elastic: it should have hysteresis, in which displacements are not fully recovered.

* All conics are “closed”, meaning that each divides its plane into two regions, with the conic as boundary of both. It is this, as we have seen, that underlies the imaginary, for a line may lie entirely in just one of the two regions.
If we have specified that there are points at infinity on that plane, they must lie on a line, and if that line traverses both regions defined by the conic on that plane, then the conic may be said to be unlocalised. Else, the conic is all local.

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