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Consider

whether or not the mathematical device called the Limit – as used, for example, in differential and integral calculus – may validly be applied to projectively defined curves, such as the conic, or the path curve.

Take a non-degenerate conic, such as shown below.

We have in mind to progressively ‘reduce’ (shrink) intervals, such as H0H1, here indicated in yellow, on black line b so as to have the corresponding (red) chords c on the conic move to become, “in the limit”, tangents to the conic, each in place, everywhere on the conic.

That is, we are to see a set of straight-line, chordal segments, “in the limit”, merge and, in some sense, sum, or integrate,
to a “smooth” curve (which is one without discontinuities)
- namely, the conic itself.

In other words, we seek the projective geometric equivalent of integration.

—There seem to be three possibilities—


Possibility 1: If the chordal segments are to “sum”, they must link end to end, and we then have that –

All the intervals on line b, such as the one between points H0 and H1 on line b, are generated
via the directrices Du and Dd, and
via the conic
and these intervals are neither independently generated, nor independently generable.

It follows at once that if, for the purposes of establishing a limit-condition in respect of one interval, we shrink that interval, then all the other intervals must correspondingly and simultaneously shrink, and given that line b and the conic remain ‘fixed as shown ’ (invariant) throughout such an operation, the only way to bring about the shrinkage of any interval is to have directrix Du approach directrix Dd on the conic.

If that is done, and if we elect to keep point H0 fixed in place (invariant) on line b, then while point H1 approaches H0, H2 will approach H1, and H3 will approach H2, .... and Hn+1 will approach Hn, .... and so on, for as many intervals as there may be.

Try it!

Click and drag either directrix, Du or Dd, above, toward the other.

In other words, all intervals on line b will approach the first interval, H0H1, and all will, “in the limit”, be zero – that is, the intervals will all be zero when Du actually coincides with, and lies in, Dd – and the intervals will all lie at point H0.

All the chords similarly must approach each other and all eventually pass through the point on the conic chosen to remain invariant, and these chords will indeed then all become tangents to the conic – but all at the same place on the conic.

So, having approached and reached the “limit”, we surely have all our tangents,
but we appear to have lost the pre-specified distribution of them, namely,
each in place, everywhere on the conic.

We conclude that a projectively-specified curve, such as a conic,
is not the result of a limiting and summing process, at any rate not one defined as above
.


Possibility 2: Intervals, with their corresponding chords and chordal segments, are defined seperately and one-at-a-time,
but on the same directrices.

Two, one black, one red, are shown below.

Once again, given invariance of the disposition of conic relative to line b, all intervals depend simultaneously and only on directrices Du and Dd, and all shrink as Du and Dd are made to approach each other – but all shrink in place.

Try it!

Click and drag either directrix, Du or Dd, above, toward the other.

Accordingly, in the limit, we do get distributed tangents, as many as we have chosen to specify, but at the expense of linkage (because the segments need not be end-to-end), so the segments, in the limit, do not automatically sum to the conic.

And we see that, if the segments are to be forced to go end-to-end in order to sum one to the next, we must have recourse to possibility (1), above, which, as we saw, does not go to a limit with the desired or anticipated effect. So neither possibility in fact works, and we may say that conics, and by extension, path curves, are not the result of a limit process.


Possibility 3: The shrinkage is to be brought about by “iterative division”.

That is, each and every interval in a set is split into at least two, equal sub-intervals, N times over, where N is an integer “made” to increase, eventually to . This procedure

  • keeps all new sub-intervals linked end-to-end for summing, and
  • appears to keep each group of fresh sub-intervals on the site of the newly-divided interval.
Thus the chordal segments should remain linked and distributed – i.e., “everywhere in place” – en route to tangency.

Now the only way to ensure the projective, geometric identity of a set of intervals is to fashion them all in the same geometric way, which here requires that all the intervals in the set be referred to the same pair of directrices. This in turn means that a given interval can, at each division, be divided by just two (2). There is no explicit, strictly-geometric way to divide an interval by three (3), or more.

So, 'by two' turns out to be both the minimum and the maximum by which an interval may be divided using a strictly geometrical method – which means, of course, a method strictly in accord with projective-geometric axioms.

But actually finding and ‘marking-in’ the successive divisions-by-two is not an especially straightforward business, as may be guaged from the diagram below, where it has been done once, for just one interval.

It involves finding

  • the polars, p0 and p1, w.r.t. the conic, of P0 and P1 in line t,
  • the pole, T, of line t itself,
  • one of the two tangents to the conic of each of P0 and P1.
  • Joining the join of these tangents to the pole T: this is the polar, phalf, of the bisector, Phalf, on line t,
  • finding Phalf on line t from the polar, phalf, and
  • projecting Phalf across the conic into the new directrix, Dhalf.

This new directrix could then be used with Du and Dd to “distribute” all the interval-bisections - which would need to happen at least times (i.e., once for each of the intervals needing bisection).

And all this has to happen again at least times for the next general bisection, then again for each of at least further general bisections.

However, just by working in this way,
we become aware of fundamental contradictions – here, and in everything so far considered —

  • We must already have the tangents we seek to find the tangents we seek.
  • We need elements derived from an already-smooth curve to smooth that curve.
  • We must have reached the limit to be able to proceed to the limit.

So even in this third, apparently most promising, case, we find that the Limiting Process is not properly available.

We may now say that shrinkage must be continuous, and based only on the approach of directrices.
Complete shrinkage must bring complete collapse of the entire transformation.

The corollary is that uncollapsed transformation must be quantised.

This is to say that, because the chords cannot go everywhere tangential, they must stay as chords, and the transformation cannot proceed "continuously",
but rather in finite, countable steps.

It must leave irreducible, “unvisitable” gaps.

In the final analysis,
the Limit Process fails for strictly axiomatic geometry
because that geometry
preserves neither absolute equality nor isotropy,
and its elements are unbroken and unbreakable.

They cannot be segmented and summed at all.

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