A copy eased into place loses its seams before its source

A compiled map agrees with its graticule except where it was copied built a method out of one observation: on a sheet whose coast was partly lifted from a map in a different projection, the drawn coast lies exactly on the coast its own graticule predicts everywhere except along the borrowed stretch, and the two ends of that stretch are corners in the departure.

Everything the method does follows from the copy having been moved rigidly — as a scaled, rotated piece, pinned at two points. The departure then rises in a near-straight line from each seam, so the seams are kinks; the piece between them is an exact similar image of its source, so the candidate library matches one of them exactly and the rest not at all.

A compiler fitting a sheet by eye does not move anything rigidly. They pin the ends and then ease the copy along its length until it looks continuous with what is on either side. That attacks all three of those properties at once, and this is what it costs.

Easing, as one number #

The model is a blend, because that is what easing is:

$$\text{drawn} = \text{host} + w(s)\,\bigl(\text{copy} - \text{host}\bigr)$$

with $w$ a smoothstep rising from nought to one across a window centred on each seam. At zero window width the blend is a step and the construction reproduces the rigid copy exactly — that is the control, and it is checked to the last digit before anything else is measured. As the window widens, the copy is pulled towards the host’s own coast just inside the seam and pushed away from it just outside.

The half-width of that window, as a share of the whole outline, is the one parameter the rest of this measurement is swept over. It is stated that way rather than in kilometres so that it can be compared with the size of the copied stretch itself, which is thirty per cent of this coast.

That last observation is worth taking slowly, because it is what makes the rest of the result possible. Easing rounds the corner and does not flatten the curve. The middle of a long copied stretch is at full weight however wide the transitions are, so the largest departure falls by only a third across the entire sweep — from 9.4 thousandths of the map’s width to 5.3 — while the corner at the seam falls by a factor of nearly five.

The corner goes smoothly and the seam goes all at once #

The corner is halved by easing over five per cent of the outline and is a fifth of its rigid value by fifty. It falls smoothly and predictably, because a smoothstep spread over a wider window has a smaller second derivative and nothing else in the construction depends on the width.

The seam location does not behave that way at all. It sits between two and five per cent of the outline from the truth from no easing up to thirty per cent of it, and then at forty per cent it jumps to twenty — a factor of 4.6 in one step of the sweep.

A quantity that degrades smoothly and a method that fails suddenly. The reason is that the finder does not need the corner to be large; it needs the corner to be the largest feature nearby. The threshold is crossed where the copy’s departure first exceeds a stated fraction of its own peak, and while the transition is narrow that crossing is sharp wherever it is. Once the transition is wider than the distance from the seam to the point where the copy’s own departure is rising fastest, the crossing slides, and it slides a long way at once. This is the same shape of failure a residual has more than one explanation catalogues from the other direction: the instrument’s assumption fails before its arithmetic does.

The source survives what the seams do not #

This is the finding, and it was not the expected one.

The sinusoidal source is named correctly at every easing up to sixty per cent of the outline and wrongly at seventy-five, where the piece is called an equal-area conic and the true source falls to fourth. The margin over the runner-up falls steadily the whole way — from 21 at five per cent easing to 1.9 at forty and 1.3 at sixty — so the confidence is draining continuously even while the verdict is right.

So the two thresholds come in the order seams first, source second, and the gap between them is nearly a factor of two.

The reason is in the shape of the two tests, and it generalises past this construction. The identification fits a similarity over a whole stretch, which is a least-squares problem in four parameters over a hundred and twenty points; a smooth deformation concentrated at the two ends of that stretch moves four numbers by very little, and what is left goes into a residual that is compared with every candidate’s residual and not with zero. The seam-finder reads a threshold crossing at one point, and there is no averaging in it at all. An estimator that integrates is robust to a local disturbance and an estimator that localises is not, which is the trade the whole of which of these numbers are the samplers is about.

Against the size of the copied stretch the numbers become a practical verdict. The stretch is thirty per cent of the coast. The seams are lost at forty per cent of easing either side of each seam, which is a transition wider than the piece it is joining; the source at seventy-five, which is a transition covering most of the map. Neither is a plausible amount of eye-fitting. A compiler who eased that far would not be joining a copied stretch to a host; they would be redrawing the coast — which is the case a map with no graticule opens and this method has never been able to reach.

What is eased out of the stretch does not leave the map #

A rigid copy leaves the rest of the coast alone, exactly: it is the host’s own line and lies on its own graticule to machine precision. Easing does not have that property, and cannot. The blend pulls the copy towards the host inside the seam and pushes the host towards the copy outside it, so ground that was never copied acquires a departure it has no reason to have. It reaches four per cent of the copied stretch’s peak at twenty per cent easing, eighteen at forty, sixty-one at seventy-five.

That is the compiler’s own work made visible, and it is the reason the method does not simply fail as the easing grows. What easing removes from one part of the coast it deposits in another, where before there was nothing at all to explain — so the total amount of unexplained departure is very nearly conserved and only its distribution changes. Where a fit leaves residuals makes the same observation about a datum transformation: a fit moves a discrepancy around a region and only a change of model removes it.

That conservation answers the question the previous measurement left open in the form it was asked — how much bending a compiler can do before a compilation looks like a projection error. A plain misfit between a conformal conic and a polyconic over this coast leaves 1.03 thousandths of the map’s width. The rigid compilation leaves 9.4, nine times as much; the most heavily eased one leaves 5.3, still five times as much.

There is no amount of easing that brings a compiled sheet down to the level of a badly fitted one, because easing redistributes a departure and does not remove it. A compiler wanting their work to read as an ordinary projection error would have to make the copy smaller, not smooth it in.

Easing and digitising error are two different failures #

The real question for a real sheet is what the two do together, since no archival map offers either alone.

They do not multiply. A rigid copy survives digitising error up to three parts in ten thousand of the map’s width and fails at one part in a thousand, which is the earlier measurement recovered. Heavy easing moves that line by one step — at forty-five per cent easing the source is lost at three parts in ten thousand rather than one in a thousand — and otherwise leaves it alone. What easing does instead is make the seam error large in every column at once.

So the two defeat the method in different ways. Noise takes the source and leaves the seams; a noisy rigid copy still has its corners in the right places, and what fails is the similarity fit, which is now matching a jagged curve. Easing takes the seams and leaves the source; a clean bent copy is still recognisably a sinusoidal piece and nobody can say where it starts. A sheet carrying both at moderate levels is readable, and one carrying both at high levels fails twice over for two unrelated reasons.

An estimator that integrates, and one that localises #

The order of the two thresholds is not a fact about coastlines. It is a fact about the two estimators, and it is worth stating in a form that outlives this construction.

A localising estimator answers a question about one place — where does this quantity first exceed a threshold — and it has no averaging in it anywhere. Its precision is set by how sharply the signal rises at that place, and a disturbance concentrated at exactly that place is the one thing it cannot survive. Easing is such a disturbance by construction: the whole of it is centred on the seam.

An integrating estimator answers a question about a stretch — which projection, of twenty, best fits these hundred and twenty points — and every point contributes. A disturbance at the two ends of the stretch moves four fitted parameters by a little and the rest goes into a residual that is judged against every rival’s residual rather than against zero. The answer is a set is the measurement that made that last point unavoidable: an identification is a comparison, so what matters is not how large the winner’s residual is but how much smaller it is than the next one’s.

Put the two side by side and several earlier results rearrange slightly. The sheet moved before it was measured found paper shrinkage absorbed almost entirely into the fitted parameters, which is the integrating estimator’s robustness seen as a weakness — it swallowed a real effect. Here the same robustness is a strength, and the two are the same property. Rounding is not noise makes the neighbouring observation about a different estimator: what an instrument survives is decided by the shape of the disturbance against the shape of the instrument, and not by the size of either.

So the practical rule is not use the robust one. It is that a compilation analysis runs two instruments with opposite failure modes, and reporting their outputs with the same confidence is what a reader should not do.

What a catalogue entry should say #

The measurement changes what the method’s answer looks like, and the change is small enough to state exactly.

The rigid model invites two crisp numbers: the copied stretch runs from here to here, and it came from a sinusoidal sheet. The sweep says the second half of that sentence is far better supported than the first. At a plausible amount of eye-fitting — say ten to twenty per cent of the outline, well inside what a compiler would do — the source is named with a margin of four to eleven over the runner-up, and the seams are located to three or four per cent of the coast, which on this outline is fifty kilometres of shoreline.

So the honest entry is a point estimate for the source and an interval for the seams:

This stretch of coast, running approximately between X and Y, is drawn from a sinusoidal sheet of the same ground; the ends are uncertain by several per cent of the outline because the join has been eased.

That is a weaker claim than the rigid model offers and a stronger one than nothing, and it is the shape a published coordinate is a result argues every derived quantity should carry — the number and the thing that decides its width, together. The tolerance decides the model is the same argument one level up: what is worth computing is set by what the answer will be used for, and a seam known to fifty kilometres is enough to identify a source sheet and not enough to register two maps against each other.

What each number was checked against #

At zero easing the construction must reproduce the rigid copy exactly. The two profiles are required to agree to within $10^{-12}$ of the map’s width at every one of the 720 points. They agree to the last bit. A difference there would mean the blend was moving the copy rather than easing it, and every threshold below would be measuring the blend’s own arithmetic.

The corner must fall monotonically as the easing widens, within a five per cent tolerance for step-to-step noise. It falls at every step. Easing is defined as spreading the transition, and a corner that grew anywhere would mean something else was happening.

The rigid copy must still be found and named, or the sweep starts from a failure rather than from the previous measurement’s result. It is: the seams at 2.1 per cent of the outline, the source named sinusoidal.

The corner must be halved somewhere inside the sweep, or the range of easing is too narrow to be about anything. It is halved at five per cent.

Both thresholds must fall inside the sweep, and the seam one must come first. They do, at forty and seventy-five per cent.

And a compiled map must depart from its graticule by more than a plain misfit does, checked as a factor of two at the rigid end and as still above the line at the widest easing. It is 9.1 times at the rigid end and 5.1 at the widest. If a compilation were below a misfit from the start there would be nothing detectable to lose, and this whole method would have been measuring a coincidence.

Where the model stops #

One shape of easing. A smoothstep is the natural model for ease it in until it looks right and it is not the only one. A compiler working to a rubber-sheet fit with several control points would leave a departure with structure at each of them, which is a different signal and probably an easier one — more corners, not fewer.

The copy is still a similarity underneath. The blend interpolates between the host and a rigidly placed copy, so the piece in the middle of the stretch is exactly the source’s own shape. A compiler who redrew the stretch by hand would leave something that is not any projection’s image, and the identification would then be fitting noise rather than a smoothly perturbed signal. Nothing here prices that, and it is the harder case.

One region, one pair of projections. Japan on a conformal conic with a sinusoidal source, which is the configuration a compiled map agrees with its graticule except where it was copied published and is kept so the two can be read together. The thresholds would move with the pair — a source more similar to the host leaves a smaller departure and loses its seams sooner — and the order of the two thresholds is what this essay claims, not their values.

And the correspondence is not known to the method but is known to the construction. The departure is measured to the nearest point of the predicted coast, so a drawn point that slid along the coast reads as zero, and easing slides points along as well as across. That hides part of the bending from the profile, in exactly the way the tangential share measured for the rigid case — so the corner numbers here are a lower bound on what a compiler actually did.

Still open: whether a real archival sheet has these two failures in the order this predicts #

The result is a claim about two estimators with different shapes, and it makes a prediction that does not need a synthetic coast to test.

On a genuine compiled sheet, the seam location should be the less trustworthy of the two outputs. If the prediction is right, an archival map should show a clear source identification for a piece whose end points can only be given to within a few per cent of the coast — and a catalogue entry saying this stretch is from a sinusoidal sheet, somewhere between here and here would be the honest form of the answer, rather than the two crisp seam positions the rigid model invites.

What would settle it is a sheet whose compilation is independently documented: a map whose source sheets are known from a publisher’s record, so the true seams are available without being inferred. Then both estimators can be scored against the same truth, and whether the seam error really is the larger of the two is a question about an archive rather than about a construction.