The sheet moved before it was measured

Nine rungs of this anchor take a set of control points off a map and fit a library of projections to them. The whole method depends on the points being where the cartographer put them.

They are not, and the reason has nothing to do with cartography. Paper is a mat of fibres laid down wet, and the fibres swell across their own length far more than along it — so a sheet taking up or giving off moisture changes size anisotropically, by two or three tenths of a per cent across the grain and about a tenth along it. Every map printed before the second half of the twentieth century was measured off a sheet that had been doing this since the day it was printed.

The shrinkage is exactly a projection parameter #

What a careless copy hides prices a photocopier that stretches one axis and finds that removing it costs more than it looks: an affine fit absorbs the entire difference between the cylindrical equal-area projections, so Gall–Peters and Behrmann become the same picture to sixteen decimal places.

This rung is the sharper version of the same trouble, and it is sharper because the shrinkage is not a nuisance to be removed. It is one of the things being measured.

An equirectangular projection with standard parallel φ₁ draws x = λ·cos φ₁ and y = φ. Changing φ₁ scales x and leaves y alone. A sheet whose grain runs along the map’s own axis shrinks x and leaves y alone. They are the same transformation, and no fit of any kind can tell them apart.

Nought point five seven degrees per per cent, and the arithmetic behind it is one line: the recovered parallel satisfies cos φ₁′ = cos φ₁·(1 − p), so to first order Δφ₁ = p·cot φ₁ — which at 45 degrees is p radians and grows without bound towards the equator.

The consequence is not that the identification fails. It is that the identification succeeds and reports a wrong parameter, with no residual to warn anybody. A map drawn at 45° on a sheet that has taken up one per cent of moisture is a map drawn at 44.43°, as far as any fit can see, and the two hypotheses are not merely hard to separate — they are the same hypothesis.

The two horns #

That leaves a choice with no third option, and both sides of it have been measured in this anchor.

Fit a similarity, and the sheet is charged to the projection. A similarity has four freedoms — a scale, a rotation and two shifts — and none of them is an anisotropic stretch. So paper shrinkage arrives in the residual exactly as though it were the wrong projection.

Fit an affine, and the sheet is removed along with the family. The affine fit’s six freedoms take the paper out completely, and by the same six freedoms it loses the ability to separate any two projections that differ by an affine map. Rung 3 measures that: within the cylindrical equal-area family the affine residual falls to zero rather than merely falling, so the family collapses to one map.

So a person identifying a projection from a paper map must either believe a residual that contains the sheet, or use a fit that cannot distinguish a family of three.

The refusal this rung needs #

A measurement of what a shrunk sheet does is only worth what the unshrunk case says, and the unshrunk case is the check.

With no shrinkage at all, both fits identify the map: the similarity fit’s residual is 1.3 × 10⁻¹⁰ and the affine fit’s 1.2 × 10⁻¹⁰, both at the parameter search’s own tolerance rather than at anything geometric. A run that failed there would be measuring the fitter and every number afterwards would be about the solver.

The second half of the refusal is that the affine fit must pay for absorbing the sheet, and it is checked on the family rung 3 established: the affine residual between Lambert’s cylindrical and Gall–Peters must be zero rather than small. It is 3.2 × 10⁻¹⁵, against 1.0 × 10⁻¹ for a similarity fit on the same pair.

Those two together are what make the two horns real. Without the first, the similarity fit’s rising residual could be a solver drifting; without the second, the affine fit would be a free improvement and there would be no dilemma to write about.

What the sheet does to the margin #

When the answer is not in the library establishes that a residual on its own is a ceremony, and that the quantity that separates a real identification from a forced one is the margin — how much worse the second-best candidate is. That is the quantity paper shrinkage destroys.

On a clean sheet, a conformal conic is identified with a margin of 3.7 × 10⁷: the runner-up is thirty-seven million times worse. At five hundredths of a per cent of differential shrinkage the margin is 76. At four tenths — the middle of the real range — it is 8.9. At 1.6 per cent it is 1.77, and a margin of 1 is where the wrong projection wins.

Nothing about the residual warns of this. The residual at four tenths of a per cent is 4.9 × 10⁻⁴, which is half a part in a thousand of the map’s width and would be a triumph in any other context. What has happened is that every candidate’s residual has risen by roughly the same amount, so the ratio between the best and the second-best has collapsed while the best stayed small.

That is the general shape of the failure and it is worth stating apart from paper: a systematic error common to all candidates destroys a margin without touching a residual. A datum shift does it, as the datum hides inside the projection’s parameters shows; so does a sheet.

The other reproductions, ranked #

A paper map has been through a chain of transformations before anybody measures it, and the sheet’s own movement is one of several. Putting them in order is worth doing because the chain’s terms behave differently.

The plate. A copper or stone plate has its own dimensional history, and printing transfers ink under pressure that stretches the sheet along the direction of travel. Both are anisotropic and both are indistinguishable from the sheet’s own shrinkage by anything measured here.

The sheet. Two or three tenths of a per cent differentially, the subject of this rung, and reversible in principle and not in practice.

The photocopier or the scanner. What a careless copy hides is about this one, and it can be a per cent or more on a bad machine — larger than the sheet, and with the advantage that the copy can be remade.

The measurement. Reading a crossing off a sheet with a rule or a digitiser is good to a few tenths of a millimetre on a sheet half a metre across, which is a part in a thousand or two per point — but random, so it averages down over many points and does not bias a parameter.

The last of those is the only one this collection’s earlier rungs modelled. The answer is a set prices what a control point’s own width does to an identification and finds that at one per cent noise a four-degree region admits ten of twenty candidates. Every term above it is systematic, does not average down, and is absorbed by the fit rather than showing in it.

Where the grain runs turns out not to matter #

A reader who has followed this far will expect the angle of the grain to be a parameter with consequences, since the shrinkage’s axes are the paper’s and the projection’s axes are the map’s.

That is a genuine simplification and it is worth having: the sheet contributes one number to the problem rather than two. The differential shrinkage matters and the grain’s angle does not, so a person estimating what a paper map’s age is worth needs to know how much the sheet moved and not which way it was cut.

The exception is the exact-absorption case above, and it is an exception about the projection rather than about the fit. An equirectangular’s parameter scales x alone, so it can absorb a shrinkage whose axes happen to be the map’s and not one at 23 degrees to them. A conic’s parameter is not an axis scaling at all, which is why the conic’s residual rises with the shrinkage while the equirectangular’s does not.

What would separate them, and why nothing here does #

If the sheet and the scale are the same transformation, is there anything that could tell them apart?

Not more control points. Where the control points are establishes the general shape of this: a parameter that is invisible in a configuration stays invisible however many points are added, and five hundred and twelve on one parallel are as blind as eight. Here the degeneracy is exact and global, so no arrangement of crossings helps.

Not a better fit. The two hypotheses produce identical predictions at every point of the map. No estimator can separate hypotheses that make the same predictions; that is not a limitation of least squares.

Something off the map, then. The sheet’s own edges, if the original trim size is known. A printed scale bar, which shrinks with the sheet and so is not affected — a measurement made with the map’s own scale bar is immune to uniform shrinkage and still wrong under anisotropic shrinkage, in one direction only. A second copy of the same map on a differently-cut sheet, whose grain runs the other way. Or the plate itself.

Every one of those is external evidence, which is the honest conclusion: the sheet’s contribution is not recoverable from the sheet. The answer is a set makes the same move one rung down, replacing a best fit with the set of candidates a control point’s own width cannot rule out; the set here is a one-parameter family of (projection parameter, shrinkage) pairs, and narrowing it needs something the map does not carry.

What a real sheet actually does #

Every number above is applied shrinkage, and the honest question is what range is real.

Paper’s dimensional response to humidity is between one and three tenths of a per cent per ten per cent of relative humidity across the grain, and about a third of that along it. A sheet that has gone from a damp printing house to a dry archive has moved by several tenths of a per cent differentially, and one that has been through a century of seasons has done it repeatedly, not always reversibly.

So the middle of the real range is the four-tenths row: a similarity residual of 4.9 × 10⁻⁴, a margin fallen from thirty-seven million to nine, and a standard parallel wrong by a quarter of a degree if the grain happens to lie along the map’s axis.

None of that is catastrophic and none of it is negligible, and the reason it has not shown up in the earlier rungs is that they measure distinguishability — how large a region two projections need before they can be told apart — rather than accuracy of the recovered parameters. A shrunk sheet leaves the first almost intact and moves the second by more than the measurement’s own precision.

Why this is a wrong question and not a practice one #

The anchor sits in the field about what is taught wrongly, and it is worth saying which part of this belongs there.

Not the paper. That paper shrinks anisotropically is well known to anybody who has handled an old sheet, and conservators measure it routinely.

What is taught wrongly is the separation. The standard account of identifying a projection from a map treats the reproduction — the scale, the rotation, the offset, and sometimes a photocopier’s stretch — as a nuisance transformation to be removed before the real work starts, and treats the projection’s parameters as the thing being recovered afterwards. Two clean steps, in order.

They are not two steps. The nuisance and the answer live in the same four numbers for a cylindrical map, and removing the first removes some of the second. Report the map, not the parameters makes a neighbouring point about publication — a set of parameters is not a map unless the convention behind them is stated — and this is the same confusion arriving at the other end of the process, where a recovered parameter is reported as though it described the cartographer’s choice when it describes the cartographer’s choice plus a century of weather.

What this does to the anchor’s own numbers #

Nine rungs of measurements were made on synthetic maps drawn by this collection’s own code, so no sheet has ever entered any of them. What the rung changes is the interpretation.

The separability extents are unaffected. Two projections that cannot be told apart finds that two conformal projections need twelve degrees of extent before their graticules separate. That is a statement about the shapes and a shrunk sheet does not change a shape’s identity, only its aspect ratio.

The margin thresholds are affected and by a lot. The rule that a margin of 1.6 means the truth is absent and 10¹³ means it is present was calibrated on clean observations. On a sheet at the middle of the real range, a present truth gives a margin of nine — which on the clean calibration would read as absent.

And the recovered parameters carry an unremovable bias whenever the projection has a parameter that scales one axis. That is the equirectangular’s standard parallel here; it is also every cylindrical’s, and it is why the family this anchor keeps returning to is the one most exposed.

What a person holding an old map should do #

The rung produces one practical rule and one honest admission, and they are worth separating from the measurements.

The rule: report the pair, not the parameter. A standard parallel recovered from a paper map is not a number; it is a curve in a two-dimensional space of (parallel, shrinkage) pairs, every point of which fits the observations equally well. Reporting the middle of that curve as though it were a measurement is the failure. Reporting it as “45.6°, on the assumption of no shrinkage, moving by 0.57° per per cent” is a complete statement and costs one clause.

The admission: for most maps the shrinkage is unknown. A sheet’s original size is rarely recorded and its grain direction almost never is, so the assumption of no shrinkage is not a conservative one — it is a guess that the sheet has not changed, which it certainly has. What can be said is the size of the effect, which the ladder above gives, and the direction, which is unknown without knowing whether the sheet is drier or damper than when it was printed.

Two ways out exist and both need something the map does not carry. A second copy printed from the same plate on a differently-cut sheet gives two shrinkages at right angles and separates them. And a known ground distance between two identified features fixes the scale directly, which turns the two unknowns into one — the same manoeuvre where the control points are uses to break a degeneracy, applied to a degeneracy nothing on the map can break.

The map that has no crossings at all #

Every rung of this anchor, including this one, is handed control points: graticule crossings whose latitude and longitude are known, read off a sheet whose accuracy is the subject.

A great many maps have no graticule. A road atlas page, a hand-drawn estate plan, a bird’s-eye town view, a great deal of what is in archives — none of them carries a single labelled crossing, and the identification method this anchor is built on has nothing to be handed.

What such a map does have is a coastline, a river, a boundary: features whose ground positions are known from elsewhere but whose correspondence with the drawn ink is not given and has to be found. That is a harder problem of a different kind, and nothing in ten rungs addresses it.