A compiled map agrees with its graticule except where it was copied

Every essay about identifying a map’s projection has assumed the map was drawn in one projection, once. A map with no graticule took the labels away and found that an outline alone recovers the projection exactly, because a scale, a rotation and a shift preserve ratios of arc length. It ended on the case none of those essays can handle: a map that was compiled rather than projected.

Compilation was the ordinary way maps were made for most of their history. A new sheet was assembled from older ones — a coast from one survey, an interior from another, a stretch of shoreline from a chart drawn by somebody else in a different projection — traced, fitted into place, and redrawn under a single graticule that looked, and was, perfectly regular. For such a map no one projection fits the whole of it, and the residual any candidate leaves is a mixture of two things with nothing obvious to separate them: error in the projection and error in the compilation.

The graticule is the one part that is not compiled #

The first thing to settle is what a graticule fit can see, and on a compiled map the answer is nothing. The graticule was drawn last, in one projection, as a frame for everything compiled into it. Every essay from a map does not say what it is onward shows that a graticule fit recovers its projection exactly, and on a compiled map it does: it names the conformal conic, with a residual at the arithmetic’s floor and a margin as large as any clean map’s. That is a correct answer to the question it was asked. The question is about the frame, and the frame is not where the compilation is.

So the evidence has to come from the content, and the content most maps carry is a coast. The graticule, once fitted, says where the coast should be: take the ground outline, put it through the fitted projection, and the result is the coast a map drawn in one projection would show. The compiled map’s coast either lies on that curve or it does not.

How a stretch is fitted by hand #

The model of compilation used here is the simplest one that is not a straw man. A stretch of coast, thirty per cent of Japan’s outline going round from the south-west, is taken from a sinusoidal sheet of the same ground, centred on the same meridian as the host sheet. It is fitted into the conformal conic map the way a draughtsman with tracing paper fits it: scaled, turned and shifted so that its two ends land on the coast either side of the gap. A scale, a rotation and a shift in the plane have four numbers, and two points pinned to two points fix four numbers, so the fit has no freedom left once the ends are chosen. The copied stretch meets the rest of the coast exactly at both ends and departs from where the graticule would have put it everywhere between.

The departure is small. At its largest it is 9.4 thousandths of the map’s width — on a sheet half a metre across, under five millimetres — and a coast is exactly the sort of line that is drawn from uncertain sources, so a departure that size would pass unremarked on any real map. It is also, as a single number, the same size as the residual a wrong projection leaves.

The residual as a profile instead of a number #

Every identification method described so far reduces a residual to one number, a root-mean-square over all the points, because a number is what a ranking needs. On a compiled map that number is the problem. A root-mean-square of a few thousandths says the coast does not fit, and says nothing about whether it does not fit everywhere a little, which is what a wrong projection does, or somewhere a lot and elsewhere not at all, which is what a compilation does.

So the coast’s residual is kept as a profile. Each drawn point is measured against the coast the graticule predicts, by its distance to the nearest point of that curve, with a sign for which side it lies on. That needs no correspondence between drawn points and ground points — nothing has to say which point of the coast is which, the difficulty a map with no graticule was about — because a nearest distance is a property of the curve, not of any labelling of it.

The profile makes the compilation plain. For seven tenths of the way round, the drawn coast lies exactly on the predicted one. At the first seam the departure starts from nothing and grows, reaches its largest a fifth of the way along the copied stretch, crosses back through zero where the two projections’ errors change sign, and shrinks to nothing at the second seam.

Why it starts from nothing is worth a sentence, because it decides everything that follows. The copied stretch was pinned to the coast at its two ends, so at each seam the drawn coast is continuous — but not smooth. The sinusoidal sheet’s coast leaves each seam in a slightly different direction from the conformal conic’s, and the drawn coast turns a corner there: 0.89 degrees at the first seam and 0.69 at the second. A corner under a degree in a drawn line is invisible to the eye, and a draughtsman easing the join would remove even that. It is the whole of what the profile has to work with near a seam, because a line leaving a point at an angle of under a degree gets away from where it should be by under a sixtieth of the distance travelled.

The size of the corner is set by the source. Copied from an equal-area conic, the stretch meets the coast at 0.32 and 0.05 degrees; from a Mercator sheet, at 0.27 and 0.32; from a Lambert cylindrical equal-area, at 11 and 17 degrees, which is a kink a compiler would have seen and smoothed by hand.

That shape is the signature, and it gives a rule with no free parameters on clean data: the seams are the two ends of the longest stretch of coast that agrees with its graticule. Applied to this map it finds them at 55.1 and 84.9 per cent of the way round, against 55.0 and 85.0 where they were made — one point of the 720 at each.

The copied stretch, cut out, names its source #

Once the seams are known, the stretch between them can be treated as a map of its own. Its two ends lie on the predicted coast, so the stretch of ground it covers is known too; and inside it the reproduction is a single scale, rotation and shift, exactly as on an uncompiled map. The argument that made an outline enough for identification applies to the piece: the true source projection’s image of that stretch of ground has the same arc-length parameterisation as the drawn stretch, and every other projection’s does not.

The piece names the sinusoidal, at the floor of the arithmetic, with a margin of 3.6 × 10¹¹ over the polyconic. The host projection comes third: over a stretch that short, a conformal conic and a sinusoidal are not so different, and it takes the exactness of the arc-length match to tell them apart at all. When the answer is not in the library established that the margin, not the residual, is the quantity that separates a correct identification from a ceremony. Here the margin is as large as on any clean map.

So a compiled map is identified piece by piece: the graticule names the frame, the profile finds the seams, and each piece between seams names its own source. That is an answer to the question a map with no graticule left open, and it is worth being exact about how far it goes.

A projection error and a compilation, told apart #

The claim was that the residual of a compiled map is a mixture of construction and compilation with no way to separate them. The separation is in the shape, and the three coasts below have residuals of about the same size.

The first coast is the refusal the rule must pass: drawn wholly in the graticule’s projection, it departs nowhere, and the finder reports that nothing was copied. The third is the refusal that matters more. Its whole coast was copied from the sinusoidal sheet and fitted to the graticule as a whole, by the best scale, rotation and shift over every point, which is what a map drawn in the wrong projection looks like to anybody holding its graticule. Its largest departure, 10.1 thousandths of the map’s width, is a little larger than the compiled coast’s 9.4, and its root-mean-square departure, 2.9 thousandths, is the same size as the compiled coast’s 1.9. But it agrees with its graticule nowhere. Its departure passes through zero at a handful of isolated points, where its error changes sign, and never stays there, so the finder reports no seams, because there is no stretch of agreement to have ends.

A single number cannot tell those three maps apart in the right way: the two wrong ones have residuals of the same size, and one of them is a coast in the wrong projection while the other is a correct coast with a patch in it. The profile tells them apart at once. A residual has more than one explanation found a datum shift hiding inside a residual the size of noise, and the lesson there was that a residual is evidence for a set of explanations rather than for one. The lesson here is the complementary one: keep the residual as a field, and the set can shrink.

What the profile cannot see #

Two limits come with measuring against the nearest point rather than a known one.

The first is that a copied stretch can be wrong along the coast as well as across it. Where the sinusoidal sheet puts a headland a little further along the shore than the conformal conic would, the drawn coast still lies on the predicted curve — the headland has slid along it — and a nearest distance reads zero. The construction knows which drawn point came from which point of the ground, and that correspondence can be used once, to measure how much of the copied stretch’s displacement lies across the coast, where the profile sees it, and how much along it, where nothing that lacks the correspondence can.

For the sinusoidal stretch only 23 per cent of the displacement lies across the coast, and for the polyconic 26 per cent. The rest is along it and invisible to a profile. For the other sources the visible share runs from a third to three quarters. A compilation’s seams are found by what little of its error happens to point across the line being compared.

The second limit is the source itself. Copied from an equal-area conic — a sheet in another conic projection of the same ground — the stretch departs by at most 1.26 thousandths of the map’s width, and from a polyconic by 1.61. Those are well under a millimetre on a half-metre sheet, and they are as far as the departure ever gets: the seams they leave are real and cannot be found by any measurement made with a pencil’s accuracy. Copied from a Mercator sheet the largest departure is 14.3 thousandths, from a Mollweide 20.3, and from a Lambert cylindrical equal-area 92.9 — a stretch so badly out of shape that a careful compiler would have noticed it by eye. Two projections that cannot be told apart measured the extent below which two projections are the same picture. The same fact decides which compilations are visible: stretches copied between similar projections leave seams that are there and cannot be seen.

Seams are harder to find than projections #

On clean data the seams are found to one point. Real coasts are digitised by hand, and the departure’s own shape makes noise costly in a specific way.

A departure that starts from nothing is, near its seam, smaller than any noise, so the first point that clearly departs is some way past the seam: the threshold divided by the rate at which the departure grows. With noise a tenth of a thousandth of the map’s width, and the agreement threshold at three times that, the seams found lie 2.8 per cent of the coast’s length from the true ones; at three tenths of a thousandth, 6.5 per cent.

The obvious repair is to use the departure’s growth rather than its size. Near a seam it grows roughly in a straight line, so a line fitted to the profile just inside the copied stretch and run back to zero puts the seam where the growth started. That halves the error at a tenth of a thousandth and cuts it by four fifths at three tenths, to 1.3 and 1.4 per cent. It costs something on clean data, where the departure is not quite straight over the forty points the line is fitted to and the extrapolation lands 0.8 per cent away. Neither rule survives a noise of a thousandth of the map’s width: they put the seams fifteen and twenty-one per cent of the coast away, which on a copied stretch thirty per cent long is no location at all.

That sets the scale of what can be done. A map with no graticule found that an outline identifies a projection through a noise of a thousandth of the map’s width and fails only at three. The seams of a stretch whose largest departure is 9.4 thousandths are lost at one. Locating a compilation’s seams to within a couple of per cent of the coast needs its largest departure to be some thirty times the digitising error, and the reason is the corner: the departure a seam leaves is under a degree’s worth of divergence, and noise hides the first part of it however large the rest becomes. On a sheet half a metre across, a careful hand digitises to a few tenths of a millimetre, which is about where the seams of a sinusoidal stretch in a conformal conic map stop being findable, and far above what an equal-area conic stretch needs.

What was assumed, and what that leaves open #

The measurement makes four assumptions, and each of them flatters the method.

The copy was fitted by a similarity. Two ends pinned, nothing else adjusted. A draughtsman fitting a stretch by eye also bends it between the ends to make it look right, and a copy through a stretched photocopy picks up exactly the distortion what a careless copy hides measures. Either makes the departure grow from the seam along a curve rather than a line, and either removes the exactness that lets the piece name its source.

There were two seams. A compiled map with many copied stretches has many, and the longest stretch of agreement is then the longest uncompiled piece, which may be short. The rule generalises to every stretch of agreement longer than a threshold, and the threshold then has to be chosen.

The ground outline is exact. Every departure here is measured against the ground curve put through the graticule’s projection, and a real coast’s ground position depends on its dataset and its generalisation. A boundary that two features share prices what a simplification tolerance does to a line, and a compiled map compared with a coastline generalised differently from its sources shows departures that are neither projection nor compilation.

The graticule is right. It was fitted first and taken as given. The sheet moved before it was measured shows that on paper even that is a curve of possible parameters rather than one, and every departure measured here would then carry the graticule’s uncertainty as well.

The answer is a set is where identification learned to report the members consistent with the evidence rather than one winner, and a compiled map sharpens what the members are. They are not projections any more but assignments: which stretch of coast is in which projection, with the seams among the unknowns, and a profile is the instrument that makes that set small.

Still open: a copy bent into place #

Everything above rests on the copied stretch having been moved as a rigid, scaled piece. The departure then grows from each seam in a nearly straight line and the piece between the seams is an exact image of its source.

A stretch fitted by eye is not moved that way. It is pinned at the ends and then eased between them until it looks continuous with its neighbours, which spreads its departure out, softens the corners of the profile at the seams, and leaves a piece that no projection in the library fits exactly. Whether the seams of a bent copy can still be located, whether the piece still names its source once the bending has been fitted away, and how much bending a compiler can do before a compilation looks like a projection error, are questions a rigid copy cannot ask.