A second pin is a measurement of the places

Two numbers about the paper buy nothing about the places counted what a flat picture of four cities can hold when some of its constraints are about the sheet rather than about the places. The count was clean. Distances and bearings, being statements about how two places stand to each other, are blind to where the picture sits, and together they can fix $2n-2$ of its $2n$ numbers. Two pins fix the other two. Every independent set factorised into a choice of pins and a choice of place constraints, and nothing was ever exchanged between the two accounts.

That essay ended on the case the count cannot reach. A real pin has an error, and so has every distance and bearing. A picture fitted to all of them by least squares has one misfit to minimise, and the fit is free to move a place to satisfy a pin or to move the registration to satisfy a distance. Whether the two accounts stay separate once everything has a tolerance is a question about where the errors go, and a fit’s covariance answers it exactly.

Where an error goes, exactly #

The fit is linear once it is written about the best flat picture. Each distance is a row saying how the gap between two places changes when they move; each bearing a row saying how the direction between them turns; each pin a row with a single one in it, saying where a place is on the sheet. Every row carries a weight, the reciprocal of its variance, and the fitted picture is the weighted least-squares solution.

A linear estimate is a weighted sum of the observations, so its covariance is a sum of one term per observation: how much that observation moves the estimate, squared, times that observation’s variance. The terms can be grouped. Adding up the ones for the pins gives the error the pins put into the picture, and adding up the others gives the error the distances and bearings put there. The two groups add, in their squares, to the whole.

Two things are read off each group. The shape is every city’s position less the centroid of the four: it is what the picture looks like, and no shift of the picture on the sheet changes it. The registration is where the pinned cities sit on the sheet, on average. The question the count left is whether the pins’ error stays in the registration and the places’ error stays in the shape.

The tolerances are stated. A distance good to one per cent of its length, a bearing to a hundredth of a radian — about 0.57 degrees — and a pin to fifty kilometres on a picture about fifteen thousand kilometres across. Every result is also given against the pins’ tolerance, since that is the one number a registration really decides.

One place pinned: the sum survives exactly #

With both pins at one place, the pins spend exactly the two numbers the distances and bearings cannot see. The fit cannot use them to improve the shape, because they carry no information about it: a picture moved as a whole satisfies every distance and bearing equally well. So the fit reproduces the pins exactly, with no residual, and moves the whole picture to wherever they say.

The measurement agrees. With London pinned, the pins put 0.00 kilometres into the shape — zero to the precision of the arithmetic, at a pin tolerance of one kilometre, of fifty and of a thousand — and the distances and bearings put 63.0. The pins’ whole error is a displacement of the entire picture, 70.7 kilometres against the sheet for pins good to fifty in each direction, and none of it is in what the picture looks like. The direct sum of the count survives the arrival of tolerances without any change.

More places pinned: the sum survives in half #

Pin a second city and the account changes. Two pinned places are four numbers, and only two of them are the translation nobody else can see. The other two are a statement that New York is a particular distance and bearing from London on the sheet — which is exactly what a distance and a bearing between them say. The fit now has two sources for the same fact and weights them against each other, so the pins’ error reaches the shape.

How much depends on the tolerance, and the curve has a peak for a clear reason. Pins far worse than the distances and bearings are nearly ignored, so their share of the shape falls to nothing as their tolerance grows. Pins far better have little error to give, so for two and three pinned places the share falls towards nothing as they become exact too. In between, the pins are good enough to be listened to and bad enough to matter: 4.6 per cent of the shape’s variance at two places, 20.7 per cent at three, both near a tolerance of forty kilometres.

Four pinned places are different at the exact end. Four pins at four places are eight numbers, a complete drawing of the picture, and pins good to a kilometre leave the distances and bearings nothing to decide. The pins then supply all of the shape’s error — a hundred per cent, and a small one, since the pins are good.

That is the half of the direct sum that fails: every place pinned after the first is a measurement of the geography, arriving on the paper’s account.

The other half survives, at any number of pins #

The error the distances and bearings put into the registration is zero with one pinned place, and zero with two, three and four. That does not weaken as pins are added, and the reason is the same blindness the count was built on. Every distance and bearing is unchanged by moving the whole picture, so none of them can say anything about where its average position is. The only rows that can are the pins, and in a weighted fit the average of the fitted pinned positions — weighted as the pins are — is forced to equal the average of the pins themselves.

So the registration is 70.7 kilometres with one pinned place, 50.0 with two, 40.8 with three and 35.4 with four: the pins’ own error averaged over however many there are, and nothing else. A cartographer’s distances could be excellent or dreadful and the picture’s place on the sheet would be the same.

Put together, the direct sum does not survive tolerances as a statement about pins and places. It survives as a statement about the average of the pins and everything else. The pins’ average stays on the paper exactly. The pins’ spread about their average is geography, and it trades with the distances and bearings like any other measurement of the places.

Weighted as they deserve, pins never make the shape worse #

A measurement of the places that has a tolerance and is weighted by it can only help, and the shape’s error shows it. Pinned at one place the shape is 63.0 kilometres whatever the pins are like. Pins at two places good to fifty kilometres bring it to 61.1; at three, 53.5; at four, 42.6. With pins good to a kilometre at all four places the shape is 1.2 kilometres, because four pins that good are a far better drawing of the four cities than a set of one-per-cent distances.

None of that contradicts the count. The count said a largest picture holds the same six place constraints with pins and without, and that is still true: the extra pins beyond the first place are not buying capacity for distances and bearings. They are distances and bearings themselves, written as positions on the sheet, and a fit that knows their tolerance counts them as such.

Held exact, a pin helps only while it is better than the shape already is #

The weighted fit is not how registration is usually done. A scanned map is registered to four corner ticks and the ticks are held exact; a compiled sheet is pinned to two corners of the sheet it continues and the corners are treated as fixed. A pin held exact has infinite weight whatever its real error.

Held exact, a pin at one place is still harmless: its error moves the whole picture and nothing else, so the shape is 63.0 kilometres whether the pins are perfect or wrong by a hundred. Held exact at more places, the pins’ real error goes straight into the shape. With two pinned places each kilometre of real pin error adds 0.71 kilometres to the shape’s error in quadrature; with three, 1.03; with four, 1.22 — at four places the pins are the whole shape, and their error is its error.

Each count therefore has a crossing. Two pins held exact beat a single pin until their real error reaches 37 kilometres; three until 46; four until 51. The single pin’s shape error is 63, so the crossings sit between 0.58 and 0.82 of it. An exact pin at a second place improves the geography only while it is more accurate than roughly two thirds of the geography’s own error, and beyond that it drags every city towards a sheet that is wrong about them.

The practical form of that is familiar to anybody who has registered a scan. Four corner ticks held exact force the map’s interior to fit them; if the ticks are good the fit is better than the map’s own drawing, and if the paper has shrunk unevenly or a tick was misread the whole interior is distorted to agree. The sheet moved before it was measured finds paper shrinkage absorbed into a projection’s fitted parameters, and this is the same mechanism measured from the other side: a registration held exact at more than one place is a claim about the geography that the geography has no way to contest.

Why the paper has an average and not a shape #

The two results fit one picture, and it is worth stating plainly because it is the part that generalises beyond four cities.

A constraint’s reach is decided by what it is blind to. Distances and bearings are blind to translation and to nothing else once both kinds are present, which is the result a mixed picture holds one more, and it is north counted. So they determine everything about the picture except two numbers, and those two numbers are, precisely, its average position. Pins are blind to nothing, so they reach everything — including the shape. A set of pins therefore has a part in the space the place constraints cannot touch, which is its average, and a part in the space the place constraints already cover, which is its spread.

The first part cannot be contested by any place measurement and cannot contaminate one. The second part is contested by every place measurement and contaminates them in proportion to how much it is trusted. That is why the average stays separate at every count and the spread does not, and why a single pinned place is the one arrangement in which the spread is empty.

It is also why the right number of exact pins is one place’s worth. A grid has an origin that is not there makes the same point about a false origin: a decision about where the whole thing sits is free exactly because nothing measured can argue with it. A second decision of that kind is not free. It is an unmeasured measurement.

Many pins are a transformation #

A set of pins at many places is how most real registrations are made, and the split says what kind of object that is. What another common point buys fitted a seven-parameter transformation between two datums through a set of common points — each one a pin at a place, stated in both systems — and found that adding points reduced the transformation’s own error while leaving untouched the distortion no seven parameters could follow. Read through the split here, the common points’ average fixes the transformation’s shift and nothing else can, and every other parameter it fits — rotation, scale — is fitted from their spread, which is a measurement of the geography carried on the paper’s account. That is why more points help those parameters and why nothing about the points can reach the distortion: the distortion is in the geography, and the points’ spread is only a second, noisier drawing of it.

The same structure is the reason the control carries the error of every network above it: a local survey held to three control stations is a picture pinned exact at three places, and the control’s error arrives in its shape by exactly the route measured above, while its position against the origin is the control’s average and nothing the local survey measured.

And it is the reason a map with no graticule is hard rather than impossible. With no pins at all a picture has no registration, and a registration is two numbers nobody can measure from the geography; everything else a map carries can in principle be read from its shape.

How the split was checked #

Pins at one place must put nothing into the shape, at tolerances of one, fifty and a thousand kilometres. They put less than a millionth of what the places put, which is the arithmetic’s zero.

The registration must take nothing from the places, at one, two, three and four pinned places. It takes less than a millionth of its whole error at every count.

Correctly weighted pins may never make the shape worse. At every count and every tolerance from one kilometre to a thousand, the shape’s error is no larger than the single pin’s 63.0 kilometres.

Trials at full nonlinearity must scatter as the linear split says. Fifteen hundred seeded sets of noisy distances, bearings and pins, fitted by iterated weighted least squares from the true picture, put the shape 61.8 kilometres off and the registration 49.8 with two pinned places, against 61.1 and 50.0 computed. With four pins held exact but really wrong by forty kilometres, fifteen hundred more fits put the shape 48.5 kilometres off and the registration 28.6, against 49.0 and 28.3.

Where the fit stops #

The picture is linearised. Every covariance is taken at the best flat picture, which is where the fit ends up when the errors are small against the picture. The trials run the full nonlinear fit and agree to about one per cent, at errors of tens of kilometres on a picture fifteen thousand across.

The tolerances are stated and uncorrelated. A real set of distances measured from one source shares a scale error, and a real registration’s corner ticks share whatever the paper did. The weights are a guess the solve believes is the general case; correlated pins would put part of their shared error into the average and part into the spread in a proportion this essay does not measure.

Four cities. The structure — average on the paper, spread in the geography — is a statement about blindness and holds for any number of places. The crossings, 0.58 to 0.82 of the single pin’s shape error, belong to this set and these tolerances.

The flat picture is taken as true. Every observation here is generated from the best flat picture itself, so the fit has no misfit beyond its stated tolerances. The real distances between four cities spread across the Earth have no flat picture that holds them, and the pair that cannot be held is not the one the geometry names measured that misfit on its own. It would sit in the distances’ and bearings’ residuals on top of everything measured here, and a pin at a second place would be one more constraint for it to be spread across.

Still open: the pins a projection supplies #

Every pin here was a stated position for one place. A map projection is a pin for every place at once: a formula that says where each city goes on the sheet, with an error that is not independent from city to city but is the projection’s own distortion, smooth and correlated across the whole picture.

Read through this split, a projection is a registration with an enormous spread, and the spread is exactly its distortion. A projection that preserves no distances pins every place to a sheet that disagrees with every distance by a pattern. Whether a projection’s distortion splits the same way — an average that leaves the geography alone and a spread that competes with it — and whether the least-distorting projection for a region is the one whose spread a weighted fit would trust least, are questions four independent pins cannot ask.