A projection pins every place, and its error does not average

A second pin is a measurement of the places fitted a flat picture of four cities to distances, bearings and stated positions, each with a tolerance, and found the stated positions split cleanly in two. Their average fixes where the picture sits on the sheet, and nothing measured between the places can contest it. Their spread about that average is a second, noisier drawing of the places, and it trades with the distances like any other measurement of the geography.

That essay ended on the extreme case. A map projection is a stated position for every place at once, and its error is not a scatter but the projection’s own distortion, one smooth function. A cartographer who knows the formula can invert it and has no error to worry about. The commoner case is a printed map that is all there is — positions read off a sheet and registered to a survey’s measured distances, the distortion unmodelled. The sheet’s positions are then pins, and the questions are how their error divides and whether a fit that uses them knows what it is doing.

A reference that is not a map #

A projection’s error needs something to be measured against, and there is no flat truth. The best flat picture is not a map found that dots fitted to a set of distances beat every projection at holding them, and how wrong a flat picture has to be found that such a picture’s error grows as the square of the region’s span. The reference here is that picture for the twenty-four cities, fitted by least squares to their great-circle distances with each weighted by the reciprocal of its own square.

Its largest relative error over the 276 pairs is 0.61 per cent, across a region 3,906 kilometres from Lisbon to Moscow. It is turned so that north at 15°E 50°N points up, and it sits 2.37 kilometres in root-mean-square from the azimuthal equidistant map centred there. That 2.37 is the limit of everything below: a projection whose error comes out at a kilometre or two is at the reference’s own departure from the sphere, and a difference between two such projections is a statement about the reference.

Seventeen projections are compared against it, in three groups. Nine world maps are drawn as they come, with Greenwich at the centre. Two conics are placed for somewhere else: the Lambert conformal and the Albers equal-area with standard parallels at 20° and 60° and their central meridian at Greenwich, which is a reasonable map of a broad mid-latitude band and a careless one of Europe. Six regional maps are placed for this region, conics on 15°E with standard parallels at 43° and 57° and azimuthals centred on 15°E 50°N.

Four of the numbers belong to the paper #

The first thing removed is the shift that takes the projection’s origin to the reference’s, which is pure convention — a grid has an origin that is not there for exactly this reason. What is left after the shift divides further.

Mercator is 661.7 kilometres off after the shift alone. A turn takes that to 660.3, because Mercator’s north is the reference’s north to within 1.73 degrees. A size takes it to 153.8, because a Mercator map of Europe is 1.56 times larger than Europe — the secant of the region’s latitude, near enough. Of the squared error, 0.4 per cent is turn, 94.2 per cent is size, and 5.4 per cent is pattern. Miller is the same story at 91.0 per cent size, and the plate carrée at 74.7.

The misplaced Lambert conic is the opposite case. With its central meridian at Greenwich and Europe’s centre fifteen degrees east, its meridians converge on a pole that is not above the region’s middle, and the whole map is turned 10.05 degrees against north at 15°E. That turn is 92.2 per cent of its squared error. Its standard parallels are too far apart for a band of twenty-four degrees, so it draws the region 4.6 per cent too small, and that is another 6.7 per cent. What is left is 21.4 kilometres, 1.1 per cent of what there was. The misplaced Albers does the same with the turn and draws the region at the right size, since an equal-area map cannot do otherwise, and leaves 65.6.

The world maps without a mean scale error — Mollweide, sinusoidal, Gall–Peters, Eckert IV — have nothing to give up to the size. Mollweide and the sinusoidal give up 30.3 and 47.5 per cent to a turn, because their meridians lean at 15°E and the reference’s north does not. Gall–Peters gives up 1.5 per cent to the turn and 6.6 to the size, and is 91.9 per cent pattern.

Every one of those shifts, turns and sizes is something a registration fits for itself. Nobody lays a printed map over a survey without sliding, turning and scaling it to agree; a Helmert transformation is those four numbers, and what another common point buys priced its seven-parameter cousin between two datums. For any use that begins with a registration, a projection’s only error over a region is what those four numbers leave.

What is left follows how much the scale varies #

A map that is everywhere too large by the same factor is a perfect map at the wrong scale. What no registration can remove is a scale that differs from place to place, because no single factor fits every place at once. That predicts what should govern the residual: not how far a projection’s scale is from one on average over the region, but how far it wanders about its own average.

It does. Across the seventeen projections the correlation between the error left and the scale’s spread is 0.992; against the size of the average scale error it is 0.516. Among the nine world maps alone the two are 0.974 and 0.394.

The sharpest case is the pair a reader would least expect. Gall–Peters is equal-area, so its average areal scale over any region is exactly one, and Mercator’s average linear scale over these cities is 1.57. Read as distortion in the usual way, Gall–Peters is the far better map of Europe. Read as pins, after the registration every printed map receives, Mercator is closer: 153.8 kilometres against 172.3, because over the cities’ twenty-two degrees of latitude Mercator’s scale has a spread of 0.136 in its logarithm and Gall–Peters’ 0.171. Mercator’s large, uniform error is almost all removable; Gall–Peters’ small one is almost all pattern.

A national grid is designed around the same quantity. The scale factor was chosen found Britain’s factor spending the free size to centre the scale’s range on one, and Chebyshev’s map is the best at its worst and not on average found the conformal map with the least range of scale over a region. Both treat the level as free and the spread as the cost; a registration makes the same judgement, in kilometres.

The order of the maps changes #

The registration also reorders the maps. Ranked by error after the shift alone, Mercator is last of seventeen, Miller sixteenth and the plate carrée fifteenth, with the two misplaced conics just above them. Ranked after shift, turn and size, the misplaced Lambert conic rises from fourteenth to seventh — 21.4 kilometres, closer to Europe than any world map by a factor of five — and the misplaced Albers rises from thirteenth to eighth. Mercator rises three places to fourteenth. Gall–Peters falls from twelfth to sixteenth, and the plate carrée falls to last, because its scale along the parallels runs from 1.27 at Athens to 2.01 at Helsinki and no size fits both.

Among the regional maps the order should not be read closely. The azimuthal equidistant and Lambert azimuthal leave 1.6 and 1.9 kilometres, the stereographic 4.1, the Albers 4.3 and the Lambert conic 5.1 — all within a factor of three of the reference’s own 2.37 — and the gnomonic 14.4, the one regional map whose scale grows fast enough from its centre to matter at this size.

What is left is one displacement drawn many times #

In a second pin is a measurement of the places each pin’s error was independent of every other. A projection’s leftover is a different kind of object, and the first figure’s right-hand panel already shows it: neighbouring cities are displaced alike.

The measure is direct: each place’s displacement multiplied as a vector by its nearest neighbour’s, summed, and divided by the sum of the squared displacements. For sixty-four seeded places the regional Lambert conic gives 0.845, the stereographic 0.873, the gnomonic 0.870, Mercator 0.882 and Mollweide 0.909. Independent errors of the same size give −0.013.

That is not a surprise, and it is the whole of what follows. A projection’s distortion is smooth, so what a registration leaves of it is smooth too. The regional conic’s scale depends on latitude alone, below one between its standard parallels and above one outside them, so once the best size is fitted its leftover is a squeeze: places north and south of the middle are drawn towards it, by about two kilometres on average, places east and west are pushed slightly away, and the displacement is largest in the south-east. Mollweide’s meridians lean by different amounts across fifty degrees of longitude, so no single turn suits them all, and its leftover is a swirl, largest in the north-west, where the places in that corner are carried 163 kilometres west on average. Neither is sixty-four errors. Each is one displacement, evaluated sixty-four times.

The fit is told the error’s size and not its shape #

A fit that uses those positions as pins is told something true and something false. The true part is the size: every pin’s tolerance can be set to the projection’s own leftover error, 4.7 kilometres a coordinate for the regional conic over sixty-four places, which is exactly what each pin’s error is in root-mean-square. The false part is everything a least-squares weight implies beyond that: that each pin’s error is independent of every other, so that the more places are pinned the more those errors cancel.

The setup is the earlier essay’s, made larger. A region’s places are fitted to their measured distances, each good to one per cent, and to a projection’s positions as pins, with a shift, turn and size fitted at the same time. The fit’s covariance states how wrong the fitted shape is, split into what the distances and the pins put there. The actual error is the distances’ share, which is stated correctly because their errors really are independent, together with what the projection’s real leftover does when pushed through the same fit — a fixed displacement, computed exactly.

Independent pin errors the size of the conic’s give 0.985. The projection does not. At eight places the conic’s leftover moves the shape 1.38 times as far as stated; at thirty-two, 2.77; at a hundred and twenty-eight, 4.30. Mollweide goes from 1.52 to 4.05 and Mercator from 2.95 to 4.76.

The climb is the arithmetic of averaging failing. The fit’s statement about the pins falls as places are added, because it believes it is averaging more independent errors: from 4.02 kilometres at eight places to 0.51 at a hundred and twenty-eight for the conic. The actual displacement the pattern produces falls too, but much more slowly — from 5.53 to 2.18 — because adding places adds more evaluations of one pattern, not more independent errors to cancel it.

One way to say how many independent pins a projection is worth is to ask how many would put the same error into the shape. At eight places the conic is worth about four; at a hundred and twenty-eight, about seven. Mollweide is worth from three and a half to eight, and Mercator from one to six. However many places a projection pins, it supplies the information of a handful, and a fit that counts it as a hundred and twenty-eight independent pins understates what the pins do by the square root of the ratio between the two counts.

Weighted at its own size, a projection can make the shape worse than nothing #

The four-city essay found one guarantee that held at every count and tolerance: a pin weighted at its true tolerance never made the fitted shape worse. That is a property of least squares with correct weights, and it needs them correct in full — errors independent as well as of the stated size. A projection’s are not.

With every pair of places measured, the regional conic’s pins halve the shape’s error at eight places: 6.63 kilometres against the distances’ 13.71. At sixteen they are a wash, 7.17 against 7.31. At twenty-four they make it worse, 5.91 against 5.30, and from there the ratio sits between 1.34 and 1.50. The distances keep improving as places are added — their number grows as the square of the places — and the pins’ damage does not shrink to match, because it is one pattern however many times it is sampled.

A sparse network shows the other side. Measured only to its six nearest neighbours, a picture’s fine detail is well fixed and its large-scale shape poorly, since error accumulates along chains of short measurements — and a projection’s leftover is large-scale too. The regional conic, whose leftover is a few kilometres, still helps the sparse network until about a hundred places. Mollweide, whose leftover is 147 kilometres, makes it worse from twenty-four places and reaches 1.81 at forty-eight and sixty-four: 15.40 kilometres where the distances alone give 8.50. Winkel tripel reaches 1.92 at a hundred and twenty-eight.

The twenty-four cities themselves show the same thing at one count. With every pair measured, the distances alone leave the shape 5.23 kilometres wrong; the regional conic’s pins bring it to 4.43, but the fit says 3.27, and the misplaced conic’s bring it to 6.27 while the fit says 4.89. On a six-nearest network the misplaced conic makes the cities’ shape 10.77 kilometres wrong against the distances’ 8.33. In every case the fit believes the pins helped.

The weight that serves the shape is looser than the honest one #

If an honest weight can do harm, the question is which weight does the most good, and the answer separates regional maps from world maps.

For the regional conic the best tolerance is three times the conic’s own error. Stated that loosely the pins give 5.74 kilometres, against 6.21 at their true size and 8.50 without them. Loosening further gives back the gain; at thirty-two times, 8.29. The optimum is broad — 5.85 at twice the honest tolerance and 5.84 at four times — and it sits about where the effective count says it should. On this network the conic’s sixty-four pins are worth about thirteen independent ones, so their weight should be cut about fivefold, which is a tolerance about 2.2 times larger.

For Mercator and Mollweide there is no good weight. No tolerance does better than the distances’ 8.50 kilometres by more than 0.02, and every tolerance near their own size does worse. With a leftover near 150 kilometres, a world map registered to a survey has nothing to add to the survey’s shape, and weighted as a pin of its own size it subtracts.

The weights are a guess the solve believes met the same rule from the other side: the fit trusts whatever weights it is given, and the weights move the answer. Here the size of the projection’s error is read exactly from its residuals. What the residuals cannot say is that the error is one pattern, because a pattern and a scatter of the same size leave residuals of the same size.

A freer registration takes the pattern out #

The pattern is smooth, and a smooth pattern is nearly a low-order polynomial. A registration is free to fit more than a shift, turn and size: an affine map adds two more numbers, a quadratic twelve in all, a cubic twenty. Each absorbs more of the projection’s error before any of it reaches the places.

Over the twenty-four cities the leftover after a cubic registration is 0.64 kilometres for the regional conic, 0.74 for Mollweide, 1.72 for Winkel tripel, 4.43 for the plate carrée and 4.50 for Mercator. The order changes a third time: Mollweide’s shear is something a cubic follows almost exactly, and Mercator’s secant growth of scale over twenty-two degrees of latitude is not.

Fitted jointly with sixty-four places on a six-nearest network, a cubic makes every projection nearly honest. The regional conic gives 3.99 kilometres against a stated 3.95, Mollweide 4.27 against 4.15, and both are about half the distances’ 8.50. Mercator and the plate carrée give 7.69 and 7.10 against stated 6.05 and 5.54 — still optimistic, by about a quarter, and still better than the distances alone.

The path there is not monotone, and the reason is the essay’s argument in one case. Under a quadratic registration Mercator’s leftover falls from 184 kilometres to 35. The fit sees a much smaller error, states each pin’s tolerance five times tighter, and the error that remains is still one pattern. The shape comes out 12.78 kilometres wrong, against 9.38 under four numbers and 8.50 with no pins at all. Removing part of a correlated error and then trusting the rest more makes the fit worse, and only removing nearly all of it makes the trust deserved.

A freer registration also spends something: a cubic keeps 108 of sixty-four places’ 128 numbers rather than 124. For a smooth pattern that is a good trade, since the numbers taken are the ones carrying it. For a map drawn by hand, whose errors really are independent, it would be information thrown away.

Why the paper takes an average and the geography takes a pattern #

The four-city essay found a pin’s average belongs to the paper because the distances and bearings are blind to it. The same reasoning covers everything above, with distances alone blind to a turn as well as a shift, and the registration chosen to take a size too. A projection’s error has a part nothing measured between the places can contest, which can be any size without harm — Mercator’s 1.56, the misplaced conic’s ten-degree turn — and a part every measurement contests. A rotation is not absorbed the way a shift is found a plane fit swallowing a datum’s rotation better than its translation, because the rotation that matters there is a change of longitude, a symmetry of the map; here the turn is absorbed outright because the registration was given it.

The second part is the geography’s, and the new fact is that it arrives as one object. Four independent pins are four statements about the places, and their errors cancel as statements do. A projection’s leftover at sixty-four places is one statement, smoothly interpolated, and does not cancel at all. A map does not say what it is ranks candidate projections by what a fit to a graticule leaves over, and a wrong candidate leaves something with a shape; a shape is exactly what a weight per pin cannot describe.

How the split was checked #

A layout that is a similarity of the reference must leave no pattern. The reference turned through seven degrees, drawn 1.4 times too large and moved three thousand kilometres leaves 6.7 × 10⁻¹³ kilometres after the registration — nothing, to the arithmetic’s precision.

Freeing more may never leave more. For every projection the error after the shift is at least that after the turn, which is at least that after the size, and each polynomial registration leaves no more than the one before it.

Independent pins must move the shape as far as the fit says. Forty sets of pin errors drawn independently at each place, the size of the regional conic’s leftover, give a ratio of 0.985. Without that control, the climb in the honesty figure could be a property of the fit rather than of the projection.

The linear account must match the full fit. Two hundred seeded fits at full nonlinearity — noisy distances, the regional conic’s real positions as pins, the similarity solved by Gauss–Newton with the places — put the twenty-four cities’ shape 4.39 kilometres off. The linear account says 4.43 actual and 3.27 stated. The full fit agrees with the actual figure, not the stated one.

Where the fit stops #

The reference is a fitted picture, not the ground. It is 0.61 per cent from the sphere at worst over the twenty-four cities and 2.37 kilometres from the azimuthal equidistant map in root-mean-square. Residuals of a kilometre or two, the regional maps’ and the cubic registrations’, are at that limit.

Distances only, with independent one per cent tolerances. Bearings would make the turn a measured quantity rather than a free one, and the four-city essay found that a bearing changes what a pin can reach. A real survey’s distances share a scale error, which a registration’s free size would then trade against.

One region, at one latitude, with seeded places spread evenly across it. The counts at which a projection’s pins start to harm a fit belong to this region, these tolerances and these networks. The direction of every effect is general: a projection’s leftover is smooth wherever its distortion is, and a smooth error never averages.

The azimuthal maps understate least, and why is not established. On the twenty-four cities with every pair measured, the gnomonic’s pins move the shape 0.96 times as far as the fit says, and the azimuthal equidistant, stereographic and Lambert azimuthal 1.16 to 1.20 — the four lowest of the seventeen. Their leftovers are symmetric about the region’s centre, and a symmetric stretch may be a shape the full set of distances fixes unusually well; nothing here tests that. Which smooth patterns a given network is blind to is the question underneath every result above.

The positions are exact. A printed map adds the hand that drew it, the paper that shrank and the eye that read it, and those errors are partly independent. The sheet moved before it was measured is the case where they are not, and where the control points are the case where the places themselves decide what a fit can see.

Still open: whether the pattern’s shape can be weighted #

A cubic registration takes a projection’s pattern out by giving the fit twenty numbers to absorb it, which works because the pattern is smooth, and spends those twenty numbers whether they carry the pattern or not.

The alternative is to tell the fit the truth: not that each pin has an error of a certain size, but that the errors are correlated, with a correlation that falls off over some distance. A covariance that says so would let the fit count a projection as the handful of independent pins it really is, without being told what the projection was. Whether the correlation length can be estimated from the residuals of one map, whether a fit weighted that way recovers the honest statement at every count of places, and whether it does better than a cubic registration when the map also carries a hand’s independent errors, are questions a weight per pin cannot ask.