Cartographic Projection
every map is wrong, and by exactly how much
Concept

Control points — where it appears

A place on a map whose ground position is known, used to recover what the map was drawn with. The identification methods here are handed graticule crossings; a map with no graticule supplies none, and the correspondence between ink and ground has to be found rather than given.

Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.

Two explanations for one residual, against the size of the region. A map in the right projection whose control coordinates are on the wrong datum, and a map fitted with the wrong projection and the right datum, both measured as a residual after the reproduction's scale, rotation and offset have been removed. The datum shift's residual is flat: it is the same 1.3e-6 at every size, because a similarity fit is very nearly what a datum shift is and it absorbs the rest. The wrong projection's grows by three orders of magnitude with the region. They are the same number below about a degree, which is where a residual stops saying anything about either of them.

A residual has more than one explanation

The method names a projection by fitting every candidate to a set of control points and taking the smallest residual. It has never been asked what else a small residual could be. A map drawn in the right projection from coordinates on the wrong datum leaves a residual of one part in a million — indistinguishable from noise, at every region size, because a similarity fit absorbs a datum shift almost exactly.

wrong · Identify A plane fit swallows a datum shift, and keeps swallowing it. A map of OSGB36's ground drawn as though it were on WGS84, at seven sheet sizes. The upper curve is how far the drawing moves — a real 99-metre error on the ground — and the lower one is what survives the best similarity between the two, which is what a fit for the projection removes for free. The absorption runs from 99.68 per cent on a 1° sheet to 85.16 on a 60° one, and what is left is 3.9 parts per million of the map's own size even then — two microns on a sheet half a metre across.

The datum hides inside the projection's parameters

Six essays recover a projection from control points with the body it was drawn on taken as known. It is not known, and the fit cannot find it: a 99-metre datum shift is absorbed to 99.7 per cent on a two-degree sheet and to 85 per cent on a hemisphere, leaving eleven parts per million of the map behind.

wrong · Identify Which rotation a projection cannot see is decided by the projection. The same rotation — 2.455 arcseconds, DHDN's polar one — applied about each of the three axes in turn, with the residual after the best plane fit drawn on a logarithmic scale. A cylindrical and a conic in their normal aspects hide the polar rotation to arithmetic noise, 3e+5 times better than either equatorial one, because a change of longitude is a symmetry of both. A pseudocylindrical hides none of them — its horizontal coordinate carries a factor in latitude, so a longitude shift is a shear rather than a translation. And an azimuthal centred on the equator hides the equatorial rotation instead.

A rotation is not absorbed the way a shift is

A datum's rotations were expected to be the part a plane fit could not swallow. They are the part it swallows best — 93 times better than the translations over a hemisphere, and 24 times better per metre moved — and the reason is that the rotation which matters is a change of longitude, which is a symmetry of the map.

wrong · Identify What a sheet does to the points before anybody measures them. The graticule crossings of a map drawn on Conformal conic, with an arrow at each one showing where the same crossing has moved to after the sheet dried — 0.1 per cent along the grain and 0.4 across it, with the grain at 23° to the map's axis, and the displacement magnified 60 times so it can be seen at all. The pattern is a stretch along one direction and a squeeze along the perpendicular, which is what an anisotropic scaling looks like. It is a property of the paper and has nothing to do with the map printed on it. A similarity fit to these points leaves 3.66e-4 of the map's own width unexplained, against 1.22e-10 on the unshrunk sheet.

The sheet moved before it was measured

Every method of reading a projection off a map takes its control points from the sheet and assumes the sheet is the one the cartographer drew. Paper shrinks across its grain three times as fast as along it, and on a map whose grain runs along its own axis that shrinkage is EXACTLY a change of standard parallel — one per cent moves the recovered parallel by 0.57 degrees with the residual sitting at the solver's floor.

wrong · Identify The whole of what an unlabelled map gives you. The outline of Japan as drawn on Conformal conic, delivered as an ordered list of page positions with nothing attached to any of them. No latitude, no longitude, no scale, no north. The question is whether a projection can be recovered from that, and it can: the correspondence between the ink and the ground is found by sweeping the starting point round the curve and both directions, and the true candidate comes back with a residual of 2.88e-14 against the runner-up's 4.57e-4.

A map with no graticule

Identifying a projection starts from control points, and a great many maps have none. Handed an outline with no labels on it at all, the method still works — and works better: the correspondence between ink and ground is recoverable exactly, because a similarity preserves ratios of arc length, and the margin on clean observations is 1.6 × 10¹⁰ against a graticule's 9.9 × 10⁶. What breaks it is noise, at three parts in a thousand.

wrong · Identify A local station's error ellipse is the national network's, arriving. Three orders of a stated survey, each a network of distances adjusted with the stations of the order above held fixed: a national network of seven stations 60 km apart, a regional one of ten stations inside one of its triangles, and a local one of five stations about a kilometre apart tied to three regional stations 10 km away. Every error ellipse is drawn at one scale in all three panels. Dashed: the ellipse each adjustment reports from its own observations. Solid: the full ellipse, with the error of every order above carried in. The national network's centre station is its origin and its eastern station fixes its bearing, so the one has no ellipse and the other's is a line; it reports its full error because nothing is above it. The local stations report 11.2 mm and carry 59.4 mm: their solid ellipses are the size of the national network's at that place.

The control carries the error of every network above it

A local survey is adjusted with its control stations held fixed, and the error ellipses it reports come from its own observations alone. On a stated three-order network a local station reports 11 millimetres and sits 59 from where the national origin puts it: 52 of them from the national order and 26 from the regional. Its own residuals can see almost none of that. What it does own outright is its lengths, and nothing of those is inherited.

practice · Reduction Pins at one place move where the picture sits; pins at four reshape it. The best flat picture of four cities, fitted to every distance with a tolerance of 1 per cent, one bearing a pair to 0.01 radians, and pins to the sheet good to 50 km. Each ellipse is the error in a city's position relative to the picture's centre — its shape, which no shift changes — drawn 30 times its size. Solid: the part the distances and bearings put there. Dashed: the part the pins put there. Pinned at London alone, the pins put 0.00 km into the shape and the places 63.0 km. Pinned at all four, the pins put 30.5 km and the places 29.7 km, and the shape as a whole is 42.6 km against 63.0.

A second pin is a measurement of the places

Counted exactly, a picture of places is a direct sum: the distances and bearings spend its shape and two pins spend where it sits on the sheet. Given tolerances, the sum survives in one form and fails in another. Pins at one place put none of their error into the shape at any tolerance, and wherever the pins are, their average takes none of the distances' error. But every place pinned after the first is a statement about the geography, and held exact it makes the shape worse once its error passes about two thirds of the shape's own.

impossibility · Embedding Ties ten kilometres long locate the control along themselves and hardly at all across. The local order of the stated survey: five stations about a kilometre apart, and the twenty-five lines from them to one another and to three control stations 10 km away. Adjusted on its own lines, the network knows where each control station lies along its ties to within a few centimetres and across them only to 179.0 mm, because five lines leaving a cluster a kilometre wide are nearly parallel ten kilometres out. The long ellipses at the control stations are that knowledge, drawn at one scale. At the local cluster, drawn at the same scale: dashed, the error the held-fixed adjustment reports for each station, 8.0 mm on its long axis at the first station; solid, the same station's own error after an equal-weight rigid fit of the free network onto the three control stations, 70.2 mm. The equal-weight fit trusts each control coordinate as much as every other, and the across-tie ones are worth almost nothing.

Three numbers are all a local survey can check of its control

Adjust a local network on its own lines, then fit it onto its three control stations, and the misfit at the control is the only place the control's error can show. It is exactly the excess the held adjustment's variance factor already carried, counted on three degrees of freedom instead of fifteen, and it catches a disturbed monument 79 times in a hundred where the held report catches it 49. Give the fit a scale and it hides 98 per cent of that excess, and lends the control's error to every length.

practice · Reduction Moved five centimetres, one station of three looks like all three, and one of five looks like itself. The local cluster at the centre, tied to three control stations (left) and to five (right) ten kilometres away, drawn on their true bearings with the distance shortened. The northern station, filled, has been moved 50 mm along its tie and nothing else is wrong. The bar at each station is the expected value of its own test for a displacement along its tie, noise-free. With three stations the three tests read 3.25, 3.22, 3.22: the data say only that the triangle is too large, and any of its corners could have made it so. With five, the moved station reads 4.31 and no other more than 2.44.

Naming the control station that moved takes a fifth

A local survey tied to distant control can tell that a monument has moved, and with three control stations it cannot say which. Each station gives the local lines one number they know well — its distance — and a shift of the whole control uses up two of them, so three stations leave one and four leave two. Five evenly spaced name a station moved five centimetres 83 times in a hundred; three name it 20. Better lines cannot stand in for the missing stations, and every station added also exposes more of the control's own published error.

practice · Reduction Told the control has an error, the test must see a larger move before it names a station. The local cluster at the centre and its 5 control stations ten kilometres away, drawn on their true bearings with the distance shortened. The shaded ellipse at each station is its published error, 53 by 31 mm on average. Outward along each tie, at the same scale, the smallest move of that station its test finds four times in five at a size of 0.001: the thick bar for a test that carries the published error, 74, 79, 78, 78, 78 mm; the thin dashed bar for a test that holds the control exact, 48, 48, 49, 48, 48 mm.

Honest about its control, the test needs eight centimetres where it needed five

A test that holds the control exact reads the control's published error as movement, and at five stations flags a good survey sixteen times in a hundred. Carry the published covariance into the adjustment and the false alarms fall to half a per cent — and the smallest move the test can find grows from 48 millimetres to 77. A five-centimetre move is then named 29 times in a hundred instead of 63, the naming rate of perfect control takes twenty-four stations, and above ten centimetres the two tests agree. What the honest test buys is a flag that means what it says.

practice · Reduction From one survey an optimistic report and a moved mark look alike. The scale on the regional network's published covariance, estimated from the residuals of one survey of five control stations, over 3,000 seeded surveys each: honest report, nothing moved, mean 0.92 and median 0.53; report twice too optimistic, mean 1.91 and median 1.43; honest report, one mark moved 5 cm, mean 4.84 and median 4.48. One survey's estimate has a standard deviation of 3.80, and a five-centimetre move along a tie adds between 3.7 and 4.2 to it: about one standard deviation, and four times what a report twice too optimistic adds.

One survey cannot tell an optimistic report from a moved mark

A local survey that carries its control's published covariance can estimate, from its own residuals, how far that covariance understates the truth. With five control stations the estimate has a standard deviation of 3.8 times the published scale, and a single monument moved five centimetres adds about as much again — four times what a report twice too optimistic would add. Run at the scale it estimated for itself, the honest test stops flooding good surveys with alarms, and names a moved mark seven times in a hundred instead of twenty-five: the move inflates the estimate and the estimate absorbs the move. The report can be checked only across surveys, and telling one twice too optimistic from an honest one takes about two hundred of them with five stations each, or fifty with twelve.

practice · Reduction Eight hundred surveys, each a poor estimate of the scale, find the patch where the published covariance is four times too optimistic. The middle one of nine agencies, by how much of the patch its map flags: a region covered by 800 local surveys at random, each tied to five control stations of a regional network whose published covariance is honest everywhere except a disc covering 15% of the region (dashed), where the control's true errors are 4 times its published variance; one survey in ten has a monument moved five centimetres. Filled dots: the 357 surveys whose own residuals put the scale above 2. Shaded: the 18 of 441 grid cells where the count of such surveys within 0.2 of the region's width exceeds what an honest region exceeds there one time in twenty, over the whole map; 18 of them lie in the disc.

An agency can map where its published covariance is optimistic

The local surveys tied to a regional network each estimate, badly, how far the network's published covariance understates the truth, and the agency that receives them can pool what they say. When the understatement is confined to one part of the region, pooling it into one number dilutes it: a patch covering 15 per cent of the region with errors four times their published size reads as 1.30 for the whole. Counted place by place, the same surveys find the patch — two agencies in three find it with 400 surveys and nearly every one with 800, flagging almost nothing outside it. Fed back as one number the scale removes little of the false alarms inside the patch and costs nearly a third of the test's power outside; fed back as a map it cuts the false alarms inside from 20 per cent to 3 and costs little outside.

practice · Reduction An office that overshoots headlands draws every coast wrong by that coast's own curvature. Four coasts at one map size: the compiled coast whose seams are counted, and three control coasts an office might trace with the truth known — an ellipse as thin as the counted coast (axis ratio 2.88), a nearly round one (1.12) and a coast of five headlands. Solid: each coast. Dashed: the same coast as an office would draw it whose every hand pushes a headland out and pulls a bay in by a fixed amount per unit of curvature, exaggerated a hundred and twenty times so that it shows. The curvature's spread round each coast, its variance in units of the coast's size: the sheet being counted 16.9, control, as thin as the target 15.6, control, nearly round 0.1, control, five headlands 29.8. The habit's size on a coast follows that spread, and the habit's slope does not change from coast to coast.

A known coast charges an office's habit, and a bending one takes it out

An office whose hands all share a habit leaves it in every tracing of every sheet, where the spread between hands cannot see it. One control sheet — a coast known independently, traced the office's usual way — reads the habit's size and restores the charge, if the habit is a drift. If the habit answers to the coast's own curvature, a size read on a rounder coast charges a thinner one at little more than half, but the same control read differently does better than charge the habit: its departures, set against its curvature, give a slope that is the office's own on every coast, and subtracting it leaves about one per cent of the habit in the mean. The slope also tells the two kinds of office apart. A coast that hardly bends teaches neither, and corrects worse than nothing.

wrong · Identify

Named alongside it

The objects these essays reach for when they reach for this one.

Least-squaresVerificationCovarianceAdjustmentResidualIdentificationNetworkRedundancyReliabilityDatumEstimatorHierarchy

All concepts