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

The thread: Measured, not named — page 11

A projection is usually called conformal because that is its name. Here it is called conformal only after the angular deformation has been computed at several hundred points and found to be zero. Essays 241 to 264 of 372.
What a seven-parameter fit's residual is made of. A published transformation accuracy is the root-mean-square residual at the common points, and here it is 1.51 metres. Almost all of it — 1.51 — is the network's own distortion, which a rigid motion and a scale cannot follow and which is present at every point of the country whether it was used in the fit or not. The transformation's own error, measured as the disagreement between the fitted parameters and the true ones over a clean grid, is 0.091 metres: 6 per cent of the quoted figure. The last bar is the control — the same fit with the distortion switched off, at 2.9e-4 metres. What the numbers refer to

What another common point buys

A seven-parameter datum fit leaves a pattern rather than noise. Six per cent of the residual it reports is the transformation's own error and the other ninety-four is distortion no seven parameters can follow — so adding common points improves a term that was already small and cannot touch the one that is quoted.

 A coast that agrees with its graticule, except where it was copied. The outline of Japan on a map whose graticule is a conformal conic projection. 30 per cent of the coast, drawn in the second ink, was copied from a sinusoidal sheet of the same ground and fitted into place so that its two ends meet the coast either side, the dots. The dashed line is the coast the graticule predicts. Everywhere else the drawn coast lies on it exactly; along the copied stretch it departs by up to 9.4 thousandths of the map's width. Nothing in the graticule shows it. What is taught wrongly

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

A map compiled from two sheets has no single projection behind it, and its graticule cannot say so. Its coast can — not as a residual, which is a number the size of a projection error, but as a profile: the drawn coast lies exactly on the one its graticule predicts, then departs from a seam, and returns at another. The seams are found to within one point in 720, and the stretch between them names the projection it was copied from with a margin of 3.6 × 10¹¹.

 Two defensible answers from the same four sights, and they are kilometres apart. Four bodies within sixty degrees of one another at azimuths of 60°, 80°, 100°, 120°, each sight carrying an independent error of about 2 km and the same common error of 3 km. The four lines are nearly parallel, so a step towards every body at once is almost exactly a step to the south. The two-unknown least-squares point takes that step and lands 3.66 km from the ship; the three-unknown fit, which solves for the common error as well, refuses it and lands 2.22 km away — its estimate of the common error being 1.87 km. Neither is wrong. They are the two ends of one exchange. Paths and directions

What the extra unknown costs where nothing can see it

Four lines of position can estimate a common error in every sight as well as the position. The estimate is honest — three kilometres put in comes back as 2.8 — and where the residual is blind, its scatter is plus or minus thirty-one, and the fix that carries it is scattered fourteen kilometres from the ship against the biased fix's two and a half. Letting the residual choose between the two is worse than either, because a test with no power is a one-in-twenty lottery on the worse answer.

 What simplifying a boundary does to the number stored beside it. One region, simplified at five tolerances, with the error in the two quantities a consumer computes from the pair. If the density was stored, the total it implies moves by exactly the area's error — -1.55 per cent at the loosest tolerance. If the total was stored, the density it implies moves the other way by the same amount. Nothing in the file says which of the two was measured and which is being derived, and the simplification is normally done by a tool that never opens the attribute table. What a machine does with it

The attribute is a claim about the geometry

Fourteen essays price what a stored coordinate means and not one asks what the number stored beside it means. A rate is a quantity divided by an area, the area belongs to the geometry, and no format records which area — so a simplification that moves the outline by nothing visible moves the implied total by 1.55 per cent, an unweighted average of densities is 4.09 per cent out, and a choropleth gives a polar square kilometre fifteen times the ink of an equatorial one.

 Two source geometries of 96 cells each, rebinned to the same three targets. Both curves start from a source of 96 cells and rebin to targets of 32, 128, 512 cells, so the count ratio is identical along them and the only difference is the shape of the source cells: gnomonic squares on a cube against rectangles in longitude and latitude. The ratio dominates — both curves fall by more than half across the range — and the shapes still separate by 25 points at the middle target. The cube loses less, because its cells are all much the same size and the lon/lat source's collapse towards the poles. What a machine does with it

The same number of cells, in two shapes

Moving a field between two cell schemes loses 18 per cent of it per cell in one geometry and 39 in another, and the earlier measurement could not say whether that was the shape of the cells or the ratio of their sizes, because changing the schemes changed both. Holding the counts settles it: the count ratio decides most of the loss, and the shape is still worth a quarter of the field.

 An error in a rotation rate is a longitude error that never stops growing. Where the prime meridian of each body has got to, if its published rotation rate is wrong by one unit in its last published decimal. Every line is straight through the origin, because the error is the rate error multiplied by the elapsed time and nothing else — there is no date after which it settles. Jupiter reaches 2279 metres at the equator after a century; Mars, whose rate is published to twelve decimals rather than seven, reaches 0.0011 metres over the same interval. What the numbers refer to

A longitude that drifts with the rotation rate

Ten essays here map bodies whose shape is the problem. A longitude is not about shape: it is a landmark plus an extrapolation over however many days have passed, and an error in the last published decimal of a rotation rate is a coordinate error that grows without bound in time.

 A bank that dries at one tide adds a bulge to the territorial sea. A stated coast 116 km long with five bays 5 km deep, and its twelve-nautical-mile limit measured from the low-water line at two tidal datums 1 m apart. A bank 10 km offshore has its crest 0.5 m below the higher datum, so it is under water at that datum and dries at the lower one, over a disc 1.0 km in radius. Dried and inside twelve miles of the coast, its low-water line becomes part of the baseline, and the limit bulges round it: 339 km² of sea. Everywhere else the two limits are drawn on top of one another, 11.1 km² apart over the whole coast. The coast is stated, not surveyed. Grids, and what a survey does

The sea is measured from a tide, and the headlands decide how much

A territorial sea is measured from the low-water line, and the low-water line is where a chart's tidal datum meets the shore — so a datum a metre lower moves it twenty metres at a headland and a kilometre across a tidal flat. On a stated coast the twelve-mile limit barely notices the flats: fifty-six square kilometres of shore are uncovered and eleven of sea are gained. A bank that dries at the lower datum adds three hundred and thirty-nine in one step.

 Three budgets for one meridian distance, seven orders of magnitude apart. Every quantity that can make a computed distance from the equator to the pole wrong, on a scale of powers of ten. The series in the third flattening, at its worst anywhere on the meridian, is out by 76 nm at four sine terms, 31 µm at three and 22.0 mm at two. The two conventions WGS84 and GRS80 put the pole 82 µm apart. The ellipsoid that best fits the Earth is known only to 15.7 cm from its axis and 563 µm from its flattening, and WGS84 puts the pole 67.2 cm away from it. What is taught wrongly

An ellipsoid computed to a nanometre is known to a decimetre

The meridian series is exact to seventy-six nanometres on WGS84, and WGS84 is exact by definition: its axis and flattening are conventions with no uncertainty at all. The ellipsoid that actually fits the Earth is a measurement, known to sixteen centimetres from equator to pole, and WGS84 puts the pole sixty-seven centimetres away from it. From its second term, the series is more accurate than anything it is used to compute.

 The triangle and the ellipse are two pictures of the same three sights. Three bodies at azimuths of 20°, 140°, 250° with independent errors of 2 km, drawn twice: as the cocked hat their lines bound, dashed, and as the 95 per cent confidence ellipse of the least-squares point, outlined. The ellipse is 4.25 by 3.78 km and covers 50.5 km²; the average hat from the same sights covers 7.06. The triangle is seven times the smaller and holds the ship a quarter of the time; the ellipse holds it 95 per cent of the time, which is what it was constructed to do. The hat is a picture of how much the sights disagree, and the ellipse is a picture of where the ship is. Paths and directions

A confidence ellipse is honest only where the sheet keeps angles

The cocked hat covers seven square kilometres and holds the ship a quarter of the time; the confidence ellipse from the same three sights covers fifty and holds it ninety-five per cent, which is what it was built to do. On a plotting sheet whose longitude is not corrected for latitude that ellipse fails in two unrelated ways — its centre is displaced, which costs coverage and depends on the navigator's own assumed position, and its shape is misread, which costs nothing but meaning and depends only on the latitude.

 Forty-two starts, forty-two answers. Every local optimum coordinate ascent reaches from 42 starts on the same region and the same criterion — a 25° spherical square at a tolerance of ±0.5% — plotted at the share of area it keeps. They run from 36.98 per cent to 67.50, a spread of 30.5 points, and no two of them are the same map: clustered on their scale fields rather than their coefficients, 42 of the 42 are distinct. The two marked starts are the ones anybody would actually use — Chebyshev's map reaches 36.98 per cent and the mean-square map 64.05, and neither is the best this search found. What each projection optimises

A criterion worth using is one whose answer is not unique

Forty-two starts on one region and one tolerance give forty-two different maps, keeping from 36.98 per cent of the area to 67.50. The two starts anybody would actually use — Chebyshev's map and the mean-square map — end 27.1 points apart, which is more than the criterion buys over either of them. And the spread is widest at exactly the tolerance where the criterion is worth using at all, because both facts are the same fact: an objective with a flat top and a sharp edge has somewhere to hide answers.

 What a folded net is accurate to, beside what the map it carries is wrong by. For a cube globe of 100 mm radius: each tolerance of cutting and folding a paper net, as a displacement on the assembled sphere, and the seam corner of each of the three face maps beside it. The fold's own error is the largest at 0.955° — 106 km at Earth scale — because a degree of dihedral error moves the far edge of a face by a degree of the face's own radius, which for a cube is 54.7°. The conformal face map's corner is 0.769°, below the whole budget of 1.000°; the gnomonic's is 20.2°, twenty times above it. The families

The map is finer than the paper it is folded from

Thirteen measurements price a polyhedral map in angles, areas and lengths, and none of them is about paper. A hand-folded net is accurate to about a degree of dihedral angle, which displaces the far edge of a cube's face by 0.955° of arc — 106 km at Earth scale — and that floor does not shrink when the globe is made larger, because a degree is a degree. Against it the conformal face map's seam corner is 0.769° on a cube and 0.0040° on an icosahedron, where it is 165 times finer than the sheet can be folded to.

 The geoid's curvature is whatever degree the model was stopped at. The Gaussian curvature a geoid model contains, as a fraction of the sphere's own, against the degree the spherical harmonic series is truncated at. It rises in proportion to the cutoff, because a degree-n undulation moves the curvature by n(n+1) − 2 times its own amplitude while the amplitude itself falls only as n to the three halves. The dashed line is the ellipsoid's whole variation from equator to pole, 1.33 per cent; the geoid overtakes it at degree 1332, a half-wavelength of 15 km. Below that resolution the ellipsoid's flattening is the larger term and above it the geoid's own bumps are. The impossibility

The geoid has a curvature only if you say where you stopped

Thirteen measurements have priced the curvature of surfaces that are stated — a sphere, an ellipsoid, a triaxial body, a surface with a hole, real relief. The surface a height actually refers to is none of them. It is an equipotential of the Earth's own gravity, it is only ever given as a series stopped at a degree, and its Gaussian curvature carries two more powers of the degree than its height does. So the omitted height converges, the omitted slope diverges as a logarithm, and the omitted curvature diverges as a power: EGM96 carries 0.36 per cent of the sphere's own curvature, EGM2008 carries 2.19, and a one-kilometre model would carry twenty.

 What reading the seven parameters under the wrong convention costs. The distance between the two published conventions' answers for the same parameter set, at the worst point of the world. The difference is 2s(r × X) exactly — twice the rotation, crossed into the position — so a parameter set with no rotations is immune and one with a large rotation is not: DHDN reaches 152.3 metres and NAD27, whose published transformation is three translations and nothing else, reaches zero. The note beside each bar is the size of that datum's rotation. What the numbers refer to

The rotation has two sign conventions

Eleven essays price a datum transformation's parameters, their fit, their residuals and their uncertainty. None asks what the numbers mean: three of the seven are published under two conventions whose rotations differ in sign, and reading one set with the other formula costs exactly twice the rotation — 55 metres on OSGB36 and 152 on DHDN.

 What a rebinning loses depends on where the target's edges are. Two grids of fixed counts, fixed shapes and fixed resolution, with the target slid across the source from perfect alignment to a full cell. Nothing about either grid changes except where its boundaries fall. The loss runs from 27.5 per cent at zero to 56.3 at half a cell — a factor of 2.04 — and the longitude-only curve returns to its starting value at a full cell to six decimal places, which is the periodicity check. A cell boundary that coincides with a target boundary loses nothing, and a grid comparison that does not say where its boundaries are has left that out. What a machine does with it

When the edges do not line up

Rung eight held the cell counts equal so that shape could be compared without the count ratio drowning it, and recorded a doubt: a longitude–latitude source shares its boundaries with a longitude–latitude target wherever their counts share a factor. The mechanism is real and worth a factor of two. It was not what the published number was made of.

 The pass that fails first stands above the pass between the two deepest basins. For a region grown from 6° to 40° of span at 38° N, the threshold at which the near-optimal aspect set first comes apart — the highest merge in the whole filtration — against the merge between the two deepest basins, which is what the earlier measurement of this quantity reported. The first is higher at every size, by a factor of 1.02 to 2.09. At every size the deciding merge is the last basin joining the optimum's own — a basin whose floor stands 0.019 to 0.191 above the best score — so the set is held together by a near-rival to the best placement, reached over higher ground than the second-best basin is. What each projection optimises

The pass that fails first is not the one that was measured

Three attempts to explain why the near-optimal aspect set comes apart when it does have fitted the height of the pass between the two deepest basins. That is not the pass that fails first. At every region size the set is held together by the last near-rival to join the best placement's own basin — a basin between 2 and 19 per cent worse than the best — over ground 1.02 to 2.09 times higher than the deep pair's pass. The two passes fall at nearly one rate, −1.42 and −1.31 powers of the region's span, so what separates them is a factor, not a law.

 What 16 bearings draw beside a river that bends. Everywhere within a 10 km walk of the centre, with a river 3 km to the north meandering ±2.0 km on a 4 km wavelength, crossable only at bridges every 2.5 km. The reachable ground is shaded — pale where it is reached directly and darker where the walk had to go by a bridge. The outline is the polygon 16 bearings draw. It misses 34.1% of the true ground, as a fan beside a straight river does, and it also CLAIMS 0.14% that is not there — ground cut off by a bend, which a chord between two neighbouring bearings passes straight over. Paths and directions

A bend in the barrier puts the drawn polygon over water

A straight river leaves the near side of a reach set convex, and that is what makes a fan of bearings fail in one direction only: every chord cuts a corner, nothing is claimed that is not there. A meander breaks that: the near side is no longer convex, and a chord between two bearings can cross water. So the drawn polygon now over-claims as well as omitting — by a seventh of a per cent, at every bend measured, and more bearings do not reduce it. The question was whether the two errors partly cancel. They do not, and the reason is not their signs: the over-claim never reaches a hundredth of the omission, because one scales with the whole set and the other only with the river.

 A picture of four places, and the one constraint that is about the sheet. The best flat picture of four world cities, with the six distances it could be asked to hold drawn between the dots and the twelve bearings implied by them. Every one of those is a statement about how two places stand to one another, so every one is blind to where the whole picture sits on the paper. The cross is a constraint of a third kind: a stated position for one place — this dot goes here — which is two numbers and is blind to nothing. Adding both of them takes the largest independent set from 6 to 8. The impossibility

Two numbers about the paper buy nothing about the places

A picture of n places has four freedoms, and a constraint holds only what its own blindness leaves: distances reach 2n − 3, bearings 2n − 3, the two together 2n − 2. Position is reached by neither, because nothing about how places stand to one another says where on the sheet to put them. A stated position does reach it, and two of them take the budget to 2n exactly. What the count conceals is that the extra two buy nothing: a largest picture holds six place constraints with the pins and six without, the same 7,596 of them, holding the same pairs at the same rates. The constraint system is a direct sum, and a pin is a different currency.

 Straight baselines across a fringe of islands. A stated coast 220 km long, a gentle undulation 10 km in amplitude, with 8 islands off it: six between 5 and 12 km offshore and two rocks at 30 and 26 km. The dashed line is the twelve-nautical-mile limit measured from the low-water line of the mainland and every island. The solid lines are the straight baselines enclosing the most water under one reading of the article — segments of at most 24 nautical miles departing at most 20° from the coast's general direction — 6 segments, the longest 44.4 km — and the solid curve is the limit measured from them. Tinted: 988 km² of water landward of the lines, now internal waters. Shaded: 68 km² of sea beyond the dashed limit, against a territorial sea of 8,122 km². Neither rock is reached. The coast is stated, not surveyed. Grids, and what a survey does

A straight baseline moves more water inward than sea outward

The seventh article of the law of the sea lets a coast fringed with islands measure its territorial sea from straight lines joining chosen points, and gives no length, no angle and no scale for the choice. On a stated coast the lines a common reading permits add sixty-eight square kilometres of sea at the outer edge and turn nine hundred and eighty-eight into internal waters. Drop the test of direction and the outer rocks come into reach: eighteen hundred added, forty-six hundred enclosed.

 Goode's composite with a strip of a generalised Eckert II, whose meridians are straight. The sinusoidal to 32.28°, a strip of the generalised Eckert II projection with r = 0.8311 from there to 49.27°, and the unscaled Mollweide beyond, both hemispheres; the strip's edges are dashed. Inside the strip every meridian is a straight line, as Eckert II's are, and it leaves the sinusoidal's meridian and joins the Mollweide's along their own tangents, so the 11.9° corner Goode's join turns at 180° of longitude is gone. The composite is equal-area throughout, and its strip is 17.0° wide and centred at 40.78°, four hundredths of a degree from Goode's own join. The families

Only Eckert II's family can be the strip, and its meridians are straight

The cubic that smooths Goode's seam is four conditions on the length of a parallel, and equal area turns them into four conditions on how the parallels are spaced. The sinusoidal's spacing is flat, so a strip can leave it only where its own spacing is momentarily flat — and of eleven named equal-area maps, only Eckert II's spacing ever turns over. One member of its family meets all four conditions exactly, over seventeen degrees centred four hundredths of a degree from Goode's join, with every meridian straight. The cubic over the same span is the same map to four parts in a hundred thousand.

 There is no such thing as the radius of the Earth at a latitude. The two principal radii of curvature of WGS84, against latitude, with the mean radius that every table prints drawn across them. The meridional radius M runs from 6335.44 km at the equator to 6399.59 at the pole; the prime-vertical radius N from 6378.14 to the same value. They differ by 42.70 km at the equator and meet only at the pole, and the single number 6,371 km lies between them at no latitude where either is right. What the numbers refer to

The radius of curvature is two numbers

Twelve essays work on the ellipsoid and every one of them takes a radius when it needs one. There are two at every point, they differ by 42.70 kilometres at the equator, and the single number every table prints is out by 5,583 parts per million on a line running north.

 The same 2-pixel road at three latitudes, zoom 5. The dark bar is the mark as drawn — 2 pixels, identical in all three panels, because that is what the stylesheet says. The pale band behind it is the ground that mark covers, drawn to one common ground scale: 9.78 kilometres at the equator, 6.92 at 45° and 1.70 at 80°. The reader sees the dark bar and is being told about the pale one. What a machine does with it

The road is drawn two pixels wide

Seven rungs measure what a screen map does to position. Nothing on a map is a point: every mark has a width, the width is chosen in pixels, and a two-pixel road covers 9.78 kilometres of ground at the equator and 1.70 at 80° north. That is a generalisation applied at a strength varying by a factor of six across one sheet, by a stylesheet with no latitude in it.

 An angle is a difference of two directions, and the error does not cancel between them. A geodesic triangle with sides of 30 km at 45° N, its first side laid off on a bearing of 45° and the second 60° round from it. Substituting a sphere carrying the geodetic latitude turns each direction at a corner by an amount that runs as the sine of twice its azimuth, so the two directions at A are turned by different amounts and the angle between them is wrong by their difference. Here the worst single direction is out by 345.79″ and the worst angle by 521.09″ — larger, not smaller. Bessel's sphere, which also corrects the longitude, keeps every angle to 6.2e-9″. What is taught wrongly

An angle is a difference, and the difference doubles the error

Substituting a sphere for the ellipsoid turns every direction at a point, and a surveyor measures angles rather than directions — so the obvious hope is that a turn common to both directions cancels in their difference. It does not cancel, because the turn is not common: it runs as the sine of twice the azimuth, and the largest angle error is exactly twice the largest direction error at a well-shaped corner. A first-order triangulation computed on the geodetic-latitude sphere carries angles nearly nine minutes of arc wrong, and its closure check passes perfectly.

 Its level belongs to the projection more than to the latitude. The absolute depth of the pass that fails first, averaged over regions of 6° to 18° span, for three projections at four latitudes. Across the twelve rows it runs from 1.80e-2 to 7.86e-2, a factor of 4.4, and most of that is between projections: Robinson 1.80e-2 to 2.06e-2 over the four latitudes, Winkel tripel 2.74e-2 to 3.81e-2 over the four latitudes, Mollweide 4.02e-2 to 7.86e-2 over the four latitudes. Within a row the depth spreads by 4 to 35 per cent over the four region sizes, so it is not a constant of the case either. The whiskers are that spread. What each projection optimises

The pass's depth belongs to the projection, and it is not a constant

Separating the fracture threshold into a pass depth and the best score a region admits raised the question of what the pass depth is. Across three projections and four latitudes it runs over a factor of 4.4, and most of that is the projection: Robinson's stays between 0.018 and 0.021 at every latitude, Mollweide's between 0.040 and 0.079. It is not a constant of anything — it falls as the region grows in ten cases of twelve — and it does not follow how much of the projection the region sees. The denominator, over the same cases, moves by half again.

 A fast stretch over one half of a gap, with nothing changed across the jet. A jet of 100 km/h blowing along the Goose Bay–Narsarsuaq gap, carrying a streak that reaches 300 km/h over a stretch about 1,000 km long, centred 500 km from the gap's midpoint on the Goose Bay side. The dotted bands are where the wind has slowed to 150, 100 and 50 km/h, and they are drawn from the jet's own cross-section — nothing about it changes across the flow. The solid outlines are the places within sixty minutes of each airfield. The least allowance at which New York is joined to London is 665.6 km of still-air radius against the plain jet's 624.7 — a move of +40.9 km, pinned between Goose Bay–Iqaluit. Paths and directions

A streak along the jet moves the floor while the crossing is flown

The diversion floor answers only to the difference between the wind on a gap's two halves, and every case so far produced that difference by moving a jet sideways. A jet carrying a streak — a thousand kilometres of core markedly faster than the rest — produces the same difference with nothing changed across the flow at all, and moves the floor by up to 43 km where the same jet at one speed along its whole length moves it by ten metres. Plotted against the half-integral difference the two geometries fall on one line: the floor cannot tell them apart. And because a streak drifts with the flow, the allowance an operator must satisfy is 30 km higher than the one the departure analysis reported.

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