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

The thread: Computed, not quoted — page 12

Scale factors, areal ratios, angular deformations, geodesic distances and the areas of countries are all computed from the projection's own formulae. None is a number recalled from a table. Essays 265 to 288 of 383.
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.

 Four spheres are wrong about a direction before anything has moved. At 45° of latitude, the worst error over twenty-four starting directions of the azimuth each substituted sphere gives for a geodesic on WGS84, against the distance to its end. Geodetic latitude is wrong by 347″ at one kilometre, and the parametric, authalic and rectifying latitudes by 174″, 116″ and 86.8″ — a half, a third and a quarter of it — flat until a few hundred kilometres. The conformal latitude is right at the start and wrong by 54.3 mas at one kilometre, 5.43″ at 100 and 54.3″ at 1,000; the geocentric latitude starts at 620 mas and joins it. Bessel's sphere, which also corrects the longitude, is exact at every distance to the arithmetic's floor of 1.2e-7″. What is taught wrongly

A bearing on a sphere is decided by its latitude, not its radius

A sphere that stands in for the ellipsoid cannot be wrong about a direction by being the wrong size, because enlarging a sphere moves no angle. What it can be wrong by is the latitude put on it. Geodetic latitude turns every bearing at 45° by up to 347 arcseconds before anything has moved; the conformal latitude starts exact and drifts 54 arcseconds in a thousand kilometres; and only Bessel's sphere, which changes the longitude too, is exact at every distance.

 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.

 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.

 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. Grids, and what a survey does

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.

 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. The impossibility

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.

 The curve that does the whole job has no kink and does not turn over. The parallel spacing along the central meridian, as a share of the equator's, for the best single smooth curve found and for three maps it is being compared with. Goode's composite is flat to its join at 40.7° and then follows the Mollweide's, and the kink where the two meet is the join every essay before this one has been about. The fitted curve has no kink, and it is not the shape the question expected either: it is flat only through the first fifteen degrees (97.2 per cent at 15°), then closes monotonically and faster than the Mollweide — 54.3 against 77.3 at 60° — and it never turns over. The families

One curve beats the three maps it would replace

Goode's composite is a sinusoidal, a Mollweide and now a splice strip between them, and the question left open was whether one smooth spacing curve could do the whole job or whether the joins are doing work no single curve can. One term does it: a curve with a single coefficient carries 6.8 degrees less angular deformation than the composite, two beat every named equal-area pseudocylindrical in the library, and the composite turns out to be worse than the Mollweide it is half made of.

 Two features, one shared boundary, simplified apart. Two neighbouring areas whose common boundary is a curve with structure at every scale — a river or a ridge, in effect — each stored with its own copy of that boundary and each simplified on its own at a tolerance of 0.01. The faint outlines are the originals and the solid ones what came back. The two copies of the shared boundary were within 0.01 of each other before the simplification and are not afterwards: 144 probe cells of 40000 now lie inside both features and 0 inside neither. What a machine does with it

A boundary that two features share

Simplifying one curve prices what a tolerance covers. Almost no boundary in a real dataset belongs to one feature: a county's edge is the next county's edge, it is stored twice, and it is simplified twice. What opens between the two answers is a region belonging to both features or to neither, and its area is not bounded by the tolerance.

 The same baseline, turned. The part of a 20 km height difference a degree-360 geoid model omits, against the direction the baseline runs, at four anisotropy ratios. A ratio of one is the model rung 9 used and is a flat line — the isotropic covariance cannot depend on a direction, by construction, which is the whole of the objection. At a ratio of two the same baseline omits 165 millimetres along the grain and 241 across it. What the numbers refer to

The correlation is not the same in every direction

Every number in the measurement before this one came out of Σ cₙ Pₙ(cos ψ) — a covariance that depends on the angular distance and nothing else. Ground has grain: at a modest anisotropy the same 20 km baseline omits 165 millimetres along it and 241 across, and the isotropic answer understates the worse direction by 18.4 per cent.

 Where the correction overtakes the sphere it is correcting for. The worst angle error of four substituted spheres against the size of the triangle, with Legendre's spherical excess on the same axes. The two that are wrong at first order in the flattening are flat in the side, because the error belongs to the latitude and not to the distance. The two that are right at first order have no such floor, so what is left is a term rising in proportion to the side. The excess rises as its square, so each pair meets exactly once, and the meeting decides which of the two corrections a computation is entitled to ignore: the geodetic latitude at 490 km, the rectifying latitude at 247 km, the conformal latitude at 23.8 km, the geocentric latitude at 17.6 km. What is taught wrongly

The correction is the smaller of the two corrections

Legendre's theorem subtracts a third of the spherical excess from each angle of a triangle and solves the rest as a plane figure. It is the correction every nineteenth-century computer applied and the one every textbook explains. Beside the error the choice of substituted sphere leaves in the same triangle it is small: on the thirty-kilometre triangle a first-order chain is made of, omitting it costs 166 millimetres in a computed side and using the geodetic-latitude sphere costs 43 metres. The two terms scale differently, so each sphere has a size at which the correction overtakes it — and for the two spheres a careful computer would have chosen, that size falls inside a real chain.

 A sea with two islands in it, and the three circles it already is. A stated sea 60° across with two islands 12° across, their centres 12° from its own, drawn on the globe and drawn again under the stereographic projection centred on the sea. Stereographic projection carries every circle on the sphere to a circle in the plane, so the flat picture is bounded by three exact circles with no fitting and no iteration: the sea's edge at radius 0.5359, and the two islands at radius 0.1060 with their centres 0.2108 from the origin. Normalised, the region's three conformal invariants are 0.2361, 0.6848, 0.1221 — three numbers, which is what a region with two holes carries and no angle-keeping map can change. The impossibility

Two islands are three numbers, and the proof does not survive them

A sea with one island at its centre has one conformal invariant and a forced best map. Put two islands in it and there are three invariants, readable in closed form because stereographic projection carries circles to circles — and the forcing is gone. The theorem that picked the map needed the region to be unchanged by rotation, not to have one hole, and a single harmonic term two hundredths in size beats the stereographic by 0.19 per cent the moment an island moves off centre.

 Two of six projections have a curve. The depth of the pass that decides when a near-optimal aspect set comes apart, against how much of its own scale variation the region sees, for six projections at nine latitudes and four region sizes each, with each projection's own power law. A curve, R² 0.8 or better: Mollweide (0.47, R² 0.81), Hammer (1.35, R² 0.95). Partial: Eckert IV (0.34, R² 0.52). No curve: Robinson (0.02, R² 0.01), Winkel tripel (-0.06, R² 0.05), Sinusoidal (0.28, R² 0.15). Pooled over all six the fit is R² 0.03, because the six sit at different levels. What each projection optimises

Six projections have two curves, and the refusal has company

Three projections at four latitudes gave three pass-depth exponents, and the question was what sets them. Six at nine latitudes say that most of them have none to set: the Hammer's pass depth rises with what the region sees at R-squared 0.95 and Mollweide's at 0.81, Eckert IV's half-follows, and Robinson's, the Winkel tripel's and the sinusoidal's do not follow it at all. Neither candidate explanation predicts a left-out projection's exponent better than guessing the average — and the sinusoidal, once the lone refusal, is one of three.

 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. Grids, and what a survey does

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.

 Twenty-four versions of one shape, and not one of them gains area. The same closed boundary rotated twenty-four times and simplified at the same tolerance. If the area error were noise the values would straddle zero and their mean would fall towards it; they do not. Every one is negative, the mean is -0.4644 per cent, and the mean is 71 standard errors from zero. A bias of that size cannot be removed by averaging over more boundaries, which is the only defence anybody has against a rounding error. What a machine does with it

A thousand features are wrong in the same direction

The area a simplification costs is unpredictable in sign for one feature. Over a population it is not: twenty-four presentations of one shape all lose area, the mean is seventy standard errors below zero, and no amount of aggregation removes it.

 a transform boundary: what the ground is doing to itself. The same 72 places as the velocity field, with the strain rate computed from the motion's own four partial derivatives rather than from its size. Each cross carries two principal rates: the long stroke is the greater extension, the barred one is shortening. The largest second invariant anywhere here is 241.0 nanostrain/yr, and the arithmetic is the same arithmetic that reads a projection's indicatrix. Drawn in Azimuthal equidistant centred on the window. What the numbers refer to

The ground has an indicatrix too

Two hundred and forty-six essays hold the Earth still while the page is measured. The ground is moving at tens of millimetres a year, and the motion is a map with four partial derivatives — so the same construction that draws Tissot's ellipse draws one for the ground, and the fastest plate in the model returns exactly nothing.

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