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

The thread: A theorem, not a limitation — page 4

No map is faithful, and this is not a shortcoming awaiting a cleverer cartographer. Gaussian curvature is intrinsic, a sphere has some and a plane has none, and that settles it. Essays 73 to 96 of 106.
Six of eighteen, where either kind alone can hold only five. London, New York, Tokyo, Sydney, drawn holding five distances exactly — the thick edges — and one bearing exactly, the arrow from London to New York. Six exact constraints, where five distances is the most any flat picture of four places can hold and five bearings is the most it can hold of those. All six are satisfied to 1.9 × 10⁻¹¹. The sixth is not free room found by luck: a picture is blind to where it sits, how it is turned and how large it is, distances see the last of those three and bearings see the second, and a picture holding both kinds is blind only to where it sits. Tokyo–Sydney, dashed, is the distance left out; the picture is wrong about it by 6.9%. The impossibility

A mixed picture holds one more, and what it is holding is north

A flat picture of n places can hold 2n − 3 distances exactly, and it can hold 2n − 3 bearings exactly, and it is tempting to read one budget spent twice. It is not one budget. A distance cannot see which way the page is turned and a bearing cannot see how large it is drawn, so a picture constrained by both kinds is blind to one freedom instead of two — and holds 2n − 2. For four cities that is six of the eighteen available where either kind alone can hold five, and nothing whatever can hold seven: nought of 31,824 choices.

 The same composite with a strip spliced in, and the meridians no longer turn a corner. Goode's composite redrawn with an equal-area strip of 8° of latitude spliced between the sinusoidal and the Mollweide cap, shaded. The strip's parallel half-length is the unique cubic matching the sinusoidal's value and slope at 36.74° and the Mollweide's at 44.74°, so every meridian arrives and leaves along one direction and the 11.2° corner the plain join turns at 60° of longitude is gone. Nothing else about the map is changed: the cap is the unmodified Mollweide, and the composite is equal-area throughout. The families

An equal-area strip removes the corner and charges nothing for it

Goode's seam matches three quantities and breaks the fourth: every meridian but the central one turns a corner there. A third map spliced between the two removes it — four conditions on one function, satisfied uniquely by a cubic — and the expected price does not arrive. The spliced composite carries less angular deformation across the seam than the plain join, not more: 34.7° against 37.7° at sixty degrees of longitude. What it does leave is the same defect one derivative up, a jump in curvature falling as the reciprocal of the strip's width.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

 One degenerate zero, nudged, becomes two ordinary ones. The direction of steepest ascent within twelve degrees of the north pole, for the sectoral harmonic alone and with two amounts of the tesseral added. On the left is one zero of index −2, a monkey saddle: three ways up and three ways down, and a Hessian that vanishes. On the right are two ordinary saddles of index −1 each, both of which the second-derivative test names correctly. Nothing has been added to the field but a term whose size can be made as small as anyone likes, and the classification changes at every nonzero value of it while the total does not change at all. Measuring distortion

The second derivative cannot classify

Seven essays have used a second derivative to measure a size. The one thing a second derivative is classically used to do is say what KIND of thing is at a point, and on a sphere that use fails: the determinant test totals six where the truth is two, its failure is confined to exactly the field it is asked about, and the count that gets it right never differentiates twice.

 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. Grids, and what a survey does

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.

 The geoid's curvature at a point has no limit; averaged over a triangle it settles. Dashed: the curvature a geoid model contains at a point, as a fraction of the sphere's own, rising in proportion to the degree the model stops at — from 0.36 per cent at degree 360 to 36 per cent at 36,000. Solid: the geoid's contribution to the mean curvature over equilateral triangles of 1, 10, 50 and 100 km, the rms of the anomaly in each one's spherical excess divided by that excess. Each rises while the model is coarser than the triangle and settles once it is finer: the 50 km triangle reads 0.64 per cent at EGM2008's degree 2190 and 0.68 per cent at 36,000; the 1 km triangle 2.19 per cent and 27 per cent. The impossibility

A triangle reads the geoid's curvature, and that one converges

The geoid's Gaussian curvature at a point has no value: its spectrum carries two more powers of the degree than the height's, and the sum never settles. A surveyor's triangle does not measure it at a point. By Gauss–Bonnet its excess is the total curvature over its area, which on the geoid is the flux of the deflection of the vertical through its three sides — a first derivative averaged along lines, and that sum converges. A fifty-kilometre triangle reads 0.64 per cent of the sphere's curvature from the geoid at EGM2008's degree and 0.68 at degree 36,000, where the curvature at a point has grown sixteenfold. The quantity nothing could measure was the one nobody measures.

 Töpfer's square root is one line of a family. The fraction of features surviving to a smaller scale, for four stated populations whose size distributions differ only in their exponent. Every one is a straight line on these axes, and the slope of each is its own exponent: 0.3, 0.5, 0.8, 1.2. Töpfer's radical law is the line at 0.5 — the square root — and it is exact for that population and for no other. The law is not a rule of thumb with exceptions; it is a theorem with a hypothesis nobody states. What a machine does with it

How many features a scale can carry

Töpfer's radical law is quoted everywhere as a rule of thumb. It is not one: it is a theorem about a size distribution with a Pareto exponent of exactly one half, exact to 1.8 per cent for that population and out by 99.4 per cent for a lognormal one.

 The same two islands, and the cut that buys most is the central one's. A sea 60° across with a large island A, 12° across, and a small one B, 6° across, on opposite sides of its centre, drawn in the plane the stereographic makes of it. Each dashed line is one way of spending a single cut — from A to the open sea, from B to the open sea, or from A to B — and the solid line is the one that buys most, as a share of what two cuts buy. Left, A 18° from the centre and B 6°: A 9% · B 78% · between 21%. Right, A 6° and B 18°: A 81% · B 12% · between 11%. The better cut is the central island's in both, whichever island is large, and a cut joining the islands buys 21% and 11%. The impossibility

One cut belongs to the island that would cost most alone

A sea with two islands pays a few per cent for being drawn whole by an angle-keeping map, and one cut buys back part of it. Which island to cut is a design choice that leaves the ground exactly where it was, and it is not decided by size: across twenty arrangements of a large island and a small one, the cut worth most goes to the island that would cost the sea most if it were alone in it — in eighteen of twenty, the other two near-ties — which is usually the more central one. A cut joining the two islands buys almost nothing, and the better cut is nearly always the longer.

 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. Grids, and what a survey does

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.

 Round a triangle the geoid's deflection has a flux and no circulation. A fifty-kilometre triangle on a stated patch of geoid whose undulations follow Kaula's spectrum between 10 and 400 km, with the deflection of the vertical drawn as strokes down the geoid's slope. Beside each side, the deflection's component across it and along it, each integrated along the side, in metres: across −0.689, −0.538, +0.499; along −0.235, +0.275, −0.040. The sum across is −0.727 m, which over the Earth's radius is −0.0235″ of the triangle's excess. The sum along is +0.000003 m — the geoid's height carried round the loop and returned to where it started. The impossibility

A levelling loop reads the circulation, and the geoid has none

A triangle of angles reads the geoid as the flux of the deflection of the vertical through its sides. A levelling loop is the other integral round the same curve, the circulation, and at the geoid that one is identically zero, because the deflection is the slope of a height. What a levelling loop run on the ground misses by is the Jacobian of gravity and height over the loop: 31 mm round a fifty-kilometre loop in a stated mountain range, none of it from the free-air gradient that changes gravity by 1,333 mGal there, uncorrelated with what the triangle reads, and two and a half times the noise of the levelling where the triangle's reading is a fiftieth of its own.

 Charged ln N a number, the count comes out right; charged two, it takes all it is offered. Two information criteria over the same seven segmentations, each shown above its own minimum. Both add the log of the residual to a charge for every number the model spends — four for each piece's placement and one for each seam's position. BIC charges ln N, 6.6 for 720 coordinates, and is least at 2 seams, the number made. AIC charges two, and is least at 6, the most it was offered: each extra seam buys more than two units of fit because its position and its projection are both chosen by search, and a charge for the numbers alone does not pay for the search. What is taught wrongly

A residual cannot count the pieces of a map

A compiled map does not say how many sheets went into it, and trying every count and keeping the best fit cannot tell. Seven pieces always fit better than two, because they contain them. A criterion that charges ln N for each number a piece spends recovers the true count in every trial, for one copied stretch and for two, until the digitising error reaches a thousandth of the map. AIC charges two and takes every piece it is offered. And an eased copy defeats the right criterion the other way: it is not a projection, so BIC buys pieces to explain it, three seams, then four, then six.

 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. Grids, and what a survey does

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.

 The two islands dearest alone are the wrong pair to cut when they sit on the same side. One of forty three-island seas, 60° across, drawn in the plane the stereographic makes of it; beside each island, what it would cost the sea if it were alone in it — P 0.97%, Q 0.72%, R 1.68%. Left: the two dearest islands, P and R, cut to the open sea, which buys 75% of what three cuts buy. Right: the best pair, Q and R, which buys 97%. The dearest islands are 69° apart as seen from the centre, and their single cuts overlap by 40 points. The impossibility

The second cut goes across the sea

With two islands and one cut, the island that would cost the sea most alone names the cut. With three islands and two cuts, the obvious rule — cut the two dearest — names the best pair in only 23 of 40 seas, and loses up to 23 points of what three cuts would buy. It fails when the two dearest sit on the same side of the sea, where their cuts do each other's work. The best pair keeps the dearest island in 37 of 40; its partner is the island across the centre from it, which is right in 32 of 40 and never costs more than 12 points. A cut joining two islands is never worth a second cut to the sea.

 Twelve cells of the grid's pole row are one map. The pole row of the aspect grid for Robinson over a 10° square region centred at 38° N: 24 pole longitudes across, 12 turns of the third angle down, with the projection's pole on the south geographic pole. There, moving the pole in longitude and turning the third angle are the same rotation, so every cell whose longitude less its turn is the same is the same map. The twelve filled cells are one map, the lowest on the row at 1.71 times the region's best score, and they score alike to 2e-11. No two of them are neighbours. The hollow cells are the maps one 15° step round the axis, the lowest at 2.33 times the best, and on the grid as it was the copies could join only through them. What each projection optimises

The pass was a map meeting itself

Four measurements fitted a power law to the depth of the pass that splits a near-optimal set of aspects, and found it tight — R-squared of 0.98 for Robinson, 0.996 along most of a family of maps. On the grid those aspects were searched over, a pole row holds every normal-aspect map twelve times at cells that are not neighbours, and in 28 of Robinson's 36 regions the published pass was two copies of one map meeting through the map one step round the axis. Joined as the one map they are, Robinson's curve falls to R-squared 0.008, and the sinusoidal's famous refusal turns out to be the one measurement the copies never reached.

 The mountain's gravity follows the ground and the root's does not. A profile 260 km long across the stated mountain patch. Top: the height of the ground. Bottom: the gravity at the ground from the terrain's own rock, a sheet at 2,670 kg/m³ (solid), and from a root at 30 km carrying the same mass with the opposite sign (dashed). Over the whole patch the rock's gravity correlates with the height at 0.99, so of its 54.0 mGal rms only 7.8 is not a function of the height under it. The root's, smoothed over thirty kilometres, correlates at −0.85, and of its 10.0 mGal 5.3 is free of the local height. A levelling loop misses by exactly that free part, which crosses the ground's contours. The impossibility

A levelling loop weighs the root, and a triangle weighs the mountain

A mountain range is carried partly by a root of light crust beneath it, and a survey that knows the ground can ask how much. Put a root thirty kilometres down under a stated range and it moves a levelling loop's misclosure by 85 per cent as much as the mountains' own rock does, and a triangle's excess by 16 per cent: the root's gravity is a smoothed copy of the ground, and a smoothed copy crosses the contours. Read against their noise, pinning the root's share to a tenth takes about 2,700 fifty-kilometre loops, a million triangles or 550 astronomic stations — and if the rock's density is unknown too, a triangle cannot tell a compensated mountain from a lighter one.

 A parent and its children, twice. An aperture-7 hexagonal hierarchy beside a square one. The heavy outline is the parent and the light ones are its children. On the right every child is wholly inside and the four of them tile the parent exactly. On the left the child lattice is turned by 19.107° relative to the parent's, only the central child is wholly inside, and 7.14% of the parent is covered by no child of its own. The two families have exactly the same total area — a hexagon cannot be tiled by smaller hexagons at any ratio at all, which is why the mismatch is a construction rather than an approximation. What a machine does with it

A cell's children do not fit inside it

Ten rungs price one cell system at one resolution, and every one of them is used hierarchically. A hexagonal hierarchy does not nest: at the aperture-seven scheme the discrete global grids use, one fourteenth of a parent is covered by no child of its own, exactly, and each of the six ring children is eleven twelfths inside.

 The straight line between two correct maps passes through an incorrect one. Both ends of this sequence are exact cartograms of four cities: the triangular, x first construction at s = 0 and the triangular, y first at s = 1, each meeting the density to arithmetic noise. The panels between them are the straight-line blend of the two, which is what an animation between two maps computes. The cells drawn solid have turned inside out — their signed area is negative, so the map has folded over itself there and two places on the sphere are drawn at one place on the page. Measuring distortion

A map that meets its target can fold

The flow a diffusion cartogram integrates is a diffeomorphism at every instant and cannot fold. Every discretisation of it can, and the point at which one does is a root of a quadratic — written down rather than searched for.

All threads