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

The thread: Measured, not named — page 13

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 289 to 312 of 372.
A week of twilights learns the refraction share, and one twilight learns almost nothing. The share of the refraction correction that is uncertain, estimated from the residuals of equal-weight fixes pooled over one to fifty-six twilights, each residual corrected exactly for its own fix's geometry; the truth is a tenth. Shaded: the middle eighty per cent of 1,500 seeded weeks for a navigator who always takes one body between 3° and 8°, with its median solid. Dashed: the median for one who takes six bodies anywhere between 4° and 70°. After one twilight the first navigator's middle eighty per cent runs from 0.000 to 0.181; after fourteen, from 0.055 to 0.131; after fifty-six, 0.079 to 0.116. The medians climb to the truth from below: 0.049 and 0.020 after one twilight, 0.092 and 0.088 after fourteen. Paths and directions

A week of twilights learns what a sight is worth

One fix cannot tell how good its low sight was, but every residual of every fix is a point on one straight line whose slope is the square of the refraction share, once each is corrected exactly for its own fix's geometry. After one twilight the estimate is zero four times in ten; after a week it is within half again of the truth nearly four times in five. And the lesson is worth little: a guess within a factor of two already costs under 4 per cent of the fix, a week's estimate 1.3, and a share that changes from night to night by half again costs more than the learning buys.

 Three exponents, three maps, and each is the only answer its specification has. The conformal maps of a 25° spherical square that minimise the mean of |ln k| raised to three exponents. The heavy contours are where the scale departs from true by half a per cent either way, so the ground between them is what a surveyor asking for that band would keep. At p = 1, the mean departure, the map keeps 52.8 per cent of the square inside the band; at p = 2, the mean square, 44.2; at p = 32, which is within a per cent of the least worst case, 32.7. Its worst departure falls the other way, from 3.25 per cent to 1.94 and 1.48. For every exponent from one upward the specification is convex and has one answer. What each projection optimises

One exponent names one map, and at tight bands a better one than the band usually finds

A specification written as a band keeps a count of area and throws the rest of the scale field away, which is why it had forty-two answers and why a tie-break could not name one. Written instead as the mean of |ln k| raised to a stated power, it is convex for every power from one up, and has one answer. At a band of a tenth of a per cent that single answer keeps more of the region than forty-one of the band's forty-two, and at half a per cent fewer than nine of them: the band's best answer leads it by 14.7 points there, and the band's own search started from it reaches within two points of that best. Only as the band approaches the least worst case does a large power take over.

 Where on the land the homolosine beats the sinusoidal, read from Goode's own meridians. The land of the world outside Antarctica on a plain latitude–longitude grid, in two-degree cells, each shaded by how much less angular deformation Goode's homolosine gives it than the plain sinusoidal does, both cut into Goode's lobes and each point read from its own lobe's central meridian (dashed); solid lines are the cuts, and the dotted parallels are Goode's join at 40.7°. The key names the two inks: one where the homolosine wins, the other where the sinusoidal does, deeper for a larger difference. Weighted by land, the homolosine averages 16.51° and the sinusoidal 17.34°. Antarctica, drawn pale, is not in this score. The families

On the land it was drawn for, the homolosine wins by its meridians

Scored over the whole sphere and cut into Goode's lobes, the homolosine loses to the plain sinusoidal; for it to win on latitude alone, a reader would have to care about the band from 32° to 61° forty per cent more than its area. The land does not do that — the band holds 1.03 times its share of the continents. Weighted by land anyway, and each point read from the meridian of the lobe it lies in, the homolosine wins: 16.51° against Craster's 17.00 and the sinusoidal's 17.34. The win is in where Goode put his meridians, which lift his map from third to first; count Antarctica as land and it falls to fourth.

 The same resolution, the grid moved, and a different answer. The same field at a fixed 18 × 9 division, with the grid slid by fractions of a cell. Nothing is lost — every cell is the same size as before and there are exactly as many of them — and the largest reported value moves over a range of 12.59 per cent. That is more than a whole halving of the resolution costs, which is 10.15 per cent on the same field. The scale effect has an excuse and this one has none. What a machine does with it

The answer depends on the cells it was counted in

Nine essays price the cell as a shape. The number reported out of it is priced nowhere: sliding a grid without changing its resolution moves the largest reported value by 12.6 per cent, which is more than halving the resolution costs.

 How long a network has before it breaks its own tolerance. The years until the worst baseline in a 400-kilometre network exceeds five millimetres plus one part per million of its length, which is an ordinary first-order specification. A boundary zone breaks it in 16 years; a plate interior takes 1819; a rigid plate never does, at any speed, because a rigid body keeps every distance it has. Bars are clipped at five thousand years. What the numbers refer to

A grid stops fitting the ground it was laid on

A hundred-kilometre baseline across a plate boundary changes by 815 millimetres in fifty years, and the same baseline on the fastest plate in the model changes by nothing at all in the same fifty. The interval before an ordinary first-order specification is broken is six years in one place and never in the other.

 How aligned a region's ellipses are, for every projection and every region. The resultant length of the doubled indicatrix orientations, over 12 projections and 9 regions. One means every ellipse in the region points the same way; zero means they are spread evenly and cancel. Three cylindrical rows are 1.00 throughout, and everything else varies down the row and across it — which is the answer to the question the number was first stated without: alignment is a property of the pair. Mercator's row is the exception that is not a measurement, because a conformal projection's indicatrix is a circle and a circle has no orientation. Measuring distortion

Whether the ellipses point the same way

Alignment was found to decide whether the average of a region's deformations is above or below the deformation of its average, and stated it at ten projections over one region. Swept over a hundred and eight pairs, the answer is that alignment belongs to the projection 28 per cent, to the region 38, and to neither 33 — and two rows of the table turn out not to be measurements at all.

 A territory, and two enclosures somebody might score instead. A stated territory at 45° north — an ellipse twenty-eight degrees long and eight wide, its long axis bearing 45° — with its latitude–longitude bounding box and the smallest cap around it, on a globe. The territory fills 40.6 per cent of the box and 28.6 per cent of the cap. Scoring the library over each names a winner: an azimuthal for the territory itself, an azimuthal for its bounding box, an azimuthal for the smallest cap around it. The cost of working from the box rather than the territory, measured as the box's winner scored on the territory against the territory's own winner, is a factor of 1.0584. What is taught wrongly

The region was a rectangle in a projection

Every verdict this collection hands out is a verdict about a region, and almost every region it has been given is a latitude–longitude box. A box discards a territory's shape, which costs a few per cent, and it records the territory's width in degrees of longitude, which costs up to a factor of 1.72 and changes which family wins on four of forty stated territories. Convert that width to arc and the box gives the right answer every time — so the comparison had a projection in it before it began.

 The level is flat through the family, climbs over its last twentieth, and jumps at the cylinder. Each member's pass depth averaged over its 36 regions. From m = 0.2 to 0.9 it stays between 0.0224 and 0.0450 with no trend. From 0.95 it rises at every step — 0.0436, 0.0844, 0.0919, 0.1082, 0.1235, 0.1388 — a factor of 3.2 to the sinusoidal. And the cylinder at m = 0 stands at 0.2506, 8.5 times its nearest neighbour: a cylindrical map's pattern slides unchanged when it is turned about the Earth's axis, and no other member's does. What each projection optimises

The sinusoidal is the edge of its family, not outside it

The sinusoidal is the last member of a family of equal-area maps that begins at Lambert's cylindrical, and it was thought to be the one projection whose pass depth has no curve. Along the family the curve comes and goes: present at m = 0 and 0.8, absent at 0.6 and 0.9. What does move smoothly is the pass depth's level. It is flat from m = 0.2 to 0.9, then rises at every step over the last twentieth, 3.2-fold, to the sinusoid — whose high ridges are the end of that rise, not an exception to the family — while the cylinder at m = 0 stands eight times above its neighbour.

 The floor against where the streak is, and the stretch of it a crossing sweeps. The diversion floor for the New York–London chain with a streak 200 km/h faster than a 100 km/h jet core, against where along the jet the streak's centre lies, computed exactly on a grid of positions 125 km apart and read between them. With the streak far off the floor is the plain jet's, 624.7 km. With it over the Goose Bay half it rises by up to 41.1; over the Narsarsuaq half it falls by up to 44.8. The bar is the stretch a streak starting at -1400 km sweeps in a four-hour crossing at 250 km/h; the floor to be met is the worst on it, 666.0 km. The shaded band either side is a forecast position error of ±200 km at one standard deviation. Paths and directions

A forecast's error in where the streak is becomes a margin

A jet streak drifting along a gap moves the diversion floor by up to forty kilometres during a crossing, and an operator knows where the streak is only from a forecast. Planned to the worst over the forecast's own crossing window, the floor needs a margin of about eight kilometres for every hundred of error in the streak's position, and stops paying for itself against simply ignoring the streak near an error of two hundred and fifty. Planned to the departure analysis alone, it is worse than ignoring the streak at any error at all.

 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.

 The land wants the join at fifty-two degrees; Goode put it at forty-one. The homolosine rejoined at every latitude from 20° to the pole, the Mollweide cap rescaled so each parallel stays continuous, each composite cut into Goode's lobes at its own best proportions. Solid: mean angular deformation over the land outside Antarctica. It falls from 17.33° at 20° to its least, 16.29°, at a join of 51.7°, and rises again to the sinusoidal's 17.34° at the pole; Goode's join at 40.7° scores 16.51°, and every join sampled from 50° to 60° is within a tenth of a degree of the best. Dashed: with Antarctica counted as land, and dotted: over the whole sphere by area — both fall all the way to the pole, where there is no cap at all. The families

The land puts the homolosine's join eleven degrees higher

Goode joined the sinusoidal to the Mollweide at 40° 44′ because that is where their parallels have the same length — a fact of geometry, not of any objective. On the land his map was drawn for, read from his own lobes, the join has an objective at last, and its best value is 51.7°: a composite joined there gains a fifth of a degree over Goode's, and every join from about 46° to 60° does nearly as well. A smooth curve fitted to the land with no join in it at all stays within three per cent of flat to 50° and then closes, gaining twice as much again. Goode found the right shape and put its knee too low; with Antarctica counted, the land wants no knee at all.

 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.

 Four cities, met — the triangular, x first construction. Every cell drawn here holds the same area of ground and a different area of page: the map has been constructed so that its areal scale factor is the density it was handed, cell by cell, over a contrast of 25.99 to one. The residual against that target is 5.1e-6, measured from the map's own derivatives rather than from the construction. What it cost is the shape: the redistribution alone reaches 139.1° of angular deformation and averages 70.4°, on top of whatever the equal-area projection under it was already doing. Measuring distortion

A map drawn to a density it was handed

Two hundred and twenty-six essays measure distortion after the fact. This one specifies it: a density is handed to a map as a boundary condition, the areal scale factor comes out equal to it to five parts in a million, and every other invariant the site owns becomes the price.

 A ground that is not deforming, read off a Mercator sheet. Every place on this map is carried by a rigid rotation of the whole Earth at forty millimetres a year — the motion that deforms nothing, and that this collection has already shown deforms nothing. The circles are the strain rate a geodesist would report from the grid coordinates alone, up to 17.4 nanostrain/yr. None of it is on the ground. It is the projection's own scale factor changing along the displacement, which is a second derivative of the map arriving in a first-order measurement. What the numbers refer to

The strain a map adds to the ground's

A ground carried rigidly at forty millimetres a year deforms nothing, and read off a Mercator sheet at sixty degrees north it reports 10.9 nanostrain a year. Along a profile through a real boundary the invented part is 0.0 per cent of the answer where the zone is loud and 884 per cent where it is quiet.

 One of these three lines does not slope. Three lengths at each zoom, for a world coordinate near the antimeridian. The pixel halves with every level, as it must. The spacing between representable double-precision values is far below it and halves with it. The spacing between representable SINGLE-precision values does not move at all — 2.0 metres at every zoom, because it is a property of the size of the number and the world does not get smaller. The two cross at zoom 17, and past it a vertex snaps to a lattice coarser than the pixels it is drawn into. What a machine does with it

The renderer runs out of numbers before the zoom does

Eleven essays price the pyramid in exact arithmetic. The pipeline that draws it carries single precision, where a world coordinate near the antimeridian quantises to two metres — at every zoom, because the number does not get smaller when the pixel does. At zoom 22 that is fifty-four pixels.

 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.

 Tonight's residuals cost the fix a steady three per cent; the typical share costs more every time the nights differ more. Six bodies a twilight, 3 of them between 3° and 8°, and tonight's refraction share the typical tenth times a factor drawn afresh each night; the horizontal axis is that factor at one standard deviation. The fix error, root-mean-square over 40,000 seeded twilights, as a percentage above that of a navigator who knows tonight's share. Dashed: weighting with the typical share, 0.8 per cent at a factor of one and a quarter, 10.3 per cent at two and 36.2 per cent at three, off the top of the chart. Solid: weighting with tonight's share estimated from tonight's own residuals, 3.6 per cent, 3.2 per cent and 1.9 per cent. Dotted: an even blend of the two, 1.2 per cent, 2.3 per cent and 2.0 per cent. Tonight's estimate beats the typical share from a factor of about 1.56, and the blend from about 1.33. Paths and directions

Tonight's residuals measure tonight's air badly and weight tonight's fix well

A week of twilights learns the typical refraction share and nothing about tonight's, and a share that changes from night to night costs more than the week buys. With the sextant's error taken from the week, one twilight's residuals estimate tonight's share alone — badly: within half again of the truth 43 times in a hundred with three bodies near the horizon, and zero 22 times. Used as a weight, that poor estimate costs the fix between two and four per cent whatever the night, so it beats the typical share once nights differ by more than half again, and a blend of the two, leaning on tonight as the nights grow more variable, beats both. A weather predictor has to explain most of the nightly change to do better.

 Two lawful systems, one sea: the lines move water between regimes and the outer limit does not move. The stated archipelago with the lawful system enclosing the most water (solid, 186,617 km² inside) and the one enclosing the least (dashed, 159,032 km²). Round each, the limit two hundred nautical miles from its baselines and from every island's coast: the two outer limits lie on top of each other everywhere but a few kilometres off the widest gaps. A straight coast stands 700 km east of the centre; the line equidistant between it and the archipelago is drawn for both systems and they coincide. Inside plus outside, the two systems account for 1,344,857 km² and 1,343,864 km² of sea. Grids, and what a survey does

The archipelago rule moves water between regimes, not between states

The eighteen baseline systems the archipelago article allows on a stated archipelago differ by 27,585 square kilometres of water inside the lines. Measured outward, as the Convention measures every zone, the difference almost vanishes: whatever a system takes inside it gives up from the zone outside, and water inside plus zone outside differs by 993 square kilometres, seven hundredths of a per cent. Against a straight neighbouring coast on any side, the equidistance line does not move at all, because a baseline's nearest point to a straight coast is always an island. Only a neighbour facing the middle of a gap feels the choice — 2,112 square kilometres at thirty kilometres off, 688 at 250.

 Goode's lobes and the lobes the land would choose, at his own count of two north and four south. The land in two-degree cells, Antarctica pale because it is not scored, with Goode's cuts dashed and the land's own solid; a short tick at the equator marks each lobe's central meridian. Goode's: edge at 180°, northern cut at 40°W, southern cuts at 100°W, 20°W and 80°E, meridians 100°W and 30°E north and 160°W, 60°W, 20°E, 140°E south — 16.51° of mean angular deformation on the land outside Antarctica. The land's, with no cut crossing more land than Goode's worst: edge at 20°W, northern cut at 165°E, southern cuts at 70°E, 155°E, 175°W, meridians 65°E and 100°W north and 25°E, 135°E, 175°E, 60°W south — 14.48°. The families

The land would move Goode's lobes before his join

Goode chose his lobes by eye: where the cuts run and where each lobe's meridian sits. Scored on the land outside Antarctica and found exactly on a five-degree grid, the land's own layout at his count of two lobes north and four south deforms it by 14.48° against his 16.51° — two degrees, more than his meridians were worth in the first place — while its cuts cross less land than his: 2,057 kilometres against 2,391, the worst of them 1,056 against the 1,946 his northern cut spends on Greenland. The land wants the map's edge in the Atlantic and its northern cut in the Pacific. His fourth southern lobe is worth four hundredths of a degree, and once the lobes are the land's the best join climbs again, to sixty degrees.

 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.

 A square switches from p = 1 to the worst case at one band; a thin ellipse passes through every exponent between. Area kept inside a band, per cent, against the band as a fraction of each region's least worst departure. Solid: the map minimising the mean of |ln k|, p = 1. Dotted: the least worst case, p = 32. Thin line: the best of twelve exponents from 1 to 32. Shaded: what a rule taking p = 1 below 0.9 and the worst case above it gives up against the best exponent. On the 25° square the best exponent is p = 1 up to 0.85 and the rule gives up at most 0.9 points. On an ellipse with axes 1 : 0.2 an intermediate exponent is best from 0.60 and the rule gives up 5.9. What each projection optimises

The band names the exponent for a compact region, and not for a thin one

A tolerance band is best served by the mean departure when it is tight and by the least worst case when it is wide. Read as a fraction of the region's own least worst departure, the band decides which, and the region's size drops out: a square, a triangle or a hexagon at 10°, 25° or 40° switches from one to the other at nine tenths, and a surveyor told only that loses at most two points of area on every compact region but the degenerate cap. A thin ellipse does not switch. From six tenths upward it is served best by exponents in between, and any rule with two exponents in it gives up five to seven points there, wherever its switch is put.

 A compacting basin, and the tilt it produces. The stated vertical velocity field — a bowl of subsidence 120 km across, with no mass leaving — drawn as circles proportional to the rate, with the tilt of the ground surface as the arrows. The subsidence is largest at the centre and the tilt is exactly zero there, because a smooth bowl has no gradient at its own bottom. The largest tilt is 125.5 nanoradians a year, on a ring at the bowl's own scale length over root two, and it is the quantity a levelling network measures. What the numbers refer to

A vertical rate needs a height system

Four rungs of this anchor measure the two-by-two horizontal tensor, because that is what a tangent chart returns. The larger signal in a subsiding basin is vertical — 126 nanoradians a year of tilt against 0.76 nanostrain a year of horizontal strain — and it is not a measurement at all until the surface it is measured against is named, because that surface is moving too.

 The query, the box it becomes, and the 14 per cent it loses. A query for everything within 400 kilometres of a place at 55° north, drawn in the stored coordinates of Web Mercator. The ring is the true answer's edge; the rectangle is the box the index is given, sized from the scale factor at the query point. The arcs outside the rectangle are true neighbours the index never returns — 13.6 per cent of the rim, reaching 5.0 per cent of the box's own half-width beyond it — and nothing downstream can tell they are missing. What a machine does with it

The query a fast path actually answers

Fifteen rungs price what a stored coordinate means and none asks what it is searched with. No index answers "within two hundred kilometres of here"; an index answers "inside this rectangle of stored coordinates", and the rectangle is built by somebody's arithmetic — which at 55° north silently drops a tenth of the true answer on a conformal projection and three fifths of it on an equal-area one.

 Four cartograms of one density. The same stated density — four cities — met four different ways, drawn on the same cells. Every panel is a correct cartogram of the same numbers: a region's page area is proportional to its mass in all four. They do not look alike, because the areal scale factor fixes one number per point and a map has four derivatives, so three degrees of freedom per point are left over and each construction spends them differently. Measuring distortion

Every density can be met and none is free

Four maps of the same data, all of them correct, charging between 57.6° and 104.4° of angular deformation for it. There is no such thing as the cartogram of a density — there is an infinite family, and somebody picked a member of it without saying so.

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