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

The thread: What survives a change of coordinates — page 4

Scale along the meridian and scale along the parallel depend on how the sphere was parameterised. The principal scales, the areal factor and the angular deformation do not, and only those are properties of the map itself. Essays 73 to 96 of 115.
Where the summits are, and where the water divides. A stated range 100 km across and 80 km long, its summits 3,000 m high and about 35 km from the western edge, with a plateau falling gently to the eastern edge. Over 40 km of the range the plateau carries a broad rise up to 18 km east of the summits, and a gorge cuts through the range in the middle of that stretch. The dashed line runs through the highest summits; the solid line is the watershed, computed from the land. Lightly shaded ground drains west. Where they part, the darker ground lies east of the summits and drains west: 331 km², reaching 18.1 km from the summits. The terrain is stated, not surveyed. Grids, and what a survey does

The highest summits stay where they are, and the watershed does not

A treaty that puts a boundary on the highest crests which divide the waters names one line where the water divides on the summits and two where a river has cut back through the range. On a stated range the two part by up to eighteen kilometres. They also differ in kind: an error of ten metres in every height moves the summits three metres and the watershed on the plateau more than a kilometre.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

 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 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.

 Weighting the low sight turns the region the fix is in. Four bodies, three of them between 44° and 58° up on one side of the ship and one 5° up behind, with each sight's error made of a sextant part of 0.5′ and a tenth of its refraction correction: 0.94, 0.95, 0.93 km for the high bodies and 2.05 km for the low one. Solid: the 95 per cent ellipse of the fit that weights each sight by its variance, 2.30 by 1.55 km. Dashed: the ellipse a navigator draws who gives every sight the same variance, 2.63 by 2.02 km, its long axis turned 20.1° from the weighted one. Dotted: where that unweighted fix actually scatters, 2.98 by 1.57 km, turned 27.7° from the ellipse drawn for it. All three are centred on one point. Paths and directions

A low sight is worth keeping only if it is weighted

A body five degrees above the horizon is seen through so much air that its refraction correction is ten times a high body's, and a tenth of that correction uncertain doubles the sight's error. Given the same weight as the others, a sight below seven degrees makes the fix worse than throwing it away; weighted by its variance it never does. The ellipse drawn without weights is turned twenty degrees from the right one, and a navigator who must guess the weights loses far less by guessing high.

 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.

 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.

 One velocity field, three frames. a transform boundary, drawn three times. The ground is the same ground and the arrows are not the same arrows: the fastest velocity anywhere in the 3 panels is 70.6 mm/yr and in the quietest panel the same places are nearly still. A frame is a choice of which rotation to subtract, and there is no measurement that picks one. Every panel has the same strain rate at every point. What the numbers refer to

A velocity needs a frame and a strain rate does not

One place on a plate boundary moves at 5.3, 7.0, 10.2 or 54.6 millimetres a year, and towards the south-east or the north-west, depending on which rotation was subtracted first. Every one of those readings reports the same strain rate, to two parts in a hundred thousand million.

 A 900 km circular accuracy at 55° north, projected. Six thousand ground positions drawn from a circular error of 900 kilometres about one place, each projected in Mercator and plotted as a displacement from the projected place. The curve is the nominal 95 per cent ellipse, computed the standard way — the ground covariance sandwiched between the projection's own derivatives. It holds 93.83 per cent of the points, the cloud is measurably longer than it along its own long axis by 5.52 per cent, and it is not symmetric: the third moment along the page's second axis is 0.727 rather than zero. Measuring distortion

The error ellipse is not an ellipse

A positional covariance pushed through a projection is the same matrix sandwich that draws an indicatrix. That is a first-order operation on a map with a second derivative, so the propagated distribution is not the ellipse the sandwich draws — and a nominal 95 per cent ellipse holds 93.06 per cent on one projection and 95.63 on another, in opposite directions, from the same input.

 The same copied stretch, dropped in rigidly and eased into place. One coast drawn twice. In the first ink the copied stretch is moved as a rigid, scaled piece and meets the rest of the coast at two corners — the dots. In the second it is pinned at the same two points and then eased along 30 per cent of the outline either side of each of them, which is what a compiler fitting a sheet by eye does. The dashed line is the coast the graticule predicts. The eased version departs from it by 6.1 thousandths of the map's width against the rigid one's 9.4: easing moves the departure about, and removes very little of it. What is taught wrongly

A copy eased into place loses its seams before its source

A stretch of coast lifted from another sheet and dropped in rigidly leaves two sharp corners where it joins, and both of them are findable. A compiler does not work that way: they pin the ends and ease the middle until it looks continuous. Easing halves the corner within five per cent of the outline and the seams survive it — until forty per cent, where the located seam jumps from four per cent of the coast out to twenty. The source projection survives twice as far, to seventy-five. The piece goes on naming where it came from after the method has lost track of which piece it is.

 Eighteen ways to close an archipelago within a hundred miles. A stated archipelago about 580 km across: three main islands near the centre (A, B and C, 23,576 km² of land between them) and thirteen smaller islands round them, 26,970 km² of land in all. The dot on each island is its outermost point, where a state drawing the widest lawful system would put its basepoint. Solid: the lawful system enclosing the most water, 186,617 km² at a water-to-land ratio of 6.96 to one, 12 segments, the longest 86.6 nautical miles. Dashed: the lawful system enclosing the least, 159,032 km², through the main islands A, B and C. Dotted: the two segments that would take in the outlier P to the south-east, 122.3 nautical miles and 100.9 nautical miles — both over the hundred-mile limit. The archipelago is stated, not surveyed. Grids, and what a survey does

Written in numbers, the archipelago rule still leaves eighteen

The article on archipelagic baselines gives its tolerances as numbers: no segment over a hundred nautical miles except three per cent of them, which may reach 125, and water between one and nine times the land. On a stated archipelago 8,799 closed systems keep the main islands inside, the ratio passes all of them and the length test eighteen, and the eighteen still differ by 27,585 square kilometres. Thin the land and the ratio does the deciding; the lower bound decides nothing; and the three-per-cent exception can be used only by a system padded to thirty-four segments.

 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.

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