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

Identifiability — where it appears

Whether a quantity can be recovered from data at all, as distinct from how precisely — lost when a nuisance transformation acts the way the quantity does. An aspect search returns a triple that is not identifiable and a distortion field that is, so the reproducible answer is the one nobody prints.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

How large a map has to be before a wrong projection stops fitting it. Three rivals fitted to a Mercator graticule centred at 45° north, over regions from 1° to 70° of half-extent. The residual is the shape difference alone, with the best scale, rotation and offset removed, and both axes are logarithmic. The horizontal rule is a fifth of a per cent of the map's width — about the width of a drawn line on a printed sheet — and where a curve is below it, no measurement of that map can tell the two projections apart, however carefully it is made.

Two projections that cannot be told apart

Over a small enough region every projection is the same picture, so the question is how small. The answer is not one number: two conformal projections need twelve degrees of extent before their graticules can be separated, and two projections with different anisotropy separate below one.

wrong · Identify The half-extent at which each rival stops fitting a Gall–Peters graticule. For each candidate, the size of region at which its best fit to a Gall–Peters map first leaves a residual of a fifth of a per cent of the map's width. Below that size the two are the same picture. The numbers are half-extents in degrees of latitude, at 20° north, with the plane affine transformation removed; a bar at 90° is a rival that never separates at all within the range searched.

What a careless copy hides

A photocopier that stretches one axis is a nuisance to remove before a projection can be identified. Removing it costs more than it looks: an affine fit absorbs the entire difference between the cylindrical equal-area projections, so Gall-Peters and Behrmann become the same picture at any size, to sixteen decimal places.

wrong · Identify The same ranking, with the right answer removed from the library. A map drawn in Mercator, fitted by every candidate except Mercator. Something still wins: Conformal conic, by a factor of 1.64 over the runner-up, leaving 0.4 per cent of the map's width unexplained. With Mercator in the library the winner's margin is 1.2e+13. The ranking always produces a name; what tells the two situations apart is how far ahead the name is.

When the answer is not in the library

Fitting twenty candidates to a map and ranking the residuals always produces a winner, which makes it a ceremony unless it can also produce a refusal. Held out of its own library, a Mercator map is named as a conformal conic, leaving 0.4 per cent of the map's width unexplained — and the quantity that tells the two situations apart is not the residual but the margin, which is 1.6 when the truth is absent and 10¹³ when it is present.

wrong · Identify One slice of the aspect objective, at the best γ. The Kavrayskiy score for Robinson over Japan, as the pole is moved over the whole sphere with the third rotation held at the value the search settled on. Dark is good. The marks are local minima of the full three-dimensional grid that happen to lie in this slice: there are 6 of them here and 58 in the cube, and a search that walks downhill from a random start reaches the best of them 7 per cent of the time.

The landscape the search walks on

The three-parameter aspect search was run and its answer recorded with a note admitting nothing proved it global. Mapping the objective finds 26 to 34 local minima for every projection and region tried, a downhill walk from a random start reaching the best of them 6 to 35 per cent of the time — and one seed from the coarse grid the search already uses reaching it in all four cases. The score is reproducible to two per cent across a sevenfold refinement; the pole it names moves 60 degrees.

choosing · Choosing What each configuration can see, and what it cannot. The smallest eigenvalue of the fit's own normal matrix — how much the residual changes for a unit move in the worst direction of parameter space — for three parameterised candidates against six configurations of the same size. A zero is not a hard fit: it is a direction the control points cannot see at all, so every value of the parameter along it gives an identical residual. 2 of 18 are at the floor of double precision, and they are not the ones a reader would guess. Where none is, the spread between the best and worst arrangement is still Infinity at a fixed point count.

Where the control points are

Five rungs fit a library to a map and ask how much to trust the winner. None asks whether the parameters are recoverable at all. On control points along one parallel an equirectangular's standard parallel is not merely hard to find — it is invisible, exactly, and five hundred and twelve points on the same parallel are as blind as eight.

wrong · Identify The set of aspects within a stated distance of the best, for Robinson over Japan. Each row takes every point of a 36 × 19 × 24 grid in the three aspect parameters that scores within (1 + t) of the best, joins neighbouring points, and identifies the pieces the exact degeneracy relates. At t = 3 it is one connected piece spanning 170° of pole; by t = 1 it has broken into 2 pieces; and by t = 0.3 the largest of them spans 11°. So it is not one valley and it is not one basin — it is a sheet that fractures.

The shape of the valley

An aspect search returns three numbers, two searches return triples that differ by a hemisphere, and the maps they produce agree. One cause is an exact degeneracy and the rest was called a valley and left unmeasured. Sampled densely, it is neither a valley nor a basin: a connected sheet spanning 170° of pole that splits in two at twice the best score and into thirteen pieces by 1.7 times it.

choosing · Choosing Where Robinson's valley breaks, against how large the region is. The threshold at which the set of near-optimal aspects stops being one connected piece, for square regions of growing size at 38° north. It falls from 2.10 at 6° to 0.27 at 30°, a factor of 7.9. The earlier measurement was made at one region and quoted "about twice the optimum"; that value belongs to a small region, and the prediction that it should fall as the region grows is what this tests.

Where the valley breaks in two

A near-optimal set is one connected sheet at a loose threshold and thirteen basins at a tight one, and the transition was explained without being tested. The explanation is a prediction about region size and projection sharpness: swept over both, the threshold falls from 2.10 to 0.27 as a region grows from 6° to 30°, three projections lie on nearly one curve, and a fourth declines to join for a reason worth having.

choosing · Choosing A plane fit swallows a datum shift, and keeps swallowing it. A map of OSGB36's ground drawn as though it were on WGS84, at seven sheet sizes. The upper curve is how far the drawing moves — a real 99-metre error on the ground — and the lower one is what survives the best similarity between the two, which is what a fit for the projection removes for free. The absorption runs from 99.68 per cent on a 1° sheet to 85.16 on a 60° one, and what is left is 3.9 parts per million of the map's own size even then — two microns on a sheet half a metre across.

The datum hides inside the projection's parameters

Six essays recover a projection from control points with the body it was drawn on taken as known. It is not known, and the fit cannot find it: a 99-metre datum shift is absorbed to 99.7 per cent on a two-degree sheet and to 85 per cent on a hemisphere, leaving eleven parts per million of the map behind.

wrong · Identify Which rotation a projection cannot see is decided by the projection. The same rotation — 2.455 arcseconds, DHDN's polar one — applied about each of the three axes in turn, with the residual after the best plane fit drawn on a logarithmic scale. A cylindrical and a conic in their normal aspects hide the polar rotation to arithmetic noise, 3e+5 times better than either equatorial one, because a change of longitude is a symmetry of both. A pseudocylindrical hides none of them — its horizontal coordinate carries a factor in latitude, so a longitude shift is a shear rather than a translation. And an azimuthal centred on the equator hides the equatorial rotation instead.

A rotation is not absorbed the way a shift is

A datum's rotations were expected to be the part a plane fit could not swallow. They are the part it swallows best — 93 times better than the translations over a hemisphere, and 24 times better per metre moved — and the reason is that the rotation which matters is a change of longitude, which is a symmetry of the map.

wrong · Identify How many projections the map could be in. The number of candidates whose residual sits below the measurement noise, against the size of the region, at four noise levels. At one per cent of the map's width — a hand-digitised graticule — a four-degree region admits ten of the twenty candidates and a forty-degree one admits exactly one. Every curve falls, none of them crosses another, and all four end at one: identification works, and what it needs is extent rather than precision.

The answer is a set

Fitting a projection to control points produces a best fit — one projection, ranked first, with a margin. A best fit without a spread is not a measurement, and the spread is free: a control point has a width, and every candidate whose residual is inside that width has not been ruled out. At one per cent noise a four-degree region admits ten of twenty candidates and a forty-degree one admits exactly one.

wrong · Identify 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.

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.

impossibility · Curvature A triangle never sees the root against its own noise; a levelling loop sees it until it is deep. For a root at each depth carrying all of the terrain's mass, the rms of each reading's response to it over 160 placements, divided by that reading's noise — the background the root sits in plus the observing error. A 25 km loop: 34.2 per cent at 10 km, 15.6 at 30, 3.9 at 60. A 50 km loop: 37.6 per cent at 10 km, 20.1 at 30, 5.5 at 60. A 100 km loop: 34.9 per cent at 10 km, 20.2 at 30, 6.1 at 60. A 50 km triangle: 3.1 per cent at 10 km, 1.0 at 30, 0.2 at 60. The triangle's response falls with depth more steeply than the loop's, which is what made their ratio look like a gauge of depth, but at every depth it is a few per cent of its noise or less.

A loop reads a root's depth from its own pattern, and a triangle reads none of it

A levelling loop and a geodetic triangle respond to a buried root in proportions that change with its depth, so their ratio looked like a gauge of how deep a range is compensated. It is not, because a triangle's response to a root at 30 km is a hundredth of its own noise. The depth is there in the loops alone: a root changes the pattern of misclosures from loop to loop as it deepens, and 1,600 loops of 50 km place a 10 km root within a factor of 1.5 three times in four and a 30 km root about half the time. A 60 km root is not placed even by 6,400, and when the depth is fitted rather than stated, such a root is read as a light shallow one carrying 25 per cent of the mass where it carries 60.

impossibility · Curvature Seven published numbers, twenty-eight actual ones. The correlation matrix of a seven-parameter fit to 64 common points over a region 9° across. Only the diagonal is ever published — seven standard deviations — and the twenty-one off-diagonal entries are not small: the strongest is ty against rx at 0.940. A translation and the rotation that mimics it over a small patch are very nearly the same parameter, so the fit cannot tell them apart and its errors in the two are locked together.

The parameters are not independent

Rung seven gives the seven parameters their own uncertainty and stops at seven numbers. There are twenty-eight, and the twenty-one nobody publishes are not small: a translation and the rotation that mimics it correlate at 0.94, the normal matrix has a condition number of 4 × 10¹⁶, and propagating from the diagonal alone overstates the transformation's uncertainty by up to a factor of thirty-six.

datums · Datum A gravity station sees a buried root against its own noise three or four times as well as a loop near the surface, and half again as well at sixty kilometres. For a root at each depth carrying all of the terrain's mass, the rms of each reading's response to it over 160 placements on the stated range, divided by that reading's noise — the background the root sits in plus the observing error. A gravity station (noise 28.6 mGal): 111.5 per cent at 10 km, 38.5 at 30, 9.0 at 60, 1.4 at 100. A 50 km loop (noise 25 mm): 37.6 per cent at 10 km, 20.1 at 30, 5.5 at 60, 0.9 at 100. A 100 km loop (noise 39 mm): 34.9 per cent at 10 km, 20.2 at 30, 6.1 at 60, 1.1 at 100. A 50 km triangle (noise 1.7″): 3.1 per cent at 10 km, 1.0 at 30, 0.2 at 60, 0.0 at 100.

Gravity is worth four loops a station, and the background still hides a deep root

Levelling loops alone place a range's compensating root at ten kilometres three times in four from 1,600 loops, and do not place one at sixty from 6,400. Gravity measured on the ground reads the root's attraction directly, and a station is worth about four loops for a shallow root and two for one at thirty kilometres. It does not place the deep root either: 1,600 stations put a sixty-kilometre root within a factor of 1.5 of its depth in fewer than one survey in five. With the background field taken away, a hundred stations place it every time. What hides the deep root is not the reading but the geoid's own mass that the range does not explain, and either half of its spectrum is enough to do it. Fitted for depth, gravity reads the deep root as too heavy, where the loops read it as too light.

impossibility · Curvature

Named alongside it

The objects these essays reach for when they reach for this one.

Least-squaresResidualVerificationInverse problemDegeneracyProjection identificationAspectCorrelationDatumEstimatorIsostasyLevelling

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