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

Reproducibility — where it appears

Whether a stated procedure, run again, returns the same answer. It has to be asked of the quantity a reader consumes rather than of the one that is easy to print: an aspect search's parameters are not reproducible and the map they produce is.

Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.

What a solved map is, as a list of numbers. A map with no formula is a list of coefficients, and this is the list. The Chebyshev map of an elongated region, 30° by 10° has its coefficients falling by a factor of 1.5e+13 from the first to the fourteenth, so a table of a dozen numbers carries the whole projection; the conformal cube face's coefficients, marked separately, fall far more slowly because the map has a singularity at each corner. How fast this line falls is exactly how portable the map is — and neither map has a name, an inverse in closed form, or a formula anybody could quote.

A map with no formula

The solved projection has no name, no formula and no closed-form inverse. It is fourteen numbers — and the rate at which those numbers fall away decides whether a map can be shipped at all: geometrically for a smooth region, and like a power for one with corners.

choosing · Condition 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 The parameters are not reproducible and the map is. The three-parameter aspect search run at three grid resolutions and compared with the finest, twice over. Compared on the numbers it returns, the answers are 65° of pole apart. Compared on what they do to the region — the root-mean-square difference in angular deformation at every sample — they are 0.29° apart, against a map whose own deformation over that region averages about a degree. The disagreement recorded as a shortfall is a disagreement about coordinates for one map.

Report the map, not the parameters

The aspect search was found returning the same score to 2.3 per cent from poles sixty degrees of latitude apart, which left the answer looking irreproducible. That reading assumed the disagreeing triples make disagreeing maps. They do not — the three answers agree on the distortion field to a quarter of the deformation the map already has.

choosing · Choosing 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 Three implementations of one projection, and three different distortions. The spread in angular deformation between linear, Catmull–Rom and Aitken interpolations of Robinson's own table, at 60° of longitude. It is exactly zero at every fifth degree, because all three pass through the tabulated rows, and it is not zero anywhere else: 37 of 87 latitudes differ by more than half a degree and the worst is 8.42°. The graticules the three draw differ by 4.2 parts per thousand of the map's own span, which is under the width of a printed line.

A projection defined by a table has an interpolation in it

One member of this library has no formula: Robinson set nineteen pairs of numbers by eye and the table is the definition. Three published interpolations of it draw graticules that differ by four parts in a thousand of the map's span and report angular deformations that differ by 8.4 degrees.

families · Families The pass between the basins, measured rather than bracketed. The height of the lowest path from one near-optimal basin to the other, for regions of growing size. It is found by sorting the score surface and joining cells in order, so it is the minimax path's own height rather than the level at which a bisection stops finding two pieces. The fitted exponent is -1.34 at a coefficient of determination of 0.986. The earlier level-set measurement on the same projection gave -1.57 and the argument predicts −1, so measuring the height directly moves the answer towards the prediction, though not onto it.

The height of the pass between two basins

One thing was left owed: the fracture threshold falls faster than the argument predicts, and the difference was supposed to be the level set's blindness to the height of the pass. Measured directly, the pass between the two deepest basins falls as the −1.34 power at R-squared 0.986 — nearer the predicted −1 than the level set's −1.57 on the same projection, and not onto it. It is not the pass that breaks the set first, and at five of six sizes the second basin is the first one's mirror twin.

choosing · Choosing Four layouts, the same five stations, the same ten distances. Every panel has five stations, all ten distances between them, the same instrument precision and the same three degrees of freedom. The ellipses are the error ellipses of the adjusted coordinates, drawn at one common exaggeration, and they are computed from the geometry and the weights alone — no observation value enters any of them. The worst semi-axis runs from 11.1 millimetres to 232, a factor of 20.9, and the difference is entirely where the marks were put.

The network's answer is decided before it is measured

An adjustment is usually judged by what it does with observations. Every quantity a specification is written about — the error ellipses, the redundancy numbers, the smallest detectable blunder — is a function of the geometry and the weights alone, and does not contain an observed value anywhere. Four layouts of five stations with the same ten distances differ by a factor of 20.9 in their worst coordinate.

practice · Reduction Forty-two starts, forty-two answers. Every local optimum coordinate ascent reaches from 42 starts on the same region and the same criterion — a 25° spherical square at a tolerance of ±0.5% — plotted at the share of area it keeps. They run from 36.98 per cent to 67.50, a spread of 30.5 points, and no two of them are the same map: clustered on their scale fields rather than their coefficients, 42 of the 42 are distinct. The two marked starts are the ones anybody would actually use — Chebyshev's map reaches 36.98 per cent and the mean-square map 64.05, and neither is the best this search found.

A criterion worth using is one whose answer is not unique

Forty-two starts on one region and one tolerance give forty-two different maps, keeping from 36.98 per cent of the area to 67.50. The two starts anybody would actually use — Chebyshev's map and the mean-square map — end 27.1 points apart, which is more than the criterion buys over either of them. And the spread is widest at exactly the tolerance where the criterion is worth using at all, because both facts are the same fact: an objective with a flat top and a sharp edge has somewhere to hide answers.

choosing · Condition 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.

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.

choosing · Choosing The second clause moves the map without moving what the first clause measures. The plateau the tie-break changes most, drawn before and after. The contours are where the scale departs from true by the stated half a per cent, so the ground between them is the ground the specification asks to be kept. The rule leaves that ground alone — 52.23 per cent of the square before and 52.54 after — and spends its freedom outside it, where the worst departure falls from 27.16 per cent to 23.77. Of the 42 plateaux this is the largest change any of them shows.

A second clause decides the half that was already decided

A specification written as a band admits forty-two maps keeping between 37.0 and 67.5 per cent of a region. Adding a tie-break — among those, take the one whose worst departure is least — is a linear programme, and it works: it makes each of the forty-two definite. It also merges three neighbours at the bottom into one, leaves 29.1 of the 30.5 points of spread in area kept, and names a single map only at the tolerance where every map keeps everything and the first clause has stopped asking for anything at all.

choosing · Condition 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.

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.

choosing · Condition 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.

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.

choosing · Choosing 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.

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.

choosing · Choosing How far the two routes end up apart. The distance between a line simplified directly at the final tolerance and the same line simplified through two intermediate products, as a multiple of the final tolerance, with a stated extra step applied to each intermediate. With nothing in between the two are the same line to the last bit, because Douglas–Peucker's outputs are nested. Rounding the intermediate to half the tolerance, smoothing it for legibility, or running it through a moving average each break that, and the last of them puts the final product 0.142 away from where a direct route would have put it — nine times the tolerance the product is published under.

Two routes to one scale

A national series is cascaded — the million is derived from the quarter-million, which was derived from the fifty — and the folklore is that the errors accumulate. They do not: Douglas–Peucker and Visvalingam both cascade to the same line the direct route produces, bit for bit, because both output a sublevel set of a per-vertex number. What breaks it is anything else in the chain, and a moving average puts the product nine tolerances away.

applied · Generalise

Named alongside it

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

OptimisationObjective functionAspectVerificationScale factorRegionToleranceDegeneracyOptimal conformalSearchSymmetryConvergence

All concepts