Ranking — where it appears
Named by 21 essays across 6 fields — each of them below, with the objects they name alongside it.
Distortion over a region
An indicatrix describes a point and every practical question is about a country. Going from one to the other means integrating, integrating means choosing a weighting, and the weighting is the step that turns a measurement into somebody's opinion.
The rule of thumb, scored
Cylindrical near the equator, conic in the middle latitudes, azimuthal at the poles. It is the most repeated piece of practical advice in cartography and it has never been run against a population of regions. Run against thirty, it is right nineteen times, and every one of its failures has the same shape.
A map does not say what it is
Every map on this site is one the site drew, from a projection it chose. Every map a reader has ever used is the other kind — a picture whose projection is a sentence in a corner, a legend, or nothing at all. The graticule is enough to recover it: fit every candidate to the crossings and rank what is left over.
The second derivative has its own ranking
Rank eight world maps by how much they stretch and Mercator comes sixth of eight. Rank the same eight by how much they bend and it comes third. The two halves of the second-order score disagree with each other more sharply than either disagrees with the first-order one — Spearman 0.45 against 0.69.
The second derivative over a region
Flexion has been measured at points and over the whole sphere, and never over a region — which is the only unit anybody chooses a projection for. Doing it finds that the second-order criterion moves the winner in one region of four, and that one projection in the library has no second derivative at all.
Nearest of many is a partition
One reach question with one source is a disc. With several sources it is a division of the whole surface, every place belonging to whichever source is nearest — and computing that division in the plane the data is stored in hands away between 0.75 and 22.16 per cent of the ground, in unbroken strips up to 1,591 kilometres across.
The second derivative is not an invariant
The first-order ladder established which quantities survive a change of coordinates and which are artefacts of the parameterisation. Asked of the second order, the answer is that flexion survives a rotation and a magnification of the page exactly, and survives nothing else: a stretch of 1.6 in one axis moves it by eleven per cent, a shear of 0.5 by thirty-five, and a shear of 3 leaves a ranking of eight world maps with a rank correlation of −0.07 to the one it started with.
The net that loses the fewest neighbours
Every one of the cube's 384 nets cuts exactly seven edges of exactly the same length, so the quantity this collection has been pricing cutting by is a constant that cannot choose between them. Measured on the reader's side — how far apart a net puts two places that touch on the globe — the best net scores 11.30 and the worst 17.97, for identical cutting.
The best flat picture is not a map
A set of dots whose separations are as nearly right as separations can be made beats every projection in this collection — by a factor of thirteen on four world cities, and by eight per cent on sixteen. The collapse between those two numbers is not about cartography. It is that a picture of n places has 2n − 3 free numbers and n(n − 1)/2 distances to spend them on.
Which projection a weighting can make best
The seven world projections nothing beats on both counts tell a reader with a preference that one of them is their answer. Four of the seven are not: they sit in dents of the front, undominated and unreachable, and no weighting of angle against area can ever put them first. Robinson can be first, on 2.8 per cent of the weight range.
The error belongs to a few of the places
The least error of any flat picture of sixteen cities is 53.3 per cent, and it is not spread over the sixteen. Six pairs hold it there and Sydney is in three of them: leave Sydney out and it falls to 25.5, while leaving out any of five other cities changes it not at all. Choosing which places to draw is worth more than any choice of how to draw them.
The ranking is not an order
A weighting can be made to put almost any projection first. Remove the weights entirely, let each of the seven criteria vote once, and the answer is worse: over the whole sphere Mercator beats the sinusoidal, the sinusoidal beats Eckert IV, and Eckert IV beats Mercator — four such circles, every margin four to three, with a Condorcet winner sitting above them all.
The average was a choice of norm
Ten projections, one region, one measured quantity, and the only free decision left is how to turn a field into a number. Over the world's scale departure the ordering at the mean and the ordering at the worst case have a rank correlation of −0.04, all ten maps change position, and the exponent that produced each answer is stated nowhere.
The maps with no family are simply better
Seven projections have no continuous symmetry, and they are almost exactly the set anybody would choose for a world map. That coincidence came with a conjecture and a test attached: their advantage should collapse under a criterion that does not care where anything is. Run, it does the opposite — 1.343 times under a uniform weighting and 1.204 under a concentration. The conjecture is refuted.
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.
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.
The area weighting was a readership all along
Twelve measurements have taken a projection comparison apart and found the comparer's own freedom in every part of it. All twelve rested on one thing none of them examined: that a square kilometre of empty ocean counts exactly as much as a square kilometre of city. That is not the absence of an assumption about the reader — it is the assumption that everybody who will ever look at the map is spread evenly over the globe. Named as one readership among five, it produces a different winner from three of the other four, and against a North Atlantic readership its choice is fifth of twenty and 33 per cent worse.
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.
An ensemble carries its own margin, if its spread is honest
A fixed margin of about eight kilometres for every hundred of forecast error covers the diversion floor in nineteen crossings of twenty, but it needs the error's size and it is spent on every crossing alike. Nineteen forecasts of the same crossing need neither. If the truth is one more draw from the spread they come from, the highest of them covers the floor met in nineteen crossings in twenty by rank alone — whatever the error, the streak or the drift — and it holds a fifth to two fifths less in hand than the fixed margin, because it spends nothing on the crossings the streak cannot reach. What it cannot do is notice that its own spread is too small: at half the true spread it covers 88 in a hundred.
A season learns the stretch from where the streak was, not from the floors
An ensemble whose members spread four fifths as widely as the forecast's real error covers the diversion floor in 94 crossings of a hundred instead of 95, and the repair is to stretch every member away from the centre by a factor learned from past crossings. Learned the way the rank histogram teaches it — from where the truth's floor fell among the members' floors — a season of a hundred crossings lands within a tenth of the right stretch one time in three, and a season of eight hundred two times in three. Learned from where the streak turned out to be, which the analysis after each crossing supplies whether the streak reached the gap or not, a hundred crossings land within a tenth 85 times in a hundred and two hundred almost always. The floor is the quantity the plan is about; it is the wrong quantity to learn the spread from.
Which of these numbers are the sampler's
Four rungs have shown that a sampled maximum understates, a sampled mean can be a report on the sampler, no arrangement of points is neutral, and refining until the answer settles proves nothing. So the collection re-measured itself. The means move by at most 1.1 per cent, the worst points by up to 19, and the rankings — which is what the essays actually argue with — do not move at all.
Named alongside it
The objects these essays reach for when they reach for this one.
PurposeVerificationEstimatorOptimisationKavrayskiy's criterionRegionRegional distortionToleranceTrade-offAspectDistortion criterionWeighting