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

Coverage — where it appears

The fraction of a region a set of pieces actually reaches, which a criterion has to check before it may compare what it scored. It is also the cheapest self-check a geometric routine has: pieces of a sphere must sum to the sphere, by a route that shares no arithmetic with the answer.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Mercator swung through every tilt, over Chile and the tropics. The regional distortion of one projection as its axis is tilted from the normal aspect at 0° to the transverse at 90°, each curve divided by its own value in the normal aspect so the two regions can share an axis. Chile is best at a tilt of 90°, a factor of 183.5 better than north-up; the tropics is best at a tilt of 0°, which is north-up. Rotating the sphere costs nothing and changes none of the projection's own properties, which makes this the cheapest improvement available and the one most often left unmade.

Fitting the aspect to the region

Choosing a projection is a choice among a few dozen named things. Choosing its aspect is a choice among a continuum, it costs nothing, it changes none of the projection's own properties, and for a long thin country it is worth a factor of 183.

choosing · Choosing A 6° query against a cube scheme, and the cells it fetches. The cells of a tangent-warped cube scheme at level 5, with the 29 cells a query of 6° radius touches shaded. The disc's own area is 16.84 cells; the count is 29, because every cell the disc's boundary crosses is fetched as well as every cell inside it. In Hilbert order those cells form six contiguous ranges of identifiers, which is six range scans, and the span from the lowest to the highest covers 91 cells against the 29 wanted. Drawn in Mollweide, with the mesh shown only near the query.

A query is a disc, and a disc is not a cell

Everything a cell system does is an address lookup except the one question anybody actually asks it: find everything within five kilometres of here. That is a disc, and the number of cells it fetches is not its area divided by a cell's — at the radii a query is really made at, it is three to seventeen times that.

applied · Cells Two features, one shared boundary, simplified apart. Two neighbouring areas whose common boundary is a curve with structure at every scale — a river or a ridge, in effect — each stored with its own copy of that boundary and each simplified on its own at a tolerance of 0.01. The faint outlines are the originals and the solid ones what came back. The two copies of the shared boundary were within 0.01 of each other before the simplification and are not afterwards: 144 probe cells of 40000 now lie inside both features and 0 inside neither.

A boundary that two features share

Simplifying one curve prices what a tolerance covers. Almost no boundary in a real dataset belongs to one feature: a county's edge is the next county's edge, it is stored twice, and it is simplified twice. What opens between the two answers is a region belonging to both features or to neither, and its area is not bounded by the tolerance.

applied · Generalise Nineteen forecasts of one crossing, and the floor each says must be met. One crossing, with the forecast's error in where the streak starts at 150 km and in its drift speed at 50 km/h. The line is the worst floor over a four-hour crossing against where the streak starts, at the assumed drift of 250 km/h. Dots: nineteen ensemble members, each drawn with its own start and drift speed and placed at the worst floor its own crossing window meets. Open circle: the truth, 660.3 km. Square: the ensemble's centre, 640.3 km. Dashed: the centre's floor plus the fixed margin that covers 95 per cent of crossings at these errors, 653.3 km. Solid: the highest member, 663.2 km — the plan the ensemble gives with no margin added.

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.

paths · Paths A season of crossings pins the stretch to a tenth only if it learns from where the streak was. The share of 400 seasons at each length whose learned stretch lies within a tenth of 1.25. From the floors' rank histogram: 25 crossings 19 per cent, 50 crossings 27 per cent, 100 crossings 32 per cent, 200 crossings 43 per cent, 400 crossings 55 per cent, 800 crossings 67 per cent. From where the streak turned out to be: 25 crossings 56, 50 crossings 72, 100 crossings 85, 200 crossings 98, 400 crossings 99, 800 crossings 100. At a hundred crossings, 85 against 32 per cent.

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.

paths · Paths After a revision narrows the ensemble, a learner that never forgets is still well short of the new stretch three hundred crossings later. The stretch each learner holds before each crossing, the median over 400 winters, from 100 crossings before a forecast revision to 300 after it. Before it the members spread 0.8 of the truth's spread and the right stretch is 1.25; after it they spread 0.6 and the right stretch is 1.667 (dashed). never forgets: 1.28 at 25 crossings, 1.34 at 100, 1.44 at 300; memory 200: 1.31 at 25 crossings, 1.44 at 100, 1.59 at 300; memory 50: 1.42 at 25 crossings, 1.62 at 100, 1.68 at 300; restarts on the date: 1.63 at 25 crossings, 1.67 at 100, 1.67 at 300. Shaded: the middle eight winters in ten for the learner with a memory of 50 crossings, 1.55–1.78 at the end.

A revised forecast is relearned in as many crossings as the learner remembers

An ensemble's stretch is a property of the forecast system, and forecast systems are revised. When a revision narrows the members from four fifths of the real spread to three fifths, a learner that weights every crossing since the start of the winter equally still holds 1.44 for a new stretch of 1.67 three hundred crossings later, and leaves the plan under the floor in 93.8 crossings of a hundred instead of 95. A learner that forgets regains ninety-five per cent coverage after about as many crossings as its memory — 31, 57, 113 and 206 for memories of 25, 50, 100 and 200 — and pays for forgetting only when nothing has changed. A memory of fifty costs nothing measurable over a winter. And a bias the whole ensemble shares, which the stretch alone reads as extra spread, is found to a kilometre by a learner that tracks the mean departure as well.

paths · Paths

Named alongside it

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

VerificationEstimatorFlow fieldMarginToleranceAveragingBiasConventionNoiseRankingAddressAggregation

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