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

Flow field — where it appears

A velocity at every place, added to whatever is travelling through it. It makes the cost of a step depend on its direction, which is what turns every symmetric question in this field into two questions with different answers.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

The shortest route and the quickest one, in a zonal jet. A craft making 20 km/h through the medium, from 40° north, 34° west to 52° north, 6° east. The great circle is 3313 km and takes 111.6 hours in this flow; the quickest track is 169 km longer — 5.1 per cent further — and takes 99.1 hours, saving 11.2 per cent of the time. The strokes are the flow at its own scale, and the whole difference is that the track bends into the helping part of it. Drawn in Lambert conformal conic.

The quickest route is not the shortest

In a medium that does nothing, length and time are the same question divided by a constant. Once the water moves, they are different questions with different answers: the quickest track sails five per cent further and arrives eleven per cent sooner, and the journey back takes four times as long as the journey out.

paths · Paths The plan, the crossing that was flown, and the one a right forecast would have given. A crossing of NaN km planned against a jet forecast to sit on the fortieth parallel, flown through a jet that is actually 6 degrees further south, and re-planned six times on the way. Committing to the plan costs 167.81 hours. Re-planning costs 165.84. A vehicle that had known where the jet was would have taken 163.21. So the wrong forecast is worth 4.60 hours and re-planning recovers 1.97 of them.

A crossing is a chain of decisions

A route is handed back as a curve, and nothing anybody flies is decided once. Re-planning is worth exactly nothing when the forecast turns out right — a sub-path of an optimal path is optimal, and the chain is the plan. When it turns out wrong it recovers 19 per cent of the cost at two decisions and 83 at eighteen, and never all of it.

paths · Paths In a jet the sixty-minute sets stop moving together. The North Atlantic airfields with a southerly jet of 200 km/h whose core, the solid line, runs north past the southern tip of Greenland; dotted lines mark where it has slowed to 150, 100 and 50 km/h. Faint dashed circles are the still-air sets of 745 km. Solid outlines are the places within sixty minutes of each airfield in the jet, computed as quickest flights through it: each is a circle moved by the wind it meets, and airfields on opposite sides of a gap meet different winds. The least allowance at which New York is joined to London moves from 624.7 km of still-air radius to 587.7 (−36.9 km), pinned between Goose Bay–Narsarsuaq, whose sets first touch at the marked point. A uniform southerly of the same speed moves it +3.0 km.

A jet moves the floor that a uniform wind cannot

A uniform wind leaves the least radius at which the North Atlantic can be crossed almost where it was, because it blows the same way on both halves of every gap between airfields. A jet does not. A southerly of 200 km/h past the southern tip of Greenland lowers that floor by 36.9 kilometres, the same jet 400 kilometres further west raises it by 23.3, and one integral of the wind along each gap predicts every such move to within a few kilometres.

paths · Paths The floor against where the streak is, and the stretch of it a crossing sweeps. The diversion floor for the New York–London chain with a streak 200 km/h faster than a 100 km/h jet core, against where along the jet the streak's centre lies, computed exactly on a grid of positions 125 km apart and read between them. With the streak far off the floor is the plain jet's, 624.7 km. With it over the Goose Bay half it rises by up to 41.1; over the Narsarsuaq half it falls by up to 44.8. The bar is the stretch a streak starting at -1400 km sweeps in a four-hour crossing at 250 km/h; the floor to be met is the worst on it, 666.0 km. The shaded band either side is a forecast position error of ±200 km at one standard deviation.

A forecast's error in where the streak is becomes a margin

A jet streak drifting along a gap moves the diversion floor by up to forty kilometres during a crossing, and an operator knows where the streak is only from a forecast. Planned to the worst over the forecast's own crossing window, the floor needs a margin of about eight kilometres for every hundred of error in the streak's position, and stops paying for itself against simply ignoring the streak near an error of two hundred and fifty. Planned to the departure analysis alone, it is worse than ignoring the streak at any error at all.

paths · Paths 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.

EstimatorVerificationMarginToleranceZermelo navigationConventionCoverageGreat circleAveragingBiasBottleneckClosed form

All concepts