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

Zermelo navigation — where it appears

Finding the quickest route through a moving medium. Its cost is the ground length divided by the speed made good along the step, which is exact while the flow is slower than the vehicle, and its answers are not the shortest routes.

Named by 5 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 ground two divisions disagree about. seven places across the world dividing the sphere twice under a steady westerly. Outbound, a place belongs to the site that can reach it soonest; inbound, to the site it can reach soonest. The pale tints are the outbound division where the two agree; the dark ground is where they do not, and it is 50.7% of the sphere — 258,530,666 km². Every site's own share of the world barely moves between the two divisions, by at most 3.8%. The two divisions assign nearly the same AMOUNT of ground to each site and assign completely different ground.

A partition under a directed cost has two versions

Dividing a surface among several sites is one question when the cost is a distance and two questions when it is not. Under a steady flow at 0.45 of a vehicle's own speed, the division by who can reach a place soonest and the division by which place can be reached soonest disagree about 50.7 per cent of the sphere — while no site's own share of the world moves by more than 3.87 points.

paths · Reach 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

Named alongside it

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

Flow fieldVerificationEstimatorGreat circleToleranceAsymmetryClosed formConventionCostOptimal controlPurposeReach set

All concepts