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

Shortest path — where it appears

The route of least length between two points on a stated surface, which is a geodesic. It is unique almost everywhere and not everywhere: the set of points it fails to be unique for is a single point on a sphere and an arc on an ellipsoid.

Named by 6 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

Every route on this site so far is in a medium that does nothing, so 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 Where two equal geodesics meet, from 15° north. A geodesic leaving at azimuth α and its mirror image at −α have the same length wherever they meet, and by symmetry they meet on the antipodal meridian. On a sphere they all meet at one point, the antipode, to 1.4e-7° — the flat line. On the ellipsoid the meeting latitude moves with the azimuth, from -15.563° to -15.009°, so the set of points with two shortest routes is an arc 61 km long, and the distance to it varies by 31 km along its own length.

Where the shortest route stops being the only one

On a sphere there is exactly one point with no shortest route from a given place: the antipode. On the ellipsoid the Earth actually is, that point is an arc — sixty-six kilometres of the antipodal meridian for a point on the equator, half a kilometre for one at 85°, and every point of it reachable by two different geodesics of exactly equal length.

paths · Paths The route at height does not lie above the route on the ground. London to Tokyo, solved at the surface and again at a stated height, with the two ground tracks compared point for point. At a cruising altitude of eleven kilometres the two part by 11.8 metres; the departure is proportional to the height, at a fitted slope of 0.997. On a sphere the same measurement returns 2.7e-9 metres, because the offset of a sphere is a sphere and the two geodesics coincide exactly. The offset of an ellipsoid is not an ellipsoid, and this is what that costs.

The shortest route is not at sea level

Ten rungs route on a surface and nothing is ever flown on one. The offset of a sphere is a sphere, so at altitude the great circle is the great circle. The offset of an ellipsoid is not an ellipsoid — its radii of curvature are M + h and N + h, which belong to no ellipsoid — so the shortest route at cruising height does not lie above the shortest route on the ground.

paths · Paths The shortest route, and the shortest route a vehicle can fly. A leg of 60 kilometres for a vehicle whose minimum turning radius is 5, arriving on a heading 120° off the line and required to leave on one -60° off it. The straight line is the geodesic; the curve is the shortest curvature-bounded path, which is Dubins's RSR — a turn, a straight, a turn — at 67.29 kilometres against 60. The second curve is the runner-up word, drawn to show that the choice between them is a real one rather than a formality.

The shortest route a vehicle can fly

Nine rungs find the shortest path under a metric and none asks whether the thing travelling can follow it. Bound the curvature and the route depends on two headings as well as two positions: the turning cost is a fixed 1.81 kilometres whatever the leg length, so it is 18 per cent of a short leg and 0.28 per cent of a long one, and the whole of it vanishes when the vehicle happens to be pointing the right way.

paths · Paths The shortest route between two regions, and the pair it runs between. Europe and the conterminous United States, with the shortest route between them: 3,581 km, attained at one point on each boundary. The pair is not a corner, not a centre and not anything a reader could name — it is wherever two edges happen to come closest, which depends on the whole shape of both. The dashed route is the pair Gall–Peters makes look nearest: 4,085 km, which is 505 kilometres long, and its two ends sit 2135 kilometres from the true pair between them.

The shortest route between two coasts

Ten rungs find the shortest path between two points. Between two regions the answer is attained at a pair of boundary points, and which pair is not the pair any page makes look nearest — a cylindrical map picks a pair 2,135 kilometres from the right one across the Atlantic, and 1,716 kilometres wrong between Chile and New Zealand, where its route is 22.5 per cent long.

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

Eleven rungs hand back a curve and stop, 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

Named alongside it

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

GeodesicGreat circleClosed formRoute planningVerificationConvergenceCostDiscretisationEllipsoidFlatteningFlow fieldOptimal control

All concepts