Paths — the ladder · Cartographic Projection

London to Tokyo on Mercator. Two routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Mercator the rhumb line departs from straight by 5.0e-9 of its own length.

The shortest route is not straight

The shortest path between two points on a sphere is an arc of a great circle, and on almost every map it is a curve. The straight line on a Mercator chart is a different route entirely, and on some journeys it is twenty-eight per cent longer.

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New York to Madrid on Mercator. Two routes. The great circle is 5768 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 5939 km — 171 km further, or 3.0 per cent. On Mercator the rhumb line departs from straight by 9.6e-16 of its own length.

Why Mercator exists

A ship can hold a compass bearing and cannot easily hold a great circle. Mercator is the answer to one question — what must a map do so that a constant bearing is a straight line — and it answers it exactly.

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London to Tokyo, seen four ways. The same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did.

The gnomonic companion

One projection turns every great circle into a straight line, and it is the only one that does. It shows less than half the sphere, distorts enormously, and was indispensable for three centuries because of that single exact property.

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London to Tokyo on Orthographic. Two routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Orthographic the rhumb line departs from straight by 2.9e-1 of its own length.

The great-circle vertex

One quantity is constant along a shortest path on a sphere, and it fixes the highest latitude that path will reach before the journey starts. That number is why polar routes exist, and it can be read off the departure bearing without tracing the route at all.

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What treating the Earth as a sphere costs, per journey. The ellipsoidal geodesic minus the spherical great circle, in kilometres, for five journeys. The correction is a few tenths of a per cent and it changes sign: a route running east–west at mid latitude is longer on the ellipsoid, and one running along a meridian is shorter, because an oblate body is fatter round the equator and flatter pole to pole. Both distances are computed — Vincenty's iteration against the haversine formula — and the ellipsoidal one is checked against a geodesic obtained by integrating its own differential equation.

Geodesics on the ellipsoid, and why they are hard

The shortest path on a flattened Earth is not a plane curve, has no closed form, and can be longer or shorter than the spherical answer depending on which way it runs. Every practical method is a series or an iteration, and the correction changes sign.

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Where the inverse problem stops converging, 3° around the antipode of London. Every target in a 6° square centred on the point diametrically opposite London, shaded by how many iterations Vincenty's inverse formula needed to find the geodesic to it. 3 of 625 — 0% — never converged at all, and the worst success took 113 steps against a handful anywhere else on Earth. The failure is not a defect in the formula: near the antipode the shortest path is nearly ambiguous, and an iteration looking for one answer is being asked which of many.

The route with no shortest path

Between a point and the point diametrically opposite there are infinitely many shortest routes and no shortest route, and the standard formula for the distance between two places stops converging in a neighbourhood of it. The failure is a property of the question rather than a defect in the answer.

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London to Tokyo in three straight legs. The great circle, and the route a plan of three constant-heading legs actually follows between waypoints on it. The two touch at the waypoints and part between them by up to 456 km, and the flown route is 253 km longer than the direct one. The headings are 50°, 109°, 149°. Two routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Orthographic the rhumb line departs from straight by 2.4e-1 of its own length.

Flying a curve in straight legs

Nobody steers a great circle, because a great circle requires the heading to change continuously. What is actually flown is a handful of constant-heading legs between waypoints on it, and the gap between plan and curve falls as the square of the number of legs.

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A route that must go round

Every route on this site so far has been free to go anywhere, and no real route is. The shortest path past a circular exclusion is two tangent great circles and an arc of the rim — a closed form that agrees with a shortest-path search to three metres in 9,598 kilometres — and it costs not the obstacle's size but the square of how far the obstacle reaches past the route.

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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.

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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.

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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.

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What the drawn line costs, projection by projection. The ground length of the page-straight route from London to Tokyo, as a percentage above the shortest route. Two of these have names. On the gnomonic, centred on the route, the drawn line IS the shortest route, at -0.0000 per cent. On Mercator it is the rhumb, matching the rhumb's own length to 1 parts per million — which is why that projection exists. On the other eight it is a curve with no name and a cost between 0.0 and 19.6 per cent.

The line drawn straight on the page is a route

Eleven essays draw the route on the map. Nobody has drawn the map's own proposal: the ground curve somebody follows by laying a ruler on the page. It has a name on exactly two projections and is 18.2 per cent long on the one where it is famous.

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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.

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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.

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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.

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