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 11 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 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

Routes are computed 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

A shortest path under a metric says nothing about 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

A shortest path is usually asked for 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

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 The shortest crossing that never goes more than 745 km from an airfield. New York to London, drawn on a conformal conic, with a circle of 745 km round each of twenty airfields on both sides of the North Atlantic. The dashed line is the great circle, 5,540 km. The solid line is the shortest route that stays inside the circles, 5,698 km — 158 km or 2.85 per cent longer — and it bends only where two circles cross: between Narsarsuaq and Keflavík; Keflavík and Vágar. Nowhere else can it bend, because anywhere else a straight cut stays inside a circle.

The rule that keeps a route near land is pinned at both ends

A crossing that may never be more than a stated distance from an airfield is refused below one radius and free above another, and each wall is set by a single feature of the ocean. Between them the price falls continuously, with corners and no jumps — and at the lower wall it falls as the square root of the allowance, so the first kilometre is worth twelve of route.

paths · Paths The quickest way out of the ice bends at the edge. A ship twelve kilometres inside a straight ice edge, making 6 km/h in the ice and 25 km/h in open water, bound for a point ten kilometres out into the water and twenty along. The dashed line is the straight route, 3.244 hours. The solid one is the quickest, 2.849 hours — 24 min sooner — and it reaches the edge only 2.55 km along, leaving the slow ice as directly as it can. At the edge it meets the perpendicular at 12.0° in the ice and leaves at 60.2° in the water, and the sines of those angles stand in the ratio of the speeds to within 4e-17.

A crossing bends by a law only a conformal chart can show

A ship leaving pack ice for open water takes its quickest route by bending at the ice edge, so that the sines of its angles to the perpendicular stand in the ratio of its two speeds. The law has nothing in it but angles, so on Mercator it can be checked with a protractor — and on a plate carrée at sixty degrees north the same crossing reads as a ship making ten kilometres an hour in the ice rather than six.

paths · Paths What 16 bearings draw beside a river that bends. Everywhere within a 10 km walk of the centre, with a river 3 km to the north meandering ±2.0 km on a 4 km wavelength, crossable only at bridges every 2.5 km. The reachable ground is shaded — pale where it is reached directly and darker where the walk had to go by a bridge. The outline is the polygon 16 bearings draw. It misses 34.1% of the true ground, as a fan beside a straight river does, and it also CLAIMS 0.14% that is not there — ground cut off by a bend, which a chord between two neighbouring bearings passes straight over.

A bend in the barrier puts the drawn polygon over water

A straight river leaves the near side of a reach set convex, and that is what makes a fan of bearings fail in one direction only: every chord cuts a corner, nothing is claimed that is not there. A meander breaks that: the near side is no longer convex, and a chord between two bearings can cross water. So the drawn polygon now over-claims as well as omitting — by a seventh of a per cent, at every bend measured, and more bearings do not reduce it. The question was whether the two errors partly cancel. They do not, and the reason is not their signs: the over-claim never reaches a hundredth of the omission, because one scales with the whole set and the other only with the river.

paths · Reach Beside a bending river, the fan's chords cross water; a route drawn from the river's own outline never does. Everywhere within a 10 km walk of the centre, shaded pale, with a river 3 km north meandering ±2.0 km on a 4 km wavelength and bridges every 2.5 km. Left: the polygon 16 bearings draw, with the ground it claims and should not in solid ink — chords spanning the bends. Averaged over where the first bearing falls, it misses 35.0% of the true ground and claims 0.15% that is not there. Right: every point reachable by a route with at most one crossing — straight out, or straight to a bridge and straight on — worked out from the river's outline and the bridges alone, with no fan and no shortest path. It misses 6.4% and claims nothing.

A reach set drawn from the river's outline is a bound again

Beside a bending river, a fan of bearings drawn with straight chords claims a strip of water, and the drawing stops being a lower bound. The fan's own numbers cannot repair it: what a bearing records about the river is where the bridges are, not where the bends are, and the best a rule built from them does is cut the over-claim by seven eighths while giving up twenty times as much ground. The river's outline repairs it exactly. Cutting each bearing at the first bank costs three and a half times the over-claim it removes; keeping the chords wherever a route over one bridge reaches costs one and a half. Drawn from the outline and the bridges with no fan at all, that route set misses 6.4 per cent of the reachable ground where the fan misses 35.

paths · Reach Beside a bending river, every reachable point is reached by a route that meets the bank at most three times. Everywhere within a 10 km walk of the centre, beside the stated river meandering ±2 km on a 4 km wavelength with bridges every 2.5 km, shaded by the fewest times a shortest route to it has to meet the bank — a contact being one run along the bank, however many corners it passes. 70.3% of the ground is seen from the centre across dry land. One contact reaches 28.0% more, 85% of it beyond the river and the rest in bays of the near bank hidden from the centre by a spur; two reach 1.18% and three the last 0.47%, 70% of those two beyond the river, in the lee of the far bank's bends beside the outer bridges. Nothing needs a fourth.

A layer is a meeting with the bank, not a corner turned

A reach set drawn in layers — rays from the centre, then from every place a route can leave the bank — ends in the exact answer, and how deep it must go depends on what a layer is. Counted by corners turned, the depth belongs to the polygon the bank is drawn as: five turns on a bank drawn every 400 metres, eighteen on one drawn every 50. Counted by meetings with the bank, it is three at every resolution. One meeting anywhere leaves 1.6 per cent of a ten-kilometre walk unreached, where rays from the bridges alone leave 6.4; two leave 0.47; three leave nothing. A longer walk needs more — up to six meetings at twenty kilometres across thirty-six rivers — and the bend's steepness, which prices the first layer, ranks the depth at 0.24.

paths · Reach

Named alongside it

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

VerificationGreat circleClosed formGeodesicRoute planningPurposeToleranceBarrierCostDiscretisationPolygonReach set

All concepts