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

Route planning — where it appears

Choosing a path between two places under whatever constraints apply, decided by trigonometry before any map is chosen to draw it on. The shortest route and the quickest route are different problems, and on a rotating body with wind the second is not decided by geometry at all.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

paths · Paths London to Tokyo, round a 15° exclusion. The direct great circle, dashed, runs through the disc. The admissible shortest route leaves it along a great circle tangent to the rim, follows the rim, and leaves along another tangent — which is the closed-form answer and is checked against a shortest-path search over the rim that knows nothing about tangents. It costs 39 kilometres on 9559, which is 0.41 per cent. The rim stretch is 293 kilometres of it, and it is the only part of the route that is not a geodesic anywhere along its length. Drawn in Orthographic.

A route that must go round

A shortest route is usually 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.

paths · Paths Exact distances from two places, and from nowhere else. The two-point equidistant projection with London and Cape Town as its centres, 87.0° apart. The light circles are drawn in the map at radii of 30°, 60°, 90°, 120° about each centre; every one of them is a true distance circle on the sphere, to 1.8e-14 relative. Between two points that are not centres the drawn distance is wrong by up to 633 per cent.

A projection written as a condition

Instead of a formula, a sentence: the distance from these two places must be exactly right. The map that satisfies it is found by intersecting two circles, it is exact to five parts in a hundred million million, and it exists over the whole sphere for a reason that belongs to the sphere rather than to the construction.

choosing · Condition Which projection wins, by the first derivative and by the second. Each column is a region, with the projections listed in the order Kavrayskiy's first-order criterion puts them in and the figure at the right of each row giving that projection's rank under the second-order criterion — flexion and skewness aggregated the same way. The rank correlations are Europe 0.83, the conterminous United States 0.81, the tropics 0.90, a cap of 30° radius 0.88, so the two orders agree broadly and disagree in detail. Where it matters is the winner: over a cap of 30° radius the choice moves from Albers equal-area conic to Lambert azimuthal equal-area, while Europe and the conterminous United States and the tropics keep theirs. A criterion that changed every answer would be suspect and one that changed none would be decoration.

The second derivative over a region

Flexion has been measured at points and over the whole sphere, and never over a region — which is the only unit anybody chooses a projection for. Doing it finds that the second-order criterion moves the winner in one region of four, and that one projection in the library has no second derivative at all.

distortion · Flexion 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

Named alongside it

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

Closed formGreat circleShortest pathToleranceVerificationConstraintPurposeCostGeodesicNavigationOptimisationBearing

All concepts