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

Navigation — where it appears

Getting from one place to another, which is a question about the ground and about time rather than about a map. The shortest route is a geodesic, the easiest to steer is a rhumb line, and the quickest is neither once wind and airspace are in the problem.

Named by 15 essays across one field — each of them below, with the objects they name alongside it.

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.

paths · Paths 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.

paths · Paths 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 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.

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 Two lines of position replace two circles thousands of kilometres across. A ship at 50° N 20° W observes two bodies at altitudes of 45.60° and 42.69°, which place it on two circles 4,938 and 5,261 km in radius, drawn faint. From an assumed position 150 km to the north-east, each body's azimuth is laid off (dashed) and its intercept marked — 12.2 km and 111 km — and a line of position drawn perpendicular to the azimuth through that point. On Mercator the two lines cross 3.28 km from the ship: each line is only its circle's tangent at the intercept, and by the time the two lines meet they have left their circles.

A line of position is Newton's method, but only on a conformal chart

An altitude of a star puts a ship on a circle five thousand kilometres across, and the intercept method replaces that circle with a straight line. On Mercator the fix those lines give is one step of Newton's method: its error falls as the square of the assumed position's, and re-running it turns 300 kilometres into 24 metres in two steps. On a sheet that does not correct longitude for latitude it keeps a fixed share of every east–west error, one less the cosine of the latitude — 36 per cent at 50° north.

paths · Position fix Three lines of position, and a cocked hat that misses the ship. Three bodies whose azimuths from the ship are 20°, 140°, 250°, each observed with an independent error of about 2 km along its azimuth, give three lines that do not meet. They bound a triangle of 7.5 km², outlined, and the ship — the dot — lies outside it. With independent errors that is the usual outcome: three times in four, whatever the triangle's shape.

A cocked hat holds the ship one time in four

Three lines of position with any error in them bound a small triangle, and the instinct is that the ship is inside. With independent errors it is inside one time in four, whatever the triangle's shape. A common error in every sight makes that almost always when the stars surround the ship and almost never when they do not. And on a sheet that forgets the latitude, the triangle keeps its size and moves off the ship altogether.

paths · Position fix Four lines of position, two unknowns, and four gaps that do not close. Four bodies at azimuths of 20°, 110°, 200°, 290° from the ship, each sight given an independent error of about 2 km along its azimuth. No point lies on all four lines. The least-squares point — the hollow dot — is the one whose squared distances to the four lines are smallest, and the dashed perpendiculars are those distances: 0.31, 2.83, 0.32, 2.83 km. The ship, the solid dot, is 3.21 km away. Three lines would have left a triangle and no residual worth the name; four leave two degrees of freedom, and the four gaps are the first quantity here that can be read as a measurement of the sights themselves.

The residual reports the error the fix was immune to

A fourth sight gives a four-line fit something three lines never had: a leftover. The leftover is a real measurement of the sights, and it is useless exactly where it is needed. One number computed from the four azimuths alone — before any star is observed — says how much of a common error reaches the residual, and it is 1 when the bodies stand at the quarters of the compass, where such an error moves the fix by nothing at all, and 0.004 when they are bunched within sixty degrees, where it moves the fix by two kilometres.

paths · Position fix Two defensible answers from the same four sights, and they are kilometres apart. Four bodies within sixty degrees of one another at azimuths of 60°, 80°, 100°, 120°, each sight carrying an independent error of about 2 km and the same common error of 3 km. The four lines are nearly parallel, so a step towards every body at once is almost exactly a step to the south. The two-unknown least-squares point takes that step and lands 3.66 km from the ship; the three-unknown fit, which solves for the common error as well, refuses it and lands 2.22 km away — its estimate of the common error being 1.87 km. Neither is wrong. They are the two ends of one exchange.

What the extra unknown costs where nothing can see it

Four lines of position can estimate a common error in every sight as well as the position. The estimate is honest — three kilometres put in comes back as 2.8 — and where the residual is blind, its scatter is plus or minus thirty-one, and the fix that carries it is scattered fourteen kilometres from the ship against the biased fix's two and a half. Letting the residual choose between the two is worse than either, because a test with no power is a one-in-twenty lottery on the worse answer.

paths · Position fix The triangle and the ellipse are two pictures of the same three sights. Three bodies at azimuths of 20°, 140°, 250° with independent errors of 2 km, drawn twice: as the cocked hat their lines bound, dashed, and as the 95 per cent confidence ellipse of the least-squares point, outlined. The ellipse is 4.25 by 3.78 km and covers 50.5 km²; the average hat from the same sights covers 7.06. The triangle is seven times the smaller and holds the ship a quarter of the time; the ellipse holds it 95 per cent of the time, which is what it was constructed to do. The hat is a picture of how much the sights disagree, and the ellipse is a picture of where the ship is.

A confidence ellipse is honest only where the sheet keeps angles

The cocked hat covers seven square kilometres and holds the ship a quarter of the time; the confidence ellipse from the same three sights covers fifty and holds it ninety-five per cent, which is what it was built to do. On a plotting sheet whose longitude is not corrected for latitude that ellipse fails in two unrelated ways — its centre is displaced, which costs coverage and depends on the navigator's own assumed position, and its shape is misread, which costs nothing but meaning and depends only on the latitude.

paths · Position fix Weighting the low sight turns the region the fix is in. Four bodies, three of them between 44° and 58° up on one side of the ship and one 5° up behind, with each sight's error made of a sextant part of 0.5′ and a tenth of its refraction correction: 0.94, 0.95, 0.93 km for the high bodies and 2.05 km for the low one. Solid: the 95 per cent ellipse of the fit that weights each sight by its variance, 2.30 by 1.55 km. Dashed: the ellipse a navigator draws who gives every sight the same variance, 2.63 by 2.02 km, its long axis turned 20.1° from the weighted one. Dotted: where that unweighted fix actually scatters, 2.98 by 1.57 km, turned 27.7° from the ellipse drawn for it. All three are centred on one point.

A low sight is worth keeping only if it is weighted

A body five degrees above the horizon is seen through so much air that its refraction correction is ten times a high body's, and a tenth of that correction uncertain doubles the sight's error. Given the same weight as the others, a sight below seven degrees makes the fix worse than throwing it away; weighted by its variance it never does. The ellipse drawn without weights is turned twenty degrees from the right one, and a navigator who must guess the weights loses far less by guessing high.

paths · Position fix Weighted by their residuals, six sights hand the low one six times its share. Six bodies seen from one ship: five between 44° and 61° up between east-north-east and east-south-east, and one 5° up due south, alone in its direction. Along each bearing, two bars: the share of the fix's weight the sight's altitude earns it, and the median share it ends with when every sight is weighted by its own residual and the fit is repeated until the weights settle. The low sight earns 4.0% and ends with 25.0%; the five good ones earn about 19% each and end with 1.8% to 3.1%. Bar length goes as the square root of the share.

Weighted by what it disagrees with, the fix trusts the sight it leans on

With six sights and two unknowns, the residuals seem to offer a way to find the bad sight without knowing the weather: weight each line by how little it disagrees, and solve again. One step of that gives a low body almost exactly the weight its altitude would, 4.5 per cent against 4.0. Every further step walks it away. Iterated until the weights settle, the fit is worse than weighting nothing in all four arrangements tried. Where the low body stands alone in its direction it ends with six times the weight it earns, because the fix leans on its line and its residual shows only 46 per cent of its error.

paths · Position fix A week of twilights learns the refraction share, and one twilight learns almost nothing. The share of the refraction correction that is uncertain, estimated from the residuals of equal-weight fixes pooled over one to fifty-six twilights, each residual corrected exactly for its own fix's geometry; the truth is a tenth. Shaded: the middle eighty per cent of 1,500 seeded weeks for a navigator who always takes one body between 3° and 8°, with its median solid. Dashed: the median for one who takes six bodies anywhere between 4° and 70°. After one twilight the first navigator's middle eighty per cent runs from 0.000 to 0.181; after fourteen, from 0.055 to 0.131; after fifty-six, 0.079 to 0.116. The medians climb to the truth from below: 0.049 and 0.020 after one twilight, 0.092 and 0.088 after fourteen.

A week of twilights learns what a sight is worth

One fix cannot tell how good its low sight was, but every residual of every fix is a point on one straight line whose slope is the square of the refraction share, once each is corrected exactly for its own fix's geometry. After one twilight the estimate is zero four times in ten; after a week it is within half again of the truth nearly four times in five. And the lesson is worth little: a guess within a factor of two already costs under 4 per cent of the fix, a week's estimate 1.3, and a share that changes from night to night by half again costs more than the learning buys.

paths · Position fix Tonight's residuals cost the fix a steady three per cent; the typical share costs more every time the nights differ more. Six bodies a twilight, 3 of them between 3° and 8°, and tonight's refraction share the typical tenth times a factor drawn afresh each night; the horizontal axis is that factor at one standard deviation. The fix error, root-mean-square over 40,000 seeded twilights, as a percentage above that of a navigator who knows tonight's share. Dashed: weighting with the typical share, 0.8 per cent at a factor of one and a quarter, 10.3 per cent at two and 36.2 per cent at three, off the top of the chart. Solid: weighting with tonight's share estimated from tonight's own residuals, 3.6 per cent, 3.2 per cent and 1.9 per cent. Dotted: an even blend of the two, 1.2 per cent, 2.3 per cent and 2.0 per cent. Tonight's estimate beats the typical share from a factor of about 1.56, and the blend from about 1.33.

Tonight's residuals measure tonight's air badly and weight tonight's fix well

A week of twilights learns the typical refraction share and nothing about tonight's, and a share that changes from night to night costs more than the week buys. With the sextant's error taken from the week, one twilight's residuals estimate tonight's share alone — badly: within half again of the truth 43 times in a hundred with three bodies near the horizon, and zero 22 times. Used as a weight, that poor estimate costs the fix between two and four per cent whatever the night, so it beats the typical share once nights differ by more than half again, and a blend of the two, leaning on tonight as the nights grow more variable, beats both. A weather predictor has to explain most of the nightly change to do better.

paths · Position fix

Named alongside it

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

EstimatorLeast-squaresGreat circleVerificationConformalityCovarianceMercatorResidualAzimuthGnomonicNoiseStandard error

All concepts