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

Bearing — where it appears

A direction measured clockwise from north, which changes along a great circle and stays fixed along a rhumb line. It is the quantity a compass gives and the quantity a great-circle route fails to hold, which is why a rhumb line was worth a projection of its own.

Named by 19 essays across 6 fields — each of them below, with the objects they name alongside it.

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.

paths · Paths 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 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.

paths · Paths 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.

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 A triangle of 25.0° excess, drawn on Mercator. The three geodesic sides as curves and the three straight sides a ruler draws, with each vertex labelled by how far the ruler's angle is from the true one. The straight-sided triangle's angles sum to exactly 180° because it lies in a plane, and the real one's sum to 205.02°, so the three errors have to account for the whole 25.02° of spherical excess between them — and they do so on this conformal projection exactly as they do on any other.

Conformal does not mean the angles are right

A conformal projection preserves angles between curves at a point. Draw a triangle on one with a ruler and its angles are wrong by degrees — and the total error is fixed before the projection is chosen, because a plane triangle sums to 180° and the real one does not.

wrong · Audit Which way Sinusoidal stretches the ground. The major axis of Tissot's indicatrix at 180 points, drawn as a stroke rather than an ellipse because the argument is the orientation. Longer strokes mark more elongated indicatrices. The direction departs from the graticule by up to 44.4°, so the scale "along the meridian" is not the scale along the direction that is actually stretched most.

Distortion has a direction

Tissot's indicatrix is an ellipse, an ellipse has an orientation, and the orientation is never reported. On most projections it is not along the meridian, on a conformal one it does not exist at all, and both facts are computable from the same four derivatives as everything else.

distortion · Tissot Grid north against true north at 52°, across a 6° zone. seven meridians drawn in the grid's own coordinates. The grid's north is straight up the page everywhere by construction; the meridians are not, and the angle between them is the convergence. It is exactly zero on the central meridian, reaches 2.36° — 142 minutes of arc — at 3° out, and changes sign across the middle. The manual's γ = Δλ sin φ is 3.0″ short of it at the zone edge.

Grid north is not north

A grid has one north, parallel everywhere on the sheet by construction. The Earth has a different one at every point. The angle between them reaches two degrees at the edge of a UTM zone, and a straight line on the grid is not a straight line on the ground either.

families · Ellipsoid A closed traverse cannot see a scale error. The same closed figure with two different errors in it. On the left every leg is 400 parts per million too long, which is roughly what forgetting the grid scale factor costs — and the figure closes to 2.3e-13 m, which is the last bit of a double rather than a measurement. Scaling every leg of a closed figure by the same factor produces a similar figure and a similar figure is still closed, so the check every specification leans on is blind to it. Every dimension in that figure is wrong: the perimeter is out by 1.17 m. On the right one angle is 20 seconds out — a far smaller disturbance in its own units — and the figure fails to close by 0.062 m. Closure tests the shape and says nothing about the size.

A traverse must close

The check every survey specification leans on cannot see the error this whole field is about. Scale every leg of a closed figure by four hundred parts per million and it closes to the last bit of a double, while every dimension in it is wrong.

practice · Reduction Three curves between 45°N and 55°N, 3026 km apart. The normal section observed from the first point, the normal section observed from the second, and the geodesic, each plotted as its distance to one side of the great-circle chord between the two ends. The two sections are 79.0 metres apart at their widest and the geodesic runs between them, 53.3 metres from the first. In LENGTH they differ by almost nothing — 2.40 millimetres over 3026 kilometres — so an observation that follows the wrong one measures the right distance along the wrong ground. On a sphere all three coincide exactly.

The normal section is not the geodesic

A theodolite at A sighted on B swings in one plane and the instrument at B sighted back swings in another, so the two observations trace different curves on the ground — 79 metres apart over 3,026 kilometres — and the shortest path is neither of them. The lengths differ by 2.4 millimetres, so the wrong curve measures the right distance along the wrong ground.

wrong · Ellipsoid Two conventions, and neither of them finds the blunder. A blunder of 350 mm added to leg 3 of a closed traverse, producing a misclosure of 350 mm — one part in 8344, which most specifications would accept. Bowditch's rule shares the misclosure out in proportion to leg length; the Transit rule shares it in proportion to each leg's component along the axis being corrected. Both close the figure exactly, so both are valid; they disagree with each other by up to 14 mm; and neither puts more than 104 mm of correction on the leg that is actually wrong. A rule for distributing a misclosure is a convention for producing consistent numbers, and it is not an instrument for finding errors.

The two ways to spread a misclosure

Bowditch's rule and the Transit rule take the same closed figure and the same misclosure and disagree about which legs were wrong. Both close it exactly, neither puts the correction on the leg that actually carries the blunder, and no measurement can settle which is right.

practice · Reduction The order the corrections go in, and what it costs. Setting out a 51.8 km line on the UTM zone 31N means turning a grid bearing into an azimuth to observe, and there are two corrections: the meridian convergence, 0.8669° here, and the arc-to-chord correction, 6.242″. Each bar carries the lateral offset it produces at the far end of the line, because a bearing error is a number nobody can picture and a sideways miss is the thing that misses. The exact arc-to-chord and the classical formula every manual gives differ by 0.0044″, which is 1.10 mm at the far end — small, real, and the reason the corrections have an order rather than a sum.

Setting out runs the chain backwards

Turning a design coordinate into something to observe on the ground means undoing the reduction chain, and undoing a chain reverses the order as well as the operations. Two of the corrections do not commute, and getting them the wrong way round misses by a millimetre at fifty kilometres.

practice · Reduction What a 20 mm closure tolerance lets each station hide. A closed traverse of seven stations in plan, each labelled with the angle blunder that would leave the closure inside a 20 millimetre tolerance. An angle error at a station rotates everything downstream of it about that station, so the closing point moves by the distance from the station to the close — and the last station before the close stands 72 metres from it and can hide 57 arcseconds, against 2.1 at the worst-placed station. The check is not insensitive; it is unevenly sensitive, and nothing in the specification says so.

What a closed figure cannot see

A closed traverse imposes exactly two conditions on its observations, so everything else is free — and the freedom is not spread evenly. At a twenty-millimetre tolerance the worst-placed station in a seven-station loop hides an angle blunder of 2.1 arcseconds and the station standing seventy-two metres from the close hides 57, because a rotation about a point near the finish moves the finish hardly at all.

practice · Reduction 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 One line cannot point both ways between New York and Tokyo. The great-circle bearing from New York to Tokyo is 333.0°, and from Tokyo back to New York 25.1°. A flat picture draws one straight line between them, whose two directions differ by exactly 180°; the two bearings differ by 180° and another 127.9°. The dashed direction is the best a straight line can do for both ends at once, and it is 63.9° from the truth at each — which is a floor on the bearing error of every flat picture that contains these two cities.

A flat picture has one direction between two places, and the Earth has two

The bearing from New York to Tokyo and the bearing back are not reverses of each other: they miss by 128 degrees. A flat picture draws one line between the two cities, so no picture containing both can have a bearing error below 64 degrees. On any set of places up to about fifty degrees across, the best flat picture sits exactly on that floor — and away from the poles, so does Mercator.

impossibility · Embedding Five bearings of four cities, held exactly. A flat picture of London, New York, Tokyo, Sydney holding five one-way bearings exactly, each drawn as an arrow from the city it is measured at: London to Tokyo, Sydney to London, New York to Tokyo, New York to Sydney, Tokyo to Sydney. Five is the most any flat picture of four places can hold, and of the 192 ways of choosing five pairs and a direction on each, this is one of eight that can be drawn with every arrow pointing forwards; all eight leave London–New York, dashed, unheld. The price is paid by the other seven bearings: the worst, London to New York, is 165.0° from the truth.

Five bearings of twelve, and only eight ways to draw them

A flat picture of four places can hold five of their twelve one-way bearings exactly, the same count as distances, and the five are found by solving equations rather than searching. The equations cannot tell a line from its reverse. Of the 192 ways to choose five, eight can be drawn with every arrow pointing forwards, all eight leave London and New York unheld, and on sixteen cities no choice made at random can be drawn at all.

impossibility · Embedding

The pair that cannot be held is not the one the geometry names

Four world cities have exactly one pair of the six that no drawable picture holds, and the obvious candidate — the pair whose bearing there and bearing back disagree most — is not it. That quantity predicts something else exactly: whether a held pair's arrow has a forced direction. It separates the two groups with a clean gap and nothing in between. And at five places there is no forbidden pair at all, so the four-place case was a statement about how little room five constraints leave rather than about the map.

impossibility · Embedding

A mixed picture holds one more, and what it is holding is north

A flat picture of n places can hold 2n − 3 distances exactly, and it can hold 2n − 3 bearings exactly, and it is tempting to read one budget spent twice. It is not one budget. A distance cannot see which way the page is turned and a bearing cannot see how large it is drawn, so a picture constrained by both kinds is blind to one freedom instead of two — and holds 2n − 2. For four cities that is six of the eighteen available where either kind alone can hold five, and nothing whatever can hold seven: nought of 31,824 choices.

impossibility · Embedding

Two numbers about the paper buy nothing about the places

A picture of n places has four freedoms, and a constraint holds only what its own blindness leaves: distances reach 2n − 3, bearings 2n − 3, the two together 2n − 2. Position is reached by neither, because nothing about how places stand to one another says where on the sheet to put them. A stated position does reach it, and two of them take the budget to 2n exactly. What the count conceals is that the extra two buy nothing: a largest picture holds six place constraints with the pins and six without, the same 7,596 of them, holding the same pairs at the same rates. The constraint system is a direct sum, and a pin is a different currency.

impossibility · Embedding

Named alongside it

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

VerificationGreat circleGeodesicToleranceMercatorEmbeddingRhumb lineVertexAzimuthConformalityConstraintDegrees of freedom

All concepts