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

Azimuth — where it appears

A direction measured as an angle clockwise from north, which on a map means clockwise from whichever north the page uses. Grid north and true north differ by the meridian convergence, so an azimuth read off a sheet and an azimuth measured on the ground are two quantities with one name.

Named by 12 essays across 5 fields — each of them below, with the objects they name alongside it.

A map for pointing, not for locating. Craig's retroazimuthal projection, centred on Mecca. The straight line from any point to the centre makes an angle with the map's vertical equal to the true initial bearing from that place to the centre — checked here at 169 points, worst disagreement 5.7e-14 degrees. The bearings printed beside each line are computed on the sphere with no projection in them.

A map that cannot be read backwards

Craig's projection answers one question exactly — lay a straight edge from any place to the centre and read the compass course, right to 6 × 10⁻¹⁴ of a degree. It pays by folding: 78°S and 48°S on the same meridian are drawn at the same point, so no inverse exists and nothing else can be read off it at all.

choosing · Condition 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 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 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 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 pairs of ten are in every picture that can be drawn. London, New York, Tokyo, Sydney, Cape Town, drawn holding seven of their ten pairs' bearings exactly — the most any flat picture of five places can hold. Of the 15,360 ways of choosing seven pairs and a direction on each, 152 give a picture whose held arrows all point forwards, and London–Tokyo and Sydney–Cape Town, drawn heavy, appear in every one of them. The three pairs this particular choice leaves out are dashed. The price is paid by the rest: the worst, New York to Sydney, is 142.1° from the truth.

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 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 Six of eighteen, where either kind alone can hold only five. London, New York, Tokyo, Sydney, drawn holding five distances exactly — the thick edges — and one bearing exactly, the arrow from London to New York. Six exact constraints, where five distances is the most any flat picture of four places can hold and five bearings is the most it can hold of those. All six are satisfied to 1.9 × 10⁻¹¹. The sixth is not free room found by luck: a picture is blind to where it sits, how it is turned and how large it is, distances see the last of those three and bearings see the second, and a picture holding both kinds is blind only to where it sits. Tokyo–Sydney, dashed, is the distance left out; the picture is wrong about it by 6.9%.

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 Four spheres are wrong about a direction before anything has moved. At 45° of latitude, the worst error over twenty-four starting directions of the azimuth each substituted sphere gives for a geodesic on WGS84, against the distance to its end. Geodetic latitude is wrong by 347″ at one kilometre, and the parametric, authalic and rectifying latitudes by 174″, 116″ and 86.8″ — a half, a third and a quarter of it — flat until a few hundred kilometres. The conformal latitude is right at the start and wrong by 54.3 mas at one kilometre, 5.43″ at 100 and 54.3″ at 1,000; the geocentric latitude starts at 620 mas and joins it. Bessel's sphere, which also corrects the longitude, is exact at every distance to the arithmetic's floor of 1.2e-7″.

A bearing on a sphere is decided by its latitude, not its radius

A sphere that stands in for the ellipsoid cannot be wrong about a direction by being the wrong size, because enlarging a sphere moves no angle. What it can be wrong by is the latitude put on it. Geodetic latitude turns every bearing at 45° by up to 347 arcseconds before anything has moved; the conformal latitude starts exact and drifts 54 arcseconds in a thousand kilometres; and only Bessel's sphere, which changes the longitude too, is exact at every distance.

wrong · Ellipsoid There is no such thing as the radius of the Earth at a latitude. The two principal radii of curvature of WGS84, against latitude, with the mean radius that every table prints drawn across them. The meridional radius M runs from 6335.44 km at the equator to 6399.59 at the pole; the prime-vertical radius N from 6378.14 to the same value. They differ by 42.70 km at the equator and meet only at the pole, and the single number 6,371 km lies between them at no latitude where either is right.

The radius of curvature is two numbers

Twelve essays work on the ellipsoid and every one of them takes a radius when it needs one. There are two at every point, they differ by 42.70 kilometres at the equator, and the single number every table prints is out by 5,583 parts per million on a line running north.

datums · Ellipsoid An angle is a difference of two directions, and the error does not cancel between them. A geodesic triangle with sides of 30 km at 45° N, its first side laid off on a bearing of 45° and the second 60° round from it. Substituting a sphere carrying the geodetic latitude turns each direction at a corner by an amount that runs as the sine of twice its azimuth, so the two directions at A are turned by different amounts and the angle between them is wrong by their difference. Here the worst single direction is out by 345.79″ and the worst angle by 521.09″ — larger, not smaller. Bessel's sphere, which also corrects the longitude, keeps every angle to 6.2e-9″.

An angle is a difference, and the difference doubles the error

Substituting a sphere for the ellipsoid turns every direction at a point, and a surveyor measures angles rather than directions — so the obvious hope is that a turn common to both directions cancels in their difference. It does not cancel, because the turn is not common: it runs as the sine of twice the azimuth, and the largest angle error is exactly twice the largest direction error at a well-shaped corner. A first-order triangulation computed on the geodetic-latitude sphere carries angles nearly nine minutes of arc wrong, and its closure check passes perfectly.

wrong · Ellipsoid

Named alongside it

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

VerificationDegrees of freedomGreat circleBearingConstraintEmbeddingNavigationToleranceConformalityEstimatorLeast-squaresAuxiliary latitude

All concepts