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

Seven essays on flat pictures of places, beginning with four cities that cannot be drawn to scale, are about distances. A distance is half of what a reader takes from a picture of where things are. The other half is direction: which way Tokyo is from New York.

Direction on a sphere has a property distance does not. The distance from New York to Tokyo is the same number as the distance from Tokyo to New York. The direction is not the same number turned round.

Two bearings for one pair #

The bearing from a place to another is the direction, measured clockwise from north, in which the shortest route leaves the first place. From New York the shortest route to Tokyo leaves heading 333 degrees — a little west of north, towards Alaska — because the great circle between them runs far into the Arctic. From Tokyo the shortest route to New York leaves heading 25 degrees, a little east of north, for the same reason seen from the other end.

On a flat sheet those two directions would be reverses: whatever way New York is from Tokyo, Tokyo is the opposite way from New York, 180 degrees round. On the sphere they are 333 and 25 — which differ by 52 degrees, or equivalently miss being reverses by 127.9 degrees.

The miss is the amount the great circle turns, relative to north, between the two ends. North at New York and north at Tokyo are different directions in space, the great circle is straight, and so its heading measured against the local north changes along the way. The great-circle vertex is about the same turning seen from the route: a great circle’s heading changes continuously, and the place where it runs due east or west is its highest latitude.

What one line can do for two bearings #

A flat picture of the two cities has one line between them. Whatever direction that line is drawn in, its direction from New York and its direction from Tokyo are exact reverses, so it can match at most one of the two bearings — or split the difference.

Splitting evenly is the best it can do if both ends count. The line then misses each bearing by half of 127.9 degrees, which is 63.9. Any other direction does better at one end and worse at the other. So 63.9 degrees is a floor: no flat picture that contains New York and Tokyo can have a worst bearing error smaller than that, whatever else it does and however the picture is rotated.

A floor for any set #

The argument works for every pair, so every set of places has a floor: the largest half-miss over all its pairs.

For the four cities London, New York, Tokyo and Sydney, the worst pair is New York and Tokyo and the floor is 63.9 degrees. For eight cities on every continent the worst pair is Cape Town and Anchorage, which miss by 147 degrees, and the floor is 73.7. For sixteen cities it is New York and Singapore, and the floor is 87.1. For five towns in Britain, the worst pair is Norwich and Plymouth, whose bearings miss being reverses by 4.3 degrees, and the floor is 2.13.

A floor is only a floor. The question worth measuring is whether any flat picture actually reaches it, since a picture has to satisfy every pair at once with a single arrangement of dots and a single rotation.

On a region, the best picture reaches the floor #

The best picture is found by moving dots until no move improves the worst bearing error — started from classical scaling, from every projection in the library and from twenty random arrangements. The worst bearing error of a picture is computed exactly: every ordered pair contributes the angle between its drawn direction and its true bearing, a rotation of the picture adds the same angle to all of them, and the best rotation centres the smallest arc of the circle that holds them all.

Shrunk to a span of 10 degrees, the four cities’ best picture has a worst bearing error of 5.68 degrees, and the floor is 5.68. At 20 degrees, 13.56 and 13.56. At 40 degrees, 39.52 and 39.52. At 56 degrees, 77.2 and 77.2. Each pair of numbers agrees to the last digit printed.

That is a strong result and it deserves stating as one. On every one of these regions, the whole of the best picture’s bearing error is the half-miss of one pair — the directional counterpart of the error belongs to a few of the places, where six pairs held a picture of distances at its worst. Every other pair is satisfied better than that, the rotation is chosen to split the worst pair evenly, and the arrangement of dots can be made to do everything else the bearings ask.

By a span of 71 degrees that stops. The best picture’s worst error is 110 degrees against a floor of 70; at the full span of the four cities it is 116 against 64. No arrangement satisfies all the pairs at once any more, and the error is no longer one pair’s business.

Mercator splits the bearings evenly #

The figure above has a third series, Mercator’s own layout of the same places, and on the smaller regions it is on the floor too. That is not a coincidence, and the reason is the oldest fact about Mercator.

Why Mercator exists is the essay about the projection built so that a line of constant bearing is straight. A straight line between two places on Mercator is therefore the rhumb line between them, and its direction on the page is the rhumb line’s constant bearing.

The figure measures where that bearing falls. To first order in the distance, the rhumb line’s bearing is exactly the mean of the great circle’s bearing out and its bearing back turned round. The two great-circle bearings miss each other by an amount proportional to the distance; the rhumb bearing misses their mean by an amount proportional to its square, so over a regional distance the miss is small — about a fortieth of the half-miss at a thousand kilometres, and less the shorter the distance.

The half-miss is the convergence of the meridians #

For a pair of places a regional distance apart, the miss has a name surveyors have used for two centuries. North at one place and north at another are different directions, and the angle between them, measured along the line joining the places, is the convergence of the meridians. To first order it is the difference in longitude times the sine of the mean latitude.

Measured against that formula, the miss between a pair’s two bearings is the convergence to four figures at a hundred kilometres from London, to two parts in a thousand at a thousand kilometres, and to three and a half per cent at four thousand. Grid north is not north meets the same angle as the difference between a grid’s single north and the ground’s, which reaches two degrees at the edge of a transverse Mercator zone.

So the floor under a regional picture’s bearings is half the largest convergence of the meridians between any two of its places. That is a quantity a navigator could compute from a gazetteer and a table of sines, before anything is drawn, and it says how well any flat picture of those places — any chart, any sketch, any diagram — can possibly show their directions.

So a straight line on Mercator splits each pair’s two bearings evenly, pair by pair, without any search at all. That is precisely the arrangement that reaches the floor, and on a region Mercator does reach it: 2.13 degrees for the British towns, the same as the floor and the same as the best picture found by search; 13.56 at a 20-degree span of the four cities; 5.68 at 10.

Where Mercator stops splitting them #

Mercator’s layout leaves the floor well before the best picture does. At a span of 40 degrees the best picture is on the floor at 39.5 degrees and Mercator’s layout is at 131.

Shrinking the four cities about their centre moves them north and round the globe. At that span New York sits at 150 degrees west and London, Tokyo and Sydney between 130 and 155 degrees east, all of them north of 43 degrees. The set straddles the 180th meridian, and Mercator’s page is cut along it, so the page draws New York at the opposite edge of the sheet from the other three and every bearing to New York points the wrong way across the page. At a span of 56 degrees London has moved to 84.8 degrees north, beyond the 84 degrees the library’s Mercator draws, and there is no layout at all.

Neither failure is about bearings. Both are about where a projection puts its seam and its edge, which is the subject of the topology essays rather than of this one, and both would vanish if the page were re-centred on the set — which is what the free picture, not being a projection, does automatically.

The best picture has no such trouble, because it is not a projection. It is a free arrangement of dots chosen for the bearings, and it can put the places wherever splitting their pairs requires.

What a chart plotter already corrects for #

The floor is not a new difficulty to anybody who has plotted a radio bearing.

A bearing taken by radio direction-finding is a great-circle bearing: the signal arrives along the shortest path, so the direction measured is the great circle’s direction at the receiver. A Mercator chart can only draw a rhumb line straight. So before a radio bearing could be ruled on the chart, navigators applied a correction to turn the great-circle bearing into the rhumb bearing between the same two places — and the correction, the conversion angle, is half the convergence of the meridians between them.

That half-convergence is exactly the floor measured here, for a single pair. The practice treated it as a correction to be applied pair by pair, which it is when one pair is being plotted. What the floor adds is the statement about a whole set: on a region, no picture can do better than the worst of those corrections, and Mercator does exactly that well. The line drawn straight on the page is a route is the essay about what that ruled line means on the ground; this one is about what it means for the directions a reader takes from it.

Which way Tokyo looks from New York on the maps people use #

The floor is the best any picture can do. The maps a reader actually meets do not try.

On a Mercator world map centred on Greenwich, the straight line from New York to Tokyo leaves New York heading 91.7 degrees on the page — a degree and a half south of due east, across the Atlantic, Europe and Asia. On a plate carrée it heads 91.3 degrees; on Robinson 91.6; on Mollweide 91.8; on the Winkel tripel 90.6. Every common world map says that Tokyo is east of New York.

The great-circle bearing from New York to Tokyo is 333 degrees, north-north-west, over the Arctic. Every one of those maps is wrong about it by between 117 and 121 degrees — nearly twice the floor of 63.9 that no picture can go under. And the short way round in a constant compass direction, the rhumb line across the Pacific, heads 267.5 degrees, a little south of due west: nearly the opposite of what the maps show, because they have been cut at the 180th meridian and the short way is not on the sheet.

One map gets the direction exactly. An azimuthal equidistant map centred on New York draws every great circle through its centre as a straight line at its true bearing, so on it Tokyo lies at 333.0 degrees. It pays at the other end: from Tokyo the same line points at 153 degrees, and the bearing back to New York is 25. That is the floor’s pair of errors put entirely on one end rather than split, 128 degrees at Tokyo and none at New York — a legitimate choice for a map made for somebody standing in New York, and no help to anybody standing anywhere else.

Set by set, at their own sizes #

The last row is the refusal, and it has to be there. Five places on the equator have great-circle bearings of exactly 90 and 270 degrees between every pair — the equator is itself a great circle, and the heading along it never turns — so every pair’s bearings are exact reverses and the floor is zero. A flat picture of five dots on a horizontal line reaches it, and the search finds that picture, to the arithmetic. A floor that could not come out at zero would be measuring something other than the sphere.

The world sets are the other end, and the numbers there are worth reading as a verdict. Sixteen cities on every continent have a floor of 87 degrees and no flat picture below 158. A worst bearing error of 158 degrees means that some place on the picture is drawn in very nearly the opposite direction from where it is. For a set of places spread over the world, a flat picture cannot tell a reader which way anything is.

Why the world sets cannot reach their floor #

On a region the pairs can be split one at a time because the pictures they ask for are compatible. On the world they are not, and the incompatibility has a shape that the floor does not see.

Take three places and walk the triangle between them, turning at each corner from the bearing in to the bearing out. On a flat sheet the three turns add up to exactly one full revolution, whatever the triangle. On the sphere they do not, and the amount by which they fail is tied to the area of the triangle — the quantity a direction carried round a loop measures on a closed circuit. A flat picture has no area to put that failure in. It has to absorb it as bearing error, and it cannot absorb it in any one pair; it is spread round the triangle.

The floor is a statement about pairs, and on a small region the triangles are small and carry almost no area, so the pairs can be satisfied independently. On the world, every triangle of places is large, and each one contributes error that no rotation and no arrangement can cancel. That accounts for the shape of the sequence above — the floor holding while triangles are small and failing as they grow — but it is an account rather than a proof, and the exact point at which the best picture leaves the floor is measured here, not derived.

The same limit, from north instead of from pairs #

North cannot be up everywhere proves that no map of the whole sphere can draw north pointing the same way everywhere on the page, by counting the zeros a field of directions on a sphere must have. That is a statement about a continuum, and it is proved by topology.

The floor here is its finite relative. It says nothing about north and nothing about fields. It says that a finite set of places, with no map in sight, has a pair of bearings that no single line can hold, and it prices the failure in degrees from the places’ own coordinates. Where the topological argument says that some place on every map must have north drawn wrong, this one says which pair of places a picture must get wrong, and by how much at least.

Where the model stops #

A bearing here is the initial great-circle bearing. It is the direction the shortest route leaves in, which is what “which way” means to anyone flying or sailing a great circle. A constant compass course to the other place is a different bearing — the rhumb bearing — and a picture scored on rhumb bearings would have no floor at all, since rhumb bearings between two places are exact reverses. Mercator draws those exactly, which is why it exists.

The sphere is a sphere. An ellipsoid changes the bearings between world cities by a fraction of a degree, and nothing here depends on that precision.

The picture is allowed one rotation and nothing else. A picture may not be mirrored, since a mirrored map reverses every bearing’s sense.

And the world sets’ best pictures are upper bounds, found by searching. The floor is exact, the regional results agree from every start, and the world results are the best the search found.

Still open: which directions a picture should keep #

Every picture here is scored on all of its pairs’ bearings, in both directions. A picture that only needs to be right about bearings from one place — a home port, a capital — has a different problem and a known answer: an azimuthal projection centred there draws every bearing from its centre exactly, and gets the rest wrong by whatever it gets them wrong by.

Between those two lies a question with the same shape as five distances of six, and never more: how many of a set’s bearings a flat picture can hold exactly, and which ones. For distances the count is 2n − 3, fixed by the degrees of freedom of dots on a plane. For bearings a flat picture has fewer freedoms — it is blind to scale — and every pair it holds exactly in one direction costs the other direction its half-miss. What the count is, and whether a picture reaching it exists for a set of world cities, is not something the floor can say.