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

Everywhere within 3,000 km of London, drawn in Lambert cylindrical

Drawn here at the parameters it defaults to, with every essay that calls it.
Everywhere within 3,000 km of London, drawn in Lambert cylindrical. The set of places exactly 3,000 kilometres from London by the shortest route, projected point by point. On the ground it is a circle — every point of it is the same distance from the centre, in every direction. On this page the longest radius from the drawn centre is 4.17 times the shortest, so a reader with a ruler measures two different distances for one ground distance depending on which way the ruler points. The two extreme radii are drawn.

The set of places exactly 3,000 kilometres from London by the shortest route, projected point by point. On the ground it is a circle — every point of it is the same distance from the centre, in every direction. On this page the longest radius from the drawn centre is 4.17 times the shortest, so a reader with a ruler measures two different distances for one ground distance depending on which way the ruler points. The two extreme radii are drawn.

It is one of 0 figures drawn by the same construction, and the drawing above is this one with nothing changed.

14 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

Everywhere within 3,000 km of London, drawn in Lambert cylindrical. The set of places exactly 3,000 kilometres from London by the shortest route, projected point by point. On the ground it is a circle — every point of it is the same distance from the centre, in every direction. On this page the longest radius from the drawn centre is 4.17 times the shortest, so a reader with a ruler measures two different distances for one ground distance depending on which way the ruler points. The two extreme radii are drawn. Paths and directions

A circle of a distance is not a circle

Eleven essays in this field have followed a route across a map. A range ring is not a route — it is the edge of a set — and drawing one exposes a failure the route essays cannot: the same ground distance comes out 4.17 times longer in one direction than another on a common projection, and 13.03 times at 70° north.

 Which of eight places is nearest, decided on the ground and decided on the page. Every cell of the window shaded by which of eight places across Europe is nearest on the ground, with the cells the page's own answer would hand to a different site drawn in the failure colour. Plate carrée misassigns 13.67 per cent of the window's ground area — 3,058 thousand square kilometres. The misassigned cells are not scattered: they lie in bands along the boundaries, which is what a systematic error looks like and what a sampled test of a few query points is least likely to find. Paths and directions

Nearest of many is a partition

One reach question with one source is a disc. With several sources it is a division of the whole surface, every place belonging to whichever source is nearest — and computing that division in the plane the data is stored in hands away between 0.75 and 22.16 per cent of the ground, in unbroken strips up to 1,591 kilometres across.

 Everywhere within 200 km of the route from London to Tokyo. The shortest route between the two places, and the set of places within 200 kilometres of it, with both edges computed on the sphere and then projected. On the ground the set holds 3.949 million square kilometres — the band 3.823 and the two end caps 0.126, which between them make one disc of the corridor's own width. Drawn in Mollweide the corridor is visibly wider at one end than the other, and the ground it stands for is not. Paths and directions

A corridor has a width the page cannot keep

Buffering a line is the most-run operation in spatial analysis and the corridor it produces is a reach set with two failures nobody separates: stroked on the page, one width covers 190 to 471 kilometres of ground along a single route; computed in closed form, the formula stops being the area at a width the route's own length fixes, and eventually claims more ground than the sphere has.

 A reach set with a cost that depends on direction. Everywhere reachable in the time it takes to cover 3000 kilometres in still air, under a steady westerly, with the still-air set drawn inside it. The anisotropic set runs from 1653 kilometres against the flow to 4261 with it — a ratio of 2.58 — while the still-air set is round to 1.023, which is the lattice's own floor and not a shape. Drawn in an azimuthal equidistant centred on the source, so every radius on the page is a ground distance and none of the shape is the projection's. Paths and directions

A reach set with a cost that depends on direction

A reach set built out of a distance is symmetric and isotropic by construction. Nothing anybody travels is: in a flow at 45 per cent of a vehicle's own speed the same vehicle gets 4,261 kilometres one way and 1,653 the other — a ratio of 2.58 — while the ground it covers grows by ten per cent.

 The set that can be reached is not the set that can reach. Two sets under a steady westerly: everywhere reachable from the marked place in a stated time, and everywhere from which the marked place can be reached in the same time. They have the same area to a fraction of a per cent — reversing a uniform flow is a reflection — and they overlap on only 32.3 per cent of their union. 67.7 per cent of the ground in one of them is not in the other. Paths and directions

The set that can be reached is not the set that can reach

The moment a cost stops being symmetric, two questions that read alike stop having the same answer. Under a flow at 45 per cent of a vehicle's own speed the set reachable from a place and the set from which the place is reachable have the same area to five significant figures and share 32 per cent of their union — so 68 per cent of the ground in one of them is not in the other.

 How far each page reorders the shapes. The number of pairs of shapes whose order on the page differs from their order on the ground, out of 36, for ten projections. Four of them put a shape other than the geodesic disc at the top — Equirectangular, Lambert azimuthal equal-area, Robinson, Miller cylindrical — which means the shape that attains the isoperimetric bound on the sphere is not the most compact thing on those sheets. The projection with none is not the equal-area one; it is whichever one's stretching happens to leave this particular set of shapes alone. Paths and directions

The most compact shape depends on the paper

A geodesic disc attains the isoperimetric bound on a sphere — its score is one, exactly, at any radius. Score the same nine regions from their images on ten projections and four of them put something else on top, an oval and its own 45° rotation come out 4.7 per cent apart, and the projection that preserves the order is not the equal-area one.

 The score does not settle at any resolution. The compactness of one stated boundary — a circle with cosine ripples at eight geometrically spaced wavenumbers, so it has structure at every scale — read at sixteen vertices up to two thousand and forty-eight. The ground score falls from 0.980 to 0.834, and it keeps falling: the boundary's length grows without bound as it is resolved while the area it encloses converges, so the quotient has no limit. The four page curves sit within a fraction of a per cent of the ground curve and of each other, which is the comparison this rung exists to make. Paths and directions

The score is not stable at any scale

One boundary, read at eight resolutions from sixteen points to two thousand and forty-eight: the compactness score falls from 0.980 to 0.834 and is still falling. Changing the projection instead moves it by 0.69 per cent. The two decisions are made by the same person on the same afternoon and only one of them is ever reported.

 The set a reach map shows, drawn from eight bearings. A geodesic disc of 4,000 km and the polygon a fan of eight bearings draws round it, on an equal-area azimuthal page centred on the disc so that the shaded ground is proportional to the ground it stands for. Every vertex of the polygon is on the true boundary and every edge between two of them is a chord, so the drawn set is inside the true one — always, at every count, for any convex reach set. The area it misses is 7.53% of 48,635,855 km², and it is not an error that care removes. It is what a finite fan is. Paths and directions

Every reach set ever drawn is too small

An isochrone is drawn by walking out along a finite number of bearings and joining the points, so its vertices are on the true boundary and its edges are chords — which puts the drawn set inside the true one, always, at every count, for any convex reach set. The deficit falls as the square of the count, and a spherical cap loses less than a circle by exactly cos t (1 + cos t)/2.

 The ground two divisions disagree about. seven places across the world dividing the sphere twice under a steady westerly. Outbound, a place belongs to the site that can reach it soonest; inbound, to the site it can reach soonest. The pale tints are the outbound division where the two agree; the dark ground is where they do not, and it is 50.7% of the sphere — 258,530,666 km². Every site's own share of the world barely moves between the two divisions, by at most 3.8%. The two divisions assign nearly the same AMOUNT of ground to each site and assign completely different ground. Paths and directions

A partition under a directed cost has two versions

Dividing a surface among several sites is one question when the cost is a distance and two questions when it is not. Under a steady flow at 0.45 of a vehicle's own speed, the division by who can reach a place soonest and the division by which place can be reached soonest disagree about 50.7 per cent of the sphere — while no site's own share of the world moves by more than 3.87 points.

 Everywhere reachable in 1 h 15 min on roads 5 km apart. The ground reachable in 1 h 15 min from a road crossing, with roads every 5 km in two directions at 80 km/h and the ground between them at 20 km/h, computed on a one-kilometre lattice. It reaches 100 km along a road and 67 km along the diagonal, and covers 18,873 km². The dashed diamond is the shape the set converges to as the roads get denser, whose area is 20,000 km². The dotted ring is what a range ring at road speed claims — 31,416 km², of which 39.9% cannot be reached — and the small dotted circle is what the ground alone would give. Paths and directions

The reach set takes the shape of the roads

On open ground a reach set is a disc. On a grid of roads it converges to the convex hull of a diamond and a disc, and once the roads are √2 times faster than the ground between them the ground's speed stops mattering at all. A range ring drawn at road speed then claims 57 per cent more ground than anybody can reach.

 What 24 bearings draw beside a river crossable at eight bridges. Everywhere within a 10 km walk of the centre, with a river 3 km to the north crossable only at bridges every 2.5 km. Short of the river the reachable ground is the disc; beyond it, each bridge reaches a half-disc of whatever walk is left, and those half-discs are 30.3% of all the reachable ground. The solid outline is the polygon a fan of 24 bearings draws: every bearing that meets the river stops at the bank, so the polygon holds none of the ground beyond it and is short by 31.6% in total. The river is drawn wider than its 80 m. Paths and directions

A drawn reach set stops at the river

A polygon drawn through a fan of bearings round a reach set is short by a chord deficit that falls as the square of the count — on ground with nothing in the way. Put a river three kilometres off, crossable at bridges, and the polygon misses 30 per cent of the reachable ground at every count from forty-eight bearings to a thousand, and a convergence check passes on it.

 What 16 bearings draw beside a river that bends. Everywhere within a 10 km walk of the centre, with a river 3 km to the north meandering ±2.0 km on a 4 km wavelength, crossable only at bridges every 2.5 km. The reachable ground is shaded — pale where it is reached directly and darker where the walk had to go by a bridge. The outline is the polygon 16 bearings draw. It misses 34.1% of the true ground, as a fan beside a straight river does, and it also CLAIMS 0.14% that is not there — ground cut off by a bend, which a chord between two neighbouring bearings passes straight over. Paths and directions

A bend in the barrier puts the drawn polygon over water

A straight river leaves the near side of a reach set convex, and that is what makes a fan of bearings fail in one direction only: every chord cuts a corner, nothing is claimed that is not there. A meander breaks that: the near side is no longer convex, and a chord between two bearings can cross water. So the drawn polygon now over-claims as well as omitting — by a seventh of a per cent, at every bend measured, and more bearings do not reduce it. The question was whether the two errors partly cancel. They do not, and the reason is not their signs: the over-claim never reaches a hundredth of the omission, because one scales with the whole set and the other only with the river.

 Beside a bending river, the fan's chords cross water; a route drawn from the river's own outline never does. Everywhere within a 10 km walk of the centre, shaded pale, with a river 3 km north meandering ±2.0 km on a 4 km wavelength and bridges every 2.5 km. Left: the polygon 16 bearings draw, with the ground it claims and should not in solid ink — chords spanning the bends. Averaged over where the first bearing falls, it misses 35.0% of the true ground and claims 0.15% that is not there. Right: every point reachable by a route with at most one crossing — straight out, or straight to a bridge and straight on — worked out from the river's outline and the bridges alone, with no fan and no shortest path. It misses 6.4% and claims nothing. Paths and directions

A reach set drawn from the river's outline is a bound again

Beside a bending river, a fan of bearings drawn with straight chords claims a strip of water, and the drawing stops being a lower bound. The fan's own numbers cannot repair it: what a bearing records about the river is where the bridges are, not where the bends are, and the best a rule built from them does is cut the over-claim by seven eighths while giving up twenty times as much ground. The river's outline repairs it exactly. Cutting each bearing at the first bank costs three and a half times the over-claim it removes; keeping the chords wherever a route over one bridge reaches costs one and a half. Drawn from the outline and the bridges with no fan at all, that route set misses 6.4 per cent of the reachable ground where the fan misses 35.

 Beside a bending river, every reachable point is reached by a route that meets the bank at most three times. Everywhere within a 10 km walk of the centre, beside the stated river meandering ±2 km on a 4 km wavelength with bridges every 2.5 km, shaded by the fewest times a shortest route to it has to meet the bank — a contact being one run along the bank, however many corners it passes. 70.3% of the ground is seen from the centre across dry land. One contact reaches 28.0% more, 85% of it beyond the river and the rest in bays of the near bank hidden from the centre by a spur; two reach 1.18% and three the last 0.47%, 70% of those two beyond the river, in the lee of the far bank's bends beside the outer bridges. Nothing needs a fourth. Paths and directions

A layer is a meeting with the bank, not a corner turned

A reach set drawn in layers — rays from the centre, then from every place a route can leave the bank — ends in the exact answer, and how deep it must go depends on what a layer is. Counted by corners turned, the depth belongs to the polygon the bank is drawn as: five turns on a bank drawn every 400 metres, eighteen on one drawn every 50. Counted by meetings with the bank, it is three at every resolution. One meeting anywhere leaves 1.6 per cent of a ten-kilometre walk unreached, where rays from the bridges alone leave 6.4; two leave 0.47; three leave nothing. A longer walk needs more — up to six meetings at twenty kilometres across thirty-six rivers — and the bend's steepness, which prices the first layer, ranks the depth at 0.24.

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