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

Partition — where it appears

A division of a surface into regions that cover it and do not overlap. Its adjacency graph is planar when the regions are simply connected, which is what makes four colours enough — and a region in two disconnected pieces is exactly the case that breaks it.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

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.

paths · Reach The cut makes 20 faces out of 14 regions and needs no more colours. The same partition on a sheet cut at the antimeridian. six of the 14 regions are drawn in two pieces, one against each edge, and they are shown darker. The sheet has 20 faces where the globe had 14 regions, and it needs exactly the same four colours — because the two pieces of a split region between them touch exactly what the region touched, so identifying them gives back the sphere's own graph, edge for edge.

Four colours, and what a cut cannot do to them

Every projection removes a set, and a map cut at the antimeridian draws six of its fourteen countries in two pieces — twenty faces where the globe had fourteen regions. The obvious guess is that a map with split countries is the exclave problem and needs a fifth colour. It needs exactly four, and the reason is that the cut adds faces and adds no edges.

impossibility · Topology 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.

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.

paths · Reach 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.

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.

paths · Reach One equidistance line, computed five ways. The line equidistant from two facing coasts — Jan Mayen and Greenland — traced by bisection along a fan of parallels, with the two distances measured on the ellipsoid, on the sphere, and with a ruler on three different pages. The basepoints are the same four in every case and the rule is the same words in every case. The sphere sits 20 metres from the ellipsoid; with a ruler on Lambert's cylindrical sits 13.2 kilometres from it, with 324 km² of seabed in between. A delimitation is a sentence about distances, and a distance is a statement about a surface.

An equidistance line belongs to a surface

A maritime boundary is very often defined as the line equidistant from two coasts — a description with no coordinate in it and no curve to choose between. It has a third ambiguity: equidistant measured how. On the ellipsoid, on the sphere, and with a ruler on three different charts, the same four basepoints give lines up to 39.6 kilometres apart and 2,816 square kilometres of seabed between them.

practice · Boundary Three boundaries, three tripoints. Three bilateral boundaries drawn through one nominal point, each described as the line between two monuments and each realised under a different convention — a geodesic, a rhumb line and a straight line on a Mercator sheet. The tripoint is defined three times, once by each pair of boundaries, and the three definitions are the three marked crossings. They are 21.33 kilometres apart at the widest and enclose 194.960 square kilometres. Under one convention throughout, the same construction puts all three crossings within 0.0 millimetres of each other — which is the refusal this figure carries, and the reason the triangle is a fact about the conventions rather than about the crossing arithmetic.

A tripoint defined three times

A tripoint is very often not a coordinate in any treaty. It is a description — the point where the boundary between A and B meets the boundary between B and C — and each of those boundaries is itself a description. So the point is defined three times, once by each pair, and under one convention throughout the three definitions agree to half a micrometre. Under three they enclose 6.69 square kilometres.

practice · Boundary A meander grows, and its boundary moves with it. Three loops of a river whose valley repeats every 5 km, drawn when its bends are gentle — 30° at most from the valley's direction — and again at the moment its necks close to the channel's own 150 m width, at 120.3°. Under accretion the boundary is the deep-water line at every moment, so it moves with the river, and the shaded ground is the ground that has changed side of it between the two moments: 83.5 km², 27.8 km² a loop. The river is a stated sine-generated curve, not a surveyed one.

A river boundary goes where the river goes, or stays where it was

A boundary that follows a river moves when the river moves, and the law of every such boundary turns on one word: whether the river moved gradually or suddenly. Over one meander's life, gradual movement changes the country of about twenty-eight square kilometres of land a loop. A single flood that cuts the loop off strands twenty-six — nearly the same land, arriving as one parcel on the wrong side of the river.

practice · Boundary Where the summits are, and where the water divides. A stated range 100 km across and 80 km long, its summits 3,000 m high and about 35 km from the western edge, with a plateau falling gently to the eastern edge. Over 40 km of the range the plateau carries a broad rise up to 18 km east of the summits, and a gorge cuts through the range in the middle of that stretch. The dashed line runs through the highest summits; the solid line is the watershed, computed from the land. Lightly shaded ground drains west. Where they part, the darker ground lies east of the summits and drains west: 331 km², reaching 18.1 km from the summits. The terrain is stated, not surveyed.

The highest summits stay where they are, and the watershed does not

A treaty that puts a boundary on the highest crests which divide the waters names one line where the water divides on the summits and two where a river has cut back through the range. On a stated range the two part by up to eighteen kilometres. They also differ in kind: an error of ten metres in every height moves the summits three metres and the watershed on the plateau more than a kilometre.

practice · Boundary A parent and its children, twice. An aperture-7 hexagonal hierarchy beside a square one. The heavy outline is the parent and the light ones are its children. On the right every child is wholly inside and the four of them tile the parent exactly. On the left the child lattice is turned by 19.107° relative to the parent's, only the central child is wholly inside, and 7.14% of the parent is covered by no child of its own. The two families have exactly the same total area — a hexagon cannot be tiled by smaller hexagons at any ratio at all, which is why the mismatch is a construction rather than an approximation.

A cell's children do not fit inside it

Ten rungs price one cell system at one resolution, and every one of them is used hierarchically. A hexagonal hierarchy does not nest: at the aperture-seven scheme the discrete global grids use, one fourteenth of a parent is covered by no child of its own, exactly, and each of the six ring children is eleven twelfths inside.

applied · Cells The same labels, placed once and placed per tile. 140 stated label boxes on a 1024-pixel page cut into 256-pixel tiles. The pale boxes are placed the same way by both rules. The dark ones are placed by the tiled renderer and suppressed by the global one — labels that should have lost a collision with something in the next tile and did not, because the tile that drew them could not see it. nine of them, against 104 labels the global rule keeps.

A label belongs to no tile

Nine rungs price the tile as a piece of geometry. A label is not geometry — it is a page object placed by collision against other page objects, and collision is a global relation while a tile is rendered alone. Cut a page into tiles and 10.3 per cent of the labels are placed differently; the buffer that closes the gap is half a tile at moderate density and a whole one when it is crowded.

applied · Screen

Named alongside it

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

VerificationAreaBoundaryPurposeConventionToleranceAnisotropyCostDiscretisationFlowGeodesicIntersection

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