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

One sentence, three readings of it

Drawn here at the parameters it defaults to, with every essay that calls it.
One sentence, three readings of it. "a straight line from the initial point on the Rio Grande to a point on the Colorado" — Treaty of Mesilla, 1853. Every curve here answers to those words. the geodesic, the rhumb line, straight on the sheet, drawn between the same two monuments on a conformal conic fitted to the segment itself. The widest pair, the geodesic against the rhumb line, are 7.53 kilometres apart at their worst and enclose 3,925 km². The shading is that ground. It is not an artefact of the drawing: the same figure of the 141st meridian shows one line, because on a meridian every one of these readings is the same curve.

"a straight line from the initial point on the Rio Grande to a point on the Colorado" — Treaty of Mesilla, 1853. Every curve here answers to those words. the geodesic, the rhumb line, straight on the sheet, drawn between the same two monuments on a conformal conic fitted to the segment itself. The widest pair, the geodesic against the rhumb line, are 7.53 kilometres apart at their worst and enclose 3,925 km². The shading is that ground. It is not an artefact of the drawing: the same figure of the 141st meridian shows one line, because on a meridian every one of these readings is the same curve.

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

12 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.

One sentence, three readings of it. "a straight line from the initial point on the Rio Grande to a point on the Colorado" — Treaty of Mesilla, 1853. Every curve here answers to those words. the geodesic, the rhumb line, straight on the sheet, drawn between the same two monuments on a conformal conic fitted to the segment itself. The widest pair, the geodesic against the rhumb line, are 7.53 kilometres apart at their worst and enclose 3,925 km². The shading is that ground. It is not an artefact of the drawing: the same figure of the 141st meridian shows one line, because on a meridian every one of these readings is the same curve. Grids, and what a survey does

One sentence, and the ground between its readings

Every land boundary in the world is defined by a sentence, and a sentence naming two monuments does not name a curve. "A straight line" between the Rio Grande and the Colorado admits at least three answers 7.53 kilometres apart at their worst; "the forty-ninth parallel" admits four, 96 kilometres apart, with 130,972 square kilometres between the extremes.

 The line a commission can actually run. A boundary described as a parallel of latitude and marked by monuments 220 kilometres apart, with the offset exaggerated 700 times so that it can be seen at all. A commission cannot run a parallel: it can set a monument, sight a straight line to the next and clear the trees between, and a straight line between two points of equal latitude is a geodesic, which passes POLEWARD of the parallel everywhere between them. So the marked line lies north of the described one, by 1176.5 metres at the middle of each of its 9 chords, and encloses 1611.2 square kilometres that the words put on the other side. At the 20-kilometre spacing used throughout, the same offset is 9.03 metres. Grids, and what a survey does

The line a commission can actually run

A boundary commission cannot run a parallel of latitude. It can sight a straight line between monuments, and a straight line between two points of equal latitude passes poleward of the parallel — by s² tan φ / 8R, which at a mile of spacing is fifty-eight millimetres and at a hundred kilometres is two hundred and twenty-six metres. The described line and the marked line are different curves, and the marked one governs.

 One meridian, two datums, and the ground between them. "the meridian line of the 141st degree of west longitude" — Anglo-Russian Convention, 1825. The line whose longitude is exactly 141° west, drawn twice: once on nad27 and once on WGS84, with the east–west separation exaggerated 3,000 times. The two are 129.0 metres apart, and remarkably constant — the shift changes by five centimetres over nine degrees of latitude — so the strip between them is a ribbon 129 metres wide and 1039 kilometres long, which is 134.0 square kilometres. The sentence has not changed. The surface the number refers to has. Grids, and what a survey does

A meridian boundary moves when its datum does

The 141st meridian is the one boundary description in this collection with no geometric ambiguity in it: every reading of it is the same curve, exactly. It has a different one. A longitude refers to a datum, the 1825 convention named none, and the line of longitude exactly 141° west sits 129.0 metres apart on NAD27 and WGS84 — a ribbon 1,039 kilometres long and 134 square kilometres in area.

 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. Grids, and what a survey does

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.

 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. Grids, and what a survey does

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.

 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. Grids, and what a survey does

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.

 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. Grids, and what a survey does

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.

 A bank that dries at one tide adds a bulge to the territorial sea. A stated coast 116 km long with five bays 5 km deep, and its twelve-nautical-mile limit measured from the low-water line at two tidal datums 1 m apart. A bank 10 km offshore has its crest 0.5 m below the higher datum, so it is under water at that datum and dries at the lower one, over a disc 1.0 km in radius. Dried and inside twelve miles of the coast, its low-water line becomes part of the baseline, and the limit bulges round it: 339 km² of sea. Everywhere else the two limits are drawn on top of one another, 11.1 km² apart over the whole coast. The coast is stated, not surveyed. Grids, and what a survey does

The sea is measured from a tide, and the headlands decide how much

A territorial sea is measured from the low-water line, and the low-water line is where a chart's tidal datum meets the shore — so a datum a metre lower moves it twenty metres at a headland and a kilometre across a tidal flat. On a stated coast the twelve-mile limit barely notices the flats: fifty-six square kilometres of shore are uncovered and eleven of sea are gained. A bank that dries at the lower datum adds three hundred and thirty-nine in one step.

 Straight baselines across a fringe of islands. A stated coast 220 km long, a gentle undulation 10 km in amplitude, with 8 islands off it: six between 5 and 12 km offshore and two rocks at 30 and 26 km. The dashed line is the twelve-nautical-mile limit measured from the low-water line of the mainland and every island. The solid lines are the straight baselines enclosing the most water under one reading of the article — segments of at most 24 nautical miles departing at most 20° from the coast's general direction — 6 segments, the longest 44.4 km — and the solid curve is the limit measured from them. Tinted: 988 km² of water landward of the lines, now internal waters. Shaded: 68 km² of sea beyond the dashed limit, against a territorial sea of 8,122 km². Neither rock is reached. The coast is stated, not surveyed. Grids, and what a survey does

A straight baseline moves more water inward than sea outward

The seventh article of the law of the sea lets a coast fringed with islands measure its territorial sea from straight lines joining chosen points, and gives no length, no angle and no scale for the choice. On a stated coast the lines a common reading permits add sixty-eight square kilometres of sea at the outer edge and turn nine hundred and eighty-eight into internal waters. Drop the test of direction and the outer rocks come into reach: eighteen hundred added, forty-six hundred enclosed.

 Eighteen ways to close an archipelago within a hundred miles. A stated archipelago about 580 km across: three main islands near the centre (A, B and C, 23,576 km² of land between them) and thirteen smaller islands round them, 26,970 km² of land in all. The dot on each island is its outermost point, where a state drawing the widest lawful system would put its basepoint. Solid: the lawful system enclosing the most water, 186,617 km² at a water-to-land ratio of 6.96 to one, 12 segments, the longest 86.6 nautical miles. Dashed: the lawful system enclosing the least, 159,032 km², through the main islands A, B and C. Dotted: the two segments that would take in the outlier P to the south-east, 122.3 nautical miles and 100.9 nautical miles — both over the hundred-mile limit. The archipelago is stated, not surveyed. Grids, and what a survey does

Written in numbers, the archipelago rule still leaves eighteen

The article on archipelagic baselines gives its tolerances as numbers: no segment over a hundred nautical miles except three per cent of them, which may reach 125, and water between one and nine times the land. On a stated archipelago 8,799 closed systems keep the main islands inside, the ratio passes all of them and the length test eighteen, and the eighteen still differ by 27,585 square kilometres. Thin the land and the ratio does the deciding; the lower bound decides nothing; and the three-per-cent exception can be used only by a system padded to thirty-four segments.

 Two lawful systems, one sea: the lines move water between regimes and the outer limit does not move. The stated archipelago with the lawful system enclosing the most water (solid, 186,617 km² inside) and the one enclosing the least (dashed, 159,032 km²). Round each, the limit two hundred nautical miles from its baselines and from every island's coast: the two outer limits lie on top of each other everywhere but a few kilometres off the widest gaps. A straight coast stands 700 km east of the centre; the line equidistant between it and the archipelago is drawn for both systems and they coincide. Inside plus outside, the two systems account for 1,344,857 km² and 1,343,864 km² of sea. Grids, and what a survey does

The archipelago rule moves water between regimes, not between states

The eighteen baseline systems the archipelago article allows on a stated archipelago differ by 27,585 square kilometres of water inside the lines. Measured outward, as the Convention measures every zone, the difference almost vanishes: whatever a system takes inside it gives up from the zone outside, and water inside plus zone outside differs by 993 square kilometres, seven hundredths of a per cent. Against a straight neighbouring coast on any side, the equidistance line does not move at all, because a baseline's nearest point to a straight coast is always an island. Only a neighbour facing the middle of a gap feels the choice — 2,112 square kilometres at thirty kilometres off, 688 at 250.

 The coast copied twelve miles out gives up the sea of every narrow bay. A stated coast 116 km long with five bays 5 km deep, 6, 10, 16, 24, 36 km wide. Dashed: the envelope of arcs, the line every point of which is twelve nautical miles from the nearest point of the coast. Solid: the coast itself moved twelve miles seaward, which dents the limit by the full 5 km over every bay. Tinted: the sea between them, 240 km² in all, 9% of a straight coast's territorial sea. Over the 6 km bay the copy lies 4.8 km inside the envelope; over the 36 km bay the two meet. The coast is stated, not surveyed. Grids, and what a survey does

Copied out to sea, a coast keeps its length and loses its bays

Before 1930 a territorial sea was drawn by following the coast in all its sinuosities, and the phrase fits two constructions. Moved twelve miles seaward, a stated coast of five bays gives up 240 square kilometres against the envelope of arcs, most of it over the narrowest bay; moved along its own normals, it ties a loop over every bay. On a fractal coast charted at finer detail the copy grows by a quarter at every level, to 480 kilometres, while the envelope settles at 133.0.

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