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

The sea is measured from a tide, and the headlands decide how much took the outer limit of the territorial sea as settled. The convention defines it as the line every point of which lies at the breadth of the sea from the nearest point of the baseline, and that essay drew it as the envelope of arcs: a circle of twelve nautical miles about every point of the low-water line, and the seaward edge of all of them together. In passing it noted that the envelope replaced an older habit of copying the coastline’s every wiggle twelve miles out.

That older habit is worth measuring, because it was not a mistake anybody could see at the time and because it is still the construction most people reach for when asked to draw a zone of fixed width off a coast. It was described in words rather than drawn by a rule: the limit follows the coast in all its sinuosities. Like the sentences in one sentence, and the ground between its readings, those words fit more than one curve. This essay draws two of them on stated coasts, sets each against the envelope, and then charts one coast at six levels of detail to see which construction notices.

Following the coast in all its sinuosities #

Until the codification conference at The Hague in 1930 there was no agreed construction for the outer edge of a belt of territorial waters. The belt was three miles wide in most of the practice of the time, it was measured from the low-water mark, and its outer edge was drawn on charts as a line that kept to the shape of the shore. The method came to be called the tracé parallèle. S. Whittemore Boggs, the geographer of the United States Department of State, proposed to the conference a construction with no shape in it at all: draw an arc of the belt’s radius about every point of the coast, and take the outer edge of the arcs. He called it the envelope of arcs, and the American delegation offered it as the one reading of “a belt measured from the coast” that a draughtsman could carry out without judgement.

The conference failed to agree a breadth for the belt, no convention on the territorial sea came out of it, and the envelope was not adopted then. It entered the law by a slower route. In 1951 the International Court of Justice, deciding the Anglo-Norwegian Fisheries case, described the tracé parallèle as drawing the outer limit by following the coast in all its sinuosities, and observed that it could be applied without difficulty to an ordinary coast that was not too broken but not to a coast like Norway’s. The 1958 Convention on the Territorial Sea wrote the envelope into its sixth article as a definition, and the 1982 Convention repeated it in its fourth: the outer limit is “the line every point of which is at a distance from the nearest point of the baseline equal to the breadth of the territorial sea.” That sentence has only one reading. The older phrase had at least two.

Two curves that follow a coast #

The plainest reading moves the coast. Take the low-water line as drawn and slide it seaward by the breadth of the sea, perpendicular to the general direction of the coast, and every bay, headland and inlet reappears twelve miles out at full size. Call it the copy. It is what a draughtsman with a tracing and a parallel rule produces, and on a straight coast it is exactly right.

The geometer’s reading moves each point instead. Every point of the low-water line is carried twelve miles along its own seaward normal — the direction straight out from the shore at that point — and where the shore turns a corner towards the land the moved points fill an arc about the corner. That is the parallel curve of the coast, the curve a lathe or a cutting tool follows when told to keep a fixed distance from a profile, and it is what “parallel to the coast” means to anybody who has done the calculus. On a straight coast it too is exactly right. On a coast with bays the three readings part company, and the stated coast of five bays shows by how much.

The copy gives up the sea of every narrow bay #

The coast is the one used for the tidal datum: a hundred and sixteen kilometres of straight shore with five bays set into it, each half an ellipse five kilometres deep, six, ten, sixteen, twenty-four and thirty-six kilometres wide, four kilometres of straight shore between them and twelve miles of it beyond each end so that no limit is cut short. Nothing about it was surveyed. A stated coast has the one property a surveyed coast lacks for this purpose: every number in it is known, so the difference between two constructions is the constructions and nothing else.

Over every bay the copy dents the limit landward by the bay’s whole depth, five kilometres. The envelope dents it by much less, because the limit over a bay is measured from the nearest land, and for a bay narrower than the breadth of the sea the nearest land to the limit over its middle is the pair of corners at its mouth. Those corners lie half the bay’s width to either side, so the limit over the middle sits $\sqrt{R^2 - w^2/4}$ out from the line of the mouth, which falls short of the full breadth R by a fifth of a kilometre for the six-kilometre bay and by a kilometre and a half for the sixteen.

So the copy honours a bay most exactly where the envelope honours it least. Over the narrowest bay it lies 4.80 kilometres inside the envelope and over the widest the two meet, because a bay thirty-six kilometres across is wider than $2\sqrt{2Rd - d^2}$, 28.1 kilometres, and its own head is the nearest land to the limit over its middle — the same threshold that decided whether a bay’s tidal flat could move the limit at all. Wide bays are part of the coast as seen from twelve miles out, and the copy and the envelope agree about them. Narrow bays are not, and the copy keeps them anyway.

The widest bay still costs fifty-eight square kilometres. Its middle is settled by its head, but its shoulders are not: partway up each side the bay’s shore is set back several kilometres from the mouth while the limit over it is still measured from the corner, so the copy dents there and the envelope does not. Across the five bays the copy gives up 240 square kilometres, nine per cent of the 2,578 a straight coast of the same length would carry. It gives up most of it, in proportion to the bay’s own area, over the narrowest.

The copy is not twelve miles from the coast #

The sign of the difference is not a property of this coast. Every point of the copy is exactly twelve miles from one point of the coast, the point it was copied from, so it lies inside that point’s circle; and the envelope is the outer edge of all the circles. A point inside one of the circles cannot be outside their edge. So the copy lies on or inside the envelope everywhere, on every coast, and the sea it draws is never more than the sea the convention draws. On the five-bay coast that was checked at every point rather than taken on trust, along with the control that must fail: a copy pushed a hundred metres further out is found outside the envelope over every bay.

The same fact read the other way says what the copy is not. Its point over the head of the narrowest bay is twelve miles from the head and only about seventeen and a half kilometres from the corners of the bay’s mouth, which are nearer. The copy lies closer than twelve miles to some part of the coast along 81 per cent of its length, by up to 4.76 kilometres. The convention’s sentence says every point of the limit is the breadth from the nearest point of the coast. The copy is the breadth from a point of the coast, the one it came from, and on most of its length that point is not the nearest.

That is the whole of the difference between the two readings, put in a word. The copy measures from a point chosen by the drawing, the one directly behind; the envelope measures from the point chosen by the geography, whichever is nearest. Where the two are the same point — on straight shore, and at the outer tips of headlands — the lines coincide. Everywhere else the copy is drawn from land that is not the closest land, and so it falls short.

The parallel curve ties a knot over every bay #

The second reading avoids the copy’s error by construction: each point is carried along its own normal, so on smooth convex shore the moved point is exactly twelve miles from its source and from nothing nearer. The trouble is the shore that is not convex. The two sides of a bay face each other, and their normals point across the bay. A point on the western side of the six-kilometre bay, carried twelve miles along its normal, crosses the bay’s mouth and lands well inland of its eastern shore.

The result is not a limit. It is a curve that crosses itself repeatedly over every bay, with loops reaching back to the coast and over the land, and it encloses no definite region of sea until somebody decides which of its pieces count. The condition for a loop is local: wherever the shore curves towards the sea more tightly than the breadth — wherever its radius of curvature, measured on the sea side, is less than twelve miles — the normals from that stretch meet before they have gone the full distance, and the moved points pass each other. The heads of the three narrowest bays curve with radii of 1.8, 5.0 and 12.8 kilometres, and the sides of every bay near its mouth more tightly still, so every bay on the coast produces one.

Trimmed, the parallel curve is the envelope #

The decision about which pieces count has a natural answer, and it turns out to be the convention’s. Keep only the outermost part of the parallel curve — at every position along the coast, the point furthest out to sea — and throw the loops away. What remains is a curve every point of which is twelve miles along a normal from some point of the coast and no nearer to any other, which is the envelope of arcs exactly. On the fractal coast drawn below, charted to three levels of detail, the outermost point of the parallel curve in each column was compared with the envelope there, and the largest difference anywhere was four metres, well inside the twenty-metre spacing at which the coast was sampled.

So the parallel curve is not a rival to the envelope in the way the copy is. It is the envelope with its construction showing: the loops are what the arcs draw on the inside, and Boggs’s instruction to take the outer edge is the instruction to cut them off. The copy has no such relationship. Its points are not on the envelope’s construction at all, except where the shore behind them happens to be straight.

Charted in finer detail, the copy grows and the envelope settles #

A stated coast of five smooth bays has one level of detail. A real coast has as many as the chart drawing it can show, and the measured length of a real coastline grows as the chart’s detail is refined — the effect a line has a length only at a scale measured on curves built to a known fractal dimension. Lewis Fry Richardson found the growth in measured coastlines and frontiers, published in 1961, and Benoît Mandelbrot read his exponents as fractal dimensions in 1967: about 1.25 for the west coast of Britain and 1.13 for the coast of Australia. A limit that keeps the coast’s shape inherits the coast’s dependence on how finely the coast was drawn.

To see how much, a coast was stated with a fractal dimension of 1.2, between those two. It is a Koch curve 120 kilometres across: every segment is replaced by four, each 0.315 of its length, the middle two bent out at fifty-four degrees, and a coin toss for each segment decides whether the bend points to the land or to the sea, so the coast has inlets and promontories at every scale. The toss is fixed by each segment’s place in the subdivision, which means each finer chart is the same coast with more detail added, keeping every feature of the coarser ones. Six levels take the segments from 37.8 kilometres to 117 metres.

The picture repeats the five-bay coast’s lesson at every scale at once. The copy follows the central inlet thirty kilometres into the land and comes out twelve miles short of where it went in, which puts the limit of the territorial sea eight kilometres up an inlet, inside the line of the open coast. It follows every smaller inlet the same way. The envelope ignores the inlets and rests on the tips of the promontories, which are the land nearest the sea. Between the two lie 905 square kilometres, thirty-four per cent of what a straight coast of the same span would carry — a far larger share than on the five-bay coast, because a fractal coast is mostly inlet.

The copy is as long as the coast #

The copy is a translation, and a translation keeps length. So the copy is exactly as long as the coastline it was copied from, at every level of detail, and the coastline’s length is the thing a fractal coast does not have.

Each level of subdivision replaces a segment by four segments of 0.315 of its length, so every level multiplies the coast’s length by 1.26 — four times 0.315, which for this dimension is the cube root of two. The copy goes with it: 480 kilometres at six levels, and nothing about the construction would stop it at sixty levels or six hundred. A territorial sea limit drawn as a copy would be twice as long on a chart with three more levels of detail, and the coordinates a state lists for it would multiply fourfold with every level.

The envelope’s length rises for two levels, while the chart is adding inlets and promontories larger than the breadth of the sea, then falls back as the third level adds promontories that fill hollows the second had left in it. From the fourth level it changes by less than a hundred metres. The reason is in the construction: an arc of twelve-mile radius cannot follow a feature much smaller than twelve miles. A promontory a few hundred metres across adds a small arc at its tip only if its tip is the nearest land to some part of the limit, and moves the limit there by no more than its own height.

The envelope is not immune to detail, and it is worth saying how it is not. Each level adds arcs about new tips, so the number of distinct pieces the envelope is made of grows — eleven at three levels, eighteen at four, twenty-six at five, forty-two at six. But each new piece is an arc about a smaller tip, and it moves the line by less than the piece before. The copy at six levels needs 4,097 vertices, one for every vertex of the coast.

The envelope’s largest move at each refinement falls from fifteen kilometres to under eight hundred metres in four levels, reaches ninety-six metres at the fifth, and stays there at the sixth. The flattening is real rather than a floor of the calculation. At the fifth and sixth levels the largest move happens at a single promontory tip where a new bump on the seaward side becomes the nearest land; the bumps at those levels are about a hundred metres high, and the envelope moves by the bump’s height and no more. Meanwhile each refinement adds more length to the copy than the one before.

In fairness to the copy, its position settles too. The coast it copies moves by about a hundred metres at the last refinement, the height of the newest bumps, so the copy moves by the same, and the sea between successive copies shrinks from 679 square kilometres at the first level to five hundredths of one at the last. The copy is not unstable in where it lies. It is unstable in what it is: a line that carries every feature of the chart it was drawn from, its length set by the chart rather than by the sea.

The parallel curve’s knots multiply #

The parallel curve fares worse than the copy under refinement, because every new inlet is a new pair of facing shores. On the fractal coast the count of its self-crossings can be made exactly, since a charted coast is a polyline with a definite number of segments.

The count grows by between ten and eighteen times a level, much faster than the coast’s four new segments for each old one, because a crossing is a pair: each new stretch of shore is facing not only its new neighbours but every stretch within twenty-four nautical miles across the water. At six levels the curve crosses itself more than half a million times. Its outermost edge, through all of it, is still the envelope — the same line at the last two levels to within a hundred metres. Half a million crossings are thrown away to get it, and the envelope is what was left over every time.

That is the honest description of why the envelope won. It was not chosen because it gives more sea, though it always does, nor because it is easier to draw. It was chosen because, alone of the three readings, its answer does not depend on how much of the coast the chart can show.

On eight coasts, the same answer #

One coast settling might be luck in the coin tosses. The construction was repeated on eight stated coasts of the same dimension and span, differing only in which side each bend falls on.

On every coast the last refinement moves the envelope by under a hundred metres and leaves its length within a few tens of metres of the length at the level before. The lengths themselves differ by twenty kilometres from coast to coast, because which side the large bends fall on decides how much of the open water the arcs have to wrap. The copy’s gain is the same ninety-nine kilometres on all eight, because the copy’s length is the coast’s, and a fractal coast’s length does not depend on which way its features face. The copy also gives up between 905 and 1,192 square kilometres against the envelope across the eight, a third or more of the sea a straight coast would carry.

The coasts are stated, and a stated coast is not a surveyed one. A real coastline’s dimension varies along it and is different at different scales, a real chart is generalised by a cartographer rather than truncated at a level of subdivision, and real practice before 1930 surely smoothed the copy by eye rather than tracing every wiggle. None of that changes the sign of anything above: the copy lies inside the envelope by a fact true of every coast, its length is the coast’s by a fact true of every translation, and the parallel curve’s loops come from facing shores, which every coast with an inlet has. What a real coast would change is the size — how much sea and how many kilometres — and that would need the real coast.

A rule that asks which land is nearest #

The pattern is the one the boundary essays keep finding, in a new place. The line a commission can actually run found a treaty’s parallel of latitude replaced on the ground by the chords a commission could sight; an equidistance line belongs to a surface found the word “equidistant” waiting for a surface to be named. Here the words “following the coast” waited for a construction, and the two constructions they admitted were the copy, which carries the coast’s detail out to sea, and the parallel curve, which ties itself in knots on it. The convention’s sentence settled the matter by naming the one thing the older phrase left out: the nearest point of the coast.

That word is also why the envelope settles. A limit measured from the nearest land sees only the land that is nearest to it, which on any coast is the outermost — the headlands, the promontory tips, the rocks — and the outermost land changes slowly as a chart improves, because new detail is overwhelmingly inside the old outline rather than beyond it. The same property made the tidal datum matter only over bays wide enough for their heads to be nearest, and made a straight baseline move far more water inward than sea outward. The rule that measures from the nearest point of a line is a rule that discards most of the line, and that is what makes it robust — as the highest summits stay where they are, and the watershed does not found a line drawn through the highest points steadier under error than one drawn wherever the water happens to divide.

It is also why the definition sits comfortably beside the later rules that replace the coast with chosen points. Straight baselines and archipelagic baselines keep only a handful of the coast’s points and throw the rest away; the envelope of arcs keeps all of them and lets the geometry decide which matter. Both are ways of not inheriting a coast’s every wiggle. The copy is the one construction that inherits all of them.

Still open: whether the envelope’s settling survives generalisation #

Every chart here was the same coast truncated at a level of subdivision, so each finer chart contained every vertex of the coarser one. A real hydrographic office does not chart that way. It generalises the coast for each scale of chart, and generalisation moves vertices as well as dropping them — a tolerance is a promise about the picture found Douglas–Peucker’s one guarantee to be a bound on how far a discarded vertex lies from the line kept, which says nothing about which side of the line the outermost tips end up on.

The envelope rests entirely on the outermost tips, so the question is whether it rests on them just as well when the coarser chart is a generalisation rather than a truncation: whether a limit measured from a coast simplified to a given tolerance lies within that tolerance of the limit from the full coast, whether simplification can move it seaward as well as landward, and whether the offices that chart the same coast at several scales are drawing territorial seas that differ by more than the tolerance they chart to. A coast whose coarse charts are truncations of its fine ones cannot ask.