An equidistance line belongs to a surface

Three rungs of this anchor price three ambiguities in a boundary description. A sentence naming two monuments leaves the curve between them undecided. A sentence naming a coordinate leaves the surface the coordinate refers to undecided. A sentence naming a parallel is realised on the ground as something else.

There is a class of description that looks immune to all three, and it is the one most maritime boundaries in the world use: the line equidistant from the two coasts. No coordinate, so no datum question. No two monuments, so no curve to choose. Just a rule, applied to two coastlines that both parties agree about.

The rule has a surface in it, and it is a large one.

What is being varied, and what is not #

This is a comparison in which almost everything is held fixed, and being precise about what changes is most of the argument.

The basepoints are the same four coordinates throughout. The rule is the same in every case: find the places where the distance to the nearer of one coast’s basepoints equals the distance to the nearer of the other’s. The tracing method is the same bisection along the same fan of parallels, to eighty halvings, which is machine precision.

The only thing that varies is what “distance” means. On the ellipsoid it is a geodesic length computed by Vincenty’s method. On the sphere it is a great-circle length. On a chart it is what a ruler measures: the straight-line distance between the two points as the page draws them.

Each of those three is a defensible reading of the same English word, and the third is not a straw man. Delimitation lines have been drawn with dividers on charts for as long as there have been delimitations, and a chart is a projection.

How the line is traced, and the axis that had to be got right #

The tracing method is worth a paragraph because it decided one of the numbers above and because a plausible version of it produces nonsense.

The quantity being solved for is the difference between the distance to the nearer basepoint of one coast and the distance to the nearer basepoint of the other. That difference is monotone across a strait — it is negative on one side and positive on the other — so a bisection finds its zero to any precision without needing a derivative, which matters because the derivative of a minimum over a set of basepoints does not exist where the nearest basepoint changes.

What the bisection needs is a direction to search along, and the direction has to cross the strait rather than run down it. The Jan Mayen pair is separated east from west, so the fan is a set of parallels and the search is in longitude. The channel pair is separated north from south, so the fan is a set of meridians and the search is in latitude.

Getting that wrong is not a small error. Tracing the north–south pair along parallels finds a crossing on almost no parallel at all, and the handful it does find are at the ends where the geometry degenerates; the first run of this figure reported a Mercator line 1,494 kilometres from the ellipsoid’s, on a strait a hundred kilometres wide. That number was a fact about which way the fan was pointed and about nothing else, and it is recorded here because a result of the same shape — a large number from a correct calculation asked the wrong question — is the failure this kind of measurement is most prone to.

Why the sphere is nearly right and a page is not #

The sphere’s twenty metres is the interesting number in that figure, because a spherical distance and an ellipsoidal distance differ by far more than twenty metres. Over the several hundred kilometres involved here they differ by kilometres — geodesics on the ellipsoid measures the gap and finds it can run either way depending on the route’s bearing.

The reason it does not show up is that an equidistance line is defined by an equality of two distances, not by their values. Replacing the ellipsoid with a sphere scales both distances by nearly the same factor at nearly the same place, and a bisection looking for the crossing of the difference sees almost nothing. Most of the error cancels before the instrument reaches it.

A page does not cancel, and the difference is directional rather than a matter of degree. A projection stretches by different amounts in different directions at the same point, so the distance from a place to a basepoint north of it and to a basepoint west of it are scaled by different factors — and the difference of two distances scaled differently does not vanish when the true difference does. The residual is the whole of the page error, and it is a function of the local anisotropy of the projection.

That is the same mechanism nearest of many is a partition measures for a Voronoi division computed in the plane, where between 0.75 and 22.16 per cent of the ground is handed to the wrong source in strips up to 1,591 kilometres across. A median line is a Voronoi boundary with two sources, so this rung is that essay’s argument applied to a case where the ground being handed over belongs to a state.

The anisotropy, written down #

The claim that the page error tracks anisotropy rather than scale can be made sharper than a mechanism, because both quantities are computable at a point and this collection computes them.

Tissot’s construction gives two principal scale factors a and b at every point of every projection. Their product is the areal factor and their ratio is the anisotropy; a conformal map has a = b everywhere by definition, so its anisotropy is one, and an equal-area cylindrical map has ab = 1 with a and b as far apart as the latitude forces. At seventy-one degrees north, Lambert’s cylindrical has a ratio of sec²φ, which is 9.5; the plate carrée has sec φ, which is 3.1; Mercator has exactly 1.

Ordered by that ratio the three pages read 1, 3.1, 9.5. Ordered by their error on the east–west pair they read 2.54, 11.7, 13.2 kilometres. The two orderings agree, and the first two ratios are close to the ratio of the first two errors — 4.6 against 3.1, which is the right order of magnitude for a quantity that also depends on the direction the strait runs.

That is not a fitted law and this rung does not claim one. What it is, is the reason a reader should expect Mercator to do well here despite everything else this collection says about it: distortion has a direction is where the two scale factors are separated from each other, and a conformal projection’s one virtue — that its two are equal — is exactly the virtue an equidistance problem rewards.

The worst page depends on which way the strait runs #

Moving the same comparison to a strait running north and south rather than east and west changes the numbers by more than it changes their order.

The mechanism is direct. Mercator stretches north–south exactly as much as it stretches east–west at every point, being conformal, so a ruler on it makes the same proportional error in every direction and much of that error cancels out of a difference of distances. Lambert’s cylindrical compresses north–south by exactly the factor it stretches east–west, so its anisotropy is the largest in the library at these latitudes and its error is the largest here.

Which means the right question about a chart used for a delimitation is not “is it a good projection” but “how anisotropic is it, at this latitude, in the direction this strait runs”. That is a purpose-before-property question of exactly the kind every projection minimises something is about, and no published ranking of projections answers it.

Why the ordering is not the usual one #

The three pages come out in an order that would surprise anybody who knows this collection’s other rankings, and it is worth saying so plainly rather than letting it pass.

Mercator is the projection this site spends most of its time explaining the costs of. Its areal exaggeration at seventy-one degrees is a factor of nine and a half, its shape of Greenland is the standard illustration of what a world map can do wrong, and no regional criterion computed anywhere on this site puts it near the top for a high-latitude region.

On this problem it is the best of the three pages by a factor of five. Nothing about its area behaviour matters, because an equidistance line is not an area; nothing about its shape at continental scale matters, because the calculation is local. What matters is one property, conformality, and Mercator has it exactly.

That is a clean instance of the rule which projection is best is written around: the ordering of projections is a function of the criterion, and a criterion nobody has computed produces an ordering nobody expects. It is also a warning about the reverse direction. A chart chosen for a delimitation because it is “a good projection for this region” has been chosen on a criterion that has nothing to do with the operation about to be performed on it.

Where the whole effect goes away #

Both the pairs above are at high or middle latitudes. Moving the same east–west geometry to eight degrees north changes the picture completely.

That collapse is the useful practical result of the rung, and it is a sharper statement than “projections are worse at high latitudes”. Every quantity in this collection is worse at high latitudes; what is specific here is that the residual after cancellation is proportional to the local anisotropy rather than to the local scale error, and anisotropy is what goes to zero at a cylindrical projection’s standard parallel.

So the rule is: a delimitation drawn with dividers on a chart is accurate to within a few hundred metres near the projection’s true-scale parallel and to tens of kilometres far from it. Which chart, and which parallel it is true on, is the whole of the design decision — and it is a decision that has to be made before any delimitation is drawn rather than assessed afterwards, because the line the dividers produce is the agreement.

The seabed, and what the numbers are worth #

The areas in these figures are the quantity a delimitation is actually about, and they are worth converting into terms somebody would negotiate over.

Three hundred and twenty-four square kilometres between two readings of one rule, on the Jan Mayen pair. Two thousand eight hundred and sixteen on the channel pair. Those are the differences between two good-faith applications of the same agreed sentence by two parties using different charts, with no dispute about the coastlines, no dispute about the basepoints and no bad arithmetic anywhere.

For scale, the exclusive economic zone extends two hundred nautical miles — 370 kilometres — so a boundary running four hundred kilometres carries something over a hundred thousand square kilometres of seabed on either side of it. The disagreements measured here are therefore one to three per cent of what is at stake, which is small as a fraction and is not small as a resource.

What makes them worth measuring rather than dismissing is that they are systematic. A page’s anisotropy points the same way along the whole line, so the error does not average out between the two ends: one party’s chart puts the line on their side for the whole of its length. That is the same property that makes the chord scallop of the line a commission can actually run worth computing at fifty-eight millimetres — a signed error over a long line is a different object from an unsigned one of the same size.

The three ambiguities are not independent #

An equidistance line has no coordinate in its definition, and the rung below this one is about coordinates. That looks like an escape and is not one.

The basepoints are coordinates. Every one of them refers to a datum, and re-expressing a basepoint between datums moves it by around a hundred metres — which moves the equidistance line by about half that, since a median line moves half as far as the basepoint that defines it. Over a delimitation of several hundred kilometres that is tens of square kilometres of seabed, from the datum alone, before any question of surface arises.

So a maritime boundary defined as an equidistance line carries at least three conventions that its text does not state: the datum of the basepoints, the surface the distances are computed on, and — if the answer is to be a curve rather than a set of points — the rule for joining the computed points into a line. Only the second is peculiar to this kind of description; the first is the previous rung’s subject and the third is the first rung’s.

The three do not simply add. The datum error moves both coasts together, so much of it cancels in the same way the sphere-against-ellipsoid difference does; the surface error does not cancel; and the joining rule contributes nothing at all when the computed points are dense and everything when they are the four turning points a treaty actually lists.

What a treaty can do about it #

The three conventions are all fixable and the fixes are of different kinds, which is worth separating because only one of them is a matter of writing more words.

The datum is fixable by naming it, and by naming the realisation as well as the datum — a point a published coordinate is a result makes about coordinates generally and which applies here with money attached.

The surface is fixable by naming it, and the naming has to be specific: “geodesic distances on the WGS84 ellipsoid” is a complete specification and “the shortest distance” is not. Modern delimitation practice does state it, which is why the numbers in this rung are a historical measurement rather than a live problem for new agreements.

The joining rule is not fixable by naming it, and it is the one nobody thinks about. A treaty lists turning points and says the boundary runs between them; what runs between them is a curve, and choosing it is the first rung of this anchor over again. Naming the surface does not choose it, because a geodesic, a rhumb line and a loxodromic arc on the ellipsoid are all defined on the same surface and are all different curves.

That last one closes a loop the anchor did not set out to close. An equidistance line escapes the curve ambiguity while it is a continuous line and reacquires it the moment it is written down as a finite list of points, which is the only form a treaty can carry it in.

What happens when there are more than two coasts #

Everything above is two coasts and one line. Real delimitation is rarely two.

A tripoint at sea is where three states’ equidistance lines meet, and each of the three is an equidistance line between a different pair. That is the same object nearest of many is a partition builds — a Voronoi division with three sources — and its boundaries meet at a point, exactly, when all three are computed the same way.

When they are not, they do not — and the same is true of three land boundaries meeting at a monument that was never placed, which is the more common case and the one the rung above is about.

The general form is worth stating because it covers both. A point defined as the intersection of two curves is defined once for each pair of curves that meet there. If three boundaries nominally concur, the point has three definitions; if all three are realised under one convention they coincide, and if they are not they form a triangle with an area. Nothing in that depends on whether the boundaries are equidistance lines, geodesics, or straight lines on somebody’s chart.