A tripoint defined three times

Four rungs of this anchor price an ambiguity in one boundary. The last of them ends with an observation about three: where three states meet, each pair has a boundary, and the point where all three meet is very often not written down anywhere.

It is described instead. The point at which the boundary between A and B meets the boundary between B and C — a definition that is a computation rather than a coordinate. And each of those two boundaries is itself a description, of the kind the first rung of this anchor found admits several readings.

So the tripoint has three definitions, one for each pair of boundaries, and they agree only if the three lines are concurrent.

The construction, and the refusal in it #

Nothing in the figure is fitted, and the geometry is pinned so that the triangle is a property of the conventions and of nothing else.

A nominal tripoint is chosen. Three lines are run through it at sixty degrees to each other, and each line’s two monuments are placed on it at equal distances either side. Under one convention throughout, all three lines pass through the nominal point exactly, so the three pairwise crossings coincide and the triangle has zero area.

Measured: three geodesics through one point meet at it to 5.5 × 10⁻¹⁰ kilometres, which is half a micrometre and is the sampling of the curves rather than anything geometric. That is the refusal, and it is what makes the 3.94 kilometres a statement about conventions rather than about the crossing arithmetic.

Getting the third convention to be genuinely third took a correction worth recording. The obvious chart to draw a straight line on is Mercator, and Mercator is the one chart that cannot be used here: why Mercator exists is the essay about the projection built so that a rhumb line is straight, so its straight line is the rhumb line, exactly. A run with conventions “geodesic, rhumb, chart” reported the same numbers as “geodesic, rhumb, rhumb” and the figure was drawing two conventions while its caption claimed three. The chart used now is Gall–Peters, whose straight line is neither of the other two.

Why a tripoint is described rather than fixed #

A reader who has not looked at boundary treaties will find it strange that the meeting point of three states is not simply given as a coordinate. It usually is not, and the reasons are practical rather than careless.

Bilateral boundaries are negotiated bilaterally. A and B agree their line, B and C agree theirs, and the two agreements are made at different times by different people and very often in different centuries; neither pair can bind the third state, so neither agreement can fix the point where all three meet. What each can do is describe its own line, and the tripoint falls out of the two descriptions as a consequence.

That sequencing is why the point ends up defined three times over rather than once. It is also why the three definitions are so often under different conventions: the A–B treaty of 1890 and the B–C treaty of 1926 were drawn with different instruments, on different charts, by people with different ideas about what a straight line is, and neither had any reason to consult the other about it.

A grid has an origin that is not there is the same shape one field over — a point that every coordinate on a national grid is measured from, which corresponds to nothing on the ground and was chosen for convenience. A tripoint is the opposite: a point that corresponds to something real and was never chosen at all.

More conventions is not more disagreement #

The natural expectation is that the more ways there are to read the three descriptions, the further apart the answers get. That is wrong, and it is wrong by a factor of two.

The mechanism is worth following because it is not an artefact. A rhumb line between two monuments straddling the tripoint misses the tripoint, by 7.4 kilometres on these arms; a straight line on a chart misses it too, by a similar amount and in a similar direction, because both curves depart from the geodesic by bowing towards the equator. Two curves that both miss in the same direction cross each other near where each crosses the geodesic, so the triangle they make with the geodesic is smaller than the one two geodesics make with a single rhumb.

Which produces a rule that has no business being true and is: a boundary settlement in which every party reads the descriptions differently can be more consistent than one in which two parties agree and the third does not. The three-convention case is the one that sounds most chaotic and is the second-best row in the table.

There is a second row worth stopping on. Three rhumb lines throughout give a spread of 24.7 metres — nearly concurrent — while every one of the three misses the nominal point by 7.4 kilometres. Consistency between the three definitions and agreement with the intended point are different properties, and a convention can have the first without the second.

The triangle grows as the square of the arms #

The monuments in the figure are six hundred and sixty-seven kilometres from the tripoint, which is a long way. Real boundary monuments are much closer, and the question is what the triangle does as they come in.

A square law is the same exponent the line a commission can actually run finds for a chorded parallel, and it has the same cause. A rhumb line and a geodesic between the same two points agree to first order and part at second, so any quantity built from the difference between them is quadratic in the separation — and here the separation is the arm.

That exponent is what makes the ambiguity manageable in practice. Real tripoint monuments sit tens of kilometres from the point rather than hundreds, so scaling from the ladder, a monument fifty kilometres out contributes about twenty-seven metres of spread and about three hundred square metres of ground. Which is a survey question rather than a diplomatic one.

The three crossings, and which one is nearest the truth #

There are four points in play in the hero figure — three crossings and the nominal point — and it is worth asking which of the four has the best claim.

None of them, is the honest answer, and the reason is that the nominal point has no standing in the descriptions. Nothing in the A–B treaty says where the tripoint is; it says where the A–B boundary runs. The nominal point in the figure is a construction convenience, the place the three lines were built to pass through, and a real tripoint has no such thing behind it.

So the three crossings are the three answers and there is no fourth to rank them against. The measured offsets from the nominal point — 8.05, 10.58 and 11.98 kilometres on the hero construction — are therefore not errors. They are the distances between the construction’s own scaffolding and the answers it produced, and the quantity with meaning is the spread between the answers rather than any one of their offsets.

That distinction matters when the same argument is run on a real tripoint, where there is no scaffolding at all. What is available is three crossings, and what has to be agreed is which of them, or what fourth point between them, the parties will accept. It is a decision rather than a calculation, which is the second time in this anchor that a geometric ambiguity has turned into one.

Where on the Earth it is worst #

The construction above is at twelve degrees north, and the latitude is not incidental.

Four hundred and thirty times, from the equator to seventy-five degrees, for identical geometry. That is a larger latitude dependence than anything else in this anchor, and it stacks with the arm law rather than replacing it — a high-latitude tripoint with distant monuments is where every one of the effects measured in these five rungs is at its largest simultaneously.

The maritime version, where there is no mark to put #

The construction above is three land boundaries and the resolution is a monument. At sea there is nothing to monument, and the same geometry produces a live problem rather than a historical one.

A tripoint at sea is where three states’ equidistance lines meet, and each of the three is an equidistance line between a different pair of coasts. An equidistance line belongs to a surface measures what one of those lines does when the surface changes: up to 39.6 kilometres of movement between two readings of the same rule. Three such lines, computed by three pairs of states on three different surfaces, meet at three points, and the triangle they make is of the same order.

It is also the same object nearest of many is a partition builds. A Voronoi division of a surface among several sources has boundaries meeting at vertices, and those vertices are exactly tripoints; that essay finds that computing the division in a plane rather than on the ground hands away between 0.75 and 22.16 per cent of the area in unbroken strips. The vertices move with the strips.

The difference from the land case is only that nothing settles it. A land tripoint is settled by a mark; a maritime one is settled by an agreement about the arithmetic, and if the three bilateral agreements specify different arithmetic there is nothing to appeal to but a fourth agreement.

What the triangle actually is #

A triangle of three points, each a valid answer to the question “where is the tripoint”, is an unusual object and it is worth being clear about what a state would do with it.

It is not disputed territory in the ordinary sense. All three points are correct applications of the agreed descriptions; there is no error anywhere, no bad survey and no bad faith. What there is, is a question whose answer depends on a convention nobody agreed because nobody noticed there was one.

The practical resolution is the same as the one every rung of this anchor arrives at: somebody puts a mark in the ground, and the mark becomes the boundary. A tripoint monument, once placed and accepted, has no ambiguity in it at all — which is why the great majority of the world’s land tripoints are monumented and the great majority of its maritime ones are not.

Where a mark cannot be placed, the resolution has to be textual, and the text has to say the thing the original descriptions did not: which curve, computed on which surface, from which coordinates. An equidistance line belongs to a surface is the rung about the second of those three, and the three together are the whole of what a boundary description has to state and usually does not.

What five rungs have measured #

The anchor set out to price the gap between a boundary agreed in words and a boundary realised on the ground. Five quantities, on real segments and real constructions:

7.53 kilometres between readings of “a straight line” over seven hundred and eighty. 58 millimetres of scallop on a chorded parallel at the spacing anybody uses, rising as the square of the spacing. 129 metres between two datums’ readings of one meridian, over a strip of 134 square kilometres. 39.6 kilometres between two surfaces’ readings of one equidistance rule, over 2,816 square kilometres of seabed. And 3.94 kilometres between three definitions of one tripoint.

Those are four orders of magnitude apart and they are not ordered by how vague the description is. The most specific description in the set — a stated longitude — carries a hundred-metre ambiguity that its own words conceal entirely, while the vaguest — “a straight line” — carries a seven-kilometre one that anybody can see coming.

The pattern is that an ambiguity is dangerous in proportion to how well the sentence hides it. Nobody signs a treaty saying “a straight line” without knowing that a straight line has to be defined; a great many have signed treaties naming a longitude without knowing that a longitude has to be referred to something.

What the five rungs share #

Reading the five together, one thing is common to all of them and it is not the geometry.

In every case the ambiguity is invisible in the sentence. “A straight line”, “the forty-ninth parallel”, “the meridian of the 141st degree”, “the line equidistant from the two coasts”, “where the boundary between A and B meets the boundary between B and C” — every one of those reads as a complete specification, and every one leaves out something that turns out to be worth square kilometres.

What is left out is always of the same kind: an operation. A curve has to be drawn between the monuments; a parallel has to be run; a longitude has to be referred to a surface; a distance has to be computed somewhere; two lines have to be crossed. The sentence names the object and the ground gets the operation, and the gap between them is the whole subject.

That is the field this anchor sits in, stated for boundaries. What a grid is made of makes the same observation about a national grid: a coordinate is the end of a chain of conventions and the conventions are not in the number. A boundary description is the same chain, run in the opposite direction — from a number to the ground rather than from the ground to a number — and it loses the same information on the way.

The one thing that would have prevented all of it #

Five rungs of measurement point at one drafting rule, and it is short enough to state.

A boundary description must name the operation as well as the object. Not “a straight line” but “the geodesic on the WGS84 ellipsoid”; not “the forty-ninth parallel” but “the parallel of 49° north on NAD83 (2011), realised as geodesic chords between monuments not more than N kilometres apart”; not “the line equidistant from the two coasts” but “the line equidistant under geodesic distance on the stated ellipsoid from the basepoints listed in the annex”.

Modern practice does much of this and it did not arrive by accident — it arrived because the ambiguities were found the expensive way, one arbitration at a time. What the five rungs add is the sizes, and the sizes are what tell a drafter which omissions matter. Naming the surface for an equidistance line is worth thousands of square kilometres. Naming the chord spacing on a parallel is worth hectares. Both are one clause, and only one of them is worth arguing about.

The measurement that says which is which is the whole reason to price an ambiguity rather than merely to notice it. The tolerance decides the model is the essay in this field about which corrections a stated tolerance permits dropping, and a boundary treaty is a document with a tolerance in it whether or not it says so.

Where the ladder stops: a boundary defined by something that moves #

Every segment here is defined by points or by coordinates, and both of those are things a treaty can list. The other half of the world’s boundaries are defined by features — a river’s thalweg, a watershed, the low-water line of a coast — and none of the five ambiguities priced above applies to them.

They have their own, and it is of a different kind: the feature moves. A river migrates, a watershed is a ridge whose crest is not a curve anybody can point at, and a coastline depends on the tide and on the date. A boundary described by a feature has no convention problem and has a problem this anchor has no instrument for, because the thing being described is not a geometric object at all.