Three conditions are one too many

A point in a plane has two degrees of freedom. One distance from a known place puts it on a circle; a second distance fixes it, up to the choice of which of two intersections. A third distance has nothing left to determine and is therefore either redundant or contradictory.

On a sphere flattened onto that plane it is contradictory, always, everywhere, by an amount that can be computed.

The count, which decides the answer before any arithmetic #

Two unknowns and three equations. Unless the equations are dependent, there is no solution.

In the plane the three equations would be dependent only if the three distances happened to be consistent with a plane triangle geometry — which is exactly what they are not, because they were measured on a sphere. The excess is the spherical excess in another guise, and it does not vanish for any point of any region of non-zero size.

That is a stronger statement than the construction is approximate. It says the condition has no solution at all, so the construction is not solving it: it is applying a rule for what to do instead, and the rule is a choice.

Chamberlin’s rule, which is a convention and not an estimator #

Wellman Chamberlin’s rule, used by the National Geographic Society for continental maps from 1946, is to take the three intersection points obtained from the conditions two at a time and place the point at their centroid.

Nothing about that is forced. Three other rules are available immediately — take the circumcentre of the triangle of candidates, take the point minimising the sum of squared distance errors, take the point minimising the worst error — and they give different maps.

The distinction the site draws here is the one what a closed figure cannot see draws for a traverse misclosure: a rule for distributing an inconsistency is a convention; a rule that claims to have found the best answer is an estimator. Chamberlin’s centroid is a convention. Least squares on the three residuals is an estimator, belongs to a subject this site does not take, and is deliberately not implemented here — the same boundary the practice field states about the Crandall rule.

The consequence for a reader is practical. Two atlases using “the Chamberlin trimetric” with different averaging rules would produce different coordinates for the same place, by a distance the next section measures, and neither would be wrong.

The residual, measured #

The spread of the three candidate positions is the quantity that measures the extra clause. Over a region the size of North America it is tens of kilometres:

and it is nowhere zero. Sampled at 169 points across the region, the smallest value anywhere is 5.8 km and the largest 233 km, at the far corner of the sampled box where the extrapolation is well outside the triangle of centres.

The assertion this figure carries is the refusal: there is no point in the region at which all three distances are simultaneously exact. That is what an over-determined system means, and it is worth testing rather than reasoning about, because a construction that silently satisfied all three would mean the code was solving a different problem from the one described.

Why not simply drop one of the three? #

The obvious response to an over-determined condition is to take two of the clauses and discard the third. That gives the two-point equidistant projection of the first two centres, which satisfies its condition exactly and has no residual at all.

What it costs is measurable, and the measurement is the argument for keeping the third clause. Building the two-point map from the first two centres and then asking it about the third — how far is the third centre from each place on the map, as drawn, against how far it really is — gives an error averaging 27 km over the same region and reaching 111 km at the north-western corner.

So the choice is between a map that is exactly right about two places and 27 km wrong about the third on average, and a map that is 22 km inconsistent about all three. Chamberlin’s construction spreads the disagreement rather than concentrating it, and that is the whole of what it buys.

The general shape of that trade will be familiar to anybody who has adjusted a survey: an inconsistency can be forced into one place or spread over everything, and neither makes it smaller. What is different here is that the inconsistency has a geometric origin rather than an observational one — nothing was measured badly, and the numbers are exact.

The law it grows by, which is cubic and not quadratic #

The residual is a curvature effect, so the obvious guess is that it grows as the square of the size of the region — that is the law how small is flat enough fits for the departure of a plane survey, and the law that governs almost every flattening error on this site.

It is cubic. Fitted over five region sizes spanning a factor of sixteen:

Doubling the region multiplies the residual by eight, and the fitted exponent is 2.99.

The two exponents are consistent rather than contradictory, and telling them apart requires knowing which quantity is being measured. That is the same distinction the indicatrix is a limit turned on Tissot’s ellipse — where the absolute departure is quadratic and the relative one linear — arriving here with an extra factor of the region’s own size.

The paradox: an unsatisfiable condition produces an excellent map #

The Chamberlin trimetric has no exact property. It is not conformal, not equal-area, not equidistant from anything, and it does not even satisfy its own defining condition. On the site’s usual grounds it should be nothing much.

Measured over the region it was built for, it is one of the best maps in the collection:

The flexion figure is the striking one. At the same point the Mercator projection bends geodesics at 0.70 radians per radian and the two-point equidistant at 1.21; the trimetric manages 0.068, which is a factor of ten better than either, and its skewness is at the level of the differencing noise.

The reason is not mysterious and is worth stating, because it is the general lesson of this rung. Three conditions that cannot all hold, resolved by averaging, distribute their disagreement evenly over the region between them. That is what a compromise projection is — compromise projections makes the same argument for the Winkel tripel and the Robinson — and Chamberlin arrived at one from a direction nobody else in the field takes: by asking for too much and dividing the failure three ways.

The same construction over a different continent #

The three centres are the projection’s parameters and there is nothing continent-specific in the code. Moving them moves the good region with them.

Sampled over Africa the residual runs from 8.3 km to 752 km, the large value again at the corner of the sampled box rather than inside the triangle. The rule for choosing centres is the one implied by both figures: put them near the corners of what has to be mapped, and do not read the map far outside them.

Four centres, and why nobody uses them #

Three conditions are one too many, so four are two too many, and the obvious question is whether the extra clauses make things worse in any way that matters.

They do not make the residual larger — the four circles’ six pairwise intersections still cluster in a small region, and averaging them gives a placement no worse than the three-centre one. What they change is that the construction stops having a rule anybody can carry out with a compass. Chamberlin’s three arcs and an eyeballed centroid are a drafting procedure; six intersections and their mean are a computation.

That is the whole reason the trimetric has three centres rather than four or five, and it is worth naming because it is a constraint from outside the mathematics. The projection was designed in 1946 to be drawn, and the number of conditions is set by what a cartographer could do at a drawing board rather than by any property of the sphere.

A modern version with no such constraint would not use more centres at all; it would minimise something over the whole region, which is the optimisation route Chebyshev’s criterion takes and which produces a different kind of object — one with no exact conditions in it anywhere.

Where the model stops #

The residual is not an error bar. It is the spread of three candidate placements, which is a measure of the inconsistency, not a bound on the distance between the plotted point and where it should be. There is no should — the condition has no solution, so the true placement does not exist to be missed.

The construction assumes the three centres form a proper triangle. Three nearly collinear centres give a triangle of vanishing area, the intersections become ill-conditioned, and the library asserts a positive area rather than dividing by something near zero.

The centroid is of three plane points, and the rule is part of the projection. Two implementations that average differently produce two different maps, and the residual is exactly the scale of the disagreement between them: any rule that returns a point inside the triangle of candidates differs from any other by at most the 22 km measured above. So the residual doubles as a bound on how much the choice of convention can matter, which is the only sense in which it is an error bar. The trap of averaging positions in the wrong plane is a real one and is the whole subject of a centroid belongs to a plane; here the objects being averaged are already plane constructions, so the plane is where they belong.

Everything here is on the sphere. On the ellipsoid the three distances would each move by up to a third of a per cent, which is far larger than the residual at continental scale — so a trimetric map of a real continent is dominated by the choice of body long before it is limited by the construction. That ordering is the same one datum shifts dwarf projection errors establishes for the ordinary projections.

What the residual is on a printed sheet #

A number in kilometres is hard to weigh without a scale attached, and the scale is what decides whether any of this is visible.

At 1:5,000,000 — the scale of a large single-sheet map of the United States — a residual of 22 km is 4.4 millimetres on the paper. At 1:20,000,000, a continental map in an atlas spread, it is 1.1 millimetres. At the 32° region where the residual reaches 636 km, drawn at 1:40,000,000, it would be sixteen millimetres, which is not a subtlety but a visibly different place.

Two consequences follow. The first is that Chamberlin’s construction is genuinely at the edge of what hand drafting could resolve at the scales it was used at, which is presumably why the eyeballed centroid was good enough for the Society’s cartographers and why nobody needed a rule with a justification. The second is that the extrapolation warning is not academic: a trimetric map read well outside its triangle of centres is wrong by distances a reader could measure with a ruler.

The same arithmetic gives the design rule. For a residual under half a millimetre at the drawn scale — about the width of a drawn line, and the tolerance the tolerance decides the model argues everything else should be chosen against — the three centres must surround the mapped region, and the region’s radius must stay under about 8° at 1:20,000,000 — the ladder above puts the residual at 10.4 km there, which is 0.52 mm on that sheet. At 16° it is 82 km and four millimetres, which is not a line width.

Who found it, and when #

Wellman Chamberlin was the National Geographic Society’s chief cartographer, and the trimetric construction is his, from 1946. It was drawn by hand for decades: the three arcs were struck with a beam compass and the point placed by eye at the centre of the little triangle they left, which is precisely the centroid rule and is a great deal easier to do with a compass than to justify.

The Society replaced it for world maps with the Robinson in 1988 and then the Winkel tripel in 1998, and kept it for continents, which the numbers above support exactly: it is a very good regional map and it has no business being extended.

What is new here is not the construction but its accounting. The residual is usually described as small; it is 22 km over North America, 636 km over a region of 32° radius, and it grows as the cube of the region’s size — three statements that can be checked, and that decide where the projection may be used without anybody having to be told.

What makes an unsatisfiable condition produce a good map #

The paradox is the essay’s most interesting result and it deserves a statement that carries beyond this construction, because the reasoning is not about trimetric projections.

Being unsatisfiable is not the same as being badly approximated. Three conditions on two unknowns have no exact solution, and that says nothing about how far the best compromise sits from satisfying all three. Here the compromise is 22 kilometres out over a continent — a fifth of a millimetre on a printed sheet — so the conditions are jointly unsatisfiable and jointly nearly satisfiable at the same time.

The size of the shortfall is the question, and it has an answer with a law in it. Growing as the cube of the region’s radius means the shortfall is negligible over a continent and disqualifying over a hemisphere, and the transition is fast. That is a much more useful statement than the conditions conflict, which is true at every size.

Which is why the counting argument decides the answer and not the verdict. Counting says there is no exact solution and therefore that a convention is needed; it says nothing about whether the convention’s output is fit for anything. The two questions are answered by different work — one by arithmetic on dimensions, the other by measuring the residual and fitting its growth.

And the general form is worth carrying. An over-determined specification is the normal case in design rather than a pathology: most useful objects are asked for more properties than they can have. What separates a good design from a bad one is not whether the specification was satisfiable but how large the compromise is over the range the object will be used in — and that is a measurement, not a count.

Both mistakes come from the same place, which is treating a count of equations as a verdict about an object.

The failure this guards against is the opposite of the usual one. The usual mistake is treating an unsatisfiable specification as satisfiable and shipping something that does not work. The mistake available here is treating it as disqualifying and discarding a construction that is excellent within its range, on the grounds of an argument that never mentioned the range.

Where the ladder goes next #

Two conditions had a solution. Three had none, and the repair was a convention. The remaining case is a condition that has a solution, satisfies it exactly, and produces something that is not a map in the ordinary sense at all — because nothing requires a conditioned construction to be one-to-one, and the next rung’s projection draws two places thousands of kilometres apart at the same point.