The third parameter, run

The aspect has three numbers, not one established what an aspect is — a rotation of the sphere, so three degrees of freedom — and measured that the third is inert for a cylindrical or conic projection and worth per cent for a pseudocylindrical one. It then wrote down what it had not done:

The three-parameter aspect search is measured and not run. Where the third rotation is live — pseudocylindricals, Winkel tripel — the figures still search two parameters. The extension is the same compass walk with two more directions and about three times the cost.

Running it produces two findings, and the second is about the sentence above rather than about projections. The third parameter is worth more than per cent: over Europe it is worth a factor of two. And the extension is not the same compass walk with two more directions — done that way it returns nothing.

What the obvious implementation returns #

The natural way to pay a debt like this is to keep the machinery and add a direction: take the two-parameter optimum, let the compass walk move γ as well as the pole, and report the improvement.

It reports nothing. Not a small gain — no gain, on the first pair tried.

The reason is visible once the profile is drawn and is not a defect of the walk. The two-parameter optimum has put the region where the projection is symmetric about the third rotation. A symmetric configuration has a vanishing derivative in the rotation that breaks the symmetry, so γ = 0 is a genuine local minimum — the score rises in both directions — and every step the walk can take makes things worse.

That is a fact about optimisation shaped by a fact about projections, and it is why the note’s estimate of the work was wrong. The cost is not three times a compass walk. It is a grid over the whole third range, from which the walks then start.

What it is worth when the grid finds it #

Searched properly, the third parameter is worth:

• 2.11× for Mollweide over Europe, at γ = −132°;
• 2.03× for Robinson over Japan, at γ = +115°;
• 1.82× for the Winkel tripel over Europe;
• 1.25× for the Winkel tripel over Japan;
• 1.0000× for Mercator over Japan.

The last of those is the control and it does more work than the others. A cylindrical projection’s distortion depends only on the rotated latitude, so turning the sphere about the projection’s own pole slides the region along a line of constant distortion and cannot change any regional score. The search is therefore required to find exactly nothing, and it does — which is what says the gains above are the projections’ rather than the search’s.

What the search’s own resolution contributes #

The inert case does one more job, and it is the reason this essay can quote a factor of two without hedging.

At the coarse settings the table uses — a ten by five grid of poles, six values of γ, three starting points — Mercator over Europe comes back at 1.021 rather than 1.000. The third rotation cannot have produced that, so it is the denser grid finding a slightly better pole than the two-parameter search did, which is the search’s own resolution showing up as a gain.

So a reported improvement of a few per cent at these settings is not evidence of anything, and the numbers worth quoting are the ones far above it. That is a calibration a control makes available and that no amount of care with the live cases would have produced.

The gain and the profile are different quantities #

Robinson over Europe is the case where the two answers can be compared, and it separates two things that a single number would run together.

Walking γ alone from the two-parameter optimum finds 1.43 at γ = −156°. Searching all three properly finds 1.87 at γ = −135° and a pole 47° away from where the two-parameter search put it. The extra 30 per cent is not in the third parameter; it is in the two the search had already fixed, which only become worth revisiting once the third is free.

That is the general shape of the difficulty and the reason a staged search is untrustworthy. Optimising two parameters, freezing them, and then optimising a third is not optimising three: it finds the best value of the third given a choice of the first two that was made without it.

What was computed, and how #

The score is the same one every other essay on this ladder uses: Kavrayskiy’s criterion over a stated region, with each candidate normalised so that its overall scale is not what is being measured, evaluated over the region’s own sample points.

The search is a grid followed by walks. The grid runs over pole longitude, pole latitude and the third rotation; the walk is the compass walk this site already had, started from the two-parameter answer and from the best few grid points, with the best of all the results kept. Starting only from the best grid point was tried and is not enough, for a reason that turns out to be common: on several pairs the two-parameter optimum scored better than every point of the three-parameter grid, so the walk began there, found γ stationary and reported nothing.

The cost is eight times the two-parameter grid at these settings, not three, and it grows as the third range is sampled more finely.

Putting that figure beside the table above shows exactly how a shortfall note can under-price itself. A quantity’s variation along a line through one point is not what an optimisation over it is worth, and the two differ here by an order of magnitude.

What an aspect is, and why it has three numbers #

An aspect is a rotation of the sphere, and a rotation of the sphere has three degrees of freedom — the two that say where the projection’s pole goes and the one that says how far the sphere is turned about it. The aspect is a free choice makes the case that using it is the cheapest improvement available to a cartographer; the aspect has three numbers, not one counts them.

The history of searching them is a history of adding one at a time. The classical treatment sweeps a single angle — tilt the axis within one plane through the region — and reports the best. This collection’s own previous rung replaced that with a search over the pole’s two coordinates and found the sweep under-reporting by factors of 2.6 and 3.4 on two regions. This rung adds the third and finds another factor of two.

Each stage found the previous one under-reporting, and each stage’s note under-priced the next. That is not a criticism of any of the three; it is what a hierarchy of nested searches does, and the only defence against it is to search the whole space at once, which is what the grid here finally does.

A quantity that varies is not a quantity that is worth optimising #

There is a distinction this rung turns on and it is worth stating on its own, because the shortfall note ran the two together and so would most readers.

The previous measurement asked: how much does the score change as γ is swept, at a fixed aspect? The answer was per cent for pseudocylindricals and nothing for cylindricals, and it is the quantity assertGammaMattersOnlySometimes still checks.

This one asks: how much lower is the best score when γ is free than when it is fixed at zero? The answer is up to 2.11.

Those are different questions and the second is not bounded by the first. A parameter whose variation along one line is a few per cent can be worth a factor of two, because freeing it changes where the other parameters’ optimum is — the search is not walking along the line the first measurement looked at, it is walking to a different part of the space entirely.

The general form is worth carrying: the sensitivity of an objective to a parameter, measured at a point, says nothing about what optimising over that parameter is worth. Sensitivity is a local derivative; optimisation is a global search; and a symmetric configuration is exactly where the two disagree most, because that is where the derivative vanishes and the improvement does not.

Where the model stops #

The search finds a local optimum from several starts and there is no proof that it is global. The score is a smooth function of three angles on a compact domain, so a global optimum exists; nothing here shows the search has found it, and the honest statement is that ten by five by six starting points and eight walks did not find anything better.

The criterion decides the answer, as it decides every ranking in this field. Kavrayskiy’s criterion weights the logarithms of both principal scales; Airy’s weights their departures from one, and the two disagree about rankings. A different criterion would move the optimum aspect and change what the third parameter is worth.

And the regions are the site’s own stated shapes. Fitting the aspect to the region shows how much the answer depends on the region’s own extent and orientation, and this rung inherits all of it: an aspect optimised for a box is optimal for a box.

What it costs to run #

The eightfold increase in evaluations is worth putting in context, because a search that costs eight times as much and finds twice the improvement is an easy trade and the note’s estimate of three times was not the reason the work waited.

At the resolution the figures use, a three-parameter search over one projection and one region is about a second — three hundred and fifty pole positions times six rotations, each scoring sixty-four sample points, plus eight compass walks. The two-parameter version is an eighth of that.

What that buys is a factor of two in a distortion criterion, on a decision a cartographer makes once per map. Set against the other choices this collection scores — a factor of 1.51 for a country’s own grid over the international zone, a factor of 2.25 for following the taught rule about families where it is wrong — it is one of the larger single improvements available, and it is entirely free at the point of use.

The generalisation #

Three statements, in ascending order of how far they travel.

A shortfall note’s estimate of its own cost is a guess made by whoever could not do the work. This one said three times the cost and the same compass walk, and both were wrong in the same direction: the work was harder because the natural implementation returns a false negative rather than because it is slow.

A staged optimisation is not an optimisation. Fixing parameters found without the later ones and then refining is a standard shortcut, and the price of it here is 30 per cent of the available gain.

And a symmetry in a model produces stationary points that hide improvements from any local method. The optimum this site had was symmetric about the parameter it was not searching, which is not a coincidence: a two-parameter optimum of a symmetric objective tends to be symmetric, and symmetric configurations are exactly the ones where the missing parameter’s derivative vanishes. Wherever a search has been narrowed by a symmetry argument, the narrowed search’s answer is the worst possible starting point for widening it.

What is still not searched #

The aspect is three numbers and all three are now searched. The choice a cartographer makes is larger than that, and the rest of it is still handled one stage at a time.

A projection has parameters of its own — a cone constant, a standard parallel, a mixing weight — and those are searched separately from the aspect wherever this collection searches them at all. The same argument this rung makes about the third rotation applies: the best cone constant given a fixed aspect is not the best cone constant, and freeing both at once may find a pair neither stage would reach.

Beyond that is the family, and beyond that the property. The full problem is a search over a projection, its parameters and its aspect together, scored against a purpose, and nothing here does that. What this rung establishes is that each stage of the staged version leaves something on the table, and that the amount is about a factor of two per stage — which is an argument for combining them rather than for adding a fourth.

Who found it, and when #

Oblique aspects are as old as the subject — Lambert gives the general rotation in 1772 — and the practical question of choosing one for a region is Nordic and Soviet twentieth-century cartography, where Kavrayskiy’s criterion comes from. Automated searches over the aspect appear from the 1990s, mostly over two parameters, and usually with the third dropped on the grounds that it is inert for the projections being searched, which is true for the cylindricals and conics that dominate national mapping.

What this rung adds is the price of that convention on the projections it is not true for, and the reason the obvious way of removing it does not work. The regions it is measured over are this site’s own stated shapes, which is the standing qualification on every optimisation here.

Why the convention is right where it came from #

The third parameter being dropped on the grounds that it is inert is worth defending before it is priced, because the convention is not lazy — it is correct in the setting that produced it.

National mapping is dominated by cylindricals and conics, and for a national grid the projection is chosen first, from a very short list, with the region’s shape already known. In that setting the third parameter is a genuine degree of freedom only in the sense that it is being used: an oblique Mercator’s azimuth is the third parameter, chosen deliberately, and nobody would call it inert.

Where it is dropped is the automated search, and there the projections being searched are usually the same short list with the aspect already partly fixed by the application. So the convention encodes a true statement about a restricted problem: for these projections, in this use, searching the third dimension finds nothing.

The price appears when the search is generalised — to azimuthal members, to compromise projections, to a library rather than a shortlist — and the convention travels with the code rather than with its justification. That is the same failure mode as every other inherited assumption in this collection: the reasoning was sound, it was about a scope, and the scope was not written down beside the rule.

Which makes the useful output of this rung a boundary rather than a correction. The two-parameter search is right for the projections whose distortion is organised about a point; it leaves a factor of about two on the table for the ones organised about a curve. That is decidable from the projection before any search is run, so the convention can be kept exactly where it holds.

Where the ladder goes next #

This rung searches an aspect. The rung before it searched a projection over a set of named candidates, and the rung above that has to search a family — which is the taught rule for choosing one, scored and found to key on latitude where the measurement keys on shape. Between them the three make one argument: every stage of the choice a cartographer makes has more freedom in it than the stage’s usual advice searches, and each stage’s freedom is worth about a factor of two.