The aspect has three numbers, not one

Rotating the sphere before projecting it costs nothing and changes nothing about the projection. The aspect is a free choice establishes that: a rotation is an isometry, so an oblique Mercator has exactly the distortion an ordinary Mercator has, at correspondingly rotated points. The distortion pattern is fixed and the world can be turned underneath it.

Fitting the aspect to the region then spends that freedom, and it ended by writing down what it could not do:

aspectSweep searches the tilt of the projection’s axis in one plane. A region whose long axis runs diagonally — Japan, Italy — has its optimum at an oblique aspect with a rotation about the new axis as well, which this sweep passes near and does not reach, so it under-reports the available gain for exactly the regions that would benefit most.

The reason given was cost: each evaluation is a full regional integration and a two-dimensional grid at the same resolution is thirty-seven times the work.

That is true and it is not the fix. The fix is a coarse grid and a local walk, which costs about twice the sweep rather than thirty-seven times it.

How many numbers an aspect actually has #

Three, and the third is inert far more often than not, which is why one plane felt like enough.

A rotation of the sphere has three degrees of freedom: two say where the projection’s pole goes and one says how far the sphere is turned about it. The site’s aspect() has always taken all three — poleLon, poleLat and gamma — and the sweep varies one of them.

The third is inert for a projection whose distortion depends only on the rotated latitude. A cylindrical projection’s scale factors are functions of latitude alone; so are a conic’s; so are an azimuthal projection’s about its own centre. Turning the sphere about the pole slides the region along a line of constant distortion, and the regional score cannot change.

It is not inert for a projection whose meridians are curved, because there the distortion depends on longitude as well.

The measurement is worth having rather than the reasoning, because the reasoning is easy to get half-right. A reader who knows that Mollweide is pseudocylindrical might expect it to behave like a cylindrical projection; its parallels are straight, after all. Its meridians are not, which is exactly the property that makes the third parameter live.

What the extra freedom is worth #

The honest way to report a search that replaces an older one is to score both with the same instrument, and that took one correction: the sweep computes its score over the points every aspect in the sweep can show, which is not the sample the search scores on. Comparing the two numbers directly compared two samples as well as two searches, and reported the search as worse on two regions where it had found the same aspect.

Re-scored properly, and with the sweep’s own answer used as one of the search’s starting points so that the search can never do worse:

Chile and Europe returning exactly 1.00 is the refusal that makes the other numbers mean something. A search that improved on every region would be a search that had changed the scoring, and the two regions where the older method is provably adequate are the ones where the new one must agree with it.

Why a diagonal region is the hard case #

The sweep searches poles along one meridian — the one 90° from the region’s own central meridian — which is exactly the set of aspects that tilt the projection’s axis towards or away from the region without turning it.

For a region whose long axis runs north–south, that set contains the answer: the transverse aspect puts the projection’s line of least distortion along the region. For a region running east–west it contains the answer too, at the other end. For a region running north-east, the best axis is a great circle at 45° to the graticule, and the pole of that great circle is not on the sweep’s meridian.

The conterminous United States is the case that surprised the measurement. It is not a diagonal region in the sense Japan is — it is wide in longitude and shallow in latitude, the classic conic case — and it gains 3.40× from the second parameter, more than Japan does. The reason is that its best axis is a great circle inclined to the equator by a few degrees and rotated in longitude, which lets one line of true scale run the length of the country instead of one parallel running across the middle of it.

What the freedom is, geometrically #

It is worth being precise about what a pole position is, because the surface figures are pictures of a space that has no obvious shape.

Choosing an aspect means choosing where the projection’s own pole lands on the Earth. That is a point on a sphere, so the space of aspects is a sphere — and every figure above draws it as a rectangle in longitude and latitude, which is itself a projection of the search space and distorts it exactly as a plate carrée distorts the world. The optimum for Japan sits at 45° of longitude from the sweep’s meridian; how far that is in angle depends on the latitude, and near the poles of the search space the rectangle exaggerates it enormously.

There is a second thing hidden by the rectangle. A pole and its antipode give the same great circle and therefore, for a cylindrical projection, the same distortion pattern — so the surface is symmetric under a half-turn, and every optimum appears twice. That symmetry is visible in the figures as a duplicated dark region, and it is a check rather than a nuisance: a search that found only one of the pair would be a search with a bug in its rotation.

The case that gains almost nothing, and why #

New Zealand is diagonal, narrow and long, which is the shape the shortfall named — and the second parameter is worth 1.11× there against Japan’s 2.60×.

The reason is that the sweep is already nearly right. New Zealand’s long axis runs about 35° from the meridian and its extent is only 13° of longitude, so the transverse aspect the sweep finds already puts a line of small distortion within a few degrees of the country’s spine, and gains a factor of 50 doing it. The remaining 11 per cent is the last few degrees of alignment.

That is the corrective to the shortfall’s own wording. It predicted trouble for diagonal regions and the trouble is real, but diagonal is not the discriminating property — a narrow diagonal region is nearly fine and a wide shallow one at mid-latitude is not.

What was computed, and how #

The score is the site’s own regional criterion, unchanged: Kavrayskiy’s, which is the root-mean-square of the logarithms of the two principal scale factors over the region, with each projection first normalised to its own best overall scale so that the comparison is scale-free.

The grid is 24 by 12 in pole longitude and latitude, and the latitude count is even for a reason found the hard way: with an odd number of divisions the equator is not on the grid, and the equator is exactly where a cylindrical projection’s optimum usually is. A grid that misses the answer and a refinement that starts from the wrong basin produced a search that lost to the thing it replaced.

The refinement is a compass walk — try four neighbours at the current step, move to the best, halve the step when nothing improves — run for at most forty rounds down to a twentieth of a degree. It is deterministic, which matters because a build must give the same figure twice.

Aspects that cannot show the whole region score infinity. A projection with a limit or a seam can be rotated until part of the region falls outside it, and a score computed over the part that remains would prefer exactly those aspects. The check is a coverage test: an aspect showing less than 98 per cent of the sampled region is refused rather than scored.

The optimum is flat, which is why this is safe to do #

A search that reported a sharp optimum would be reporting something a cartographer cannot use, because the aspect of a real map is constrained by things the criterion does not know about — where the graticule should look sensible, which way the sheet is printed, what the neighbouring sheets do.

The criterion is smooth and its optimum is flat. Over Japan, moving the pole ten degrees from the best position costs a few per cent of the gain; the difference between the sweep’s answer and the search’s is a factor of 2.6 because they are in different basins, not because one is finely tuned. That is the practically important shape of the result: the search is worth running because it finds the right basin, and having found it, the exact position is a matter of taste.

The same flatness is what makes the coarse grid legitimate, and the two facts are the same fact seen twice.

The cost, which is the objection the shortfall raised #

The original note was right that a two-dimensional grid at the sweep’s resolution is thirty-seven times the sweep. The measured cost of the search as built is about twice, and the difference is where the resolution goes.

The sweep uses 25 tilts at a fine regional sampling. The search uses 288 grid points at a coarse sampling — nine rings of samples rather than fourteen — followed by a walk of at most 240 evaluations at the same coarse sampling. The coarse sampling is adequate because the criterion is smooth in the pole position: it is an integral over a region, and integrals do not have narrow features. The walk then refines the position rather than the score.

That is the general shape of the answer to a grid is too expensive: use the grid to find the basin and something else to find the bottom.

Where the model stops #

One projection at a time. The search optimises the aspect of a given projection. Optimising over projection and aspect together is a different problem, and the honest version of it would have to say what the space of projections is — which is the question every projection minimises something is about and the essay before this one answers for the conformal case by dropping the idea of a projection family altogether.

The criterion is fixed. Kavrayskiy’s criterion is one of several; Airy’s weights differently and a second-order criterion weights differently again. The optimum’s position moves with the criterion, and how much is not measured here.

The third parameter is measured as inert or live, not optimised. Where it is live, the search does not vary it: the figures above use two parameters throughout. A three-parameter search over a pseudocylindrical projection is a straightforward extension of the same walk and would take about three times as long, and nothing in this essay reports its result because it has not been run.

One more figure than the search needs. The surfaces are drawn at 24 by 12 because that is what the search uses; a picture at four times the resolution would be prettier and would say the same thing, and the figure deliberately shows the grid the answer was actually found on.

A region is a shape, not a country. Japan and New Zealand here are ellipses on the sphere with stated axes and orientations, not coastlines. The gain figures are therefore statements about diagonal regions of that size, and a real boundary would change them by whatever the shape difference is worth.

Who found it, and when #

The oblique aspect is old — Lambert’s 1772 memoir gives the oblique cases of his own projections — and the idea of fitting one to a region is much younger. Laborde’s oblique Mercator for Madagascar, from 1926, is the first well-known instance of a national grid whose axis was chosen to lie along the country rather than along a meridian, and the Swiss and Hungarian oblique systems followed the same reasoning.

The optimisation literature is post-war and mostly numerical: Snyder’s space oblique Mercator of 1978 solves a harder version of the same problem for a moving satellite track, and the minimum-error projections of Canters and others in the 1990s optimise over parameter families with a regional criterion in exactly the way this essay does over aspects.

What is not in that literature is a statement of how much the aspect alone is worth on ordinary regions, which is what the figures above measure, and the answer — a factor of two to three on a badly oriented region, nothing on a well oriented one — is the sort of number that only appears when somebody runs the search twice.

Which families have three numbers, and why #

The count is not the same for every projection, and the rule that decides it is a symmetry argument worth stating plainly, because it says in advance where the search for a third number is worth running.

The third number is a rotation of the projection’s own frame about the axis pointing at the reader. Whether it does anything depends entirely on whether the projection is invariant under that rotation.

An azimuthal projection is. Its distortion depends only on angular distance from the centre — that is what makes the whole family one function of one variable — so rotating the page about the centre carries every point to another point with identical scale factors. The third number turns the sheet and changes nothing measurable. An azimuthal aspect really does have two numbers, and a search over three would spend a dimension discovering that one of them is free.

A cylindrical projection is not. Its distortion depends on distance from a great circle, not from a point, so the orientation of that circle is a genuine degree of freedom: the same centre with the axis running east–west and running north–south give different maps of the same region. Here the third number is the one that lets the low-distortion band be laid along a diagonal country, which is exactly the case the measurements above find worth the most.

A conic sits between them and has three for the same reason a cylinder does — its distortion is organised about a small circle, and a small circle has an orientation.

So the rule is: count the numbers by asking what the projection’s distortion is a function of. A function of distance from a point gives two; a function of distance from a curve gives three. That is decidable from the projection’s definition before any search is run, and it explains the pattern in the results — the family that gains almost nothing from the third number is precisely the family whose symmetry makes it free.

It also says what a search that finds a flat third dimension has actually discovered. On an azimuthal projection that flatness is exact and structural. On a cylindrical one it is a fact about the region — a region round enough to have no preferred direction — and the two look identical in the output.

Where the ladder goes next #

The aspect is settled: two numbers, sometimes three, searched properly, with the gain measured against the thing it replaces.

The choosing field has one other place where a number was borrowed rather than computed, and it is older and more visible: the count of sheets an atlas needs, which multiplies an ideal by the thinnest covering density of the plane. The sphere is not a plane, and at the counts a coarse atlas has it is not even close.