Fitting the aspect to the region

The aspect is a free choice makes the case that rotating a projection’s axis costs nothing and changes none of its properties. What it did not do is measure what rotating it is worth.

The dimension the catalogue leaves out #

A cartographer choosing a projection is usually choosing from a list: Mercator, Albers, Lambert conformal conic, Robinson, and perhaps thirty more. That is a choice among a few dozen discrete options.

Every one of those options carries three continuous parameters nobody lists — where the projection’s axis points, and how the sheet is rotated about it. Rotating the sphere before projecting is an isometry, so the projection’s distortion pattern is unchanged; it has simply been moved somewhere else on the globe. The pattern is rigid and its placement is free.

That makes the real search space a few dozen discrete choices crossed with a three-dimensional continuum, and the continuum is the part that is almost never searched.

The sweep #

The measurement here searches one dimension of it: the tilt of the projection’s axis, from the normal aspect at 0° through to the transverse at 90°, in the plane through the region’s own centre.

At each tilt the projection is rebuilt as a rotation composed with the ordinary forward map, and scored by the area-weighted root-mean-square departure of the two principal scale factors from unity — Kavrayskiy’s criterion, the same one the regional essay uses, with each candidate normalised so the geometric mean of its areal factor over the region is one.

Chile is 39° tall and 10° wide, and the transverse aspect of Mercator puts its line of true scale straight down the country’s long axis. The result is a scale-free criterion of 0.0008 against 0.1465 for the normal aspect. That is not a marginal improvement; it is the difference between a usable map and an unusable one, obtained by changing no formula and adding no complexity.

The tropics are the control. An equatorial band is exactly what the normal aspect exists for, and the sweep finds no improvement at all — its best tilt is 0.0° and its gain is 1.00.

The control is the whole point #

Without a region that prefers the normal aspect, this measurement would prove nothing. A sweep that reported an improvement everywhere would be reporting a bug — most likely one in which the tilted projections were being scored more leniently than the upright ones.

Which is exactly what happened. The first version of the sweep scored each aspect over whatever part of the region that projection could reach, and duly announced that the best way to map an equatorial band is the transverse aspect. A transverse Mercator cannot show the ground 90° from its central meridian at all, so a third of the tropics silently left the average and the remaining two thirds were the good two thirds.

The repair is that every aspect is scored over the same set of points: the points every candidate in the sweep can show. That drops the tropics’ coverage to 88% and leaves the comparison honest, and the answer immediately becomes 0.0° with a gain of 1.00.

The general form of the error is worth naming because it is not specific to projections. A score computed over a different sample is not the same score, and any optimisation whose candidates differ in what they can evaluate will find the candidate that evaluates least.

What the shape of the curve says #

The curves are as informative as their minima.

Chile’s curve falls steeply and monotonically from 0° to 90°, by more than two orders of magnitude, with no local minimum. Every degree of tilt is an improvement, so a cartographer who tilts the axis part of the way gets part of the benefit — there is no threshold to cross.

The tropics’ curve is nearly flat, varying by about 1% across the whole sweep before the coverage limit cuts it off. For that region the aspect genuinely does not matter, and knowing that is as useful as knowing that it does for Chile.

Europe’s and the United States’ curves fall by factors of 17 and 3.8 respectively, and the difference between those two numbers is the difference in shape: Europe spans 50° of longitude and 35° of latitude while the conterminous United States spans 59° and 26°, so Europe is closer to square in ground terms at its latitude and gains more from being wrapped round a meridian.

Why the transverse case keeps winning #

Three of the four regions want the maximum tilt, and the reason is structural rather than a property of these particular countries.

A cylindrical projection has a line of true scale — its own equator — and its distortion grows with distance from that line. The normal aspect puts that line on the Earth’s equator, which is a good place for it only if the region straddles the equator. For anything else, the best available line is one that runs through the region, and a great circle through a region at any latitude is reached by tilting the axis.

So for a cylindrical projection the sweep is really asking: where should the line of true scale go? And the answer is nearly always “through the middle of the region”, which for a region not on the equator means a tilt.

That is what a national grid is, stated as an optimisation. A transverse Mercator centred on a country’s own meridian is the 90° end of this sweep, and the reason so many countries use one is that the sweep has this shape for almost every country.

What the rotated map actually looks like #

A factor of 183 in a scale-free criterion is an abstraction, and the underlying change is concrete.

Nothing about the projection has changed. The band of low distortion is the same band, the growth away from it follows the same law, and every claim in the site’s audit of Mercator holds unchanged — because rotation is an isometry of the sphere and a rotated projection has exactly the distortion the normal one has at correspondingly rotated points.

What has changed is which ground the band lands on, and for a country that is the only question.

The gain is predictable from the bounding box #

The three gains — 183, 17 and 3.8 — are not four measurements with a story attached. They follow from each region’s own extent, and the estimate takes no sweep at all.

A cylindrical projection’s scale factor is the secant of the distance from its line of true scale, so a criterion built on the logarithms of the principal scales is governed by the mean of |ln sec d| over the region, with d the angular distance from that line. The normal aspect puts the line on the equator, so d is the latitude; the transverse aspect puts it on the region’s own central meridian, so d is the ground distance east or west of it — the longitude offset times the cosine of the latitude.

Taking each region’s bounding box and comparing the two means:

Two of the three agree closely and the third is the right order. The gain is the ratio of how far the region sits from the equator to how wide it is in ground terms, read through a secant, and a cartographer can compute it from four numbers before deciding whether the sweep is worth running.

That also says which regions the answer is large for, without listing countries. A region is worth rotating in proportion to how far it is from the equator and in inverse proportion to how wide it is — so the extreme case is a narrow country at high latitude, and the null case is a wide band on the equator. Chile and the tropics are not two examples; they are the two ends of one ratio.

What the sweep does not search #

Three things, and each is a real limitation rather than a simplification.

The other two rotation parameters. The axis can be tilted in any plane, not only the one through the region’s centre, and the sheet can be rotated about the axis afterwards. For a region whose long axis runs north-east — most of Chile does not, but Italy and Japan do — the optimum is an oblique aspect that this one-dimensional sweep passes near and does not reach.

The projection. Holding Mercator fixed and sweeping the aspect answers where should this projection point, not which projection. Doing both at once is a mixed discrete–continuous problem, and the honest observation is that the continuous part is worth more: the factor of 183 available from rotating Mercator over Chile is far larger than the difference between any two projections in the library over the same region.

The interruptions. Cutting the sheet is another free parameter and it is not in the search either.

The criterion. Kavrayskiy’s is one of several, and the criteria disagree. The location of the optimum is much less sensitive to the criterion than its depth is, because every reasonable criterion is minimised by putting the line of true scale through the region — but “much less sensitive” is not “insensitive”, and a sweep quoted without its criterion is a sweep quoting somebody’s opinion.

The cost of the sweep, and what it forced #

Each evaluation is a full regional integration, and 37 of them per curve at build time is a real cost on a page that draws three sweeps.

The sampling is therefore deliberately coarse — 196 points per region rather than 1,600 — and the position of the optimum was checked once against the fine sampling rather than on every build. That is an explicit trade and it is recorded in the generator, because a figure whose sampling was quietly reduced until the build was fast enough is a figure whose numbers nobody can reproduce.

The coarse sampling is defensible here for a specific reason: the quantity being read off the curve is the location of the minimum, which is stable, rather than its depth, which is not. A criterion evaluated at 196 points has a standard error of a few per cent on its value and a fraction of a degree on the position of a minimum this steep.

Two regions the sweep gets wrong, and why #

Honesty about a one-dimensional search means naming the cases it mishandles.

A region straddling the equator but not centred on it — Indonesia, say, or equatorial Africa — has an optimum at a small non-zero tilt that this sweep finds correctly and reports as unimpressive, because the gain is small. The correct reading is that the aspect does not matter much there, and a cartographer who reads the small gain as “the sweep failed” will go looking for something that is not there.

A region whose long axis runs diagonally is the real failure. Japan runs north-east; Italy runs north-west; Chile happens to run almost exactly north–south, which is why it is the demonstration case. For a diagonal region the optimum is an oblique aspect with a rotation about the new axis as well as a tilt of it, and this sweep passes near that optimum without reaching it, so it under-reports the available gain.

The fix is a two- or three-parameter search, which is the same calculation run more times, and the reason it is not run here is build time: each evaluation is a full regional integration and a two-dimensional grid at the same resolution is thirty-seven times the cost. That is a recorded shortfall rather than a limitation of the idea.

The recommendation, stated plainly #

For a region not straddling the equator, and a projection with a line or point of true scale:

1. Work out where the region’s own long axis runs, as a great circle.
2. Rotate the projection’s axis so that its line of true scale lies along that great circle.
3. Only then argue about which projection.

Doing the third step first, which is what a catalogue encourages, is optimising the smaller term. That is the question behind the question in its most practical form: which projection is best is a question about a list, and where should this projection point is a question about the region, and the second one is worth more.

What it costs to be wrong about this #

The factor of 183 is a scale-free criterion and criteria are abstractions, so it is worth converting into something a map’s reader would notice.

Over Chile, the normal-aspect Mercator’s principal scale factors run from about 1.07 at the northern edge to about 1.79 at the southern one — the country is drawn 67% larger at one end than at the other, and every shape in it is a circle of the correct proportions at the wrong size. The transverse aspect over the same ground holds the scale to within a fraction of a per cent from end to end.

A reader comparing the area of two provinces on the first map is out by up to 70% and has no way to know it, because a conformal projection gives no local warning: nothing is squashed, nothing is sheared, the coastline is the right shape everywhere. The two ways a map is wrong are independent, and a conformal projection at the wrong aspect fails only the second one, silently.

That is the strongest practical argument for searching the aspect. The distortion an unrotated projection imposes on a country is the kind a reader cannot see, and the remedy is free.

Two regions that both want the transverse aspect and want it by very different amounts.

What was computed here #

For each region, 37 aspects were constructed as rotations of the sphere composed with the projection’s forward map, with the rotation pole placed 90° from the region’s central meridian so that the tilt swings the axis in the plane through the region’s centre. Each was scored by Kavrayskiy’s area-weighted criterion after normalising its overall scale, over the subset of the region’s sample points that every aspect in the sweep can display.

Three claims are asserted: a region 39° tall and 10° wide must prefer a tilt above 60°; an equatorial band must prefer a tilt below 15°; and the tall region’s gain must exceed 1.5, because a true statement about a 3% improvement is not worth an essay.

What the pictures cannot show #

Each curve is divided by its own value at north-up so that two regions with very different absolute distortion can share an axis. That makes the shapes comparable and the depths not: Chile’s best score of 0.0008 and the tropics’ 0.0252 are thirty times apart in absolute terms and both appear at 1.0 on the left of the plot.

The figures also stop at the coverage limit rather than continuing with a partial score, so the tropics’ curve ends before 90°. That truncation is the honest form of the failure the sweep was rewritten to avoid, and it is visible as a curve that stops rather than as a number that lies.

Who found it, and when #

Oblique aspects are as old as the azimuthal projections, which have always taken a centre. What took much longer was the recognition that the same freedom belongs to every projection: Lambert’s 1772 memoir has the transverse Mercator, and treating the aspect as a parameter to be fitted to a region rather than chosen from a short list of conventional cases is a twentieth-century habit that arrived with national grids.

The oblique Mercator fitted to a country’s own axis was worked out several times independently — Rosenmund for Switzerland in 1903, Laborde for Madagascar in 1928, Hotine in 1946 — which is what happens when the answer follows from the question and the question keeps being asked.

Where this goes next #

The objective throughout has been an average over an area. Some maps are not about an area at all, and for those the whole apparatus points at the wrong thing.