Whether the ellipses point the same way

The previous rung established that the average of a region’s indicatrices is not the indicatrix of its average, and closed on a second finding it could not follow up: whether the average is more or less deformed than the average deformation is decided by whether the region’s ellipses point the same way. Three cylindrical projections returned an alignment of exactly 1.0000 and sat above the diagonal; seven others returned 0.23 to 0.57 and sat below it.

That was ten projections over one region — a single row of a table, and a single row cannot say whose property the number is. It might belong to the projection, so that Mollweide’s ellipses are crossed wherever they are measured. It might belong to the region, so that any projection’s ellipses are crossed over the world and aligned over Japan. Or it might belong to the pair, in which case neither can be quoted without the other.

What alignment is, exactly #

An indicatrix is an ellipse and an ellipse has an axis, which the tissot ladder measures and almost nobody reports. An axis is a line rather than a direction — turning it through 180° gives the same ellipse — so averaging a set of them has to be done on the doubled angle, which is the standard treatment of axial data.

The alignment is then the length of the resultant of those doubled angles, weighted by the region’s own sampling weights. It is one when every ellipse points the same way, zero when they are spread evenly and cancel, and anything between when they are partly agreed.

The reason it matters is that it decides whether deformations reinforce or cancel when they are averaged. Two ellipses at right angles, each stretched two-to-one, average to a circle; two pointing the same way average to an ellipse stretched two-to-one.

The answer: all three, and a third belongs to neither #

Twelve projections against nine regions is a hundred and eight numbers, and a two-way table has a standard decomposition: how much of its variation is explained by the row, how much by the column, and how much by neither.

So the question has no single answer and the three shares are all substantial. The region is the largest single factor, which is worth noticing on its own: alignment is more nearly a property of where the region is than of which map it is seen through, which is the reverse of what almost every distortion measure on this site does.

And a third of it is the interaction. That is the part which says knowing the projection and knowing the region is worth more than knowing each — and it is the part that makes any single-row table, including the one this rung is paying off, unquotable outside the row it was measured in.

Reading the table down a column #

Down a column, the table asks what a region does to every projection at once, and two columns behave very differently from the rest.

Japan and Britain are aligned for nearly everything. Eleven of the twelve rows return above 0.99 over Japan. These are small regions — a few degrees across — and inside a small region a projection has not had room to turn its ellipses very far, whatever its global behaviour. Alignment is nearly one for any projection over any sufficiently small region, and the size of “sufficiently” is set by how fast the projection’s principal direction rotates.

The world and a 30° cap are aligned for almost nothing. Over the world every non-cylindrical projection returns 0.23 to 0.57, and over a polar cap the numbers fall further, to 0.001 for Mercator and 0.042 for Mollweide. A cap centred on the pole is the extreme case for a reason that has nothing to do with distortion: the principal directions of an azimuthally symmetric arrangement point radially, and a set of radial directions covering every bearing has a resultant of exactly zero.

That is the whole content of the column effect, and it is a fact about geometry rather than about maps: a region with a symmetry averages its own directions away. Which is why the region accounts for more of the variation than the projection does.

Two rows that are not measurements #

The plan for this rung expected the cylindrical projections to return 1.000 everywhere, on the reasoning that their indicatrix axes lie along the meridian and the parallel at every point. Three of the five do — the Lambert cylindrical, the plate carrée and Miller, at 1.000000000 in all nine regions. Two do not, and each fails for a different reason that is worth more than the rule they break.

Mercator has no orientation to be aligned. It is conformal: its indicatrix is a circle at every point, its axis ratio is 1.000000000000 everywhere, and a circle has no major axis. The eigenvector the arithmetic returns is whichever way the last bits fall, so its row of the table — 0.475, 0.141, 0.885, 0.056 — is noise dressed as a measurement. The essay that introduced the indicatrix’s direction says the direction does not exist on a conformal map; this is what happens when a summary is computed anyway.

Gall–Peters turns its axis over at 45°. It is equal-area and secant, so the meridian scale exceeds the parallel scale inside its standard parallels and the parallel scale exceeds the meridian outside them. The major axis is north–south below 45° and east–west above it, turning through a right angle exactly at the standard parallel.

The consequence is exact and it shows in the table:

Every region entirely on one side of the standard parallel returns exactly one; every region that crosses it returns less. From a projection whose ellipses are never at an angle to anything.

Which is the more interesting failure #

The Gall–Peters row is the one worth taking seriously, because the Mercator row is a computation that should not have been made and this one is a real property of a real map.

It says that a summary over a region can be destroyed by a feature of the projection inside that region rather than by anything about the region itself. A region straddling a standard parallel contains ellipses at right angles to each other, so any average of them cancels — and the resulting average indicatrix is rounder than every ellipse it was computed from, which is the failure the previous rung named and this one localises.

That generalises past this projection. Any secant construction has the same turnover somewhere, and what a standard parallel buys is bought by putting the scale error on both sides of a line where the scale is exactly right. The line is invisible in every summary statistic, it is where the axis flips, and a region chosen to be centred on it is the worst possible region for an average.

The instrument, and why the doubling matters #

One methodological note, because it is the kind of thing that produces a wrong table without producing an error.

An ellipse axis is a line, not an arrow. An ellipse whose major axis points north-east is identical to one pointing south-west, so averaging the raw angles of a set of ellipses is averaging quantities that are not comparable: two identical ellipses reported at 45° and 225° would average to 135°, which is at right angles to both.

Doubling the angle before averaging fixes it exactly — 90° and 450° are the same direction — and halving the result at the end recovers the axis. The resultant length of the doubled angles is then a number between zero and one with the meaning used throughout this essay, and it is the standard treatment of axial data in every field that has any.

The check that it is working is the three cylindrical rows. A projection whose ellipses lie along the graticule everywhere must return exactly one, and the Lambert cylindrical, plate carrée and Miller return 1.000000000 in all nine regions — nine numbers that could have come back at anything and came back at the only value the geometry permits.

What alignment predicts, over the whole table #

The claim being paid off is that alignment decides the sign of a gap. Over 108 pairs rather than ten, the correlation is 0.551 — a real relationship and a long way from a law.

What survives without exception is narrower than the claim as it was written, and sharper:

• Thirty-four pairs are aligned to 1.000000000, and not one of them is below zero. The smallest gap among them is exactly 0.000000. If every ellipse points the same way there is nothing to cancel, so the average cannot be rounder than its members.
• Twenty pairs are above 0.99 without being exactly one, and eleven of those are below zero — by up to 0.1036 of a degree. So the cancellation begins the moment the ellipses stop agreeing exactly, not at some threshold near agreement.
• Twenty-six pairs are below 0.6, and twenty-five of those are below zero, by up to 29.67 degrees.

The plan for this rung asserted the boundary at 0.99 and the measurement moved it to exactly one, which is the more interesting place for it to be: alignment is not a tolerance quantity in the region where it matters most.

The middle is where the correlation of 0.551 comes from, and the reason is that alignment measures whether the axes agree and not whether the magnitudes do. A region with perfectly aligned ellipses of wildly different eccentricities behaves differently from one with aligned ellipses that are all alike, and the alignment number cannot tell them apart. That is a second quantity, it is not measured here, and naming it is what this rung can honestly leave for another.

What this does to the rankings #

The previous rung found that all eight world projections change rank between the four means. This rung says when that can happen and when it cannot.

If a region’s ellipses are perfectly aligned, every mean of them has the same axes, so the four averages differ only in their sizes and the rank order is nearly stable. If they are crossed, the means differ in shape as well, and the arithmetic mean in particular is pulled towards a circle by cancellation — so a projection whose ellipses disagree looks better under the arithmetic mean than under the others, and better than it deserves.

Over the world, where the alignments run from 0.23 to 0.57 for everything that is not cylindrical, that pull is available to every candidate in different amounts, and the ranking is unstable for exactly that reason. Over Japan, where every alignment is above 0.99, the four means agree with each other and the ranking is the same under all of them.

So the stability of a distortion ranking is predictable from the alignment before the ranking is computed — which is a cheaper check than computing four rankings and comparing them, and it says the same thing.

Reading a regional distortion figure after this #

Distortion over a region established that a regional summary needs a weighting and that the weighting is an opinion. This adds a second thing every regional summary needs and never states.

Two additions to the reader’s checklist, and one link. The operation decides the coordinate system is the same argument for a different quantity: the summary that matters depends on what is being done with it.

A summary ellipse is only a summary if the ellipses agreed. Where the alignment is high, the average indicatrix is a fair picture of what the region’s ellipses look like. Where it is low, the average is rounder than any of them and reports a map as gentler than it is at every point of the region.

And the alignment is cheap. It is one number, computed from quantities the summary already has, and printing it beside the average would say whether the average means anything. No published regional distortion figure this site has seen carries it.

The rule of thumb that follows is the one the table makes obvious: an average indicatrix is worth drawing when the alignment is above about 0.9, and below that the honest summary is the average of the deformations rather than the deformation of the average — because averaging numbers cannot cancel and averaging ellipses can.

Why a single row could not have answered this #

It is worth being explicit about what went wrong the first time, because the shape of the mistake is common and cheap to avoid.

The number was computed correctly, for ten projections, over the world. Every value in that row was right. What the row could not carry is that the world is the least aligned region in the whole table — a set of ellipses covering every bearing has a resultant near zero almost regardless of the map, and eight of the ten values were therefore reporting a property of the world rather than of the projections listed beside them.

A sweep is the cure and it is not free: a hundred and eight pairs is ten times the arithmetic, over regions that had to be chosen. But the cost is arithmetic and the alternative is a conclusion, and this collection has met the same trade twice before. A rule of thumb read off thirty regions had to be scored on forty-five different ones before it was evidence about anything, and the second scoring changed the verdict on seven regions. The second derivative over a region found the winner moving in one region of four for exactly this reason.

The pattern is that a quantity measured over one region is a measurement of that region as much as of the projection, and the only way to find out which is to change the region. Nothing about the first measurement looks wrong; the row is internally consistent, the numbers agree with the pictures, and the conclusion is a third of a conclusion.

What a variance decomposition is for #

The split into three shares is the rung’s method as much as its result, and it is worth stating why that particular arithmetic answers the question.

A number quoted for a projection is an implicit claim that the projection is what it varies with. Publishing an alignment figure per projection asserts that the quantity belongs to the projection — otherwise the table’s rows would not be projections. That claim is testable and had not been tested.

The decomposition tests it by asking how much of the variation each factor accounts for. If the projection owned the quantity, its share would be most of the variation and the region’s would be small. Here the region’s share is larger than the projection’s, and a third belongs to the interaction — which is the pair rather than either member, and cannot be attributed to a row or a column of any table.

An interaction term that large is the finding. It means the quantity is not merely region-dependent in a way a second table could fix; there is no factorisation into a projection effect and a region effect, so no table indexed by one of them can carry it. The only honest presentation is per pair.

And the decomposition is cheap and general. Any quantity this collection reports per projection could be swept over regions and split the same way, and the split would say immediately whether the per-projection table is a summary or a fiction. That is a check worth running on a table before publishing it, and it costs one sweep.

What this rung establishes #

Alignment belongs to the pair. 28.4 per cent of its variation is the projection’s, 38.2 per cent the region’s, and 33.4 per cent belongs to neither — so quoting it for a projection without naming the region, which is what the previous rung’s table did, is quoting a third of a number.

It predicts the sign of the gap it was claimed to, at a correlation of 0.551 over 108 pairs — without exception at exact alignment, where all thirty-four pairs are at or above zero and the smallest is exactly zero, and loosely everywhere else, where a second quantity the alignment cannot see takes over.

And two rows of the table are not measurements. A conformal projection has no indicatrix orientation, so its alignment is arithmetic noise; a secant equal-area projection turns its major axis through ninety degrees at its standard parallel, so its alignment is one for a region on either side and as little as 0.113 for a region across it. Both were found by sweeping a number that had been reported once.