Six projections have two curves, and the refusal has company

The pass’s depth belongs to the projection measured the depth of the pass that decides when a near-optimal aspect set comes apart on three projections at four latitudes, and found its level set mostly by the projection. It also fitted each projection’s depth against how much of its own scale variation the region sees, and the fits were poor to middling — which raises the question behind them. For which projections does the pass depth follow the region at all, and where it does, what sets the rate?

Six projections at nine latitudes each answer the first half, and the answer changes what the second half can ask.

What a pass depth is, in one paragraph #

Choosing a projection’s aspect means choosing where its axis points, which is three numbers. The landscape the search walks on maps the objective that choice is made against and finds it neither a valley nor a basin but a set of basins separated by passes. The set of aspects that serve a region nearly as well as the best one is a single connected sheet when nearly as well is read loosely and several separated basins when it is read tightly, and where the valley breaks in two measures the threshold where it comes apart. The pass that fails first is not the one that was measured found that the pass doing the separating is the last good placement to join the best one, and the first break is mostly its denominator separated the threshold into that pass’s own depth and the best score the region admits. The pass depth is that first part: the height a walk between two near-optimal aspects has to climb before the two stop counting as one answer.

Two curves, one half-curve and three without #

Plotted against how much of its own pattern the region sees, each projection’s pass depth either follows it or does not, and the six divide cleanly.

The Hammer and Mollweide follow it. The Hammer’s depth rises as the 1.35 power of what the region sees, at $R^2$ 0.95 over nine latitudes; Mollweide’s as the 0.47 power at 0.81. Lambert’s cylindrical, measured the same way beside them, has a curve too, at 1.15 and 0.95.

Eckert IV half-follows it, at 0.34 with $R^2$ 0.52: the depth rises on average and scatters widely about the rise.

Robinson, the Winkel tripel and the sinusoidal do not follow it at all. Their fits have $R^2$ of 0.01, 0.05 and 0.15, and their exponents — 0.02, −0.06 and 0.28 — are the slopes of clouds.

Pooled over all six the fit explains nothing, $R^2$ 0.03, because the six sit at six different levels: the mean pass depth runs from 0.021 for Robinson to 0.139 for the sinusoidal. The level, as the earlier measurement found for three, is the projection’s; the rate, for most of them, is nothing.

The first version of this comparison found five projections on tight curves with exponents inside a quarter of one another, and the sinusoidal alone without one. Those five curves were made of a defect in the aspect grid — the pole rows’ copies of each map, whose meeting it recorded as a pass; the pass was a map meeting itself sets it out — and the sinusoidal was the one projection the defect hardly touched. Corrected, the picture above is what the six projections actually do.

An exponent without a curve moves with the points under it #

The earlier measurement on three projections used four latitudes, and the question it left was framed around the three exponents those four points gave. Fitting the same three through five latitudes and then nine shows what those exponents were.

A power law fitted through n points has n − 2 degrees of freedom left, so four points leave two and nine leave seven, and a four-point slope is about twice as uncertain as a nine-point one before any question of whether a curve is there. Where there is a curve, adding points refines the slope: Mollweide’s moves from 0.56 to 0.47. Where there is none, adding points moves it anywhere: the Winkel tripel’s goes from 0.38 to −0.06, changing sign, and Robinson’s wanders about zero. An exponent is a property of a curve, and a slope fitted through scatter is a property of the particular points.

That is the practical warning in this figure. A comparison of exponents across projections is only a comparison if each exponent belongs to a curve, and the only way to know is to report the fit’s quality beside it and to see whether the slope survives more points.

So there is no band. Two exponents belong to curves and they differ by a factor of three; the other four describe scatter. The question what sets each projection’s exponent has, on this evidence, two cases to explain rather than six — and a prior question, which is why four of six projections have no rate at all.

Cylindrical is a measurement, not a label #

The earlier account named two candidates for what might set a projection’s curve. The first was the one that had already caused trouble elsewhere: rotating a cylindrical projection’s aspect moves its map in a way that rotating a pseudocylindrical’s does not.

A label cannot be correlated with anything. What is needed is the quantity the label stands for, and it is available directly. A projection is cylindrical exactly when its scale depends on latitude alone, so the share of its scale field’s variance that survives removing the latitude average is zero for a cylindrical map and positive for everything else. That number is continuous, computable from the projection with no region anywhere in it, and larger the further a map’s meridians bend away from straight.

The measurement also corrects something the label hides. A pseudocylindrical projection’s parallels are straight, but its meridians are not, so its principal scale does vary along a parallel and its longitudinal share is not zero: Robinson reads 0.080, Eckert IV 0.142, the Winkel tripel 0.275, Mollweide 0.447, the Hammer 0.581 and the sinusoidal 0.606. Only a genuinely cylindrical projection reads zero. Cylinders, cones and planes argues that the taxonomy of developable surfaces says almost nothing about the properties anybody chooses on, and this is the same point in a different field: the useful version of cylindrical here is a number between zero and one.

And the number does not sort the curves. The two projections with a curve sit at 0.447 and 0.581; the sinusoidal, with none, at 0.606, between them and the top; Robinson, with none, at 0.080; and Lambert’s cylindrical, with a curve, at zero. Whatever decides whether a projection’s pass depth follows the region, it is not how far the projection is from cylindrical.

A correlation that is strong and predicts nothing #

With a number to correlate against, the exponents can be tested against it anyway, and against the second candidate — the smoothness of the scale field, made definite as the range of the field divided by its largest gradient, the length in radians over which it turns over.

Both candidates correlate with the exponents, and one of them strongly once the sinusoidal is set aside. It means very little, for a reason the previous section already gave: four of the six points are slopes of scatter, and a correlation among them is driven mostly by the Hammer, whose exponent is large and whose longitudinal share is high. A correlation fitted on six points and reported on the same six is not a claim anybody can act on. The claim worth having is that a projection’s rate can be predicted without running the aspect search at all, and that has a standard test.

All three predictors are worse than knowing nothing about the projection at all. That is the answer to the question this comparison set out to settle, and it is the same answer the first version reached — for a clearer reason. Then, the exponents were thought to cluster so tightly that there was nothing left to predict. Now, most of them are not exponents of anything, and a predictor asked to reproduce the slope of a cloud has nothing stable to aim at.

The refusal has company #

The sinusoidal’s pass depth scatters at every latitude, and how much of the projection’s pattern the region sees runs from about one per cent at the equator to ten per cent at 66°, so the explanatory variable has ample range. There is simply no relationship. That was the finding the first version of this comparison built its title on, and it survives the correction unchanged, because the defect that made the other curves hardly touched this one.

What changed is that it is no longer alone. Robinson’s pass depth is as indifferent to the region as the sinusoidal’s, and more tightly so: it moves by less than a factor of two across a tenfold range of what the region sees, where the Hammer’s moves by sixty. The Winkel tripel is the same. So the refusal is not a property of the sinusoidal’s pointed pole, or of equal-area pseudocylindricals, or of anything that marks the sinusoidal out; Robinson shares none of those and refuses too.

That turns the question round. The first version asked why one projection declines a curve the others follow. On the corrected evidence the curve is the exception: three projections of six decline it, and the ones that follow it — the Hammer, Mollweide, and Lambert’s cylindrical beside them — are an azimuthal-derived oval, an equal-area oval and a cylinder, with nothing obvious in common.

The level is the firmer quantity #

If the rate is mostly absent, the level is what the six projections actually differ in, and it can be set out with the scale-field numbers beside it:

The levels span a factor of twelve, from Robinson’s 0.021 to Lambert’s cylindrical’s 0.251, far more than any fitted rate distinguishes. That is the earlier three-projection finding — the projection sets the level — holding on seven, with a wider range.

The scale spread column is how far the projection’s own scale factor ranges over the whole sphere, and it orders the levels better than anything else here: the two largest spreads, Lambert’s cylindrical’s and the sinusoidal’s, belong to the two highest levels, and the smallest spread, the Winkel tripel’s, to the second-lowest. The middle of the order is looser — Mollweide has the third-largest spread and only the fifth-highest level — and ranked, the two columns agree with a rank correlation of 0.61. The exception is Robinson, whose spread is middling and whose level is the lowest of all, and it is a telling exception, because Robinson was designed by eye to keep its distortion from concentrating anywhere; a spread over the whole sphere does not see how evenly the distortion is laid out, and Robinson’s is laid out more evenly than any other here.

That is a lead rather than a result, and it is worth being plain about how weak a lead. A rank correlation of 0.61 or better on seven points arises by chance in one ordering in twelve — counted exactly over all 5,040 orderings — which is suggestive and nowhere near decisive, and the one projection it fails on is the one every earlier measurement on this ground used. Seven points ranked, with a named exception, is the kind of relationship that failed the leave-one-out test above when it was asked of the exponents, and it has not been asked of the levels. What it does say is that the level, unlike the rate, varies enough across projections for a predictor to have something to predict — a factor of twelve rather than a scatter about zero — and that a measure of the scale field’s relief, how unevenly it rises, is the candidate worth trying.

It also says what the choice between these projections costs a search. A level twelve times higher means ridges between good placements twelve times higher over a best score that varies far less, so the near-optimal set of Lambert’s cylindrical or the sinusoidal is broken up at tolerances where Robinson’s is still one piece. For a small region, Robinson’s good placements are nearly interchangeable and easy to move between; the sinusoidal’s are nearly as good and separated by high ground.

What each number was compared against #

The fits must be reproducible before they can be revised. The exponents at four latitudes are recomputed from the same regions, not quoted, and the comparison with nine latitudes is a comparison of fits through nested sets of the same measurements.

Each projection must fit its own curve better than the pooled fit does, or there is only one population and this comparison is about nothing. All six do, against a pooled $R^2$ of 0.03, because the six sit at different levels.

The longitudinal share must be zero for a cylindrical projection and not for the others, or it is not measuring the class it stands in for. A Lambert cylindrical gives $5 \times 10^{-23}$; the six here give 0.080 to 0.606.

A curve is called a curve at $R^2$ 0.8 or better, and the classification is not sensitive to the line: the two projections above it sit at 0.81 and 0.95, and the next below at 0.52.

And the pass is the corrected one throughout. Every depth is the highest merge in the filtration, with both of the aspect grid’s ways of holding one map twice joined before anything is counted.

What six projections cannot settle #

Six is still few. A predictor that fails leave-one-out on six points has not been shown to be useless; it has been shown to be unproved. With two genuine curves among six, the question of what sets a curve’s rate cannot even be posed properly until more projections with curves are found.

The region shape is fixed. Every region here is a square of a stated span at a stated latitude. The pooled score abandons a region is the essay about what happens when the region stops being one box, and nothing here varies the shape.

And every fit belongs to the aspect grid it was measured on. The search behind each pass depth runs over 24 pole longitudes, 13 pole latitudes and 12 rotations, and the pass was a map meeting itself is the sharpest reminder on this ground of what a grid can put into a measurement. Whether the Hammer’s and Mollweide’s curves survive a finer grid, and whether any of the other four acquires one, is not measured here.

Still open: what the projections with a curve have in common #

On the corrected evidence the pass depth follows the region for the Hammer, Mollweide and Lambert’s cylindrical, and not for Robinson, the Winkel tripel or the sinusoidal. The first version of this question asked whether the sinusoidal was the edge of the field or outside it; that framing assumed the curve was the rule, and it is not.

The question that replaces it is what the three with a curve share. It is not their distance from cylindrical, which runs from zero to 0.58 among them. It might be something about the landscape rather than the projection: a projection whose highest merge is a genuine ridge between two real placements would give a depth that follows the region, and one whose highest merge is a small pocket of poor aspects would give a depth that follows nothing. That is a count the aspect sweep can make directly, of how deep each basin lies below the pass that ends it, and it would say for each projection whether its first pass is a ridge or a pocket. If the three with a curve turn out to be the three whose first pass is a ridge, the curve was never a property of the projection at all, only of which kind of merge happens to be highest in its landscape.