The sinusoidal is the edge of its family, not outside it

Six projections have one band and one refusal measured, for six world maps, how the depth of the pass that decides when a near-optimal set of aspects comes apart grows with how much of the projection’s own variation a region sees. Five of them followed a power law closely, with exponents inside a factor of 1.26. The sinusoidal did not: its pass depth was flat across a ninefold range of what the region sees, a fitted exponent of 0.26 at an $R^2$ of 0.15.

Two readings were left standing. Either the sinusoidal is one projection behaving oddly, and the task is to find what is odd about it; or there is a class of projections whose aspect sets fracture at a depth that does not depend on the region, and the band of five is the special case. The two candidates for the odd thing — how much of the scale field varies with longitude, and how gently the field turns — were tested and failed: leave a projection out and predict its exponent from either, and the prediction is worse than guessing the average.

The cheapest test that distinguishes the readings is a family. The sinusoidal is the end member of a one-parameter family of ordinary equal-area maps, and running the same measurement along the parameter asks whether the curve fades gradually, vanishes at a point, or holds until the last member.

What the pass depth is #

The quantity has a history on this ground and it is worth restating, because the family’s result is about its behaviour rather than its value. Choosing a projection’s aspect for a region — where to put its pole and how to turn it — is a search over three angles, and the score being minimised is a distortion statistic over the region. The aspects that score within a stated tolerance of the best form a set, and where the valley breaks in two found that set to be one connected sheet at a loose tolerance and many separate basins at a tight one. The tolerance at which it first comes apart is a property of the region and the projection together.

The height of the pass between two basins measured that as the height of the saddle between the two deepest basins, and the pass that fails first is not the one that was measured found that the saddle that actually decides the fracture is usually a different one, between two shallow basins far from the optimum, standing well above the deep pair’s. Its depth is the pass depth here: how far above the best score the first pass to split the set stands. For the five named projections it grows as a power of how much of the projection’s own scale variation the region sees, and the power is what the comparison called the exponent. The basins have widths as well as depths is the reminder that a depth is one number about a landscape with more in it.

A family from a cylinder to a sinusoid #

The family is built on an auxiliary angle, the device Karlheinz Wagner used from 1932 onwards to derive new pseudocylindricals from old ones. Each member replaces latitude φ by an angle θ with sin θ = m sin φ, then draws the parallel of latitude φ as a straight line at height θ / m with length proportional to cos θ. The area of a thin band between two parallels is then the band’s area on the sphere, so every member is equal-area. At m = 1, θ is φ and the map is the sinusoidal exactly — to the last bit of the arithmetic, which is the first control. As m shrinks, θ approaches m sin φ, the height of a parallel approaches sin φ and its length approaches the equator’s: the map tends to Lambert’s cylindrical equal-area.

In between are ordinary flat-polar pseudocylindricals. The member at $m = \sqrt{3}/2$ is Wagner I, also called Kavrayskiy VI, up to an area-preserving stretch. Their poles are lines whose length is $\sqrt{1 - m^2}$ of the equator’s, so the pole shrinks slowly at first and then fast: 80 per cent of the equator at m = 0.6, 44 per cent at 0.9, 14 per cent at 0.99, and a point at 1.

A family chosen this way is unusual in one respect. Any two equal-area maps are related by an area-preserving deformation of the sheet — every equal-area map is every other one is the construction — so the members here are one map slid continuously into another, and nothing about the property they share changes along the way. What changes is only how the area is traded for shape. That makes the family a clean instrument: a change in the pass-depth curve along it is a change in the trade, not in what the maps preserve. It also crosses the boundary the textbook taxonomy draws, since its first member is a cylindrical and the rest are pseudocylindricals, and cylinders, cones and planes is the case that the boundary says little about what a projection does.

Every member goes through exactly the measurement the six named projections went through: nine latitudes for the region’s centre, four region sizes at each, an aspect search on a grid of 24 pole longitudes, 12 pole latitudes and 12 rotations, the depth of the first pass that splits the near-optimal set, and a power law fitted through the nine points.

The curve holds until the last twentieth #

From the cylindrical end to m = 0.95, every member has a curve: $R^2$ of 0.973 or more, and 0.996 or more below m = 0.9. The pass depth rises with how much of the field the region sees, as it did for the five named projections, and the power law fits it about as tightly as it fitted them.

Then the curve comes apart. At m = 0.97 the points still rise but scatter, $R^2$ 0.753. At 0.98, 0.670. At 0.99, 0.289, and at 0.995 and 1 the fit is no better than a flat line.

So the answer to the question the earlier comparison left is neither of the two shapes it named. The refusal does not vanish at a point, since the members at 0.99 and 0.995 refuse as firmly as the sinusoidal. It does not fade gradually across the family either: the whole of the change happens between m = 0.95 and 0.99, a twenty-fifth of the parameter’s range, after nine tenths of the family has behaved like the band. The sinusoidal is the last member of a short stretch of projections without a curve. It is the edge of the family’s behaviour rather than something outside it, and the stretch it ends is narrow enough that a comparison choosing projections for variety would almost never land a second member in it.

The exponent falls the whole way #

The exponent does not wait for the fit to fail. It falls from 1.17 at the cylindrical end, slowly at first — 1.16 at m = 0.4 — then faster: 1.10 at 0.6, 0.94 at 0.8, 0.81 at 0.9 and 0.74 at 0.95. The five named projections of the earlier comparison sit between 0.92 and 1.16, and the family runs through their band between about 0.4 and 0.8 and out below it.

That changes what the band was. Six projections chosen for variety landed five exponents inside a factor of 1.26, and the natural reading was that exponents cluster. One family, varied continuously, covers a factor of 1.6 among members whose curves are all good. The band of five was a property of the five projections chosen, not a limit on what the measurement can return. The constant belongs to the projection, not to the problem found the constant in front of each curve to be a fact about the projection; the family says the exponent is too, and that it can be moved at will by moving along a family.

The members at 0.97 and 0.98, where the fit is breaking, carry exponents of 0.44 and 0.40. They are not measurements of a curve that exists. They are the slope of the least-squares line through points that no longer lie on one, and they drift towards the sinusoidal’s 0.26 because the points are drifting towards flat.

Near the end the depth follows the search, not the region #

The raw points say how the curve fails. At m = 0.9 the mean pass depth rises steadily from the lowest latitude to the highest: a region centred further from the equator sees more of the projection’s variation and the near-optimal set holds together longer. At the sinusoidal the depth jumps between neighbouring latitudes — high at 8°, low at 16°, high again at 23° — with no trend beneath the jumps.

The centres fall into two interleaved sets, each spaced about fifteen degrees apart, and fifteen degrees is the spacing of the aspect search’s pole latitudes. A depth that alternates with the grid’s own period is a depth set by where the region’s best aspects fall against the searched ones, not by the region. That is the second reading of the refusal made visible: at the end of the family the pass that fails first is decided by something other than what the region sees, and the thing deciding it is close to the scale of the search.

The grid is part of the measurement #

A fit whose points alternate with the grid invites the obvious test: search on a finer grid and see whether the curve comes back.

It does not come back. On the grid the comparison used the sinusoidal’s $R^2$ is 0.150. With the rotations doubled it is 0.092; with the grid doubled in every direction, 0.036, and the alternation is gone, leaving a depth flat between 0.11 and 0.16 at every latitude. On one intermediate grid, 36 by 18 by 12, it reaches 0.626, a partial curve. The member at m = 0.9 has a curve on the grid the comparison used and on the doubled one, $R^2$ of 0.996 and 0.997.

So the refusal survives resolution: on three of four grids the sinusoidal’s pass depth does not follow the region, and the finest grid makes it flatter rather than steeper. But the instability of the fit at the edge is itself a finding. A member well inside the family gives the same verdict on every grid tried; the sinusoidal gives verdicts from 0.04 to 0.63. That is what an edge looks like in a measurement: the quantity being fitted is small and the search’s own granularity is large beside it.

The doubled grid also moves the exponent of a member that has a curve: at m = 0.9 from 0.81 to 0.99, nearly a quarter. The presence of a curve is robust to the grid and its slope is not. Every pass-depth exponent measured on this ground, including the five in the named band, is a statement about the aspect grid it was measured on, and exponents measured on different grids cannot be compared.

Nothing about the scale field marks the edge #

The earlier comparison’s two candidates can be tested again along the family, where they vary continuously. The share of the scale field’s variance that belongs to longitude rises from zero at the cylindrical end to 0.417 at m = 0.9, 0.552 at 0.95 and 0.606 at the sinusoidal. The field’s turning length — its range divided by its steepest gradient — rises from 0.364 to 0.538 and 0.760 radians over the same members. The longitude share does most of its rising before the curve fails, between m = 0.8 and 0.95; the turning length changes fastest exactly where it fails.

Neither can be the cause, and the named projections say why. The Hammer’s longitude share is 0.581, within a hundredth of the members at 0.97 and 0.98, and its turning length is 0.821, longer than the sinusoidal’s; the Mollweide’s are 0.447 and 0.763. Both have good curves — the Hammer’s is the tightest of the six. A quantity whose value at the family’s edge is matched or exceeded by projections that behave like the band is not what marks the edge. The pole’s length is the third candidate the family suggests, and it fails the same way: the Mollweide and the Hammer have pointed poles, like the sinusoidal, and curves.

Where a pseudocylindrical puts its error found that the sinusoidal carries the largest angular deformation at high latitudes of any equal-area pseudocylindrical in the library, because its meridians arrive at a point pole at their full curvature. That is a property its near neighbours in the family share almost as strongly, and it may be the right kind of candidate: something about the shape of the deformation near the poles rather than about the field’s statistics over the whole map. What it would need to predict is a change concentrated in the last twentieth of the family, and it has not been measured in a form that could.

What each number was compared against #

The end members must be the named maps. The member at m = 1 is the sinusoidal to the last bit of the arithmetic, and its curve must be the sinusoidal’s own: exponent and $R^2$ identical to twelve decimal places. The member at m = 0 must give Lambert’s cylindrical’s curve, and does to within 0.02 in the exponent.

Every member must be equal-area. The areal scale must be one at every sampled point, and departs from it by less than 10⁻⁸ for every member tested.

The change must be real at the grid used. Members at m ≤ 0.9 must each have $R^2$ above 0.95, and members at m ≥ 0.99 each below 0.35; the measurement would have no edge to locate otherwise.

The verdict at the edge must be tested at more than one resolution. Four grids for the sinusoidal and two for a member well inside the family, reported in full rather than summarised.

Where the family stops #

One family. A different family ending at the sinusoidal — one keeping the pole pointed and varying the spacing of the parallels, say — might put its edge somewhere else, or nowhere. The result is that along this family the change is confined to its last twentieth.

One grid throughout. Every curve along the family is measured on the grid the earlier comparison used, so that its numbers sit beside those. The grid controls say the verdict at the edge is stable in kind on three of four grids and that exponents are not stable in size.

Nine latitudes and four sizes. The same design as the six-projection comparison. More sizes would test the power law more severely; they would not move a curve’s presence, which is what the family is about.

Still open: what the last twentieth does to the aspect search #

The family confines the change to m between 0.95 and 0.99 and shows, in the raw points, that inside that stretch the pass depth stops following the region and starts following the search. That points at the aspect search’s landscape rather than at the projection’s statistics. Near the sinusoidal the near-optimal aspects may form many shallow basins separated by passes of nearly equal depth, so that which pass fails first is decided by small differences the grid resolves unevenly; further in, the landscape may have a few deep basins whose separation grows with the region in the way the power law describes.

The measurement that would settle it is a count rather than a depth: how many basins the near-optimal set holds at each member and each region, and how that count changes along the family. A count that stays constant as the region grows would produce a flat pass depth by itself, which is the possibility the earlier comparison named and could not test. Whether the basin count jumps where the curve fails, and whether it does so at the same m on every grid, are questions a depth alone cannot ask.