Cartographic Projection
every map is wrong, and by exactly how much
Theme

The thread: Computed, not quoted — page 13

Scale factors, areal ratios, angular deformations, geodesic distances and the areas of countries are all computed while the figure is drawn. None is a number recalled from a table. Essays 289 to 299 of 299.
The doubling ladder is the one sequence that cannot see it. The largest departure of a small circle from its own indicatrix that a grid of n latitudes over 10° to 70° north finds on the Robinson projection. The filled marks are 4, 8, 16, 32, 64 and 128 — the doubling ladder every convergence study runs — and they rise smoothly to about 4.66e-4 with the increments halving, which is what a convergent first-order sequence looks like. The open marks are grids whose samples land on the projection's five-degree table entries. They report 1.28e-2, twenty-seven times higher, and whether a grid does that is decided by whether n is a multiple of four. Measuring distortion

A refinement that stops moving

Doubling the sample and watching the answer settle is how every quadrature in every field is checked. On the Robinson projection the doubling ladder — 4, 8, 16, 32, 64, 128 — converges beautifully, with its increments halving at every step, on a limit that is wrong by a factor of twenty-seven. Whether a grid finds the answer is decided by whether n is a multiple of four.

 Which of this collection's numbers are the sampler's. How much each of two published quantities moves when the sampler behind it goes from twenty samples a side to sixty, for eight projections over the whole sphere. The Kavrayskiy numbers — the summary means the rankings are built from — move by at most 1.15 per cent. The worst-point angular deformations move by up to 19.0 per cent, all in the same direction, because they are maxima over a sample and a maximum over a sample is a lower bound. The split is clean, and it says which numbers here need refining and which do not. Measuring distortion

Which of these numbers are the sampler's

Four rungs have shown that a sampled maximum understates, a sampled mean can be a report on the sampler, no arrangement of points is neutral, and refining until the answer settles proves nothing. So the collection re-measured itself. The means move by at most 1.1 per cent, the worst points by up to 19, and the rankings — which is what the essays actually argue with — do not move at all.

 One number changed, and the whole map moved. Two cartograms of the same density differing in one bump's weight — London's, raised from 6 to 9 — with an arrow at each sample showing how far the second puts that place from where the first does. Every other density in the specification is identical. The largest movement is 0.255 of the rectangle's half-height and it is not at London: it is about forty-five degrees away, and places on the far side of the world move by a tenth of it. Measuring distortion

One number changed and the whole map moved

Raise one bump's weight in a density specification, leave every other number identical, and solve again. London's own value is what changed; London moves 0.054 and Delhi moves 0.226 — four times as far, with its own number untouched. The largest displacement anywhere is thirty degrees from the change, and the antipodal band still moves a fifth of the peak.

 Two special lines, and they are not the same line. Each projection's true-scale parallels, found by solving for a parallel scale of exactly one, and the parallels along which its second-order failure vanishes, found by minimising the flexion. Five of eleven put them more than five degrees apart, and Gall–Peters puts them forty-five. The pattern is tangency: a projection with one standard parallel has both lines there, and a secant one has its true-scale lines moved off the centre while the bending zero stays. Measuring distortion

The lines where the bending vanishes

Every projection has a line printed in its margin — the standard parallel, where the scale is exactly one. It has a second special line nobody prints: the one along which a geodesic is drawn straight to second order. On Gall–Peters they are forty-five degrees apart, and the rule turns out to be tangency — a tangent construction puts both lines at its point of contact and a secant one moves only the first.

 Densifying helps one and hurts the other. The standard deviation of the shoelace area, and the bias of the perimeter, against the number of vertices on the same 100-metre circle with the same 5-centimetre noise on each. The area's spread falls from 9.9 m² to 2.0 — the closed form says it goes as the square root of the vertex count's reciprocal, because a vertex's influence on the area is the vector between its two neighbours and densifying shortens it. The perimeter's bias rises from 0.07 mm to 261, a factor of 3697, because every leg contributes its own σ²/d and shorter legs contribute more. Measuring distortion

The area is unbiased and the perimeter is not

A boundary measured from noisy vertices comes out long, always, by σ²/d on every leg. The area enclosed by the same vertices comes out exactly right, because a shoelace is bilinear and the cross terms vanish. So densifying a boundary makes its area five times more precise and its perimeter three thousand times more wrong, and every compactness score computed from it falls short.

 Two maps with the same ellipses. Mollweide, and the same projection with the sheet turned 37° on the desk. Every ellipse is the same size and the same shape as its counterpart, and its angle to its own graticule is the same — the two indicatrix fields agree to 1.8e-12, which is arithmetic noise. The rotation differs by 37° at every point. So the classical description of a map's distortion is exactly blind to the difference between these two, and the difference is the whole of what a surveyor calls convergence. Measuring distortion

The fourth number the ellipse does not carry

A Jacobian has four independent entries and an indicatrix reports three. The missing one is the rigid rotation in the polar decomposition A = R·S, the indicatrix is exactly S, and R is what turns north into grid north: for the transverse Mercator it agrees with the survey formula for convergence to 5 × 10⁻¹⁰ degrees, from a different library and a different derivative.

 One map, two samples, two answers. The same Miller cylindrical projection carrying two sets of sample points of the same size. On the left the points are uniform over the SPHERE — equal ground area between them, which is what every mean in this collection integrates against. On the right they are uniform over the PAGE, which is what a raster, a pixel loop or any figure that walks its own canvas produces. The right-hand set crowds where the map stretches, and the mean angular deformation it returns is 18.0° against the left-hand set's 7.2°. Measuring distortion

The sample was drawn on the page

Every mean in this collection integrates over the sphere, because that is where the ground is. A raster, a pixel loop and any figure that walks its own canvas integrate over the page instead, and the difference is exactly the covariance between the quantity being measured and the map's own area distortion — 7.2° of mean angular deformation on Miller becoming 18.0°.

 What a reprojection does, drawn where it does it. The angular deformation of the map that takes Sinusoidal's page to Gall–Peters's — which is what a reprojection is — shaded over the sphere and drawn on Mollweide so that equal ground areas are equal page areas. Both parents are equal-area, so the composite preserves area exactly; its mean angular deformation is 55.5° against 38.9° and 32.2° for the two maps separately. Composing two maps did not average their shape errors; it made a larger one. Measuring distortion

Two indicatrices do not make a third

Reprojecting is composing, and the composite's indicatrix is not a function of its parents'. Mollweide followed by Hammer is gentler than either map over eighty per cent of the sphere; the sinusoidal followed by Gall–Peters is worse than both everywhere. What separates them is one angle, and it is the number rung ten showed the ellipse does not carry.

 Six regions, before and after. Six circular regions of unequal size on the equal-area rectangle, and the same six after the cartogram of four cities has been solved. Each drawn area is exactly its base area times the density it was asked for. The four that grow stay recognisably round; the two that shrink are drawn out into shapes that no longer resemble what they were, and nothing in the construction chose to treat them differently. Measuring distortion

A cartogram keeps the shapes it inflates

Seven rungs build cartograms and none reads one back. Reading means recognising a region and dividing its drawn area by its base area, and the construction is against the reader twice: the correlation between how much a region grows and how much of its shape it keeps is −0.996, and the base area a reader has to divide by is the map the cartogram replaced.

 One current, two maps, and only one of them conserves it. A flow that has no sources or sinks anywhere on the sphere — it is the perpendicular of a stream function's gradient, so its divergence is zero by construction — drawn on two projections. The arrows on the right are the images of the same ground velocities as the ones on the left. On the equal-area map the drawn field is still divergence-free; on the other it is not, and the arrows drawn in the warning colour are where a reader measuring the picture would find a source or a sink that is not there. Measuring distortion

A current drawn on a page has sources

Seven rungs project a scalar field and ask what the page does to its gradient. A wind or a current is the other half of what gets mapped, and the operator that matters for it is the divergence — which is preserved by an equal-area map exactly, by no other map at all, and by a conformal map least of anybody's expectation.

 Two places, one written coordinate. The lattice a coordinate written to 5 decimal places lives on, at 45° of latitude, where a cell is 1.11 metres north to south. Two real places a fifth of a cell apart are drawn as open marks and the single value they both round to as a filled one. The distance between them, computed from what was written, is exactly zero — an error a noise model cannot produce, since independent noise never puts two different points in the same place. Measuring distortion

Rounding is not noise

Eight rungs treat a coordinate's error as noise that averages down. A published coordinate has a second error that does not: it is deterministic, it is shared between every point in the same cell, and at five centimetres apart ninety-three per cent of pairs come out as the same place — which no amount of independent noise can produce.

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