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

Sampling — the ladder

6 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. The areal factor over 0° to 60° north, and the points it is measured at. Mercator's areal factor shaded over 0° to 60° north, from 1.0001 to 3.8473, with the 12 × 12 grid of cell centres a regional measurement uses drawn on top of it. The cross marks where the quantity is actually largest, found by a search that is allowed to leave the sample; the ring marks the largest value the sample contains. The grid's answer is 3.4639 and the real one is 4.0000, short by 13.40% — and the reason is visible in the picture, because no cell centre is ever on an edge.

    The worst point is not on the grid

    Every maximum distortion this collection has printed is a maximum over a sample, and a maximum over a sample is a lower bound. On Mercator over a sixty-degree band a twelve-by-twelve grid reports 3.464 where the answer is exactly 4, and the shortfall does not go away with refinement so much as decay at a rate that says where the extreme is hiding.

    rung 1 · distortion
  2. One of these averages exists. Mercator's area-weighted mean areal factor, and its Kavrayskiy number, against how close the sampled band comes to the pole. The first is artanh(sin Φ)/sin Φ in closed form and has no limit: it passes 7.04 at a tenth of a degree from the pole and keeps going, gaining a fixed amount every time the remaining gap is halved. The second settles by 85° and does not move again. Both are published as summary distortion figures for the same map.

    A mean that does not exist can still be printed

    Mercator's area-weighted mean areal factor is artanh(sin Φ)/sin Φ, and it has no limit. A sampler asked for it returns the logarithm of its own sample count plus 1.512 — measured slope 1.001 against ln n — so the number is a property of the person who computed it. The Kavrayskiy number for the same map over the same sphere settles at 0.52124 and is a number.

    rung 2 · distortion
  3. Three ways to cover a sphere with 900 points. The three samplers this collection's numbers are computed with, each with about 900 points, drawn on the Mollweide projection so that equal areas on the sphere are equal areas on the page and the crowding is the samplers' rather than the map's. The graticule piles points at the poles; the equal-area rings space them evenly by area and unevenly by distance; the Fibonacci lattice trades a little of each. None of them is equal-area, and no finite set is.

    There is no equal-area lattice on a sphere

    Every number in this collection is an average over a point set, and the three point sets available all fail to be equal-area in different ways. The equal-area ring sampler this site has used since its early essays is the one whose outermost ring sits half a step inside the rim — which is how a measured scale spread once came in 3.42 parts in a thousand below a proved bound.

    rung 3 · distortion
  4. 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.

    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.

    rung 4 · distortion
  5. 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.

    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.

    rung 5 · distortion
  6. 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°.

    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°.

    rung 6 · distortion

All ladders