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

Rebinning — where it appears

Moving a field from one cell scheme to another by averaging over the overlap areas, which conserves the total exactly and conserves nothing else. What it loses per cell is set mostly by how much coarser the target is and partly by how uniform the source cells are.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

The piece two schemes share, clipped rather than assumed. A cell of a gnomonic cube, whose four edges are great-circle arcs because a straight line on a gnomonic face is one, against a cell of a longitude–latitude grid, whose north and south edges are parallels and are not. Their overlap is neither a rectangle nor a spherical polygon of any standard kind, and it is 5965687 km² of the cube cell's 5965687 km² — 100.0 per cent. Computing it needs the arc of one boundary intersected with the plane of the other, which is three equations and two roots, and it is exact.

Cells that are rectangles in no coordinate

The previous rung measured what moving a field between two cell schemes costs, and did it between two schemes whose cells are longitude–latitude rectangles — which is what made every overlap a rectangle with a closed-form area. The schemes anybody actually argues about have cells that are rectangles in no coordinate, and their overlaps have to be clipped.

applied · Cells Two source geometries of 96 cells each, rebinned to the same three targets. Both curves start from a source of 96 cells and rebin to targets of 32, 128, 512 cells, so the count ratio is identical along them and the only difference is the shape of the source cells: gnomonic squares on a cube against rectangles in longitude and latitude. The ratio dominates — both curves fall by more than half across the range — and the shapes still separate by 25 points at the middle target. The cube loses less, because its cells are all much the same size and the lon/lat source's collapse towards the poles.

The same number of cells, in two shapes

Moving a field between two cell schemes loses 18 per cent of it per cell in one geometry and 39 in another, and the earlier measurement could not say whether that was the shape of the cells or the ratio of their sizes, because changing the schemes changed both. Holding the counts settles it: the count ratio decides most of the loss, and the shape is still worth a quarter of the field.

applied · Cells What a rebinning loses depends on where the target's edges are. Two grids of fixed counts, fixed shapes and fixed resolution, with the target slid across the source from perfect alignment to a full cell. Nothing about either grid changes except where its boundaries fall. The loss runs from 27.5 per cent at zero to 56.3 at half a cell — a factor of 2.04 — and the longitude-only curve returns to its starting value at a full cell to six decimal places, which is the periodicity check. A cell boundary that coincides with a target boundary loses nothing, and a grid comparison that does not say where its boundaries are has left that out.

When the edges do not line up

Rung eight held the cell counts equal so that shape could be compared without the count ratio drowning it, and recorded a doubt: a longitude–latitude source shares its boundaries with a longitude–latitude target wherever their counts share a factor. The mechanism is real and worth a factor of two. It was not what the published number was made of.

applied · Cells A parent and its children, twice. An aperture-7 hexagonal hierarchy beside a square one. The heavy outline is the parent and the light ones are its children. On the right every child is wholly inside and the four of them tile the parent exactly. On the left the child lattice is turned by 19.107° relative to the parent's, only the central child is wholly inside, and 7.14% of the parent is covered by no child of its own. The two families have exactly the same total area — a hexagon cannot be tiled by smaller hexagons at any ratio at all, which is why the mismatch is a construction rather than an approximation.

A cell's children do not fit inside it

Ten rungs price one cell system at one resolution, and every one of them is used hierarchically. A hexagonal hierarchy does not nest: at the aperture-seven scheme the discrete global grids use, one fourteenth of a parent is covered by no child of its own, exactly, and each of the six ring children is eleven twelfths inside.

applied · Cells

Named alongside it

The objects these essays reach for when they reach for this one.

AggregationCell systemAreaConservationDiscrete global gridCellClippingInterpolationResolutionSpherical polygonValidationAliasing

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