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

Tangent-warped cube cells, shaded by area

Drawn here at the parameters it defaults to, with every essay that calls it.
Tangent-warped cube cells, shaded by area. The tangent-warped cube at level 2, drawn on Mollweide with each cell shaded by its own measured area. The largest cell is 1.21 times the smallest. Every area is computed with the spherical polygon formula from the cell's own boundary, not from the scheme's intentions, and the shading is what the numbers say rather than what the mesh looks like.

The tangent-warped cube at level 2, drawn on Mollweide with each cell shaded by its own measured area. The largest cell is 1.21 times the smallest. Every area is computed with the spherical polygon formula from the cell's own boundary, not from the scheme's intentions, and the shading is what the numbers say rather than what the mesh looks like.

It is drawn by cell-figure with show: "cell-scheme" — one member of a family of 5 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

10 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

The same address length, a tenth of the area. Every cell of a lon/lat quadtree at level 4 carries an identifier of the same length. The heavy curve is each cell's area as a fraction of the largest, against its latitude: a polar cell is 10.2 times smaller than an equatorial one. The light curve is the inverse of the cell's aspect ratio, which falls from 1.00 near the equator to 0.10 at the top — the cells stop being anything like square long before they stop being usable. What a machine does with it

An address is an area

A cell identifier does not name a place, it names a region — so its precision is an area rather than a length. On the obvious lon/lat scheme that area varies by a factor of 10 at level 4 and 163 at level 8, and the factor doubles with every level: the same identifier length means less ground the further north it is used.

 A hexagonal tiling of the sphere, and its pentagons. 362 cells — 350 hexagons and 12 pentagons, the pentagons marked — drawn on Orthographic. The twelve are not a defect of the construction and cannot be removed by subdividing further: Euler's formula requires exactly twelve however many hexagons there are. Each pentagon here has 0.52 times the area of an average hexagon, so a count aggregated over these cells has twelve entries that mean something different from all the others. What a machine does with it

Hexagons cannot tile the sphere

Hexagons are the best cell shape a plane offers and the sphere will not take them. Euler's formula forces exactly twelve pentagons into any such tiling — twelve at 42 cells and twelve at 642, while the hexagon count rises twenty-one-fold — and each of the twelve is measurably smaller than the hexagons around it.

 Equal area or steady shape, and not both. Four cell schemes plotted by how much their cells vary in area and how far from square the worst of them is. The bottom-left corner is the scheme that has both, and it is empty: the equal-area cube holds area to 1.003 and has the most elongated cells, the tangent-warped cube has the tightest shapes and lets area vary by 1.20, and the lon/lat scheme is off the scale on both. Neither axis can be driven to one while the other stays there. What a machine does with it

A cell system trades area for shape

A grid can hold every cell to exactly the same area or hold every cell nearly square, and the measurement says it cannot do both: the equal-area cube's areas agree to a part in a thousand and its worst cell is 1.29 times as long as it is wide, while the tangent-warped cube holds shape to 1.19 and lets area vary by 20 per cent.

 A 6° query against a cube scheme, and the cells it fetches. The cells of a tangent-warped cube scheme at level 5, with the 29 cells a query of 6° radius touches shaded. The disc's own area is 16.84 cells; the count is 29, because every cell the disc's boundary crosses is fetched as well as every cell inside it. In Hilbert order those cells form six contiguous ranges of identifiers, which is six range scans, and the span from the lowest to the highest covers 91 cells against the 29 wanted. Drawn in Mollweide, with the mesh shown only near the query. What a machine does with it

A query is a disc, and a disc is not a cell

Everything a cell system does is an address lookup except the one question anybody actually asks it: find everything within five kilometres of here. That is a disc, and the number of cells it fetches is not its area divided by a cell's — at the radii a query is really made at, it is three to seventeen times that.

 One field, one round trip between two cell schemes. Left: a stated field binned into an equal-angle grid of 36 by 18 cells. Right: the same field after being rebinned into an equal-area grid of 30 by 15 offset by six degrees of longitude, and rebinned back. Every step is exact area-weighted averaging, the total is preserved to 2 × 10⁻¹⁶, and the root-mean-square difference between the two pictures is 0.144 on a field whose own standard deviation is 0.370. What a machine does with it

The same data on two grids

Five essays have addressed, queried and ordered cells within one scheme and nobody has moved a number between two. Doing it exactly — area-weighted, both directions — preserves the total to 2 × 10⁻¹⁶ and loses 39 per cent of the field's own standard deviation in a single round trip; six round trips leave 23 per cent of its variance. The quantity that would reveal the damage is the one that never moves.

 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. What a machine does with it

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.

 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. What a machine does with it

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.

 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. What a machine does with it

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.

 The same resolution, the grid moved, and a different answer. The same field at a fixed 18 × 9 division, with the grid slid by fractions of a cell. Nothing is lost — every cell is the same size as before and there are exactly as many of them — and the largest reported value moves over a range of 12.59 per cent. That is more than a whole halving of the resolution costs, which is 10.15 per cent on the same field. The scale effect has an excuse and this one has none. What a machine does with it

The answer depends on the cells it was counted in

Nine essays price the cell as a shape. The number reported out of it is priced nowhere: sliding a grid without changing its resolution moves the largest reported value by 12.6 per cent, which is more than halving the resolution costs.

 The same tiling, turned. The cube's eight vertices, ringed, and eight cities, with the aspect of the cell each city falls in written beside it. The two panels are the same tiling: the same cells, the same areas, the same shapes, in the same numbers. Only where they sit has changed, and the eight cities get cells whose areas spread by 1.26 in one and 1.39 in the other. Drawn on Mollweide, in which equal ground areas are equal page areas. What a machine does with it

The orientation is a policy

A polyhedral cell system has three free angles nobody scores. They cannot improve it: rotating the solid rotates every cell rigidly, so the distribution of cell areas is identical for every orientation there is. What they decide is who stands on the bad cells — and the eight cities measured here get a spread of cell area of 1.00 under the best turn and 1.50 under the worst.

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