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

The thread: The trade-off is forced — page 6

Conformal means the principal scales are equal; equal-area means their product is one. Both at once forces them both to one, which is an isometry, which cannot exist. The trade-off is two lines of algebra. Essays 121 to 122 of 122.
The same count, scattered two ways. 22 dots in every cell of a 30-cell covering, drawn on Mercator. The upper panel places them uniformly on the GROUND — uniform in longitude and in the sine of latitude, which is what uniform on a sphere means — and the lower places them uniformly on the PAGE, which is what a drawing routine handed a polygon does. Both panels carry exactly the same number of dots in exactly the same regions, so both are honest as totals. They are different pictures, and a reader reads a dot map by density. Measuring distortion

A dot map's density is partly the projection's

A dot map carries the right number of dots in every region whichever way it is drawn, so it is honest as a total under both placements. It cannot be honest as a density under both: ground on a uniform field reads 0.099 of its equatorial density at 72° north on Mercator, and scattering inside the polygon on the page moves 64.3 per cent of a cell's dots into its northern half without one of them leaving the cell.

 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.

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