Cartographic Projection
every map is wrong, and by exactly how much
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The thread: Computed, not quoted — page 15

Scale factors, areal ratios, angular deformations, geodesic distances and the areas of countries are all computed from the projection's own formulae. None is a number recalled from a table. Essays 337 to 338 of 338.
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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