Cartographic Projection
every map is wrong, and by exactly how much
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The figure library — page 2

Families 25 to 29 of 29, in the same order, each drawn at its own defaults.
A curve built to have dimension 1.2619. The generator replaces every segment with four of equal length at headings 0, +60.0°, −60.0° and 0. Closing the displacement fixes the length ratio at 0.33335, and four copies at that ratio give a dimension of exactly log 4 / log(2 + 2 cos θ) = 1.2619. Nothing here is measured yet: this is the construction the measurement will be checked against. Drawn at depth 6, which is 4096 segments, with the second-level shape shown faint beneath it.

simplify-figure

8 essays a transform boundary: what the ground is doing to itself. The same 72 places as the velocity field, with the strain rate computed from the motion's own four partial derivatives rather than from its size. Each cross carries two principal rates: the long stroke is the greater extension, the barred one is shortening. The largest second invariant anywhere here is 241.0 nanostrain/yr, and the arithmetic is the same arithmetic that reads a projection's indicatrix. Drawn in Azimuthal equidistant centred on the window.

strain-figure

5 essays Four steps between a tape and a drawing. A slope distance of 76.895 km measured at 3.2° on ground 220 m above sea level, reduced to the British National Grid, with every step drawn on a logarithmic scale in millimetres. The slope reduction is the largest by a wide margin at 119.90 m and is the one everybody applies. The grid reduction is 30.23 m. The fourth bar is not a step at all: it is what using the levelled height where the height above the ellipsoid is wanted costs, with a geoid separation of 48.5 m — 583 mm, which is 1.9% of the grid reduction it sits beside and 29 times the tolerance the job closes to. Reading a correction's importance off its share of the chain is how it gets dropped.

survey-figure

16 figures · 25 essays A band field, largest in the mid latitudes, on a map that reads it true. The stated field v = 20 + 70 exp(−((|φ| − 45°)/18°)²), shaded over cells of equal ground area on an equal-area projection, so the ink a value gets is proportional to the ground it covers. It runs from 20.9 to 78.8 and its area-weighted mean over the sphere is 47.257. Nothing has been misread yet: this is the number every other figure in this ladder is measured against, and it is a property of the field rather than of any map.

thematic-figure

4 essays How each projection escapes being one to one. Eighteen projections, and the theorem allows no fourth column. A map of the whole sphere either leaves ground undrawn, or draws one place as a curve, or is cut so that one ground curve appears twice — and 11 of the eighteen do more than one of those. The three columns are a solid angle, a count of places and a page length, which is why they are three columns rather than one score: there is no rate at which a hemisphere converts into a pole.

topology-figure

8 essays