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

Three criteria, three maps, and the tolerance band each one encloses

Drawn here at the parameters it defaults to, with every essay that calls it.
Three criteria, three maps, and the tolerance band each one encloses. A spherical square of 25° circumradius under three conformal maps, with the two contours at ±0.5% of true scale drawn heavy and the region between them the area each map keeps inside that tolerance. Chebyshev's map holds 36.5 per cent of the area there, the mean-square map 45.4, and the map fitted to the tolerance itself 60.1. The faint lines are further multiples of the tolerance, and they say where each map spends what it is not counting: the third map's outermost contours are much further out than the other two's.

A spherical square of 25° circumradius under three conformal maps, with the two contours at ±0.5% of true scale drawn heavy and the region between them the area each map keeps inside that tolerance. Chebyshev's map holds 36.5 per cent of the area there, the mean-square map 45.4, and the map fitted to the tolerance itself 60.1. The faint lines are further multiples of the tolerance, and they say where each map spends what it is not counting: the third map's outermost contours are much further out than the other two's.

It is one of 4 figures drawn by the same construction, and the drawing above is this one with nothing changed.

2 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.

Three criteria, three maps, and the tolerance band each one encloses. A spherical square of 25° circumradius under three conformal maps, with the two contours at ±0.5% of true scale drawn heavy and the region between them the area each map keeps inside that tolerance. Chebyshev's map holds 36.5 per cent of the area there, the mean-square map 45.4, and the map fitted to the tolerance itself 60.1. The faint lines are further multiples of the tolerance, and they say where each map spends what it is not counting: the third map's outermost contours are much further out than the other two's. What each projection optimises

The map that keeps the most ground inside a tolerance

A surveyor does not want the smallest worst case or the smallest mean square. They want as much of the territory as possible within a stated tolerance of true scale and are indifferent to how far outside the rest goes. That is a third criterion, it is answered by a third map, and on a 25° square at half a per cent it keeps 60.1 per cent of the area where the mean-square map keeps 45.4 and Chebyshev's 36.5 — bought by sending a tail out to 5.7 per cent where Chebyshev's worst is 3.0.

 Forty-two starts, forty-two answers. Every local optimum coordinate ascent reaches from 42 starts on the same region and the same criterion — a 25° spherical square at a tolerance of ±0.5% — plotted at the share of area it keeps. They run from 36.70 per cent to 65.15, a spread of 28.4 points, and no two of them are the same map: clustered on their scale fields rather than their coefficients, 42 of the 42 are distinct. The two marked starts are the ones anybody would actually use — Chebyshev's map reaches 37.43 per cent and the mean-square map 60.14, and neither is the best this search found. What each projection optimises

A criterion worth using is one whose answer is not unique

Forty-two starts on one region and one tolerance give forty-two different maps, keeping from 36.70 per cent of the area to 65.15. The two starts anybody would actually use — Chebyshev's map and the mean-square map — end 22.7 points apart, which is more than the criterion buys over either of them. And the spread is widest at exactly the tolerance where the criterion is worth using at all, because both facts are the same fact: an objective with a flat top and a sharp edge has somewhere to hide answers.

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