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

Robustness — where it appears

How little an answer moves when a few of the values it was computed from are wrong or extreme. A median is robust and a mean is not, because the median depends only on which side of it each value lies; a criterion that charges departures by their size rather than their square inherits the same indifference to outliers.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Two explanations for one residual, against the size of the region. A map in the right projection whose control coordinates are on the wrong datum, and a map fitted with the wrong projection and the right datum, both measured as a residual after the reproduction's scale, rotation and offset have been removed. The datum shift's residual is flat: it is the same 1.3e-6 at every size, because a similarity fit is very nearly what a datum shift is and it absorbs the rest. The wrong projection's grows by three orders of magnitude with the region. They are the same number below about a degree, which is where a residual stops saying anything about either of them.

A residual has more than one explanation

The method names a projection by fitting every candidate to a set of control points and taking the smallest residual. It has never been asked what else a small residual could be. A map drawn in the right projection from coordinates on the wrong datum leaves a residual of one part in a million — indistinguishable from noise, at every region size, because a similarity fit absorbs a datum shift almost exactly.

wrong · Identify Three exponents, three maps, and each is the only answer its specification has. The conformal maps of a 25° spherical square that minimise the mean of |ln k| raised to three exponents. The heavy contours are where the scale departs from true by half a per cent either way, so the ground between them is what a surveyor asking for that band would keep. At p = 1, the mean departure, the map keeps 53.7 per cent of the square inside the band; at p = 2, the mean square, 44.2; at p = 32, which is within a per cent of the least worst case, 32.7. Its worst departure falls the other way, from 3.26 per cent to 1.95 and 1.48. For every exponent from one upward the specification is convex and has one answer.

One exponent names one map, and at tight bands a better one

A specification written as a band keeps a count of area and throws the rest of the scale field away, which is why it had forty-two answers and why a tie-break could not name one. Written instead as the mean of |ln k| raised to a stated power, it is convex for every power from one up, and has one answer. At a band of a tenth of a per cent that single answer keeps more of the region than any of the band's forty-two, and the band's own search started from it beats them all. Only as the band approaches the least worst case does a large power take over.

choosing · Condition

Named alongside it

The objects these essays reach for when they reach for this one.

ConfoundingControl pointsDatumDiagnosisEvidenceFitIdentificationMinimaxObjective functionOptimal conformalOptimisationProjection

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