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

Diagnosis — where it appears

Deciding which of several explanations produced an observed misfit, rather than only how large the misfit is. A residual's size rarely separates the candidates; its pattern — where it is zero, where it grows, and what it is proportional to — usually does.

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

Also named here as evidence — the same set of essays touches all of them, so they are one junction rather than several.

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 A coast that agrees with its graticule, except where it was copied. The outline of Japan on a map whose graticule is a conformal conic projection. 30 per cent of the coast, drawn in the second ink, was copied from a sinusoidal sheet of the same ground and fitted into place so that its two ends meet the coast either side, the dots. The dashed line is the coast the graticule predicts. Everywhere else the drawn coast lies on it exactly; along the copied stretch it departs by up to 9.4 thousandths of the map's width. Nothing in the graticule shows it.

A compiled map agrees with its graticule except where it was copied

A map compiled from two sheets has no single projection behind it, and its graticule cannot say so. Its coast can — not as a residual, which is a number the size of a projection error, but as a profile: the drawn coast lies exactly on the one its graticule predicts, then departs from a seam, and returns at another. The seams are found to within one point in 720, and the stretch between them names the projection it was copied from with a margin of 3.6 × 10¹¹.

wrong · Identify

Named alongside it

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

ConfoundingEvidenceIdentificationResidualSimilarity transformationControl pointsDatumFitGraticuleProjectionProjection identificationRobustness

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