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

Affine — where it appears

A linear map of the plane with a shift, which is a similarity plus an independent second scale and a shear. It absorbs anisotropic paper shrinkage exactly, and by the same freedom it stops being able to separate projections that differ from each other by an affine map.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

OSGB36 then DHDN, done four ways. How far each route lands from the truth — applying the two transformations one after the other — at 54.5° north, 2.0° west. Composing them into one affine map is exact, because two affine maps compose into an affine map and nothing is dropped. Describing that composition with seven parameters again leaves 0.19 millimetres, because two linearised rotations compose into something with a symmetric part that seven numbers cannot hold. Adding the fourteen published numbers in pairs — which is what a chain is usually done with — leaves 7.4 millimetres. Logarithmic from a micrometre.

A chain of transformations does not close

Two datum transformations applied one after the other are not the sum of their fourteen published numbers. The gap is seven millimetres in Britain, doing the two in the other order moves the answer twenty, and the rotation matrix everybody prints is not a rotation.

datums · Datum What a sheet does to the points before anybody measures them. The graticule crossings of a map drawn on Conformal conic, with an arrow at each one showing where the same crossing has moved to after the sheet dried — 0.1 per cent along the grain and 0.4 across it, with the grain at 23° to the map's axis, and the displacement magnified 60 times so it can be seen at all. The pattern is a stretch along one direction and a squeeze along the perpendicular, which is what an anisotropic scaling looks like. It is a property of the paper and has nothing to do with the map printed on it. A similarity fit to these points leaves 3.66e-4 of the map's own width unexplained, against 1.22e-10 on the unshrunk sheet.

The sheet moved before it was measured

Nine rungs take control points off a map and assume the sheet they came from is the sheet the cartographer drew. Paper shrinks across its grain three times as fast as along it, and on a map whose grain runs along its own axis that shrinkage is EXACTLY a change of standard parallel — one per cent moves the recovered parallel by 0.57 degrees with the residual sitting at the solver's floor.

wrong · Identify The whole of what an unlabelled map gives you. The outline of Japan as drawn on Conformal conic, delivered as an ordered list of page positions with nothing attached to any of them. No latitude, no longitude, no scale, no north. The rung's question is whether a projection can be recovered from that, and it can: the correspondence between the ink and the ground is found by sweeping the starting point round the curve and both directions, and the true candidate comes back with a residual of 2.88e-14 against the runner-up's 4.57e-4.

A map with no graticule

Ten rungs are handed control points, and a great many maps have none. Handed an outline with no labels on it at all, the method still works — and works better: the correspondence between ink and ground is recoverable exactly, because a similarity preserves ratios of arc length, and the margin on clean observations is 1.6 × 10¹⁰ against a graticule's 9.9 × 10⁶. What breaks it is noise, at three parts in a thousand.

wrong · Identify

Named alongside it

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

Control pointsDegeneracyEstimatorIdentificationMarginReproductionResidualScale factorSimilarity transformationToleranceVerificationClosed form

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