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

Georeferencing — where it appears

Recovering the coordinate system a drawn map was made in, from points whose ground positions are known. What it can recover is decided by the model's own symmetries rather than by how many points were measured.

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

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

A plane fit swallows a datum shift, and keeps swallowing it. A map of OSGB36's ground drawn as though it were on WGS84, at seven sheet sizes. The upper curve is how far the drawing moves — a real 99-metre error on the ground — and the lower one is what survives the best similarity between the two, which is what a fit for the projection removes for free. The absorption runs from 99.68 per cent on a 1° sheet to 85.16 on a 60° one, and what is left is 3.9 parts per million of the map's own size even then — two microns on a sheet half a metre across.

The datum hides inside the projection's parameters

Six essays recover a projection from control points with the body it was drawn on taken as known. It is not known, and the fit cannot find it: a 99-metre datum shift is absorbed to 99.7 per cent on a two-degree sheet and to 85 per cent on a hemisphere, leaving eleven parts per million of the map behind.

wrong · Identify Which rotation a projection cannot see is decided by the projection. The same rotation — 2.455 arcseconds, DHDN's polar one — applied about each of the three axes in turn, with the residual after the best plane fit drawn on a logarithmic scale. A cylindrical and a conic in their normal aspects hide the polar rotation to arithmetic noise, 3e+5 times better than either equatorial one, because a change of longitude is a symmetry of both. A pseudocylindrical hides none of them — its horizontal coordinate carries a factor in latitude, so a longitude shift is a shear rather than a translation. And an azimuthal centred on the equator hides the equatorial rotation instead.

A rotation is not absorbed the way a shift is

The previous rung expected a datum's rotations to be the part a plane fit could not swallow. They are the part it swallows best — 93 times better than the translations over a hemisphere, and 24 times better per metre moved — and the reason is that the rotation which matters is a change of longitude, which is a symmetry of the map.

wrong · Identify

Named alongside it

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

ConfoundingControl pointsDatumEllipsoidIdentifiabilityIdentificationResidualSimilarity fitCone constantDhdnFalse originOSGB36

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