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

Confounding — where it appears

Two variables changing together in an experiment, so that the effect measured cannot be attributed to either. It is the reason a comparison between two cell geometries has to hold the cell counts, and the reason a family of cuts has to hold the surface area.

Named by 6 essays across 4 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 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 Seven cuts of the same size, in different places on one body. The body's own colatitude and longitude, with the seven windows drawn on it. Each covers the same surface area to 0.24 per cent — the longitude extent is divided by sin θ and a scale is then solved per window — so their sizes are held and only their positions differ. Each is shaded by the areal spread of the conformal map solved on it, listed beside the grid and running from 1.03 to 1.36. The darker windows are the worse ones, and they are the ones over the body's lobes.

Cuts of the same size in different places

Where a body is cut decides how well it can be mapped, by a factor of ten — established with five windows of five different sizes, so *where* and *how much* were confounded and the factor could have been entirely about extent. Held to the same surface area to a quarter of a per cent, the answer survives at a factor of 1.32, and what predicts it is the curvature the window encloses.

datums · Bodies Two source geometries of 96 cells each, rebinned to the same three targets. Both curves start from a source of 96 cells and rebin to targets of 32, 128, 512 cells, so the count ratio is identical along them and the only difference is the shape of the source cells: gnomonic squares on a cube against rectangles in longitude and latitude. The ratio dominates — both curves fall by more than half across the range — and the shapes still separate by 25 points at the middle target. The cube loses less, because its cells are all much the same size and the lon/lat source's collapse towards the poles.

The same number of cells, in two shapes

Moving a field between two cell schemes loses 18 per cent of it per cell in one geometry and 39 in another, and the earlier measurement could not say whether that was the shape of the cells or the ratio of their sizes, because changing the schemes changed both. Holding the counts settles it: the count ratio decides most of the loss, and the shape is still worth a quarter of the field.

applied · Cells How aligned a region's ellipses are, for every projection and every region. The resultant length of the doubled indicatrix orientations, over 12 projections and 9 regions. One means every ellipse in the region points the same way; zero means they are spread evenly and cancel. Three cylindrical rows are 1.00 throughout, and everything else varies down the row and across it — which is the answer to the question the number was first stated without: alignment is a property of the pair. Mercator's row is the exception that is not a measurement, because a conformal projection's indicatrix is a circle and a circle has no orientation.

Whether the ellipses point the same way

The previous rung found that alignment decides whether the average of a region's deformations is above or below the deformation of its average, and stated it at ten projections over one region. Swept over a hundred and eight pairs, the answer is that alignment belongs to the projection 28 per cent, to the region 38, and to neither 33 — and two rows of the table turn out not to be measurements at all.

distortion · Tissot

Named alongside it

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

Control pointsDatumIdentificationResidualConformalityEllipsoidGeoreferencingIdentifiabilitySimilarity fitAggregationAngular deformationAnisotropy

All concepts