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

Common point — where it appears

A place whose coordinates are published in two datums, and therefore one of the points a transformation between them is fitted to. Adding more of them reduces the transformation's own error and not the network distortion that dominates the residual, so they are worth far less than a residual table suggests.

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

What each configuration can see, and what it cannot. The smallest eigenvalue of the fit's own normal matrix — how much the residual changes for a unit move in the worst direction of parameter space — for three parameterised candidates against six configurations of the same size. A zero is not a hard fit: it is a direction the control points cannot see at all, so every value of the parameter along it gives an identical residual. 2 of 18 are at the floor of double precision, and they are not the ones a reader would guess. Where none is, the spread between the best and worst arrangement is still Infinity at a fixed point count.

Where the control points are

Five rungs fit a library to a map and ask how much to trust the winner. None asks whether the parameters are recoverable at all. On control points along one parallel an equirectangular's standard parallel is not merely hard to find — it is invisible, exactly, and five hundred and twelve points on the same parallel are as blind as eight.

wrong · Identify What a seven-parameter fit's residual is made of. A published transformation accuracy is the root-mean-square residual at the common points, and here it is 1.51 metres. Almost all of it — 1.51 — is the network's own distortion, which a rigid motion and a scale cannot follow and which is present at every point of the country whether it was used in the fit or not. The transformation's own error, measured as the disagreement between the fitted parameters and the true ones over a clean grid, is 0.091 metres: 6 per cent of the quoted figure. The last bar is the control — the same fit with the distortion switched off, at 2.9e-4 metres.

What another common point buys

Rung three finds that a seven-parameter datum fit leaves a pattern rather than noise. Six per cent of the residual it reports is the transformation's own error and the other ninety-four is distortion no seven parameters can follow — so adding common points improves a term that was already small and cannot touch the one that is quoted.

datums · Datum What reading the seven parameters under the wrong convention costs. The distance between the two published conventions' answers for the same parameter set, at the worst point of the world. The difference is 2s(r × X) exactly — twice the rotation, crossed into the position — so a parameter set with no rotations is immune and one with a large rotation is not: DHDN reaches 152.3 metres and NAD27, whose published transformation is three translations and nothing else, reaches zero. The note beside each bar is the size of that datum's rotation.

The rotation has two sign conventions

Eleven essays price a datum transformation's parameters, their fit, their residuals and their uncertainty. None asks what the numbers mean: three of the seven are published under two conventions whose rotations differ in sign, and reading one set with the other formula costs exactly twice the rotation — 55 metres on OSGB36 and 152 on DHDN.

datums · Datum

Named alongside it

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

ConditioningDatumError budgetHelmert transformationLeast-squaresResidualVerificationCandidateCone constantConventionEPSGGeodetic datum

All concepts