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

Compilation — where it appears

A map assembled from pieces of other maps rather than projected once from the ground. Each copied stretch keeps the projection of the sheet it came from and is fitted into place by eye, so the result has no single projection behind it and its graticule cannot say so.

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

The same copied stretch, dropped in rigidly and eased into place. One coast drawn twice. In the first ink the copied stretch is moved as a rigid, scaled piece and meets the rest of the coast at two corners — the dots. In the second it is pinned at the same two points and then eased along 30 per cent of the outline either side of each of them, which is what a compiler fitting a sheet by eye does. The dashed line is the coast the graticule predicts. The eased version departs from it by 6.1 thousandths of the map's width against the rigid one's 9.4: easing moves the departure about, and removes very little of it.

A copy eased into place loses its seams before its source

A stretch of coast lifted from another sheet and dropped in rigidly leaves two sharp corners where it joins, and both of them are findable. A compiler does not work that way: they pin the ends and ease the middle until it looks continuous. Easing halves the corner within five per cent of the outline and the seams survive it — until forty per cent, where the located seam jumps from four per cent of the coast out to twenty. The source projection survives twice as far, to seventy-five. The piece goes on naming where it came from after the method has lost track of which piece it is.

wrong · Identify Charged ln N a number, the count comes out right; charged two, it takes all it is offered. Two information criteria over the same seven segmentations, each shown above its own minimum. Both add the log of the residual to a charge for every number the model spends — four for each piece's placement and one for each seam's position. BIC charges ln N, 6.6 for 720 coordinates, and is least at 2 seams, the number made. AIC charges two, and is least at 6, the most it was offered: each extra seam buys more than two units of fit because its position and its projection are both chosen by search, and a charge for the numbers alone does not pay for the search.

A residual cannot count the pieces of a map

A compiled map does not say how many sheets went into it, and trying every count and keeping the best fit cannot tell. Seven pieces always fit better than two, because they contain them. A criterion that charges ln N for each number a piece spends recovers the true count in every trial, for one copied stretch and for two, until the digitising error reaches a thousandth of the map. AIC charges two and takes every piece it is offered. And an eased copy defeats the right criterion the other way: it is not a projection, so BIC buys pieces to explain it, three seams, then four, then six.

wrong · Identify

Named alongside it

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

EstimatorProjection identificationResidualSeamSimilarityDegrees of freedomDigitisingLeast-squaresOutlineSearchToleranceVerification

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