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

Registration — where it appears

Placing a map or a set of positions onto a sheet by fitting a transformation to points whose places are known in both. A shift, a turn and a size are the four numbers nearly every registration fits for itself; whatever the fitted transformation cannot follow is left over, and is read as the map's own error.

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

Pins at one place move where the picture sits; pins at four reshape it. The best flat picture of four cities, fitted to every distance with a tolerance of 1 per cent, one bearing a pair to 0.01 radians, and pins to the sheet good to 50 km. Each ellipse is the error in a city's position relative to the picture's centre — its shape, which no shift changes — drawn 30 times its size. Solid: the part the distances and bearings put there. Dashed: the part the pins put there. Pinned at London alone, the pins put 0.00 km into the shape and the places 63.0 km. Pinned at all four, the pins put 30.5 km and the places 29.7 km, and the shape as a whole is 42.6 km against 63.0.

A second pin is a measurement of the places

Counted exactly, a picture of places is a direct sum: the distances and bearings spend its shape and two pins spend where it sits on the sheet. Given tolerances, the sum survives in one form and fails in another. Pins at one place put none of their error into the shape at any tolerance, and wherever the pins are, their average takes none of the distances' error. But every place pinned after the first is a statement about the geography, and held exact it makes the shape worse once its error passes about two thirds of the shape's own.

impossibility · Embedding Mercator's positions for twenty-four cities are mostly a size, and the rest is one pattern. Twenty-four European cities from Lisbon to Moscow. Solid dots: the best flat picture of their 276 great-circle distances, north up at 15°E 50°N. Lines run from each dot to where Mercator puts the city. Left, with only the sheet's origin moved: the projection is 662 km off in root-mean-square, because it draws the region 1.56 times too large. Right, after the four numbers any registration fits for itself — a shift, a turn of 1.73° and that size — what is left is 154 km, drawn 4 times its length: 5.4 per cent of the squared error, and it is a smooth swirl rather than an error at each city.

A projection pins every place, and its error does not average

A projection used as a stated position for every place in a region splits the way a set of pins does: a shift, a turn and a size that any registration takes for itself, and a pattern that competes with the geography. Over twenty-four European cities 94 per cent of Mercator's error is size. What is left is one smooth displacement, so a fit told its size but not its shape says the pins move the picture as little as a quarter as far as they do — and with the weight a pin of that size deserves, past a few dozen places the projection makes the shape worse than no projection at all.

impossibility · Embedding

Named alongside it

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

Control pointsCovarianceEmbeddingLeast-squaresWeightingAffine transformationConstraintDegrees of freedomDistortion distributionOver determinedScale spreadSimilarity transformation

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