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

Rule of thumb — where it appears

A short rule for choosing a projection or a parameter without computing anything. Two of them are scored here against the optimum, and both collapse on regions whose long axis is oblique, because both read a bounding box.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The taught rule against the measurement, on 30 regions. Each cell is a region built to order — a box of the stated height and width-to-height ratio, centred at the stated latitude — with the family that actually scores best over it, and whether that is what the rule says. Every family is given its own parameters for the region: the conic its cone constant, the cylindrical its standard parallel and the choice of a normal or transverse axis, the azimuthal its centre. The rule is right on 19 of 30, and where it is wrong it is wrong in one direction: it keys on latitude, and what decides the answer is shape.

The rule of thumb, scored

Cylindrical near the equator, conic in the middle latitudes, azimuthal at the poles. It is the most repeated piece of practical advice in cartography and it has never been run against a population of regions. Run against thirty, it is right nineteen times, and every one of its failures has the same shape.

wrong · Audit Two rules, three populations. The taught rule keys on latitude; the replacement keys on shape. On the thirty regions the replacement was read from it scores 83 per cent against the taught rule's 63. On forty-five different regions built the same way it scores 91 — higher, not lower, so the generalisation the shortfall doubted is real. On the seven named regions this collection actually uses, both rules score 29 per cent, and following either costs a mean factor of 14.6. The third bar of each group puts the taught rule's polar clause back into the shape rule, which is what the out-of-sample failures ask for: it repairs four of them, breaks two that were right, and reaches 43 of 45.

The rule scored out of sample

A replacement rule was read off thirty regions and scored on the same thirty, and this collection recorded that as not being evidence about any other thirty. It is: on forty-five different regions the rule scores 91 per cent against the 83 it managed at home. What it cannot do is the seven regions the collection actually uses, where both it and the rule it replaced name the winner twice out of seven and cost a mean factor of 14.6.

wrong · Audit Every library projection over Europe, on the two axes it can be wrong on. Each dot is one projection, scored over Europe on the two independent failures: how much it turns angles and how much it changes areas. The lower-left corner is the isometry that does not exist. The line joins the five projections nothing beats on both counts — the rest are inside it, and a reader who prefers either failure to the other should still not choose one of them, whatever weighting they hold.

The projections that are beaten on both counts

A rule of thumb has been scored over thirty regions and then forty-five. The same populations answer a harder question nobody has put: which library members are never the right answer at all. Two are beaten outright on both criteria everywhere, one is on no regional front in any population — and it is on the world's.

wrong · Audit Töpfer's square root is one line of a family. The fraction of features surviving to a smaller scale, for four stated populations whose size distributions differ only in their exponent. Every one is a straight line on these axes, and the slope of each is its own exponent: 0.3, 0.5, 0.8, 1.2. Töpfer's radical law is the line at 0.5 — the square root — and it is exact for that population and for no other. The law is not a rule of thumb with exceptions; it is a theorem with a hypothesis nobody states.

How many features a scale can carry

Töpfer's radical law is quoted everywhere as a rule of thumb. It is not one: it is a theorem about a size distribution with a Pareto exponent of exactly one half, exact to 1.8 per cent for that population and out by 99.4 per cent for a lognormal one.

applied · Generalise A territory, and two enclosures somebody might score instead. A stated territory at 45° north — an ellipse twenty-eight degrees long and eight wide, its long axis bearing 45° — with its latitude–longitude bounding box and the smallest cap around it, on a globe. The territory fills 40.6 per cent of the box and 28.6 per cent of the cap. Scoring the library over each names a winner: an azimuthal for the territory itself, an azimuthal for its bounding box, an azimuthal for the smallest cap around it. The cost of working from the box rather than the territory, measured as the box's winner scored on the territory against the territory's own winner, is a factor of 1.0584.

The region was a rectangle in a projection

Every verdict this collection hands out is a verdict about a region, and almost every region it has been given is a latitude–longitude box. A box discards a territory's shape, which costs a few per cent, and it records the territory's width in degrees of longitude, which costs up to a factor of 1.72 and changes which family wins on four of forty stated territories. Convert that width to arc and the box gives the right answer every time — so the comparison had a projection in it before it began.

wrong · Audit

Named alongside it

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

Projection selectionPurposeRegionAspectAuditAzimuthalBounding boxConicDistortion criterionGeneralisationKavrayskiy's criterionOptimisation

All concepts