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

Projection selection — where it appears

Choosing which projection to draw a map in, which is a question about purpose before it is a question about properties. Naming a property leaves a family rather than a projection, and picking within the family needs a region and a criterion.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

6 projections of the same sphere. The same graticule under Equirectangular, Mercator, Mollweide, Sinusoidal, Robinson, Winkel tripel. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve.

Which projection is best

An incomplete question, and the incompleteness is the answer. Every projection preserves something and destroys something else, so the comparison worth making is between a projection and a purpose, not between two projections.

choosing · Choosing Mercator swung through every tilt, over Chile and the tropics. The regional distortion of one projection as its axis is tilted from the normal aspect at 0° to the transverse at 90°, each curve divided by its own value in the normal aspect so the two regions can share an axis. Chile is best at a tilt of 90°, a factor of 183.5 better than north-up; the tropics is best at a tilt of 0°, which is north-up. Rotating the sphere costs nothing and changes none of the projection's own properties, which makes this the cheapest improvement available and the one most often left unmade.

Fitting the aspect to the region

Choosing a projection is a choice among a few dozen named things. Choosing its aspect is a choice among a continuum, it costs nothing, it changes none of the projection's own properties, and for a long thin country it is worth a factor of 183.

choosing · Choosing The scale along the Madrid–Tokyo corridor. The scale factor of four projections along the great circle from Madrid to Tokyo, each normalised to its own average over the route so the comparison is of variation rather than of size. The oblique Mercator whose own equator is laid along the corridor holds the scale to 0 parts per million; Mercator varies by 118.7% over the same line. A corridor is a curve, not a region, and the projection an area criterion picks is not the one a curve wants.

Choosing for a line, not a region

Every criterion in this subject integrates over an area. A pipeline, a railway or a coastal survey is a curve, and the projection an area criterion picks for it is not the one it should have — measurably, by a factor of five thousand.

choosing · Choosing Sheets for a tolerance, from Chebyshev's bound and a covering. For each stated tolerance on the scale error, the cap radius at which the best possible conformal projection just meets it — sec²(ρ/2) − 1 = tolerance, which is Chebyshev's bound and has no fitting in it — and then the number of such caps needed to cover the sphere at the packing density a real arrangement achieves. One part in a thousand costs 1210 sheets of 403 kilometres radius. The slope is -0.989: a factor of ten in what the job will accept is a factor of ten in the atlas.

How many sheets an atlas needs

A tolerance on the scale error inverts, through Chebyshev's bound, into a sheet radius — and a covering problem turns the radius into a count. One part in a thousand costs 1,210 sheets of 403 kilometres radius, the count goes as the reciprocal of the tolerance exactly, and the projection multiplies it by anything from one to fifty-six.

choosing · Choosing Which of eight places is nearest, decided on the ground and decided on the page. Every cell of the window shaded by which of eight places across Europe is nearest on the ground, with the cells the page's own answer would hand to a different site drawn in the failure colour. Plate carrée misassigns 13.67 per cent of the window's ground area — 3,058 thousand square kilometres. The misassigned cells are not scattered: they lie in bands along the boundaries, which is what a systematic error looks like and what a sampled test of a few query points is least likely to find.

Nearest of many is a partition

One reach question with one source is a disc. With several sources it is a division of the whole surface, every place belonging to whichever source is nearest — and computing that division in the plane the data is stored in hands away between 0.75 and 22.16 per cent of the ground, in unbroken strips up to 1,591 kilometres across.

paths · Reach 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 aspect of Mercator over Japan, and the line the old sweep searched. The regional distortion of Mercator over Japan for every position of the projection's pole — darker is better — with the meridian the site's one-dimensional sweep searches drawn on it. The sweep's best is a gain of 24.4× over the normal aspect; the two-parameter search finds 61.4×, which is 2.51 times better again, at a pole 45° of longitude away from anything the sweep could reach. The shortfall recorded when the sweep was written said this would happen for a region whose long axis runs diagonally, and this is the measurement of it.

The aspect has three numbers, not one

The site's aspect search has swung the projection's axis through one plane for a long time, and recorded that a region whose long axis runs diagonally has its optimum somewhere that plane never reaches. Searching the whole sphere of pole positions finds 2.6 times more improvement over Japan and 3.4 over the conterminous United States.

choosing · Choosing 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 The quickest way out of the ice bends at the edge. A ship twelve kilometres inside a straight ice edge, making 6 km/h in the ice and 25 km/h in open water, bound for a point ten kilometres out into the water and twenty along. The dashed line is the straight route, 3.244 hours. The solid one is the quickest, 2.849 hours — 24 min sooner — and it reaches the edge only 2.55 km along, leaving the slow ice as directly as it can. At the edge it meets the perpendicular at 12.0° in the ice and leaves at 60.2° in the water, and the sines of those angles stand in the ratio of the speeds to within 4e-17.

A crossing bends by a law only a conformal chart can show

A ship leaving pack ice for open water takes its quickest route by bending at the ice edge, so that the sines of its angles to the perpendicular stand in the ratio of its two speeds. The law has nothing in it but angles, so on Mercator it can be checked with a protractor — and on a plate carrée at sixty degrees north the same crossing reads as a ship making ten kilometres an hour in the ice rather than six.

paths · Paths Two regions, three answers. Britain and New Zealand, 166° apart, with the pole of the best oblique conic under each of three objectives. Pooling the samples and taking an area-weighted score puts the pole in one place; refusing to let either region be worse than the other puts it somewhere else. The regions are drawn on Mollweide so that equal ground areas are equal page areas.

The pooled score abandons a region

Every criterion in this subject optimises for one region. An atlas is several, and pooling their samples into one area-weighted score is what everybody does — which on Britain and New Zealand serves Britain 1.2 times worse than it could be served alone and New Zealand 125 times worse. The worst-case objective makes them equal at 33 and 59, and the cost of sharing rises with separation from 1.4 to 59.

choosing · Choosing Five readerships, four winners, one library. The best projection under each stated readership, with the share of the sphere carrying half that readership's weight. The criterion is Kavrayskiy's throughout, every map is in its normal aspect, and the library is the same twenty. Four different projections win. The flat weighting at the top is the one every earlier comparison used, and naming it as a readership rather than as the absence of one is the whole of what this changes: it is the assumption that everybody who will ever read the map is spread evenly over the globe.

The area weighting was a readership all along

Twelve measurements have taken a projection comparison apart and found the comparer's own freedom in every part of it. All twelve rested on one thing none of them examined: that a square kilometre of empty ocean counts exactly as much as a square kilometre of city. That is not the absence of an assumption about the reader — it is the assumption that everybody who will ever look at the map is spread evenly over the globe. Named as one readership among five, it produces a different winner from three of the other four, and against a North Atlantic readership its choice is fifth of twenty and 33 per cent worse.

wrong · Audit 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.

PurposeAspectRegional distortionKavrayskiy's criterionOptimisationRegionVerificationMercatorObjective functionObliqueRankingRule of thumb

All concepts