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

Plate carrée — where it appears

Longitude across and latitude up, plotted on ordinary axes — a projection with no exact property and the one a lon/lat file is implicitly in. Its page coordinates are the degrees themselves, so simplifying on its page and simplifying in degree space are one operation rather than two.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

How Equirectangular distorts, by latitude. Angular deformation and areal scale plotted against latitude along the meridian at 0°. On Equirectangular the angular deformation reaches 114.2° and the areal factor reaches 11.5.

The plate carrée, the projection nobody chooses

Plotting latitude against longitude on ordinary axes is a projection. It preserves nothing, its angular deformation reaches 108° and its areal error eightfold, and it is probably the most widely produced map in the world because it is what happens when nobody decides anything.

wrong · Audit Two readings of the same triangle, and the band between them. The same three vertices, joined two ways: with straight lines in the plane the coordinates are stored in, and along the ground. Every dot is a point the two readings disagree about — inside on one and outside on the other. The band covers 29.5 per cent of the polygon, which is 4241 thousand square kilometres, and it is not an error in either reading: the file does not say which one it means. Densifying the stored boundary removes it, which is the only fix there is. Drawn in Equirectangular.

Inside is a claim about the edges

Whether a point is inside a polygon is not a property of the point and the polygon. It is a property of the plane the edges were understood to be straight in, and between two readings of the same file there is a band of disagreement — 29 per cent of one triangle's area, 4.2 million square kilometres, and the file does not say which reading it means.

applied · Dataset 10,000 observations of one place, averaged 120 times. Left: 400 single observations of the same place, with a 60-kilometre standard deviation east and north, drawn across ±400 km. They are scattered about the truth and their average is unbiased on the ground. Right: 120 independent averages of 10,000 such observations each, computed on the page and taken back to the ground, drawn across ±1.8 km. The cloud is tight, as averaging ten thousand things should make it, and it is not centred on the cross: it sits 489 m away, against 403 m predicted by the projection's second derivative alone.

The average of noisy positions moves

Average sixty thousand scattered observations of one place on a Mercator map and the answer is 404 metres too far north — at every sample size, because it is a bias and not noise. The same average on the Lambert cylindrical equal-area is 404 metres too far south, the two being ½ (σ²/R) tan φ and its exact negative, and on the plate carrée it is not displaced at all.

distortion · Precision Two lines of position replace two circles thousands of kilometres across. A ship at 50° N 20° W observes two bodies at altitudes of 45.60° and 42.69°, which place it on two circles 4,938 and 5,261 km in radius, drawn faint. From an assumed position 150 km to the north-east, each body's azimuth is laid off (dashed) and its intercept marked — 12.2 km and 111 km — and a line of position drawn perpendicular to the azimuth through that point. On Mercator the two lines cross 3.28 km from the ship: each line is only its circle's tangent at the intercept, and by the time the two lines meet they have left their circles.

A line of position is Newton's method, but only on a conformal chart

An altitude of a star puts a ship on a circle five thousand kilometres across, and the intercept method replaces that circle with a straight line. On Mercator the fix those lines give is one step of Newton's method: its error falls as the square of the assumed position's, and re-running it turns 300 kilometres into 24 metres in two steps. On a sheet that does not correct longitude for latitude it keeps a fixed share of every east–west error, one less the cosine of the latitude — 36 per cent at 50° north.

paths · Position fix Three lines of position, and a cocked hat that misses the ship. Three bodies whose azimuths from the ship are 20°, 140°, 250°, each observed with an independent error of about 2 km along its azimuth, give three lines that do not meet. They bound a triangle of 7.5 km², outlined, and the ship — the dot — lies outside it. With independent errors that is the usual outcome: three times in four, whatever the triangle's shape.

A cocked hat holds the ship one time in four

Three lines of position with any error in them bound a small triangle, and the instinct is that the ship is inside. With independent errors it is inside one time in four, whatever the triangle's shape. A common error in every sight makes that almost always when the stars surround the ship and almost never when they do not. And on a sheet that forgets the latitude, the triangle keeps its size and moves off the ship altogether.

paths · Position fix The triangle and the ellipse are two pictures of the same three sights. Three bodies at azimuths of 20°, 140°, 250° with independent errors of 2 km, drawn twice: as the cocked hat their lines bound, dashed, and as the 95 per cent confidence ellipse of the least-squares point, outlined. The ellipse is 4.25 by 3.78 km and covers 50.5 km²; the average hat from the same sights covers 7.06. The triangle is seven times the smaller and holds the ship a quarter of the time; the ellipse holds it 95 per cent of the time, which is what it was constructed to do. The hat is a picture of how much the sights disagree, and the ellipse is a picture of where the ship is.

A confidence ellipse is honest only where the sheet keeps angles

The cocked hat covers seven square kilometres and holds the ship a quarter of the time; the confidence ellipse from the same three sights covers fifty and holds it ninety-five per cent, which is what it was built to do. On a plotting sheet whose longitude is not corrected for latitude that ellipse fails in two unrelated ways — its centre is displaced, which costs coverage and depends on the navigator's own assumed position, and its shape is misread, which costs nothing but meaning and depends only on the latitude.

paths · Position fix 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.

VerificationConformalityNavigationToleranceAzimuthCovarianceEstimatorGreat circleLeast-squaresMercatorQuadratic lawAreal factor

All concepts