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

Web Mercator — where it appears

The projection almost every online map is drawn in, which applies spherical Mercator formulae to ellipsoidal latitudes and is therefore not conformal. Its measured maximum angular deformation is 0.3848 degrees against a noise floor of 1.6 × 10⁻⁶ for the genuinely conformal projections.

Named by 19 essays across 4 fields — each of them below, with the objects they name alongside it.

The tile pyramid, four levels down to quadkey 120. The world as a square, quartered three times. Each level's tile is exactly half the width of its parent, so a tile is four tiles at the next level and never needs resampling to serve one — the property the whole scheme rests on, and one that holds only because the projected world is square. Level 3 has 64 tiles at 19567.9 metres per pixel, which at 51.5° north is 12181.3 metres of ground per pixel rather than the number the scheme publishes.

A screen map is a pyramid of tiles

The scheme every slippy map runs on is a coordinate system with three integers and one projection, and almost all of it is forced. A square world is what makes the quadtree work, the levels are exact powers of two, and the published resolution — 156,543 metres per pixel at zoom zero — is a distance on the ground at exactly one latitude.

applied · Screen Every projection in the library, measured against both properties. Maximum angular deformation across the bottom, maximum areal error up the side, both on logarithmic scales and both measured over several hundred sample points rather than taken from the projection's description. Conformal projections lie on the left edge, equal-area ones along the bottom, and the corner where both vanish is empty because a projection there would be an isometry of the sphere onto the plane. All 26 are plotted; the four named here are Mercator, Mercator (ellipsoidal), Web Mercator, Gall–Peters.

Web Mercator is not conformal

It carries almost every map on the internet, it is named after the projection whose entire purpose is preserving angles, and it does not preserve angles. The machinery here found that without being told to look.

wrong · Audit What "1:136,495" means at each latitude, at zoom 12. A screen map at zoom 12 prints one scale for the whole world. The curve is how much larger in scale the map really is, measured from the projection's own derivatives rather than from a formula: at 60° it is 1.98 times, so the map labelled 1:136,495 is a 1:68,765 map. The hollow marks are sec φ, the textbook answer. They do not sit on the curve — the worst gap is 9949 parts per million at 85° — because Web Mercator puts a geodetic latitude into a spherical formula, and the same spherical Mercator measured the same way reproduces sec φ exactly.

The scale of a screen map is not one number

A zoom level prints one scale for the whole world, and the map is at that scale along exactly one line. At 60° north the picture labelled 1:136,495 is a 1:68,765 map — and the factor is not quite sec φ either, because the projection puts a geodetic latitude into a spherical formula.

applied · Screen Why there are two latitudes, at 45°. A meridian section of an ellipsoid with the flattening exaggerated 12× — at the true value of 1/298 this outline would be indistinguishable from a circle. Geodetic latitude is the angle the surface normal makes with the equatorial plane; geocentric latitude is the angle the radius makes with it. The two lines meet the plane at different points and the angles differ by 11.55 arcminutes on the real WGS84 ellipsoid, which is about 21 km of ground distance.

The Earth is a sphere, and when it is not

Every essay before this one treated the Earth as a ball, and said so. The flattening is one part in three hundred, which is nothing for a distance, everything for a latitude, and exactly enough to make the most-used projection in the world fail the property in its own name.

wrong · Ellipsoid The cut at 85.0511°, and the alternatives that are not square. Mercator's northing runs to infinity at the pole, so a tiling has to stop somewhere, and the latitude is not a rounding: 85.0511° is where the northing equals half the world's width, which is the only cut that makes the projected world a square. The bars are what other cuts would give — at 89° the world is 1.51 times as tall as it is wide, and a single square root tile cannot cover it. The price is 0.373 per cent of the Earth's surface, 1,901,487 square kilometres in two caps, computed from 2πR²(1 − sin φ) rather than estimated.

The square costs the poles

The cut at 85.0511287798° is the most-quoted number in web mapping and is almost never derived. It is where Mercator's northing equals half the world's width — the condition for a square — and it drops 1,901,487 square kilometres. A two-tile root would have reached 89.786° and dropped 3,558.

applied · Screen Every projection in the library, measured against both properties. Maximum angular deformation across the bottom, maximum areal error up the side, both on logarithmic scales and both measured over several hundred sample points rather than taken from the projection's description. Conformal projections lie on the left edge, equal-area ones along the bottom, and the corner where both vanish is empty because a projection there would be an isometry of the sphere onto the plane. All 26 are plotted; the six named here are Mercator, Web Mercator, Gall–Peters, Mollweide, Equirectangular, Stereographic.

Measuring instead of naming

A projection is called conformal because that is its name. Running the definition as a computation over several hundred points takes about twenty lines, catches a projection the whole internet uses, and is almost never done.

distortion · Tissot One pixel at zoom 11: 76.4 m projected, 47.6 m at 51.5°. The grid squares are pixels, at their own size. The open mark is the stored coordinate and the filled one is where it is drawn: rounding moves it 20.2 metres, against a worst case of 33.6 — half a pixel's diagonal, which is the whole of the bound. The second point sits 43 metres away, 0.58 of a pixel east and 0.70 north, so whether the two are drawn as one dot or two is decided by where the tile grid happens to fall: they merge for 12 per cent of the possible offsets, against the 12 per cent the two fractions predict.

The pixel is a place with a size

Drawing a coordinate rounds it to a pixel, which moves it by up to half a diagonal — 16.8 metres at zoom 12. Whether two points 43 metres apart appear as two dots is not a property of the data at all: they merge for 12 per cent of the positions the tile grid could take, and the closed form predicts 12.4.

applied · Screen Two square worlds: Mercator's areas against an equal-area scheme's angles. Both projections give a world exactly as wide as it is tall, so either could carry a quadtree of square tiles. Mercator keeps every angle and inflates area by sec²φ — 132-fold at 85°. The cylindrical equal-area scheme whose world is square has standard parallels at ±55.65° — the solution of π cos²φ₀ = 1 — and keeps every area exactly, at a cost of nothing there and 145° of angular deformation at the edges. At the standard parallel itself Mercator's areal factor is 3.142, which is π, because the square-world condition and sec²φ are the same equation.

The pyramid did not have to be Mercator

The usual defence is that a quadtree needs a square world and Mercator supplies one. So does the cylindrical equal-area with standard parallels at ±55.654° — the solution of π cos²φ₀ = 1 — and it needs no polar cut at all. What Mercator actually buys is conformality, and the price of giving it up is 13.8° of shear at 60° north.

applied · Screen A straight line stored in Web Mercator, and where it really goes. Two points 5570 kilometres apart, joined by a straight segment in the plane the file's coordinates are in. Drawn on that plane it is a straight line and looks like the route; on the ground it is the curve marked measured, and the geodesic between the same two endpoints is the other one. The profile below is the distance between them along the line, reaching 718 kilometres at 49 per cent of the way across. Nothing here is an error in the data: both endpoints are exact, and the whole of the discrepancy is the word straight.

A straight segment is a claim about a plane

Two exact endpoints, joined by a straight line in the plane the file is stored in. On the ground the line is 718 kilometres from the route it claims between New York and London, and 2,961 between London and Tokyo. The departure grows as the square of the length — fitted exponent 2.001 — so a stated tolerance costs vertices as a square root.

applied · Dataset The scale asked for and the scale the pyramid has, at 0°. A tiling scheme exists only at integer zoom levels, a factor of two apart in resolution, so a request for any scale between them is answered by the nearest rung. The ratio runs from 0.707 to 1.405 — 1/√2 to √2 — and repeats identically at every doubling, which is four times that in area. A request for 1:10,000 is served at zoom 16, which is 1:8,531; A request for 1:25,000 is served at zoom 14, which is 1:34,124; A request for 1:50,000 is served at zoom 13, which is 1:68,247. Nothing anywhere reports it, because the map that arrives is a perfectly good map of something.

Zoom is a ladder

A tiling scheme exists only at integer zoom levels a factor of two apart, so a request for 1:25,000 is answered with 1:34,124 — 36 per cent coarser, and 86 per cent coarser in area. The mismatch runs from 1/√2 to √2 and repeats identically at every doubling, and nothing anywhere reports it, because the map that arrives is a perfectly good map of something.

applied · Screen The error of a spherical formula is twice the flattening. The worst angular deformation of the spherical Mercator formulae applied to each body's own latitudes, against that body's flattening, on logarithmic axes. The points are measured by differencing the projection; the line is 2f radians, which is not fitted. The two agree to a third of a per cent for Mercury, the Earth and Mars, and depart by 3.3 and 5.1 per cent for Jupiter and Saturn, whose flattenings are too large for the first term to be the whole story. The site's headline number — 0.3848° on Earth — is an instance of this law rather than a fact about Web Mercator.

The same projection on a different body

A projection's distortion does not depend on the body's size at all — only on its flattening — so Web Mercator's 0.3848° of angular deformation is not a fact about Web Mercator. It is 2f radians, and the same mistake made on Mars measures 0.6765°, on Jupiter 7.68°, and on a body scaled ten times larger than Mars the identical number to twelve figures.

datums · Bodies Every projection's angular deformation, and the tolerance that judges it. The whole library on one logarithmic axis, with the tolerance drawn as a line. The population is bimodal: the projections that satisfy the condition sit at 2.09e-6 and below, the ones that do not at 3.85e-1 and above, and there is nothing between. The tolerance could be moved anywhere in that gap — a factor of 1.84e+5 — without changing one verdict. Larger marks are projections that claim the property.

The tolerance that decides the verdict

Eight rungs of this ladder hand out verdicts, and every one rests on a tolerance chosen once, in the site's second phase, at sixty times a measured noise floor. Swept, it decides nothing: the population is bimodal, the tolerance sits in a gap 238,000 times wide for conformality and 646 million times wide for equal area, and the verdict with the least room is a passing one whose margin is the arithmetic's rather than the map's.

wrong · Audit Equal bands of screen, unequal bands of ground. A section through a camera pitched 60° from straight down, at the height that makes an untilted view exactly the flat map — 1152 pixels above the plane for a 768-pixel screen and a 36.87° field of view. The rays are drawn at equal spacings up the screen and the blocks below them are the ground each of those equal bands covers: 1.75 plane units per pixel at the bottom of the frame and 16.88 at the top, a factor of 9.7. Nothing about this is a projection of the sphere; it is a photograph of a map, and the map underneath it is still Web Mercator.

A tilted view has no zoom level

Every essay about the screen so far assumes the map lies flat on it, which was true until about 2015. Pitch the camera sixty degrees and one frame asks for 3.6 zoom levels at once, a square tile covers ground four and a half times deeper than it is wide, and the pyramid has one integer per tile to answer with.

applied · Screen One unit of a vector tile, in metres of ground. A vector tile's coordinates are integers on a lattice 4096 units across the tile, and the tile halves at every level, so one unit is a distance that halves too: 5.48 m at z10 and 0.086 m at z16, at 55°. It is also a different distance at every latitude, by cos φ, because the tile is in Web Mercator — the same factor that makes a grid metre a different quantity of ground at every latitude, arriving in the file format rather than in the projection.

A vector tile has an integer grid

Six essays on this ladder treat a vector tile as the thing a raster tile is not: geometry, resolution-free, styled at draw time. Its coordinates are integers on a lattice 4,096 units across a tile, the tile halves at every level, and at 55° north one unit is 88 metres at zoom 6 and 21 millimetres at zoom 18.

applied · Screen The same 2-pixel road at three latitudes, zoom 5. The dark bar is the mark as drawn — 2 pixels, identical in all three panels, because that is what the stylesheet says. The pale band behind it is the ground that mark covers, drawn to one common ground scale: 9.78 kilometres at the equator, 6.92 at 45° and 1.70 at 80°. The reader sees the dark bar and is being told about the pale one.

The road is drawn two pixels wide

Seven rungs measure what a screen map does to position. Nothing on a map is a point: every mark has a width, the width is chosen in pixels, and a two-pixel road covers 9.78 kilometres of ground at the equator and 1.70 at 80° north. That is a generalisation applied at a strength varying by a factor of six across one sheet, by a stylesheet with no latitude in it.

applied · Screen One of these three lines does not slope. Three lengths at each zoom, for a world coordinate near the antimeridian. The pixel halves with every level, as it must. The spacing between representable double-precision values is far below it and halves with it. The spacing between representable SINGLE-precision values does not move at all — 2.0 metres at every zoom, because it is a property of the size of the number and the world does not get smaller. The two cross at zoom 17, and past it a vertex snaps to a lattice coarser than the pixels it is drawn into.

The renderer runs out of numbers before the zoom does

Eleven essays price the pyramid in exact arithmetic. The pipeline that draws it carries single precision, where a world coordinate near the antimeridian quantises to two metres — at every zoom, because the number does not get smaller when the pixel does. At zoom 22 that is fifty-four pixels.

applied · Screen

The query a fast path actually answers

Fifteen rungs price what a stored coordinate means and none asks what it is searched with. No index answers "within two hundred kilometres of here"; an index answers "inside this rectangle of stored coordinates", and the rectangle is built by somebody's arithmetic — which at 55° north silently drops a tenth of the true answer on a conformal projection and three fifths of it on an equal-area one.

applied · Dataset

A tolerance in map units is not a tolerance

A snapping tolerance is a number, and the number is in whatever units the file is in. Five map units on Web Mercator is 4.97 metres of ground at the equator and 0.87 at eighty degrees — so a rule that merges two features three metres apart merges them everywhere below 52.8° north and refuses everywhere above it, in one pass, over one dataset, with nothing recording where the boundary is.

applied · Dataset

A mean that does not exist can still be printed

Mercator's area-weighted mean areal factor is artanh(sin Φ)/sin Φ, and it has no limit. A sampler asked for it returns the logarithm of its own sample count plus 1.512 — measured slope 1.001 against ln n — so the number is a property of the person who computed it. The Kavrayskiy number for the same map over the same sphere settles at 0.52124 and is a number.

distortion · Sampling

Named alongside it

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

VerificationToleranceConformalityScale factorZoom levelConventionTile pyramidGround resolutionQuantisationMercatorAngular deformationClosed form

All concepts