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

Graticule — where it appears

The network of meridians and parallels drawn on a map, which is a set of coordinate lines rather than anything on the ground. A projection's identification is fitted to its crossings, and the quantities that depend on it — h and k — are exactly the ones that do not describe the map.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

Construction against property. Every projection in the library, sorted by the shape it was notionally rolled from and by what it actually preserves. The families do not line up with the properties: cylindrical and pseudocylindrical and pseudoazimuthal and azimuthal and conic each contain projections of more than one kind, which is why the cylindrical–conic–azimuthal taxonomy answers a question nobody has. Three entries are picked out: Mercator, Albers equal-area conic, Orthographic.

Cylinders, cones and planes

The standard taxonomy sorts projections by the shape of paper they were notionally rolled from. It is memorable, it is how everyone is taught, and it says almost nothing about the properties anyone actually chooses on.

families · Families The indicatrix at 30°, 40° on Gall–Peters. A circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.083 and b = 0.923; their product is the areal factor 1.000; and the maximum angular deformation is 9.16°. h and k are shown too, and depend on the coordinates rather than on the map.

What survives a change of coordinates

The scale along the meridian is a property of the map and the grid together. The principal scale factors are a property of the map alone. Only the second kind describes the projection, and the two are routinely quoted as though they were the same thing.

distortion · Tissot London to Tokyo, seen four ways. The same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did.

The aspect is a free choice

A projection's distortion pattern is fixed relative to its own axis, and where that axis points is entirely up to the cartographer. Rotating it is the cheapest available improvement and it is the one most often left unmade.

families · Families London to Tokyo, seen four ways. The same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did.

The gnomonic companion

One projection turns every great circle into a straight line, and it is the only one that does. It shows less than half the sphere, distorts enormously, and was indispensable for three centuries because of that single exact property.

paths · Paths 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 What a pole line buys, and what it costs. four pseudocylindrical projections placed by the two numbers the decision moves. Across the bottom, the angular deformation averaged over the whole sphere, where the pole-line projections — Eckert IV and Robinson — are the better maps. Up the side, on a log axis, the factor by which the last parallel is stretched, where they are worse by more than tenfold: 49× at best against 4.6× at worst for a pole drawn as a point. A pole line represents one point of the globe by a line of the map, and that is the price.

Where a pseudocylindrical puts its error

Every projection in this family has to decide what to do at the pole, which on the globe is a point where every meridian meets. Drawing it as a point and drawing it as a line are the two answers, and the trade between them is measurable in both directions.

families · Families Which way Sinusoidal stretches the ground. The major axis of Tissot's indicatrix at 180 points, drawn as a stroke rather than an ellipse because the argument is the orientation. Longer strokes mark more elongated indicatrices. The direction departs from the graticule by up to 44.4°, so the scale "along the meridian" is not the scale along the direction that is actually stretched most.

Distortion has a direction

Tissot's indicatrix is an ellipse, an ellipse has an orientation, and the orientation is never reported. On most projections it is not along the meridian, on a conformal one it does not exist at all, and both facts are computable from the same four derivatives as everything else.

distortion · Tissot "Within 0.01°" on the ground, at five latitudes. The same condition drawn at five latitudes, all at one scale. A degree of latitude is a fixed distance — 1106 metres here, varying by less than one per cent from equator to pole — while a degree of longitude collapses as cos φ, from 1113 metres to 289 at 75°. So the "circle" is an ellipse of 3.86:1 there, and it encloses 26 per cent of the ground the same condition covers on the equator. Even on the equator it is not round: M is smaller than N by the flattening, so the shape is 6694 parts per million shorter north–south than east–west.

A degree is not a unit of length

"Within 0.01 degrees" is a condition anybody can write and no instrument can measure. On the ground it is an ellipse — 1,106 metres north–south and 558 east–west at 60° — and even on the equator it is not a circle, because the meridian's radius of curvature is smaller than the parallel's by the flattening.

applied · Dataset A 20° shape across the antimeridian, in the space where the numbers live. Longitude runs across the page from −180° to 180°, which is where the failure is: the shape is one rectangle on the ground and two pieces in the numbering, and every operation that treats longitude as a real number sees the two. The bounding box comes out 359° wide instead of 20°, the planar area comes out 17 times too large because the shoelace encloses the complement, and the midpoint of a segment from one edge to the other lands 20015 kilometres away — the antipode of where it belongs. The true area, from the closed form, is 4,920,667 square kilometres.

The antimeridian is a cut in the numbers

A twenty-degree box across 180° has a bounding box of 359.4°, a planar area seventeen times too large, and a midpoint 20,015 kilometres from where it belongs — which is the antipode, exactly. Moving the cut moves the failure and never removes it, because a circle cannot be numbered by an interval.

applied · Dataset Every candidate fitted to one map's graticule, ranked by what is left over. The map is drawn in Conformal conic over a region 40° tall centred at 45° north, and the projection is not told to the fit. Each candidate is evaluated at the same 121 graticule crossings, its own parameters are searched, the best plane similarity between its output and the picture is removed, and what remains is drawn as a proportion of the map's own width on a logarithmic scale. Conformal conic fits to 1.3e-10, which is the arithmetic's floor; the next candidate is 3.7e+7 times worse.

A map does not say what it is

Every map on this site is one the site drew, from a projection it chose. Every map a reader has ever used is the other kind — a picture whose projection is a sentence in a corner, a legend, or nothing at all. The graticule is enough to recover it: fit every candidate to the crossings and rank what is left over.

wrong · Identify How large a map has to be before a wrong projection stops fitting it. Three rivals fitted to a Mercator graticule centred at 45° north, over regions from 1° to 70° of half-extent. The residual is the shape difference alone, with the best scale, rotation and offset removed, and both axes are logarithmic. The horizontal rule is a fifth of a per cent of the map's width — about the width of a drawn line on a printed sheet — and where a curve is below it, no measurement of that map can tell the two projections apart, however carefully it is made.

Two projections that cannot be told apart

Over a small enough region every projection is the same picture, so the question is how small. The answer is not one number: two conformal projections need twelve degrees of extent before their graticules can be separated, and two projections with different anisotropy separate below one.

wrong · Identify Two conventions for one indicatrix field on Mercator. The left panel is the convention nearly every published field uses: every ellipse drawn at the same area on the paper, so the picture carries the shape and throws the size away. The right panel draws the image of the same ground circle at every place, so the drawn area IS the areal scale factor. On Mercator the axis ratio spans 1.0000 and the areal factor spans 14.93, so the left panel shows nothing the other one shows. Neither caption states which is which, on any published map this collection has found.

The ellipses are a sample, drawn at a size somebody chose

Thirteen essays measure with the indicatrix and none audits it as an instrument. A published field has a gauge nobody states and a placement nobody states: on Mercator the standard convention draws twenty-five identical circles while the areal factor runs over a factor of 14.9, and the average a reader takes off any of these fields is between 22 and 64 per cent too high.

distortion · Tissot One map, two samples, two answers. The same Miller cylindrical projection carrying two sets of sample points of the same size. On the left the points are uniform over the SPHERE — equal ground area between them, which is what every mean in this collection integrates against. On the right they are uniform over the PAGE, which is what a raster, a pixel loop or any figure that walks its own canvas produces. The right-hand set crowds where the map stretches, and the mean angular deformation it returns is 18.0° against the left-hand set's 7.2°.

The sample was drawn on the page

Every mean in this collection integrates over the sphere, because that is where the ground is. A raster, a pixel loop and any figure that walks its own canvas integrate over the page instead, and the difference is exactly the covariance between the quantity being measured and the map's own area distortion — 7.2° of mean angular deformation on Miller becoming 18.0°.

distortion · Sampling

Named alongside it

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

VerificationAngular deformationConformalityEqual-areaMercatorAspectAzimuthalConventionPrincipal scale factorsToleranceAnisotropyAreal factor

All concepts