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

Principal scale factors — where it appears

The largest and smallest scale factors at a point, which are the semi-axes of Tissot's indicatrix and survive any change of coordinates. They are the singular values of the projection's Jacobian, and their ratio is what decides conformality and their product what decides equal area.

Named by 21 essays across 6 fields — each of them below, with the objects they name alongside it.

The indicatrix at 45°, 55° on Mercator. A circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.743 and b = 1.743; their product is the areal factor 3.040; and the maximum angular deformation is 0.00°. h and k are shown too, and depend on the coordinates rather than on the map.

Tissot's indicatrix

A tiny circle on the sphere becomes an ellipse on the map, and the ellipse's two axes are the whole story. Almost every published indicatrix is drawn without them, which discards the content and keeps the decoration.

distortion · Tissot 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 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 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 five named here are Mercator, Stereographic, Gall–Peters, Mollweide, Winkel tripel.

The trade-off is two lines

Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument.

impossibility · Curvature A 1,000 km bar on Mercator, drawn at 0° and read elsewhere. The same length of paper, carried up the map. Each pair of bars is the ground distance that length actually spans at that latitude — the filled bar along the parallel, the outline along the meridian. At 75° the reading along the parallel is 259 km against the 1,000 the bar claims, an error of 74 per cent. The two readings agree everywhere, because Mercator is conformal — so one number per latitude corrects any measurement taken off it.

A scale bar is right in one place

The bar in the corner of a world map is a picture of a distance, and it is a true picture along one line. On Mercator it reads 500 kilometres for a thousand at 60° north — and on an equal-area map it reads 500 one way and 2,000 the other, so the projection recommended for measuring is the one on which no single correction exists.

applied · Screen The least distortion possible over a 20° region. Scale factor along a radius of the cap, each projection normalised to unit scale at the centre. Chebyshev's criterion names the projection whose scale is constant on the boundary as the conformal map of least scale variation, and for a cap that is the stereographic projection centred on it — reaching exactly sec²(ρ/2) = 1.0311 at the rim, marked. Every other conformal projection drawn here rises past that line before it gets there. This is the only figure on the site showing an optimum rather than a comparison.

Scale distortion is the third failure

A map's stated scale is its scale at one place. Everywhere else it differs, by a factor that no conformal projection can avoid and that neither of the two usual distortion measures reports. It is the failure everybody uses and nobody counts.

distortion · Tissot The scale factor across Europe, from the middle outwards. Each band is the range the larger principal scale factor takes on the ring at that distance from the centre of Europe: 0 is the middle, 1 the frontier. two of the three projections drawn are conformal, and every one of those reaches its maximum at the right-hand edge, because the logarithm of a conformal map's scale factor is subharmonic. None of them reaches a maximum inside the region.

Where the worst point is

The largest scale error on a conformal map of a country is always on the frontier, never inside it, whatever the country's shape and whichever conformal projection was chosen. It is a theorem rather than a tendency, and it is the reason Chebyshev's criterion works.

distortion · Tissot 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 The image of a circle on Mercator, against its own indicatrix. A circle of three radii on the ground at 30°, 40°, projected exactly — the solid curve — against the ellipse the indicatrix predicts for it, dashed. The filled dot is the image of the circle's centre and the hollow one is the centre of area of what was actually drawn, which is not the same point. The departure runs from 3.14 per cent at 4° to 15.80 per cent at 16°, so it grows in proportion to the radius rather than to its square: halving the circle halves the relative error and does not quarter it.

The indicatrix is a limit

Tissot's ellipse describes an infinitesimal circle, and every published one is drawn finite. The error of the description falls as the radius rather than as its square — halving the circle halves the lie — and on the Robinson projection at a table entry it does not fall at all: sixty metres and six kilometres are both 1.17 per cent wrong.

distortion · Tissot The azimuthal family, as one function of one variable. Every azimuthal projection is a rule for how far from the centre to draw a point at angular distance ρ, and nothing else. five named projections are plotted as their radial functions — tan ρ, 2 tan(ρ/2), ρ, 2 sin(ρ/2), sin ρ — and the two dashed curves are the solutions of the two conditions, integrated from the centre with no more than "true scale at the centre" to start from. They land on the stereographic and the Lambert azimuthal to 8e-10. The names are the answers to the equations, not descriptions of viewpoints.

The azimuthal family is one function

Five azimuthal projections are taught as five viewpoints — from the centre, from the far pole, from infinity. They are one projection with five choices of how far out to draw a point at angular distance ρ, and each named property is a differential equation in that one function: f′ = f/sin ρ integrates to the stereographic projection and f f′ = sin ρ to Lambert's, from nothing but true scale at the centre.

families · Families Everywhere within 3,000 km of London, drawn in Lambert cylindrical. The set of places exactly 3,000 kilometres from London by the shortest route, projected point by point. On the ground it is a circle — every point of it is the same distance from the centre, in every direction. On this page the longest radius from the drawn centre is 4.17 times the shortest, so a reader with a ruler measures two different distances for one ground distance depending on which way the ruler points. The two extreme radii are drawn.

A circle of a distance is not a circle

Eleven essays in this field have followed a route across a map. A range ring is not a route — it is the edge of a set — and drawing one exposes a failure the route essays cannot: the same ground distance comes out 4.17 times longer in one direction than another on a common projection, and 13.03 times at 70° north.

paths · Reach 5 of 400 nearest-neighbour queries change answer in the plane. 40 sites and 400 queries over -20° to 40° east and 35° to 70° north, each query answered twice — once by geodesic distance and once by straight-line distance in the stored plane. They agree 98.8 per cent of the time, which is why the operation survives, and the 5 that differ are marked. The mechanism is not that the plane is wrong by a lot but that its scale factor varies: over this region it spans 139 per cent, and every disagreement is a contest closer than that — the worst margin measured is 6.4 per cent. The furthest a wrong answer is from the right one is 24 kilometres.

Nearest is a question about the metric

Asked in the plane the data is stored in, a nearest-neighbour query returns a different site for 5 of 400 queries over Europe on Web Mercator — and 103 of 400 on the plate carrée, with the wrong answer up to 309 kilometres further away. Every disagreement is a contest closer than the region's own scale spread, and over a city there are none.

applied · Dataset Everywhere Albers is exactly right, and the band round it. The set on which both principal scale factors are one — the only ground where a ruler on this map, at the map's own stated scale, measures the true distance in every direction. It is the parallels at 20.000° and 60.000°, drawn as a curve, with the band within 0.01 of true scale shaded round it. That band is 2.673 per cent of the sphere, and it narrows as ε as the tolerance tightens. The curve at its centre has no width at all, and no tolerance makes it have one.

The places where a map is exactly right

Nine essays on this ladder say a map cannot be right everywhere. None asks where it IS right — and the answer is a curve, a pair of curves, or two isolated places, never a patch. Measured across fourteen projections the set's neighbourhood shrinks with an exponent of 0.48, 1.0 or 2.0, and the value the impossibility forbids is 0.

impossibility · Curvature One instrument, one covariance, drawn where Mollweide puts it. The same measurement is made at every point: 5 m east by 5 m north, uncorrelated, which on the ground is a circle. Each ellipse is that covariance pushed through the projection's own Jacobian and drawn 26,000 times life size. On Mollweide the axis ratio reaches 4.15, so an instrument that is equally good in every direction is drawn as though it were not.

An error ellipse is an indicatrix

A positional covariance pushed through a projection is the same matrix sandwich that produces Tissot's indicatrix, so the error ellipse drawn on a map and the distortion ellipse drawn beside it are the same ellipse. A five-metre circular accuracy is drawn at an axis ratio of 3.04 on one common projection — and the same projection draws a genuinely lopsided 304 by 100 metre error as a perfect circle.

distortion · Precision Vesta, with its own coordinate lines. Vesta drawn in an orthographic projection — every figure here is a projection, including this one — with the parametric coordinate lines its published coordinates are written in. Its three semi-axes are 286.3, 278.6 and 223.2 km, so the equator is an ellipse rather than a circle and the body has no axis of revolution to define a latitude against. The surface's own radius runs from 223.2 to 286.2 km, a ratio of 1.282, and the outward normal departs from the direction out of the centre by up to 14.10°.

A map of a body with three axes

Every projection on this site is written in a latitude, and a body with three unequal axes has none: the coordinate lines are not perpendicular, Lambert's equal-area formula spreads areas by 28 per cent on Vesta, and the equal-area map that does work has a different one on every meridian.

datums · Bodies Four cities, met — the triangular, x first construction. Every cell drawn here holds the same area of ground and a different area of page: the map has been constructed so that its areal scale factor is the density it was handed, cell by cell, over a contrast of 25.99 to one. The residual against that target is 5.1e-6, measured from the map's own derivatives rather than from the construction. What it cost is the shape: the redistribution alone reaches 139.1° of angular deformation and averages 70.4°, on top of whatever the equal-area projection under it was already doing.

A map drawn to a density it was handed

Two hundred and twenty-six essays measure distortion after the fact. This one specifies it: a density is handed to a map as a boundary condition, the areal scale factor comes out equal to it to five parts in a million, and every other invariant the site owns becomes the price.

distortion · Cartogram

Every density can be met and none is free

Four maps of the same data, all of them correct, charging between 57.6° and 104.4° of angular deformation for it. There is no such thing as the cartogram of a density — there is an infinite family, and somebody picked a member of it without saying so.

distortion · Cartogram

Everything else on the page pays for the areas

A cartogram gets one quantity exactly right and every other reading a page supports is collateral. A ruler on it is out by 35 per cent after the most generous calibration available, and ten of sixty triples of places change which one is in the middle.

distortion · Cartogram

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

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

The fourth number the ellipse does not carry

A Jacobian has four independent entries and an indicatrix reports three. The missing one is the rigid rotation in the polar decomposition A = R·S, the indicatrix is exactly S, and R is what turns north into grid north: for the transverse Mercator it agrees with the survey formula for convergence to 5 × 10⁻¹⁰ degrees, from a different library and a different derivative.

distortion · Tissot

Named alongside it

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

Angular deformationConformalityScale factorEqual-areaJacobianVerificationTissot's indicatrixToleranceMercatorAnisotropyAreal factorClosed form

All concepts