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

Angular deformation against latitude, four projections

Drawn here at the parameters it defaults to, with every essay that calls it.
Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, mollweide, winkelTripel, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both.

The same quantity for mercator, gallPeters, mollweide, winkelTripel, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both.

It is drawn by distortion-figure with show: "distortion-compare" — one member of a family of 7 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

12 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, winkelTripel, robinson, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both. What each projection optimises

Every projection minimises something

A projection is the solution to an optimisation problem, and naming the objective explains more than naming the family. Some objectives are exact constraints, some are least-squares fits, and one is a table of numbers a man adjusted until it looked right.

 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. What is taught wrongly

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.

 Angular deformation around three standard parallels. Three equal-area cylindrical projections differing only in where they are exact — standard parallels at equator, 30°, 45°. Each has zero angular deformation at its own standard parallel and grows away from it in both directions. Choosing a standard parallel is choosing which latitudes to treat well, and there is no choice that treats them all well. The families

What a standard parallel buys

A standard parallel is a line where the projection is exact. Choosing one does not reduce the distortion — it decides where the distortion is zero and lets everything grow away from it.

 How much each projection inflates a cell, by latitude. Five patches of the sphere, each 20° by 10°, and the factor by which each projection enlarges them relative to the equatorial one. An equal-area projection sits flat on 1. Mercator reaches 15.4× at 70°, which is the mechanism behind every complaint about the size of Greenland. What is taught wrongly

Mercator against Peters

The most-argued question in cartography, conducted almost entirely without anyone measuring anything. Both projections are exactly what they claim, each destroys what the other keeps, and the numbers are computable in either direction.

 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 impossibility

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.

 Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, mollweide, winkelTripel, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both. Measuring distortion

The two ways a map is wrong

Angles and areas fail independently. A projection can be perfect about one and catastrophic about the other, and a single number summarising "how distorted" a map is has already thrown away the distinction that matters.

 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. What a machine does with it

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.

 Four ways to build the same equal-area projection. The cylindrical equal-area projection's own areal scale factor, measured from its Jacobian against the metric of the body it is drawn for. On a sphere with the spherical formula it is one everywhere, which is the control. Feed the same formula the geodetic latitude a coordinate actually carries and measure against the ellipsoid it refers to, and it is 1.00674 on the equator and 0.99332 at 88° — a spread of 1.34 per cent, on a projection whose entire purpose is that there is no spread. Rescaling to make the totals agree does not repair it. The authalic northing q/2 does, exactly. What is taught wrongly

Equal-area on the wrong body

The site's headline is that Web Mercator puts geodetic latitudes into a spherical conformal formula and stops being conformal. The same sentence is true with "equal-area" in it and nobody says it: the areal factor is 1.00674 at the equator, 0.99332 at 88°, and averages to almost exactly one — so every check that adds up areas passes while every cell is wrong.

 The half-extent at which each rival stops fitting a Gall–Peters graticule. For each candidate, the size of region at which its best fit to a Gall–Peters map first leaves a residual of a fifth of a per cent of the map's width. Below that size the two are the same picture. The numbers are half-extents in degrees of latitude, at 20° north, with the plane affine transformation removed; a bar at 90° is a rival that never separates at all within the range searched. What is taught wrongly

What a careless copy hides

A photocopier that stretches one axis is a nuisance to remove before a projection can be identified. Removing it costs more than it looks: an affine fit absorbs the entire difference between the cylindrical equal-area projections, so Gall-Peters and Behrmann become the same picture at any size, to sixteen decimal places.

 five planes a dataset might be stored in, scored on three operations. Each candidate measured over -10° to 30° east and 35° to 60° north: the worst areal error, the worst angular deformation, and the spread of the scale factor, which are what an area query, a shape and a distance respectively depend on. The best plane for areas is Gall–Peters, for shapes Lambert conformal conic, and for distances Lambert conformal conic — three different answers, and no fourth candidate would collapse them, because a projection exact in two of these columns has a = b = 1 everywhere and is the isometry Gauss's theorem forbids. area of a polygon costs 3.06× too large in the wrong plane; drawing a line between two points costs 194 km from the ground it claims. What a machine does with it

The operation decides the coordinate system

Five candidate planes over one region, scored on the three things a spatial operation depends on. The conformal conic wins shape and distance and is 11.7 per cent out on area; the equal-area member is exact on area and 38.9° out on shape. No candidate is exact in two columns, and no candidate ever will be, because one that was would be an isometry.

 An equal-area map nobody would publish. Mollweide, with a row-dependent horizontal displacement applied to the page afterwards. That plane map has Jacobian determinant 1 everywhere, so every areal scale factor is untouched: the worst departure from 1 anywhere sampled here is 6.0e-12, which is the arithmetic's own floor. It satisfies the equal-area condition exactly and completely, and it is a ruin. The angular deformation at 30°E 20°N has gone from 11.0° to 60.5°. What each projection optimises

Every equal-area map is every other one

Take Mollweide and slide every row of the page sideways by an amount that depends on the row. The result satisfies the equal-area condition to 6 × 10⁻¹², exactly as well as Mollweide does, and it is a ruin — the angular deformation at one ordinary point has gone from 11° to 60°. Equal-area is one equation, and one equation leaves a whole function free.

 The same tolerance, applied in two orders, at 65°. The faint line is the region's boundary as built, 1025 vertices across 400 km of ground. Both pipelines were given the same tolerance of 2000 m on the ground. Simplifying in degrees and then projecting keeps 311 of them; projecting into Mercator and then simplifying keeps 129; doing it on the ground itself, which no pipeline does, keeps 129. The two drawn lines separate by 1883 m, which is 94 per cent of the tolerance that was supposed to bound the whole operation. What a machine does with it

Simplification does not commute with the projection

A pipeline either simplifies the geometry and then projects it, or projects it and then simplifies. Both orders are in use, neither is recorded, and given the same tolerance in ground metres they keep different vertices — 129 of them on the ground, 367 in degree space at 80°, and 459 on an equal-area page.

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