Cartographic Projection
every map is wrong, and by exactly how much
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The thread: The trade-off is forced — page 1

Conformal means the principal scales are equal; equal-area means their product is one. Both at once forces them both to one, which is an isometry, which cannot exist. The trade-off is two lines of algebra. Essays 1 to 24 of 112.
Angular deformation against latitude, four projections. The same quantity for Mercator, Gall–Peters, Winkel tripel, 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.

 6 projections of the same sphere. The same graticule under Equirectangular, Mercator, Mollweide, Sinusoidal, Robinson, Winkel tripel. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve. What each projection optimises

Which projection is best

An incomplete question, and the incompleteness is the answer. Every projection preserves something and destroys something else, so the comparison worth making is between a projection and a purpose, not between two projections.

 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.

 6 projections of the same sphere. The same graticule under Robinson, Winkel tripel, Mollweide, Mercator, Gall–Peters, Eckert IV. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve. What each projection optimises

Compromise projections

A projection that preserves nothing exactly can distort everything less than one that preserves something exactly. For a general-purpose world map that is the right trade, and it is why the two most widely used ones today have no exact property at all.

 Fitting a sphere to 10° of latitude at 54.5°. The Earth's Gaussian radius of curvature across the band, with two candidate spheres drawn against it. The global mean radius is 6371.0 kilometres and sits 2203 parts per million away from the ground here — 2.2 metres in every kilometre measured. The best local radius is 6385.0 kilometres and has no bias at all by construction, leaving 325 parts per million of spread that no sphere can remove, because the curvature varies across the band and a sphere's does not. The gain is a factor of 6.9. What the numbers refer to

A datum is fitted to a region

Forty countries adopted forty ellipsoids, and the usual explanation is that measurement was poor. It was not — a figure fitted to one country beats the global one over that country by a factor of seven, and what it removes is a systematic bias of two metres in every kilometre.

 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. What a machine does with 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.

 One 20° × 10° cell at 60°, under four projections. The same patch of the sphere, drawn by four projections and scaled to fit. The shapes differ and so do the areas: the figure under each panel is the areal inflation relative to the same cell at the equator, so 1.00 means the projection treats the two fairly. What is taught wrongly

The projection that shows true size

There is no such thing, and the phrase hides a real question. Equal-area projections preserve area and destroy shape; nothing preserves size in the sense the phrase implies; and the tools that make the point best are not maps at all.

 Angular deformation against latitude, four projections. The same quantity for Mercator, Gall–Peters, Mollweide, Winkel tripel, 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.

 The scale factor was chosen, and the choice is a V. The worst departure of the grid scale factor from unity across a region 9° wide and 9° tall, for every scale factor between 0.998108 and 1.0002, on a central meridian of 2°W. A tangent projection touches at one meridian and is too big everywhere else, at 1987 ppm. Scaling the whole grid down slides that interval until it straddles unity, and the minimum sits at 1/√k_max = 0.999008102, where the worst departure is 993 ppm — a factor of 2.00, which is the most this construction can buy and is reached exactly. The published value marked beside it was chosen the same way, for this region, in the 1930s. Grids, and what a survey does

The scale factor was chosen

A grid's scale factor is not a constant of nature. Britain's 0.9996012717 is the reciprocal square root of the worst distortion a tangent projection would have had over the country, it halves the worst-case error exactly, and the halving is the most that construction can ever buy.

 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. What a machine does with it

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.

 What 4 parts per million of network strain leaves behind. Each arrow is where the best-fitting seven-parameter transformation leaves a marker, over 36 markers laid out as a grid across OSGB36's ground. The RMS residual is 1.64 metres and the worst is 3.00 metres. Seven parameters span the constant and linear parts of a displacement field; this one is quadratic, so no choice of the seven can reach it. The arrows are drawn 27× life size. What the numbers refer to

Where a fit leaves residuals

Seven parameters can carry a rigid motion and a size exactly. A triangulation network is neither, so the best possible transformation between two datums leaves metres on the table — in a pattern, not as noise — and which seven parameters come out depends on where the markers were.

 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 distortion

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.

 The least distortion possible over a 30° 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) = 1.0718 at the rim, marked. Every other conformal projection drawn here rises past that line before it gets there. It shows an optimum rather than a comparison. The impossibility

Total curvature and the scale rule

The impossibility has a size. A region covering a fraction of the sphere carries a fixed amount of curvature that any flat map must absorb, and for a circular region the least distortion any conformal projection can achieve is a closed form nobody can beat.

 The least distortion possible over a 30° 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) = 1.0718 at the rim, marked. Every other conformal projection drawn here rises past that line before it gets there. It shows an optimum rather than a comparison. What each projection optimises

Chebyshev's criterion

The only optimality theorem in the subject. Among all conformal projections of a region, the one with least scale variation is the one whose scale factor is constant on the boundary — a criterion with a proof, a unique answer, and a test anyone can run.

 Sinusoidal, cut into 6 lobes. six lobes, cut through the oceans so each continent stays whole. Each lobe is the same projection about its own central meridian, so each point is near a line where the shape distortion vanishes: the mean angular deformation over the mapped world falls from 38.6° uninterrupted to 17.8° here. What is given up is that the map is no longer one surface — the graticule stops at every cut because there is nothing on the other side of it. What each projection optimises

Giving up continuity

Cutting a map into lobes really does reduce the distortion, by a factor that can be measured. What is paid is that the map stops being one surface — and the size of the tear is the number that pictures of interrupted maps never carry.

 What grid spacing a tolerance buys. The worst bilinear interpolation error across OSGB36's ground, against the spacing of the table it is interpolated from, both axes logarithmic. The smooth shift falls by exactly four for every halving, which is what second-order interpolation does. Adding a 30 centimetre ripple 2 degrees across — three parts in a thousand of the 99 metre shift it rides on — moves the spacing needed for 20 millimetres from 1° to 0.125°, which is 64 times as many nodes. What the numbers refer to

When a formula is not enough

National mapping agencies distribute datum shifts as tables of numbers on a grid rather than as parameters. What decides the spacing that table needs is not the size of the shift — a hundred-metre shift tabulates coarsely, and a thirty-centimetre ripple riding on it does not.

 A feature across nine tiles at zoom 4, and the nine labels it gets. A rectangle from -10° to 30° east and 36° to 62° north, drawn in the projected plane with the tile boundaries over it. A renderer that draws each tile from the geometry inside that tile puts the label at the centroid of the piece, marked hollow; the whole feature's centroid is the filled mark. The furthest piece's label is 1863 kilometres from it. The pieces' areas sum to the whole to 2.2e-16 of a relative part, so the clipping is exact and the displacement is the operation rather than an error in it. What a machine does with it

A tile is drawn without its neighbours

Independence is what makes the scheme scale: one request touches one square of the world. It also means a feature crossing nine tiles is nine features, each labelled at its own centroid — up to 1,863 kilometres from the whole one — and the displacement is bounded by the size of the feature rather than the size of a tile, so it grows as the reader zooms out.

 What a grid of one's own is worth. For every candidate central meridian, the best worst-case distortion achievable over the region once the scale factor has also been optimised — so each point on the curve is already the bottom of its own V. The minimum is at 2.9°W with k₀ = 0.9993009, giving 700 ppm. The dashed line is UTM zone 30, whose meridian and scale factor were chosen for no region in particular, at 1059 ppm. The national grid is better by a factor of 1.51 — which is the whole of the answer to why a country publishes a grid rather than using the zones, and it is a smaller factor than the argument is usually made to sound. Grids, and what a survey does

Designing a grid for one region

A country with a grid of its own beats the international zone that would otherwise carry it by a factor of 1.51. That is worth having and it is much less than the argument is usually made to sound, and the difference between the two numbers is the whole case for and against national grids.

 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.

 Both zones are right, and one of each is not. A 20 km baseline straddling the boundary between UTM zones 31 and 32 at 52°N, computed from grid coordinates three ways, on logarithmic bars. Taking both ends in zone 31 and taking both in zone 32 give answers 0 nanometres apart — the resolution a double has left after carrying a six-figure easting rather than a disagreement — so a job may use either and the overlap belt every zone publishes exists to let it. Taking each end from the zone it nominally belongs to gives 392 km, because the two eastings are measured from meridians six degrees apart and subtracting them measures nothing at all. The same ground point is 706 km east in one zone and 294 km east in the other, and both coordinates are correct. Grids, and what a survey does

Where two zones meet

A twenty-kilometre baseline across a zone boundary is computed correctly in either zone — the two answers differ by nanometres — and taking one end from each returns 392 kilometres. Both zones are right and the seam between them is not a small error but a category one.

 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. The families

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.

 A grid scaled to the ground, and where it stops being one. A surface coordinate system on the British National Grid: the grid multiplied by 1.0004044, the reciprocal of the combined factor at a project origin 120 m above the ellipsoid, so that a grid distance equals a ground distance there. It is exact at the origin by construction and wrong everywhere else, and the two directions are not comparable — the axis is logarithmic across five decades. 60 km due east the set-out distance is out by 5559 mm, and the same distance due north by 20.5 mm. The scale factor of a transverse Mercator depends on the easting and almost not at all on the northing, so a system sold as good "within a few kilometres" has a useful area that is a strip rather than a circle. Grids, and what a survey does

A grid scaled to the ground is not a map

Construction sites work in coordinates where a tape reading equals a computed distance, which is achieved by multiplying the national grid by a constant. The result is exact at one point, degrades east–west forty times faster than north–south, and is not a map projection at all.

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