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

Scale — where it appears

The ratio between a distance on the map and the same distance on the ground, which is a field rather than a number. It depends on the point and, except where the map is conformal, on the direction as well.

Named by 12 essays across 6 fields — each of them below, with the objects they name alongside it.

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.

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.

choosing · Choosing Two explanations for one residual, against the size of the region. A map in the right projection whose control coordinates are on the wrong datum, and a map fitted with the wrong projection and the right datum, both measured as a residual after the reproduction's scale, rotation and offset have been removed. The datum shift's residual is flat: it is the same 1.3e-6 at every size, because a similarity fit is very nearly what a datum shift is and it absorbs the rest. The wrong projection's grows by three orders of magnitude with the region. They are the same number below about a degree, which is where a residual stops saying anything about either of them.

A residual has more than one explanation

The method names a projection by fitting every candidate to a set of control points and taking the smallest residual. It has never been asked what else a small residual could be. A map drawn in the right projection from coordinates on the wrong datum leaves a residual of one part in a million — indistinguishable from noise, at every region size, because a similarity fit absorbs a datum shift almost exactly.

wrong · Identify The least error a flat picture can have, against how much sphere it spans. The same configuration of places, shrunk about its own centroid so that every bearing is kept and only the span changes, with the least worst-case relative error of the best flat picture at each size. Both axes are logarithmic. The fitted slope over the rows below ninety degrees is 2.0246: the error falls as the SQUARE of the diameter. That is Gauss's theorem arriving as a number for a finite set — curvature is a second derivative, so its first effect on a distance is quadratic in the separation — and it is why a county fits on a sheet and a hemisphere does not.

How wrong a flat picture has to be

A determinant proves no flat picture of four places is exact and leaves the size of the failure where nobody can read it. Measured directly, the least error falls as the square of how much sphere the places span — fitted exponent 2.0088 — and the same exponent comes back from five different arrangements while the constant in front of it moves by a factor of seventeen.

impossibility · Embedding A radius of 37.7 km makes a flat picture of four cities exact. Each row is one pair of London, New York, Tokyo, Sydney. The bar is the range of distances between two places of radius 37.7 km — the true centre-to-centre distance, plus or minus 75.4 km — and the solid dot is where the best picture for places of that size draws the pair. Every dot is inside its bar, and the pairs at the bars' ends are the ones that fix the radius. The hollow dots are the picture that is best about ratios instead: its worst pair is 103.3 km off, so it would need places of 51.6 km. The best picture of points is not the best picture of places.

A place with a size can be drawn to scale

Points on a sphere have distances no flat picture holds. Places are not points: give every place a radius and the distance between two of them becomes a range, and some flat picture is right about every range once the radius passes a threshold — 38 kilometres for London, New York, Tokyo and Sydney, six metres for five towns in Britain, growing as the cube of the set.

impossibility · Embedding A curve built to have dimension 1.2619. The generator replaces every segment with four of equal length at headings 0, +60.0°, −60.0° and 0. Closing the displacement fixes the length ratio at 0.33335, and four copies at that ratio give a dimension of exactly log 4 / log(2 + 2 cos θ) = 1.2619. Nothing here is measured yet: this is the construction the measurement will be checked against. Drawn at depth 5, which is 1024 segments, with the second-level shape shown faint beneath it.

A line has a length only at a scale

A curve given by a formula can be sampled as finely as the picture needs. A map is not that: the geometry that reaches the page has been through an algorithm whose job is to throw most of it away. The first thing that goes is the idea that the line had a length.

applied · Generalise Six of eighteen, where either kind alone can hold only five. London, New York, Tokyo, Sydney, drawn holding five distances exactly — the thick edges — and one bearing exactly, the arrow from London to New York. Six exact constraints, where five distances is the most any flat picture of four places can hold and five bearings is the most it can hold of those. All six are satisfied to 1.9 × 10⁻¹¹. The sixth is not free room found by luck: a picture is blind to where it sits, how it is turned and how large it is, distances see the last of those three and bearings see the second, and a picture holding both kinds is blind only to where it sits. Tokyo–Sydney, dashed, is the distance left out; the picture is wrong about it by 6.9%.

A mixed picture holds one more, and what it is holding is north

A flat picture of n places can hold 2n − 3 distances exactly, and it can hold 2n − 3 bearings exactly, and it is tempting to read one budget spent twice. It is not one budget. A distance cannot see which way the page is turned and a bearing cannot see how large it is drawn, so a picture constrained by both kinds is blind to one freedom instead of two — and holds 2n − 2. For four cities that is six of the eighteen available where either kind alone can hold five, and nothing whatever can hold seven: nought of 31,824 choices.

impossibility · Embedding What a folded net is accurate to, beside what the map it carries is wrong by. For a cube globe of 100 mm radius: each tolerance of cutting and folding a paper net, as a displacement on the assembled sphere, and the seam corner of each of the three face maps beside it. The fold's own error is the largest at 0.955° — 106 km at Earth scale — because a degree of dihedral error moves the far edge of a face by a degree of the face's own radius, which for a cube is 54.7°. The conformal face map's corner is 0.769°, below the whole budget of 1.000°; the gnomonic's is 20.2°, twenty times above it.

The map is finer than the paper it is folded from

Thirteen measurements price a polyhedral map in angles, areas and lengths, and none of them is about paper. A hand-folded net is accurate to about a degree of dihedral angle, which displaces the far edge of a cube's face by 0.955° of arc — 106 km at Earth scale — and that floor does not shrink when the globe is made larger, because a degree is a degree. Against it the conformal face map's seam corner is 0.769° on a cube and 0.0040° on an icosahedron, where it is 165 times finer than the sheet can be folded to.

families · Polyhedral Straight baselines across a fringe of islands. A stated coast 220 km long, a gentle undulation 10 km in amplitude, with 8 islands off it: six between 5 and 12 km offshore and two rocks at 30 and 26 km. The dashed line is the twelve-nautical-mile limit measured from the low-water line of the mainland and every island. The solid lines are the straight baselines enclosing the most water under one reading of the article — segments of at most 24 nautical miles departing at most 20° from the coast's general direction — 6 segments, the longest 44.4 km — and the solid curve is the limit measured from them. Tinted: 988 km² of water landward of the lines, now internal waters. Shaded: 68 km² of sea beyond the dashed limit, against a territorial sea of 8,122 km². Neither rock is reached. The coast is stated, not surveyed.

A straight baseline moves more water inward than sea outward

The seventh article of the law of the sea lets a coast fringed with islands measure its territorial sea from straight lines joining chosen points, and gives no length, no angle and no scale for the choice. On a stated coast the lines a common reading permits add sixty-eight square kilometres of sea at the outer edge and turn nine hundred and eighty-eight into internal waters. Drop the test of direction and the outer rocks come into reach: eighteen hundred added, forty-six hundred enclosed.

practice · Boundary 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 The seam corner falls faster than the paper and crosses it at 720 faces. For an icosahedron globe of 100 mm radius with every face divided ν ways, the gnomonic face map's worst seam corner against what a hand-folded paper net is accurate to. On twenty faces the corner is 9.19°, 12.8 times the whole budget of 0.716°. The corner falls as ν to the power -1.97 and the fold's error as ν to the -0.97, so the two lines converge; the budget stops falling once the crease, a fixed quarter-millimetre radius of paper, is larger than the fold. The corner passes under it at ν = 6, 720 faces, where it is 0.315° against 0.321°.

The seam falls faster than the paper

Divide each face of an icosahedron ν ways and two things shrink at different rates. The fold's error falls as 1/ν, because it is the face's own radius times a degree. The gnomonic seam corner falls as 1/ν², because it needs both a tilt between faces and a distance along their edge. So the gnomonic, 12.8 times the whole assembly budget on twenty faces, goes under it at 720 faces on a 100 mm globe. No globe of any size needs more than 5,120, and what it costs is paper folded out of sight: 5.8 per cent of the sheet at that crossing.

families · Polyhedral Three selection rules, and what each one keeps. Keeping one feature in ten from a stated population whose size distribution has a Pareto exponent of a half — the exponent Töpfer's law is a theorem about. Keeping the largest carries 99.99 per cent of the total size and inflates the median feature by a factor of 95. A random sample keeps the median to 1.068 and carries 5.0 per cent of the total. The two rules are right about different things and there is no rule that is right about both, because the total lives in the tail and the median does not.

Which features survive is not a sample

How many features a scale can carry is answered and treats the population as a number. Which ones survive is a different question: keeping one feature in ten carries 99.99 per cent of the total length and inflates the median feature by a factor of 95, and the shape of the size distribution survives both exactly.

applied · Generalise How far the two routes end up apart. The distance between a line simplified directly at the final tolerance and the same line simplified through two intermediate products, as a multiple of the final tolerance, with a stated extra step applied to each intermediate. With nothing in between the two are the same line to the last bit, because Douglas–Peucker's outputs are nested. Rounding the intermediate to half the tolerance, smoothing it for legibility, or running it through a moving average each break that, and the last of them puts the final product 0.142 away from where a direct route would have put it — nine times the tolerance the product is published under.

Two routes to one scale

A national series is cascaded — the million is derived from the quarter-million, which was derived from the fifty — and the folklore is that the errors accumulate. They do not: Douglas–Peucker and Visvalingam both cascade to the same line the direct route produces, bit for bit, because both output a sublevel set of a per-vertex number. What breaks it is anything else in the chain, and a moving average puts the product nine tolerances away.

applied · Generalise

Named alongside it

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

ToleranceVerificationClosed formGeneralisationPurposeEmbeddingError budgetExponentSimplificationConstraintConvergenceConvergence rate

All concepts