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

Design — where it appears

The deliberate choice of a projection's parameters for a stated region and purpose, as opposed to adopting a named projection at its defaults. A grid designed for one country can beat every named candidate, and the margin is the measure of what naming costs.

Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.

The scale spread of every grid Britain could have adopted. A grid is conformal by requirement, so its angular deformation is zero everywhere and the whole design problem is the spread of its one remaining number: the largest scale factor over the region divided by the smallest, in parts per million. The grid's own scale factor does not enter — multiplying every scale by a constant leaves the ratio alone, which is why it is chosen last. The bottom bar is Chebyshev's optimum, the conformal map of this region whose scale is constant on its boundary, which no map of any family can beat; the adopted grid sits 1.98 times above it. Measured over a stated box rather than a coastline, because a coastline would put the vendor's generalisation into the answer.

The best grid a country could have had

A national grid is a conformal map chosen for one region, so its whole design problem is one number: the spread of its scale factor. That number has a theoretical floor, this site can now compute it, and the adopted grid turns out to be either exactly optimal or half as good again — depending entirely on which box the country is declared to be.

practice · Grid The bound was spherical, and the country is not. Three numbers per region, all in parts per million of scale spread. The first is the Chebyshev bound computed on the sphere. The second is that same optimal map used on the ellipsoid, which is what adopting it would actually deliver. The third is the bound with the ellipsoid-to-sphere factor put into the boundary condition, which is the real optimum. The penalty for using the spherical answer reaches 1.22 times — while a named candidate barely moves, because an optimal map has cancelled its own variation and has nothing left to hide a new one in.

The bound on the body the country is on

The best conformal grid a country could have had was computed against a bound solved on a sphere, with a note saying the flattening was second order and unquantified. It is second order for a named projection — every family's best candidate moves by under 0.7 per cent — and it is 22 per cent for the bound, because an optimal map has already cancelled its own variation and has nothing left to hide a new one in.

practice · Grid Which scale fields a map could have, and which are only wishes. Liouville's equation — the Laplacian of log k equals 1/k² on the page — is the whole condition for a conformal map of a unit sphere to have a stated scale factor. The first two rows are the scale fields of real projections and they satisfy it to the differencing step. The rest are requests a designer might write, and every one of them fails — except one, which turns out to be a projection somebody already found. Asking for no distortion anywhere fails by exactly one, which is the curvature of the sphere.

Not every distortion can be asked for

Six essays have written projections as conditions and asked how much freedom a condition leaves. The reverse question has never been put: a cartographer knows what distortion they want, so can they ask for it? For a conformal map the answer is a single equation, it is the Theorema Egregium in disguise, and asking for no distortion anywhere fails it by exactly the curvature of the sphere.

choosing · Condition The widest zone a tolerance of 690 parts per million allows. At each latitude, the half-width at which a transverse Mercator grid with its scale factor rebalanced for that width reaches 690 ppm at its worst point. That tolerance is the one UTM actually meets at the equator, so the curve passes through UTM's own 3° there — and rises to 37.0° at 85° north, because a degree of longitude covers cos φ of the ground and the scale error goes as the square of the ground width. Six degrees is the answer at one latitude.

Sixty zones was a decision about one latitude

Twelve rungs price a grid, a zone, an origin and a reference, and every one of them works inside a single zone. The number of zones has never been asked about: six degrees meets its tolerance at the equator and is loose everywhere else, so a system spending the same tolerance evenly would use 51 zones at the equator and 8 at 82° — and UTM's worst error is 981 parts per million, not the 400 always quoted.

practice · Grid What a cut buys. The mean angular deformation of the interrupted sinusoidal against the total length of cut the interruption spends, for lobe counts from one to twenty-four. Goode's interruption — the one actually printed — is the marked point: it spends 100 thousand kilometres and returns 18.0°, where the even-lobed curve returns 9.2° for the same length. It is not on the frontier and it was never trying to be: its cuts are placed to keep continents whole.

What a cut buys

Six rungs count cuts and none measures one. A cut is a curve on the sphere with a length in kilometres, the shape distortion it removes is a falling function of that length, and the interruption everybody prints spends a hundred thousand kilometres to reach a figure the even-lobed curve reaches with sixty.

impossibility · Topology How large a blunder has to be before the test notices. A blunder of increasing size put into the least-checked observation of a braced quadrilateral, with the standardised residual it produces. The horizontal line is the critical value the test uses, and the vertical one is the minimal detectable bias — δ₀σ/√r, which is 87 millimetres for this observation and is computed from the network's DESIGN, before any observation is made. Below it nothing is flagged; above it everything is. The observation's redundancy number is 0.144, so it is checked by a seventh of an observation and hides six-sevenths of whatever is wrong with it.

The blunder the network cannot see

A least-squares adjustment has no concept of a mistake. The smallest blunder its test will find in the least-checked leg of a braced quadrilateral is 87 millimetres, and by the time it fires a station has moved by nearly ten times the accuracy the same adjustment reports for it.

practice · Reduction Four layouts, the same five stations, the same ten distances. Every panel has five stations, all ten distances between them, the same instrument precision and the same three degrees of freedom. The ellipses are the error ellipses of the adjusted coordinates, drawn at one common exaggeration, and they are computed from the geometry and the weights alone — no observation value enters any of them. The worst semi-axis runs from 11.1 millimetres to 232, a factor of 20.9, and the difference is entirely where the marks were put.

The network's answer is decided before it is measured

Nine rungs measure what an adjustment does with observations. Every quantity a specification is written about — the error ellipses, the redundancy numbers, the smallest detectable blunder — is a function of the geometry and the weights alone, and does not contain an observed value anywhere. Four layouts of five stations with the same ten distances differ by a factor of 20.9 in their worst coordinate.

practice · Reduction The same tiling, turned. The cube's eight vertices, ringed, and eight cities, with the aspect of the cell each city falls in written beside it. The two panels are the same tiling: the same cells, the same areas, the same shapes, in the same numbers. Only where they sit has changed, and the eight cities get cells whose areas spread by 1.26 in one and 1.39 in the other. Drawn on Mollweide, in which equal ground areas are equal page areas.

The orientation is a policy

A polyhedral cell system has three free angles nobody scores. They cannot improve it: rotating the solid rotates every cell rigidly, so the distribution of cell areas is identical for every orientation there is. What they decide is who stands on the bad cells — and the eight cities measured here get a spread of cell area of 1.00 under the best turn and 1.50 under the worst.

applied · Cells Six regions, before and after. Six circular regions of unequal size on the equal-area rectangle, and the same six after the cartogram of four cities has been solved. Each drawn area is exactly its base area times the density it was asked for. The four that grow stay recognisably round; the two that shrink are drawn out into shapes that no longer resemble what they were, and nothing in the construction chose to treat them differently.

A cartogram keeps the shapes it inflates

Seven rungs build cartograms and none reads one back. Reading means recognising a region and dividing its drawn area by its base area, and the construction is against the reader twice: the correlation between how much a region grows and how much of its shape it keeps is −0.996, and the base area a reader has to divide by is the map the cartogram replaced.

distortion · Cartogram

Named alongside it

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

ConformalityLeast-squaresNational GridPurposeScale factorScale spreadAdjustmentAreaBlunderCentral meridianChebyshev's criterionEqual-area

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