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

Measurement — where it appears

A number obtained by a stated procedure applied to a stated object, as distinct from one recalled or asserted. The rule this collection runs on is that a projection's property is measured before it is printed, so *conformal* is a result rather than a label.

Named by 10 essays across 6 fields — each of them below, with the objects they name alongside it.

What a 3-figure grid reference names. The square is the 100-metre cell a 3-figure reference stands for, and the filled dot is the point the reference was made from. A reference is TRUNCATED rather than rounded, so the digits name the square's south-west corner — the open dot — which here is 9.1 metres west and 82.8 metres south of the point. Rounding would have landed on the nearest corner instead, 9.1 metres away east–west. The truncation is deliberate: it is what makes a shorter reference a larger square containing the same point.

A grid reference names a square

This ladder has priced everything about a grid except how a coordinate on it is written. A grid reference is truncated rather than rounded, so it names the south-west corner of a square rather than a point in it — and a population of references is displaced half a cell each way, which is a bias rather than scatter and does not average out.

practice · Grid The same six contours on Mercator and Lambert cylindrical. The value of a harmonic sum with a summit and a basin in the northern mid-latitudes travels with the point, so the set of points at a stated level is the same set on every map and each contour is exactly right on both panels. Everything a reader measures from them is not: the spacing between neighbouring contours, their lengths, and the area between two of them all change from one panel to the other, and the two panels are the same field.

The contour is right and the reading is wrong

There is exactly one thing about a field that no projection can get wrong: which points share a value. The contour lines on any two maps of the same field are the same set of points. Every quantity a reader takes off them — the spacing, the length, the area between two of them, the hypsometric curve — is not, and the two kinds of map get different ones wrong.

distortion · Gradient Bigger triangles, and how much bigger depends on what is fixed. How well a survey can resolve Gaussian curvature, against the side of its triangles, under three things being held fixed. One triangle: the accuracy improves as the inverse SQUARE of the side, fitted exponent -2.000. A chain of fixed length, which is what every great arc was: bigger triangles mean fewer of them, and the exponent is -1.503 — exactly three halves. A network covering a fixed area: -0.999, exactly one. The trade depends on what a survey is short of.

How many triangles it takes

Pricing one triangle left the fact that a survey observes hundreds unaccounted for. Averaging n of them divides the noise by √n, and n is not free: a chain of fixed length holds fewer big triangles than small ones, so the accuracy improves as the side to the power three halves rather than two. Struve's 141 triangles of forty kilometres are worth exactly Gauss's seven of eighty-five.

impossibility · Curvature Every projection's angular deformation, and the tolerance that judges it. The whole library on one logarithmic axis, with the tolerance drawn as a line. The population is bimodal: the projections that satisfy the condition sit at 2.09e-6 and below, the ones that do not at 3.85e-1 and above, and there is nothing between. The tolerance could be moved anywhere in that gap — a factor of 1.84e+5 — without changing one verdict. Larger marks are projections that claim the property.

The tolerance that decides the verdict

Every verdict handed out about a projection's claimed property rests on a tolerance chosen once, long ago, at sixty times a measured noise floor. Swept, it decides nothing: the population is bimodal, the tolerance sits in a gap 238,000 times wide for conformality and 646 million times wide for equal area, and the verdict with the least room is a passing one whose margin is the arithmetic's rather than the map's.

wrong · Audit A chain of stations, its positions' uncertainty and its baselines'. Nine stations held at the left-hand one, with every neighbour, second neighbour and third neighbour observed. The filled ellipses are each station's own uncertainty, which grows without limit as the chain runs away from the point it is held at — 7.5 mm at the near end and 574 mm at the far. The open ellipses above each leg are the uncertainty of the baseline, drawn at the same scale: they hardly grow at all, because almost everything that is wrong with one end is wrong with the other in the same direction.

The difference of two coordinates

Three essays give a single coordinate a width. Every practical use of one is a difference of two — a distance, a bearing, a movement, an area — and the width of a difference is not the two widths combined, because the errors are not independent. Far from its datum a one-leg baseline is six times more certain than the positions it joins.

distortion · Precision 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 The same deflection, from a compact mass and from a broad root. Each curve is the deflection of the vertical across a mass buried 8 km down, with the mass solved so that all of them peak at 10″. The narrow one is the buried sphere the earlier measurement used; the broad ones are crustal roots 40 and 160 km wide, modelled as that mass spread along a line. They agree where it matters most and disagree everywhere else: the signal is 30 km wide for the compact body and 188 km for the widest, which is the difference the plumb line feels as it descends.

A mountain is not a buried sphere

The plumb line's drift was measured over a compact buried body and grows as the 0.69 power of the column's height — an exponent that is a statement about how quickly a buried sphere's field weakens with distance rather than about mountains. Spread the same mass into a crustal root and the exponent climbs to 0.84, while the proportionality to the deflection survives exactly.

datums · Height The picture is kept, at four tolerances. One closed curve of 3001 vertices, simplified at four tolerances. Douglas–Peucker's promise holds in every panel: no discarded vertex is further than ε from the line drawn in its place, measured at 0.1158 against 0.128 in the last. The picture survives. The enclosed area does not: it falls by 5.43 per cent, and it falls rather than wandering, because cutting a corner takes area off and never puts it back.

A tolerance is a promise about the picture

Douglas–Peucker guarantees exactly one thing: no vertex it discarded is further than ε from the line drawn in its place. It says nothing about the enclosed area, nothing about which side of the boundary a point ends up on, and nothing about whether the curve still fails to cross itself — and all three are what the geometry is usually being asked.

applied · Generalise A territory, and two enclosures somebody might score instead. A stated territory at 45° north — an ellipse twenty-eight degrees long and eight wide, its long axis bearing 45° — with its latitude–longitude bounding box and the smallest cap around it, on a globe. The territory fills 40.6 per cent of the box and 28.6 per cent of the cap. Scoring the library over each names a winner: an azimuthal for the territory itself, an azimuthal for its bounding box, an azimuthal for the smallest cap around it. The cost of working from the box rather than the territory, measured as the box's winner scored on the territory against the territory's own winner, is a factor of 1.0584.

The region was a rectangle in a projection

Every verdict this collection hands out is a verdict about a region, and almost every region it has been given is a latitude–longitude box. A box discards a territory's shape, which costs a few per cent, and it records the territory's width in degrees of longitude, which costs up to a factor of 1.72 and changes which family wins on four of forty stated territories. Convert that width to arc and the box gives the right answer every time — so the comparison had a projection in it before it began.

wrong · Audit Two places, one written coordinate. The lattice a coordinate written to 5 decimal places lives on, at 45° of latitude, where a cell is 1.11 metres north to south. Two real places a fifth of a cell apart are drawn as open marks and the single value they both round to as a filled one. The distance between them, computed from what was written, is exactly zero — an error a noise model cannot produce, since independent noise never puts two different points in the same place.

Rounding is not noise

Eight rungs treat a coordinate's error as noise that averages down. A published coordinate has a second error that does not: it is deterministic, it is shared between every point in the same cell, and at five centimetres apart ninety-three per cent of pairs come out as the same place — which no amount of independent noise can produce.

distortion · Precision

Named alongside it

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

VerificationConformalityConventionPrecisionToleranceAveragingClosed formEqual-areaError budgetGeneralisationInvariantNoise

All concepts