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 7 essays across 4 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

The rung below proves no flat picture of four places is exact and leaves the size of the failure to a determinant nobody can read. 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 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

Every measurement on this site so far has been of a curve given by a formula, sampled as finely as the picture needed. 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 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 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

The rung below answers how many features a scale can carry 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.

GeneralisationClosed formSimplificationToleranceConvergenceLegibilityPurposeQuadratic lawResolutionVerificationAggregationAspect

All concepts