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

Curvature — where it appears

How sharply a surface bends, measured intrinsically as the product of the two principal curvatures at a point. Gauss's theorem says it can be computed without leaving the surface, so a sphere's cannot be made zero by any map, and that is where the whole impossibility comes from.

Named by 9 essays across 7 fields — each of them below, with the objects they name alongside it.

Three distances, two degrees of freedom. The Chamberlin trimetric construction with its three centres marked. A point is placed by intersecting circles of its true distance from each centre — but three circles in a plane do not meet in a point, and the three candidate positions obtained by taking the conditions two at a time are 17.0 km apart at the sample point. They are drawn here magnified 120 times about their own centroid, which is where Chamberlin's rule puts the point. The residual is not an error in the arithmetic; it is the amount by which the sphere refuses to be a plane, and it grows as the cube of the size of the region.

Three conditions are one too many

Two distances fix a point in a plane and a third has no freedom left to be satisfied with. Chamberlin's trimetric construction averages the three positions that satisfy two conditions each, and the spread between them — never zero anywhere, 22 km over North America, growing as the cube of the region — is the price of the extra clause.

choosing · Condition Everywhere Albers is exactly right, and the band round it. The set on which both principal scale factors are one — the only ground where a ruler on this map, at the map's own stated scale, measures the true distance in every direction. It is the parallels at 20.000° and 60.000°, drawn as a curve, with the band within 0.01 of true scale shaded round it. That band is 2.673 per cent of the sphere, and it narrows as ε as the tolerance tightens. The curve at its centre has no width at all, and no tolerance makes it have one.

The places where a map is exactly right

Nine essays on this ladder say a map cannot be right everywhere. None asks where it IS right — and the answer is a curve, a pair of curves, or two isolated places, never a patch. Measured across fourteen projections the set's neighbourhood shrinks with an exponent of 0.48, 1.0 or 2.0, and the value the impossibility forbids is 0.

impossibility · Curvature Where a ridge, a valley and the pass between them curves the other way. The Gaussian curvature of a stated terrain over a 80-kilometre window, with the sign shown by the colour and the size by the ink. Positive on the summits and in the hollows, negative everywhere between — which is most of it: 79 per cent of the curved area. The extremes are 2.2e+4 times the Earth's own curvature, which is the quantity every argument on a sphere takes to be the curvature of the thing being mapped.

Where the surface curves the other way

The impossibility of a perfect map is usually argued on surfaces whose curvature is positive everywhere. The surface a map of the ground actually depicts is not one of them: a stated terrain is saddle-shaped over 79 per cent of its curved area, its curvature runs to twenty-two thousand times the Earth's own, and even a single smooth hill is concave over 89 per cent of itself.

impossibility · Curvature The shortest route, and the shortest route a vehicle can fly. A leg of 60 kilometres for a vehicle whose minimum turning radius is 5, arriving on a heading 120° off the line and required to leave on one -60° off it. The straight line is the geodesic; the curve is the shortest curvature-bounded path, which is Dubins's RSR — a turn, a straight, a turn — at 67.29 kilometres against 60. The second curve is the runner-up word, drawn to show that the choice between them is a real one rather than a formality.

The shortest route a vehicle can fly

A shortest path under a metric says nothing about whether the thing travelling can follow it. Bound the curvature and the route depends on two headings as well as two positions: the turning cost is a fixed 1.81 kilometres whatever the leg length, so it is 18 per cent of a short leg and 0.28 per cent of a long one, and the whole of it vanishes when the vehicle happens to be pointing the right way.

paths · Paths The indicatrix is a limit, and here it is being taken. The axis ratio of the image of a circle of radius ρ, measured from the picture rather than computed from the derivatives, against the ratio the derivatives give. It converges, and it converges as ρ² — each halving of the radius quarters the departure. At the smallest radius drawn the two agree to 7.6e-4 relative. This is what "the indicatrix is a limit" means when it is working: there is a number for the ratio to settle on, and it settles.

The indicatrix at a point that has none

Ten essays on this ladder take the derivative for granted. Rung three proved the indicatrix is a limit and every figure since has assumed the limit exists. There are points on the maps this collection draws where it does not, and they are not exotic: the horizon of a gnomonic map, and every corner of every polyhedral net.

distortion · Tissot The part of a request no conformal map can supply — scale falling with distance from the centre. The difference between the requested scale field and the nearest achievable one, over the patch, with the sign shown by colour. The RMS is 1.81e-1 in the logarithm of the scale and the worst single point is 4.91e-1. This is not an error: it is the part of the request that no conformal map of any kind can grant, and its SHAPE is the answer — it says where the request was impossible, which a single number cannot.

The nearest map to an impossible request

Rung seven found that not every distortion can be asked for. It never asked what happens when one is asked for anyway — and the answer has a shape: the achievable fields are the solutions of an elliptic equation, a request is a point off that set, and the nearest point leaves a residual whose floor is the curvature rather than the size of the ask.

choosing · Condition Where the summits are, and where the water divides. A stated range 100 km across and 80 km long, its summits 3,000 m high and about 35 km from the western edge, with a plateau falling gently to the eastern edge. Over 40 km of the range the plateau carries a broad rise up to 18 km east of the summits, and a gorge cuts through the range in the middle of that stretch. The dashed line runs through the highest summits; the solid line is the watershed, computed from the land. Lightly shaded ground drains west. Where they part, the darker ground lies east of the summits and drains west: 331 km², reaching 18.1 km from the summits. The terrain is stated, not surveyed.

The highest summits stay where they are, and the watershed does not

A treaty that puts a boundary on the highest crests which divide the waters names one line where the water divides on the summits and two where a river has cut back through the range. On a stated range the two part by up to eighteen kilometres. They also differ in kind: an error of ten metres in every height moves the summits three metres and the watershed on the plateau more than a kilometre.

practice · Boundary Twenty-four versions of one shape, and not one of them gains area. The same closed boundary rotated twenty-four times and simplified at the same tolerance. If the area error were noise the values would straddle zero and their mean would fall towards it; they do not. Every one is negative, the mean is -0.4644 per cent, and the mean is 71 standard errors from zero. A bias of that size cannot be removed by averaging over more boundaries, which is the only defence anybody has against a rounding error.

A thousand features are wrong in the same direction

The area a simplification costs is unpredictable in sign for one feature. Over a population it is not: twenty-four presentations of one shape all lose area, the mean is seventy standard errors below zero, and no amount of aggregation removes it.

applied · Generalise An office that overshoots headlands draws every coast wrong by that coast's own curvature. Four coasts at one map size: the compiled coast whose seams are counted, and three control coasts an office might trace with the truth known — an ellipse as thin as the counted coast (axis ratio 2.88), a nearly round one (1.12) and a coast of five headlands. Solid: each coast. Dashed: the same coast as an office would draw it whose every hand pushes a headland out and pulls a bay in by a fixed amount per unit of curvature, exaggerated a hundred and twenty times so that it shows. The curvature's spread round each coast, its variance in units of the coast's size: the sheet being counted 16.9, control, as thin as the target 15.6, control, nearly round 0.1, control, five headlands 29.8. The habit's size on a coast follows that spread, and the habit's slope does not change from coast to coast.

A known coast charges an office's habit, and a bending one takes it out

An office whose hands all share a habit leaves it in every tracing of every sheet, where the spread between hands cannot see it. One control sheet — a coast known independently, traced the office's usual way — reads the habit's size and restores the charge, if the habit is a drift. If the habit answers to the coast's own curvature, a size read on a rounder coast charges a thinner one at little more than half, but the same control read differently does better than charge the habit: its departures, set against its curvature, give a slope that is the office's own on every coast, and subtracting it leaves about one per cent of the habit in the mean. The slope also tells the two kinds of office apart. A coast that hardly bends teaches neither, and corrects worse than nothing.

wrong · Identify

Named alongside it

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

ConformalityLeast-squaresResidualToleranceAreaBiasConditionConstraintConventionGeneralisationOptimisationTheorema Egregium

All concepts