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

Reach — where it appears

The set of places within a stated travel cost of somewhere — a distance, a time, a fuel load — which is a two-dimensional region rather than a route. Its quantities are area, boundary length and membership, none of which a projection preserves in the way it can preserve a route's position.

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

Everywhere within 3,000 km of London, drawn in Lambert cylindrical. The set of places exactly 3,000 kilometres from London by the shortest route, projected point by point. On the ground it is a circle — every point of it is the same distance from the centre, in every direction. On this page the longest radius from the drawn centre is 4.17 times the shortest, so a reader with a ruler measures two different distances for one ground distance depending on which way the ruler points. The two extreme radii are drawn.

A circle of a distance is not a circle

Eleven essays in this field have followed a route across a map. A range ring is not a route — it is the edge of a set — and drawing one exposes a failure the route essays cannot: the same ground distance comes out 4.17 times longer in one direction than another on a common projection, and 13.03 times at 70° north.

paths · Reach Which of eight places is nearest, decided on the ground and decided on the page. Every cell of the window shaded by which of eight places across Europe is nearest on the ground, with the cells the page's own answer would hand to a different site drawn in the failure colour. Plate carrée misassigns 13.67 per cent of the window's ground area — 3,058 thousand square kilometres. The misassigned cells are not scattered: they lie in bands along the boundaries, which is what a systematic error looks like and what a sampled test of a few query points is least likely to find.

Nearest of many is a partition

One reach question with one source is a disc. With several sources it is a division of the whole surface, every place belonging to whichever source is nearest — and computing that division in the plane the data is stored in hands away between 0.75 and 22.16 per cent of the ground, in unbroken strips up to 1,591 kilometres across.

paths · Reach Everywhere within 200 km of the route from London to Tokyo. The shortest route between the two places, and the set of places within 200 kilometres of it, with both edges computed on the sphere and then projected. On the ground the set holds 3.949 million square kilometres — the band 3.823 and the two end caps 0.126, which between them make one disc of the corridor's own width. Drawn in Mollweide the corridor is visibly wider at one end than the other, and the ground it stands for is not.

A corridor has a width the page cannot keep

Buffering a line is the most-run operation in spatial analysis and the corridor it produces is a reach set with two failures nobody separates: stroked on the page, one width covers 190 to 471 kilometres of ground along a single route; computed in closed form, the formula stops being the area at a width the route's own length fixes, and eventually claims more ground than the sphere has.

paths · Reach How far each page reorders the shapes. The number of pairs of shapes whose order on the page differs from their order on the ground, out of 36, for ten projections. Four of them put a shape other than the geodesic disc at the top — Equirectangular, Lambert azimuthal equal-area, Robinson, Miller cylindrical — which means the shape that attains the isoperimetric bound on the sphere is not the most compact thing on those sheets. The projection with none is not the equal-area one; it is whichever one's stretching happens to leave this particular set of shapes alone.

The most compact shape depends on the paper

A geodesic disc attains the isoperimetric bound on a sphere — its score is one, exactly, at any radius. Score the same nine regions from their images on ten projections and four of them put something else on top, an oval and its own 45° rotation come out 4.7 per cent apart, and the projection that preserves the order is not the equal-area one.

paths · Reach The score does not settle at any resolution. The compactness of one stated boundary — a circle with cosine ripples at eight geometrically spaced wavenumbers, so it has structure at every scale — read at sixteen vertices up to two thousand and forty-eight. The ground score falls from 0.980 to 0.834, and it keeps falling: the boundary's length grows without bound as it is resolved while the area it encloses converges, so the quotient has no limit. The four page curves sit within a fraction of a per cent of the ground curve and of each other, which is the comparison this rung exists to make.

The score is not stable at any scale

One boundary, read at eight resolutions from sixteen points to two thousand and forty-eight: the compactness score falls from 0.980 to 0.834 and is still falling. Changing the projection instead moves it by 0.69 per cent. The two decisions are made by the same person on the same afternoon and only one of them is ever reported.

paths · Reach The set a reach map shows, drawn from eight bearings. A geodesic disc of 4,000 km and the polygon a fan of eight bearings draws round it, on an equal-area azimuthal page centred on the disc so that the shaded ground is proportional to the ground it stands for. Every vertex of the polygon is on the true boundary and every edge between two of them is a chord, so the drawn set is inside the true one — always, at every count, for any convex reach set. The area it misses is 7.53% of 48,635,855 km², and it is not an error that care removes. It is what a finite fan is.

Every reach set ever drawn is too small

An isochrone is drawn by walking out along a finite number of bearings and joining the points, so its vertices are on the true boundary and its edges are chords — which puts the drawn set inside the true one, always, at every count, for any convex reach set. The deficit falls as the square of the count, and a spherical cap loses less than a circle by exactly cos t (1 + cos t)/2.

paths · Reach The ground two divisions disagree about. seven places across the world dividing the sphere twice under a steady westerly. Outbound, a place belongs to the site that can reach it soonest; inbound, to the site it can reach soonest. The pale tints are the outbound division where the two agree; the dark ground is where they do not, and it is 50.7% of the sphere — 258,530,666 km². Every site's own share of the world barely moves between the two divisions, by at most 3.8%. The two divisions assign nearly the same AMOUNT of ground to each site and assign completely different ground.

A partition under a directed cost has two versions

Dividing a surface among several sites is one question when the cost is a distance and two questions when it is not. Under a steady flow at 0.45 of a vehicle's own speed, the division by who can reach a place soonest and the division by which place can be reached soonest disagree about 50.7 per cent of the sphere — while no site's own share of the world moves by more than 3.87 points.

paths · Reach

Named alongside it

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

AreaBoundaryClosed formToleranceVerificationGeneralisationGreat circlePurposeAnisotropyCatchmentEqual-areaEstimator

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