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

Spherical cap — where it appears

The region of a sphere cut off by a plane, which is the simplest region a projection can be optimised over. Its optimal conformal map has a closed form, which makes it the case every numerical optimum can be checked against.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

The least distortion possible over a 30° region. Scale factor along a radius of the cap, each projection normalised to unit scale at the centre. Chebyshev's criterion names the projection whose scale is constant on the boundary as the conformal map of least scale variation, and for a cap that is the stereographic projection centred on it — reaching exactly sec²(ρ/2) = 1.0718 at the rim, marked. Every other conformal projection drawn here rises past that line before it gets there. This is the only figure on the site showing an optimum rather than a comparison.

Total curvature and the scale rule

The impossibility has a size. A region covering a fraction of the sphere carries a fixed amount of curvature that any flat map must absorb, and for a circular region the least distortion any conformal projection can achieve is a closed form nobody can beat.

impossibility · Curvature How large a patch can be treated as flat, at 10 parts per million. The smallest scale distortion any map of a circular patch can have, against the radius of the patch, on logarithmic axes. The line is straight with a slope of 2.00: the error grows as the SQUARE of the size, so a patch ten times wider is a hundred times worse. A tolerance of 10 ppm is reached at a radius of 40.3 km — 81 km across — and that is the number behind the boundary between plane surveying and geodesy.

How small is flat enough

A builder works in plane coordinates and a national mapping agency does not, and the line between them is not a convention. The unavoidable error of treating a patch of the Earth as flat grows as the square of its size, and the size at any stated tolerance is a number.

impossibility · Curvature London to Tokyo, round a 15° exclusion. The direct great circle, dashed, runs through the disc. The admissible shortest route leaves it along a great circle tangent to the rim, follows the rim, and leaves along another tangent — which is the closed-form answer and is checked against a shortest-path search over the rim that knows nothing about tangents. It costs 39 kilometres on 9559, which is 0.41 per cent. The rim stretch is 293 kilometres of it, and it is the only part of the route that is not a geodesic anywhere along its length. Drawn in Orthographic.

A route that must go round

Every route on this site so far has been free to go anywhere, and no real route is. The shortest path past a circular exclusion is two tangent great circles and an arc of the rim — a closed form that agrees with a shortest-path search to three metres in 9,598 kilometres — and it costs not the obstacle's size but the square of how far the obstacle reaches past the route.

paths · Paths Sheets for a tolerance, from Chebyshev's bound and a covering. For each stated tolerance on the scale error, the cap radius at which the best possible conformal projection just meets it — sec²(ρ/2) − 1 = tolerance, which is Chebyshev's bound and has no fitting in it — and then the number of such caps needed to cover the sphere at the packing density a real arrangement achieves. One part in a thousand costs 1210 sheets of 403 kilometres radius. The slope is -0.989: a factor of ten in what the job will accept is a factor of ten in the atlas.

How many sheets an atlas needs

A tolerance on the scale error inverts, through Chebyshev's bound, into a sheet radius — and a covering problem turns the radius into a count. One part in a thousand costs 1,210 sheets of 403 kilometres radius, the count goes as the reciprocal of the tolerance exactly, and the projection multiplies it by anything from one to fifty-six.

choosing · Choosing One length, three corrugations. A straight segment shortened by a factor of 0.9, then wiggled across its own direction until its length is back to what it was. The wiggle's amplitude is what buys the length and the number of wiggles is free, so all three curves have exactly the target length while the third stays 16 times closer to the shortened segment than the first. That is the mechanism: the family converges to a map that is not isometric, while every member of it is.

Impossible in two derivatives, possible in one

The impossibility this whole collection rests on computes a second derivative, so it is a statement about maps that have two. Take one away and it is false: a corrugation restores an exact length while converging to the map that does not, and iterating it gives a flattening whose derivative converges and whose curvature runs to half a million.

impossibility · Curvature 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 How thinly a sphere can be covered by a few equal caps. The covering density of the best arrangement of n equal caps found for each n — the total area of the caps divided by the sphere's, so a value of one would be a perfect tiling with no overlap. The horizontal line is 2π/√27 = 1.2092, the thinnest covering density of the PLANE by equal discs, which this site has used for the sphere since its first atlas essay. It is wrong in both directions: at 2 caps the sphere is covered more thinly than any plane can be, because a cap may be a hemisphere, and at every count from 3 upwards more thickly — 1.5092 at 3, and 1.3377 at 14. The ringed points are the four counts whose optimum is proved: 2 at 90.00°, 4 at 70.53°, 6 at 54.74°, 12 at 37.38°. Everything else is an upper bound from a search, drawn as one, and the bound loosens as the count rises — the search reaches the proved optimum to 3.4 per cent at these counts and has no such check anywhere else.

The sphere is not the plane at small counts

The site's atlas arithmetic multiplies an ideal sheet count by 2π/√27, the thinnest covering density of the plane. At the four counts whose optimal covering of the sphere is a theorem the plane's number is 21 per cent high at two caps and 9, 5 and 2 per cent low at four, six and twelve — wrong in both directions, and the direction changes with the count.

choosing · Choosing How thinly a sphere can be covered by a few equal caps. The covering density of the best arrangement of n equal caps found for each n — the total area of the caps divided by the sphere's, so a value of one would be a perfect tiling with no overlap. The horizontal line is 2π/√27 = 1.2092, the thinnest covering density of the PLANE by equal discs, which this site has used for the sphere since its first atlas essay. It is wrong in both directions: at 2 caps the sphere is covered more thinly than any plane can be, because a cap may be a hemisphere, and at every count from 3 upwards more thickly — 1.5092 at 3, and 1.3377 at 14. The ringed points are the four counts whose optimum is proved: 2 at 90.00°, 4 at 70.53°, 6 at 54.74°, 12 at 37.38°. Everything else is an upper bound from a search, drawn as one, and the bound loosens as the count rises — the search reaches the proved optimum to 3.4 per cent at these counts and has no such check anywhere else.

The sphere is not the plane at small counts

The site's atlas arithmetic multiplies an ideal sheet count by 2π/√27, the thinnest covering density of the plane. At the four counts whose optimal covering of the sphere is a theorem the plane's number is 21 per cent high at two caps and 9, 5 and 2 per cent low at four, six and twelve — wrong in both directions, and the direction changes with the count.

choosing · Choosing 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

Named alongside it

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

Chebyshev's boundClosed formLower boundToleranceBoundaryCoveringOptimisationQuadratic lawRepresentative fractionSheet layoutAreaGaussian curvature

All concepts