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

Chebyshev's bound — where it appears

The least scale variation any conformal projection of a stated region can achieve, which Chebyshev's criterion characterises rather than merely bounds. Computed on the sphere it is not the answer for a country, because the country is on an ellipsoid, and the difference is about a fifth.

Named by 6 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 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 The minimum-distortion conformal map of an elongated region, 30° by 10°. The conformal projection of an elongated region, 30° by 10° whose scale is constant on the boundary, which is Chebyshev's criterion, obtained by fitting eight terms of a series rather than by choosing a named projection. The scale factor runs from 0.98640 to 1.00000, a spread of 1.01379; on the boundary itself the largest departure from constancy is 2.13e-7 in the log, which is what the fit achieved and not what it was told. Each dot is an interior sample shaded by its own departure from the boundary's scale.

Solving for the map instead of choosing it

Chebyshev's criterion has sat on this site since its second phase with one case it could be applied to: the spherical cap, whose answer is the stereographic projection. For any other region the site stated the criterion and stopped. It is a linear least-squares fit, and the fitted map beats every named projection over the region it was fitted to.

choosing · Condition 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 bound was spherical, and the country is not. Three numbers per region, all in parts per million of scale spread. The first is the Chebyshev bound computed on the sphere. The second is that same optimal map used on the ellipsoid, which is what adopting it would actually deliver. The third is the bound with the ellipsoid-to-sphere factor put into the boundary condition, which is the real optimum. The penalty for using the spherical answer reaches 1.22 times — while a named candidate barely moves, because an optimal map has cancelled its own variation and has nothing left to hide a new one in.

The bound on the body the country is on

The best conformal grid a country could have had was computed against a bound solved on a sphere, with a note saying the flattening was second order and unquantified. It is second order for a named projection — every family's best candidate moves by under 0.7 per cent — and it is 22 per cent for the bound, because an optimal map has already cancelled its own variation and has nothing left to hide a new one in.

practice · Grid

Named alongside it

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

Spherical capLower boundClosed formNational GridOptimal conformalToleranceBoundaryChebyshev's criterionConformalityCoveringLeast-squaresRegional distortion

All concepts