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

Sheet layout — where it appears

How a map series divides the world into printable pieces, a covering problem whose count is set by the tolerance before the projection. It is a covering problem, and the count follows from the tolerance the sheets must hold rather than from the projection they are drawn in.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

American polyconic. The graticule of the American polyconic projection at 30° of longitude and 15° of latitude. a different tangent cone for every parallel — true to scale along all of them, and along the central meridian. It is neither conformal nor equal-area.

The projections that gave up being one thing

The polyconic is built from a different cone for every parallel, which means it is built from no cone at all. It preserves nothing the usual tests look for, it has an exact property neither of them measures, and its sheets do not fit together — a defect discovered in the field rather than at the drawing board.

families · Families 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 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 widest zone a tolerance of 690 parts per million allows. At each latitude, the half-width at which a transverse Mercator grid with its scale factor rebalanced for that width reaches 690 ppm at its worst point. That tolerance is the one UTM actually meets at the equator, so the curve passes through UTM's own 3° there — and rises to 37.0° at 85° north, because a degree of longitude covers cos φ of the ground and the scale error goes as the square of the ground width. Six degrees is the answer at one latitude.

Sixty zones was a decision about one latitude

Twelve rungs price a grid, a zone, an origin and a reference, and every one of them works inside a single zone. The number of zones has never been asked about: six degrees meets its tolerance at the equator and is loose everywhere else, so a system spending the same tolerance evenly would use 51 zones at the equator and 8 at 82° — and UTM's worst error is 981 parts per million, not the 400 always quoted.

practice · Grid Two regions, three answers. Britain and New Zealand, 166° apart, with the pole of the best oblique conic under each of three objectives. Pooling the samples and taking an area-weighted score puts the pole in one place; refusing to let either region be worse than the other puts it somewhere else. The regions are drawn on Mollweide so that equal ground areas are equal page areas.

The pooled score abandons a region

Fifteen rungs optimise for one region. An atlas is several, and pooling their samples into one area-weighted score is what everybody does — which on Britain and New Zealand serves Britain 1.2 times worse than it could be served alone and New Zealand 125 times worse. The worst-case objective makes them equal at 33 and 59, and the cost of sharing rises with separation from 1.4 to 59.

choosing · Choosing

Named alongside it

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

OptimisationSpherical capToleranceChebyshev's boundClosed formCoveringLower boundProjection selectionPurposeRepresentative fractionZoneAggregation

All concepts