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

Nominal scale — where it appears

The ratio a map prints for itself, which on a screen map is the equatorial one and is true along exactly one line. It is true along one line and wrong everywhere else, which is why a printed scale bar is a statement about one place on the sheet.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

The least distortion possible over a 20° 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) = 1.0311 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.

Scale distortion is the third failure

A map's stated scale is its scale at one place. Everywhere else it differs, by a factor that no conformal projection can avoid and that neither of the two usual distortion measures reports. It is the failure everybody uses and nobody counts.

distortion · Tissot 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 The scale asked for and the scale the pyramid has, at 0°. A tiling scheme exists only at integer zoom levels, a factor of two apart in resolution, so a request for any scale between them is answered by the nearest rung. The ratio runs from 0.707 to 1.405 — 1/√2 to √2 — and repeats identically at every doubling, which is four times that in area. A request for 1:10,000 is served at zoom 16, which is 1:8,531; A request for 1:25,000 is served at zoom 14, which is 1:34,124; A request for 1:50,000 is served at zoom 13, which is 1:68,247. Nothing anywhere reports it, because the map that arrives is a perfectly good map of something.

Zoom is a ladder

A tiling scheme exists only at integer zoom levels a factor of two apart, so a request for 1:25,000 is answered with 1:34,124 — 36 per cent coarser, and 86 per cent coarser in area. The mismatch runs from 1/√2 to √2 and repeats identically at every doubling, and nothing anywhere reports it, because the map that arrives is a perfectly good map of something.

applied · Screen The same totals, drawn as symbols, on two pages. A band field, largest in the mid latitudes carried by each region and drawn as a circle whose area is proportional to the total. Every symbol is right: a total is a total wherever it is drawn, and the two maps carry identical numbers. What differs is the ground under each symbol. On Mercator the symbol at 60° sits on a region drawn 4.0 times larger than it would be on an equal-area sheet, so the density a reader forms — symbol against region — is out by that factor.

A symbol has a size on the page and an area on the ground

A proportional symbol is right as a total wherever it is drawn — a count is a count. Read against the region beneath it, which is how a reader forms a density, it is wrong by exactly the reciprocal of the areal factor: 0.083 at 73° north on Mercator, a factor of twelve, with the correction available as one multiplication that no atlas makes.

distortion · Thematic

Named alongside it

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

Representative fractionScale factorMercatorPrincipal scale factorsToleranceAngular deformationArea weightingAreal factorAzimuthalConformalityConventionCurvature

All concepts