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

Scale distortion — where it appears

The departure of a projection's scale factor from unity, which is the third way a map can be wrong after shape and area. It cannot be zero everywhere on any map of a curved body, and it is the quantity Chebyshev's criterion minimises the variation of.

Named by 5 essays across 4 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 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 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 The shortest route between two regions, and the pair it runs between. Europe and the conterminous United States, with the shortest route between them: 3,581 km, attained at one point on each boundary. The pair is not a corner, not a centre and not anything a reader could name — it is wherever two edges happen to come closest, which depends on the whole shape of both. The dashed route is the pair Gall–Peters makes look nearest: 4,085 km, which is 505 kilometres long, and its two ends sit 2135 kilometres from the true pair between them.

The shortest route between two coasts

Ten rungs find the shortest path between two points. Between two regions the answer is attained at a pair of boundary points, and which pair is not the pair any page makes look nearest — a cylindrical map picks a pair 2,135 kilometres from the right one across the Atlantic, and 1,716 kilometres wrong between Chile and New Zealand, where its route is 22.5 per cent long.

paths · Paths One equidistance line, computed five ways. The line equidistant from two facing coasts — Jan Mayen and Greenland — traced by bisection along a fan of parallels, with the two distances measured on the ellipsoid, on the sphere, and with a ruler on three different pages. The basepoints are the same four in every case and the rule is the same words in every case. The sphere sits 20 metres from the ellipsoid; with a ruler on Lambert's cylindrical sits 13.2 kilometres from it, with 324 km² of seabed in between. A delimitation is a sentence about distances, and a distance is a statement about a surface.

An equidistance line belongs to a surface

A maritime boundary is very often defined as the line equidistant from two coasts — a description with no coordinate in it and no curve to choose between. It has a third ambiguity: equidistant measured how. On the ellipsoid, on the sphere, and with a ruler on three different charts, the same four basepoints give lines up to 39.6 kilometres apart and 2,816 square kilometres of seabed between them.

practice · Boundary

Named alongside it

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

BoundaryToleranceAreaAzimuthalClosed formEquidistanceGeodesicGreat circlePurposeScale factorVerificationAngular deformation

All concepts