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

Affine transformation — where it appears

A linear map of the plane followed by a translation, which carries straight lines to straight lines and parallel lines to parallel lines. It has six parameters against a similarity's four, and the two extra ones are exactly what separates the members of the cylindrical equal-area family from each other.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The half-extent at which each rival stops fitting a Gall–Peters graticule. For each candidate, the size of region at which its best fit to a Gall–Peters map first leaves a residual of a fifth of a per cent of the map's width. Below that size the two are the same picture. The numbers are half-extents in degrees of latitude, at 20° north, with the plane affine transformation removed; a bar at 90° is a rival that never separates at all within the range searched.

What a careless copy hides

A photocopier that stretches one axis is a nuisance to remove before a projection can be identified. Removing it costs more than it looks: an affine fit absorbs the entire difference between the cylindrical equal-area projections, so Gall-Peters and Behrmann become the same picture at any size, to sixteen decimal places.

wrong · Identify The same map on four differently prepared pages. Mollweide at 30°E 20°N, then the same projection with a rotation and a magnification applied to the page, then with one axis stretched by 1.6, then with a shear of 0.5. The similarity changes nothing: flexion, skewness, ω and the anisotropy are identical to every printed figure, and only the last column — the same turning measured per unit of page arc rather than per unit of ground arc — moves, by exactly the magnification. The stretch and the shear change all of them, and flexion by 35 per cent.

The second derivative is not an invariant

The first-order ladder established which quantities survive a change of coordinates and which are artefacts of the parameterisation. Asked of the second order, the answer is that flexion survives a rotation and a magnification of the page exactly, and survives nothing else: a stretch of 1.6 in one axis moves it by eleven per cent, a shear of 0.5 by thirty-five, and a shear of 3 leaves a ranking of eight world maps with a rank correlation of −0.07 to the one it started with.

distortion · Flexion

Named alongside it

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

AnisotropySimilarity transformationCoordinate independenceCylindrical equal-areaDistortion criterionEqual-areaFlexionGall–PetersIdentifiabilityInvariantLeast-squaresNuisance parameter

All concepts