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

Distortion measures — where it appears

The quantities a map can be scored on — areal factor, angular deformation, scale spread, and the second-order pair — which rank projections differently. The ones that survive a change of coordinates are the two principal scale factors, the areal factor and the angular deformation, and no others do.

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

Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, mollweide, winkelTripel, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both.

The two ways a map is wrong

Angles and areas fail independently. A projection can be perfect about one and catastrophic about the other, and a single number summarising "how distorted" a map is has already thrown away the distinction that matters.

distortion · Tissot Which projection wins, by the first derivative and by the second. Each column is a region, with the projections listed in the order Kavrayskiy's first-order criterion puts them in and the figure at the right of each row giving that projection's rank under the second-order criterion — flexion and skewness aggregated the same way. The rank correlations are Europe 0.83, the conterminous United States 0.81, the tropics 0.90, a cap of 30° radius 0.88, so the two orders agree broadly and disagree in detail. Where it matters is the winner: over a cap of 30° radius the choice moves from Albers equal-area conic to Lambert azimuthal equal-area, while Europe and the conterminous United States and the tropics keep theirs. A criterion that changed every answer would be suspect and one that changed none would be decoration.

The second derivative over a region

Flexion has been measured at points and over the whole sphere, and never over a region — which is the only unit anybody chooses a projection for. Doing it finds that the second-order criterion moves the winner in one region of four, and that one projection in the library has no second derivative at all.

distortion · Flexion

Named alongside it

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

Angular deformationAreal scaleCorrelationFlexionIndependenceInterpolationKavrayskiy's criterionMercatorRankingRegional distortionRobinsonRoute planning

All concepts