Cartographic Projection
every map is wrong, and by exactly how much
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The thread: What survives a change of coordinates

Scale along the meridian and scale along the parallel depend on how the sphere was parameterised. The principal scales, the areal factor and the angular deformation do not, and only those are properties of the map itself.
a = 1.74b = 1.74an infinitesimal circle, projectedmeasured at this pointh1.7434scale along the meridiank1.7434scale along the parallela1.7434larger principal scaleb1.7434smaller principal scalea·b3.0396areal scale factorω0.00°maximum angular deformationdashed: undistorteddrawn in Mercator Measuring distortion

Tissot's indicatrix

A tiny circle on the sphere becomes an ellipse on the map, and the ellipse's two axes are the whole story. Almost every published indicatrix is drawn without them, which discards the content and keeps the decoration.

 a = 1.08b = 0.92an infinitesimal circle, projectedmeasured at this pointh1.0834scale along the meridiank0.9231scale along the parallela1.0834larger principal scaleb0.9231smaller principal scalea·b1.0000areal scale factorω9.16°maximum angular deformationdashed: undistorteddrawn in Gall–Peters Measuring distortion

What survives a change of coordinates

The scale along the meridian is a property of the map and the grid together. The principal scale factors are a property of the map alone. Only the second kind describes the projection, and the two are routinely quoted as though they were the same thing.

 0°20°40°60°80°MercatorGall–PetersMollweideWinkellatitudeangular deformationalong a meridian Measuring distortion

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.

 Orthographicgreat circle bows 9.0e-2Gnomonicgreat circle bows not at allStereographicgreat circle bows 9.3e-2Azimuthal equidistantgreat circle bows 1.5e-1solid: shortest · dashed: constant bearingone pair of routes The families

The aspect is a free choice

A projection's distortion pattern is fixed relative to its own axis, and where that axis points is entirely up to the cartographer. Rotating it is the cheapest available improvement and it is the one most often left unmade.

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