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

The impossibility

Gaussian curvature is intrinsic, a sphere has some and a plane has none, so no map can be faithful. A theorem rather than an engineering limit.
planeK = 0 — unrolls flatcylinderK = 0 — unrolls flatconeK = 0 — unrolls flatsphereK = 1.00torusK = 1.79pseudosphereK = -1.00K computed at each centretwo routes, one answer

No map is faithful

Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently.

 wrappedunrolled — every distance unchanged→K = 0 on bothcircumference 2π = width 2π

What can be unrolled

A cylinder is obviously curved and is intrinsically flat, so it lays out on a table with nothing stretched. A sphere is not. The distinction is exactly zero Gaussian curvature, and it is why paper tubes exist and paper globes do not.

 10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured

The trade-off is two lines

Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument.

All essays