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

Derivative — where it appears

The rate at which a map's output changes with its input, which is where every distortion quantity on this site lives. Two constructions can agree on every drawn position and disagree entirely here, because interpolation controls values and does not control slopes.

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

The indicatrix is a limit, and here it is being taken. The axis ratio of the image of a circle of radius ρ, measured from the picture rather than computed from the derivatives, against the ratio the derivatives give. It converges, and it converges as ρ² — each halving of the radius quarters the departure. At the smallest radius drawn the two agree to 7.6e-4 relative. This is what "the indicatrix is a limit" means when it is working: there is a number for the ratio to settle on, and it settles.

The indicatrix at a point that has none

Ten essays on this ladder take the derivative for granted. Rung three proved the indicatrix is a limit and every figure since has assumed the limit exists. There are points on the maps this collection draws where it does not, and they are not exotic: the horizon of a gnomonic map, and every corner of every polyhedral net.

distortion · Tissot Three implementations of one projection, and three different distortions. The spread in angular deformation between linear, Catmull–Rom and Aitken interpolations of Robinson's own table, at 60° of longitude. It is exactly zero at every fifth degree, because all three pass through the tabulated rows, and it is not zero anywhere else: 37 of 87 latitudes differ by more than half a degree and the worst is 8.42°. The graticules the three draw differ by 4.2 parts per thousand of the map's own span, which is under the width of a printed line.

A projection defined by a table has an interpolation in it

One member of this library has no formula: Robinson set nineteen pairs of numbers by eye and the table is the definition. Three published interpolations of it draw graticules that differ by four parts in a thousand of the map's span and report angular deformations that differ by 8.4 degrees.

families · Families

Named alongside it

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

Angle deficitAngular deformationAnisotropyCompromise projectionConformalityConvergenceCurvatureDefinitionGnomonicImplementationIndicatrixInterpolation

All concepts