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

Reprojection — where it appears

Carrying a coordinate from one projection's plane into another's, which is a map of the plane and needs no knowledge of the body. Between two equal-area projections it has an areal factor of one and an angular deformation of up to 168 degrees, which is what one condition leaves free.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

The circle a reprojection is about to turn into an ellipse. A circle drawn on the Mollweide plane, and its image under the map that carries that plane to Mercator — normalised so each pair has the same area, because the whole map can be rescaled and the shape is what is being shown. The dashed circle is what an undistorted reprojection would leave. This is Tissot's construction with a plane in place of the sphere, and it is the right picture for a reprojection because both ends are pictures. Worst angular deformation over the sampled points: 81.2°. Drawn on the source plane, in Mollweide.

A projection between two projections

Every distortion measured on this site so far compares a map with the sphere. The operation a machine actually performs compares a map with another map — and that map has its own two principal scales, its own areal factor and its own angular deformation, none of which is the difference of the two it was built from.

distortion · Tissot An equal-area map nobody would publish. Mollweide, with a row-dependent horizontal displacement applied to the page afterwards. That plane map has Jacobian determinant 1 everywhere, so every areal scale factor is untouched: the worst departure from 1 anywhere sampled here is 6.0e-12, which is the arithmetic's own floor. It satisfies the equal-area condition exactly and completely, and it is a ruin. The angular deformation at 30°E 20°N has gone from 11.0° to 60.5°.

Every equal-area map is every other one

Take Mollweide and slide every row of the page sideways by an amount that depends on the row. The result satisfies the equal-area condition to 6 × 10⁻¹², exactly as well as Mollweide does, and it is a ruin — the angular deformation at one ordinary point has gone from 11° to 60°. Equal-area is one equation, and one equation leaves a whole function free.

choosing · Condition The convergence order across an edge, against the edge's own orientation. Each curve is one kernel, fitted the same way as every other convergence order on this site: the root-mean-square error against the grid spacing, in logs, over five refinements. At 27° they read 0.78, 0.58, 0.60, which is the measurement already published here — and 27° is one point. Turn the edge onto a parallel and the curves collapse, and the nearest-neighbour one goes negative, which is the fit's way of saying the error is not falling at all. A single number for "the order across an edge" is a number about the edge that was measured.

One edge is not an edge

The three resampling kernels were measured across a discontinuity and came out at 0.78, 0.58 and 0.60 — one straight edge at 27° to the graticule. Across thirteen edges the same kernels span 0.19 to 0.87, the ranking between them reverses, and for an edge lying along a parallel the error does not fall with refinement at all.

applied · Dataset What a reprojection does, drawn where it does it. The angular deformation of the map that takes Sinusoidal's page to Gall–Peters's — which is what a reprojection is — shaded over the sphere and drawn on Mollweide so that equal ground areas are equal page areas. Both parents are equal-area, so the composite preserves area exactly; its mean angular deformation is 55.5° against 38.9° and 32.2° for the two maps separately. Composing two maps did not average their shape errors; it made a larger one.

Two indicatrices do not make a third

Reprojecting is composing, and the composite's indicatrix is not a function of its parents'. Mollweide followed by Hammer is gentler than either map over eighty per cent of the sphere; the sinusoidal followed by Gall–Peters is worse than both everywhere. What separates them is one angle, and it is the number rung ten showed the ellipse does not carry.

distortion · Tissot

Named alongside it

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

Angular deformationAreal factorCompositionConformalityEqual-areaJacobianResamplingTissot's indicatrixAliasingAnisotropyClosed formCondition

All concepts