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

Sinusoidal — where it appears

The equal-area pseudocylindrical whose parallels are true to scale and whose meridians are sine curves. Every point's east–west scale is therefore exactly its ground value, and the shape error depends only on distance from the central meridian, which is why interruption helps this projection more than any other.

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

What a pole line buys, and what it costs. four pseudocylindrical projections placed by the two numbers the decision moves. Across the bottom, the angular deformation averaged over the whole sphere, where the pole-line projections — Eckert IV and Robinson — are the better maps. Up the side, on a log axis, the factor by which the last parallel is stretched, where they are worse by more than tenfold: 49× at best against 4.6× at worst for a pole drawn as a point. A pole line represents one point of the globe by a line of the map, and that is the price.

Where a pseudocylindrical puts its error

Every projection in this family has to decide what to do at the pole, which on the globe is a point where every meridian meets. Drawing it as a point and drawing it as a line are the two answers, and the trade between them is measurable in both directions.

families · Families What a cut buys. The mean angular deformation of the interrupted sinusoidal against the total length of cut the interruption spends, for lobe counts from one to twenty-four. Goode's interruption — the one actually printed — is the marked point: it spends 100 thousand kilometres and returns 18.0°, where the even-lobed curve returns 9.2° for the same length. It is not on the frontier and it was never trying to be: its cuts are placed to keep continents whole.

What a cut buys

Six rungs count cuts and none measures one. A cut is a curve on the sphere with a length in kilometres, the shape distortion it removes is a falling function of that length, and the interruption everybody prints spends a hundred thousand kilometres to reach a figure the even-lobed curve reaches with sixty.

impossibility · Topology

Named alongside it

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

Angular deformationEqual-areaGoode homolosinePseudocylindricalTrade-offCompromiseContinuityDesignDiscontinuityGraticuleInterruptionPole line

All concepts