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

Straight line — where it appears

A description that names no curve on a sphere. A geodesic, a rhumb line and a straight line on some stated sheet all answer to it, they differ by kilometres over hundreds, and a page's straight line can lie outside the other two rather than between them.

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

What the drawn line costs, projection by projection. The ground length of the page-straight route from London to Tokyo, as a percentage above the shortest route. Two of these have names. On the gnomonic, centred on the route, the drawn line IS the shortest route, at -0.0000 per cent. On Mercator it is the rhumb, matching the rhumb's own length to 1 parts per million — which is why that projection exists. On the other eight it is a curve with no name and a cost between 0.0 and 19.6 per cent.

The line drawn straight on the page is a route

Eleven essays draw the route on the map. Nobody has drawn the map's own proposal: the ground curve somebody follows by laying a ruler on the page. It has a name on exactly two projections and is 18.2 per cent long on the one where it is famous.

paths · Paths One sentence, three readings of it. "a straight line from the initial point on the Rio Grande to a point on the Colorado" — Treaty of Mesilla, 1853. Every curve here answers to those words. the geodesic, the rhumb line, straight on the sheet, drawn between the same two monuments on a conformal conic fitted to the segment itself. The widest pair, the geodesic against the rhumb line, are 7.53 kilometres apart at their worst and enclose 3,925 km². The shading is that ground. It is not an artefact of the drawing: the same figure of the 141st meridian shows one line, because on a meridian every one of these readings is the same curve.

One sentence, and the ground between its readings

Every land boundary in the world is defined by a sentence, and a sentence naming two monuments does not name a curve. "A straight line" between the Rio Grande and the Colorado admits at least three answers 7.53 kilometres apart at their worst; "the forty-ninth parallel" admits four, 96 kilometres apart, with 130,972 square kilometres between the extremes.

practice · Boundary

Named alongside it

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

Great circlePurposeAreaAzimuthalBoundaryChartConventionExcessGeodesicGnomonicInverseMercator

All concepts