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

Chart — where it appears

A map made for navigation, on which the drawn line is meant to be followed. Only two projections make a page-straight line a route with a name: Mercator, where it is the rhumb, and the gnomonic, where it is the shortest path.

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

Ground north, on a map that has the pole on it. Every arrow points along its own meridian, towards the pole, and the pole is at the centre. Walking once anticlockwise round any loop enclosing it turns the arrow once anticlockwise as well: the index is 1, counted as a winding number with no distance anywhere in the calculation. A field like this cannot be combed flat. There is no way to choose a page direction for north at every point of the neighbourhood without the choice tearing somewhere, and the somewhere is the point in the middle.

North cannot be up everywhere

Ground north is a field of arrows on the sphere, and a field of arrows on a sphere must vanish somewhere. The failure is not measured, it is counted: the indices of the zeros sum to two, obtained here as a winding number in seven different charts with no distance anywhere in the calculation, and it is the same two that Gauss–Bonnet gets by integrating curvature.

impossibility · Topology 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

Named alongside it

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

AzimuthalContinuityConvergence angleCritical pointEuler characteristicExcessGnomonicGradientGreat circleIndex of a zeroInvariantInverse

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