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

Loxodrome — where it appears

A path crossing every meridian at the same angle, which is what a fixed compass bearing traces. Mercator draws it straight, which is the property that projection was built for, and it is longer than the great circle and easier to steer.

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

New York to Madrid on Mercator. Two routes. The great circle is 5768 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 5939 km — 171 km further, or 3.0 per cent. On Mercator the rhumb line departs from straight by 9.6e-16 of its own length.

Why Mercator exists

A ship can hold a compass bearing and cannot easily hold a great circle. Mercator is the answer to one question — what must a map do so that a constant bearing is a straight line — and it answers it exactly.

paths · Paths London to Tokyo in three straight legs. The great circle, and the route a plan of three constant-heading legs actually follows between waypoints on it. The two touch at the waypoints and part between them by up to 456 km, and the flown route is 253 km longer than the direct one. The headings are 50°, 109°, 149°. Two routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Orthographic the rhumb line departs from straight by 2.4e-1 of its own length.

Flying a curve in straight legs

Nobody steers a great circle, because a great circle requires the heading to change continuously. What is actually flown is a handful of constant-heading legs between waypoints on it, and the gap between plan and curve falls as the square of the number of legs.

paths · Paths

Named alongside it

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

BearingGreat circleMercatorNavigationRhumb lineComposite methodConformalityCross-track errorCylinderGnomonicQuadratic lawRoute planning

All concepts