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

Transverse — where it appears

An aspect in which a projection's axis is rotated by a right angle, so that its line of true scale runs along a meridian rather than the equator. Transverse Mercator is the basis of almost every national grid, because a country is usually taller than it is wide.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

London to Tokyo, seen four ways. The same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did.

The aspect is a free choice

A projection's distortion pattern is fixed relative to its own axis, and where that axis points is entirely up to the cartographer. Rotating it is the cheapest available improvement and it is the one most often left unmade.

families · Families Transverse Mercator. The graticule of the Transverse Mercator projection at 30° of longitude and 15° of latitude. the only projection on which a constant compass bearing is a straight line. It is conformal.

Transverse Mercator and the series that computes it

The projection most of the world's survey data lives in has no closed form. What every national grid actually computes is a truncated power series in the flattening, and how far it can be trusted is an engineering parameter rather than a property of the projection.

families · Ellipsoid The sixty zones, each six degrees wide. Every zone is a separate transverse Mercator projection about its own central meridian, so the world is covered by sixty maps rather than one. Zone 31 is picked out, running from 0° to 6° with its axis on 3°. Coordinates do not carry across a zone boundary — a point on either side of one has two entirely different eastings, and nothing in the numbers says which zone they belong to.

UTM and the zone system

Sixty separate maps of the world, each six degrees wide, each with a scale factor of 0.9996 chosen so the projection is wrong everywhere and less wrong at the edges. Every constant in the definition is a measured trade rather than a convention.

families · Ellipsoid Mercator swung through every tilt, over Chile and the tropics. The regional distortion of one projection as its axis is tilted from the normal aspect at 0° to the transverse at 90°, each curve divided by its own value in the normal aspect so the two regions can share an axis. Chile is best at a tilt of 90°, a factor of 183.5 better than north-up; the tropics is best at a tilt of 0°, which is north-up. Rotating the sphere costs nothing and changes none of the projection's own properties, which makes this the cheapest improvement available and the one most often left unmade.

Fitting the aspect to the region

Choosing a projection is a choice among a few dozen named things. Choosing its aspect is a choice among a continuum, it costs nothing, it changes none of the projection's own properties, and for a long thin country it is worth a factor of 183.

choosing · Choosing

Named alongside it

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

MercatorAspectNational GridUTMZoneEllipsoidObliqueScale factorAzimuthalBoundaryConformalityConvergence

All concepts