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

Triangulation — where it appears

Fixing positions by measuring the angles of a chain of triangles from a single measured baseline. It was the method every national survey was built on before distances could be measured directly, and its angles are differences of directions, which is what decides its sensitivity to the surface they are computed on.

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

The same patch, pinned six ways. Two vertices have to be held or the conformal energy has a similarity's worth of null space. Which two turns out to decide two of the three numbers reported. The median angular deformation is the same to 8 per cent across all six — that is the map. The areal spread runs from 2.68 to 7.44, so the 3.07 reported for this body was a statement about its corners. And pinning two adjacent vertices, which fixes the similarity through a very short lever, ruins the worst point without touching the median: a badly conditioned constraint pays for its scale in one corner.

The map depends on where it was cut

Solving the discrete conformal equations on a triangulated body gave an areal spread of 3.07, with the number possibly belonging to the patch, the boundary and the two pinned vertices rather than to the surface. It belongs to the pins: hold a different pair and it runs from 2.68 to 7.44, while the typical angular deformation does not move at all.

datums · Bodies An angle is a difference of two directions, and the error does not cancel between them. A geodesic triangle with sides of 30 km at 45° N, its first side laid off on a bearing of 45° and the second 60° round from it. Substituting a sphere carrying the geodetic latitude turns each direction at a corner by an amount that runs as the sine of twice its azimuth, so the two directions at A are turned by different amounts and the angle between them is wrong by their difference. Here the worst single direction is out by 345.79″ and the worst angle by 521.09″ — larger, not smaller. Bessel's sphere, which also corrects the longitude, keeps every angle to 6.2e-9″.

An angle is a difference, and the difference doubles the error

Substituting a sphere for the ellipsoid turns every direction at a point, and a surveyor measures angles rather than directions — so the obvious hope is that a turn common to both directions cancels in their difference. It does not cancel, because the turn is not common: it runs as the sine of twice the azimuth, and the largest angle error is exactly twice the largest direction error at a well-shaped corner. A first-order triangulation computed on the geodetic-latitude sphere carries angles nearly nine minutes of arc wrong, and its closure check passes perfectly.

wrong · Ellipsoid

Named alongside it

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

Angular deformationAreal scaleAuxiliary latitudeAzimuthBoundaryConditioningConformal latitudeConformalityConstraintDiscrete conformal mapGaugeGeodesic

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