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 5 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 Where the correction overtakes the sphere it is correcting for. The worst angle error of four substituted spheres against the size of the triangle, with Legendre's spherical excess on the same axes. The two that are wrong at first order in the flattening are flat in the side, because the error belongs to the latitude and not to the distance. The two that are right at first order have no such floor, so what is left is a term rising in proportion to the side. The excess rises as its square, so each pair meets exactly once, and the meeting decides which of the two corrections a computation is entitled to ignore: the geodetic latitude at 490 km, the rectifying latitude at 247 km, the conformal latitude at 23.8 km, the geocentric latitude at 17.6 km.

The correction is the smaller of the two corrections

Legendre's theorem subtracts a third of the spherical excess from each angle of a triangle and solves the rest as a plane figure. It is the correction every nineteenth-century computer applied and the one every textbook explains. Beside the error the choice of substituted sphere leaves in the same triangle it is small: on the thirty-kilometre triangle a first-order chain is made of, omitting it costs 166 millimetres in a computed side and using the geodetic-latitude sphere costs 43 metres. The two terms scale differently, so each sphere has a size at which the correction overtakes it — and for the two spheres a careful computer would have chosen, that size falls inside a real chain.

wrong · Ellipsoid Along a meridian a good sphere's error adds up triangle by triangle; a bad base does not. A strip of thirty-kilometre equilateral triangles running north from 45°, each side computed from the last by Legendre's theorem on a substituted sphere. The scale error of the side just computed, against how many triangles lie between it and the measured base. The conformal sphere adds 7.90 ppm a triangle and reaches 158.1 ppm after 20; the geocentric, 158.9 ppm; the authalic, −52.4 ppm; the rectifying, 0.29 ppm. Shaded: three standard deviations of the scale error one-second angles leave, 3.96 ppm a triangle, growing as the square root of the count. Dashed: a base measured 8 ppm long, which stays 8 ppm at every triangle.

A strip of triangles carries the sphere's error to the next base

One triangle cannot tell a substituted sphere from a bad base: both leave the computed side a few parts per million long. A triangulation is not one triangle but a strip, each computed side the next one's base, and along it the two come apart completely. A bad base is one factor, the same at every triangle. A sphere's error is made afresh in each triangle and the strip adds it up, and what it adds up to at the far end is exactly the change in that sphere's own meridian scale between the two ends — 158 parts per million for the conformal sphere over twenty triangles north from 45°, predicted to half a part. Along a parallel it is nothing; at a check base three triangles on it is already three times the noise of one-second angles.

wrong · Ellipsoid Adjusted to close on the second base, a sphere's drift of hundreds of ppm becomes a bow of under one. A strip of 40 thirty-kilometre equilateral triangles north from 45°, computed on the conformal sphere, between a base at each end. Unadjusted (dashed), the side error climbs to 315.0 ppm at the second base. Adjusted to close there, each triangle's share of the misclosure equal, what is left (solid) is the sphere's stretch less a straight line between the bases: 0.63 ppm at its largest, at triangle 23. Shaded: three standard deviations of what one-second angles leave in the adjusted strip, 37.6 ppm at the middle and nothing at either base.

An adjustment hides the sphere only in a strip of equal triangles

A strip computed on a substituted sphere drifts by the sphere's own stretch, 315 parts per million over forty triangles north from 45°, and an adjustment that closes it on a second base takes almost all of that away: in a strip of equal triangles what is left is the stretch less a straight line, 0.63 parts per million, a third-order bow sixty times below the noise of the old angles. A real chain is not made of equal triangles. The adjustment spreads the misclosure by how uncertain each triangle's angles are, the sphere put it in by how far north each triangle reaches, and the difference is a first-order residual that never clears the noise at any one side and that the old network's own geometry predicts exactly — so a re-observation matched against it reads the sphere at seven standard deviations.

wrong · Ellipsoid

Named alongside it

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

Auxiliary latitudeConformal latitudeSurvey networkVerificationGeodetic latitudeSpherical approximationSpherical excessGeodesicLeast-squaresScale factorToleranceAdjustment

All concepts