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

Vincenty's formulae — where it appears

The iterative algorithm for distances and azimuths on an ellipsoid published in 1975 and still the most widely implemented. It fails to converge for nearly antipodal points, which is not a defect of the code but the cut locus appearing from the inside.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

What treating the Earth as a sphere costs, per journey. The ellipsoidal geodesic minus the spherical great circle, in kilometres, for five journeys. The correction is a few tenths of a per cent and it changes sign: a route running east–west at mid latitude is longer on the ellipsoid, and one running along a meridian is shorter, because an oblate body is fatter round the equator and flatter pole to pole. Both distances are computed — Vincenty's iteration against the haversine formula — and the ellipsoidal one is checked against a geodesic obtained by integrating its own differential equation.

Geodesics on the ellipsoid, and why they are hard

The shortest path on a flattened Earth is not a plane curve, has no closed form, and can be longer or shorter than the spherical answer depending on which way it runs. Every practical method is a series or an iteration, and the correction changes sign.

paths · Paths Where the inverse problem stops converging, 3° around the antipode of London. Every target in a 6° square centred on the point diametrically opposite London, shaded by how many iterations Vincenty's inverse formula needed to find the geodesic to it. 3 of 625 — 0% — never converged at all, and the worst success took 113 steps against a handful anywhere else on Earth. The failure is not a defect in the formula: near the antipode the shortest path is nearly ambiguous, and an iteration looking for one answer is being asked which of many.

The route with no shortest path

Between a point and the point diametrically opposite there are infinitely many shortest routes and no shortest route, and the standard formula for the distance between two places stops converging in a neighbourhood of it. The failure is a property of the question rather than a defect in the answer.

paths · Paths Three curves between 45°N and 55°N, 3026 km apart. The normal section observed from the first point, the normal section observed from the second, and the geodesic, each plotted as its distance to one side of the great-circle chord between the two ends. The two sections are 79.0 metres apart at their widest and the geodesic runs between them, 53.3 metres from the first. In LENGTH they differ by almost nothing — 2.40 millimetres over 3026 kilometres — so an observation that follows the wrong one measures the right distance along the wrong ground. On a sphere all three coincide exactly.

The normal section is not the geodesic

A theodolite at A sighted on B swings in one plane and the instrument at B sighted back swings in another, so the two observations trace different curves on the ground — 79 metres apart over 3,026 kilometres — and the shortest path is neither of them. The lengths differ by 2.4 millimetres, so the wrong curve measures the right distance along the wrong ground.

wrong · Ellipsoid The route at height does not lie above the route on the ground. London to Tokyo, solved at the surface and again at a stated height, with the two ground tracks compared point for point. At a cruising altitude of eleven kilometres the two part by 11.8 metres; the departure is proportional to the height, at a fitted slope of 0.997. On a sphere the same measurement returns 2.7e-9 metres, because the offset of a sphere is a sphere and the two geodesics coincide exactly. The offset of an ellipsoid is not an ellipsoid, and this is what that costs.

The shortest route is not at sea level

Routes are computed on a surface and nothing is ever flown on one. The offset of a sphere is a sphere, so at altitude the great circle is the great circle. The offset of an ellipsoid is not an ellipsoid — its radii of curvature are M + h and N + h, which belong to no ellipsoid — so the shortest route at cruising height does not lie above the shortest route on the ground.

paths · Paths One equidistance line, computed five ways. The line equidistant from two facing coasts — Jan Mayen and Greenland — traced by bisection along a fan of parallels, with the two distances measured on the ellipsoid, on the sphere, and with a ruler on three different pages. The basepoints are the same four in every case and the rule is the same words in every case. The sphere sits 20 metres from the ellipsoid; with a ruler on Lambert's cylindrical sits 13.2 kilometres from it, with 324 km² of seabed in between. A delimitation is a sentence about distances, and a distance is a statement about a surface.

An equidistance line belongs to a surface

A maritime boundary is very often defined as the line equidistant from two coasts — a description with no coordinate in it and no curve to choose between. It has a third ambiguity: equidistant measured how. On the ellipsoid, on the sphere, and with a ruler on three different charts, the same four basepoints give lines up to 39.6 kilometres apart and 2,816 square kilometres of seabed between them.

practice · Boundary Four spheres are wrong about a direction before anything has moved. At 45° of latitude, the worst error over twenty-four starting directions of the azimuth each substituted sphere gives for a geodesic on WGS84, against the distance to its end. Geodetic latitude is wrong by 347″ at one kilometre, and the parametric, authalic and rectifying latitudes by 174″, 116″ and 86.8″ — a half, a third and a quarter of it — flat until a few hundred kilometres. The conformal latitude is right at the start and wrong by 54.3 mas at one kilometre, 5.43″ at 100 and 54.3″ at 1,000; the geocentric latitude starts at 620 mas and joins it. Bessel's sphere, which also corrects the longitude, is exact at every distance to the arithmetic's floor of 1.2e-7″.

A bearing on a sphere is decided by its latitude, not its radius

A sphere that stands in for the ellipsoid cannot be wrong about a direction by being the wrong size, because enlarging a sphere moves no angle. What it can be wrong by is the latitude put on it. Geodetic latitude turns every bearing at 45° by up to 347 arcseconds before anything has moved; the conformal latitude starts exact and drifts 54 arcseconds in a thousand kilometres; and only Bessel's sphere, which changes the longitude too, is exact at every distance.

wrong · Ellipsoid

Named alongside it

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

EllipsoidGeodesicFlatteningGreat circleNumerical integrationVerificationBearingClairaut's relationRadius of curvatureSpherical approximationSurface normalAntipode

All concepts