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

Vertex — where it appears

A stored point of a geometry, and separately the highest latitude a great circle reaches. In the second sense it is predicted from the departure bearing alone by Clairaut's relation, which is what makes a great-circle route computable without integrating anything.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

London to Tokyo on Mercator. 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 Mercator the rhumb line departs from straight by 5.0e-9 of its own length.

The shortest route is not straight

The shortest path between two points on a sphere is an arc of a great circle, and on almost every map it is a curve. The straight line on a Mercator chart is a different route entirely, and on some journeys it is twenty-eight per cent longer.

paths · Paths London to Tokyo on Orthographic. 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 2.9e-1 of its own length.

The great-circle vertex

One quantity is constant along a shortest path on a sphere, and it fixes the highest latitude that path will reach before the journey starts. That number is why polar routes exist, and it can be read off the departure bearing without tracing the route at all.

paths · Paths The angles of a triangle, and the curvature inside it. A triangle on the sphere with sides that are great-circle arcs. Its three angles sum to 212.26°, overshooting the flat 180° by 32.26°. Integrating the curvature over the interior gives 0.56306 against an excess of 0.56306 — the same number by two routes that share no arithmetic. That is Gauss–Bonnet, and it is how a flatlander measures the curvature of a world it cannot step outside.

Measuring curvature from inside

A flatlander with a ruler can find out that its world is round, by drawing a triangle and adding the angles. Gauss–Bonnet turns that into an exact statement, and the total curvature of a closed surface turns out to be a number obtained by counting.

impossibility · Curvature 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 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 A triangle of 25.0° excess, drawn on Mercator. The three geodesic sides as curves and the three straight sides a ruler draws, with each vertex labelled by how far the ruler's angle is from the true one. The straight-sided triangle's angles sum to exactly 180° because it lies in a plane, and the real one's sum to 205.02°, so the three errors have to account for the whole 25.02° of spherical excess between them — and they do so on this conformal projection exactly as they do on any other.

Conformal does not mean the angles are right

A conformal projection preserves angles between curves at a point. Draw a triangle on one with a ruler and its angles are wrong by degrees — and the total error is fixed before the projection is chosen, because a plane triangle sums to 180° and the real one does not.

wrong · Audit Two features, one shared boundary, simplified apart. Two neighbouring areas whose common boundary is a curve with structure at every scale — a river or a ridge, in effect — each stored with its own copy of that boundary and each simplified on its own at a tolerance of 0.01. The faint outlines are the originals and the solid ones what came back. The two copies of the shared boundary were within 0.01 of each other before the simplification and are not afterwards: 144 probe cells of 40000 now lie inside both features and 0 inside neither.

A boundary that two features share

Three rungs simplify one curve and price what a tolerance covers. Almost no boundary in a real dataset belongs to one feature: a county's edge is the next county's edge, it is stored twice, and it is simplified twice. What opens between the two answers is a region belonging to both features or to neither, and its area is not bounded by the tolerance.

applied · Generalise

Named alongside it

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

BearingGeodesicGreat circleMercatorRhumb lineClairaut's relationEllipsoidGauss–Bonnet theoremSpherical excessToleranceTopologyVerification

All concepts