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

Cross-track — where it appears

The perpendicular distance from a point to a great circle, and the quantity that decides whether an obstacle blocks a route at all. It is computed from the cross product of the two endpoints' unit vectors, and it decides whether a great-circle route needs a detour at all.

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

London to Tokyo, round a 15° exclusion. The direct great circle, dashed, runs through the disc. The admissible shortest route leaves it along a great circle tangent to the rim, follows the rim, and leaves along another tangent — which is the closed-form answer and is checked against a shortest-path search over the rim that knows nothing about tangents. It costs 39 kilometres on 9559, which is 0.41 per cent. The rim stretch is 293 kilometres of it, and it is the only part of the route that is not a geodesic anywhere along its length. Drawn in Orthographic.

A route that must go round

Every route on this site so far has been free to go anywhere, and no real route is. The shortest path past a circular exclusion is two tangent great circles and an arc of the rim — a closed form that agrees with a shortest-path search to three metres in 9,598 kilometres — and it costs not the obstacle's size but the square of how far the obstacle reaches past the route.

paths · Paths A straight line stored in Web Mercator, and where it really goes. Two points 5570 kilometres apart, joined by a straight segment in the plane the file's coordinates are in. Drawn on that plane it is a straight line and looks like the route; on the ground it is the curve marked measured, and the geodesic between the same two endpoints is the other one. The profile below is the distance between them along the line, reaching 718 kilometres at 49 per cent of the way across. Nothing here is an error in the data: both endpoints are exact, and the whole of the discrepancy is the word straight.

A straight segment is a claim about a plane

Two exact endpoints, joined by a straight line in the plane the file is stored in. On the ground the line is 718 kilometres from the route it claims between New York and London, and 2,961 between London and Tokyo. The departure grows as the square of the length — fitted exponent 2.001 — so a stated tolerance costs vertices as a square root.

applied · Dataset

Named alongside it

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

GeodesicGnomonicGreat circleVerificationClosed formConstraintConventionDensificationInterpolationNavigationOptimisationQuadratic law

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