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

The shortest crossing that never goes more than 745 km from an airfield

Drawn here at the parameters it defaults to, with every essay that calls it.
The shortest crossing that never goes more than 745 km from an airfield. New York to London, drawn on a conformal conic, with a circle of 745 km round each of twenty airfields on both sides of the North Atlantic. The dashed line is the great circle, 5,540 km. The solid line is the shortest route that stays inside the circles, 5,698 km — 158 km or 2.85 per cent longer — and it bends only where two circles cross: between Narsarsuaq and Keflavík; Keflavík and Vágar. Nowhere else can it bend, because anywhere else a straight cut stays inside a circle.

New York to London, drawn on a conformal conic, with a circle of 745 km round each of twenty airfields on both sides of the North Atlantic. The dashed line is the great circle, 5,540 km. The solid line is the shortest route that stays inside the circles, 5,698 km — 158 km or 2.85 per cent longer — and it bends only where two circles cross: between Narsarsuaq and Keflavík; Keflavík and Vágar. Nowhere else can it bend, because anywhere else a straight cut stays inside a circle.

It is one of 11 figures drawn by the same construction, and the drawing above is this one with nothing changed.

2 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

The shortest crossing that never goes more than 745 km from an airfield. New York to London, drawn on a conformal conic, with a circle of 745 km round each of twenty airfields on both sides of the North Atlantic. The dashed line is the great circle, 5,540 km. The solid line is the shortest route that stays inside the circles, 5,698 km — 158 km or 2.85 per cent longer — and it bends only where two circles cross: between Narsarsuaq and Keflavík; Keflavík and Vágar. Nowhere else can it bend, because anywhere else a straight cut stays inside a circle. Paths and directions

The rule that keeps a route near land is pinned at both ends

A crossing that may never be more than a stated distance from an airfield is refused below one radius and free above another, and each wall is set by a single feature of the ocean. Between them the price falls continuously, with corners and no jumps — and at the lower wall it falls as the square root of the allowance, so the first kilometre is worth twelve of route.

 In a wind every circle moves upwind, and the route moves with them. New York to London with the sixty-minute allowance at a stated one-engine speed of 745 km/h. In still air every airfield's circle has a radius of 745 km, drawn faint and dashed. In a northerly of 150 km/h the places from which each airfield can be reached in sixty minutes form a circle of the same radius moved 150 km upwind, drawn solid. The dashed route is the shortest the still-air rule allows, 5,698 km; from one point of it the quickest diversion in this wind takes 68.7 minutes. The solid route stays inside the moved circles, 5,765 km, and its longest diversion takes 60.8 minutes — the sixty the moved circles stand for, to within what the Earth's curvature does to a wind that keeps one bearing. Paths and directions

A diversion allowance in a wind is the same circle, moved upwind

A diversion rule written in minutes is turned into a distance by assuming still air. In a uniform wind the places from which an airfield can be reached in sixty minutes are not a stretched region but the still-air circle moved upwind by the wind's run. A 150 km/h northerly makes the still-air route across the North Atlantic need a diversion of 68.7 minutes, a southerly makes it 54.0 — and the least radius at which the ocean can be crossed barely moves.

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