Atlas

The flight arcs are great circles: the shortest path between two points on a sphere, which is why a flight from Manchester to Tokyo heads up over Scandinavia and Siberia. Each distance comes from the haversine formula, which stays accurate even for cities a few streets apart. The arcs are drawn by projecting many points along each great circle onto the map.

Day and night come from where the Sun is straight overhead. Its latitude is the solar declination (between 23.4° N and 23.4° S through the year), and its longitude follows the Earth's turn, 15° an hour. Everywhere more than 90° from that point is in night; the bands at 96°, 102° and 108° mark civil, nautical and astronomical twilight. It redraws every minute.

a = sin²(Δφ/2) + cos φ₁ cos φ₂ sin²(Δλ/2)φ latitude, λ longitude

d = 2R · asin(√a), R ≈ 6371 kmthe haversine great-circle distance

sin δ = sin ε sin λₛ, ε ≈ 23.44°solar declination from the Sun's ecliptic longitude λₛ

One small formula on a sphere explains both why flights curve and where tonight begins.