Great-Circle Navigation: How Ships Actually Cross a Sphere
📍 Navigation
Question
Why do the 'straightest' routes on a map look curved?
The shortest path between two points on a sphere is a great circle - a circle whose centre is the Earth's centre. On a flat map (the Mercator projection) great circles look curved, so the 'straight line' you draw on the map is not the shortest route. Ships and planes navigate by great circles, not by rhumb lines (lines of constant compass bearing). The same spherical geometry gives spherical excess: a triangle on a sphere has angles that add to more than 180°, and that excess is proportional to the triangle's area.
The math
Great-circle distance d = R × Δσ, where Δσ is the central angle between the two points. Spherical excess E = A + B + C − 180°; the triangle's area = E × R² (E in radians). On a flat plane E is always 0 - on the Earth it is not.
Why it matters
This is how ships and planes actually cross the globe. The flat-map intuition (the straight line is shortest) is wrong: the shortest route is a curve on the map. Every long-haul flight and ocean crossing is a great circle, and the navigation math only works on a sphere.