Navigation on Earth

Differential Geometry · Axiom Academy

REAL WORLD Navigation on Earth Why your flight path curves and how differential geometry guides GPS Ever wonder why airplane routes look curved on a map? When you fly from New York to Tokyo, the flight path arcs dramatically northward through Canada and Alaska. Are pilots taking a detour? Spoiler: The curved path is actually the shortest route! This is differential geometry in action—on a sphere, straight lines aren't what they seem on flat maps. Great Circles: The Sphere's "Straight Lines" In differential geometry, a geodesic is the shortest path between two points on a curved surface. On a sphere (like Earth), geodesics are segments of great circles —circles whose center is the center of the sphere. Why are great circles special? Imagine slicing Earth with a plane. What determines if the circle formed is a great circle? On a sphere of radius R , the distance along a great circle between two points is determined by the central angle θ between them: Where the central angle comes from the dot product of position vectors: Interactive: Calculate Your Distance Move the slider to see how latitude affects great circle distances: Map Projections and Distortion The reason great circles look curved on flat maps is that it's mathematically impossible to flatten a sphere without distortion. This is a fundamental theorem in differential geometry!

This is the written version of the interactive lesson above. See the full Differential Geometry course.