gen-define-the-great-circle-distance-and-give-one-short-worked-e-aa04

Define the great-circle distance and give one short worked example.

# Great-Circle Distance ## Definition The **great-circle distance** is the shortest distance between two points on the surface of a sphere, measured along the surface. It's the length of the arc along the "great circle" — a circle that cuts the sphere into two equal hemispheres (its center coincides with the sphere's center). For Earth navigation, this is calculated using the **haversine formula**: $$d = 2r \arcsin\left(\sqrt{\sin^2\left(\frac{\phi_2-\phi_1}{2}\right) + \cos(\phi_1)\cos(\phi_2)\sin^2\left(\frac{\lambda_2-\lambda_1}{2}\right)}\right)$$ where: - $d$ = great-circle distance - $r$ = radius of sphere (Earth ≈ 6,371 km) - $\phi_1, \phi_2$ = latitudes of points 1 and 2 - $\lambda_1, \lambda_2$ = longitudes of points 1 and 2 ## Worked Example **Find the distance from New York City to London:** - NYC: 40.7°N, 74.0°W - London: 51.5°N, 0.1°W Converting to radians and applying the formula: - $\phi_1 = 0.711$, $\lambda_1 = -1.291$ - $\phi_2 = 0.899$, $\lambda_2 = -0.002$ $$d = 2(6371) \arcsin\left(\sqrt{\sin^2(0.094) + \cos(0.711)\cos(0.899)\sin^2(0.645)}\right)$$ $$d = 12742 \arcsin(0.434) \approx \boxed{5,570 \text{ km}}$$