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Explain the great-circle distance as you would to a capable student, in 3-4 sentences.

A great-circle distance is the shortest path between two points on the surface of a sphere, like Earth. Imagine stretching a string tightly between two cities on a globe—the string naturally follows a curve that's part of a "great circle," which is any circle that divides the sphere into two equal halves (like the equator). This path is shorter than other routes along the surface because it cuts directly through the sphere's curvature rather than taking a more indirect path. Airlines use great-circle routes to minimize flight distances, which is why a flight from New York to Tokyo often passes over Alaska rather than following a straight line on a flat map.