Bactra Review    Geometrical Methods of Mathematical Physics

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Consider the North Pole and the South Pole of a globe of radius R. Normally, the distance between them is 2R, in accordance with the Pythagorean formula. If you are constrained to paths on the surface of the sphere, however, the shortest distance is half a great circle, (pi)R. In fact, the shortest path between any two points is longer than it would be if one could leave the sphere.