ABSTRACT

But on a sphere, none of these impressions are true. The shortest path joining two points on a sphere is called a geodesic and it is an arc of a great circle defined by the intersection of the sphere and a plane through its center and distances are measured in arc length. Geodesic arcs are the “straight lines” of spheres. On a sphere, every pair of distinct great circles intersect, not once but twice, at points opposite one another called antipodal points. There isn’t even a unique shortest path between these opposite points, either. We see this on a globe where there are an infinite number of geodesics (longitudinal lines are arcs of  great circles) that pass through the north and south poles; all of them are the “shortest path” between these two antipodal points.