No, the shortest path does not always lie on a plane. I'm pretty sure that the geodesic flow on an ellipsoid has been studied quite extensively. I've seen pictures of geodesics, which show no particular patterns. Typically there 3 simple closed geodesics (through the major axes) and no more; that is inconsistent with all geodesics lying in a plane. But those things are based on partially remembered "authority" which I don't want to look up right now. Here's a more direct way one can see, for the special case of ellipsoids that are surfaces of revolution: Geodesics on a surface of revolution satisfy the principle of conservation of angular momentum about the axis of revolution, when traversed at unit speed. For any given value of angular momentum, there is a band on the ellipsoid (might be empty, is the whole ellipsoid <==> ngular momentum is 0) where there are unit tangent vectors with the given angular momentum. In the interior of the band, there are two solutions; at the two circles that form the boundary of the band, there is only one soution at each point, but these circles are not geodesics; the geodesics go to the boundary on one "sheet" of solutions, then go out on the other "sheet". Each geodesic oscillates, back and forth, from one boundary curve to the other, with a certain period; the period depends on the value of angular momentum. Usually it has an irrational relation the period of a meridian circle, and such geodesics are dense in the band. This is roughly the pattern of how thread or kite string winds up on a round spool as it gets a slight bulge in the middle. Bill Thurston On Jun 26, 2011, at 1:16 PM, Stephen B. Gray wrote:
A and B are points on an ellipsoid E. S is the shortest path on the ellipsoid's surface between A and B .
Does S always lie in a plane?
Steve Gray
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