I think by taking an extreme case one can see that geodesics are not typically planar. E.g., consider an ellipsoid of revolution, with semiaxes 1, 1, and 10^9. Near the xy-plane this will be very close to a vertical (circular) cylinder. So the geodesic from (1,0,0) to the point nearest (-1,0,1) will be very close to a helix. --Dan << 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?

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Sometimes the brain has a mind of its own.