I am trying to solve the 'ant and honey problem on a spherical bowl' to find the shortest route between two points on a sphere when the path is constrained by not being allowed to pass higher than a certain latitude (so interrupting some great circles connecting the two points). I intuitively think that the shortest route in this case would be to follow the great circle between the two points until the latitude is reached beyond which you cannot go; then walk along the latitude until the great circle is picked up again; then follow it down to the destination. But I have not been able to prove that this is the shortest route. If the following triangle exists then my proposed solution is not the shortest route, but if it doesn't exist then it is the shortest route. Here is the question: Can a triangle be drawn on the surface of a sphere which has points A, A',B with known lengths A-A' and A'-B and included angle AA'B greater than 90deg for which the following property holds |AB - AA'| < |A'B|
OK thanks ! By the way I discovered that my assumed shortest path is only a close approximation to the true shortest path so the proof I was attempting based on the inequality is invalid! This is the shortest path when the great circle from A to H is disrupted by a latitude restriction imposed by the missing top of the bowl : Choose the great circle from A which just touches the rim circle tangentially at one point C. If the azimuthal angle ø of point C around the rim from A is < 90° the shortest path from A to H is then 2x the distance A to C along the great circle then C to point ø=90° along the latitude (the rim). If C 's azimuthal angle ø≥ 90° the shortest route is 2x the distance along the great circle connecting A with the point on the rim 90° from A. Problem solved!