On Tue, Dec 7, 2010 at 5:58 PM, Henry Baker <hbaker1@pipeline.com> wrote:
I'm still trying to solve for the optimum path around a triangle, subject to the acceleration constraint |a|<=1.
By "around a triangle", it seems you mean any path that passes through the three vertices of the triangle; is this correct? Also, I think you have an additional implied constraint that the route be cyclic, that is, the velocity at the end of the path is the same as the velocity at the beginning. Otherwise the best solution for the degenerate triangle would be to start at one end of the line segment with velocity 2, decelerate until you reach the other end with velocity zero, and then continue to accelerate in the same direction until you return to the original vertex, again with speed 2, but now in the direction away from, rather than towards, the other vertex. The Mathematical Intelligencer had an article a month or two back about the optimal way to run the bases in baseball. That is, the fastest path with bounded acceleration starting at one vertex of a square with velocity 0, passing through the other three vertices, and returning to the original vertex (though without the requirement that the final velocity be 0). I think much of this article is relevant to your triangle problem. Andy