[math-fun] racing around a triangle
I'm still trying to solve for the optimum path around a triangle, subject to the acceleration constraint |a|<=1. On a line segment of length 1 (one kind of degenerate triangle) on the x-axis centered at the origin, one guess at an optimal path would be to start from the origin towards 1/2 with a velocity of 1 and decelerate at the constant rate of -1, achieving 1/2 after t=1 with velocity 0, and continuing to accelerate back towards the origin, reaching it at time 2 where we achieve velocity -1 & immediately start decelerating with acceleration of +1, achieving -1/2 at time 3, and continue accelerating at +1 to reach the origin again at time 4, for a complete cycle. x = x0+v0*t+a0*t^2/2 v = v0+a0*t v1 = v0-1*t=0 => t=v0 x0 = 0 x1 = 0+v0*t+a0*t^2/2 = 0+v0*v0-1*v0^2/2 = v0^2-v0^2/2 = v0^2/2 = 1/2 => v0=1 => t=1 However, if we were to simply go in a circle around this segment as a diameter with velocity |v|=1, then it would take time 2pi=6.28 to complete a cycle. Although the direction of the acceleration is constantly changing, its absolute value is still always |a|=1. So it would seem that we can go around some circles faster than simply following the circumcircle. Is there any way to see how to compute an optimum trajectory around an arbitrary triangle?
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
participants (2)
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Andy Latto -
Henry Baker