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?